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 prompt,context,A,B,C,D,E,answer
 "A $2.00 \mathrm{~kg}$ particle moves along an $x$ axis in one-dimensional motion while a conservative force along that axis acts on it. The potential energy $U(x)$ associated with the force is plotted in Fig. 8-10a. That is, if the particle were placed at any position between $x=0$ and $x=7.00 \mathrm{~m}$, it would have the plotted value of $U$. At $x=6.5 \mathrm{~m}$, the particle has velocity $\vec{v}_0=(-4.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$
-From Figure, determine the particle's speed at $x_1=4.5 \mathrm{~m}$.","If a force is conservative, it is possible to assign a numerical value for the potential at any point and conversely, when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken, contributing to the mechanical energy and the overall conservation of energy. The work done by a conservative force is equal to the negative of change in potential energy during that process. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = at+v_0 & [1]\\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\\ \end{align} where: * is the particle's initial position * is the particle's final position * is the particle's initial velocity * is the particle's final velocity * is the particle's acceleration * is the time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; \begin{align} \mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\\ \end{align} in magnitudes, \begin{align} v & = at+v_0 \,, & [1] \\\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. A conservative force depends only on the position of the object. Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. (Velocity is on the y-axis and time on the x-axis. If the force is not conservative, then defining a scalar potential is not possible, because taking different paths would lead to conflicting potential differences between the start and end points. If the velocity or positions change non-linearly over time, such as in the example shown in the figure, then differentiation provides the correct solution. thumb|360px|v vs t graph for a moving particle under a non-uniform acceleration a. In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken.HyperPhysics - Conservative force Equivalently, if a particle travels in a closed loop, the total work done (the sum of the force acting along the path multiplied by the displacement) by a conservative force is zero. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the polar unit vectors. For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the start of the slide to the end is independent of the shape of the slide; it only depends on the vertical displacement of the child. ==Mathematical description== A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions: # The curl of F is the zero vector: \vec{ abla} \times \vec{F} = \vec{0}. where in two dimensions this reduces to: \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0 # There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place: W \equiv \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0. The Lennard-Jones model describes the potential intermolecular energy V between two particles based on the outlined principles. The mean potential energy per particle is negative. Accordingly, some authors classify the magnetic force as conservative,For example, : ""In general, a force which depends explicitly upon the velocity of the particle is not conservative. Specific potential energy is potential energy of an object per unit of mass of that object. Suppose a particle starts at point A, and there is a force F acting on it. In mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B. Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Therefore, the slope of the curve gives the change in position divided by the change in time, which is the definition of the average velocity for that interval of time on the graph. Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. ",-3.8,1000,-45.0,3.0,20.2,D
+From Figure, determine the particle's speed at $x_1=4.5 \mathrm{~m}$.","If a force is conservative, it is possible to assign a numerical value for the potential at any point and conversely, when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken, contributing to the mechanical energy and the overall conservation of energy. The work done by a conservative force is equal to the negative of change in potential energy during that process. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = at+v_0 & [1]\\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\\ \end{align} where: * is the particle's initial position * is the particle's final position * is the particle's initial velocity * is the particle's final velocity * is the particle's acceleration * is the time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; \begin{align} \mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\\ \end{align} in magnitudes, \begin{align} v & = at+v_0 \,, & [1] \\\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. A conservative force depends only on the position of the object. Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. (Velocity is on the y-axis and time on the x-axis. If the force is not conservative, then defining a scalar potential is not possible, because taking different paths would lead to conflicting potential differences between the start and end points. If the velocity or positions change non-linearly over time, such as in the example shown in the figure, then differentiation provides the correct solution. thumb|360px|v vs t graph for a moving particle under a non-uniform acceleration a. In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken.HyperPhysics - Conservative force Equivalently, if a particle travels in a closed loop, the total work done (the sum of the force acting along the path multiplied by the displacement) by a conservative force is zero. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the polar unit vectors. For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the start of the slide to the end is independent of the shape of the slide; it only depends on the vertical displacement of the child. ==Mathematical description== A force field F, defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions: # The curl of F is the zero vector: \vec{ abla} \times \vec{F} = \vec{0}. where in two dimensions this reduces to: \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 0 # There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place: W \equiv \oint_C \vec{F} \cdot \mathrm{d}\vec r = 0. The Lennard-Jones model describes the potential intermolecular energy V between two particles based on the outlined principles. The mean potential energy per particle is negative. Accordingly, some authors classify the magnetic force as conservative,For example, : ""In general, a force which depends explicitly upon the velocity of the particle is not conservative. Specific potential energy is potential energy of an object per unit of mass of that object. Suppose a particle starts at point A, and there is a force F acting on it. In mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B. Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Therefore, the slope of the curve gives the change in position divided by the change in time, which is the definition of the average velocity for that interval of time on the graph. Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. ",-3.8,1000,"""-45.0""",3.0,20.2,D
 "A playful astronaut releases a bowling ball, of mass $m=$ $7.20 \mathrm{~kg}$, into circular orbit about Earth at an altitude $h$ of $350 \mathrm{~km}$.
-What is the mechanical energy $E$ of the ball in its orbit?","Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r} since in circular motion, Newton's 2nd Law of motion can be taken to be G \frac{M m}{r^2}\ = \frac{m v^2}{r} ==Conversion== Today, many technological devices convert mechanical energy into other forms of energy or vice versa. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced. ===Satellite=== thumb|plot of kinetic energy K, gravitational potential energy, U and mechanical energy E_\text{mechanical} versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, (by virtue of its motion) and gravitational potential energy, U, (by virtue of its position within the Earth's gravitational field; Earth's mass is M). Specific mechanical energy is the mechanical energy of an object per unit of mass. The relations are used. p= \frac{h^2}{\mu} = a(1-{e^2}) = r_{p}(1+e) where * p\,\\! is the conic section semi-latus rectum. * r_p\,\\! is distance at periastron of the body from the center of mass. v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)} where *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\\! is the length of the semi-major axis. === Orbital Mechanics === When calculating the specific mechanical energy of a satellite in orbit around a celestial body, the mass of the satellite is assumed to be negligible: \mu = G(M + m) \approx GM where M is the mass of the celestial body. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is defined as: \epsilon= \epsilonk+\epsilonp where * \epsilonk is the specific kinetic energy * \epsilonp it the specific potential energy == Astrodynamics == In the gravitational two-body problem, the specific mechanical energy of one body \epsilon is given as: \begin{align} \epsilon &= \frac{v^2}{2} - \frac{\mu}{r} = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right) = -\frac{\mu}{2a} \end{align} where * v\,\\! is the orbital speed of the body; relative to center of mass. * r\,\\! is the orbital distance between the body and center of mass; * \mu = {G}(m_1 + m_2)\,\\! is the standard gravitational parameter of the bodies; * h\,\\! is the specific relative angular momentum of the same body referenced to the center of mass. The gravitational potential energy of an object is equal to the weight W of the object multiplied by the height h of the object's center of gravity relative to an arbitrary datum: U = W h The potential energy of an object can be defined as the object's ability to do work and is increased as the object is moved in the opposite direction of the direction of the force. Specific potential energy is potential energy of an object per unit of mass of that object. Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. Though energy cannot be created or destroyed, it can be converted to another form of energy. ===Swinging pendulum=== 200px|thumb|A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The change in potential energy moving from the surface (a distance R from the center) to a height h above the surface is \begin{align} \Delta U &= \frac{GMm}{R}-\frac{GMm}{R+h} \\\ &= \frac{GMm}{R}\left(1-\frac{1}{1+h/R}\right). \end{align} If h/R is small, as it must be close to the surface where g is constant, then this expression can be simplified using the binomial approximation \frac{1}{1+h/r} \approx 1-\frac{h}{R} to \begin{align} \Delta U &\approx \frac{GMm}{R}\left[1-\left(1-\frac{h}{R}\right)\right] \\\ \Delta U &\approx \frac{GMmh}{R^2}\\\ \Delta U &\approx m\left(\frac{GM}{R^2}\right)h. \end{align} As the gravitational field is g = GM / R^2, this reduces to \Delta U \approx mgh. The gravitational potential energy is the potential energy an object has because it is within a gravitational field. thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. In a mechanical system like a swinging pendulum subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible, energy passes back and forth between kinetic and potential energy but never leaves the system. The force between a point mass, M, and another point mass, m, is given by Newton's law of gravitation: Extract of page 10 F = \frac {GMm}{r^2} To get the total work done by an external force to bring point mass m from infinity to the final distance R (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement: W = \int_\infty^R \frac {GMm}{r^2}dr = -\left . \frac{G M m}{r} \right|_{\infty}^{R} Because \lim_{r\to \infty} \frac{1}{r} = 0, the total work done on the object can be written as: Extract of page 143 In the common situation where a much smaller mass m is moving near the surface of a much larger object with mass M, the gravitational field is nearly constant and so the expression for gravitational energy can be considerably simplified. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity. In an elastic collision, mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. ",-214,29.36,0.01961,132.9,11000,A
-Let the disk in Figure start from rest at time $t=0$ and also let the tension in the massless cord be $6.0 \mathrm{~N}$ and the angular acceleration of the disk be $-24 \mathrm{rad} / \mathrm{s}^2$. What is its rotational kinetic energy $K$ at $t=2.5 \mathrm{~s}$ ?,"The rotational energy depends on the moment of inertia for the system, I . Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Knowledge of the Euler angles as function of time t and the initial coordinates \mathbf{r}(0) determine the kinematics of the rigid rotor. === Classical kinetic energy === The following text forms a generalization of the well- known special case of the rotational energy of an object that rotates around one axis. The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The classical kinetic energy T of the rigid rotor can be expressed in different ways: * as a function of angular velocity * in Lagrangian form * as a function of angular momentum * in Hamiltonian form. The kinetic energy T of the linear rigid rotor is given by 2T = \mu R^2 \left[\dot{\theta}^2 + (\dot\varphi\,\sin\theta)^2\right] = \mu R^2 \begin{pmatrix}\dot{\theta} & \dot{\varphi}\end{pmatrix} \begin{pmatrix} 1 & 0 \\\ 0 & \sin^2\theta \\\ \end{pmatrix} \begin{pmatrix}\dot{\theta} \\\ \dot{\varphi}\end{pmatrix} = \mu \begin{pmatrix}\dot{\theta} & \dot{\varphi}\end{pmatrix} \begin{pmatrix} h_\theta^2 & 0 \\\ 0 & h_\varphi^2 \\\ \end{pmatrix} \begin{pmatrix}\dot{\theta} \\\ \dot{\varphi}\end{pmatrix}, where h_\theta = R\, and h_\varphi= R\sin\theta\, are scale (or Lamé) factors. Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion: :E_\mathrm{translational} = \tfrac{1}{2} m v^2 In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity, \omega , takes the role of the linear velocity, v. As the Earth has a sidereal rotation period of 23.93 hours, it has an angular velocity of .Launching From Florida: Life in the Fast Lane!, NASA The Earth has a moment of inertia, I = .Moment of inertia--Earth, Wolfram Therefore, it has a rotational kinetic energy of . Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed:Resnick, R. and Halliday, D. (1966) PHYSICS, Equation 12-11 :E_\mathrm{rotational} = \tfrac{1}{2} I \omega^2 where : \omega \ is the angular velocity : I \ is the moment of inertia around the axis of rotation : E \ is the kinetic energy The mechanical work required for or applied during rotation is the torque times the rotation angle. – Page 37 of 45 (graphic) For a miniature disc with a diameter of 8 cm (radius of 4 cm), the speed ratio of outer to inner data edge is 1.6. The rotor is rigid if R is independent of time. For a stationary open rotor with no outer duct, such as a helicopter in hover, the power required to produce a given thrust is: :P = \sqrt{\frac{T^3}{2 \rho A}} where: * T is the thrust * \rho is the density of air (or other medium) * A is the area of the rotor disc * P is power A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a Rankine disk actuator. Note that a different rotation matrix would result from a different choice of Euler angle convention used. ==== Lagrange form ==== Backsubstitution of the expression of \boldsymbol{\omega} into T gives the kinetic energy in Lagrange form (as a function of the time derivatives of the Euler angles). The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow). A linear (data reading and writing) speeds of 2.4 times higher can be reached at the outer disc edge with the same angular (rotation) speed. Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond (anharmonicity in the potential). == Arbitrarily shaped rigid rotor == An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in field-free space R3, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored). In matrix-vector notation, 2 T = \begin{pmatrix} \dot{\alpha} & \dot{\beta} & \dot{\gamma} \end{pmatrix} \; \mathbf{g} \; \begin{pmatrix} \dot{\alpha} \\\ \dot{\beta} \\\ \dot{\gamma}\\\ \end{pmatrix}, where \mathbf{g} is the metric tensor expressed in Euler angles--a non-orthogonal system of curvilinear coordinates-- \mathbf{g}= \begin{pmatrix} I_1 \sin^2\beta \cos^2\gamma+I_2\sin^2\beta\sin^2\gamma+I_3\cos^2\beta & (I_2-I_1) \sin\beta\sin\gamma\cos\gamma & I_3\cos\beta \\\ (I_2-I_1) \sin\beta\sin\gamma\cos\gamma & I_1\sin^2\gamma+I_2\cos^2\gamma & 0 \\\ I_3\cos\beta & 0 & I_3 \\\ \end{pmatrix}. ==== Angular momentum form ==== Often the kinetic energy is written as a function of the angular momentum \mathbf{L} of the rigid rotor. (Note: The corresponding eigenvalue equation gives the Schrödinger equation for the rigid rotor in the form that it was solved for the first time by Kronig and Rabi (for the special case of the symmetric rotor). thumb|An actuator disk accelerating a fluid flow from right to left In fluid dynamics, momentum theory or disk actuator theory is a theory describing a mathematical model of an ideal actuator disk, such as a propeller or helicopter rotor, by W.J.M. Rankine (1865), Alfred George Greenhill (1888) and (1889). This disc creates a flow around the rotor. The instantaneous power of an angularly accelerating body is the torque times the angular velocity. ",0.38,2688,90.0, 135.36,0.18162,C
-"A food shipper pushes a wood crate of cabbage heads (total mass $m=14 \mathrm{~kg}$ ) across a concrete floor with a constant horizontal force $\vec{F}$ of magnitude $40 \mathrm{~N}$. In a straight-line displacement of magnitude $d=0.50 \mathrm{~m}$, the speed of the crate decreases from $v_0=0.60 \mathrm{~m} / \mathrm{s}$ to $v=0.20 \mathrm{~m} / \mathrm{s}$. What is the increase $\Delta E_{\text {th }}$ in the thermal energy of the crate and floor?","When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material. The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. The Ergun equation, derived by the Turkish chemical engineer Sabri Ergun in 1952, expresses the friction factor in a packed column as a function of the modified Reynolds number. ==Equation== f_p = \frac {150}{Gr_p}+1.75 where f_p and Gr_p are defined as f_p = \frac{\Delta p}{L} \frac{D_p}{\rho v_s^2} \left(\frac{\epsilon^3}{1-\epsilon}\right) and Gr_p = \frac{\rho v_s D_p}{(1-\epsilon)\mu} = \frac{Re}{(1-\epsilon)}; where: Gr_p is the modified Reynolds number, f_p is the packed bed friction factor \Delta p is the pressure drop across the bed, L is the length of the bed (not the column), D_p is the equivalent spherical diameter of the packing, \rho is the density of fluid, \mu is the dynamic viscosity of the fluid, v_s is the superficial velocity (i.e. the velocity that the fluid would have through the empty tube at the same volumetric flow rate) \epsilon is the void fraction (porosity) of the bed. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. Specific mechanical energy is the mechanical energy of an object per unit of mass. The heat loss due to linear thermal bridging (H_{TB}) is a physical quantity used when calculating the energy performance of buildings. This is an efficient way of increasing the rate, since the alternative way of doing so is by increasing either the heat transfer coefficient (which depends on the nature of materials being used and the conditions of use) or the temperature gradient (which depends on the conditions of use). In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, \partial Q/\partial t, is proportional to the rate of change of its temperature, \partial u/\partial t. One of the experimentally obtained equations for heat transfer coefficient for the fin surface for low wind velocities is: k=2.11 v^{0.71} \theta^{0.44} a^{-0.14} where k= Fin surface heat transfer coefficient [W/m2K ] a=fin length [mm] v=wind velocity [km/hr] θ=fin pitch [mm] Another equation for high fluid velocities, obtained from experiments conducted by Gibson, is k= 241.7[0.0247-0.00148(a^{0.8}/\theta^{0.4})] v^{0.73} where k=Fin surface heat transfer coefficient[W/m2K ] a=Fin length[mm] θ=Fin pitch[mm] v=Wind velocity[km/hr] A more accurate equation for fin surface heat transfer coefficient is: k_{avg} = (2.47-2.55/\theta^{0.4}) v^{0.9} 0.0872 \theta + 4.31 where k (avg)= Fin surface heat transfer coefficient[W/m2K ] θ=Fin pitch[mm] v=Wind velocity[km/hr] All these equations can be used to evaluate average heat transfer coefficient for various fin designs. == Design == The momentum conservation equation for this case is given as follows: {\partial(\rho v)\over\partial t} + v abla . (\rho v) = - abla P + abla . \tau + F + \rho g This is used in combination with the continuity equation. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. Applying the law of conservation of energy to a small element of the medium centered at x, one concludes that the rate at which heat accumulates at a given point x is equal to the derivative of the heat flow at that point, negated. The energy equation is also needed, which is: {\partial (\rho E)\over\partial t} + abla.[v(\rho E + p)] = abla.[k_{eff} abla T- \Sigma_j h_j J_j +(\tau.v)]. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. The heat equation is a consequence of Fourier's law of conduction (see heat conduction). Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. It appears in both United Kingdom and Irish methodologies. ==Calculation== The calculation of the heat loss due to linear thermal bridging is relatively simple, given by the formula below: :H_{TB} = y \sum A_{exp} In the formula, y = 0.08 if Accredited Construction details used, and y = 0.15 otherwise, and \sum A_{exp} is the sum of all the exposed areas of the building envelope, ==References== Category:Energy economics Category:Thermodynamic properties This results in velocity profiles and temperature profiles for various surfaces and this knowledge can be used to design the fin. == References == Category:Unit operations Category:Transport phenomena Category:Heat transfer Thus the rate of heat flow into V is also given by the surface integral q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS where n(x) is the outward pointing normal vector at x. When solved as a scalar equation, it can be used to calculate the temperatures at the fin and cylinder surfaces, by reducing to: abla^2 T + {\overset{.}{q}\over k} = {1 \over \alpha} {\partial T\over\partial t} Where: q = internal heat generation = 0 (in this case). * The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x: \kappa = 1/(\rho c_p). ",2.89,72,22.2,0.139,-1.0,C
+What is the mechanical energy $E$ of the ball in its orbit?","Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r} since in circular motion, Newton's 2nd Law of motion can be taken to be G \frac{M m}{r^2}\ = \frac{m v^2}{r} ==Conversion== Today, many technological devices convert mechanical energy into other forms of energy or vice versa. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced. ===Satellite=== thumb|plot of kinetic energy K, gravitational potential energy, U and mechanical energy E_\text{mechanical} versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, (by virtue of its motion) and gravitational potential energy, U, (by virtue of its position within the Earth's gravitational field; Earth's mass is M). Specific mechanical energy is the mechanical energy of an object per unit of mass. The relations are used. p= \frac{h^2}{\mu} = a(1-{e^2}) = r_{p}(1+e) where * p\,\\! is the conic section semi-latus rectum. * r_p\,\\! is distance at periastron of the body from the center of mass. v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)} where *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\\! is the length of the semi-major axis. === Orbital Mechanics === When calculating the specific mechanical energy of a satellite in orbit around a celestial body, the mass of the satellite is assumed to be negligible: \mu = G(M + m) \approx GM where M is the mass of the celestial body. The elastic potential energy equation is used in calculations of positions of mechanical equilibrium. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is defined as: \epsilon= \epsilonk+\epsilonp where * \epsilonk is the specific kinetic energy * \epsilonp it the specific potential energy == Astrodynamics == In the gravitational two-body problem, the specific mechanical energy of one body \epsilon is given as: \begin{align} \epsilon &= \frac{v^2}{2} - \frac{\mu}{r} = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right) = -\frac{\mu}{2a} \end{align} where * v\,\\! is the orbital speed of the body; relative to center of mass. * r\,\\! is the orbital distance between the body and center of mass; * \mu = {G}(m_1 + m_2)\,\\! is the standard gravitational parameter of the bodies; * h\,\\! is the specific relative angular momentum of the same body referenced to the center of mass. The gravitational potential energy of an object is equal to the weight W of the object multiplied by the height h of the object's center of gravity relative to an arbitrary datum: U = W h The potential energy of an object can be defined as the object's ability to do work and is increased as the object is moved in the opposite direction of the direction of the force. Specific potential energy is potential energy of an object per unit of mass of that object. Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. Though energy cannot be created or destroyed, it can be converted to another form of energy. ===Swinging pendulum=== 200px|thumb|A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The change in potential energy moving from the surface (a distance R from the center) to a height h above the surface is \begin{align} \Delta U &= \frac{GMm}{R}-\frac{GMm}{R+h} \\\ &= \frac{GMm}{R}\left(1-\frac{1}{1+h/R}\right). \end{align} If h/R is small, as it must be close to the surface where g is constant, then this expression can be simplified using the binomial approximation \frac{1}{1+h/r} \approx 1-\frac{h}{R} to \begin{align} \Delta U &\approx \frac{GMm}{R}\left[1-\left(1-\frac{h}{R}\right)\right] \\\ \Delta U &\approx \frac{GMmh}{R^2}\\\ \Delta U &\approx m\left(\frac{GM}{R^2}\right)h. \end{align} As the gravitational field is g = GM / R^2, this reduces to \Delta U \approx mgh. The gravitational potential energy is the potential energy an object has because it is within a gravitational field. thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. In a mechanical system like a swinging pendulum subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible, energy passes back and forth between kinetic and potential energy but never leaves the system. The force between a point mass, M, and another point mass, m, is given by Newton's law of gravitation: Extract of page 10 F = \frac {GMm}{r^2} To get the total work done by an external force to bring point mass m from infinity to the final distance R (for example the radius of Earth) of the two mass points, the force is integrated with respect to displacement: W = \int_\infty^R \frac {GMm}{r^2}dr = -\left . \frac{G M m}{r} \right|_{\infty}^{R} Because \lim_{r\to \infty} \frac{1}{r} = 0, the total work done on the object can be written as: Extract of page 143 In the common situation where a much smaller mass m is moving near the surface of a much larger object with mass M, the gravitational field is nearly constant and so the expression for gravitational energy can be considerably simplified. The energy is potential as it will be converted into other forms of energy, such as kinetic energy and sound energy, when the object is allowed to return to its original shape (reformation) by its elasticity. In an elastic collision, mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. In physical sciences, mechanical energy is the sum of potential energy and kinetic energy. ",-214,29.36,"""0.01961""",132.9,11000,A
+Let the disk in Figure start from rest at time $t=0$ and also let the tension in the massless cord be $6.0 \mathrm{~N}$ and the angular acceleration of the disk be $-24 \mathrm{rad} / \mathrm{s}^2$. What is its rotational kinetic energy $K$ at $t=2.5 \mathrm{~s}$ ?,"The rotational energy depends on the moment of inertia for the system, I . Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Knowledge of the Euler angles as function of time t and the initial coordinates \mathbf{r}(0) determine the kinematics of the rigid rotor. === Classical kinetic energy === The following text forms a generalization of the well- known special case of the rotational energy of an object that rotates around one axis. The rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The classical kinetic energy T of the rigid rotor can be expressed in different ways: * as a function of angular velocity * in Lagrangian form * as a function of angular momentum * in Hamiltonian form. The kinetic energy T of the linear rigid rotor is given by 2T = \mu R^2 \left[\dot{\theta}^2 + (\dot\varphi\,\sin\theta)^2\right] = \mu R^2 \begin{pmatrix}\dot{\theta} & \dot{\varphi}\end{pmatrix} \begin{pmatrix} 1 & 0 \\\ 0 & \sin^2\theta \\\ \end{pmatrix} \begin{pmatrix}\dot{\theta} \\\ \dot{\varphi}\end{pmatrix} = \mu \begin{pmatrix}\dot{\theta} & \dot{\varphi}\end{pmatrix} \begin{pmatrix} h_\theta^2 & 0 \\\ 0 & h_\varphi^2 \\\ \end{pmatrix} \begin{pmatrix}\dot{\theta} \\\ \dot{\varphi}\end{pmatrix}, where h_\theta = R\, and h_\varphi= R\sin\theta\, are scale (or Lamé) factors. Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion: :E_\mathrm{translational} = \tfrac{1}{2} m v^2 In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity, \omega , takes the role of the linear velocity, v. As the Earth has a sidereal rotation period of 23.93 hours, it has an angular velocity of .Launching From Florida: Life in the Fast Lane!, NASA The Earth has a moment of inertia, I = .Moment of inertia--Earth, Wolfram Therefore, it has a rotational kinetic energy of . Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed:Resnick, R. and Halliday, D. (1966) PHYSICS, Equation 12-11 :E_\mathrm{rotational} = \tfrac{1}{2} I \omega^2 where : \omega \ is the angular velocity : I \ is the moment of inertia around the axis of rotation : E \ is the kinetic energy The mechanical work required for or applied during rotation is the torque times the rotation angle. – Page 37 of 45 (graphic) For a miniature disc with a diameter of 8 cm (radius of 4 cm), the speed ratio of outer to inner data edge is 1.6. The rotor is rigid if R is independent of time. For a stationary open rotor with no outer duct, such as a helicopter in hover, the power required to produce a given thrust is: :P = \sqrt{\frac{T^3}{2 \rho A}} where: * T is the thrust * \rho is the density of air (or other medium) * A is the area of the rotor disc * P is power A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a Rankine disk actuator. Note that a different rotation matrix would result from a different choice of Euler angle convention used. ==== Lagrange form ==== Backsubstitution of the expression of \boldsymbol{\omega} into T gives the kinetic energy in Lagrange form (as a function of the time derivatives of the Euler angles). The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow). A linear (data reading and writing) speeds of 2.4 times higher can be reached at the outer disc edge with the same angular (rotation) speed. Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond (anharmonicity in the potential). == Arbitrarily shaped rigid rotor == An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in field-free space R3, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored). In matrix-vector notation, 2 T = \begin{pmatrix} \dot{\alpha} & \dot{\beta} & \dot{\gamma} \end{pmatrix} \; \mathbf{g} \; \begin{pmatrix} \dot{\alpha} \\\ \dot{\beta} \\\ \dot{\gamma}\\\ \end{pmatrix}, where \mathbf{g} is the metric tensor expressed in Euler angles--a non-orthogonal system of curvilinear coordinates-- \mathbf{g}= \begin{pmatrix} I_1 \sin^2\beta \cos^2\gamma+I_2\sin^2\beta\sin^2\gamma+I_3\cos^2\beta & (I_2-I_1) \sin\beta\sin\gamma\cos\gamma & I_3\cos\beta \\\ (I_2-I_1) \sin\beta\sin\gamma\cos\gamma & I_1\sin^2\gamma+I_2\cos^2\gamma & 0 \\\ I_3\cos\beta & 0 & I_3 \\\ \end{pmatrix}. ==== Angular momentum form ==== Often the kinetic energy is written as a function of the angular momentum \mathbf{L} of the rigid rotor. (Note: The corresponding eigenvalue equation gives the Schrödinger equation for the rigid rotor in the form that it was solved for the first time by Kronig and Rabi (for the special case of the symmetric rotor). thumb|An actuator disk accelerating a fluid flow from right to left In fluid dynamics, momentum theory or disk actuator theory is a theory describing a mathematical model of an ideal actuator disk, such as a propeller or helicopter rotor, by W.J.M. Rankine (1865), Alfred George Greenhill (1888) and (1889). This disc creates a flow around the rotor. The instantaneous power of an angularly accelerating body is the torque times the angular velocity. ",0.38,2688,"""90.0""", 135.36,0.18162,C
+"A food shipper pushes a wood crate of cabbage heads (total mass $m=14 \mathrm{~kg}$ ) across a concrete floor with a constant horizontal force $\vec{F}$ of magnitude $40 \mathrm{~N}$. In a straight-line displacement of magnitude $d=0.50 \mathrm{~m}$, the speed of the crate decreases from $v_0=0.60 \mathrm{~m} / \mathrm{s}$ to $v=0.20 \mathrm{~m} / \mathrm{s}$. What is the increase $\Delta E_{\text {th }}$ in the thermal energy of the crate and floor?","When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material. The equation is much simpler and can help to understand better the physics of the materials without focusing on the dynamic of the heat transport process. The Ergun equation, derived by the Turkish chemical engineer Sabri Ergun in 1952, expresses the friction factor in a packed column as a function of the modified Reynolds number. ==Equation== f_p = \frac {150}{Gr_p}+1.75 where f_p and Gr_p are defined as f_p = \frac{\Delta p}{L} \frac{D_p}{\rho v_s^2} \left(\frac{\epsilon^3}{1-\epsilon}\right) and Gr_p = \frac{\rho v_s D_p}{(1-\epsilon)\mu} = \frac{Re}{(1-\epsilon)}; where: Gr_p is the modified Reynolds number, f_p is the packed bed friction factor \Delta p is the pressure drop across the bed, L is the length of the bed (not the column), D_p is the equivalent spherical diameter of the packing, \rho is the density of fluid, \mu is the dynamic viscosity of the fluid, v_s is the superficial velocity (i.e. the velocity that the fluid would have through the empty tube at the same volumetric flow rate) \epsilon is the void fraction (porosity) of the bed. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. Specific mechanical energy is the mechanical energy of an object per unit of mass. The heat loss due to linear thermal bridging (H_{TB}) is a physical quantity used when calculating the energy performance of buildings. This is an efficient way of increasing the rate, since the alternative way of doing so is by increasing either the heat transfer coefficient (which depends on the nature of materials being used and the conditions of use) or the temperature gradient (which depends on the conditions of use). In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, \partial Q/\partial t, is proportional to the rate of change of its temperature, \partial u/\partial t. One of the experimentally obtained equations for heat transfer coefficient for the fin surface for low wind velocities is: k=2.11 v^{0.71} \theta^{0.44} a^{-0.14} where k= Fin surface heat transfer coefficient [W/m2K ] a=fin length [mm] v=wind velocity [km/hr] θ=fin pitch [mm] Another equation for high fluid velocities, obtained from experiments conducted by Gibson, is k= 241.7[0.0247-0.00148(a^{0.8}/\theta^{0.4})] v^{0.73} where k=Fin surface heat transfer coefficient[W/m2K ] a=Fin length[mm] θ=Fin pitch[mm] v=Wind velocity[km/hr] A more accurate equation for fin surface heat transfer coefficient is: k_{avg} = (2.47-2.55/\theta^{0.4}) v^{0.9} 0.0872 \theta + 4.31 where k (avg)= Fin surface heat transfer coefficient[W/m2K ] θ=Fin pitch[mm] v=Wind velocity[km/hr] All these equations can be used to evaluate average heat transfer coefficient for various fin designs. == Design == The momentum conservation equation for this case is given as follows: {\partial(\rho v)\over\partial t} + v abla . (\rho v) = - abla P + abla . \tau + F + \rho g This is used in combination with the continuity equation. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. Applying the law of conservation of energy to a small element of the medium centered at x, one concludes that the rate at which heat accumulates at a given point x is equal to the derivative of the heat flow at that point, negated. The energy equation is also needed, which is: {\partial (\rho E)\over\partial t} + abla.[v(\rho E + p)] = abla.[k_{eff} abla T- \Sigma_j h_j J_j +(\tau.v)]. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. The heat equation is a consequence of Fourier's law of conduction (see heat conduction). Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. It appears in both United Kingdom and Irish methodologies. ==Calculation== The calculation of the heat loss due to linear thermal bridging is relatively simple, given by the formula below: :H_{TB} = y \sum A_{exp} In the formula, y = 0.08 if Accredited Construction details used, and y = 0.15 otherwise, and \sum A_{exp} is the sum of all the exposed areas of the building envelope, ==References== Category:Energy economics Category:Thermodynamic properties This results in velocity profiles and temperature profiles for various surfaces and this knowledge can be used to design the fin. == References == Category:Unit operations Category:Transport phenomena Category:Heat transfer Thus the rate of heat flow into V is also given by the surface integral q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS where n(x) is the outward pointing normal vector at x. When solved as a scalar equation, it can be used to calculate the temperatures at the fin and cylinder surfaces, by reducing to: abla^2 T + {\overset{.}{q}\over k} = {1 \over \alpha} {\partial T\over\partial t} Where: q = internal heat generation = 0 (in this case). * The coefficient κ(x) is the inverse of specific heat of the substance at x × density of the substance at x: \kappa = 1/(\rho c_p). ",2.89,72,"""22.2""",0.139,-1.0,C
 "While you are operating a Rotor (a large, vertical, rotating cylinder found in amusement parks), you spot a passenger in acute distress and decrease the angular velocity of the cylinder from $3.40 \mathrm{rad} / \mathrm{s}$ to $2.00 \mathrm{rad} / \mathrm{s}$ in $20.0 \mathrm{rev}$, at constant angular acceleration. (The passenger is obviously more of a ""translation person"" than a ""rotation person."")
-What is the constant angular acceleration during this decrease in angular speed?","In physics, angular acceleration refers to the time rate of change of angular velocity. Therefore, the instantaneous angular acceleration α of the particle is given by : \alpha = \frac{d}{dt} \left(\frac{v_{\perp}}{r}\right). The angular velocity satisfies equations of motion known as Euler's equations (with zero applied torque, since by assumption the rotor is in field-free space). The angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as: : \omega = \frac{d\phi}{dt} = \frac{v_\perp}{r}. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin. The vector \boldsymbol{\omega} = (\omega_x, \omega_y, \omega_z) on the left hand side contains the components of the angular velocity of the rotor expressed with respect to the body-fixed frame. Angular acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α). Thus the formula for Advance ratio is \mu = \frac {V}{u} = \frac {V}{\Omega\cdot R} where Omega (Ω) is the rotor's angular velocity, and R is the rotor radius (about the length of one rotor blade)Jackson, Dave. The SI unit of angular velocity is radians per second, Extract of page 27 with the radian being a dimensionless quantity, thus the SI units of angular velocity may be listed as s−1. The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Therefore, the orbital angular acceleration is the vector \boldsymbol\alpha defined by : \boldsymbol\alpha = \frac{d}{dt} \left(\frac{\mathbf r \times \mathbf v}{r^2}\right). Angular acceleration then may be termed a pseudoscalar, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes. === Particle in three dimensions === In three dimensions, the orbital angular acceleration is the rate at which three-dimensional orbital angular velocity vector changes with time. : Quantity correspondence Angular frequency \omega Frequency u = \omega/{2\pi} 2π rad/s 1 Hz 1 rad/s ≈ 0.159155 Hz 1 rad/s ≈ 9.5493 rpm 0.1047 rad/s ≈ 1 rpm == Coherent units == A use of the unit radian per second is in calculation of the power transmitted by a shaft. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. The radian per second is defined as the angular frequency that results in the angular displacement increasing by one radian every second. * Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity. 300px|right|thumb|Rotation of a rigid body P about a fixed axis O. Angular displacement of a body is the angle (in radians, degrees or turns) through which a point revolves around a centre or a specified axis in a specified sense. The instantaneous angular velocity ω at any point in time is given by : \omega = \frac{v_{\perp}}{r}, where r is the distance from the origin and v_{\perp} is the cross-radial component of the instantaneous velocity (i.e. the component perpendicular to the position vector), which by convention is positive for counter-clockwise motion and negative for clockwise motion. Angular velocity is usually represented by the symbol omega (, sometimes ). In two dimensions, angular acceleration is a number with plus or minus sign indicating orientation, but not pointing in a direction. For rigid bodies, angular acceleration must be caused by a net external torque. ",-0.0301,635.7,260.0,-233,209.1,A
+What is the constant angular acceleration during this decrease in angular speed?","In physics, angular acceleration refers to the time rate of change of angular velocity. Therefore, the instantaneous angular acceleration α of the particle is given by : \alpha = \frac{d}{dt} \left(\frac{v_{\perp}}{r}\right). The angular velocity satisfies equations of motion known as Euler's equations (with zero applied torque, since by assumption the rotor is in field-free space). The angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as: : \omega = \frac{d\phi}{dt} = \frac{v_\perp}{r}. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin. The vector \boldsymbol{\omega} = (\omega_x, \omega_y, \omega_z) on the left hand side contains the components of the angular velocity of the rotor expressed with respect to the body-fixed frame. Angular acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α). Thus the formula for Advance ratio is \mu = \frac {V}{u} = \frac {V}{\Omega\cdot R} where Omega (Ω) is the rotor's angular velocity, and R is the rotor radius (about the length of one rotor blade)Jackson, Dave. The SI unit of angular velocity is radians per second, Extract of page 27 with the radian being a dimensionless quantity, thus the SI units of angular velocity may be listed as s−1. The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The same equations for the angular speed can be obtained reasoning over a rotating rigid body. Therefore, the orbital angular acceleration is the vector \boldsymbol\alpha defined by : \boldsymbol\alpha = \frac{d}{dt} \left(\frac{\mathbf r \times \mathbf v}{r^2}\right). Angular acceleration then may be termed a pseudoscalar, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes. === Particle in three dimensions === In three dimensions, the orbital angular acceleration is the rate at which three-dimensional orbital angular velocity vector changes with time. : Quantity correspondence Angular frequency \omega Frequency u = \omega/{2\pi} 2π rad/s 1 Hz 1 rad/s ≈ 0.159155 Hz 1 rad/s ≈ 9.5493 rpm 0.1047 rad/s ≈ 1 rpm == Coherent units == A use of the unit radian per second is in calculation of the power transmitted by a shaft. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. The radian per second is defined as the angular frequency that results in the angular displacement increasing by one radian every second. * Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity. 300px|right|thumb|Rotation of a rigid body P about a fixed axis O. Angular displacement of a body is the angle (in radians, degrees or turns) through which a point revolves around a centre or a specified axis in a specified sense. The instantaneous angular velocity ω at any point in time is given by : \omega = \frac{v_{\perp}}{r}, where r is the distance from the origin and v_{\perp} is the cross-radial component of the instantaneous velocity (i.e. the component perpendicular to the position vector), which by convention is positive for counter-clockwise motion and negative for clockwise motion. Angular velocity is usually represented by the symbol omega (, sometimes ). In two dimensions, angular acceleration is a number with plus or minus sign indicating orientation, but not pointing in a direction. For rigid bodies, angular acceleration must be caused by a net external torque. ",-0.0301,635.7,"""260.0""",-233,209.1,A
 "A living room has floor dimensions of $3.5 \mathrm{~m}$ and $4.2 \mathrm{~m}$ and a height of $2.4 \mathrm{~m}$.
-What does the air in the room weigh when the air pressure is $1.0 \mathrm{~atm}$ ?","This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\\ \end{align} where: *\rho, air density (kg/m3)In the SI unit system. Air returns from the room at ceiling level or the maximum allowable height above the occupied zone. Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. The partial pressure of dry air p_\text{d} is found considering partial pressure, resulting in: p_\text{d} = p - p_\text{v} where p simply denotes the observed absolute pressure. ==Variation with altitude== thumb|upright=2.0|Standard atmosphere: , , ===Troposphere=== To calculate the density of air as a function of altitude, one requires additional parameters. At 101.325kPa (abs) and , air has a density of approximately , which is about that of water, according to the International Standard Atmosphere (ISA). * At 70°F and 14.696psi, dry air has a density of 0.074887lb/ft3. Air density is a property used in many branches of science, engineering, and industry, including aeronautics;Olson, Wayne M. (2000) AFFTC-TIH-99-01, Aircraft Performance FlightICAO, Manual of the ICAO Standard Atmosphere (extended to 80 kilometres (262 500 feet)), Doc 7488-CD, Third Edition, 1993, .Grigorie, T.L., Dinca, L., Corcau J-I. and Grigorie, O. (2010) Aircrafts' Altitude Measurement Using Pressure Information:Barometric Altitude and Density Altitude gravimetric analysis;A., Picard, R.S., Davis, M., Gläser and K., Fujii (CIPM-2007) Revised formula for the density of moist air the air-conditioningS. * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa: ==Humid air== thumb|right|400px|Effect of temperature and relative humidity on air density The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. thumb|350px|alt=Diagram of underfloor air distribution showing cool, fresh air moving through the underfloor plenum and supplied via floor diffusers and desktop vents. Therefore: * At IUPAC standard temperature and pressure (0°C and 100kPa), dry air has a density of approximately 1.2754kg/m3. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. At 101.325 kPa (abs) and 20 °C (68 °F), air has a density of approximately , according to the International Standard Atmosphere (ISA). thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as U: \begin{align} p &= p(U) e^{-\frac{h - U}{H_\text{TP}}} = p_0 \left(1 - \frac{L U}{T_0}\right)^\frac{g M}{R L} e^{-\frac{h - U}{H_\text{TP}}} \\\ \rho &= \rho(U) e^{-\frac{h - U}{H_\text{TP}}} = \rho_0 \left(1 - \frac{L U}{T_0}\right)^{\frac{g M}{R L} - 1} e^{-\frac{h - U}{H_\text{TP}}} \end{align} ==Composition== ==See also== *Air *Atmospheric drag *Lighter than air *Density *Atmosphere of Earth *International Standard Atmosphere *U.S. Standard Atmosphere *NRLMSISE-00 ==Notes== ==References== ==External links== *Conversions of density units ρ by Sengpielaudio *Air density and density altitude calculations and by Richard Shelquist *Air density calculations by Sengpielaudio (section under Speed of sound in humid air) *Air density calculator by Engineering design encyclopedia *Atmospheric pressure calculator by wolfdynamics *Air iTools - Air density calculator for mobile by JSyA *Revised formula for the density of moist air (CIPM-2007) by NIST Category:Atmospheric thermodynamics A The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Also, the investigation of energy saving has shown that this amount varies for buildings located in different climates, suggesting further studies should investigate this factor prior to designing a suitable HVAC system. ==Applications== Underfloor air distribution is frequently used in office buildings, particularly highly-reconfigurable and open plan offices where raised floors are desirable for cable management. Room air distribution is characterizing how air is introduced to, flows through, and is removed from spaces.Fundamentals volume of the ASHRAE Handbook, Atlanta, GA, USA, 2005 HVAC airflow in spaces generally can be classified by two different types: mixing (or dilution) and displacement. ==Mixing systems== Mixing systems generally supply air such that the supply air mixes with the room air so that the mixed air is at the room design temperature and humidity. Displacement room airflow presents an opportunity to improve both the thermal comfort and indoor air quality (IAQ) of the occupied space. In architecture, construction, and real estate, floor area, floor space, or floorspace is the area (measured as square feet or square metres) taken up by a building or part of it. At a certain plane in the room, the airflow rate returned to the UZ is equal to the supply air. thumb|upright=1.25|Different air masses which affect North America as well as other continents, tend to be separated by frontal boundaries In meteorology, an air mass is a volume of air defined by its temperature and humidity. ",4.979,0.72,2.0,418,0.14,D
-"An astronaut whose height $h$ is $1.70 \mathrm{~m}$ floats ""feet down"" in an orbiting space shuttle at distance $r=6.77 \times 10^6 \mathrm{~m}$ away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head?","It is calculated as the distance between the centre of gravity of a ship and its metacentre. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth (but the latter is small enough to be negligible for most purposes); the total (the apparent gravity) is about 0.5% greater at the poles than at the Equator.""Curious About Astronomy?"", Cornell University, retrieved June 2007 Although the symbol is sometimes used for standard gravity, (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). Acceleration due to gravity, acceleration of gravity or gravity acceleration may refer to: *Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general *Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth *Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth ==See also== *g-force, the acceleration of a body relative to free-fall The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. The metacentric height (GM) is a measurement of the initial static stability of a floating body. As the ship heels over, the centre of gravity generally remains fixed with respect to the ship because it just depends on the position of the ship's weight and cargo, but the surface area increases, increasing BMφ. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships. ==Metacentre== When a ship heels (rolls sideways), the centre of buoyancy of the ship moves laterally. Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level (assumed zero potential) that represents the work done by lifting one unit mass one unit distance through a region in which the acceleration of gravity is uniformly 9.80665 m/s2. thumb|upright=1.8|A simplified spacecraft system. thumb|upright=1.6|Ship stability diagram showing centre of gravity (G), centre of buoyancy (B), and metacentre (M) with ship upright and heeled over to one side. * Cardiovascular events and changes occurring during spaceflight: these are due to body fluids shift and redistribution, heart rhythm disturbances and decrease in maximal exercise capacity in the micro gravity environment. Geopotential height may be obtained from normalizing geopotential by the acceleration of gravity: :{H} = \frac{\Phi}{g_{0}}\ = \frac{1}{g_{0}}\int_0^Z\ g(\phi,Z)\,dZ where g_0 = 9.80665 m/s2, the standard gravity at mean sea level. This position meant that a person's legs experienced only one sixth of their weight, which was the equivalent of being on the lunar surface. The value of defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. thumb|250px|A test subject being suited up for studies on the Reduced Gravity Walking Simulator. This illusion will alter the astronaut's perception of the orienting force of gravity and then lose spatial direction. ""G"", is the center of gravity. While objects are weightless in space, an astronaut has to be familiar with an object's forces of inertia and understand how the object will respond to simple motions to avoid losing it in space. Geopotential height (altitude) differs from geometric (tapeline) height but remains a historical convention in aeronautics as the altitude used for calibration of aircraft barometric altimeters. ==Definition== Geopotential is the gravitational potential energy per unit mass at elevation Z: :\Phi(Z) = \int_0^Z\ g(\phi,Z)\,dZ where g(\phi,Z) is the acceleration due to gravity, \phi is latitude, and Z is the geometric elevation. Because of this any future medical criteria for commercial spaceflight participants needs to focus specifically on the detrimental effects of rapidly changing gravitational levels, and which individuals will be capable of tolerating this. The centre of gravity of the ship is commonly denoted as point G or CG. ",-4.37 ,157.875,37.9,2.26,3.29527,A
+What does the air in the room weigh when the air pressure is $1.0 \mathrm{~atm}$ ?","This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\\ \end{align} where: *\rho, air density (kg/m3)In the SI unit system. Air returns from the room at ceiling level or the maximum allowable height above the occupied zone. Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. The partial pressure of dry air p_\text{d} is found considering partial pressure, resulting in: p_\text{d} = p - p_\text{v} where p simply denotes the observed absolute pressure. ==Variation with altitude== thumb|upright=2.0|Standard atmosphere: , , ===Troposphere=== To calculate the density of air as a function of altitude, one requires additional parameters. At 101.325kPa (abs) and , air has a density of approximately , which is about that of water, according to the International Standard Atmosphere (ISA). * At 70°F and 14.696psi, dry air has a density of 0.074887lb/ft3. Air density is a property used in many branches of science, engineering, and industry, including aeronautics;Olson, Wayne M. (2000) AFFTC-TIH-99-01, Aircraft Performance FlightICAO, Manual of the ICAO Standard Atmosphere (extended to 80 kilometres (262 500 feet)), Doc 7488-CD, Third Edition, 1993, .Grigorie, T.L., Dinca, L., Corcau J-I. and Grigorie, O. (2010) Aircrafts' Altitude Measurement Using Pressure Information:Barometric Altitude and Density Altitude gravimetric analysis;A., Picard, R.S., Davis, M., Gläser and K., Fujii (CIPM-2007) Revised formula for the density of moist air the air-conditioningS. * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa: ==Humid air== thumb|right|400px|Effect of temperature and relative humidity on air density The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. thumb|350px|alt=Diagram of underfloor air distribution showing cool, fresh air moving through the underfloor plenum and supplied via floor diffusers and desktop vents. Therefore: * At IUPAC standard temperature and pressure (0°C and 100kPa), dry air has a density of approximately 1.2754kg/m3. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. At 101.325 kPa (abs) and 20 °C (68 °F), air has a density of approximately , according to the International Standard Atmosphere (ISA). thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as U: \begin{align} p &= p(U) e^{-\frac{h - U}{H_\text{TP}}} = p_0 \left(1 - \frac{L U}{T_0}\right)^\frac{g M}{R L} e^{-\frac{h - U}{H_\text{TP}}} \\\ \rho &= \rho(U) e^{-\frac{h - U}{H_\text{TP}}} = \rho_0 \left(1 - \frac{L U}{T_0}\right)^{\frac{g M}{R L} - 1} e^{-\frac{h - U}{H_\text{TP}}} \end{align} ==Composition== ==See also== *Air *Atmospheric drag *Lighter than air *Density *Atmosphere of Earth *International Standard Atmosphere *U.S. Standard Atmosphere *NRLMSISE-00 ==Notes== ==References== ==External links== *Conversions of density units ρ by Sengpielaudio *Air density and density altitude calculations and by Richard Shelquist *Air density calculations by Sengpielaudio (section under Speed of sound in humid air) *Air density calculator by Engineering design encyclopedia *Atmospheric pressure calculator by wolfdynamics *Air iTools - Air density calculator for mobile by JSyA *Revised formula for the density of moist air (CIPM-2007) by NIST Category:Atmospheric thermodynamics A The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Also, the investigation of energy saving has shown that this amount varies for buildings located in different climates, suggesting further studies should investigate this factor prior to designing a suitable HVAC system. ==Applications== Underfloor air distribution is frequently used in office buildings, particularly highly-reconfigurable and open plan offices where raised floors are desirable for cable management. Room air distribution is characterizing how air is introduced to, flows through, and is removed from spaces.Fundamentals volume of the ASHRAE Handbook, Atlanta, GA, USA, 2005 HVAC airflow in spaces generally can be classified by two different types: mixing (or dilution) and displacement. ==Mixing systems== Mixing systems generally supply air such that the supply air mixes with the room air so that the mixed air is at the room design temperature and humidity. Displacement room airflow presents an opportunity to improve both the thermal comfort and indoor air quality (IAQ) of the occupied space. In architecture, construction, and real estate, floor area, floor space, or floorspace is the area (measured as square feet or square metres) taken up by a building or part of it. At a certain plane in the room, the airflow rate returned to the UZ is equal to the supply air. thumb|upright=1.25|Different air masses which affect North America as well as other continents, tend to be separated by frontal boundaries In meteorology, an air mass is a volume of air defined by its temperature and humidity. ",4.979,0.72,"""2.0""",418,0.14,D
+"An astronaut whose height $h$ is $1.70 \mathrm{~m}$ floats ""feet down"" in an orbiting space shuttle at distance $r=6.77 \times 10^6 \mathrm{~m}$ away from the center of Earth. What is the difference between the gravitational acceleration at her feet and at her head?","It is calculated as the distance between the centre of gravity of a ship and its metacentre. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth (but the latter is small enough to be negligible for most purposes); the total (the apparent gravity) is about 0.5% greater at the poles than at the Equator.""Curious About Astronomy?"", Cornell University, retrieved June 2007 Although the symbol is sometimes used for standard gravity, (without a suffix) can also mean the local acceleration due to local gravity and centrifugal acceleration, which varies depending on one's position on Earth (see Earth's gravity). Acceleration due to gravity, acceleration of gravity or gravity acceleration may refer to: *Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general *Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth *Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth ==See also== *g-force, the acceleration of a body relative to free-fall The standard acceleration due to gravity (or standard acceleration of free fall), sometimes abbreviated as standard gravity, usually denoted by or , is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. The metacentric height (GM) is a measurement of the initial static stability of a floating body. As the ship heels over, the centre of gravity generally remains fixed with respect to the ship because it just depends on the position of the ship's weight and cargo, but the surface area increases, increasing BMφ. Although the actual acceleration of free fall on Earth varies according to location, the above standard figure is always used for metrological purposes. Hence, a sufficiently, but not excessively, high metacentric height is considered ideal for passenger ships. ==Metacentre== When a ship heels (rolls sideways), the centre of buoyancy of the ship moves laterally. Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level (assumed zero potential) that represents the work done by lifting one unit mass one unit distance through a region in which the acceleration of gravity is uniformly 9.80665 m/s2. thumb|upright=1.8|A simplified spacecraft system. thumb|upright=1.6|Ship stability diagram showing centre of gravity (G), centre of buoyancy (B), and metacentre (M) with ship upright and heeled over to one side. * Cardiovascular events and changes occurring during spaceflight: these are due to body fluids shift and redistribution, heart rhythm disturbances and decrease in maximal exercise capacity in the micro gravity environment. Geopotential height may be obtained from normalizing geopotential by the acceleration of gravity: :{H} = \frac{\Phi}{g_{0}}\ = \frac{1}{g_{0}}\int_0^Z\ g(\phi,Z)\,dZ where g_0 = 9.80665 m/s2, the standard gravity at mean sea level. This position meant that a person's legs experienced only one sixth of their weight, which was the equivalent of being on the lunar surface. The value of defined above is a nominal midrange value on Earth, originally based on the acceleration of a body in free fall at sea level at a geodetic latitude of 45°. thumb|250px|A test subject being suited up for studies on the Reduced Gravity Walking Simulator. This illusion will alter the astronaut's perception of the orienting force of gravity and then lose spatial direction. ""G"", is the center of gravity. While objects are weightless in space, an astronaut has to be familiar with an object's forces of inertia and understand how the object will respond to simple motions to avoid losing it in space. Geopotential height (altitude) differs from geometric (tapeline) height but remains a historical convention in aeronautics as the altitude used for calibration of aircraft barometric altimeters. ==Definition== Geopotential is the gravitational potential energy per unit mass at elevation Z: :\Phi(Z) = \int_0^Z\ g(\phi,Z)\,dZ where g(\phi,Z) is the acceleration due to gravity, \phi is latitude, and Z is the geometric elevation. Because of this any future medical criteria for commercial spaceflight participants needs to focus specifically on the detrimental effects of rapidly changing gravitational levels, and which individuals will be capable of tolerating this. The centre of gravity of the ship is commonly denoted as point G or CG. ",-4.37 ,157.875,"""37.9""",2.26,3.29527,A
 "If the particles in a system all move together, the com moves with them-no trouble there. But what happens when they move in different directions with different accelerations? Here is an example.
 
-The three particles in Figure are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are $F_1=6.0 \mathrm{~N}, F_2=12 \mathrm{~N}$, and $F_3=14 \mathrm{~N}$. What is the acceleration of the center of mass of the system?","And any system of particles that move under Newtonian gravitation as if they are a rigid body must do so in a central configuration. In Euler's three-body problem we assume that the two centres of attraction are stationary. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. In celestial mechanics and the mathematics of the -body problem, a central configuration is a system of point masses with the property that each mass is pulled by the combined gravitational force of the system directly towards the center of mass, with acceleration proportional to its distance from the center. The central-force problem concerns an ideal situation (a ""one-body problem"") in which a single particle is attracted or repelled from an immovable point O, the center of force.Goldstein, p. 71; Landau and Lifshitz, p. 30; Whittaker, p. Thus, the equation of motion for r can be written in the form \mu \ddot{\mathbf{r}} = \mathbf{F} where \mu is the reduced mass \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2} As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. ==Qualitative properties== ===Planar motion=== thumb|right|alt=The image shows a yellow disc with three vectors. In an extended modern sense, a three- body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles. ==Mathematical description== The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions \mathbf{r_i} = (x_i, y_i, z_i) of three gravitationally interacting bodies with masses m_i: :\begin{align} \ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\\ \ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\\ \ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}. \end{align} where G is the gravitational constant. In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. This is the only central configuration for these masses that does not lie in a lower- dimensional subspace. ==Dynamics== Under Newton's law of universal gravitation, bodies placed at rest in a central configuration will maintain the configuration as they collapse to a collision at their center of mass. An additional mass (which may be zero) is placed at the center of the system. Systems of bodies in a two-dimensional central configuration can orbit stably around their center of mass, maintaining their relative positions, with circular orbits around the center of mass or in elliptical orbits with the center of mass at a focus of the ellipse. Those equations are an accurate description of a particular form of the three-body problem. Therefore, both bodies are accelerated if a force is present between them; there is no perfectly immovable center of force. If either center were absent, the particle would move on one of these ellipses, as a solution of the Kepler problem. Special cases of these generalized problems include Darboux's problemDarboux JG, Archives Néerlandaises des Sciences (ser. 2), 6, 371-376 and Velde's problem.Velde (1889) Programm der ersten Höheren Bürgerschule zu Berlin ==Overview and history== Euler's three-body problem is to describe the motion of a particle under the influence of two centers that attract the particle with central forces that decrease with distance as an inverse-square law, such as Newtonian gravity or Coulomb's law. The simplest generalization of Euler's three-body problem is to add a third center of force midway between the original two centers, that exerts only a linear Hooke force (confer Hooke's law). The simplest bodies or elements of a multibody system were treated by Newton (free particle) and Euler (rigid body). Each multibody system formulation may lead to a different mathematical appearance of the equations of motion while the physics behind is the same. In other words, a central force must act along the line joining O with the present position of the particle. Moreover, the motion of three bodies is generally non-repeating, except in special cases. Magnus, Dynamics of multibody systems, Springer Verlag, Berlin (1978). In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. ", 1.16,28,2283.63,3.00,0.18162,A
-"An asteroid, headed directly toward Earth, has a speed of $12 \mathrm{~km} / \mathrm{s}$ relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed $v_f$ when it reaches Earth's surface.","Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 is an Aten near-Earth asteroid less than 20 meters in diameter crudely estimated to have passed roughly 6500 km above the surface of Earth on 31 March 2004. The estimated 4 to 6 meter sized body made one of the closest known approaches to Earth. == Description == On 31 March 2004, around 15:35 UTC, the asteroid is crudely estimated to have passed within approximately 1 Earth radius () or 6,400 kilometers of the surface of the Earth (or 2.02 from Earth's center). Due to this elongated orbit, the Aten asteroid and near-Earth asteroid also classifies as Earth-crosser, Venus-crosser and Mercury-grazer. 1989 VA was the first asteroid discovered with such a small semi-major axis (0.728 AU, about the same as Venus), breaking 2100 Ra-Shalom's distance record (0.832 AU), which had held for over a decade. This asteroid orbits the Sun with a short orbital period at a distance of 0.3–1.2 AU once every 227 days. is a very eccentric, stony asteroid and near-Earth object, approximately 1 kilometer in diameter. is a sub-kilometer asteroid that orbits near Mars's Lagrangian point, on average trailing 60° behind it. It passed closest approach to Earth on 3 March 2016 05:17 UT at a distance of and was quickly approaching the glare of the Sun thus preventing further optical observations. == 2021 approach == It was recovered on 17 February 2021 by Pan-STARRS when the uncertainty in the asteroid's sky position covered about 1.2° of the sky. is an asteroid and near-Earth object approximately in diameter. Another, larger near-Earth asteroid, 2004 FH passed just two weeks prior to . is a near-Earth asteroid estimated to be roughly in diameter. It remained the asteroid with the smallest known semi-major axis for five years until the discovery of (0.683 AU), which was the first asteroid discovered closer to the Sun than Venus. Due to its eccentric orbit, is also a Mars-crosser, crossing the orbit of the Red Planet at 1.66 AU. == 2016 discovery == It was first observed by the Mount Lemmon Survey on 28 February 2016, when the asteroid was about from Earth and had a solar elongation of 174°. By early February 2021 the asteroid was brighter than apparent magnitude 24, which still placed it near the limiting magnitude of even the best automated astronomical surveys. With an exceptionally high eccentricity of 0.59, it was the most eccentric Aten asteroid known at the time of discovery, more eccentric than previously discovered Aten, 3753 Cruithne. The formerly poorly known trajectory of this asteroid was further complicated by close approaches to Venus and Mercury. It was not until (0.277 AU) was discovered that an Aten asteroid with a lower perihelion was found. 's eccentric orbit takes it out past the Earth, where it has encounters of about 0.15 to 0.20 AU about every 3 to 5 years around October–November. While listed on the Sentry Risk Table, virtual clones of the asteroid that fit the uncertainty in the known trajectory showed 116 potential impacts between 2054 and 2109. On 26 March 2010, it may have come within 0.0825 AU (12.3 million km) of Earth, but with an uncertainty parameter of 9, the orbit is poorly determined. It was the eighth Aten asteroid discovered. The combination of a small semi- major axis and high eccentricity made the first Aten asteroid discovered to get closer to the Sun (0.295 AU) than Mercury ever does. 2340 Hathor (the second Aten discovered, in 1976) had the smallest perihelion (0.464 AU) earlier, which was about the same distance as Mercury's aphelion (0.467 AU). Its orbit has an eccentricity of 0.40 and an inclination of 7° with respect to the ecliptic. ",2,-7.5,2.74,0.0384,1.60,E
-"The huge advantage of using the conservation of energy instead of Newton's laws of motion is that we can jump from the initial state to the final state without considering all the intermediate motion. Here is an example. In Figure, a child of mass $m$ is released from rest at the top of a water slide, at height $h=8.5 \mathrm{~m}$ above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.","The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Classically, conservation of energy was distinct from conservation of mass. The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. The kinetic energy, as determined by the velocity, is converted to potential energy as it reaches the same height as the initial ball and the cycle repeats. thumb|An idealized Newton's cradle with five balls when there are no energy losses and there is always a small separation between the balls, except for when a pair is colliding thumb|Newton's cradle three-ball swing in a five-ball system. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. Many physicists at that time, such as Newton, held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum: :\sum_{i} m_i v_i was the conserved vis viva. All enforce the conservation of energy and momentum. The caloric theory maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable. Thus, conservation of energy (total, including material or rest energy) and conservation of mass (total, not just rest) are one (equivalent) law. The Newton's cradle is a device that demonstrates the conservation of momentum and the conservation of energy with swinging spheres. He showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle. Energy conservation has been a foundational physical principle for about two hundred years. For systems that include large gravitational fields, general relativity has to be taken into account; thus mass–energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity. == Formulation and examples == The law of conservation of mass can only be formulated in classical mechanics, in which the energy scales associated with an isolated system are much smaller than mc^2, where m is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, and c is the speed of light. Given the stationary- action principle, conservation of energy can be rigorously proven by Noether's theorem as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time. Some say that this behavior demonstrates the conservation of momentum and kinetic energy in elastic collisions. ===Mechanical equivalent of heat=== A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist; that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings.Planck, M. (1923/1927). The law of conservation of mass and the analogous law of conservation of energy were finally generalized and unified into the principle of mass–energy equivalence, described by Albert Einstein's famous formula E = mc^2. The energy conservation law is a consequence of the shift symmetry of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Engineers such as John Smeaton, Peter Ewart, , Gustave-Adolphe Hirn, and Marc Seguin recognized that conservation of momentum alone was not adequate for practical calculation and made use of Leibniz's principle. In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so the quantity can neither be added nor be removed. In reality, the conservation of mass only holds approximately and is considered part of a series of assumptions in classical mechanics. ",0.011,13,8.8,4.979,0.132,B
-"Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.","* If one has $1000 invested for 1 year at a 7-day SEC yield of 2%, then: :(0.02 × $1000 ) / 365 ~= $0.05479 per day. * If one has $1000 invested for 30 days at a 7-day SEC yield of 5%, then: :(0.05 × $1000 ) / 365 ~= $0.137 per day. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. :Multiply by 365/7 to give the 7-day SEC yield. It is also referred to as the 7-day Annualized Yield. The calculation is performed as follows: :Take the net interest income earned by the fund over the last 7 days and subtract 7 days of management fees. It is important to note that the 7-day SEC yield is only an estimate of the fund's actual yield, and may not necessarily reflect the yield that an investor would receive if they held the fund for a longer period of time. ==Examples== The examples assume interest is withdrawn as it is earned and not allowed to compound. To calculate approximately how much interest one might earn in a money fund account, take the 7-day SEC yield, multiply by the amount invested, divide by the number of days in the year, and then multiply by the number of days in question. The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. Multiply by 30 days to yield $4.11 in interest. *For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The E-M rule could thus be written also as : t \approx \frac{70}{r} \times \frac{198}{200-r} or t \approx \frac{72}{r} \times \frac{192}{200-r} In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. ===Padé approximant=== The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: : t \approx \frac{69.3}{r} \times \frac{600+4r}{600+r} which simplifies to: : t \approx \frac{207900+1386r}{3000r+5r^2} ==Derivation== ===Periodic compounding === For periodic compounding, future value is given by: :FV = PV \cdot (1+r)^t where PV is the present value, t is the number of time periods, and r stands for the interest rate per time period. :Divide that dollar amount by the average size of the fund's investments over the same 7 days. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. == Using the rule to estimate compounding periods == To estimate the number of periods required to double an original investment, divide the most convenient ""rule-quantity"" by the expected growth rate, expressed as a percentage. Similarly, replacing the ""R"" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum. The 7-day SEC Yield is a measure of performance in the interest rates of money market mutual funds offered by US mutual fund companies. The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). 100 r is r written as a percentage. ",1.41, 7.0,-2.0,0.5,7.25,E
-"Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \mathrm{~L}$ of a dye solution with a concentration of $1 \mathrm{~g} / \mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \mathrm{~L} / \mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \%$ of its original value.","Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. If one adds 1 litre of water to this solution, the salt concentration is reduced. Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. D_t=\left [ \frac{V}{Q} \right ] \cdot \ln \left [ \frac{C_\text{initial}}{C_\text{ending}}\right ] Sometimes the equation is also written as: \ln \left [ \frac{C_\text{ending}}{C_\text{initial}}\right ] \quad = {-}\frac{Q}{V} \cdot (t_\text{ending} - t_\text{initial}) where t_\text{initial} = 0 *Dt = time required; the unit of time used is the same as is used for Q *V = air or gas volume of the closed space or room in cubic feet, cubic metres or litres *Q = ventilation rate into or out of the room in cubic feet per minute, cubic metres per hour or litres per second *Cinitial = initial concentration of a vapor inside the room measured in ppm *Cfinal = final reduced concentration of the vapor inside the room in ppm ==Dilution ventilation equation== The basic room purge equation can be used only for purge scenarios. The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). Mathematically this relationship can be shown by equation: c_1 V_1 = c_2 V_2 where *c1 = initial concentration or molarity *V1 = initial volume *c2 = final concentration or molarity *V2 = final volume .... ==Basic room purge equation== The basic room purge equation is used in industrial hygiene. The mean free time for a molecule in a fluid is the average time between collisions. Time of concentration is useful in predicting flow rates that would result from hypothetical storms, which are based on statistically derived return periods through IDF curves.Sherman, C. (1931): Frequency and intensity of excessive rainfall at Boston, Massachusetts, Transactions, American Society of Civil Engineers, 95, 951–960. (pdf) For many (often economic) reasons, it is important for engineers and hydrologists to be able to accurately predict the response of a watershed to a given rain event. The concentration of this admixture should be small and the gradient of this concentration should be also small. Lake retention time (also called the residence time of lake water, or the water age or flushing time) is a calculated quantity expressing the mean time that water (or some dissolved substance) spends in a particular lake. In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). upright=1.35|thumb|Diluting a solution by adding more solvent Dilution is the process of decreasing the concentration of a solute in a solution, usually simply by mixing with more solvent like adding more water to the solution. In a scenario where a liquid continuously evaporates from a container in a ventilated room, a differential equation has to be used: \frac{dC}{dt} = \frac{G - Q' C}{V} where the ventilation rate has been adjusted by a mixing factor K: Q' = \frac{Q}{K} *C = concentration of a gas *G = generation rate *V = room volume *Q′ = adjusted ventilation rate of the volume ==Welding== The dilution in welding terms is defined as the weight of the base metal melted divided by the total weight of the weld metal. The solutions on the left are more dilute, compared to the more concentrated solutions on the right. The equation can only be applied when the purged volume of vapor or gas is replaced with ""clean"" air or gas. He introduced several mechanisms of diffusion and found rate constants from experimental data. ",26.9,313,0.66,460.5,14,D
-A certain vibrating system satisfies the equation $u^{\prime \prime}+\gamma u^{\prime}+u=0$. Find the value of the damping coefficient $\gamma$ for which the quasi period of the damped motion is $50 \%$ greater than the period of the corresponding undamped motion.,"For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the system's equation of motion is : m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt{k m} or : c_c = 2 m \sqrt{\frac{k}{m}} = 2m \omega_n where : \omega_n = \sqrt{\frac{k}{m}} is the natural frequency of the system. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^{-\lambda t} \cos(\omega t - \varphi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\varphi is the phase angle at ; *\omega is the angular frequency. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. == Derivation == Using the natural frequency of a harmonic oscillator \omega_n = \sqrt{{k}/{m}} and the definition of the damping ratio above, we can rewrite this as: : \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. ==Damped sine wave== thumb|350px|Plot of a damped sinusoidal wave represented as the function y(t) = e^{- t} \cos(2 \pi t) A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. * Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. If the system has n degrees of freedom un and is under application of m damping forces. * Q factor: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio definition == thumb|400px|upright=1.3|The effect of varying damping ratio on a second-order system. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that : \zeta = \frac{1}{2 Q} = { \alpha \over \omega_n }. Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: thumb|Phase portrait of damped oscillator, with increasing damping strength. *Damped harmonic motion, see animation (right). The equation of motion is \ddot x + 2\gamma \dot x + \omega^2 x = 0. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. It corresponds to the underdamped case of damped second-order systems, or underdamped second-order differential equations.Douglas C. Giancoli (2000). Critically damped systems have a damping ratio of exactly 1, or at least very close to it. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Low damping materials may be utilized in musical instruments where sustained mechanical vibration and acoustic wave propagation is desired. ",0.08,0.249,11.0,1.4907,2.72,D
+The three particles in Figure are initially at rest. Each experiences an external force due to bodies outside the three-particle system. The directions are indicated, and the magnitudes are $F_1=6.0 \mathrm{~N}, F_2=12 \mathrm{~N}$, and $F_3=14 \mathrm{~N}$. What is the acceleration of the center of mass of the system?","And any system of particles that move under Newtonian gravitation as if they are a rigid body must do so in a central configuration. In Euler's three-body problem we assume that the two centres of attraction are stationary. Together with Euler's collinear solutions, these solutions form the central configurations for the three-body problem. In celestial mechanics and the mathematics of the -body problem, a central configuration is a system of point masses with the property that each mass is pulled by the combined gravitational force of the system directly towards the center of mass, with acceleration proportional to its distance from the center. The central-force problem concerns an ideal situation (a ""one-body problem"") in which a single particle is attracted or repelled from an immovable point O, the center of force.Goldstein, p. 71; Landau and Lifshitz, p. 30; Whittaker, p. Thus, the equation of motion for r can be written in the form \mu \ddot{\mathbf{r}} = \mathbf{F} where \mu is the reduced mass \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2} As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. ==Qualitative properties== ===Planar motion=== thumb|right|alt=The image shows a yellow disc with three vectors. In an extended modern sense, a three- body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles. ==Mathematical description== The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions \mathbf{r_i} = (x_i, y_i, z_i) of three gravitationally interacting bodies with masses m_i: :\begin{align} \ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\\ \ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\\ \ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}. \end{align} where G is the gravitational constant. In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation. This is the only central configuration for these masses that does not lie in a lower- dimensional subspace. ==Dynamics== Under Newton's law of universal gravitation, bodies placed at rest in a central configuration will maintain the configuration as they collapse to a collision at their center of mass. An additional mass (which may be zero) is placed at the center of the system. Systems of bodies in a two-dimensional central configuration can orbit stably around their center of mass, maintaining their relative positions, with circular orbits around the center of mass or in elliptical orbits with the center of mass at a focus of the ellipse. Those equations are an accurate description of a particular form of the three-body problem. Therefore, both bodies are accelerated if a force is present between them; there is no perfectly immovable center of force. If either center were absent, the particle would move on one of these ellipses, as a solution of the Kepler problem. Special cases of these generalized problems include Darboux's problemDarboux JG, Archives Néerlandaises des Sciences (ser. 2), 6, 371-376 and Velde's problem.Velde (1889) Programm der ersten Höheren Bürgerschule zu Berlin ==Overview and history== Euler's three-body problem is to describe the motion of a particle under the influence of two centers that attract the particle with central forces that decrease with distance as an inverse-square law, such as Newtonian gravity or Coulomb's law. The simplest generalization of Euler's three-body problem is to add a third center of force midway between the original two centers, that exerts only a linear Hooke force (confer Hooke's law). The simplest bodies or elements of a multibody system were treated by Newton (free particle) and Euler (rigid body). Each multibody system formulation may lead to a different mathematical appearance of the equations of motion while the physics behind is the same. In other words, a central force must act along the line joining O with the present position of the particle. Moreover, the motion of three bodies is generally non-repeating, except in special cases. Magnus, Dynamics of multibody systems, Springer Verlag, Berlin (1978). In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. ", 1.16,28,"""2283.63""",3.00,0.18162,A
+"An asteroid, headed directly toward Earth, has a speed of $12 \mathrm{~km} / \mathrm{s}$ relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed $v_f$ when it reaches Earth's surface.","Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 is an Aten near-Earth asteroid less than 20 meters in diameter crudely estimated to have passed roughly 6500 km above the surface of Earth on 31 March 2004. The estimated 4 to 6 meter sized body made one of the closest known approaches to Earth. == Description == On 31 March 2004, around 15:35 UTC, the asteroid is crudely estimated to have passed within approximately 1 Earth radius () or 6,400 kilometers of the surface of the Earth (or 2.02 from Earth's center). Due to this elongated orbit, the Aten asteroid and near-Earth asteroid also classifies as Earth-crosser, Venus-crosser and Mercury-grazer. 1989 VA was the first asteroid discovered with such a small semi-major axis (0.728 AU, about the same as Venus), breaking 2100 Ra-Shalom's distance record (0.832 AU), which had held for over a decade. This asteroid orbits the Sun with a short orbital period at a distance of 0.3–1.2 AU once every 227 days. is a very eccentric, stony asteroid and near-Earth object, approximately 1 kilometer in diameter. is a sub-kilometer asteroid that orbits near Mars's Lagrangian point, on average trailing 60° behind it. It passed closest approach to Earth on 3 March 2016 05:17 UT at a distance of and was quickly approaching the glare of the Sun thus preventing further optical observations. == 2021 approach == It was recovered on 17 February 2021 by Pan-STARRS when the uncertainty in the asteroid's sky position covered about 1.2° of the sky. is an asteroid and near-Earth object approximately in diameter. Another, larger near-Earth asteroid, 2004 FH passed just two weeks prior to . is a near-Earth asteroid estimated to be roughly in diameter. It remained the asteroid with the smallest known semi-major axis for five years until the discovery of (0.683 AU), which was the first asteroid discovered closer to the Sun than Venus. Due to its eccentric orbit, is also a Mars-crosser, crossing the orbit of the Red Planet at 1.66 AU. == 2016 discovery == It was first observed by the Mount Lemmon Survey on 28 February 2016, when the asteroid was about from Earth and had a solar elongation of 174°. By early February 2021 the asteroid was brighter than apparent magnitude 24, which still placed it near the limiting magnitude of even the best automated astronomical surveys. With an exceptionally high eccentricity of 0.59, it was the most eccentric Aten asteroid known at the time of discovery, more eccentric than previously discovered Aten, 3753 Cruithne. The formerly poorly known trajectory of this asteroid was further complicated by close approaches to Venus and Mercury. It was not until (0.277 AU) was discovered that an Aten asteroid with a lower perihelion was found. 's eccentric orbit takes it out past the Earth, where it has encounters of about 0.15 to 0.20 AU about every 3 to 5 years around October–November. While listed on the Sentry Risk Table, virtual clones of the asteroid that fit the uncertainty in the known trajectory showed 116 potential impacts between 2054 and 2109. On 26 March 2010, it may have come within 0.0825 AU (12.3 million km) of Earth, but with an uncertainty parameter of 9, the orbit is poorly determined. It was the eighth Aten asteroid discovered. The combination of a small semi- major axis and high eccentricity made the first Aten asteroid discovered to get closer to the Sun (0.295 AU) than Mercury ever does. 2340 Hathor (the second Aten discovered, in 1976) had the smallest perihelion (0.464 AU) earlier, which was about the same distance as Mercury's aphelion (0.467 AU). Its orbit has an eccentricity of 0.40 and an inclination of 7° with respect to the ecliptic. ",2,-7.5,"""2.74""",0.0384,1.60,E
+"The huge advantage of using the conservation of energy instead of Newton's laws of motion is that we can jump from the initial state to the final state without considering all the intermediate motion. Here is an example. In Figure, a child of mass $m$ is released from rest at the top of a water slide, at height $h=8.5 \mathrm{~m}$ above the bottom of the slide. Assuming that the slide is frictionless because of the water on it, find the child's speed at the bottom of the slide.","The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Classically, conservation of energy was distinct from conservation of mass. The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. The kinetic energy, as determined by the velocity, is converted to potential energy as it reaches the same height as the initial ball and the cycle repeats. thumb|An idealized Newton's cradle with five balls when there are no energy losses and there is always a small separation between the balls, except for when a pair is colliding thumb|Newton's cradle three-ball swing in a five-ball system. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. Many physicists at that time, such as Newton, held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum: :\sum_{i} m_i v_i was the conserved vis viva. All enforce the conservation of energy and momentum. The caloric theory maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable. Thus, conservation of energy (total, including material or rest energy) and conservation of mass (total, not just rest) are one (equivalent) law. The Newton's cradle is a device that demonstrates the conservation of momentum and the conservation of energy with swinging spheres. He showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle. Energy conservation has been a foundational physical principle for about two hundred years. For systems that include large gravitational fields, general relativity has to be taken into account; thus mass–energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity. == Formulation and examples == The law of conservation of mass can only be formulated in classical mechanics, in which the energy scales associated with an isolated system are much smaller than mc^2, where m is the mass of a typical object in the system, measured in the frame of reference where the object is at rest, and c is the speed of light. Given the stationary- action principle, conservation of energy can be rigorously proven by Noether's theorem as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time. Some say that this behavior demonstrates the conservation of momentum and kinetic energy in elastic collisions. ===Mechanical equivalent of heat=== A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind cannot exist; that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings.Planck, M. (1923/1927). The law of conservation of mass and the analogous law of conservation of energy were finally generalized and unified into the principle of mass–energy equivalence, described by Albert Einstein's famous formula E = mc^2. The energy conservation law is a consequence of the shift symmetry of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Engineers such as John Smeaton, Peter Ewart, , Gustave-Adolphe Hirn, and Marc Seguin recognized that conservation of momentum alone was not adequate for practical calculation and made use of Leibniz's principle. In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass cannot change, so the quantity can neither be added nor be removed. In reality, the conservation of mass only holds approximately and is considered part of a series of assumptions in classical mechanics. ",0.011,13,"""8.8""",4.979,0.132,B
+"Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.","* If one has $1000 invested for 1 year at a 7-day SEC yield of 2%, then: :(0.02 × $1000 ) / 365 ~= $0.05479 per day. * If one has $1000 invested for 30 days at a 7-day SEC yield of 5%, then: :(0.05 × $1000 ) / 365 ~= $0.137 per day. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. :Multiply by 365/7 to give the 7-day SEC yield. It is also referred to as the 7-day Annualized Yield. The calculation is performed as follows: :Take the net interest income earned by the fund over the last 7 days and subtract 7 days of management fees. It is important to note that the 7-day SEC yield is only an estimate of the fund's actual yield, and may not necessarily reflect the yield that an investor would receive if they held the fund for a longer period of time. ==Examples== The examples assume interest is withdrawn as it is earned and not allowed to compound. To calculate approximately how much interest one might earn in a money fund account, take the 7-day SEC yield, multiply by the amount invested, divide by the number of days in the year, and then multiply by the number of days in question. The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. Multiply by 30 days to yield $4.11 in interest. *For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The E-M rule could thus be written also as : t \approx \frac{70}{r} \times \frac{198}{200-r} or t \approx \frac{72}{r} \times \frac{192}{200-r} In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. ===Padé approximant=== The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: : t \approx \frac{69.3}{r} \times \frac{600+4r}{600+r} which simplifies to: : t \approx \frac{207900+1386r}{3000r+5r^2} ==Derivation== ===Periodic compounding === For periodic compounding, future value is given by: :FV = PV \cdot (1+r)^t where PV is the present value, t is the number of time periods, and r stands for the interest rate per time period. :Divide that dollar amount by the average size of the fund's investments over the same 7 days. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. == Using the rule to estimate compounding periods == To estimate the number of periods required to double an original investment, divide the most convenient ""rule-quantity"" by the expected growth rate, expressed as a percentage. Similarly, replacing the ""R"" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum. The 7-day SEC Yield is a measure of performance in the interest rates of money market mutual funds offered by US mutual fund companies. The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). 100 r is r written as a percentage. ",1.41, 7.0,"""-2.0""",0.5,7.25,E
+"Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \mathrm{~L}$ of a dye solution with a concentration of $1 \mathrm{~g} / \mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \mathrm{~L} / \mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \%$ of its original value.","Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. If one adds 1 litre of water to this solution, the salt concentration is reduced. Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. D_t=\left [ \frac{V}{Q} \right ] \cdot \ln \left [ \frac{C_\text{initial}}{C_\text{ending}}\right ] Sometimes the equation is also written as: \ln \left [ \frac{C_\text{ending}}{C_\text{initial}}\right ] \quad = {-}\frac{Q}{V} \cdot (t_\text{ending} - t_\text{initial}) where t_\text{initial} = 0 *Dt = time required; the unit of time used is the same as is used for Q *V = air or gas volume of the closed space or room in cubic feet, cubic metres or litres *Q = ventilation rate into or out of the room in cubic feet per minute, cubic metres per hour or litres per second *Cinitial = initial concentration of a vapor inside the room measured in ppm *Cfinal = final reduced concentration of the vapor inside the room in ppm ==Dilution ventilation equation== The basic room purge equation can be used only for purge scenarios. The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). Mathematically this relationship can be shown by equation: c_1 V_1 = c_2 V_2 where *c1 = initial concentration or molarity *V1 = initial volume *c2 = final concentration or molarity *V2 = final volume .... ==Basic room purge equation== The basic room purge equation is used in industrial hygiene. The mean free time for a molecule in a fluid is the average time between collisions. Time of concentration is useful in predicting flow rates that would result from hypothetical storms, which are based on statistically derived return periods through IDF curves.Sherman, C. (1931): Frequency and intensity of excessive rainfall at Boston, Massachusetts, Transactions, American Society of Civil Engineers, 95, 951–960. (pdf) For many (often economic) reasons, it is important for engineers and hydrologists to be able to accurately predict the response of a watershed to a given rain event. The concentration of this admixture should be small and the gradient of this concentration should be also small. Lake retention time (also called the residence time of lake water, or the water age or flushing time) is a calculated quantity expressing the mean time that water (or some dissolved substance) spends in a particular lake. In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). upright=1.35|thumb|Diluting a solution by adding more solvent Dilution is the process of decreasing the concentration of a solute in a solution, usually simply by mixing with more solvent like adding more water to the solution. In a scenario where a liquid continuously evaporates from a container in a ventilated room, a differential equation has to be used: \frac{dC}{dt} = \frac{G - Q' C}{V} where the ventilation rate has been adjusted by a mixing factor K: Q' = \frac{Q}{K} *C = concentration of a gas *G = generation rate *V = room volume *Q′ = adjusted ventilation rate of the volume ==Welding== The dilution in welding terms is defined as the weight of the base metal melted divided by the total weight of the weld metal. The solutions on the left are more dilute, compared to the more concentrated solutions on the right. The equation can only be applied when the purged volume of vapor or gas is replaced with ""clean"" air or gas. He introduced several mechanisms of diffusion and found rate constants from experimental data. ",26.9,313,"""0.66""",460.5,14,D
+A certain vibrating system satisfies the equation $u^{\prime \prime}+\gamma u^{\prime}+u=0$. Find the value of the damping coefficient $\gamma$ for which the quasi period of the damped motion is $50 \%$ greater than the period of the corresponding undamped motion.,"For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the system's equation of motion is : m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt{k m} or : c_c = 2 m \sqrt{\frac{k}{m}} = 2m \omega_n where : \omega_n = \sqrt{\frac{k}{m}} is the natural frequency of the system. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^{-\lambda t} \cos(\omega t - \varphi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\varphi is the phase angle at ; *\omega is the angular frequency. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. == Derivation == Using the natural frequency of a harmonic oscillator \omega_n = \sqrt{{k}/{m}} and the definition of the damping ratio above, we can rewrite this as: : \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. ==Damped sine wave== thumb|350px|Plot of a damped sinusoidal wave represented as the function y(t) = e^{- t} \cos(2 \pi t) A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. * Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. If the system has n degrees of freedom un and is under application of m damping forces. * Q factor: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio definition == thumb|400px|upright=1.3|The effect of varying damping ratio on a second-order system. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that : \zeta = \frac{1}{2 Q} = { \alpha \over \omega_n }. Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: thumb|Phase portrait of damped oscillator, with increasing damping strength. *Damped harmonic motion, see animation (right). The equation of motion is \ddot x + 2\gamma \dot x + \omega^2 x = 0. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. It corresponds to the underdamped case of damped second-order systems, or underdamped second-order differential equations.Douglas C. Giancoli (2000). Critically damped systems have a damping ratio of exactly 1, or at least very close to it. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Low damping materials may be utilized in musical instruments where sustained mechanical vibration and acoustic wave propagation is desired. ",0.08,0.249,"""11.0""",1.4907,2.72,D
 "Find the value of $y_0$ for which the solution of the initial value problem
 $$
 y^{\prime}-y=1+3 \sin t, \quad y(0)=y_0
 $$
-remains finite as $t \rightarrow \infty$","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. We are trying to find a formula for y(t) that satisfies these two equations. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Another solution is given by : y_s(x) = 0 . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. Starting with \varphi_0(t)=0, we iterate :\varphi_{k+1}(t)=\int_0^t (1+(\varphi_k(s))^2)\,ds so that \varphi_n(t) \to y(t): :\varphi_1(t)=\int_0^t (1+0^2)\,ds = t :\varphi_2(t)=\int_0^t (1+s^2)\,ds = t + \frac{t^3}{3} :\varphi_3(t)=\int_0^t \left(1+\left(s + \frac{s^3}{3}\right)^2\right)\,ds = t + \frac{t^3}{3} + \frac{2t^5}{15} + \frac{t^7}{63} and so on. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. ",0.7071067812, -2.5,2.0,272.8,0.24995,B
+remains finite as $t \rightarrow \infty$","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. We are trying to find a formula for y(t) that satisfies these two equations. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Another solution is given by : y_s(x) = 0 . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. Starting with \varphi_0(t)=0, we iterate :\varphi_{k+1}(t)=\int_0^t (1+(\varphi_k(s))^2)\,ds so that \varphi_n(t) \to y(t): :\varphi_1(t)=\int_0^t (1+0^2)\,ds = t :\varphi_2(t)=\int_0^t (1+s^2)\,ds = t + \frac{t^3}{3} :\varphi_3(t)=\int_0^t \left(1+\left(s + \frac{s^3}{3}\right)^2\right)\,ds = t + \frac{t^3}{3} + \frac{2t^5}{15} + \frac{t^7}{63} and so on. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. ",0.7071067812, -2.5,"""2.0""",272.8,0.24995,B
 "A certain spring-mass system satisfies the initial value problem
 $$
 u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0
 $$
 where $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.
-Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$.","The modified KdV–Burgers equation is a nonlinear partial differential equationAndrei D. Polyanin, Valentin F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, second edition, p 1041 CRC PRESS :u_t+u_{xxx}-\alpha u^2\,u_x - \beta u_{xx}=0. ==See also== *Burgers' equation *Korteweg–de Vries equation *modified KdV equation ==References== #Graham W. Griffiths William E. Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation :u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x = \frac16 (6u_{xxxx}+60uu_{xx}+45u_x^2+40u^3)_x. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. Unnormalized KdV equation is a nonlinear partial differential equationAndrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0 ==References== #Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. The case f(u) = 3u2 is the original Korteweg–De Vries equation. ==References== * Category:Partial differential equations The term for 0 < k < u, k even, may be simplified using the properties of the gamma function to :\operatorname E(T^k)= u^{\frac{k}{2}} \, \prod_{i=1}^{k/2} \frac{2i-1}{ u - 2i} \qquad k\text{ even},\quad 0 For a t-distribution with u degrees of freedom, the expected value is 0 if u>1, and its variance is \frac{ u}{ u-2} if u>2. Also, \omega(u)-e^{-\gamma} oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. In mathematics, a generalized Korteweg–De Vries equation is the nonlinear partial differential equation :\partial_t u + \partial_x^3 u + \partial_x f(u) = 0.\, The function f is sometimes taken to be f(u) = uk+1/(k+1) + u for some positive integer k (where the extra u is a ""drift term"" that makes the analysis a little easier). This may also be written as :f(t) = \frac{1}{\sqrt{ u}\,\mathrm{B} (\frac{1}{2}, \frac{ u}{2})} \left(1+\frac{t^2} u \right)^{-( u+1)/2}, where B is the Beta function. For t > 0, :F(t) = \int_{-\infty}^t f(u)\,du = 1 - \tfrac{1}{2} I_{x(t)}\left(\tfrac{ u}{2}, \tfrac{1}{2}\right), where :x(t) = \frac{ u}. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). Here a, b, and k are parameters. An alternative formula, valid for t^2 < u, is :\int_{-\infty}^t f(u)\,du = \tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}( u+1) \right)} {\sqrt{\pi u}\,\Gamma \left(\tfrac{ u}{2}\right)} \, {}_2F_1 \left( \tfrac{1}{2}, \tfrac{1}{2}( u+1); \tfrac{3}{2}; -\tfrac{t^2}{ u} \right), where 2F1 is a particular case of the hypergeometric function. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. 300px|thumbnail|Graph of the Buchstab function ω(u) from u = 1 to u = 4\. The Buchstab function (or Buchstab's function) is the unique continuous function \omega: \R_{\ge 1}\rightarrow \R_{>0} defined by the delay differential equation :\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2, :{\frac{d}{du}} (u\omega(u))=\omega(u-1), \qquad u\ge 2. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. #Eryk Infeld and George Rowlands, Nonlinear Waves, Solitons and Chaos,Cambridge 2000 #Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 #Dongming Wang, Elimination Practice, Imperial College Press 2004 # David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 # George Articolo Partial Differential Equations and Boundary Value Problems with Maple V Academic Press 1998 Category:Nonlinear partial differential equations #Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000 #Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 #Dongming Wang, Elimination Practice,Imperial College Press 2004 # David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 # George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 Category:Nonlinear partial differential equations It is the first equation in a hierarchy of integrable equations with the Lax operator : \partial_x^3 + 2u\partial_x + u_x, . The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. ==Applications== The Buchstab function is used to count rough numbers. ",5.5,0.9830,25.6773,131,399,C
+Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$.","The modified KdV–Burgers equation is a nonlinear partial differential equationAndrei D. Polyanin, Valentin F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, second edition, p 1041 CRC PRESS :u_t+u_{xxx}-\alpha u^2\,u_x - \beta u_{xx}=0. ==See also== *Burgers' equation *Korteweg–de Vries equation *modified KdV equation ==References== #Graham W. Griffiths William E. Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation :u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x = \frac16 (6u_{xxxx}+60uu_{xx}+45u_x^2+40u^3)_x. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. Unnormalized KdV equation is a nonlinear partial differential equationAndrei D. Polyanin,Valentin F. Zaitsev, HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITION p540-542 CRC PRESS u_{t}+\alpha*u_{xxx}+\beta*u*u_{x}=0 ==References== #Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer. The case f(u) = 3u2 is the original Korteweg–De Vries equation. ==References== * Category:Partial differential equations The term for 0 < k < u, k even, may be simplified using the properties of the gamma function to :\operatorname E(T^k)= u^{\frac{k}{2}} \, \prod_{i=1}^{k/2} \frac{2i-1}{ u - 2i} \qquad k\text{ even},\quad 0 For a t-distribution with u degrees of freedom, the expected value is 0 if u>1, and its variance is \frac{ u}{ u-2} if u>2. Also, \omega(u)-e^{-\gamma} oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. In mathematics, a generalized Korteweg–De Vries equation is the nonlinear partial differential equation :\partial_t u + \partial_x^3 u + \partial_x f(u) = 0.\, The function f is sometimes taken to be f(u) = uk+1/(k+1) + u for some positive integer k (where the extra u is a ""drift term"" that makes the analysis a little easier). This may also be written as :f(t) = \frac{1}{\sqrt{ u}\,\mathrm{B} (\frac{1}{2}, \frac{ u}{2})} \left(1+\frac{t^2} u \right)^{-( u+1)/2}, where B is the Beta function. For t > 0, :F(t) = \int_{-\infty}^t f(u)\,du = 1 - \tfrac{1}{2} I_{x(t)}\left(\tfrac{ u}{2}, \tfrac{1}{2}\right), where :x(t) = \frac{ u}. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). Here a, b, and k are parameters. An alternative formula, valid for t^2 < u, is :\int_{-\infty}^t f(u)\,du = \tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}( u+1) \right)} {\sqrt{\pi u}\,\Gamma \left(\tfrac{ u}{2}\right)} \, {}_2F_1 \left( \tfrac{1}{2}, \tfrac{1}{2}( u+1); \tfrac{3}{2}; -\tfrac{t^2}{ u} \right), where 2F1 is a particular case of the hypergeometric function. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. 300px|thumbnail|Graph of the Buchstab function ω(u) from u = 1 to u = 4\. The Buchstab function (or Buchstab's function) is the unique continuous function \omega: \R_{\ge 1}\rightarrow \R_{>0} defined by the delay differential equation :\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2, :{\frac{d}{du}} (u\omega(u))=\omega(u-1), \qquad u\ge 2. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. #Eryk Infeld and George Rowlands, Nonlinear Waves, Solitons and Chaos,Cambridge 2000 #Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 #Dongming Wang, Elimination Practice, Imperial College Press 2004 # David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 # George Articolo Partial Differential Equations and Boundary Value Problems with Maple V Academic Press 1998 Category:Nonlinear partial differential equations #Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000 #Saber Elaydi,An Introduction to Difference Equationns, Springer 2000 #Dongming Wang, Elimination Practice,Imperial College Press 2004 # David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 # George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 Category:Nonlinear partial differential equations It is the first equation in a hierarchy of integrable equations with the Lax operator : \partial_x^3 + 2u\partial_x + u_x, . The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. ==Applications== The Buchstab function is used to count rough numbers. ",5.5,0.9830,"""25.6773""",131,399,C
 "Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.
-Determine $T$ if $r=7 \%$.","The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). 100 r is r written as a percentage. *For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. The E-M rule could thus be written also as : t \approx \frac{70}{r} \times \frac{198}{200-r} or t \approx \frac{72}{r} \times \frac{192}{200-r} In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. ===Padé approximant=== The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: : t \approx \frac{69.3}{r} \times \frac{600+4r}{600+r} which simplifies to: : t \approx \frac{207900+1386r}{3000r+5r^2} ==Derivation== ===Periodic compounding === For periodic compounding, future value is given by: :FV = PV \cdot (1+r)^t where PV is the present value, t is the number of time periods, and r stands for the interest rate per time period. Similarly, replacing the ""R"" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%. * P = principal deposit * r = rate of return (monthly) * M = monthly deposit, and * t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} and adding all recurring deposits over the total period t (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month) : M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} recognizing the geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the closed-form formula (common ratio :1/(1+r)) we obtain: P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. Let RSt be the simple rate of return on the security from t − 1 to t. Accumulation functions for simple and compound interest are a(t)=1 + r t a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} If n t = 1, then these two functions are the same. ===Continuous compounding=== As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The continuously compounded rate of return or instantaneous rate of return RCt obtained during that period is : RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ). For continuous compounding, 69 gives accurate results for any rate, since ln(2) is about 69.3%; see derivation below. A practical estimate for reverse calculation of the rate of return when the exact date and amount of each recurring deposit is not known, a formula that assumes a uniform recurring monthly deposit over the period, is:http://moneychimp.com/features/portfolio_performance_calculator.htm ""recommended by The Four Pillars of Investing and The Motley Fool"" r=\left(\frac{P'-P-\sum{M}}{P+\sum{M}/2}\right)^{1/t} or r=\left(\frac{P'-\sum{M}/2}{P+\sum{M}/2}\right)^{1/t}-1 ==See also== * Credit card interest * Exponential growth * Fisher equation * Interest * Interest rate * Rate of return * Rate of return on investment * Real versus nominal value (economics) * Yield curve ==References== Category:Interest Category:Exponentials Category:Mathematical finance Category:Actuarial science it:Anatocismo When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. == Using the rule to estimate compounding periods == To estimate the number of periods required to double an original investment, divide the most convenient ""rule-quantity"" by the expected growth rate, expressed as a percentage. Compound interest is standard in finance and economics. For every three percentage points away from 8%, the value of 72 could be adjusted by 1: : t \approx \frac{72 + (r - 8)/3}{r} or, for the same result: : t \approx \frac{70 + (r - 2)/3}{r} Both of these equations simplify to: : t \approx \frac{208}{3r} + \frac{1}{3} Note that \frac{208}{3} is quite close to 69.3. ===E-M rule=== The Eckart–McHale second- order rule (the E-M rule) provides a multiplicative correction for the rule of 69.3 that is very accurate for rates from 0% to 20%, whereas the rule is normally only accurate at the lowest end of interest rates, from 0% to about 5%. Thus, continuing the above nominal example, the final value of the investment expressed in real terms is :P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}. For periodic compounding, the exact doubling time for an interest rate of r percent per period is :t = \frac{\ln(2)}{\ln(1+r/100)}\approx \frac{72}{r}, where t is the number of periods required. ",0.00024,1.5377,0.2115,9.90,52,D
-"A mass weighing $2 \mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position $u$ of the mass at any time $t$. Find the frequency of the motion.","The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. Since the inertia of the beam can be found from its mass, the spring constant can be calculated. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. In the case of =1, the considered problem has a closed solution: y(\tau )=\left[\frac{4}{3}\tau (1-\tau) -\frac{4}{3}\tau \left( 1+2 \tau\ln (1-\tau )+2\ln (1-\tau )\right)\right]\ . ==References== Category:Mechanical vibrations thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. We assume dimensionless displacements of the string and dimensionless time : thumb|240px|Massless string and a moving mass - mass trajectory. : y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ , where st is the static deflection in the middle of the string. More complex problems can be solved by the finite element method or space-time finite element method. massless load inertial load thumb|321px|Vibrations of a string under a moving massless force (v=0.1c); c is the wave speed. thumb|321px|Vibrations of a string under a moving massless force (v=0.5c); c is the wave speed. thumb|321px|Vibrations of a string under a moving inertial force (v=0.1c); c is the wave speed. thumb|321px|Vibrations of a string under a moving inertial force (v=0.5c); c is the wave speed. It has been the basis of all the most significant > experiments on gravitation ever since. ==Torsional harmonic oscillators== Definition of terms Term Unit Definition \theta\, rad Angle of deflection from rest position I\, kg m2 Moment of inertia C\, joule s rad−1 Angular damping constant \kappa\, N m rad−1 Torsion spring constant \tau\, \mathrm{N\,m}\, Drive torque f_n\, Hz Undamped (or natural) resonant frequency T_n\, s Undamped (or natural) period of oscillation \omega_n\, \mathrm{rad\,s^{-1}}\, Undamped resonant frequency in radians f\, Hz Damped resonant frequency \omega\, \mathrm{rad\,s^{-1}}\, Damped resonant frequency in radians \alpha\, \mathrm{s^{-1}}\, Reciprocal of damping time constant \phi\, rad Phase angle of oscillation L\, m Distance from axis to where force is applied Torsion balances, torsion pendulums and balance wheels are examples of torsional harmonic oscillators that can oscillate with a rotational motion about the axis of the torsion spring, clockwise and counterclockwise, in harmonic motion. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. If the damping is low, this can be obtained by measuring the natural resonant frequency of the balance, since the moment of inertia of the balance can usually be calculated from its geometry, so: :\kappa = (2\pi f_n)^2 I\, In measuring instruments, such as the D'Arsonval ammeter movement, it is often desired that the oscillatory motion die out quickly so the steady state result can be read off. *The torsion pendulum used in torsion pendulum clocks is a wheel-shaped weight suspended from its center by a wire torsion spring. Inertial load in numerical models is described in Unexpected property of differential equations that govern the motion of the mass particle travelling on the string, Timoshenko beam, and Mindlin plate is described in. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. In the case of inertial moving load, the analytical solutions are unknown. Made in the USA of cherry wood and power coated steel, it simulates a walking motion, using see-saw oscillations from 0-10mm, with a frequency of 0 to 15.5 Hz. The equation of motion is increased by the term related to the inertia of the moving load. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. The kinetic energy is given by: :T=\frac{1}{2}mv^2 where v is the velocity of the mass. ",0.33333333,6.283185307,0.03,9.30,0.7854,E
-"If $\mathbf{x}=\left(\begin{array}{c}2 \\ 3 i \\ 1-i\end{array}\right)$ and $\mathbf{y}=\left(\begin{array}{c}-1+i \\ 2 \\ 3-i\end{array}\right)$, find $(\mathbf{y}, \mathbf{y})$.","More precisely, given two sets of variables represented as coordinate vectors and y, then each equation of the system can be written y^TA_ix=g_i, where, is an integer whose value ranges from 1 to the number of equations, each A_i is a matrix, and each g_i is a real number. There are several possible ways to compute these quantities for a given implicit curve. XHJTA-FM is a radio station on 94.3 FM in Irapuato, Guanajuato. 300px|thumb|Cassini ovals: (1) a=1.1, c=1 (above), (2) a=c=1 (middle), (3) a=1, c=1.05 (below) 300px|thumb|Implicit curve: \sin(x+y)-\cos(xy)+1=0 In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa. is a passenger railway station located in the city of Himeji, Hyōgo Prefecture, Japan, operated by West Japan Railway Company (JR West). ==Lines== Yobe Station is served by the Kishin Line, and is located 6.1 kilometers from the terminus of the line at . ==Station layout== The station consists of two ground-level opposed side platforms connected by a level crossing. Finally, we calculate the value c via linear interpolation of c_{0} and c_{1} In practice, a trilinear interpolation is identical to two bilinear interpolation combined with a linear interpolation: :c \approx l\left( b(c_{000}, c_{010}, c_{100}, c_{110}),\, b(c_{001}, c_{011}, c_{101}, c_{111})\right) ===Alternative algorithm=== An alternative way to write the solution to the interpolation problem is :f(x, y, z) \approx a_0 + a_1 x + a_2 y + a_3 z + a_4 x y + a_5 x z + a_6 y z + a_7 x y z where the coefficients are found by solving the linear system :\begin{align} \begin{bmatrix} 1 & x_0 & y_0 & z_0 & x_0 y_0 & x_0 z_0 & y_0 z_0 & x_0 y_0 z_0 \\\ 1 & x_1 & y_0 & z_0 & x_1 y_0 & x_1 z_0 & y_0 z_0 & x_1 y_0 z_0 \\\ 1 & x_0 & y_1 & z_0 & x_0 y_1 & x_0 z_0 & y_1 z_0 & x_0 y_1 z_0 \\\ 1 & x_1 & y_1 & z_0 & x_1 y_1 & x_1 z_0 & y_1 z_0 & x_1 y_1 z_0 \\\ 1 & x_0 & y_0 & z_1 & x_0 y_0 & x_0 z_1 & y_0 z_1 & x_0 y_0 z_1 \\\ 1 & x_1 & y_0 & z_1 & x_1 y_0 & x_1 z_1 & y_0 z_1 & x_1 y_0 z_1 \\\ 1 & x_0 & y_1 & z_1 & x_0 y_1 & x_0 z_1 & y_1 z_1 & x_0 y_1 z_1 \\\ 1 & x_1 & y_1 & z_1 & x_1 y_1 & x_1 z_1 & y_1 z_1 & x_1 y_1 z_1 \end{bmatrix}\begin{bmatrix} a_0 \\\ a_1 \\\ a_2 \\\ a_3 \\\ a_4 \\\ a_5 \\\ a_6 \\\ a_7 \end{bmatrix} = \begin{bmatrix} c_{000} \\\ c_{100} \\\ c_{010} \\\ c_{110} \\\ c_{001} \\\ c_{101} \\\ c_{011} \\\ c_{111} \end{bmatrix}, \end{align} yielding the result :\begin{align} a_0 ={} &\frac{-c_{000} x_1 y_1 z_1 + c_{001} x_1 y_1 z_0 + c_{010} x_1 y_0 z_1 - c_{011} x_1 y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{ c_{100} x_0 y_1 z_1 - c_{101} x_0 y_1 z_0 - c_{110} x_0 y_0 z_1 + c_{111} x_0 y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_1 ={} &\frac{ c_{000} y_1 z_1 - c_{001} y_1 z_0 - c_{010} y_0 z_1 + c_{011} y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} y_1 z_1 + c_{101} y_1 z_0 + c_{110} y_0 z_1 - c_{111} y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_2 ={} &\frac{ c_{000} x_1 z_1 - c_{001} x_1 z_0 - c_{010} x_1 z_1 + c_{011} x_1 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} x_0 z_1 + c_{101} x_0 z_0 + c_{110} x_0 z_1 - c_{111} x_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_3 ={} &\frac{ c_{000} x_1 y_1 - c_{001} x_1 y_1 - c_{010} x_1 y_0 + c_{011} x_1 y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} x_0 y_1 + c_{101} x_0 y_1 + c_{110} x_0 y_0 - c_{111} x_0 y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_4 ={} &\frac{-c_{000} z_1 + c_{001} z_0 + c_{010} z_1 - c_{011} z_0 + c_{100} z_1 - c_{101} z_0 - c_{110} z_1 + c_{111} z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_5 = &\frac{-c_{000} y_1 + c_{001} y_1 + c_{010} y_0 - c_{011} y_0 + c_{100} y_1 - c_{101} y_1 - c_{110} y_0 + c_{111} y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_6 ={} &\frac{-c_{000} x_1 + c_{001} x_1 + c_{010} x_1 - c_{011} x_1 + c_{100} x_0 - c_{101} x_0 - c_{110} x_0 + c_{111} x_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_7 ={} &\frac{ c_{000} - c_{001} - c_{010} + c_{011} - c_{100} + c_{101} + c_{110} - c_{111}}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}. \end{align} ==See also== * Linear interpolation * Bilinear interpolation * Tricubic interpolation * Radial interpolation * Tetrahedral interpolation * Spherical Linear Interpolation ==External links== *pseudo-code from NASA, describes an iterative inverse trilinear interpolation (given the vertices and the value of C find Xd, Yd and Zd). Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. 'Systems of bilinear equations'. 250px|thumb|right|Scalar multiplication of a vector by a factor of 3 stretches the vector out. 250px|thumb|right|The scalar multiplications −a and 2a of a vector a In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Vector \mathbf t(x_0,y_0,z_0) is a tangent vector of the curve at point (x_0,y_0,z_0). 300px|thumb|Intersection curve between a sphere and a cylinder Examples: (1)\quad x+y+z-1=0 \ ,\ x-y+z-2=0 ::is a line. (2)\quad x^2+y^2+z^2-4=0 \ , \ x+y+z-1=0 ::is a plane section of a sphere, hence a circle. (3)\quad x^2+y^2-1=0 \ , \ x+y+z-1=0 ::is an ellipse (plane section of a cylinder). (4)\quad x^2+y^2+z^2-16=0 \ , \ (y-y_0)^2+z^2-9=0 ::is the intersection curve between a sphere and a cylinder. One method is to use implicit differentiation to compute the derivatives of y with respect to x. For a real scalar and matrix: : \lambda = 2, \quad \mathbf{A} =\begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix} : 2 \mathbf{A} = 2 \begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix} = \begin{pmatrix} 2 \\!\cdot\\! a & 2 \\!\cdot\\! b \\\ 2 \\!\cdot\\! c & 2 \\!\cdot\\! d \\\ \end{pmatrix} = \begin{pmatrix} a \\!\cdot\\! 2 & b \\!\cdot\\! 2 \\\ c \\!\cdot\\! 2 & d \\!\cdot\\! 2 \\\ \end{pmatrix} = \begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix}2= \mathbf{A}2. For quaternion scalars and matrices: : \lambda = i, \quad \mathbf{A} = \begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix} : i\begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix} = \begin{pmatrix} i^2 & 0 \\\ 0 & ij \\\ \end{pmatrix} = \begin{pmatrix} -1 & 0 \\\ 0 & k \\\ \end{pmatrix} e \begin{pmatrix} -1 & 0 \\\ 0 & -k \\\ \end{pmatrix} = \begin{pmatrix} i^2 & 0 \\\ 0 & ji \\\ \end{pmatrix} = \begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix}i\,, where are the quaternion units. The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along x, then along y, and finally along z, produces the same value. ) :(P3) until the distance between the points (x_{j+1},y_{j+1}),\, (x_j,y_j) is small enough. Next, we perform linear interpolation between c_{000} and c_{100} to find c_{00}, c_{001} and c_{101} to find c_{01}, c_{011} and c_{111} to find c_{11}, c_{010} and c_{110} to find c_{10}. First we interpolate along x (imagine we are ""pushing"" the face of the cube defined by C_{0jk} to the opposing face, defined by C_{1jk}), giving: : \begin{align} c_{00} &= c_{000} (1 - x_\text{d}) + c_{100} x_\text{d} \\\ c_{01} &= c_{001} (1 - x_\text{d}) + c_{101} x_\text{d} \\\ c_{10} &= c_{010} (1 - x_\text{d}) + c_{110} x_\text{d} \\\ c_{11} &= c_{011} (1 - x_\text{d}) + c_{111} x_\text{d} \end{align} Where c_{000} means the function value of (x_0, y_0, z_0). There are several ways to arrive at trilinear interpolation, which is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators. ==Method== right|thumb|Eight corner points on a cube surrounding the interpolation point C right|thumb|Depiction of 3D interpolation thumb|A geometric visualisation of trilinear interpolation. Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension (Tetralinear, Pentalinear, ...). A normal vector to the curve at the point is given by : \mathbf{n}(x_0,y_0) = (F_x(x_0,y_0), F_y(x_0,y_0)) (here written as a row vector). === Curvature === For readability of the formulas, the arguments (x_0,y_0) are omitted. ",5.828427125,2500,-21.2,0.54,16,E
+Determine $T$ if $r=7 \%$.","The function f(r) is shown to be accurate in the approximation of t for a small, positive interest rate when r=.08 (see derivation below). f(.08)\approx1.03949, and we therefore approximate time t as: : t=\bigg(\frac{\ln2}{r}\bigg)f(.08) \approx \frac{.72}{r} Written as a percentage: : \frac{.72}{r}=\frac{72}{100r} This approximation increases in accuracy as the compounding of interest becomes continuous (see derivation below). 100 r is r written as a percentage. *For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200; an exact calculation gives ln(2)/ln(1+0.09) = 8.0432 years. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. The E-M rule could thus be written also as : t \approx \frac{70}{r} \times \frac{198}{200-r} or t \approx \frac{72}{r} \times \frac{192}{200-r} In these variants, the multiplicative correction becomes 1 respectively for r=2 and r=8, the values for which the rules of 70 and 72 are most accurate. ===Padé approximant=== The third-order Padé approximant gives a more accurate answer over an even larger range of r, but it has a slightly more complicated formula: : t \approx \frac{69.3}{r} \times \frac{600+4r}{600+r} which simplifies to: : t \approx \frac{207900+1386r}{3000r+5r^2} ==Derivation== ===Periodic compounding === For periodic compounding, future value is given by: :FV = PV \cdot (1+r)^t where PV is the present value, t is the number of time periods, and r stands for the interest rate per time period. Similarly, replacing the ""R"" in R/200 on the third line with 2.02 gives 70 on the numerator, showing the rule of 70 is most accurate for periodically compounded interests around 2%. * P = principal deposit * r = rate of return (monthly) * M = monthly deposit, and * t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} and adding all recurring deposits over the total period t (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month) : M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} recognizing the geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the closed-form formula (common ratio :1/(1+r)) we obtain: P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. Let RSt be the simple rate of return on the security from t − 1 to t. Accumulation functions for simple and compound interest are a(t)=1 + r t a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} If n t = 1, then these two functions are the same. ===Continuous compounding=== As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The continuously compounded rate of return or instantaneous rate of return RCt obtained during that period is : RC_{t}=\ln\left (\frac{P_{t}}{P_{t-1}}\right ). For continuous compounding, 69 gives accurate results for any rate, since ln(2) is about 69.3%; see derivation below. A practical estimate for reverse calculation of the rate of return when the exact date and amount of each recurring deposit is not known, a formula that assumes a uniform recurring monthly deposit over the period, is:http://moneychimp.com/features/portfolio_performance_calculator.htm ""recommended by The Four Pillars of Investing and The Motley Fool"" r=\left(\frac{P'-P-\sum{M}}{P+\sum{M}/2}\right)^{1/t} or r=\left(\frac{P'-\sum{M}/2}{P+\sum{M}/2}\right)^{1/t}-1 ==See also== * Credit card interest * Exponential growth * Fisher equation * Interest * Interest rate * Rate of return * Rate of return on investment * Real versus nominal value (economics) * Yield curve ==References== Category:Interest Category:Exponentials Category:Mathematical finance Category:Actuarial science it:Anatocismo When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). As another example, if one wants to know the number of periods it takes for the initial value to rise by 50%, replace the constant 2 with 1.5. == Using the rule to estimate compounding periods == To estimate the number of periods required to double an original investment, divide the most convenient ""rule-quantity"" by the expected growth rate, expressed as a percentage. Compound interest is standard in finance and economics. For every three percentage points away from 8%, the value of 72 could be adjusted by 1: : t \approx \frac{72 + (r - 8)/3}{r} or, for the same result: : t \approx \frac{70 + (r - 2)/3}{r} Both of these equations simplify to: : t \approx \frac{208}{3r} + \frac{1}{3} Note that \frac{208}{3} is quite close to 69.3. ===E-M rule=== The Eckart–McHale second- order rule (the E-M rule) provides a multiplicative correction for the rule of 69.3 that is very accurate for rates from 0% to 20%, whereas the rule is normally only accurate at the lowest end of interest rates, from 0% to about 5%. Thus, continuing the above nominal example, the final value of the investment expressed in real terms is :P_t^{real} = P_t \cdot \frac{PL_{t-1}}{PL_t}. For periodic compounding, the exact doubling time for an interest rate of r percent per period is :t = \frac{\ln(2)}{\ln(1+r/100)}\approx \frac{72}{r}, where t is the number of periods required. ",0.00024,1.5377,"""0.2115""",9.90,52,D
+"A mass weighing $2 \mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, determine the position $u$ of the mass at any time $t$. Find the frequency of the motion.","The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. Since the inertia of the beam can be found from its mass, the spring constant can be calculated. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. In the case of =1, the considered problem has a closed solution: y(\tau )=\left[\frac{4}{3}\tau (1-\tau) -\frac{4}{3}\tau \left( 1+2 \tau\ln (1-\tau )+2\ln (1-\tau )\right)\right]\ . ==References== Category:Mechanical vibrations thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. We assume dimensionless displacements of the string and dimensionless time : thumb|240px|Massless string and a moving mass - mass trajectory. : y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ , where st is the static deflection in the middle of the string. More complex problems can be solved by the finite element method or space-time finite element method. massless load inertial load thumb|321px|Vibrations of a string under a moving massless force (v=0.1c); c is the wave speed. thumb|321px|Vibrations of a string under a moving massless force (v=0.5c); c is the wave speed. thumb|321px|Vibrations of a string under a moving inertial force (v=0.1c); c is the wave speed. thumb|321px|Vibrations of a string under a moving inertial force (v=0.5c); c is the wave speed. It has been the basis of all the most significant > experiments on gravitation ever since. ==Torsional harmonic oscillators== Definition of terms Term Unit Definition \theta\, rad Angle of deflection from rest position I\, kg m2 Moment of inertia C\, joule s rad−1 Angular damping constant \kappa\, N m rad−1 Torsion spring constant \tau\, \mathrm{N\,m}\, Drive torque f_n\, Hz Undamped (or natural) resonant frequency T_n\, s Undamped (or natural) period of oscillation \omega_n\, \mathrm{rad\,s^{-1}}\, Undamped resonant frequency in radians f\, Hz Damped resonant frequency \omega\, \mathrm{rad\,s^{-1}}\, Damped resonant frequency in radians \alpha\, \mathrm{s^{-1}}\, Reciprocal of damping time constant \phi\, rad Phase angle of oscillation L\, m Distance from axis to where force is applied Torsion balances, torsion pendulums and balance wheels are examples of torsional harmonic oscillators that can oscillate with a rotational motion about the axis of the torsion spring, clockwise and counterclockwise, in harmonic motion. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. If the damping is low, this can be obtained by measuring the natural resonant frequency of the balance, since the moment of inertia of the balance can usually be calculated from its geometry, so: :\kappa = (2\pi f_n)^2 I\, In measuring instruments, such as the D'Arsonval ammeter movement, it is often desired that the oscillatory motion die out quickly so the steady state result can be read off. *The torsion pendulum used in torsion pendulum clocks is a wheel-shaped weight suspended from its center by a wire torsion spring. Inertial load in numerical models is described in Unexpected property of differential equations that govern the motion of the mass particle travelling on the string, Timoshenko beam, and Mindlin plate is described in. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. In the case of inertial moving load, the analytical solutions are unknown. Made in the USA of cherry wood and power coated steel, it simulates a walking motion, using see-saw oscillations from 0-10mm, with a frequency of 0 to 15.5 Hz. The equation of motion is increased by the term related to the inertia of the moving load. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. The kinetic energy is given by: :T=\frac{1}{2}mv^2 where v is the velocity of the mass. ",0.33333333,6.283185307,"""0.03""",9.30,0.7854,E
+"If $\mathbf{x}=\left(\begin{array}{c}2 \\ 3 i \\ 1-i\end{array}\right)$ and $\mathbf{y}=\left(\begin{array}{c}-1+i \\ 2 \\ 3-i\end{array}\right)$, find $(\mathbf{y}, \mathbf{y})$.","More precisely, given two sets of variables represented as coordinate vectors and y, then each equation of the system can be written y^TA_ix=g_i, where, is an integer whose value ranges from 1 to the number of equations, each A_i is a matrix, and each g_i is a real number. There are several possible ways to compute these quantities for a given implicit curve. XHJTA-FM is a radio station on 94.3 FM in Irapuato, Guanajuato. 300px|thumb|Cassini ovals: (1) a=1.1, c=1 (above), (2) a=c=1 (middle), (3) a=1, c=1.05 (below) 300px|thumb|Implicit curve: \sin(x+y)-\cos(xy)+1=0 In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa. is a passenger railway station located in the city of Himeji, Hyōgo Prefecture, Japan, operated by West Japan Railway Company (JR West). ==Lines== Yobe Station is served by the Kishin Line, and is located 6.1 kilometers from the terminus of the line at . ==Station layout== The station consists of two ground-level opposed side platforms connected by a level crossing. Finally, we calculate the value c via linear interpolation of c_{0} and c_{1} In practice, a trilinear interpolation is identical to two bilinear interpolation combined with a linear interpolation: :c \approx l\left( b(c_{000}, c_{010}, c_{100}, c_{110}),\, b(c_{001}, c_{011}, c_{101}, c_{111})\right) ===Alternative algorithm=== An alternative way to write the solution to the interpolation problem is :f(x, y, z) \approx a_0 + a_1 x + a_2 y + a_3 z + a_4 x y + a_5 x z + a_6 y z + a_7 x y z where the coefficients are found by solving the linear system :\begin{align} \begin{bmatrix} 1 & x_0 & y_0 & z_0 & x_0 y_0 & x_0 z_0 & y_0 z_0 & x_0 y_0 z_0 \\\ 1 & x_1 & y_0 & z_0 & x_1 y_0 & x_1 z_0 & y_0 z_0 & x_1 y_0 z_0 \\\ 1 & x_0 & y_1 & z_0 & x_0 y_1 & x_0 z_0 & y_1 z_0 & x_0 y_1 z_0 \\\ 1 & x_1 & y_1 & z_0 & x_1 y_1 & x_1 z_0 & y_1 z_0 & x_1 y_1 z_0 \\\ 1 & x_0 & y_0 & z_1 & x_0 y_0 & x_0 z_1 & y_0 z_1 & x_0 y_0 z_1 \\\ 1 & x_1 & y_0 & z_1 & x_1 y_0 & x_1 z_1 & y_0 z_1 & x_1 y_0 z_1 \\\ 1 & x_0 & y_1 & z_1 & x_0 y_1 & x_0 z_1 & y_1 z_1 & x_0 y_1 z_1 \\\ 1 & x_1 & y_1 & z_1 & x_1 y_1 & x_1 z_1 & y_1 z_1 & x_1 y_1 z_1 \end{bmatrix}\begin{bmatrix} a_0 \\\ a_1 \\\ a_2 \\\ a_3 \\\ a_4 \\\ a_5 \\\ a_6 \\\ a_7 \end{bmatrix} = \begin{bmatrix} c_{000} \\\ c_{100} \\\ c_{010} \\\ c_{110} \\\ c_{001} \\\ c_{101} \\\ c_{011} \\\ c_{111} \end{bmatrix}, \end{align} yielding the result :\begin{align} a_0 ={} &\frac{-c_{000} x_1 y_1 z_1 + c_{001} x_1 y_1 z_0 + c_{010} x_1 y_0 z_1 - c_{011} x_1 y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{ c_{100} x_0 y_1 z_1 - c_{101} x_0 y_1 z_0 - c_{110} x_0 y_0 z_1 + c_{111} x_0 y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_1 ={} &\frac{ c_{000} y_1 z_1 - c_{001} y_1 z_0 - c_{010} y_0 z_1 + c_{011} y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} y_1 z_1 + c_{101} y_1 z_0 + c_{110} y_0 z_1 - c_{111} y_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_2 ={} &\frac{ c_{000} x_1 z_1 - c_{001} x_1 z_0 - c_{010} x_1 z_1 + c_{011} x_1 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} x_0 z_1 + c_{101} x_0 z_0 + c_{110} x_0 z_1 - c_{111} x_0 z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_3 ={} &\frac{ c_{000} x_1 y_1 - c_{001} x_1 y_1 - c_{010} x_1 y_0 + c_{011} x_1 y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)} +{} \\\ &\frac{-c_{100} x_0 y_1 + c_{101} x_0 y_1 + c_{110} x_0 y_0 - c_{111} x_0 y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_4 ={} &\frac{-c_{000} z_1 + c_{001} z_0 + c_{010} z_1 - c_{011} z_0 + c_{100} z_1 - c_{101} z_0 - c_{110} z_1 + c_{111} z_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_5 = &\frac{-c_{000} y_1 + c_{001} y_1 + c_{010} y_0 - c_{011} y_0 + c_{100} y_1 - c_{101} y_1 - c_{110} y_0 + c_{111} y_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_6 ={} &\frac{-c_{000} x_1 + c_{001} x_1 + c_{010} x_1 - c_{011} x_1 + c_{100} x_0 - c_{101} x_0 - c_{110} x_0 + c_{111} x_0}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}, \\\\[4pt] a_7 ={} &\frac{ c_{000} - c_{001} - c_{010} + c_{011} - c_{100} + c_{101} + c_{110} - c_{111}}{(x_0 - x_1) (y_0 - y_1) (z_0 - z_1)}. \end{align} ==See also== * Linear interpolation * Bilinear interpolation * Tricubic interpolation * Radial interpolation * Tetrahedral interpolation * Spherical Linear Interpolation ==External links== *pseudo-code from NASA, describes an iterative inverse trilinear interpolation (given the vertices and the value of C find Xd, Yd and Zd). Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. 'Systems of bilinear equations'. 250px|thumb|right|Scalar multiplication of a vector by a factor of 3 stretches the vector out. 250px|thumb|right|The scalar multiplications −a and 2a of a vector a In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Vector \mathbf t(x_0,y_0,z_0) is a tangent vector of the curve at point (x_0,y_0,z_0). 300px|thumb|Intersection curve between a sphere and a cylinder Examples: (1)\quad x+y+z-1=0 \ ,\ x-y+z-2=0 ::is a line. (2)\quad x^2+y^2+z^2-4=0 \ , \ x+y+z-1=0 ::is a plane section of a sphere, hence a circle. (3)\quad x^2+y^2-1=0 \ , \ x+y+z-1=0 ::is an ellipse (plane section of a cylinder). (4)\quad x^2+y^2+z^2-16=0 \ , \ (y-y_0)^2+z^2-9=0 ::is the intersection curve between a sphere and a cylinder. One method is to use implicit differentiation to compute the derivatives of y with respect to x. For a real scalar and matrix: : \lambda = 2, \quad \mathbf{A} =\begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix} : 2 \mathbf{A} = 2 \begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix} = \begin{pmatrix} 2 \\!\cdot\\! a & 2 \\!\cdot\\! b \\\ 2 \\!\cdot\\! c & 2 \\!\cdot\\! d \\\ \end{pmatrix} = \begin{pmatrix} a \\!\cdot\\! 2 & b \\!\cdot\\! 2 \\\ c \\!\cdot\\! 2 & d \\!\cdot\\! 2 \\\ \end{pmatrix} = \begin{pmatrix} a & b \\\ c & d \\\ \end{pmatrix}2= \mathbf{A}2. For quaternion scalars and matrices: : \lambda = i, \quad \mathbf{A} = \begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix} : i\begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix} = \begin{pmatrix} i^2 & 0 \\\ 0 & ij \\\ \end{pmatrix} = \begin{pmatrix} -1 & 0 \\\ 0 & k \\\ \end{pmatrix} e \begin{pmatrix} -1 & 0 \\\ 0 & -k \\\ \end{pmatrix} = \begin{pmatrix} i^2 & 0 \\\ 0 & ji \\\ \end{pmatrix} = \begin{pmatrix} i & 0 \\\ 0 & j \\\ \end{pmatrix}i\,, where are the quaternion units. The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along x, then along y, and finally along z, produces the same value. ) :(P3) until the distance between the points (x_{j+1},y_{j+1}),\, (x_j,y_j) is small enough. Next, we perform linear interpolation between c_{000} and c_{100} to find c_{00}, c_{001} and c_{101} to find c_{01}, c_{011} and c_{111} to find c_{11}, c_{010} and c_{110} to find c_{10}. First we interpolate along x (imagine we are ""pushing"" the face of the cube defined by C_{0jk} to the opposing face, defined by C_{1jk}), giving: : \begin{align} c_{00} &= c_{000} (1 - x_\text{d}) + c_{100} x_\text{d} \\\ c_{01} &= c_{001} (1 - x_\text{d}) + c_{101} x_\text{d} \\\ c_{10} &= c_{010} (1 - x_\text{d}) + c_{110} x_\text{d} \\\ c_{11} &= c_{011} (1 - x_\text{d}) + c_{111} x_\text{d} \end{align} Where c_{000} means the function value of (x_0, y_0, z_0). There are several ways to arrive at trilinear interpolation, which is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators. ==Method== right|thumb|Eight corner points on a cube surrounding the interpolation point C right|thumb|Depiction of 3D interpolation thumb|A geometric visualisation of trilinear interpolation. Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension (Tetralinear, Pentalinear, ...). A normal vector to the curve at the point is given by : \mathbf{n}(x_0,y_0) = (F_x(x_0,y_0), F_y(x_0,y_0)) (here written as a row vector). === Curvature === For readability of the formulas, the arguments (x_0,y_0) are omitted. ",5.828427125,2500,"""-21.2""",0.54,16,E
 "15. Consider the initial value problem
 $$
 4 y^{\prime \prime}+12 y^{\prime}+9 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-4 .
 $$
-Determine where the solution has the value zero.","Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. Furthermore, for a given x ot=0, this is the unique solution going through (x,y(x)). ==Failure of uniqueness== Consider the differential equation : y'(x)^2 = 4y(x) . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. If p' = 0 it means that y' = p = c = constant, and the general solution of this new equation is: : y_c(x) = c \cdot x + c^2 where c is determined by the initial value. Hence, : y_s(x) = -\tfrac{1}{4} \cdot x^2 \,\\! is tangent to every member of the one- parameter family of solutions : y_c(x) = c \cdot x + c^2 \,\\! of this Clairaut equation: : y(x) = x \cdot y' + (y')^2. \,\\! ==See also== * Chandrasekhar equation * Chrystal's equation * Caustic (mathematics) * Envelope (mathematics) * Initial value problem * Picard–Lindelöf theorem ==Bibliography== * Category:Differential equations Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Another solution is given by : y_s(x) = 0 . Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Now we shall check when these solutions are singular solutions. Uniqueness fails for these solutions on the interval c_1\leq x\leq c_2, and the solutions are singular, in the sense that the second derivative fails to exist, at x=c_1 and x=c_2. ==Further example of failure of uniqueness== The previous example might give the erroneous impression that failure of uniqueness is directly related to y(x)=0. The general solution to this equation is : y(x)= C x^{-2} . If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. # Check that the solution is consistent with step 2. ",131,0.2553,83.81,3.07, 0.4,E
-A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?,"If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. So the present value of the drawdowns is equal to the present value of the repayments, given the APR as the interest rate. For example, consider a 30-year loan of $200,000 with a stated APR of 10.00%, i.e., 10.0049% APR or the EAR equivalent of 10.4767%. There are two primary methods of borrowing money to buy a car: direct and indirect. thumb|279px|Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. If the $1000 one-time fees are taken into account then the yearly interest rate paid is effectively equal to 10.31%. The monthly payments, using APR, would be $1755.87. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). * APR is also an abbreviation for ""Annual Principal Rate"" which is sometimes used in the auto sales in some countries where the interest is calculated based on the ""Original Principal"" not the ""Current Principal Due"", so as the Current Principal Due decreases, the interest due does not. ==Rate format== An effective annual interest rate of 10% can also be expressed in several ways: * 0.7974% effective monthly interest rate, because 1.00797412=1.1 * 9.569% annual interest rate compounded monthly, because 12×0.7974=9.569 * 9.091% annual rate in advance, because (1.1-1)÷1.1=0.09091 These rates are all equivalent, but to a consumer who is not trained in the mathematics of finance, this can be confusing. In that case the formula becomes: :: S -A = R (1 + \mathrm{APR}/100)^{-t_N} + \sum_{k=1}^N A_k (1 + \mathrm{APR}/100)^{-t_k} :where: :: S is the borrowed amount or principal amount. Over 85% of new cars and half of used cars are financed (as opposed to being paid for in a lump sum with cash). If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated. Typically, the indirect auto lender will set an interest rate, known as the ""buy rate"". Using the improved notation of directive 2008/48/EC, the basic equation for calculation of APR in the EU is: :: \sum_{i=1}^M C_i (1 + \mathrm{APR}/100)^{-t_i} = \sum_{j=1}^N D_j (1 + \mathrm{APR}/100)^{-s_j} :where: :: M is the total number of drawdowns paid by the lender :: N is the total number of repayments paid by the borrower :: i is the sequence number of a drawdown paid by the lender :: j is the sequence number of a repayment paid by the borrower :: Ci is the cash flow amount for drawdown number i :: Dj is the cash flow amount for repayment number j :: ti is the interval, expressed in years and fractions of a year, between the date of the first drawdown* and the date of drawdown i :: sj is the interval, expressed in years and fractions of a year, between the date of the first drawdown* and the date of repayment j. In this equation the left side is the present value of the drawdowns made by the lender and the right side is the present value of the repayments made by the borrower. Suppose that the complete amount including the interest is withdrawn after exactly one year. Buy Here Pay Here financing accounts for 6% of the total financing market. Consumers can, of course, use the nominal interest rate and any costs on the loan (or savings account) and compute the APR themselves, for instance using one of the calculators on the internet. An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. ",38, 258.14,4.49,0.0000092,61,B
+Determine where the solution has the value zero.","Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. Furthermore, for a given x ot=0, this is the unique solution going through (x,y(x)). ==Failure of uniqueness== Consider the differential equation : y'(x)^2 = 4y(x) . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. If p' = 0 it means that y' = p = c = constant, and the general solution of this new equation is: : y_c(x) = c \cdot x + c^2 where c is determined by the initial value. Hence, : y_s(x) = -\tfrac{1}{4} \cdot x^2 \,\\! is tangent to every member of the one- parameter family of solutions : y_c(x) = c \cdot x + c^2 \,\\! of this Clairaut equation: : y(x) = x \cdot y' + (y')^2. \,\\! ==See also== * Chandrasekhar equation * Chrystal's equation * Caustic (mathematics) * Envelope (mathematics) * Initial value problem * Picard–Lindelöf theorem ==Bibliography== * Category:Differential equations Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Another solution is given by : y_s(x) = 0 . Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Now we shall check when these solutions are singular solutions. Uniqueness fails for these solutions on the interval c_1\leq x\leq c_2, and the solutions are singular, in the sense that the second derivative fails to exist, at x=c_1 and x=c_2. ==Further example of failure of uniqueness== The previous example might give the erroneous impression that failure of uniqueness is directly related to y(x)=0. The general solution to this equation is : y(x)= C x^{-2} . If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. # Check that the solution is consistent with step 2. ",131,0.2553,"""83.81""",3.07, 0.4,E
+A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?,"If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. So the present value of the drawdowns is equal to the present value of the repayments, given the APR as the interest rate. For example, consider a 30-year loan of $200,000 with a stated APR of 10.00%, i.e., 10.0049% APR or the EAR equivalent of 10.4767%. There are two primary methods of borrowing money to buy a car: direct and indirect. thumb|279px|Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. If the $1000 one-time fees are taken into account then the yearly interest rate paid is effectively equal to 10.31%. The monthly payments, using APR, would be $1755.87. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). * APR is also an abbreviation for ""Annual Principal Rate"" which is sometimes used in the auto sales in some countries where the interest is calculated based on the ""Original Principal"" not the ""Current Principal Due"", so as the Current Principal Due decreases, the interest due does not. ==Rate format== An effective annual interest rate of 10% can also be expressed in several ways: * 0.7974% effective monthly interest rate, because 1.00797412=1.1 * 9.569% annual interest rate compounded monthly, because 12×0.7974=9.569 * 9.091% annual rate in advance, because (1.1-1)÷1.1=0.09091 These rates are all equivalent, but to a consumer who is not trained in the mathematics of finance, this can be confusing. In that case the formula becomes: :: S -A = R (1 + \mathrm{APR}/100)^{-t_N} + \sum_{k=1}^N A_k (1 + \mathrm{APR}/100)^{-t_k} :where: :: S is the borrowed amount or principal amount. Over 85% of new cars and half of used cars are financed (as opposed to being paid for in a lump sum with cash). If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated. Typically, the indirect auto lender will set an interest rate, known as the ""buy rate"". Using the improved notation of directive 2008/48/EC, the basic equation for calculation of APR in the EU is: :: \sum_{i=1}^M C_i (1 + \mathrm{APR}/100)^{-t_i} = \sum_{j=1}^N D_j (1 + \mathrm{APR}/100)^{-s_j} :where: :: M is the total number of drawdowns paid by the lender :: N is the total number of repayments paid by the borrower :: i is the sequence number of a drawdown paid by the lender :: j is the sequence number of a repayment paid by the borrower :: Ci is the cash flow amount for drawdown number i :: Dj is the cash flow amount for repayment number j :: ti is the interval, expressed in years and fractions of a year, between the date of the first drawdown* and the date of drawdown i :: sj is the interval, expressed in years and fractions of a year, between the date of the first drawdown* and the date of repayment j. In this equation the left side is the present value of the drawdowns made by the lender and the right side is the present value of the repayments made by the borrower. Suppose that the complete amount including the interest is withdrawn after exactly one year. Buy Here Pay Here financing accounts for 6% of the total financing market. Consumers can, of course, use the nominal interest rate and any costs on the loan (or savings account) and compute the APR themselves, for instance using one of the calculators on the internet. An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. ",38, 258.14,"""4.49""",0.0000092,61,B
 "Consider the initial value problem
 $$
 y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0
 $$
 where $k$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coefficient (or resistance).
-Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.","* Determine the system steady-state gain k=A_0with k=\lim_{t\to\infty} \dfrac{y(t)}{x(t)} * Calculate r=\dfrac{t_{25}}{t_{75}} P=-18.56075\,r+\dfrac{0.57311}{r-0.20747}+4.16423 X=14.2797\,r^3-9.3891\,r^2+0.25437\,r+1.32148 * Determine the two time constants \tau_2=T_2=\dfrac{t_{75}-t_{25}}{X\,(1+1/P)} \tau_1=T_1=\dfrac{T_2}{P} * Calculate the transfer function of the identified system within the Laplace-domain G(s) = \dfrac{k}{(1+s\,T_1)\cdot(1+s\,T_2)} ====Phase margin==== thumbnail|280px|Figure 5: Bode gain plot to find phase margin; scales are logarithmic, so labeled separations are multiplicative factors. Here damping ratio is greater than one. ==Properties== thumb|Typical second order transient system properties Transient response can be quantified with the following properties. In particular, the unit step response of the system is: :S(t) = \left(\frac {A_0} {1+ \beta A_0}\right)\left(1 - e^{- \rho t} \ \frac { \sin \left( \mu t + \phi \right)}{ \sin \phi}\right)\ , which simplifies to :S(t) = 1 - e^{- \rho t} \ \frac { \sin \left( \mu t + \phi \right)}{ \sin \phi} when A0 tends to infinity and the feedback factor β is one. * Determine the time-spans t_{25}and t_{75}where the step response reaches 25% and 75% of the steady state output value. : \Delta = e^{- \rho t_S }\text{ or }t_S = \frac { \ln \frac{1}{\Delta} } { \rho } = \tau_2 \frac {2 \ln \frac{1} { \Delta} } { 1 + \frac { \tau_2 } { \tau_1} } \approx 2 \tau_2 \ln \frac{1} { \Delta}, where the τ1 ≫ τ2 is applicable because of the overshoot control condition, which makes τ1 = αβAOL τ2. The final value of the step response is 1, so the exponential is the actual overshoot itself. Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. Its step response is of the same form: an exponential decay toward the new equilibrium value. Notice that the damping of the response is set by ρ, that is, by the time constants of the open-loop amplifier. The equation reads as :\frac{d^2y}{d\zeta^2} =(y^2-\zeta^2)e^{-\delta^{-1/3}(y+\gamma \zeta)} subjected to the boundary conditions : \begin{align} \zeta\rightarrow -\infty : &\quad \frac{dy}{d\zeta}=-1,\\\ \zeta\rightarrow \infty : &\quad \frac{dy}{d\zeta}=1 \end{align} where \delta is the reduced or rescaled Damköhler number and \gamma is the ratio of excess heat conducted to one side of the reaction sheet to the total heat generated in the reaction zone. In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. For \delta>\delta_E with |\gamma|<1, the equation possess two solutions, of which one is an unstable solution. The impulse response and step response are transient responses to a specific input (an impulse and a step, respectively). This forward amplifier has unit step response :S_{OL}(t) = A_0(1 - e^{-t / \tau}), an exponential approach from 0 toward the new equilibrium value of A0. The time dependence of the amplifier is easy to discover by switching variables to s = jω, whereupon the gain becomes: : A_{FB} = \frac {A_0} { \tau_1 \tau_2 } \; \cdot \; \frac {1} {s^2 +s \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) + \frac {1+ \beta A_0} {\tau_1 \tau_2}} The poles of this expression (that is, the zeros of the denominator) occur at: :2s = - \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) \pm \sqrt { \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right) ^2 -\frac {4 \beta A_0 } {\tau_1 \tau_2 } }, which shows for large enough values of βA0 the square root becomes the square root of a negative number, that is the square root becomes imaginary, and the pole positions are complex conjugate numbers, either s+ or s−; see Figure 2: : s_{\pm} = -\rho \pm j \mu, with : \rho = \frac {1}{2} \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right ), and : \mu = \frac {1} {2} \sqrt { \frac {4 \beta A_0} { \tau_1 \tau_2} - \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2 }. The result for maximum step response Smax is: :S_\max= 1 + \exp \left( - \pi \frac { \rho }{ \mu } \right). It is clear the overshoot is zero if μ = 0, which is the condition: : \frac {4 \beta A_0} { \tau_1 \tau_2} = \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2. Comparing the top panel (α = 4) with the lower panel (α = 0.5) shows lower values for α increase the rate of response, but increase overshoot. As an example of this formula, if the settling time condition is tS = 8 τ2. The step response can be described by the following quantities related to its time behavior, *overshoot *rise time *settling time *ringing In the case of linear dynamic systems, much can be inferred about the system from these characteristics. The transient response is not necessarily tied to abrupt events but to any event that affects the equilibrium of the system. If there is a second pole, then as the closed-loop time constant approaches the time constant of the second pole, a two-pole analysis is needed. ===Two-pole amplifiers=== In the case that the open-loop gain has two poles (two time constants, τ1, τ2), the step response is a bit more complicated. ",2.8108,524,131.0,0.648004372,0.22222222,A
-"If a series circuit has a capacitor of $C=0.8 \times 10^{-6} \mathrm{~F}$ and an inductor of $L=0.2 \mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.","In the case of the series RLC circuit, the damping factor is given by :\zeta = \frac{\, R \,}{2} \sqrt{ \frac{C}{\, L \,} \,} = \frac{1}{\ 2 Q\ } ~. For the parallel circuit, the attenuation is given byNilsson and Riedel, p. 286. : \alpha = \frac{1}{\,2\,R\,C\,} and the damping factor is consequently :\zeta = \frac{1}{\,2\,R\,} \sqrt{\frac{L}{C}~}\,~. The value of the damping factor determines the type of transient that the circuit will exhibit.Irwin, pp. 217–220. ===Transient response=== thumb|350px|Plot showing underdamped and overdamped responses of a series RLC circuit to a voltage input step of 1 V. Rearranging for the case where is known – capacitance: : C = \frac{~\alpha + \beta~}{R\,\alpha\,\beta} \,, inductance (total of circuit and load): : L = \frac{R}{\,\alpha + \beta~} \,, initial terminal voltage of capacitor: : V_0 = \frac{\,-I_0 R\,\alpha\,\beta~}{\alpha + \beta} \left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,. ==See also== *RC circuit *RL circuit *Linear circuit == Footnotes == ==References== ==Bibliography== * * * * * Category:Analog circuits Category:Electronic filter topology For the case of the series RLC circuit these two parameters are given by:Agarwal and Lang, p. 641. :\begin{align} \alpha &= \frac{R}{\, 2L \,} \\\ \omega_0 &= \frac{1}{\, \sqrt{L\,C\,} \,} \;. \end{align} A useful parameter is the damping factor, , which is defined as the ratio of these two; although, sometimes is not used, and is referred to as damping factor instead; hence requiring careful specification of one's use of that term.Agarwal and Lang, p. 646. : \zeta \equiv \frac{\alpha}{\, \omega_0 \,} \;. These equations show that a series RC circuit has a time constant, usually denoted being the time it takes the voltage across the component to either rise (across the capacitor) or fall (across the resistor) to within of its final value. If the inductance is known, then the remaining parameters are given by the following – capacitance: : C = \frac{1}{~L\,\alpha\,\beta\,~} \,, resistance (total of circuit and load): : R = L\,(\,\alpha + \beta\,) \,, initial terminal voltage of capacitor: : V_0 = -I_0 L\,\alpha\,\beta\,\left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,. Parallel RC, series L circuit with resistance in parallel with the capacitor In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This results in the linear differential equation :C\frac{dV}{dt} + \frac{V}{R}=0 \,, where is the capacitance of the capacitor. Series RL, parallel C circuit with resistance in series with the inductor is the standard model for a self-resonant inductor A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding and its self- capacitance. thumb|350px|A series RLC network (in order): a resistor, an inductor, and a capacitor An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. With complex impedances: :\begin{align} I_R &= \frac{V_\mathrm{in}}{R} \\\ I_C &= j\omega C V_\mathrm{in}\,. \end{align} This shows that the capacitor current is 90° out of phase with the resistor (and source) current. The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. As a result, : \omega_\mathrm{d} \approx \omega_0 \,. === Voltage multiplier === In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit : I = \frac{V}{R}\,. As a result, \sigma = 0 and the impedance becomes :Z_C = \frac{1}{j\omega C} = - \frac{j}{\omega C} \,. ==Series circuit== By viewing the circuit as a voltage divider, the voltage across the capacitor is: :V_C(s) = \frac{\frac{1}{Cs}}{R + \frac{1}{Cs}}V_\mathrm{in}(s) = \frac{1}{1 + RCs}V_\mathrm{in}(s) and the voltage across the resistor is: :V_R(s) = \frac{R}{R + \frac{1}{Cs}}V_\mathrm{in}(s) = \frac{RCs}{1 + RCs}V_\mathrm{in}(s)\,. ===Transfer functions=== The transfer function from the input voltage to the voltage across the capacitor is :H_C(s) = \frac{ V_C(s) }{ V_\mathrm{in}(s) } = \frac{ 1 }{ 1 + RCs } \,. The current through the resistor must be equal in magnitude (but opposite in sign) to the time derivative of the accumulated charge on the capacitor. Solving this equation for yields the formula for exponential decay: :V(t)=V_0 e^{-\frac{t}{RC}} \,, where is the capacitor voltage at time . A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. The article next gives the analysis for the series RLC circuit in detail. Considering the expression for again, when :R \ll \frac{1}{\omega C}\,, so :\begin{align} I &\approx \frac{V_\mathrm{in}}\frac{1}{j\omega C} \\\ V_\mathrm{in} &\approx \frac{I}{j\omega C} = V_C \,.\end{align} Now, :\begin{align} V_R &= IR = C\frac{dV_C}{dt}R \\\ V_R &\approx RC\frac{dV_{in}}{dt}\,, \end{align} which is a differentiator across the resistor. RLC parallel circuit – the voltage source powering the circuit – the current admitted through the circuit – the equivalent resistance of the combined source, load, and components – the inductance of the inductor component – the capacitance of the capacitor component The properties of the parallel RLC circuit can be obtained from the duality relationship of electrical circuits and considering that the parallel RLC is the dual impedance of a series RLC. This article considers the RC circuit, in both series and parallel forms, as shown in the diagrams below. == Natural response == 200px|thumb|right| RC circuit The simplest RC circuit consists of a resistor and a charged capacitor connected to one another in a single loop, without an external voltage source. ",1000,4.4,0.264,25.6773,0.132,A
-"If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\prime \prime}+2 y^{\prime}+t e^t y=0$ and if $W\left(y_1, y_2\right)(1)=2$, find the value of $W\left(y_1, y_2\right)(5)$.","In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. The Battle of the Ypres–Comines Canal was a battle of the Second World War fought between the British Expeditionary Force (BEF) and German Army Group B during the BEF's retreat to Dunkirk in 1940. In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis function. Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Second Anglo-Sikh War was a military conflict between the Sikh Empire and the East India Company that took place in 1848 and 1849. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. Later developments indicated that the MFS can be used to solve partial differential equations with variable coefficients.C.M. Fan, C.S. Chen, J. Monroe, The method of fundamental solutions for solving convection- diffusion equations with variable coefficients, Advances in Applied Mathematics and Mechanics. 1 (2009) 215–230 The MFS has proved particularly effective for certain classes of problems such as inverse,Y.C. Hon, T. Wei, The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. A differential equation can be homogeneous in either of two respects. Introducing elliptic coordinates, : x = a \cosh \xi \cos \eta, : y = a \sinh \xi \sin \eta, the potential energy can be written as : V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)}, and the kinetic energy as : T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right). Terso Solutions, Inc., located in Madison, Wisconsin, USA, is the developer and distributor of an automated system for storage and distribution of high value research reagents and medical supplies. It follows that, if is a solution, so is , for any (non-zero) constant . Denoting the sum of the χ functions by Y, : Y = \chi_{1}(\varphi_{1}) + \chi_{2}(\varphi_{2}) + \cdots + \chi_{s}(\varphi_{s}), the kinetic energy can be written as : T = \frac{1}{2} Y F. Sci. 7 (2005) 119–132 unbounded domain, and free-boundary problems.A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics. 9 (1998) 69–95. Retrieved on July 7, 2008. thumb|right|300px|Terso Solutions, Madison, WI, USA. == History == Developed initially as an on-site inventory supplier for Promega products, the privately held Terso Solutions, Inc. was spun off from Promega Corporation in 2005. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s,R. Mathon, R.L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis. (1977) 638–650. followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications. The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems.Z. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method, Computational Mechanics. 44 (2009) 757–763.W. Chen, J. Lin, F. Wang, Regularized meshless method for nonhomogeneous problems , Engineering Analysis with Boundary Elements. 35 (2011) 253–257.W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary , Engineering Analysis with Boundary Elements. 34 (2010) 530–532.JIANG Xin-rong, CHEN Wen, Method of fundamental solution and boundary knot method for helmholtz equations: a comparative study, Chinese Journal of Computational Mechanics, 28:3(2011) 338–344 (in Chinese) In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability.M.A. Golberg, C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications. 5 (1994) 57–61.M. a. Golberg, C.S. Chen, H. Bowman, H. Power, Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method, Computational Mechanics. 21 (1998) 141–148. ",3.0,30,5.41,0.08,32,D
+Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.","* Determine the system steady-state gain k=A_0with k=\lim_{t\to\infty} \dfrac{y(t)}{x(t)} * Calculate r=\dfrac{t_{25}}{t_{75}} P=-18.56075\,r+\dfrac{0.57311}{r-0.20747}+4.16423 X=14.2797\,r^3-9.3891\,r^2+0.25437\,r+1.32148 * Determine the two time constants \tau_2=T_2=\dfrac{t_{75}-t_{25}}{X\,(1+1/P)} \tau_1=T_1=\dfrac{T_2}{P} * Calculate the transfer function of the identified system within the Laplace-domain G(s) = \dfrac{k}{(1+s\,T_1)\cdot(1+s\,T_2)} ====Phase margin==== thumbnail|280px|Figure 5: Bode gain plot to find phase margin; scales are logarithmic, so labeled separations are multiplicative factors. Here damping ratio is greater than one. ==Properties== thumb|Typical second order transient system properties Transient response can be quantified with the following properties. In particular, the unit step response of the system is: :S(t) = \left(\frac {A_0} {1+ \beta A_0}\right)\left(1 - e^{- \rho t} \ \frac { \sin \left( \mu t + \phi \right)}{ \sin \phi}\right)\ , which simplifies to :S(t) = 1 - e^{- \rho t} \ \frac { \sin \left( \mu t + \phi \right)}{ \sin \phi} when A0 tends to infinity and the feedback factor β is one. * Determine the time-spans t_{25}and t_{75}where the step response reaches 25% and 75% of the steady state output value. : \Delta = e^{- \rho t_S }\text{ or }t_S = \frac { \ln \frac{1}{\Delta} } { \rho } = \tau_2 \frac {2 \ln \frac{1} { \Delta} } { 1 + \frac { \tau_2 } { \tau_1} } \approx 2 \tau_2 \ln \frac{1} { \Delta}, where the τ1 ≫ τ2 is applicable because of the overshoot control condition, which makes τ1 = αβAOL τ2. The final value of the step response is 1, so the exponential is the actual overshoot itself. Using the equations above, the amount of overshoot can be found by differentiating the step response and finding its maximum value. Its step response is of the same form: an exponential decay toward the new equilibrium value. Notice that the damping of the response is set by ρ, that is, by the time constants of the open-loop amplifier. The equation reads as :\frac{d^2y}{d\zeta^2} =(y^2-\zeta^2)e^{-\delta^{-1/3}(y+\gamma \zeta)} subjected to the boundary conditions : \begin{align} \zeta\rightarrow -\infty : &\quad \frac{dy}{d\zeta}=-1,\\\ \zeta\rightarrow \infty : &\quad \frac{dy}{d\zeta}=1 \end{align} where \delta is the reduced or rescaled Damköhler number and \gamma is the ratio of excess heat conducted to one side of the reaction sheet to the total heat generated in the reaction zone. In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. For \delta>\delta_E with |\gamma|<1, the equation possess two solutions, of which one is an unstable solution. The impulse response and step response are transient responses to a specific input (an impulse and a step, respectively). This forward amplifier has unit step response :S_{OL}(t) = A_0(1 - e^{-t / \tau}), an exponential approach from 0 toward the new equilibrium value of A0. The time dependence of the amplifier is easy to discover by switching variables to s = jω, whereupon the gain becomes: : A_{FB} = \frac {A_0} { \tau_1 \tau_2 } \; \cdot \; \frac {1} {s^2 +s \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) + \frac {1+ \beta A_0} {\tau_1 \tau_2}} The poles of this expression (that is, the zeros of the denominator) occur at: :2s = - \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right) \pm \sqrt { \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right) ^2 -\frac {4 \beta A_0 } {\tau_1 \tau_2 } }, which shows for large enough values of βA0 the square root becomes the square root of a negative number, that is the square root becomes imaginary, and the pole positions are complex conjugate numbers, either s+ or s−; see Figure 2: : s_{\pm} = -\rho \pm j \mu, with : \rho = \frac {1}{2} \left( \frac {1} {\tau_1} + \frac {1} {\tau_2} \right ), and : \mu = \frac {1} {2} \sqrt { \frac {4 \beta A_0} { \tau_1 \tau_2} - \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2 }. The result for maximum step response Smax is: :S_\max= 1 + \exp \left( - \pi \frac { \rho }{ \mu } \right). It is clear the overshoot is zero if μ = 0, which is the condition: : \frac {4 \beta A_0} { \tau_1 \tau_2} = \left( \frac {1} {\tau_1} - \frac {1} {\tau_2} \right)^2. Comparing the top panel (α = 4) with the lower panel (α = 0.5) shows lower values for α increase the rate of response, but increase overshoot. As an example of this formula, if the settling time condition is tS = 8 τ2. The step response can be described by the following quantities related to its time behavior, *overshoot *rise time *settling time *ringing In the case of linear dynamic systems, much can be inferred about the system from these characteristics. The transient response is not necessarily tied to abrupt events but to any event that affects the equilibrium of the system. If there is a second pole, then as the closed-loop time constant approaches the time constant of the second pole, a two-pole analysis is needed. ===Two-pole amplifiers=== In the case that the open-loop gain has two poles (two time constants, τ1, τ2), the step response is a bit more complicated. ",2.8108,524,"""131.0""",0.648004372,0.22222222,A
+"If a series circuit has a capacitor of $C=0.8 \times 10^{-6} \mathrm{~F}$ and an inductor of $L=0.2 \mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.","In the case of the series RLC circuit, the damping factor is given by :\zeta = \frac{\, R \,}{2} \sqrt{ \frac{C}{\, L \,} \,} = \frac{1}{\ 2 Q\ } ~. For the parallel circuit, the attenuation is given byNilsson and Riedel, p. 286. : \alpha = \frac{1}{\,2\,R\,C\,} and the damping factor is consequently :\zeta = \frac{1}{\,2\,R\,} \sqrt{\frac{L}{C}~}\,~. The value of the damping factor determines the type of transient that the circuit will exhibit.Irwin, pp. 217–220. ===Transient response=== thumb|350px|Plot showing underdamped and overdamped responses of a series RLC circuit to a voltage input step of 1 V. Rearranging for the case where is known – capacitance: : C = \frac{~\alpha + \beta~}{R\,\alpha\,\beta} \,, inductance (total of circuit and load): : L = \frac{R}{\,\alpha + \beta~} \,, initial terminal voltage of capacitor: : V_0 = \frac{\,-I_0 R\,\alpha\,\beta~}{\alpha + \beta} \left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,. ==See also== *RC circuit *RL circuit *Linear circuit == Footnotes == ==References== ==Bibliography== * * * * * Category:Analog circuits Category:Electronic filter topology For the case of the series RLC circuit these two parameters are given by:Agarwal and Lang, p. 641. :\begin{align} \alpha &= \frac{R}{\, 2L \,} \\\ \omega_0 &= \frac{1}{\, \sqrt{L\,C\,} \,} \;. \end{align} A useful parameter is the damping factor, , which is defined as the ratio of these two; although, sometimes is not used, and is referred to as damping factor instead; hence requiring careful specification of one's use of that term.Agarwal and Lang, p. 646. : \zeta \equiv \frac{\alpha}{\, \omega_0 \,} \;. These equations show that a series RC circuit has a time constant, usually denoted being the time it takes the voltage across the component to either rise (across the capacitor) or fall (across the resistor) to within of its final value. If the inductance is known, then the remaining parameters are given by the following – capacitance: : C = \frac{1}{~L\,\alpha\,\beta\,~} \,, resistance (total of circuit and load): : R = L\,(\,\alpha + \beta\,) \,, initial terminal voltage of capacitor: : V_0 = -I_0 L\,\alpha\,\beta\,\left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,. Parallel RC, series L circuit with resistance in parallel with the capacitor In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This results in the linear differential equation :C\frac{dV}{dt} + \frac{V}{R}=0 \,, where is the capacitance of the capacitor. Series RL, parallel C circuit with resistance in series with the inductor is the standard model for a self-resonant inductor A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding and its self- capacitance. thumb|350px|A series RLC network (in order): a resistor, an inductor, and a capacitor An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. With complex impedances: :\begin{align} I_R &= \frac{V_\mathrm{in}}{R} \\\ I_C &= j\omega C V_\mathrm{in}\,. \end{align} This shows that the capacitor current is 90° out of phase with the resistor (and source) current. The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. As a result, : \omega_\mathrm{d} \approx \omega_0 \,. === Voltage multiplier === In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit : I = \frac{V}{R}\,. As a result, \sigma = 0 and the impedance becomes :Z_C = \frac{1}{j\omega C} = - \frac{j}{\omega C} \,. ==Series circuit== By viewing the circuit as a voltage divider, the voltage across the capacitor is: :V_C(s) = \frac{\frac{1}{Cs}}{R + \frac{1}{Cs}}V_\mathrm{in}(s) = \frac{1}{1 + RCs}V_\mathrm{in}(s) and the voltage across the resistor is: :V_R(s) = \frac{R}{R + \frac{1}{Cs}}V_\mathrm{in}(s) = \frac{RCs}{1 + RCs}V_\mathrm{in}(s)\,. ===Transfer functions=== The transfer function from the input voltage to the voltage across the capacitor is :H_C(s) = \frac{ V_C(s) }{ V_\mathrm{in}(s) } = \frac{ 1 }{ 1 + RCs } \,. The current through the resistor must be equal in magnitude (but opposite in sign) to the time derivative of the accumulated charge on the capacitor. Solving this equation for yields the formula for exponential decay: :V(t)=V_0 e^{-\frac{t}{RC}} \,, where is the capacitor voltage at time . A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. The article next gives the analysis for the series RLC circuit in detail. Considering the expression for again, when :R \ll \frac{1}{\omega C}\,, so :\begin{align} I &\approx \frac{V_\mathrm{in}}\frac{1}{j\omega C} \\\ V_\mathrm{in} &\approx \frac{I}{j\omega C} = V_C \,.\end{align} Now, :\begin{align} V_R &= IR = C\frac{dV_C}{dt}R \\\ V_R &\approx RC\frac{dV_{in}}{dt}\,, \end{align} which is a differentiator across the resistor. RLC parallel circuit – the voltage source powering the circuit – the current admitted through the circuit – the equivalent resistance of the combined source, load, and components – the inductance of the inductor component – the capacitance of the capacitor component The properties of the parallel RLC circuit can be obtained from the duality relationship of electrical circuits and considering that the parallel RLC is the dual impedance of a series RLC. This article considers the RC circuit, in both series and parallel forms, as shown in the diagrams below. == Natural response == 200px|thumb|right| RC circuit The simplest RC circuit consists of a resistor and a charged capacitor connected to one another in a single loop, without an external voltage source. ",1000,4.4,"""0.264""",25.6773,0.132,A
+"If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\prime \prime}+2 y^{\prime}+t e^t y=0$ and if $W\left(y_1, y_2\right)(1)=2$, find the value of $W\left(y_1, y_2\right)(5)$.","In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. The Battle of the Ypres–Comines Canal was a battle of the Second World War fought between the British Expeditionary Force (BEF) and German Army Group B during the BEF's retreat to Dunkirk in 1940. In scientific computation and simulation, the method of fundamental solutions (MFS) is a technique for solving partial differential equations based on using the fundamental solution as a basis function. Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Second Anglo-Sikh War was a military conflict between the Sikh Empire and the East India Company that took place in 1848 and 1849. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. Later developments indicated that the MFS can be used to solve partial differential equations with variable coefficients.C.M. Fan, C.S. Chen, J. Monroe, The method of fundamental solutions for solving convection- diffusion equations with variable coefficients, Advances in Applied Mathematics and Mechanics. 1 (2009) 215–230 The MFS has proved particularly effective for certain classes of problems such as inverse,Y.C. Hon, T. Wei, The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES Comput. A differential equation can be homogeneous in either of two respects. Introducing elliptic coordinates, : x = a \cosh \xi \cos \eta, : y = a \sinh \xi \sin \eta, the potential energy can be written as : V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)}, and the kinetic energy as : T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right). Terso Solutions, Inc., located in Madison, Wisconsin, USA, is the developer and distributor of an automated system for storage and distribution of high value research reagents and medical supplies. It follows that, if is a solution, so is , for any (non-zero) constant . Denoting the sum of the χ functions by Y, : Y = \chi_{1}(\varphi_{1}) + \chi_{2}(\varphi_{2}) + \cdots + \chi_{s}(\varphi_{s}), the kinetic energy can be written as : T = \frac{1}{2} Y F. Sci. 7 (2005) 119–132 unbounded domain, and free-boundary problems.A.K. G. Fairweather, The method of fundamental solutions for elliptic boundary value problems, Advances in Computational Mathematics. 9 (1998) 69–95. Retrieved on July 7, 2008. thumb|right|300px|Terso Solutions, Madison, WI, USA. == History == Developed initially as an on-site inventory supplier for Promega products, the privately held Terso Solutions, Inc. was spun off from Promega Corporation in 2005. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s,R. Mathon, R.L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM Journal on Numerical Analysis. (1977) 638–650. followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications. The MFS then gradually became a useful tool for the solution of a large variety of physical and engineering problems.Z. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method, Computational Mechanics. 44 (2009) 757–763.W. Chen, J. Lin, F. Wang, Regularized meshless method for nonhomogeneous problems , Engineering Analysis with Boundary Elements. 35 (2011) 253–257.W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary , Engineering Analysis with Boundary Elements. 34 (2010) 530–532.JIANG Xin-rong, CHEN Wen, Method of fundamental solution and boundary knot method for helmholtz equations: a comparative study, Chinese Journal of Computational Mechanics, 28:3(2011) 338–344 (in Chinese) In the 1990s, M. A. Golberg and C. S. Chen extended the MFS to deal with inhomogeneous equations and time-dependent problems, greatly expanding its applicability.M.A. Golberg, C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications. 5 (1994) 57–61.M. a. Golberg, C.S. Chen, H. Bowman, H. Power, Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method, Computational Mechanics. 21 (1998) 141–148. ",3.0,30,"""5.41""",0.08,32,D
 "Consider the initial value problem
 $$
 5 u^{\prime \prime}+2 u^{\prime}+7 u=0, \quad u(0)=2, \quad u^{\prime}(0)=1
 $$
-Find the smallest $T$ such that $|u(t)| \leq 0.1$ for all $t>T$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Assuming W_c is nonsingular (if and only if the system is controllable), the minimum energy control is then : u(t) = -B^*e^{A^*(t_1-t)}W_c^{-1}(t_1)[e^{A(t_1-t_0)}x_0-x_1]. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Also, \omega(u)-e^{-\gamma} oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. In control theory, the minimum energy control is the control u(t) that will bring a linear time invariant system to a desired state with a minimum expenditure of energy. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. ==Applications== The Buchstab function is used to count rough numbers. Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. 300px|thumbnail|Graph of the Buchstab function ω(u) from u = 1 to u = 4\. We are trying to find a formula for y(t) that satisfies these two equations. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. One seeks an input u(t) so that the system will be in the state x_1 at time t_1, and for any other input \bar{u}(t), which also drives the system from x_0 to x_1 at time t_1, the energy expenditure would be larger, i.e., : \int_{t_0}^{t_1} \bar{u}^*(t) \bar{u}(t) dt \ \geq \ \int_{t_0}^{t_1} u^*(t) u(t) dt. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. ",71,-273,4943.0,1.2,14.5115,E
+Find the smallest $T$ such that $|u(t)| \leq 0.1$ for all $t>T$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Assuming W_c is nonsingular (if and only if the system is controllable), the minimum energy control is then : u(t) = -B^*e^{A^*(t_1-t)}W_c^{-1}(t_1)[e^{A(t_1-t_0)}x_0-x_1]. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Also, \omega(u)-e^{-\gamma} oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. In control theory, the minimum energy control is the control u(t) that will bring a linear time invariant system to a desired state with a minimum expenditure of energy. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.p. 131, Cheer and Goldston 1990. ==Applications== The Buchstab function is used to count rough numbers. Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. 300px|thumbnail|Graph of the Buchstab function ω(u) from u = 1 to u = 4\. We are trying to find a formula for y(t) that satisfies these two equations. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. One seeks an input u(t) so that the system will be in the state x_1 at time t_1, and for any other input \bar{u}(t), which also drives the system from x_0 to x_1 at time t_1, the energy expenditure would be larger, i.e., : \int_{t_0}^{t_1} \bar{u}^*(t) \bar{u}(t) dt \ \geq \ \int_{t_0}^{t_1} u^*(t) u(t) dt. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. ",71,-273,"""4943.0""",1.2,14.5115,E
 "Consider the initial value problem
 $$
 y^{\prime}=t y(4-y) / 3, \quad y(0)=y_0
 $$
-Suppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. ",15,16,260.0,-3.141592,3.29527,E
+Suppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. ",15,16,"""260.0""",-3.141592,3.29527,E
 "25. Consider the initial value problem
 $$
 2 y^{\prime \prime}+3 y^{\prime}-2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-\beta,
 $$
 where $\beta>0$.
-Find the smallest value of $\beta$ for which the solution has no minimum point.","Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. (See figure at top of page.) x3 \+ 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] Local maximum at x = −1−/3, local minimum at x = −1+/3, global maximum at x = 2 and global minimum at x = −4. This is illustrated by the function :f(x,y)= x^2+y^2(1-x)^3,\qquad x,y \in \R, whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0\. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem A function need not have a least fixed point, but if it does then the least fixed point is unique. ==Examples== With the usual order on the real numbers, the least fixed point of the real function f(x) = x2 is x = 0 (since the only other fixed point is 1 and 0 < 1). From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. Then the second derivative test provides a sufficient condition for the point to be a local maximum or local minimum. ==Search techniques== Local search or hill climbing methods for solving optimization problems start from an initial configuration and repeatedly move to an improving neighboring configuration. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. If the first derivative exists everywhere, it can be equated to zero; if the function has an unbounded domain, for a point to be a local optimum it is necessary that it satisfy this equation. The definition of global minimum point also proceeds similarly. This function has no global maximum or minimum. x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. cos(x) Infinitely many global maxima at 0, ±2, ±4, ..., and infinitely many global minima at ±, ±3, ±5, .... 2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum. cos(3x)/x with Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. thumb|right|220px|Attraction basins around locally optimal points thumb|right|233px|Polynomial of degree 4: the trough on the right is a local minimum and the one on the left is the global minimum. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative 2x. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. thumb|150px|The function f(x) = x2 − 4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2 − /2. However, not all critical points are extrema. ",22,8.99,0.318,38,2,E
+Find the smallest value of $\beta$ for which the solution has no minimum point.","Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. (See figure at top of page.) x3 \+ 3x2 − 2x + 1 defined over the closed interval (segment) [−4,2] Local maximum at x = −1−/3, local minimum at x = −1+/3, global maximum at x = 2 and global minimum at x = −4. This is illustrated by the function :f(x,y)= x^2+y^2(1-x)^3,\qquad x,y \in \R, whose only critical point is at (0,0), which is a local minimum with f(0,0) = 0\. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem A function need not have a least fixed point, but if it does then the least fixed point is unique. ==Examples== With the usual order on the real numbers, the least fixed point of the real function f(x) = x2 is x = 0 (since the only other fixed point is 1 and 0 < 1). From the sign of the second derivative, we can see that −1 is a local maximum and +1 is a local minimum. Then the second derivative test provides a sufficient condition for the point to be a local maximum or local minimum. ==Search techniques== Local search or hill climbing methods for solving optimization problems start from an initial configuration and repeatedly move to an improving neighboring configuration. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. If the first derivative exists everywhere, it can be equated to zero; if the function has an unbounded domain, for a point to be a local optimum it is necessary that it satisfy this equation. The definition of global minimum point also proceeds similarly. This function has no global maximum or minimum. x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. cos(x) Infinitely many global maxima at 0, ±2, ±4, ..., and infinitely many global minima at ±, ±3, ±5, .... 2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum. cos(3x)/x with Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. thumb|right|220px|Attraction basins around locally optimal points thumb|right|233px|Polynomial of degree 4: the trough on the right is a local minimum and the one on the left is the global minimum. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative 2x. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability. thumb|150px|The function f(x) = x2 − 4 has two fixed points, shown as the intersection with the blue line; its least one is at 1/2 − /2. However, not all critical points are extrema. ",22,8.99,"""0.318""",38,2,E
 "Consider the initial value problem
 $$
 y^{\prime}=t y(4-y) /(1+t), \quad y(0)=y_0>0 .
 $$
-If $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. We are trying to find a formula for y(t) that satisfies these two equations. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons. ==Linear system== ===Discrete time=== A linear matrix difference equation of the homogeneous (having no constant term) form X_{t+1}=AX_t has closed form solution X_t=A^tX_0 predicated on the vector X_0 of initial conditions on the individual variables that are stacked into the vector; X_0 is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. We use their notation, and assume that the unknown function is u, and that we have a known solution u_n at time t_n. Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. Here the constants c_1, \dots , c_k are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition x_t Is known. ===Continuous time=== A differential equation system of the first order with n variables stacked in a vector X is :\frac{dX}{dt}=AX. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad u(t_0)=u_0, after linearizing locally about a fixed or local state u^*: : L = \frac{\partial f}{\partial u}(u^*); \qquad N = f(u) - L u. The characteristic equation of this dynamic equation is \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, whose solutions are the characteristic values \lambda_1,\dots , \lambda_k; these are used in the solution equation :x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. ",4.8,17,2.84367,131,-32,C
+If $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. We are trying to find a formula for y(t) that satisfies these two equations. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons. ==Linear system== ===Discrete time=== A linear matrix difference equation of the homogeneous (having no constant term) form X_{t+1}=AX_t has closed form solution X_t=A^tX_0 predicated on the vector X_0 of initial conditions on the individual variables that are stacked into the vector; X_0 is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. We use their notation, and assume that the unknown function is u, and that we have a known solution u_n at time t_n. Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. Here the constants c_1, \dots , c_k are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition x_t Is known. ===Continuous time=== A differential equation system of the first order with n variables stacked in a vector X is :\frac{dX}{dt}=AX. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad u(t_0)=u_0, after linearizing locally about a fixed or local state u^*: : L = \frac{\partial f}{\partial u}(u^*); \qquad N = f(u) - L u. The characteristic equation of this dynamic equation is \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, whose solutions are the characteristic values \lambda_1,\dots , \lambda_k; these are used in the solution equation :x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. ",4.8,17,"""2.84367""",131,-32,C
 "28. A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
-If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped.","If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). Some heights are difficult to verify due to lack of documentation and are approximated. ==List of the highest falls survived without a parachute== Name Image Distance of fall Date Notes and References Feet Meters Vesna Vulović 33,330 10160 1972 Flight attendant from Serbia who was the sole survivor of an airplane bombing mid-air. The mass m0 used in the fall is 80 kg. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. thumb|250px|The climber will fall about the same height h in both cases, but they will be subjected to a greater force at position 1, due to the greater fall factor. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. This article attempts to list of the highest falls survived without a parachute. We first state an equation for this quantity and describe its interpretation, and then show its derivation and how it can be put into a more convenient form. ===Equation for the impact force and its interpretation=== When modeling the rope as an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by: :F_{max} = mg + \sqrt{(mg)^2 + 2mghk}, where mg is the climber's weight, h is the fall height and k is the spring constant of the portion of the rope that is in play. thumb|The aftermath of a hypervelocity impact, with a projectile the same size as the one that impacted for scale Hypervelocity is very high velocity, approximately over 3,000 meters per second (6,700 mph, 11,000 km/h, 10,000 ft/s, or Mach 8.8). When climbing from the ground up, the maximum possible fall factor is 1, since any greater fall would mean that the climber hit the ground. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. The amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. The mass of the displaced fluid can be expressed in terms of the density and its volume, . We will see below that when varying the height of the fall while keeping the fall factor fixed, the quantity hk stays constant. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). As a numerical example, consider a fall of 20 feet that occurs with 10 feet of rope out (i.e., the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below—a factor 2 fall). This fall produces far more force on the climber and the gear than if a similar 20 foot fall had occurred 100 feet above the belayer. Therefore, the weight of the displaced fluid can be expressed as . ",144, 13.45,0.0182,0.6321205588,35,B
+If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped.","If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). Some heights are difficult to verify due to lack of documentation and are approximated. ==List of the highest falls survived without a parachute== Name Image Distance of fall Date Notes and References Feet Meters Vesna Vulović 33,330 10160 1972 Flight attendant from Serbia who was the sole survivor of an airplane bombing mid-air. The mass m0 used in the fall is 80 kg. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. thumb|250px|The climber will fall about the same height h in both cases, but they will be subjected to a greater force at position 1, due to the greater fall factor. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. This article attempts to list of the highest falls survived without a parachute. We first state an equation for this quantity and describe its interpretation, and then show its derivation and how it can be put into a more convenient form. ===Equation for the impact force and its interpretation=== When modeling the rope as an undamped harmonic oscillator (HO) the impact force Fmax in the rope is given by: :F_{max} = mg + \sqrt{(mg)^2 + 2mghk}, where mg is the climber's weight, h is the fall height and k is the spring constant of the portion of the rope that is in play. thumb|The aftermath of a hypervelocity impact, with a projectile the same size as the one that impacted for scale Hypervelocity is very high velocity, approximately over 3,000 meters per second (6,700 mph, 11,000 km/h, 10,000 ft/s, or Mach 8.8). When climbing from the ground up, the maximum possible fall factor is 1, since any greater fall would mean that the climber hit the ground. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. The amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. The mass of the displaced fluid can be expressed in terms of the density and its volume, . We will see below that when varying the height of the fall while keeping the fall factor fixed, the quantity hk stays constant. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). As a numerical example, consider a fall of 20 feet that occurs with 10 feet of rope out (i.e., the climber has placed no protection and falls from 10 feet above the belayer to 10 feet below—a factor 2 fall). This fall produces far more force on the climber and the gear than if a similar 20 foot fall had occurred 100 feet above the belayer. Therefore, the weight of the displaced fluid can be expressed as . ",144, 13.45,"""0.0182""",0.6321205588,35,B
 "A home buyer can afford to spend no more than $\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
-Determine the maximum amount that this buyer can afford to borrow.","Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. For a 24-month loan, the denominator is 300. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. The expansion P\approx P_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 . ====Example of mortgage payment==== For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find: T=30 I=0.045 which gives X=\frac{1}{2}IT=.675 so that P\approx P_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is P=\$608.02 so the approximation is an overestimate of about a sixth of a percent. ===Investing: monthly deposits=== Given a principal (initial) deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. For example, for interest rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives: = PMT(0.06/12, 25 * 12, -150000, 0, 0) = $966.45 ====Approximate formula for monthly payment==== A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (I<8\% and terms T=10–30 years), the monthly note rate is small compared to 1: r << 1 so that the \ln(1+r)\approx r which yields a simplification so that c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = IT c_0\equiv \frac{P}{n} . Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. During the second month the borrower has use of two $1000 (2/3) amounts and so the payment should be $1000 plus two $10 interest fees. ",1.61,21,0.36,"89,034.79",1.25,D
+Determine the maximum amount that this buyer can afford to borrow.","Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. For a 24-month loan, the denominator is 300. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. The expansion P\approx P_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 . ====Example of mortgage payment==== For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find: T=30 I=0.045 which gives X=\frac{1}{2}IT=.675 so that P\approx P_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is P=\$608.02 so the approximation is an overestimate of about a sixth of a percent. ===Investing: monthly deposits=== Given a principal (initial) deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. For example, for interest rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives: = PMT(0.06/12, 25 * 12, -150000, 0, 0) = $966.45 ====Approximate formula for monthly payment==== A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (I<8\% and terms T=10–30 years), the monthly note rate is small compared to 1: r << 1 so that the \ln(1+r)\approx r which yields a simplification so that c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = IT c_0\equiv \frac{P}{n} . Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. During the second month the borrower has use of two $1000 (2/3) amounts and so the payment should be $1000 plus two $10 interest fees. ",1.61,21,"""0.36""","89,034.79",1.25,D
 "A spring is stretched 6 in by a mass that weighs $8 \mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ and is acted on by an external force of $4 \cos 2 t \mathrm{lb}$.
-If the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.","As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. The effective mass of the spring can be determined by finding its kinetic energy. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. thumb|Phase portrait of damped oscillator, with increasing damping strength. The forcing amplitude increases from \gamma=0.20 to \gamma=0.65. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. For the case with linearly distributed load of maximum intensity q_0, :M_{\mathrm{right}}^{\mathrm{fixed}} = \int_{0}^{L} q_0 \frac{x}{L} dx \frac{ x^2 (L-x)}{L^2} = \frac{q_0 L^2}{20} :M_{\mathrm{left}}^{\mathrm{fixed}} = \int_{0}^{L} \left \\{ - q_0 \frac{x}{L} dx \frac{x (L-x)^2}{L^2} \right \\} = - \frac{q_0 L^2}{30} == See also == * Moment distribution method * Statically Indeterminate * Slope deflection method * Matrix method == References == * Category:Structural analysis *Damped harmonic motion, see animation (right). The lower overhanging side is unstable – i.e. the dashed-line parts in the figures of the frequency response – and cannot be realized for a sustained time. The restoring force provided by the nonlinear spring is then \alpha x + \beta x^3. The dashed parts of the frequency response are unstable. In a real spring–mass system, the spring has a non-negligible mass m. ",0.00539,420,4.0,35.2,-0.041,C
+If the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.","As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. The effective mass of the spring can be determined by finding its kinetic energy. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. thumb|Phase portrait of damped oscillator, with increasing damping strength. The forcing amplitude increases from \gamma=0.20 to \gamma=0.65. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. For the case with linearly distributed load of maximum intensity q_0, :M_{\mathrm{right}}^{\mathrm{fixed}} = \int_{0}^{L} q_0 \frac{x}{L} dx \frac{ x^2 (L-x)}{L^2} = \frac{q_0 L^2}{20} :M_{\mathrm{left}}^{\mathrm{fixed}} = \int_{0}^{L} \left \\{ - q_0 \frac{x}{L} dx \frac{x (L-x)^2}{L^2} \right \\} = - \frac{q_0 L^2}{30} == See also == * Moment distribution method * Statically Indeterminate * Slope deflection method * Matrix method == References == * Category:Structural analysis *Damped harmonic motion, see animation (right). The lower overhanging side is unstable – i.e. the dashed-line parts in the figures of the frequency response – and cannot be realized for a sustained time. The restoring force provided by the nonlinear spring is then \alpha x + \beta x^3. The dashed parts of the frequency response are unstable. In a real spring–mass system, the spring has a non-negligible mass m. ",0.00539,420,"""4.0""",35.2,-0.041,C
 "A recent college graduate borrows $\$ 100,000$ at an interest rate of $9 \%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.
-Assuming that this payment schedule can be maintained, when will the loan be fully paid?","If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Dividing by loan time period t will then give the equivalent simple interest rate. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. A graduated payment loan typically involves negative amortization, and is intended for students in the case of student loans, and homebuyers in the case of real estate, who currently have moderate incomes and anticipate their income will increase over the next 5–10 years. However this would contradict the primary assumption upon which the ""continuous payment"" model is based: namely that the annual payment rate is defined as: :M_a=\lim_{N\to\infty}N\cdot x(N) \, Since it is of course impossible for an investor to make an infinitely small payment infinite times per annum, a bank or other lending institution wishing to offer ""continuous payment"" annuities or mortgages would in practice have to choose a large but finite value of N (annual frequency of payments) such that the continuous time formula will always be correct to within some minimal pre-specified error margin. For a theoretical continuous payment savings annuity we can only calculate an annual rate of payment: :M_a=\frac{500000 \times 12\%}{e^{0.12\cdot 10}-1}=25860.77 At this point there is a temptation to simply divide by 12 to obtain a monthly payment. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. In a (theoretical) continuous-repayment mortgage the payment interval is narrowed indefinitely until the discrete interval process becomes continuous and the fixed interval payments become—in effect—a literal cash ""flow"" at a fixed annual rate. As with many similar examples the discrete interval problem and its solution is closely approximated by calculations based on the continuous repayment model - Dr Hahn's solution for interest rate is 40.8% as compared to the 41.6% calculated above. ==Period of a loan== If a borrower can afford an annual repayment rate Ma, then we can re-arrange the formula for calculating Ma to obtain an expression for the time period T of a given loan P0: : \begin{align} & M_a = \frac{P_0 r}{1-e^{-rT}} \\\\[8pt] \Rightarrow & T = \frac{1}{r}\ln\frac{M_a}{M_a-P_0 r} = -\frac{1}{r}\ln\left(1 - \frac{P_0 r}{M_a} \right) \end{align} ==Minimum payment ratio== The minimum payment ratio of a loan is the ratio of minimum possible payment rate to actual payment rate. For example, consider a $100 loan which must be repaid after one month, plus 5%, plus a $10 fee. \, Another way to calculate balance due P(t) on a continuous-repayment loan is to subtract the future value (at time t) of the payment stream from the future value of the loan (also at time t): :P(t)=P_0 e^{rt}-\frac{M_a}{r}(e^{rt}-1). Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. Beware of extremely long repayment periods, as generally speaking, the longer the term, the more you will owe because the interest accrues over a long period of time. The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated. The sum of these interest and principal payments must equal the cumulative fixed payments at time t i.e. Mat. Evaluating the first integral on the right we obtain an expression for I(t), the interest paid: :I(t)=M_at-\frac{M_a(e^{rt}-1)}{re^{rT}} Unsurprisingly the second integral evaluates to P0 − P(t) and therefore: :I(t)=M_at-P_0+P(t) \, The reader may easily verify that this expression is algebraically identical to the one above. ==Loan cost factor== The cost of a loan is simply the annual rate multiplied by loan period: : C = M_aT= \frac{P_0 rT}{1-e^{-rT}} Let s = rT. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity. ",-59.24, 135.36,3.52,0.85,-0.38,B
+Assuming that this payment schedule can be maintained, when will the loan be fully paid?","If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Dividing by loan time period t will then give the equivalent simple interest rate. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. A graduated payment loan typically involves negative amortization, and is intended for students in the case of student loans, and homebuyers in the case of real estate, who currently have moderate incomes and anticipate their income will increase over the next 5–10 years. However this would contradict the primary assumption upon which the ""continuous payment"" model is based: namely that the annual payment rate is defined as: :M_a=\lim_{N\to\infty}N\cdot x(N) \, Since it is of course impossible for an investor to make an infinitely small payment infinite times per annum, a bank or other lending institution wishing to offer ""continuous payment"" annuities or mortgages would in practice have to choose a large but finite value of N (annual frequency of payments) such that the continuous time formula will always be correct to within some minimal pre-specified error margin. For a theoretical continuous payment savings annuity we can only calculate an annual rate of payment: :M_a=\frac{500000 \times 12\%}{e^{0.12\cdot 10}-1}=25860.77 At this point there is a temptation to simply divide by 12 to obtain a monthly payment. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. In a (theoretical) continuous-repayment mortgage the payment interval is narrowed indefinitely until the discrete interval process becomes continuous and the fixed interval payments become—in effect—a literal cash ""flow"" at a fixed annual rate. As with many similar examples the discrete interval problem and its solution is closely approximated by calculations based on the continuous repayment model - Dr Hahn's solution for interest rate is 40.8% as compared to the 41.6% calculated above. ==Period of a loan== If a borrower can afford an annual repayment rate Ma, then we can re-arrange the formula for calculating Ma to obtain an expression for the time period T of a given loan P0: : \begin{align} & M_a = \frac{P_0 r}{1-e^{-rT}} \\\\[8pt] \Rightarrow & T = \frac{1}{r}\ln\frac{M_a}{M_a-P_0 r} = -\frac{1}{r}\ln\left(1 - \frac{P_0 r}{M_a} \right) \end{align} ==Minimum payment ratio== The minimum payment ratio of a loan is the ratio of minimum possible payment rate to actual payment rate. For example, consider a $100 loan which must be repaid after one month, plus 5%, plus a $10 fee. \, Another way to calculate balance due P(t) on a continuous-repayment loan is to subtract the future value (at time t) of the payment stream from the future value of the loan (also at time t): :P(t)=P_0 e^{rt}-\frac{M_a}{r}(e^{rt}-1). Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. Beware of extremely long repayment periods, as generally speaking, the longer the term, the more you will owe because the interest accrues over a long period of time. The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. If the consumer pays the loan off early, the effective interest rate achieved will be significantly higher than the APR initially calculated. The sum of these interest and principal payments must equal the cumulative fixed payments at time t i.e. Mat. Evaluating the first integral on the right we obtain an expression for I(t), the interest paid: :I(t)=M_at-\frac{M_a(e^{rt}-1)}{re^{rT}} Unsurprisingly the second integral evaluates to P0 − P(t) and therefore: :I(t)=M_at-P_0+P(t) \, The reader may easily verify that this expression is algebraically identical to the one above. ==Loan cost factor== The cost of a loan is simply the annual rate multiplied by loan period: : C = M_aT= \frac{P_0 rT}{1-e^{-rT}} Let s = rT. Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity. ",-59.24, 135.36,"""3.52""",0.85,-0.38,B
 "Consider the initial value problem
 $$
 y^{\prime}+\frac{1}{4} y=3+2 \cos 2 t, \quad y(0)=0
 $$
-Determine the value of $t$ for which the solution first intersects the line $y=12$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. We are trying to find a formula for y(t) that satisfies these two equations. The condition of intersection is : ys(x) = yc(x). In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Hence, : y_s(x) = -\tfrac{1}{4} \cdot x^2 \,\\! is tangent to every member of the one- parameter family of solutions : y_c(x) = c \cdot x + c^2 \,\\! of this Clairaut equation: : y(x) = x \cdot y' + (y')^2. \,\\! ==See also== * Chandrasekhar equation * Chrystal's equation * Caustic (mathematics) * Envelope (mathematics) * Initial value problem * Picard–Lindelöf theorem ==Bibliography== * Category:Differential equations Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. ",0.396,8.8,650000.0,10.065778,0.166666666,D
-"An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years.","Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. Over one month, :\frac{0.1299 \times \$2500}{12} = \$27.06 interest is due (rounded to the nearest cent). Simple interest applied over 3 months would be :\frac{0.1299 \times \$2500 \times 3}{12} = \$81.19 If the card holder pays off only interest at the end of each of the 3 months, the total amount of interest paid would be :\frac{0.1299 \times \$2500}{12} \times 3 = \$27.06\text{ per month} \times 3\text{ months} =\$81.18 which is the simple interest applied over 3 months, as calculated above. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.52 = $2.25. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. At the end of one year, 1,000 × 20% = 200 BRL interest is credited to the account. * P = principal deposit * r = rate of return (monthly) * M = monthly deposit, and * t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} and adding all recurring deposits over the total period t (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month) : M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} recognizing the geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the closed-form formula (common ratio :1/(1+r)) we obtain: P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. Suppose that the complete amount including the interest is withdrawn after exactly one year. The amount of interest paid (every six months) is the disclosed interest rate divided by two and multiplied by the principal. The principal remaining after the first month is P_1=(1+r)P - c, that is, the initial amount plus interest less the payment. With monthly payments, the monthly interest is paid out of each payment and so should not be compounded, and an annual rate of 12·r would make more sense. For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple annual interest rate is 12.99% per annum, applied monthly, so the frequency of applying interest is 12 per year. Compounding quarterly yields $1.00×1.254 = $2.4414..., and so on. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). If the whole loan is repaid after one month then P_1=0, so P=\frac{c}{1+r} After the second month P_2=(1+r) P_1 - c is left, so P_2=(1+r)((1+r)P-c)-c If the whole loan was repaid after two months, P_2 = 0, so P = \frac{c}{1+r}+\frac{c}{(1+r)^2} This equation generalizes for a term of n months, P = c \sum\limits_{j=1}^n \frac{1}{(1+r)^j} . The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. In total, the investor therefore now holds: :\$10\,000 + \$300 = \left(1 + \frac{r}{n}\right) \cdot B = \left(1 + \frac{6\%}{2}\right) \times \$10\,000 and so earns a coupon at the end of the next 6 months of: :\begin{align}\frac {r \cdot B \cdot m}{n} &= \frac {6\% \times \left(\$10\,000 + \$300\right)}{2}\\\ &= \frac {6\% \times \left(1 + \frac{6\%}{2}\right) \times \$10\,000}{2}\\\ &=\$309\end{align} Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of: :\begin{align}\$10,000 + \$300 + \$309 &= \$10\,000 + \frac {6\% \times \$10,000}{2} + \frac {6\% \times \left( 1 + \frac {6\%}{2}\right) \times \$10\,000}{2}\\\ &= \$10\,000 \times \left(1 + \frac{6\%}{2}\right)^2\end{align} and the investor earned in total: :\begin{align}\$10\,000 \times \left(1 + \frac {6\%}{2}\right)^2 - \$10\,000\\\ = \$10\,000 \times \left( \left( 1 + \frac {6\%}{2}\right)^2 - 1\right)\end{align} The formula for the annual equivalent compound interest rate is: :\left(1 + \frac{r}{n}\right)^n - 1 where :r is the simple annual rate of interest :n is the frequency of applying interest For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is: :\left(1 + \frac{6\%}{2}\right)^2 - 1 = 1.03^2 - 1 = 6.09\% ===Other formulations=== The outstanding balance Bn of a loan after n regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid p at the end of each period: :B_{n} = \big( 1 + r \big) B_{n - 1} - p, where :i = simple annual loan rate in decimal form (for example, 10% = 0.10. A compounding instrument adds the previously accrued interest to the principal each period, applying compound interest. == External links == * What is Accrued Interest ? ",2283.63,0.082,1.91,6.1,35.2,A
+Determine the value of $t$ for which the solution first intersects the line $y=12$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. We are trying to find a formula for y(t) that satisfies these two equations. The condition of intersection is : ys(x) = yc(x). In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. By integrating both sides, any function satisfying the differential equation must also satisfy the integral equation :y(t) - y(t_0) = \int_{t_0}^t f(s,y(s)) \, ds. If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Hence, : y_s(x) = -\tfrac{1}{4} \cdot x^2 \,\\! is tangent to every member of the one- parameter family of solutions : y_c(x) = c \cdot x + c^2 \,\\! of this Clairaut equation: : y(x) = x \cdot y' + (y')^2. \,\\! ==See also== * Chandrasekhar equation * Chrystal's equation * Caustic (mathematics) * Envelope (mathematics) * Initial value problem * Picard–Lindelöf theorem ==Bibliography== * Category:Differential equations Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. ",0.396,8.8,"""650000.0""",10.065778,0.166666666,D
+"An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years.","Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. Over one month, :\frac{0.1299 \times \$2500}{12} = \$27.06 interest is due (rounded to the nearest cent). Simple interest applied over 3 months would be :\frac{0.1299 \times \$2500 \times 3}{12} = \$81.19 If the card holder pays off only interest at the end of each of the 3 months, the total amount of interest paid would be :\frac{0.1299 \times \$2500}{12} \times 3 = \$27.06\text{ per month} \times 3\text{ months} =\$81.18 which is the simple interest applied over 3 months, as calculated above. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.52 = $2.25. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. At the end of one year, 1,000 × 20% = 200 BRL interest is credited to the account. * P = principal deposit * r = rate of return (monthly) * M = monthly deposit, and * t = time, in months The compound interest for each deposit is: M'=M(1+r)^{t} and adding all recurring deposits over the total period t (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month) : M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}} recognizing the geometric series: M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}} and applying the closed-form formula (common ratio :1/(1+r)) we obtain: P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as \text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y} where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling. thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. Suppose that the complete amount including the interest is withdrawn after exactly one year. The amount of interest paid (every six months) is the disclosed interest rate divided by two and multiplied by the principal. The principal remaining after the first month is P_1=(1+r)P - c, that is, the initial amount plus interest less the payment. With monthly payments, the monthly interest is paid out of each payment and so should not be compounded, and an annual rate of 12·r would make more sense. For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple annual interest rate is 12.99% per annum, applied monthly, so the frequency of applying interest is 12 per year. Compounding quarterly yields $1.00×1.254 = $2.4414..., and so on. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). If the whole loan is repaid after one month then P_1=0, so P=\frac{c}{1+r} After the second month P_2=(1+r) P_1 - c is left, so P_2=(1+r)((1+r)P-c)-c If the whole loan was repaid after two months, P_2 = 0, so P = \frac{c}{1+r}+\frac{c}{(1+r)^2} This equation generalizes for a term of n months, P = c \sum\limits_{j=1}^n \frac{1}{(1+r)^j} . The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. In total, the investor therefore now holds: :\$10\,000 + \$300 = \left(1 + \frac{r}{n}\right) \cdot B = \left(1 + \frac{6\%}{2}\right) \times \$10\,000 and so earns a coupon at the end of the next 6 months of: :\begin{align}\frac {r \cdot B \cdot m}{n} &= \frac {6\% \times \left(\$10\,000 + \$300\right)}{2}\\\ &= \frac {6\% \times \left(1 + \frac{6\%}{2}\right) \times \$10\,000}{2}\\\ &=\$309\end{align} Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of: :\begin{align}\$10,000 + \$300 + \$309 &= \$10\,000 + \frac {6\% \times \$10,000}{2} + \frac {6\% \times \left( 1 + \frac {6\%}{2}\right) \times \$10\,000}{2}\\\ &= \$10\,000 \times \left(1 + \frac{6\%}{2}\right)^2\end{align} and the investor earned in total: :\begin{align}\$10\,000 \times \left(1 + \frac {6\%}{2}\right)^2 - \$10\,000\\\ = \$10\,000 \times \left( \left( 1 + \frac {6\%}{2}\right)^2 - 1\right)\end{align} The formula for the annual equivalent compound interest rate is: :\left(1 + \frac{r}{n}\right)^n - 1 where :r is the simple annual rate of interest :n is the frequency of applying interest For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is: :\left(1 + \frac{6\%}{2}\right)^2 - 1 = 1.03^2 - 1 = 6.09\% ===Other formulations=== The outstanding balance Bn of a loan after n regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid p at the end of each period: :B_{n} = \big( 1 + r \big) B_{n - 1} - p, where :i = simple annual loan rate in decimal form (for example, 10% = 0.10. A compounding instrument adds the previously accrued interest to the principal each period, applying compound interest. == External links == * What is Accrued Interest ? ",2283.63,0.082,"""1.91""",6.1,35.2,A
 "A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
-If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground.","If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. The resulting outcome depends on the properties of the drop, the surface, and the surrounding fluid, which is most commonly a gas. == On a dry solid surface == When a liquid drop strikes a dry solid surface, it generally spreads on the surface, and then will retract if the impact is energetic enough to cause the drop to spread out more than it would generally spread due to its static receding contact angle. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. This outcome is representative of impact of small, low-velocity drops onto smooth wetting surfaces. The specific outcome of the impact depends mostly upon the drop size, velocity, surface tension, viscosity, and also upon the surface roughness and the contact angle between the drop and the surface.Rioboo, Romain, Cameron Tropea, and Marco Marengo. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. ""Surface phenomena: Contact time of a bouncing drop."" In fluid dynamics, drop impact occurs when a drop of liquid strikes a solid or liquid surface. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). The formula for terminal velocity (V)] appears on p. [52], equation (127). Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). ""Phenomena of liquid drop impact on solid and liquid surfaces."" The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. So instead of m use the reduced mass m_r = m-\rho V in this and subsequent formulas. Dividing both sides by m gives \frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right). If the impact energy is high enough, the jet rises to the point where it pinches off, sending one or more droplets upward out of the surface. ==See also== *Splash (fluid mechanics) == References == Category:Fluid dynamics ""Outcomes from a drop impact on solid surfaces."" At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. ",0.75, 11.58,1068.0,0.0408,3857,B
-"A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.","The effective mass of the spring can be determined by finding its kinetic energy. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the system. Pulling the spring to a greater length causes it to exert a force that brings the spring back toward its equilibrium length. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. In a real spring–mass system, the spring has a non-negligible mass m. alt=mass connected to the ground with a spring and damper in parallel|thumb|Classic model used for deriving the equations of a mass spring damper model The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. The amount of force can be determined by multiplying the spring constant, characteristic of the spring, by the amount of stretch, also known as Hooke's Law. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. The stiffness of the spring, its spring coefficient, \kappa\, in N·m/radian^2, along with the balance wheel's moment of inertia, I\, in kg·m2, determines the wheel's oscillation period T\,. An idealized spring exerts a force proportional to the amount of deformation of the spring from its equilibrium length, exerted in a direction oppose the deformation. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. The balance spring provides a restoring torque that limits and reverses the motion of the balance so it oscillates back and forth. ",56,0.2244,6.07,12,0.66666666666,B
+If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground.","If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. The resulting outcome depends on the properties of the drop, the surface, and the surrounding fluid, which is most commonly a gas. == On a dry solid surface == When a liquid drop strikes a dry solid surface, it generally spreads on the surface, and then will retract if the impact is energetic enough to cause the drop to spread out more than it would generally spread due to its static receding contact angle. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. This outcome is representative of impact of small, low-velocity drops onto smooth wetting surfaces. The specific outcome of the impact depends mostly upon the drop size, velocity, surface tension, viscosity, and also upon the surface roughness and the contact angle between the drop and the surface.Rioboo, Romain, Cameron Tropea, and Marco Marengo. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. ""Surface phenomena: Contact time of a bouncing drop."" In fluid dynamics, drop impact occurs when a drop of liquid strikes a solid or liquid surface. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). The formula for terminal velocity (V)] appears on p. [52], equation (127). Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). ""Phenomena of liquid drop impact on solid and liquid surfaces."" The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. So instead of m use the reduced mass m_r = m-\rho V in this and subsequent formulas. Dividing both sides by m gives \frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right). If the impact energy is high enough, the jet rises to the point where it pinches off, sending one or more droplets upward out of the surface. ==See also== *Splash (fluid mechanics) == References == Category:Fluid dynamics ""Outcomes from a drop impact on solid surfaces."" At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. ",0.75, 11.58,"""1068.0""",0.0408,3857,B
+"A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.","The effective mass of the spring can be determined by finding its kinetic energy. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the system. Pulling the spring to a greater length causes it to exert a force that brings the spring back toward its equilibrium length. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. In a real spring–mass system, the spring has a non-negligible mass m. alt=mass connected to the ground with a spring and damper in parallel|thumb|Classic model used for deriving the equations of a mass spring damper model The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. The amount of force can be determined by multiplying the spring constant, characteristic of the spring, by the amount of stretch, also known as Hooke's Law. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. The stiffness of the spring, its spring coefficient, \kappa\, in N·m/radian^2, along with the balance wheel's moment of inertia, I\, in kg·m2, determines the wheel's oscillation period T\,. An idealized spring exerts a force proportional to the amount of deformation of the spring from its equilibrium length, exerted in a direction oppose the deformation. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. The balance spring provides a restoring torque that limits and reverses the motion of the balance so it oscillates back and forth. ",56,0.2244,"""6.07""",12,0.66666666666,B
 "Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.
-Consider a water tank in the form of a right circular cylinder that is $3 \mathrm{~m}$ high above the outlet. The radius of the tank is $1 \mathrm{~m}$ and the radius of the circular outlet is $0.1 \mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.","The results confirm the correctness of Torricelli's law very well. ==Discharge and time to empty a cylindrical vessel== Assuming that a vessel is cylindrical with fixed cross-sectional area A, with orifice of area A_A at the bottom, then rate of change of water level height dh/dt is not constant. From Torricelli's law, the rate of outflow is :\frac{dV}{dt} = A_A v = A_A \sqrt{2gh}, From these two equations, : \begin{align} A_A \sqrt{2gh} &= \pi r^2 c \\\ \Rightarrow \quad h &= \frac{\pi^2 c^2}{2g A_A^2} r^4. \end{align} Thus, the radius of the container should change in proportion to the quartic root of its height, r \propto \sqrt[4]{h}. More precisely, : \Delta t = \frac{A}{A_A} \sqrt{\frac{2}{g}} (\sqrt{h_1} - \sqrt{h_2}) where \Delta t is the time taken by the water level to fall from the height of h_1 to height of h_2. ==Torricelli's original derivation== thumb|Figures from Evangelista Torricelli's Opera Geometrica (1644) describing the derivation of his famous outflow formula: (a) One tube filled up with water from A to B. (b) In two connected tubes the water lift up to the same height. (c) When the tube C is removed, the water should rise up to the height D. Due to friction effects the water only rises to the point C. Evangelista Torricelli's original derivation can be found in the second book 'De motu aquarum' of his 'Opera Geometrica' (see A. Malcherek: History of the Torricelli Principle and a New Outflow Theory,Journal of Hydraulic Engineering 142(11),1-7,2016,https://doi.org/10.1061/(ASCE)HY.1943-7900.0001232)): He starts a tube AB (Figure (a)) filled up with water to the level A. Torricelli's law can only be applied when viscous effects can be neglected which is the case for water flowing out through orifices in vessels. ===Experimental verification: Spouting can experiment=== thumb|Experiment to determine the trajectory of an outflowing jet: Vertical rods are adjusted so they are nearly touching the jet. Since the water level is H-h above the orifice, the horizontal efflux velocity v = \sqrt{2g(H-h)}, as given by Torricelli's law. Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The water volume in the vessel is changing due to the discharge \dot{V} out of the vessel: :\frac{dV}{dt} = A \frac{dh}{dt} = \dot{V} = A_A v_A = A_A \sqrt{2gh} \quad \Rightarrow \quad A \frac{dh}{\sqrt{h}} = A_A \sqrt{2g} \; dt Integrating both sides and re- arranging, we obtain : T = \frac{A}{A_A} \sqrt{\frac{2H}{g}}, where H is the initial height of the water level and T is the total time taken to drain all the water and hence empty the vessel. If a tank with volume V with cross section A and height H, so that V = AH, is fully filled, then the time to drain all the water is : T = \frac{V}{A_A} \sqrt{\frac{2}{gH}}. Furthermore y_1 - y_2 is equal to the height h of the liquid's surface over the opening: :\frac{{v_1}^2}{2} + g h = \frac{{v_2}^2}{2} The velocity of the surface v_1 can by related to the outflow velocity v_2 by the continuity equation v_1 A = v_2 A_A, where A_A is the orifice's cross section and A is the (cylindrical) vessel's cross section. Lastly, we can re-arrange the above equation to determine the height of the water level h(t) as a function of time t as : h(t) = H \left(1 - \frac{t}{T} \right)^2, where H is the height of the container while T is the discharge time as given above. ===Discharge experiment, coefficient of discharge=== The dicharge theory can be tested by measuring the emptying time T or time series of the water level h(t) within the cylindrical vessel. Torricelli's law is obtained as a special case when the opening A_A is very small relative to the horizontal cross-section of the container A_1: :v_A = \sqrt{2gh}. This is normally done by introducing a discharge coefficient which relates the discharge to the orifice's cross- section and Torricelli's law: : {\dot {V}}_{\text{real}}=\mu A_A v_A \quad \text{with} \quad \mu = \frac{A_C}{A_A} For low viscosity liquids (such as water) flowing out of a round hole in a tank, the discharge coefficient is in the order of 0.65. The law states that the speed v of efflux of a fluid through a sharp-edged hole at the bottom of the tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. v = \sqrt{2gh}, where g is the acceleration due to gravity. The low pressure on sides on the cylinder is needed to provide the centripetal acceleration of the flow: :\frac{\partial p}{\partial r}=\frac{\rho V^2}{L} \,, where is the radius of curvature of the flow. Thus we find the maximum speed in the flow, , in the low pressure on the sides of the cylinder. Here it was also shown that the outflow velocity is predicted extremeliy well by Torricelli's law and that no velocity correction (like a ""coefficient of velocity"") is needed. In order to derive Torricelli's formula the first point with no index is taken at the liquid's surface, and the second just outside the opening. As given by the Torricelli's law, the rate of efflux through the hole depends on the height of the water; and as the water level diminishes, the discharge is not uniform. The fluid exit velocity is greater further down the tube.Spouting cylinder fluid flow. The instantaneous rate of change in water volume is :\frac{dV}{dt} = A \frac{dh}{dt} = \pi r^2 c. We want to find the radius such that the water level has a constant rate of decrease, i.e. dh/dt = c. ",+4.1,0.16,130.41,3.51,1.5,C
-"Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. For example, for the equation (a<0), the stationary solution is , which is obtained for the initial condition . Suppose we have an ordinary differential equation in the complex domain. It can also be shown that J_\alpha^2(x)+Y_{\alpha}^2(x)=\frac{8\cos(\alpha\pi)}{\pi^2} \int_0^\infty K_{2\alpha}(2x\sinh t)\, dt, only when || < and but not when . Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):Abramowitz and Stegun, p. 375, 9.6.3, 9.6.5. \begin{align} J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\\\[5pt] Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \frac{2}{\pi}e^{-\frac{\alpha\pi i}{2}}K_\alpha(z). \end{align} and are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, p. 374, 9.6.1. x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left(x^2 + \alpha^2 \right)y = 0. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the order of the Bessel function. ",−2,-242.6,0.23333333333,20.2,6,A
-"If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$.","Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Third Solution (, also known as Russicum) is a 1988 Italian crime-thriller film written and directed by Pasquale Squitieri and starring Treat Williams.VV.AA. Variety Film Reviews, Volume 18. In mathematical physics, the Degasperis–Procesi equation : \displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx} is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: :\displaystyle u_t - u_{xxt} + 2\kappa u_x + (b+1)u u_x = b u_x u_{xx} + u u_{xxx}, where \kappa and b are real parameters (b=3 for the Degasperis–Procesi equation). Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation : u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. thumb|300px|Time-Temperature-Transformation diagram for two steels: one with 0.4% wt. C (red line) and one with 0.4% wt. C and 2% weight Mn (green line). Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics In the case \kappa=0 the solution splits into an infinite linear combination of peakons (as previously conjectured). ==Geometric formulation== In the spatially periodic case, the Camassa–Holm equation can be given the following geometric interpretation. Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. The fact that the equations admits solutions of this type was discovered by Camassa and Holm and these considerations were subsequently put on a firm mathematical basis. Then u is a solution to the Camassa–Holm equation with \kappa=0, if and only if the path t\mapsto\varphi_t\in\mathrm{Diff}(S^1) is a geodesic on \mathrm{Diff}(S^1) with respect to the right-invariant H^1 metric. As a result, κ is the phase speed for the long-wave limit of k approaching zero, and the Camassa–Holm equation is (if κ is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation. ==Hamiltonian structure== Introducing the momentum m as :m = u - u_{xx} + \kappa, \, then two compatible Hamiltonian descriptions of the Camassa–Holm equation are: : \begin{align} m_t &= -\mathcal{D}_1 \frac{\delta \mathcal{H}_1}{\delta m} & & \text{ with }& \mathcal{D}_1 &= m \frac{\partial}{\partial x} + \frac{\partial}{\partial x} m & \text{ and } \mathcal{H}_1 &= \frac{1}{2} \int u^2 + \left(u_x\right)^2\; \text{d}x, \\\ m_t &= -\mathcal{D}_2 \frac{\delta \mathcal{H}_2}{\delta m} & & \text{ with }& \mathcal{D}_2 &= \frac{\partial}{\partial x} - \frac{\partial^3}{\partial x^3} & \text{ and } \mathcal{H}_2 &= \frac{1}{2} \int u^3 + u \left(u_{x}\right)^2 - \kappa u^2\; \text{d}x. \end{align} ==Integrability== The Camassa–Holm equation is an integrable system. These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.Lundmark & Szmigielski 2003, 2005 When \kappa > 0 the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as \kappa tends to zero.Matsuno 2005a, 2005b == Discontinuous solutions == The Degasperis–Procesi equation (with \kappa=0) is formally equivalent to the (nonlocal) hyperbolic conservation law : \partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \frac{3 u^2}{2} \right] = 0, where G(x) = \exp(-|x|), and where the star denotes convolution with respect to x. By the Sobolev embedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x. == Notes == == References == * * * * * * * * * * * * * * * * * * == Further reading == * * * * * * * * * * * * * * * * * Category:Mathematical physics Category:Solitons Category:Partial differential equations Category:Equations of fluid dynamics For general \kappa, the Camassa–Holm equation corresponds to the geodesic equation of a similar right- invariant metric on the universal central extension of \mathrm{Diff}(S^1), the Virasoro group. ==See also== *Degasperis–Procesi equation *Hunter–Saxton equation ==Notes== ==References== * * * * * * * * * * * * * * * * * * * * * * * ==Further reading== ; Peakon solutions * ; Water wave theory * * ; Existence, uniqueness, wellposedness, stability, propagation speed, etc. * * * * ; Travelling waves * ; Integrability structure (symmetries, hierarchy of soliton equations, conservations laws) and differential-geometric formulation * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Category:Partial differential equations Category:Equations of fluid dynamics Category:Integrable systems Category:Solitons \, The equation was introduced by Roberto Camassa and Darryl Holm as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons. In the soliton region c>2 the solutions splits into a finite linear combination solitons. In the region 0 the solution is asymptotically given by a modulated sine function whose amplitude decays like t^{-1/2}. ",0.3359,0.2307692308,1.602,0.65625,4.946,E
- Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.,"University Library of Tromso, Ravnetrykk No. 29. , pp. 195–202. by exposing natural radium-226 to neutrons to produce radium-227, which decays with a 42-minute half-life to actinium-227. Actinium-227 (half-life 21.8 years) in turn decays via thorium-227 (half-life 18.7 days) to radium-223. Radium-223 (223Ra, Ra-223) is an isotope of radium with an 11.4-day half-life. 209P/LINEAR is a periodic comet with an orbital period of 5.1 years. Plutonium-244 (244Pu) is an isotope of plutonium that has a half-life of 80 million years. This article concerns the period 229 BC – 220 BC. ==References== __NOTOC__ Year 226 BC was a year of the pre-Julian Roman calendar. Its half-life of 80 million years ensured its circulation across the solar system before its extinction, and indeed, 244Pu has not yet been found in matter other than meteorites. Scandium-44 (44Sc) is a radioactive isotope of scandium that decays by positron emission to stable 44Ca with a half-life of 4.042 hours. 44Sc can be obtained as a daughter radionuclide of long-lived 44Ti (t1/2 60.4 a) from 44Ti /44Sc generator or can be produced by nuclear reaction 44Ca ( p, n)44Sc in small cyclotrons. Although radium-223 and its decay products also emit beta and gamma radiation, over 95% of the decay energy is in the form of alpha radiation. The amount of 244Pu in the pre-Solar nebula (4.57×109 years ago) was estimated as 0.8% the amount of 238U..As the age of the Earth is about 57 half-lives of 244Pu, the amount of plutonium-244 left should be very small; Hoffman et al. estimated its content in the rare-earth mineral bastnasite as = 1.0×10−18 g/g, which corresponded to the content in the Earth crust as low as 3×10−25 g/g (i.e. the total mass of plutonium-244 in Earth's crust is about 9 g). This is longer than any of the other isotopes of plutonium and longer than any other actinide isotope except for the three naturally abundant ones: uranium-235 (704 million years), uranium-238 (4.468 billion years), and thorium-232 (14.05 billion years). The principal use of radium-223, as a radiopharmaceutical to treat metastatic cancers in bone, takes advantage of its chemical similarity to calcium, and the short range of the alpha radiation it emits. ==Origin and preparation== Although radium-223 is naturally formed in trace amounts by the decay of uranium-235, it is generally made artificially,Bruland O.S., Larsen R.H. (2003). The recommended regimen is six treatments of 55 kBq/kg (1.5 μCi/kg), repeated at 4-week intervals. === Mechanism of action === The use of radium-223 to treat metastatic bone cancer relies on the ability of alpha radiation from radium-223 and its short-lived decay products to kill cancer cells. The 44Ti /44Sc generator represents a secular equilibrium system with a half-life ratio between parent and daughter of ca. 130 000. Radium revisited. This decay path makes it convenient to prepare radium-223 by ""milking"" it from an actinium-227 containing generator or ""cow"", similar to the moly cows widely used to prepare the medically important isotope technetium-99m. 223Ra itself decays to 219Rn (half-life 3.96 s), a short-lived gaseous radon isotope, by emitting an alpha particle of 5.979 MeV. ==Medical use== The pharmaceutical product and medical use of radium-223 against skeletal metastases was invented by Roy H. Larsen, Gjermund Henriksen and Øyvind S. Bruland""Preparation and use of radium-223 to target calcified tissues for pain palliation, bone cancer therapy, and bone surface conditioning"" US 6635234 and has been developed by the former Norwegian company Algeta ASA, in a partnership with Bayer, under the trade name Xofigo (formerly Alpharadin), and is distributed as a solution containing radium-223 chloride (1100 kBq/ml), sodium chloride, and other ingredients for intravenous injection. M.Pruszynski, A.Majkowska-Pilip, N.Loktionova, E.Eppard, F.Roesch Category:Isotopes of scandium Presence of 244Pu fission tracks can be established by using the initial ratio of 244Pu to 238U (Pu/U)0 at a time T0 = years, when Xe formation first began in meteorites, and by considering how the ratio of Pu/U fission tracks varies over time. In fact, by analyzing data from Earth's mantle which indicates that about 30% of the existing fissiogenic xenon is attributable to 244Pu decay, the timing of Earth's formation can be inferred to have occurred nearly 50–70 million years following the formation of the Solar System. Radium is preferentially absorbed by bone by virtue of its chemical similarity to calcium, with most radium-223 that is not taken up by the bone being cleared, primarily via the gut, and excreted. Radionuclides such as 244Pu undergo decay to produce fissiogenic (i.e., arising from fission) xenon isotopes that can then be used to time the events of the early solar system. ",479, 672.4,2500.0,67,-3.5,B
-"A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \mathrm{~min}$.","The solution is 9 grams of sodium chloride (NaCl) dissolved in water, to a total volume of 1000 ml (weight per unit volume). The state (1, 2), for example, is impossible to reach from an initial state of (0, 0), since (1, 2) has both jugs partially full, and no reversible action is possible from this state. === Jug with initial water === thumb|Starting with 9 liters in the 12-liter jug, the solution for 5 liters is plotted in red on the left, and the solution for 4 liters is plotted in blue on the right. Other assumptions of these problems may include that no water can be spilled, and that each step pouring water from a source jug to a destination jug stops when either the source jug is empty or the destination jug is full, whichever happens first. ==Standard example== The standard puzzle of this kind works with three jugs of capacity 8, 5 and 3 liters. The solver must pour the water so that the first and second jugs both contain 4 units, and the third is empty. Since normal saline contains 9 grams of NaCl, the concentration is 9 grams per litre divided by 58.4 grams per mole, or 0.154 mole per litre. It is most commonly used as a sterile 9 g of salt per litre (0.9%) solution, known as normal saline. In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. Saline (also known as saline solution) is a mixture of sodium chloride (salt) and water. Puzzles of this type ask how many steps of pouring water from one jug to another (until either one jug becomes empty or the other becomes full) are needed to reach a goal state, specified in terms of the volume of liquid that must be present in some jug or jugs. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal. Therefore, there are only two solutions to the problem: :(5, 0) ↔ (0, 5) ↔ (12, 5) ↔ (9, 8) ↔ (9, 0) :(5, 0) ↔ (5, 8) ↔ (12, 1) ↔ (0, 1) ↔ (1, 0) ↔ (1, 8) ↔ (9, 0) For the 4 liter question, since 4\equiv 0 \\! \mod \\!4, one irreversible action is necessary at the start of the solution; It could be simply pouring the whole 9 liters of water from the 12-liter jug to the sink (0,0), or fully fill it to 12 liters from the tap (12,0). Salt water chlorination is a process that uses dissolved salt (1000–36,000 ppm or 1–36 g/L) for the chlorination of swimming pools and hot tubs. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. For example, if one jug that holds 8 liters is empty and the other jug that hold 12 liters has 9 liters of water in it to begin with, then with a source (tap) and a drain (sink), these two jugs can measure volumes of 9 liters, 5 liters, 1 liter, as well as 12 liters, 8 liters, 4 liters and 0 liters. The answer is that the mixtures will be of equal purity. At its simplest, this figure is the result of dividing the lake volume by the flow in or out of the lake. The graph shows two ways to obtain 4 liters using 3-liter and 5-liter jugs, and a water source and sink on a Cartesian grid with diagonal lines of slope −1 (such that x+y=const. on these diagonal lines, which represent pouring water from one jug to the other jug). However, if the osmotic coefficient (a correction for non-ideal solutions) is taken into account, then the saline solution is much closer to isotonic. The question is then posed—which of the two mixtures is purer? The solution still applies no matter how many cups of any sizes and compositions are exchanged, or how little or much stirring at any point in time is done to any barrel, as long as at the end each barrel has the same amount of liquid. However, as the ideal saline concentration of a salt-chlorinated pool is very low (<3,500ppm, the threshold for human perception of salt by taste; seawater is about ten times this concentration), damage usually occurs due to improperly-maintained pool chemistry or improper maintenance of the electrolytic cell. ",-0.38,0.2553,7.42,0.2553,1.3,C
+Consider a water tank in the form of a right circular cylinder that is $3 \mathrm{~m}$ high above the outlet. The radius of the tank is $1 \mathrm{~m}$ and the radius of the circular outlet is $0.1 \mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.","The results confirm the correctness of Torricelli's law very well. ==Discharge and time to empty a cylindrical vessel== Assuming that a vessel is cylindrical with fixed cross-sectional area A, with orifice of area A_A at the bottom, then rate of change of water level height dh/dt is not constant. From Torricelli's law, the rate of outflow is :\frac{dV}{dt} = A_A v = A_A \sqrt{2gh}, From these two equations, : \begin{align} A_A \sqrt{2gh} &= \pi r^2 c \\\ \Rightarrow \quad h &= \frac{\pi^2 c^2}{2g A_A^2} r^4. \end{align} Thus, the radius of the container should change in proportion to the quartic root of its height, r \propto \sqrt[4]{h}. More precisely, : \Delta t = \frac{A}{A_A} \sqrt{\frac{2}{g}} (\sqrt{h_1} - \sqrt{h_2}) where \Delta t is the time taken by the water level to fall from the height of h_1 to height of h_2. ==Torricelli's original derivation== thumb|Figures from Evangelista Torricelli's Opera Geometrica (1644) describing the derivation of his famous outflow formula: (a) One tube filled up with water from A to B. (b) In two connected tubes the water lift up to the same height. (c) When the tube C is removed, the water should rise up to the height D. Due to friction effects the water only rises to the point C. Evangelista Torricelli's original derivation can be found in the second book 'De motu aquarum' of his 'Opera Geometrica' (see A. Malcherek: History of the Torricelli Principle and a New Outflow Theory,Journal of Hydraulic Engineering 142(11),1-7,2016,https://doi.org/10.1061/(ASCE)HY.1943-7900.0001232)): He starts a tube AB (Figure (a)) filled up with water to the level A. Torricelli's law can only be applied when viscous effects can be neglected which is the case for water flowing out through orifices in vessels. ===Experimental verification: Spouting can experiment=== thumb|Experiment to determine the trajectory of an outflowing jet: Vertical rods are adjusted so they are nearly touching the jet. Since the water level is H-h above the orifice, the horizontal efflux velocity v = \sqrt{2g(H-h)}, as given by Torricelli's law. Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The water volume in the vessel is changing due to the discharge \dot{V} out of the vessel: :\frac{dV}{dt} = A \frac{dh}{dt} = \dot{V} = A_A v_A = A_A \sqrt{2gh} \quad \Rightarrow \quad A \frac{dh}{\sqrt{h}} = A_A \sqrt{2g} \; dt Integrating both sides and re- arranging, we obtain : T = \frac{A}{A_A} \sqrt{\frac{2H}{g}}, where H is the initial height of the water level and T is the total time taken to drain all the water and hence empty the vessel. If a tank with volume V with cross section A and height H, so that V = AH, is fully filled, then the time to drain all the water is : T = \frac{V}{A_A} \sqrt{\frac{2}{gH}}. Furthermore y_1 - y_2 is equal to the height h of the liquid's surface over the opening: :\frac{{v_1}^2}{2} + g h = \frac{{v_2}^2}{2} The velocity of the surface v_1 can by related to the outflow velocity v_2 by the continuity equation v_1 A = v_2 A_A, where A_A is the orifice's cross section and A is the (cylindrical) vessel's cross section. Lastly, we can re-arrange the above equation to determine the height of the water level h(t) as a function of time t as : h(t) = H \left(1 - \frac{t}{T} \right)^2, where H is the height of the container while T is the discharge time as given above. ===Discharge experiment, coefficient of discharge=== The dicharge theory can be tested by measuring the emptying time T or time series of the water level h(t) within the cylindrical vessel. Torricelli's law is obtained as a special case when the opening A_A is very small relative to the horizontal cross-section of the container A_1: :v_A = \sqrt{2gh}. This is normally done by introducing a discharge coefficient which relates the discharge to the orifice's cross- section and Torricelli's law: : {\dot {V}}_{\text{real}}=\mu A_A v_A \quad \text{with} \quad \mu = \frac{A_C}{A_A} For low viscosity liquids (such as water) flowing out of a round hole in a tank, the discharge coefficient is in the order of 0.65. The law states that the speed v of efflux of a fluid through a sharp-edged hole at the bottom of the tank filled to a depth h is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height h, i.e. v = \sqrt{2gh}, where g is the acceleration due to gravity. The low pressure on sides on the cylinder is needed to provide the centripetal acceleration of the flow: :\frac{\partial p}{\partial r}=\frac{\rho V^2}{L} \,, where is the radius of curvature of the flow. Thus we find the maximum speed in the flow, , in the low pressure on the sides of the cylinder. Here it was also shown that the outflow velocity is predicted extremeliy well by Torricelli's law and that no velocity correction (like a ""coefficient of velocity"") is needed. In order to derive Torricelli's formula the first point with no index is taken at the liquid's surface, and the second just outside the opening. As given by the Torricelli's law, the rate of efflux through the hole depends on the height of the water; and as the water level diminishes, the discharge is not uniform. The fluid exit velocity is greater further down the tube.Spouting cylinder fluid flow. The instantaneous rate of change in water volume is :\frac{dV}{dt} = A \frac{dh}{dt} = \pi r^2 c. We want to find the radius such that the water level has a constant rate of decrease, i.e. dh/dt = c. ",+4.1,0.16,"""130.41""",3.51,1.5,C
+"Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. For example, for the equation (a<0), the stationary solution is , which is obtained for the initial condition . Suppose we have an ordinary differential equation in the complex domain. It can also be shown that J_\alpha^2(x)+Y_{\alpha}^2(x)=\frac{8\cos(\alpha\pi)}{\pi^2} \int_0^\infty K_{2\alpha}(2x\sinh t)\, dt, only when || < and but not when . Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ):Abramowitz and Stegun, p. 375, 9.6.3, 9.6.5. \begin{align} J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\\\[5pt] Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \frac{2}{\pi}e^{-\frac{\alpha\pi i}{2}}K_\alpha(z). \end{align} and are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, p. 374, 9.6.1. x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left(x^2 + \alpha^2 \right)y = 0. Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the order of the Bessel function. ",−2,-242.6,"""0.23333333333""",20.2,6,A
+"If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$.","Using the new variables, the u and w functions can be expressed by equivalent functions χ and ω. The Third Solution (, also known as Russicum) is a 1988 Italian crime-thriller film written and directed by Pasquale Squitieri and starring Treat Williams.VV.AA. Variety Film Reviews, Volume 18. In mathematical physics, the Degasperis–Procesi equation : \displaystyle u_t - u_{xxt} + 2\kappa u_x + 4u u_x = 3 u_x u_{xx} + u u_{xxx} is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs: :\displaystyle u_t - u_{xxt} + 2\kappa u_x + (b+1)u u_x = b u_x u_{xx} + u u_{xxx}, where \kappa and b are real parameters (b=3 for the Degasperis–Procesi equation). Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation : u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. thumb|300px|Time-Temperature-Transformation diagram for two steels: one with 0.4% wt. C (red line) and one with 0.4% wt. C and 2% weight Mn (green line). Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics In the case \kappa=0 the solution splits into an infinite linear combination of peakons (as previously conjectured). ==Geometric formulation== In the spatially periodic case, the Camassa–Holm equation can be given the following geometric interpretation. Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs. The fact that the equations admits solutions of this type was discovered by Camassa and Holm and these considerations were subsequently put on a firm mathematical basis. Then u is a solution to the Camassa–Holm equation with \kappa=0, if and only if the path t\mapsto\varphi_t\in\mathrm{Diff}(S^1) is a geodesic on \mathrm{Diff}(S^1) with respect to the right-invariant H^1 metric. As a result, κ is the phase speed for the long-wave limit of k approaching zero, and the Camassa–Holm equation is (if κ is non-zero) a model for one-directional wave propagation like the Korteweg–de Vries equation. ==Hamiltonian structure== Introducing the momentum m as :m = u - u_{xx} + \kappa, \, then two compatible Hamiltonian descriptions of the Camassa–Holm equation are: : \begin{align} m_t &= -\mathcal{D}_1 \frac{\delta \mathcal{H}_1}{\delta m} & & \text{ with }& \mathcal{D}_1 &= m \frac{\partial}{\partial x} + \frac{\partial}{\partial x} m & \text{ and } \mathcal{H}_1 &= \frac{1}{2} \int u^2 + \left(u_x\right)^2\; \text{d}x, \\\ m_t &= -\mathcal{D}_2 \frac{\delta \mathcal{H}_2}{\delta m} & & \text{ with }& \mathcal{D}_2 &= \frac{\partial}{\partial x} - \frac{\partial^3}{\partial x^3} & \text{ and } \mathcal{H}_2 &= \frac{1}{2} \int u^3 + u \left(u_{x}\right)^2 - \kappa u^2\; \text{d}x. \end{align} ==Integrability== The Camassa–Holm equation is an integrable system. These ODEs can be solved explicitly in terms of elementary functions, using inverse spectral methods.Lundmark & Szmigielski 2003, 2005 When \kappa > 0 the soliton solutions of the Degasperis–Procesi equation are smooth; they converge to peakons in the limit as \kappa tends to zero.Matsuno 2005a, 2005b == Discontinuous solutions == The Degasperis–Procesi equation (with \kappa=0) is formally equivalent to the (nonlocal) hyperbolic conservation law : \partial_t u + \partial_x \left[\frac{u^2}{2} + \frac{G}{2} * \frac{3 u^2}{2} \right] = 0, where G(x) = \exp(-|x|), and where the star denotes convolution with respect to x. By the Sobolev embedding theorem, this means in particular that the weak solutions of the Camassa–Holm equation must be continuous with respect to x. == Notes == == References == * * * * * * * * * * * * * * * * * * == Further reading == * * * * * * * * * * * * * * * * * Category:Mathematical physics Category:Solitons Category:Partial differential equations Category:Equations of fluid dynamics For general \kappa, the Camassa–Holm equation corresponds to the geodesic equation of a similar right- invariant metric on the universal central extension of \mathrm{Diff}(S^1), the Virasoro group. ==See also== *Degasperis–Procesi equation *Hunter–Saxton equation ==Notes== ==References== * * * * * * * * * * * * * * * * * * * * * * * ==Further reading== ; Peakon solutions * ; Water wave theory * * ; Existence, uniqueness, wellposedness, stability, propagation speed, etc. * * * * ; Travelling waves * ; Integrability structure (symmetries, hierarchy of soliton equations, conservations laws) and differential-geometric formulation * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Category:Partial differential equations Category:Equations of fluid dynamics Category:Integrable systems Category:Solitons \, The equation was introduced by Roberto Camassa and Darryl Holm as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter κ is positive and the solitary wave solutions are smooth solitons. In the soliton region c>2 the solutions splits into a finite linear combination solitons. In the region 0 the solution is asymptotically given by a modulated sine function whose amplitude decays like t^{-1/2}. ",0.3359,0.2307692308,"""1.602""",0.65625,4.946,E
+ Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.,"University Library of Tromso, Ravnetrykk No. 29. , pp. 195–202. by exposing natural radium-226 to neutrons to produce radium-227, which decays with a 42-minute half-life to actinium-227. Actinium-227 (half-life 21.8 years) in turn decays via thorium-227 (half-life 18.7 days) to radium-223. Radium-223 (223Ra, Ra-223) is an isotope of radium with an 11.4-day half-life. 209P/LINEAR is a periodic comet with an orbital period of 5.1 years. Plutonium-244 (244Pu) is an isotope of plutonium that has a half-life of 80 million years. This article concerns the period 229 BC – 220 BC. ==References== __NOTOC__ Year 226 BC was a year of the pre-Julian Roman calendar. Its half-life of 80 million years ensured its circulation across the solar system before its extinction, and indeed, 244Pu has not yet been found in matter other than meteorites. Scandium-44 (44Sc) is a radioactive isotope of scandium that decays by positron emission to stable 44Ca with a half-life of 4.042 hours. 44Sc can be obtained as a daughter radionuclide of long-lived 44Ti (t1/2 60.4 a) from 44Ti /44Sc generator or can be produced by nuclear reaction 44Ca ( p, n)44Sc in small cyclotrons. Although radium-223 and its decay products also emit beta and gamma radiation, over 95% of the decay energy is in the form of alpha radiation. The amount of 244Pu in the pre-Solar nebula (4.57×109 years ago) was estimated as 0.8% the amount of 238U..As the age of the Earth is about 57 half-lives of 244Pu, the amount of plutonium-244 left should be very small; Hoffman et al. estimated its content in the rare-earth mineral bastnasite as = 1.0×10−18 g/g, which corresponded to the content in the Earth crust as low as 3×10−25 g/g (i.e. the total mass of plutonium-244 in Earth's crust is about 9 g). This is longer than any of the other isotopes of plutonium and longer than any other actinide isotope except for the three naturally abundant ones: uranium-235 (704 million years), uranium-238 (4.468 billion years), and thorium-232 (14.05 billion years). The principal use of radium-223, as a radiopharmaceutical to treat metastatic cancers in bone, takes advantage of its chemical similarity to calcium, and the short range of the alpha radiation it emits. ==Origin and preparation== Although radium-223 is naturally formed in trace amounts by the decay of uranium-235, it is generally made artificially,Bruland O.S., Larsen R.H. (2003). The recommended regimen is six treatments of 55 kBq/kg (1.5 μCi/kg), repeated at 4-week intervals. === Mechanism of action === The use of radium-223 to treat metastatic bone cancer relies on the ability of alpha radiation from radium-223 and its short-lived decay products to kill cancer cells. The 44Ti /44Sc generator represents a secular equilibrium system with a half-life ratio between parent and daughter of ca. 130 000. Radium revisited. This decay path makes it convenient to prepare radium-223 by ""milking"" it from an actinium-227 containing generator or ""cow"", similar to the moly cows widely used to prepare the medically important isotope technetium-99m. 223Ra itself decays to 219Rn (half-life 3.96 s), a short-lived gaseous radon isotope, by emitting an alpha particle of 5.979 MeV. ==Medical use== The pharmaceutical product and medical use of radium-223 against skeletal metastases was invented by Roy H. Larsen, Gjermund Henriksen and Øyvind S. Bruland""Preparation and use of radium-223 to target calcified tissues for pain palliation, bone cancer therapy, and bone surface conditioning"" US 6635234 and has been developed by the former Norwegian company Algeta ASA, in a partnership with Bayer, under the trade name Xofigo (formerly Alpharadin), and is distributed as a solution containing radium-223 chloride (1100 kBq/ml), sodium chloride, and other ingredients for intravenous injection. M.Pruszynski, A.Majkowska-Pilip, N.Loktionova, E.Eppard, F.Roesch Category:Isotopes of scandium Presence of 244Pu fission tracks can be established by using the initial ratio of 244Pu to 238U (Pu/U)0 at a time T0 = years, when Xe formation first began in meteorites, and by considering how the ratio of Pu/U fission tracks varies over time. In fact, by analyzing data from Earth's mantle which indicates that about 30% of the existing fissiogenic xenon is attributable to 244Pu decay, the timing of Earth's formation can be inferred to have occurred nearly 50–70 million years following the formation of the Solar System. Radium is preferentially absorbed by bone by virtue of its chemical similarity to calcium, with most radium-223 that is not taken up by the bone being cleared, primarily via the gut, and excreted. Radionuclides such as 244Pu undergo decay to produce fissiogenic (i.e., arising from fission) xenon isotopes that can then be used to time the events of the early solar system. ",479, 672.4,"""2500.0""",67,-3.5,B
+"A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \mathrm{~min}$.","The solution is 9 grams of sodium chloride (NaCl) dissolved in water, to a total volume of 1000 ml (weight per unit volume). The state (1, 2), for example, is impossible to reach from an initial state of (0, 0), since (1, 2) has both jugs partially full, and no reversible action is possible from this state. === Jug with initial water === thumb|Starting with 9 liters in the 12-liter jug, the solution for 5 liters is plotted in red on the left, and the solution for 4 liters is plotted in blue on the right. Other assumptions of these problems may include that no water can be spilled, and that each step pouring water from a source jug to a destination jug stops when either the source jug is empty or the destination jug is full, whichever happens first. ==Standard example== The standard puzzle of this kind works with three jugs of capacity 8, 5 and 3 liters. The solver must pour the water so that the first and second jugs both contain 4 units, and the third is empty. Since normal saline contains 9 grams of NaCl, the concentration is 9 grams per litre divided by 58.4 grams per mole, or 0.154 mole per litre. It is most commonly used as a sterile 9 g of salt per litre (0.9%) solution, known as normal saline. In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. Saline (also known as saline solution) is a mixture of sodium chloride (salt) and water. Puzzles of this type ask how many steps of pouring water from one jug to another (until either one jug becomes empty or the other becomes full) are needed to reach a goal state, specified in terms of the volume of liquid that must be present in some jug or jugs. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal. Therefore, there are only two solutions to the problem: :(5, 0) ↔ (0, 5) ↔ (12, 5) ↔ (9, 8) ↔ (9, 0) :(5, 0) ↔ (5, 8) ↔ (12, 1) ↔ (0, 1) ↔ (1, 0) ↔ (1, 8) ↔ (9, 0) For the 4 liter question, since 4\equiv 0 \\! \mod \\!4, one irreversible action is necessary at the start of the solution; It could be simply pouring the whole 9 liters of water from the 12-liter jug to the sink (0,0), or fully fill it to 12 liters from the tap (12,0). Salt water chlorination is a process that uses dissolved salt (1000–36,000 ppm or 1–36 g/L) for the chlorination of swimming pools and hot tubs. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. For example, if one jug that holds 8 liters is empty and the other jug that hold 12 liters has 9 liters of water in it to begin with, then with a source (tap) and a drain (sink), these two jugs can measure volumes of 9 liters, 5 liters, 1 liter, as well as 12 liters, 8 liters, 4 liters and 0 liters. The answer is that the mixtures will be of equal purity. At its simplest, this figure is the result of dividing the lake volume by the flow in or out of the lake. The graph shows two ways to obtain 4 liters using 3-liter and 5-liter jugs, and a water source and sink on a Cartesian grid with diagonal lines of slope −1 (such that x+y=const. on these diagonal lines, which represent pouring water from one jug to the other jug). However, if the osmotic coefficient (a correction for non-ideal solutions) is taken into account, then the saline solution is much closer to isotonic. The question is then posed—which of the two mixtures is purer? The solution still applies no matter how many cups of any sizes and compositions are exchanged, or how little or much stirring at any point in time is done to any barrel, as long as at the end each barrel has the same amount of liquid. However, as the ideal saline concentration of a salt-chlorinated pool is very low (<3,500ppm, the threshold for human perception of salt by taste; seawater is about ten times this concentration), damage usually occurs due to improperly-maintained pool chemistry or improper maintenance of the electrolytic cell. ",-0.38,0.2553,"""7.42""",0.2553,1.3,C
 "A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.
-If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years.","The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual growth rate * Compound interest * Dollar cost averaging * Economic value added * Effective annual rate * Effective interest rate * Expected return * Holding period return * Internal rate of return * Modified Dietz method * Net present value * Rate of profit * Return of capital * Return on assets * Return on capital * Returns (economics) * Simple Dietz method * Time value of money * Time-weighted return * Value investing * Yield ==Notes== ==References== ==Further reading== * A. A. Groppelli and Ehsan Nikbakht. Assuming returns are reinvested however, due to the effect of compounding, the relationship between a rate of return r, and a return R over a length of time t is: :1 + R = (1 + r)^t which can be used to convert the return R to a compound rate of return r: :r = (1 + R)^\frac {1}{t} - 1 = \sqrt[t]{1 + R} - 1 For example, a 33.1% return over 3 months is equivalent to a rate of: :\sqrt[3]{1.331} - 1 = 10\% per month with reinvestment. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). If the value of the investment at the end of the second period is C, the holding period return in the second period is: :R_2 = \frac {C-B}{B} Multiplying together the growth factors in each period 1 + R_1 and 1 + R_2: :(1 + R_1)(1 + R_2) = \left( 1 + \frac {B-A}{A} \right) \left( 1 + \frac {C-B}{B} \right) = \left( \frac {B}{A}\right) \left( \frac {C}{B} \right) = \frac {C}{A} :(1 + R_1)(1 + R_2) - 1 = \frac {C}{A} - 1 = \frac {C-A}{A} is the holding period return over the two successive periods. As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) − 1), assuming reinvestment at the end of the first year. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. The return on the deposit over the year in yen terms is therefore: :\frac{1,346,400 - 1,200,000}{1,200,000} = 12.2\% This is the rate of return experienced either by an investor who starts with yen, converts to dollars, invests in the USD deposit, and converts the eventual proceeds back to yen; or for any investor, who wishes to measure the return in Japanese yen terms, for comparison purposes. ====Annualization==== Without any reinvestment, a return R over a period of time t corresponds to a rate of return r: :r = \frac {R}{t} For example, let us suppose that 20,000 USD is returned on an initial investment of 100,000 USD. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. * The final investment value of $103.02 compared with the initial investment of $100 means the return is $3.02 or 3.02%. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. There may also be more than one real solution to the equation, requiring some interpretation to determine the most appropriate one. ====Money-weighted return over multiple sub-periods==== Note that the money- weighted return over multiple sub-periods is generally not equal to the result of combining together the money-weighted returns within the sub-periods using the method described above, unlike time-weighted returns. ===Comparing ordinary return with logarithmic return=== The value of an investment is doubled if the return r = +100%, that is, if r_{\mathrm{log}} = ln($200 / $100) = ln(2) = 69.3%. Over 4 years, this translates into an overall return of: :1.05^4-1=21.55\% Example #2 Volatile rates of return, including losses Year 1 Year 2 Year 3 Year 4 Rate of return 50% −20% 30% −40% Geometric average at end of year 50% 9.5% 16% −1.6% Capital at end of year $150.00 $120.00 $156.00 $93.60 Dollar profit/(loss) ($6.40) The geometric average return over the 4-year period was −1.64%. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). The return over the five-year period is $19.90 × 91.314 / $1,000 − 1 = 81.71% * Geometric average annual total return with reinvestment = ($19.90 × 91.314 / $1,000) ^ (1 / 5) − 1 = 12.69% * An investor who did not reinvest would have received total distributions (cash payments) of $5.78 per share. Accumulation functions for simple and compound interest are a(t)=1 + r t a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} If n t = 1, then these two functions are the same. ===Continuous compounding=== As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72. * The continuously compounded rate of return in this example is: :\ln\left(\frac{103.02}{100}\right) = 2.98\%. Suppose the value of the investment at the beginning is A, and at the end of the first period is B. The rate of return is 4,000 / 100,000 = 4% per year. Cash flow example on $1,000 investment Year 1 Year 2 Year 3 Year 4 Dollar return $100 $55 $60 $50 ROI 10% 5.5% 6% 5% ==Uses== * Rates of return are useful for making investment decisions. ",2.8108,3930,0.249,3.42,+116.0,B
+If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years.","The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual growth rate * Compound interest * Dollar cost averaging * Economic value added * Effective annual rate * Effective interest rate * Expected return * Holding period return * Internal rate of return * Modified Dietz method * Net present value * Rate of profit * Return of capital * Return on assets * Return on capital * Returns (economics) * Simple Dietz method * Time value of money * Time-weighted return * Value investing * Yield ==Notes== ==References== ==Further reading== * A. A. Groppelli and Ehsan Nikbakht. Assuming returns are reinvested however, due to the effect of compounding, the relationship between a rate of return r, and a return R over a length of time t is: :1 + R = (1 + r)^t which can be used to convert the return R to a compound rate of return r: :r = (1 + R)^\frac {1}{t} - 1 = \sqrt[t]{1 + R} - 1 For example, a 33.1% return over 3 months is equivalent to a rate of: :\sqrt[3]{1.331} - 1 = 10\% per month with reinvestment. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). If the value of the investment at the end of the second period is C, the holding period return in the second period is: :R_2 = \frac {C-B}{B} Multiplying together the growth factors in each period 1 + R_1 and 1 + R_2: :(1 + R_1)(1 + R_2) = \left( 1 + \frac {B-A}{A} \right) \left( 1 + \frac {C-B}{B} \right) = \left( \frac {B}{A}\right) \left( \frac {C}{B} \right) = \frac {C}{A} :(1 + R_1)(1 + R_2) - 1 = \frac {C}{A} - 1 = \frac {C-A}{A} is the holding period return over the two successive periods. As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) − 1), assuming reinvestment at the end of the first year. The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. The return on the deposit over the year in yen terms is therefore: :\frac{1,346,400 - 1,200,000}{1,200,000} = 12.2\% This is the rate of return experienced either by an investor who starts with yen, converts to dollars, invests in the USD deposit, and converts the eventual proceeds back to yen; or for any investor, who wishes to measure the return in Japanese yen terms, for comparison purposes. ====Annualization==== Without any reinvestment, a return R over a period of time t corresponds to a rate of return r: :r = \frac {R}{t} For example, let us suppose that 20,000 USD is returned on an initial investment of 100,000 USD. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 4, and t = 6: A=1500\times\left(1+\frac{0.043}{4}\right)^{4\times 6}\approx 1938.84 So the amount A after 6 years is approximately $1,938.84. Then the balance after 6 years is found by using the formula above, with P = 1500, r = 0.043 (4.3%), n = 1/2 (the interest is compounded every two years), and t = 6 : A=1500\times(1+(0.043\times 2))^{\frac{6}{2}}\approx 1921.24 So, the balance after 6 years is approximately $1,921.24. * The final investment value of $103.02 compared with the initial investment of $100 means the return is $3.02 or 3.02%. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. There may also be more than one real solution to the equation, requiring some interpretation to determine the most appropriate one. ====Money-weighted return over multiple sub-periods==== Note that the money- weighted return over multiple sub-periods is generally not equal to the result of combining together the money-weighted returns within the sub-periods using the method described above, unlike time-weighted returns. ===Comparing ordinary return with logarithmic return=== The value of an investment is doubled if the return r = +100%, that is, if r_{\mathrm{log}} = ln($200 / $100) = ln(2) = 69.3%. Over 4 years, this translates into an overall return of: :1.05^4-1=21.55\% Example #2 Volatile rates of return, including losses Year 1 Year 2 Year 3 Year 4 Rate of return 50% −20% 30% −40% Geometric average at end of year 50% 9.5% 16% −1.6% Capital at end of year $150.00 $120.00 $156.00 $93.60 Dollar profit/(loss) ($6.40) The geometric average return over the 4-year period was −1.64%. * A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1). The return over the five-year period is $19.90 × 91.314 / $1,000 − 1 = 81.71% * Geometric average annual total return with reinvestment = ($19.90 × 91.314 / $1,000) ^ (1 / 5) − 1 = 12.69% * An investor who did not reinvest would have received total distributions (cash payments) of $5.78 per share. Accumulation functions for simple and compound interest are a(t)=1 + r t a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} If n t = 1, then these two functions are the same. ===Continuous compounding=== As n, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of , where is a mathematical constant that is the base of the natural logarithm. The Summa de arithmetica of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound interest to double, one should divide the interest rate into 72. * The continuously compounded rate of return in this example is: :\ln\left(\frac{103.02}{100}\right) = 2.98\%. Suppose the value of the investment at the beginning is A, and at the end of the first period is B. The rate of return is 4,000 / 100,000 = 4% per year. Cash flow example on $1,000 investment Year 1 Year 2 Year 3 Year 4 Dollar return $100 $55 $60 $50 ROI 10% 5.5% 6% 5% ==Uses== * Rates of return are useful for making investment decisions. ",2.8108,3930,"""0.249""",3.42,+116.0,B
 "Consider the initial value problem
 $$
 y^{\prime \prime}+2 a y^{\prime}+\left(a^2+1\right) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 .
 $$
-For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). We are trying to find a formula for y(t) that satisfies these two equations. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. The solution is X(t) = e^{At}X(0) with e^{At} = e^{at}\begin{bmatrix} 1 & at \\\ 0 & 1 \end{bmatrix} . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Third, we consider the case where |t| = \frac{1}{2}. Second, we consider the case where |t|>\frac{1}{2}. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Suppose we have an ordinary differential equation in the complex domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In this case, \begin{cases} 4 \det A - (\operatorname{tr} A)^2 > 0 \\\ \operatorname{tr} A < 0 \end{cases}. The origin is a source, with integral curves of form y = cx^{b/a} ** Similarly for a, b < 0. ** If a > 0 > b or a < 0 < b, then \det A < 0, and the origin is a saddle point. with integral curves of form y = cx^{-|b/a|}. * \begin{bmatrix} a & 1 \\\ 0 & a \end{bmatrix} where a eq 0. If the derivative at is exactly 1 or −1, then more information is needed in order to decide stability. ",-75,24,2.0,7,1.8763,E
+For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). We are trying to find a formula for y(t) that satisfies these two equations. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. The solution is X(t) = e^{At}X(0) with e^{At} = e^{at}\begin{bmatrix} 1 & at \\\ 0 & 1 \end{bmatrix} . In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. Third, we consider the case where |t| = \frac{1}{2}. Second, we consider the case where |t|>\frac{1}{2}. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Just as for =1, the case of the largest absolute value being 1 needs to be investigated further — the Jacobian matrix test is inconclusive. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Suppose we have an ordinary differential equation in the complex domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In this case, \begin{cases} 4 \det A - (\operatorname{tr} A)^2 > 0 \\\ \operatorname{tr} A < 0 \end{cases}. The origin is a source, with integral curves of form y = cx^{b/a} ** Similarly for a, b < 0. ** If a > 0 > b or a < 0 < b, then \det A < 0, and the origin is a saddle point. with integral curves of form y = cx^{-|b/a|}. * \begin{bmatrix} a & 1 \\\ 0 & a \end{bmatrix} where a eq 0. If the derivative at is exactly 1 or −1, then more information is needed in order to decide stability. ",-75,24,"""2.0""",7,1.8763,E
 "Consider the initial value problem
 $$
 y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0,
 $$
 where $\gamma$ is the damping coefficient (or resistance).
-Find the time $t_1$ at which the solution attains its maximum value.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Springer-Verlag, Berlin, 2008. x+506 pp. * Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. A strong maximum principle for parabolic equations. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. The essence of the maximum principle is the simple observation that if each eigenvalue is positive (which amounts to a certain formulation of ""ellipticity"" of the differential equation) then the above equation imposes a certain balancing of the directional second derivatives of the solution. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. We are trying to find a formula for y(t) that satisfies these two equations. After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. The characteristic equation of this dynamic equation is \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, whose solutions are the characteristic values \lambda_1,\dots , \lambda_k; these are used in the solution equation :x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Here the constants c_1, \dots , c_k are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition x_t Is known. ===Continuous time=== A differential equation system of the first order with n variables stacked in a vector X is :\frac{dX}{dt}=AX. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. If the final state x(T) is not fixed (i.e., its differential variation is not zero), it must also be that These four conditions in (1)-(4) are the necessary conditions for an optimal control. On the maximum principle of optimal control in economic processes, 1969 (Trondheim, NTH, Sosialøkonomisk institutt https://www.worldcat.org/title/on-the-maximum-principle-of-optimal-control-in- economic-processes/oclc/23714026) ===Textbooks=== * * Evans, Lawrence C. Partial differential equations. Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. The sign of c is relevant, as also seen in the one-dimensional case; for instance the solutions to are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions. == See also == * Maximum modulus principle * Hopf maximum principle ==Notes== ==References== ===Research articles=== * Calabi, E. This differential equation cannot be solved exactly. ",-50,1.5377,0.0029,30,2.3613,E
+Find the time $t_1$ at which the solution attains its maximum value.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Springer-Verlag, Berlin, 2008. x+506 pp. * Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. A strong maximum principle for parabolic equations. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. The essence of the maximum principle is the simple observation that if each eigenvalue is positive (which amounts to a certain formulation of ""ellipticity"" of the differential equation) then the above equation imposes a certain balancing of the directional second derivatives of the solution. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. We are trying to find a formula for y(t) that satisfies these two equations. After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. The characteristic equation of this dynamic equation is \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, whose solutions are the characteristic values \lambda_1,\dots , \lambda_k; these are used in the solution equation :x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Here the constants c_1, \dots , c_k are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition x_t Is known. ===Continuous time=== A differential equation system of the first order with n variables stacked in a vector X is :\frac{dX}{dt}=AX. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. If the final state x(T) is not fixed (i.e., its differential variation is not zero), it must also be that These four conditions in (1)-(4) are the necessary conditions for an optimal control. On the maximum principle of optimal control in economic processes, 1969 (Trondheim, NTH, Sosialøkonomisk institutt https://www.worldcat.org/title/on-the-maximum-principle-of-optimal-control-in- economic-processes/oclc/23714026) ===Textbooks=== * * Evans, Lawrence C. Partial differential equations. Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. The sign of c is relevant, as also seen in the one-dimensional case; for instance the solutions to are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions. == See also == * Maximum modulus principle * Hopf maximum principle ==Notes== ==References== ===Research articles=== * Calabi, E. This differential equation cannot be solved exactly. ",-50,1.5377,"""0.0029""",30,2.3613,E
 "Consider the initial value problem
 $$
 y^{\prime}+\frac{2}{3} y=1-\frac{1}{2} t, \quad y(0)=y_0 .
 $$
-Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in MATLAB. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Another solution is given by : y_s(x) = 0 . The solution y_s is tangent to every curve y_c(x) at the point (c,0). We are trying to find a formula for y(t) that satisfies these two equations. Colors vary from blue to yellow with time. thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in Mathematica. As such, these points satisfy x = 0. ==Using equations== If the curve in question is given as y= f(x), the y-coordinate of the y-intercept is found by calculating f(0). The condition of intersection is : ys(x) = yc(x). If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Functions which are undefined at x = 0 have no y-intercept. The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1). As such, these points satisfy y=0. We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. We calculate the derivatives: : y_c'(-2 c) = c \,\\! : y_s'(-2 c) = -\tfrac{1}{2} x |_{x = -2 c} = c. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ",0.7812,−1.642876,1.44,-57.2,-994.3,B
-"A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the decay rate $r$.","The decay constant is \frac{\ln(2)}{t_{1/2}} where ""t_{1/2}"" is the half-life of the radioactive material of interest. ==Example== The decay correct might be used this way: a group of 20 animals is injected with a compound of interest on a Monday at 10:00 a.m. (A simple way to check if you are using the decay correct formula right is to put in the value of the half-life in place of ""t"". The exponential time-constant for the process is \tau = R \, C , so the half-life is R \, C \, \ln(2) . For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. == Solution of the differential equation == The equation that describes exponential decay is :\frac{dN}{dt} = -\lambda N or, by rearranging (applying the technique called separation of variables), :\frac{dN}{N} = -\lambda dt. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where is the quantity at time , is the initial quantity, that is, the quantity at time . == Measuring rates of decay== === Mean lifetime === If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. If the radiation measured has dropped by half between the 4 hour sample and the 24 hour sample we might think that the concentration of compound in that organ has dropped by half; but applying the decay correct we see that the concentration is 0.5*2.82 so it has actually increased by 40% in that period. ==References== Category:Radioactivity A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. After a day, the decay heat falls to 0.4%, and after a week, it will be only 0.2%. After a day, the decay heat falls to 0.4%, and after a week it will be only 0.2%. This value is in the denominator of the decay correcting fraction, so it is the same as multiplying the numerator by its inverse ( {1 \over 0.3546} ), which is 2.82. Some minerals that contain thorium include apatite, sphene, zircon, allanite, monazite, pyrochlore, thorite, and xenotime. ==Decay== Thorium-232 has a half-life of 14 billion years and mainly decays by alpha decay to radium-228 with a decay energy of 4.0816 MeV. The intermediates in the thorium-232 decay chain are all relatively short- lived; the longest-lived intermediate decay products are radium-228 and thorium-228, with half lives of 5.75 years and 1.91 years, respectively. Substitute 12.7 (hours, the half- life of copper-64) for t_{1/2} , giving {0.693 \over 12.7} = 0.0546. Decay energy is usually quoted in terms of the energy units MeV (million electronvolts) or keV (thousand electronvolts): : Q \text{ [MeV]} = -931.5 \Delta M \text{ [Da]},~~(\text{where }\Delta M = \Sigma M_\text{products} - \Sigma M_\text{reactants}). Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. The units of the decay constant are s−1. === Derivation of the mean lifetime === Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, \tau, (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. The following table lists the intermediate decay products in the thorium-232 decay chain: nuclide decay mode half-life (a=year) energy released, MeV product of decay 232Th α 1.4 a 4.081 228Ra 228Ra β− 5.75 a 0.046 228Ac 228Ac β− 6.15 h 2.134 228Th 228Th α 1.9116 a 5.520 224Ra 224Ra α 3.6319 d 5.789 220Rn 220Rn α 55.6 s 6.405 216Po 216Po α 0.145 s 6.906 212Pb 212Pb β− 10.64 h 0.569 212Bi 212Bi β− 64.06% α 35.94% 60.55 min 2.252 6.207 212Po 208Tl 212Po α 299 ns 8.954 208Pb 208Tl β− 3.053 min 4.999 208Pb 208Pb stable . . . ===Rare decay modes=== Although thorium-232 mainly decays by alpha decay, it also undergoes spontaneous fission 1.1% of the time. The half-life can be written in terms of the decay constant, or the mean lifetime, as: :t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2). The total decay rate of the quantity N is given by the sum of the decay routes; thus, in the case of two processes: :-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N. Therefore, the mean lifetime \tau is equal to the half-life divided by the natural log of 2, or: : \tau = \frac{t_{1/2}}{\ln (2)} \approx 1.44 \cdot t_{1/2}. The decay energy is the energy change of a nucleus having undergone a radioactive decay. Quantitatively, at the moment of reactor shutdown, decay heat from these radioactive sources is still 6.5% of the previous core power if the reactor has had a long and steady power history. ",6.6,0.1591549431,12.0,0.02828,-9.54,D
-"Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly poured, and $1 \mathrm{~min}$ later has cooled to $190^{\circ} \mathrm{F}$ in a room at $70^{\circ} \mathrm{F}$, determine when the coffee reaches a temperature of $150^{\circ} \mathrm{F}$.","Rather, using today's terms, Newton noted after some mathematical manipulation that the rate of temperature change of a body is proportional to the difference in temperatures between the body and its surroundings. In that case, Newton's law only approximates the result when the temperature difference is relatively small. In the study of heat transfer, Newton's law of cooling is a physical law which states that > The rate of heat loss of a body is directly proportional to the difference > in the temperatures between the body and its environment. When the heat transfer coefficient is independent, or relatively independent, of the temperature difference between object and environment, Newton's law is followed. The equation to describe this change in (relatively uniform) temperature inside the object, is the simple exponential one described in Newton's law of cooling expressed in terms of temperature difference (see below). In the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature differences. This second system of measurement led Newton to derive his law of convective heat transfer, also known as Newton's law of cooling. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. When stated in terms of temperature differences, Newton's law (with several further simplifying assumptions, such as a low Biot number and a temperature-independent heat capacity) results in a simple differential equation expressing temperature-difference as a function of time. Radiative cooling is better described by the Stefan–Boltzmann law in which the heat transfer rate varies as the difference in the 4th powers of the absolute temperatures of the object and of its environment. == Mathematical formulation of Newton's law == The statement of Newton's law used in the heat transfer literature puts into mathematics the idea that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. Analytic methods for handling these problems, which may exist for simple geometric shapes and uniform material thermal conductivity, are described in the article on the heat equation. == Application of Newton's law of transient cooling == Simple solutions for transient cooling of an object may be obtained when the internal thermal resistance within the object is small in comparison to the resistance to heat transfer away from the object's surface (by external conduction or convection), which is the condition for which the Biot number is less than about 0.1. Typically, this type of analysis leads to simple exponential heating or cooling behavior (""Newtonian"" cooling or heating) since the internal energy of the body is directly proportional to its temperature, which in turn determines the rate of heat transfer into or out of it. By comparison to Newton's original data, they concluded that his measurements (from 1692 to 1693) had been ""quite accurate"". == Relationship to mechanism of cooling== Convection cooling is sometimes said to be governed by ""Newton's law of cooling."" This final simplest version of the law, given by Newton himself, was partly due to confusion in Newton's time between the concepts of heat and temperature, which would not be fully disentangled until much later.History of Newton's cooling law In 2020, Maruyama and Moriya repeated Newton's experiments with modern apparatus, and they applied modern data reduction techniques. This characteristic decay of the temperature-difference is also associated with Newton's law of cooling. == Historical background == Isaac Newton published his work on cooling anonymously in 1701 as ""Scala graduum Caloris. thumb|A small cup of ice coffee from 85°C Bakery Café. 85 °C Bakery Cafe, also brand-named 85 Cafe, 85 °C Daily Cafe, or 85 Degrees C (), is a Taiwanese international chain of retailers selling coffee, tea, and cakes, as well as desserts, smoothies, fruit juices, souvenirs, and bakery products. Newton's law is most closely obeyed in purely conduction-type cooling. Another situation that does not obey Newton's law is radiative heat transfer. In heat conduction, Newton's Law is generally followed as a consequence of Fourier's law. The name ""85°C"" refers to Wu's belief that is the optimal temperature to serve coffee. A temperature coefficient describes the relative change of a physical property that is associated with a given change in temperature. A correction to Newton's law concerning convection for larger temperature differentials by including an exponent, was made in 1817 by Dulong and Petit. ",2,9.90,0.375, 6.07,16,D
-"Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. The Newmark-beta method is a method of numerical integration used to solve certain differential equations. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Using the extended mean value theorem, the Newmark-\beta method states that the first time derivative (velocity in the equation of motion) can be solved as, :\dot{u}_{n+1}=\dot{u}_n+ \Delta t~\ddot{u}_\gamma \, where :\ddot{u}_\gamma = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1 therefore :\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Beta regression is a form of regression which is used when the response variable, y, takes values within (0, 1) and can be assumed to follow a beta distribution. Beta (Translation: Son) is a 1992 Indian Hindi drama film, directed by Indra Kumar and written by Naushir Khatau and Kamlesh Pandey. Suppose we have an ordinary differential equation in the complex domain. Average constant acceleration (Middle point rule) (\gamma=0.5 and \beta=0.25 ) is unconditionally stable. ==References== Category:Numerical differential equations This solution has a branchpoint at x=c, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant c). In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. Any given solution y(x) of this equation may well have singularities at various points (i.e. points at which it is not a regular holomorphic function, such as branch points, essential singularities or poles). An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. For example the equation : \frac{dy}{dx} = \frac{1}{2y} has solution y=\sqrt{x-c} for any constant c. ",0.321, -1,76.0,7,11,B
+Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in MATLAB. By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Another solution is given by : y_s(x) = 0 . The solution y_s is tangent to every curve y_c(x) at the point (c,0). We are trying to find a formula for y(t) that satisfies these two equations. Colors vary from blue to yellow with time. thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in Mathematica. As such, these points satisfy x = 0. ==Using equations== If the curve in question is given as y= f(x), the y-coordinate of the y-intercept is found by calculating f(0). The condition of intersection is : ys(x) = yc(x). If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Functions which are undefined at x = 0 have no y-intercept. The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1). As such, these points satisfy y=0. We solve : c \cdot x + c^2 = y_c(x) = y_s(x) = -\tfrac{1}{4} x^2 to find the intersection point, which is (-2c , -c^2). Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. We calculate the derivatives: : y_c'(-2 c) = c \,\\! : y_s'(-2 c) = -\tfrac{1}{2} x |_{x = -2 c} = c. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ",0.7812,−1.642876,"""1.44""",-57.2,-994.3,B
+"A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the decay rate $r$.","The decay constant is \frac{\ln(2)}{t_{1/2}} where ""t_{1/2}"" is the half-life of the radioactive material of interest. ==Example== The decay correct might be used this way: a group of 20 animals is injected with a compound of interest on a Monday at 10:00 a.m. (A simple way to check if you are using the decay correct formula right is to put in the value of the half-life in place of ""t"". The exponential time-constant for the process is \tau = R \, C , so the half-life is R \, C \, \ln(2) . For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. == Solution of the differential equation == The equation that describes exponential decay is :\frac{dN}{dt} = -\lambda N or, by rearranging (applying the technique called separation of variables), :\frac{dN}{N} = -\lambda dt. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where is the quantity at time , is the initial quantity, that is, the quantity at time . == Measuring rates of decay== === Mean lifetime === If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. If the radiation measured has dropped by half between the 4 hour sample and the 24 hour sample we might think that the concentration of compound in that organ has dropped by half; but applying the decay correct we see that the concentration is 0.5*2.82 so it has actually increased by 40% in that period. ==References== Category:Radioactivity A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. After a day, the decay heat falls to 0.4%, and after a week, it will be only 0.2%. After a day, the decay heat falls to 0.4%, and after a week it will be only 0.2%. This value is in the denominator of the decay correcting fraction, so it is the same as multiplying the numerator by its inverse ( {1 \over 0.3546} ), which is 2.82. Some minerals that contain thorium include apatite, sphene, zircon, allanite, monazite, pyrochlore, thorite, and xenotime. ==Decay== Thorium-232 has a half-life of 14 billion years and mainly decays by alpha decay to radium-228 with a decay energy of 4.0816 MeV. The intermediates in the thorium-232 decay chain are all relatively short- lived; the longest-lived intermediate decay products are radium-228 and thorium-228, with half lives of 5.75 years and 1.91 years, respectively. Substitute 12.7 (hours, the half- life of copper-64) for t_{1/2} , giving {0.693 \over 12.7} = 0.0546. Decay energy is usually quoted in terms of the energy units MeV (million electronvolts) or keV (thousand electronvolts): : Q \text{ [MeV]} = -931.5 \Delta M \text{ [Da]},~~(\text{where }\Delta M = \Sigma M_\text{products} - \Sigma M_\text{reactants}). Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. The units of the decay constant are s−1. === Derivation of the mean lifetime === Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime, \tau, (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly. The following table lists the intermediate decay products in the thorium-232 decay chain: nuclide decay mode half-life (a=year) energy released, MeV product of decay 232Th α 1.4 a 4.081 228Ra 228Ra β− 5.75 a 0.046 228Ac 228Ac β− 6.15 h 2.134 228Th 228Th α 1.9116 a 5.520 224Ra 224Ra α 3.6319 d 5.789 220Rn 220Rn α 55.6 s 6.405 216Po 216Po α 0.145 s 6.906 212Pb 212Pb β− 10.64 h 0.569 212Bi 212Bi β− 64.06% α 35.94% 60.55 min 2.252 6.207 212Po 208Tl 212Po α 299 ns 8.954 208Pb 208Tl β− 3.053 min 4.999 208Pb 208Pb stable . . . ===Rare decay modes=== Although thorium-232 mainly decays by alpha decay, it also undergoes spontaneous fission 1.1% of the time. The half-life can be written in terms of the decay constant, or the mean lifetime, as: :t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2). The total decay rate of the quantity N is given by the sum of the decay routes; thus, in the case of two processes: :-\frac{dN(t)}{dt} = N\lambda _1 + N\lambda _2 = (\lambda _1 + \lambda _2)N. Therefore, the mean lifetime \tau is equal to the half-life divided by the natural log of 2, or: : \tau = \frac{t_{1/2}}{\ln (2)} \approx 1.44 \cdot t_{1/2}. The decay energy is the energy change of a nucleus having undergone a radioactive decay. Quantitatively, at the moment of reactor shutdown, decay heat from these radioactive sources is still 6.5% of the previous core power if the reactor has had a long and steady power history. ",6.6,0.1591549431,"""12.0""",0.02828,-9.54,D
+"Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly poured, and $1 \mathrm{~min}$ later has cooled to $190^{\circ} \mathrm{F}$ in a room at $70^{\circ} \mathrm{F}$, determine when the coffee reaches a temperature of $150^{\circ} \mathrm{F}$.","Rather, using today's terms, Newton noted after some mathematical manipulation that the rate of temperature change of a body is proportional to the difference in temperatures between the body and its surroundings. In that case, Newton's law only approximates the result when the temperature difference is relatively small. In the study of heat transfer, Newton's law of cooling is a physical law which states that > The rate of heat loss of a body is directly proportional to the difference > in the temperatures between the body and its environment. When the heat transfer coefficient is independent, or relatively independent, of the temperature difference between object and environment, Newton's law is followed. The equation to describe this change in (relatively uniform) temperature inside the object, is the simple exponential one described in Newton's law of cooling expressed in terms of temperature difference (see below). In the case of heat transfer by thermal radiation, Newton's law of cooling holds only for very small temperature differences. This second system of measurement led Newton to derive his law of convective heat transfer, also known as Newton's law of cooling. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. When stated in terms of temperature differences, Newton's law (with several further simplifying assumptions, such as a low Biot number and a temperature-independent heat capacity) results in a simple differential equation expressing temperature-difference as a function of time. Radiative cooling is better described by the Stefan–Boltzmann law in which the heat transfer rate varies as the difference in the 4th powers of the absolute temperatures of the object and of its environment. == Mathematical formulation of Newton's law == The statement of Newton's law used in the heat transfer literature puts into mathematics the idea that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings. Analytic methods for handling these problems, which may exist for simple geometric shapes and uniform material thermal conductivity, are described in the article on the heat equation. == Application of Newton's law of transient cooling == Simple solutions for transient cooling of an object may be obtained when the internal thermal resistance within the object is small in comparison to the resistance to heat transfer away from the object's surface (by external conduction or convection), which is the condition for which the Biot number is less than about 0.1. Typically, this type of analysis leads to simple exponential heating or cooling behavior (""Newtonian"" cooling or heating) since the internal energy of the body is directly proportional to its temperature, which in turn determines the rate of heat transfer into or out of it. By comparison to Newton's original data, they concluded that his measurements (from 1692 to 1693) had been ""quite accurate"". == Relationship to mechanism of cooling== Convection cooling is sometimes said to be governed by ""Newton's law of cooling."" This final simplest version of the law, given by Newton himself, was partly due to confusion in Newton's time between the concepts of heat and temperature, which would not be fully disentangled until much later.History of Newton's cooling law In 2020, Maruyama and Moriya repeated Newton's experiments with modern apparatus, and they applied modern data reduction techniques. This characteristic decay of the temperature-difference is also associated with Newton's law of cooling. == Historical background == Isaac Newton published his work on cooling anonymously in 1701 as ""Scala graduum Caloris. thumb|A small cup of ice coffee from 85°C Bakery Café. 85 °C Bakery Cafe, also brand-named 85 Cafe, 85 °C Daily Cafe, or 85 Degrees C (), is a Taiwanese international chain of retailers selling coffee, tea, and cakes, as well as desserts, smoothies, fruit juices, souvenirs, and bakery products. Newton's law is most closely obeyed in purely conduction-type cooling. Another situation that does not obey Newton's law is radiative heat transfer. In heat conduction, Newton's Law is generally followed as a consequence of Fourier's law. The name ""85°C"" refers to Wu's belief that is the optimal temperature to serve coffee. A temperature coefficient describes the relative change of a physical property that is associated with a given change in temperature. A correction to Newton's law concerning convection for larger temperature differentials by including an exponent, was made in 1817 by Dulong and Petit. ",2,9.90,"""0.375""", 6.07,16,D
+"Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. We are trying to find a formula for y(t) that satisfies these two equations. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. The Newmark-beta method is a method of numerical integration used to solve certain differential equations. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Using the extended mean value theorem, the Newmark-\beta method states that the first time derivative (velocity in the equation of motion) can be solved as, :\dot{u}_{n+1}=\dot{u}_n+ \Delta t~\ddot{u}_\gamma \, where :\ddot{u}_\gamma = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1 therefore :\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Beta regression is a form of regression which is used when the response variable, y, takes values within (0, 1) and can be assumed to follow a beta distribution. Beta (Translation: Son) is a 1992 Indian Hindi drama film, directed by Indra Kumar and written by Naushir Khatau and Kamlesh Pandey. Suppose we have an ordinary differential equation in the complex domain. Average constant acceleration (Middle point rule) (\gamma=0.5 and \beta=0.25 ) is unconditionally stable. ==References== Category:Numerical differential equations This solution has a branchpoint at x=c, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant c). In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Exponential integrators are a class of numerical methods for the solution of ordinary differential equations, specifically initial value problems. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. Any given solution y(x) of this equation may well have singularities at various points (i.e. points at which it is not a regular holomorphic function, such as branch points, essential singularities or poles). An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. For example the equation : \frac{dy}{dx} = \frac{1}{2y} has solution y=\sqrt{x-c} for any constant c. ",0.321, -1,"""76.0""",7,11,B
 "Consider the initial value problem (see Example 5)
 $$
 y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=2, \quad y^{\prime}(0)=\beta
 $$
 where $\beta>0$.
-Determine the smallest value of $\beta$ for which $y_m \geq 4$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. For the closest approximation, the optimum values for \alpha and \beta are \alpha_0 = \frac{2 \cos \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.960433870103... and \beta_0 = \frac{2 \sin \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.397824734759..., giving a maximum error of 3.96%. \alpha\,\\! \beta\,\\! thumb|The locus of points that give the same value in the algorithm, for different values of alpha and beta The alpha max plus beta min algorithm is a high-speed approximation of the square root of the sum of two squares. In the shortest common supersequence problem, two sequences X and Y are given, and the task is to find a shortest possible common supersequence of these sequences. The Newmark-beta method is a method of numerical integration used to solve certain differential equations. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Beta regression is a form of regression which is used when the response variable, y, takes values within (0, 1) and can be assumed to follow a beta distribution. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Using the extended mean value theorem, the Newmark-\beta method states that the first time derivative (velocity in the equation of motion) can be solved as, :\dot{u}_{n+1}=\dot{u}_n+ \Delta t~\ddot{u}_\gamma \, where :\ddot{u}_\gamma = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1 therefore :\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}. Thus, :u_{n+1}=u_n + \Delta t~\dot{u}_n+\begin{matrix} \frac 1 2 \end{matrix} \Delta t^2~\ddot{u}_\beta where again :\ddot{u}_\beta = (1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}~~~~0\leq 2\beta\leq 1 The discretized structural equation becomes \begin{aligned} &\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}\\\ &u;_{n+1}=u_n + \Delta t~\dot{u}_n + \frac{\Delta t^2}{2}\left((1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}\right)\\\ &M;\ddot{u}_{n+1} + C\dot{u}_{n+1} + f^{\textrm{int}}(u_{n+1}) = f_{n+1}^{\textrm{ext}} \, \end{aligned} Explicit central difference scheme is obtained by setting \gamma=0.5 and \beta=0 Average constant acceleration (Middle point rule) is obtained by setting \gamma=0.5 and \beta=0.25 == Stability Analysis == A time-integration scheme is said to be stable if there exists an integration time-step \Delta t_0 > 0 so that for any \Delta t \in (0, \Delta t_0], a finite variation of the state vector q_n at time t_n induces only a non-increasing variation of the state- vector q_{n+1} calculated at a subsequent time t_{n+1}. Largest error (%) Mean error (%) 1/1 1/2 11.80 8.68 1/1 1/4 11.61 3.20 1/1 3/8 6.80 4.25 7/8 7/16 12.50 4.91 15/16 15/32 6.25 3.08 \alpha_0 \beta_0 3.96 2.41 800px|centre ==Improvements== When \alpha < 1, |z| becomes smaller than \mathbf{Max} (which is geometrically impossible) near the axes where \mathbf{Min} is near 0. We are trying to find a formula for y(t) that satisfies these two equations. Average constant acceleration (Middle point rule) (\gamma=0.5 and \beta=0.25 ) is unconditionally stable. ==References== Category:Numerical differential equations A lower \alpha and higher \beta can therefore increase precision further. In general, the closer the observed y values are to the (a, b) extremes, the more significant the choice of link function. ",1.22,16.3923,14.0,71,0.444444444444444 ,B
+Determine the smallest value of $\beta$ for which $y_m \geq 4$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. For the closest approximation, the optimum values for \alpha and \beta are \alpha_0 = \frac{2 \cos \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.960433870103... and \beta_0 = \frac{2 \sin \frac{\pi}{8}}{1 + \cos \frac{\pi}{8}} = 0.397824734759..., giving a maximum error of 3.96%. \alpha\,\\! \beta\,\\! thumb|The locus of points that give the same value in the algorithm, for different values of alpha and beta The alpha max plus beta min algorithm is a high-speed approximation of the square root of the sum of two squares. In the shortest common supersequence problem, two sequences X and Y are given, and the task is to find a shortest possible common supersequence of these sequences. The Newmark-beta method is a method of numerical integration used to solve certain differential equations. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Beta regression is a form of regression which is used when the response variable, y, takes values within (0, 1) and can be assumed to follow a beta distribution. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Using the extended mean value theorem, the Newmark-\beta method states that the first time derivative (velocity in the equation of motion) can be solved as, :\dot{u}_{n+1}=\dot{u}_n+ \Delta t~\ddot{u}_\gamma \, where :\ddot{u}_\gamma = (1 - \gamma)\ddot{u}_n + \gamma \ddot{u}_{n+1}~~~~0\leq \gamma \leq 1 therefore :\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}. Thus, :u_{n+1}=u_n + \Delta t~\dot{u}_n+\begin{matrix} \frac 1 2 \end{matrix} \Delta t^2~\ddot{u}_\beta where again :\ddot{u}_\beta = (1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}~~~~0\leq 2\beta\leq 1 The discretized structural equation becomes \begin{aligned} &\dot{u}_{n+1}=\dot{u}_n + (1 - \gamma) \Delta t~\ddot{u}_n + \gamma \Delta t~\ddot{u}_{n+1}\\\ &u;_{n+1}=u_n + \Delta t~\dot{u}_n + \frac{\Delta t^2}{2}\left((1 - 2\beta)\ddot{u}_n + 2\beta\ddot{u}_{n+1}\right)\\\ &M;\ddot{u}_{n+1} + C\dot{u}_{n+1} + f^{\textrm{int}}(u_{n+1}) = f_{n+1}^{\textrm{ext}} \, \end{aligned} Explicit central difference scheme is obtained by setting \gamma=0.5 and \beta=0 Average constant acceleration (Middle point rule) is obtained by setting \gamma=0.5 and \beta=0.25 == Stability Analysis == A time-integration scheme is said to be stable if there exists an integration time-step \Delta t_0 > 0 so that for any \Delta t \in (0, \Delta t_0], a finite variation of the state vector q_n at time t_n induces only a non-increasing variation of the state- vector q_{n+1} calculated at a subsequent time t_{n+1}. Largest error (%) Mean error (%) 1/1 1/2 11.80 8.68 1/1 1/4 11.61 3.20 1/1 3/8 6.80 4.25 7/8 7/16 12.50 4.91 15/16 15/32 6.25 3.08 \alpha_0 \beta_0 3.96 2.41 800px|centre ==Improvements== When \alpha < 1, |z| becomes smaller than \mathbf{Max} (which is geometrically impossible) near the axes where \mathbf{Min} is near 0. We are trying to find a formula for y(t) that satisfies these two equations. Average constant acceleration (Middle point rule) (\gamma=0.5 and \beta=0.25 ) is unconditionally stable. ==References== Category:Numerical differential equations A lower \alpha and higher \beta can therefore increase precision further. In general, the closer the observed y values are to the (a, b) extremes, the more significant the choice of link function. ",1.22,16.3923,"""14.0""",71,0.444444444444444 ,B
 "A home buyer can afford to spend no more than $\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
-Determine the total interest paid during the term of the mortgage.","Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. If the length of the periods are equal (monthly payments) then the summations can be simplified using the formula for a geometric series. For a theoretical continuous payment savings annuity we can only calculate an annual rate of payment: :M_a=\frac{500000 \times 12\%}{e^{0.12\cdot 10}-1}=25860.77 At this point there is a temptation to simply divide by 12 to obtain a monthly payment. ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. For example, for interest rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives: = PMT(0.06/12, 25 * 12, -150000, 0, 0) = $966.45 ====Approximate formula for monthly payment==== A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (I<8\% and terms T=10–30 years), the monthly note rate is small compared to 1: r << 1 so that the \ln(1+r)\approx r which yields a simplification so that c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = IT c_0\equiv \frac{P}{n} . thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. So (for example) the time constant when the interest rate is 10% is 10 years and the period of a home loan should be determined - within the bounds of affordability - as a minimum multiple of this if the objective is to minimise interest paid on the loan. ==Mortgage difference and differential equation== The conventional difference equation for a mortgage loan is relatively straightforward to derive - balance due in each successive period is the previous balance plus per period interest less the per period fixed payment. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. ","102,965.21",29.9,0.84,4.0,25.6773,A
+Determine the total interest paid during the term of the mortgage.","Each payment accumulates compound interest from time of deposit to the end of the mortgage timespan at which point the sum of the payments with their accumulated interest equals the value of the loan with interest compounded over the entire timespan. When interest is continuously compounded, use \delta=n\ln{\left(1+\frac{r}{n}\right)}, where \delta is the interest rate on a continuous compounding basis, and r is the stated interest rate with a compounding frequency n. ===Monthly amortized loan or mortgage payments=== The interest on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The total compound interest generated is the final value minus the initial principal: I=P\left(1+\frac{r}{n}\right)^{nt}-P ====Example 1==== Suppose a principal amount of $1,500 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Loan (P) Period (T) Annual payment rate (Ma) Initial estimate: 2 ln(MaT/P)/T 10000 3 6000 39.185778% Newton–Raphson iterations n r(n) f[r(n)] f'[r(n)] 0 39.185778% −229.57 4444.44 1 44.351111% 21.13 5241.95 2 43.948044% 0.12 5184.06 3 43.945798% 0 5183.74 ==Present value and future value formulae== Corresponding to the standard formula for the present value of a series of fixed monthly payments, we have already established a time continuous analogue: :P_v(t)=\frac{M_a}{r}(1-e^{-rt}). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P0 as P(t)=P_0 e ^ {rt}. ===Force of interest=== As the number of compounding periods n tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest \delta. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. The amount of interest received can be calculated by subtracting the principal from this amount. 1921.24-1500=421.24 The interest is less compared with the previous case, as a result of the lower compounding frequency. ===Accumulation function=== Since the principal P is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. If the length of the periods are equal (monthly payments) then the summations can be simplified using the formula for a geometric series. For a theoretical continuous payment savings annuity we can only calculate an annual rate of payment: :M_a=\frac{500000 \times 12\%}{e^{0.12\cdot 10}-1}=25860.77 At this point there is a temptation to simply divide by 12 to obtain a monthly payment. ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. For example, for interest rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives: = PMT(0.06/12, 25 * 12, -150000, 0, 0) = $966.45 ====Approximate formula for monthly payment==== A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (I<8\% and terms T=10–30 years), the monthly note rate is small compared to 1: r << 1 so that the \ln(1+r)\approx r which yields a simplification so that c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}} which suggests defining auxiliary variables Y\equiv n r = IT c_0\equiv \frac{P}{n} . thumb|300px|Effective interest rates thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. Starting in one month's time he decides to make equal monthly payments into an account that pays interest at 12% per annum compounded monthly. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. So (for example) the time constant when the interest rate is 10% is 10 years and the period of a home loan should be determined - within the bounds of affordability - as a minimum multiple of this if the objective is to minimise interest paid on the loan. ==Mortgage difference and differential equation== The conventional difference equation for a mortgage loan is relatively straightforward to derive - balance due in each successive period is the previous balance plus per period interest less the per period fixed payment. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. ","102,965.21",29.9,"""0.84""",4.0,25.6773,A
 "Find the fundamental period of the given function:
-$$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$","Period 2 is the first period in the periodic table from which periodic trends can be drawn. The constant function , where is independent of , is periodic with any period, but lacks a fundamental period. Since we are calculating a sine series, a_n=0\ \quad \forall n Now, b_n= \frac{2}{\pi} \int_0^\pi \cos(x)\sin(nx)\,\mathrm{d}x = \frac{2n((-1)^n+1)}{\pi(n^2-1)}\quad \forall n\ge 2 When n is odd, b_n=0 When n is even, b_n={4n \over \pi(n^2-1)} thus b_{2k}={8k \over \pi(4k^2-1)} With the special case b_1=0, hence the required Fourier sine series is \cos(x) = {{8 \over \pi} \sum_{n=1}^{\infty} {n \over(4n^2-1)}\sin(2nx)} Category:Fourier series In a quantum mechanical description of atomic structure, this period corresponds to the filling of the second () shell, more specifically its 2s and 2p subshells. The fundamental series is a set of spectral lines in a set caused by transition between d and f orbitals in atoms. Example Calculate the half range Fourier sine series for the function f(x)=\cos(x) where 0. This is a list of some well-known periodic functions. A period 2 element is one of the chemical elements in the second row (or period) of the periodic table of the chemical elements. Period 2 only has two metals (lithium and beryllium) of eight elements, less than for any subsequent period both by number and by proportion. The second period contains the elements lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon. Year 186 (CLXXXVI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. Period 2 elements (carbon, nitrogen, oxygen, fluorine and neon) obey the octet rule in that they need eight electrons to complete their valence shell (lithium and beryllium obey duet rule, boron is electron deficient.), where at most eight electrons can be accommodated: two in the 2s orbital and six in the 2p subshell. ==Periodic trends== thumb|Calculated atomic radii of period 2 elements in picometers. In mathematics, a half range Fourier series is a Fourier series defined on an interval [0,L] instead of the more common [-L,L], with the implication that the analyzed function f(x), x\in[0,L] should be extended to [-L,0] as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). Period 1, which only contains two elements (hydrogen and helium), is too small to draw any conclusive trends from it, especially because the two elements behave nothing like other s-block elements. The fundamental series was described as badly-named. thumb|The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... Since the function is defined for all , one can still take the limit as x approaches 1, and this is the definition of the Abel sum: \lim_{x\rightarrow 1^{-}}\sum_{n=1}^\infty n(-x)^{n-1} = \lim_{x\rightarrow 1^{-}}\frac{1}{(1+x)^2} = \frac14. ===Euler and Borel=== right|thumb|Euler summation to − . All period 2 elements completely obey the Madelung rule; in period 2, lithium and beryllium fill the 2s subshell, and boron, carbon, nitrogen, oxygen, fluorine, and neon fill the 2p subshell. They became known as the fundamental series. In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. In 1909 W. M. Hicks produced approximate formulas for the various series and noticed that this series had a simpler formula than the others and thus called it the ""fundamental series"" and used the letter F. In the case where an = bn = (−1)n, the terms of the Cauchy product are given by the finite diagonal sums \begin{array}{rcl} c_n & = &\displaystyle \sum_{k=0}^n a_k b_{n-k}=\sum_{k=0}^n (-1)^k (-1)^{n-k} \\\\[1em] & = &\displaystyle \sum_{k=0}^n (-1)^n = (-1)^n(n+1). \end{array} The product series is then \sum_{n=0}^\infty(-1)^n(n+1) = 1-2+3-4+\cdots. ",0.6321205588,4,7.136, -194,-0.10,B
-"A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900?","thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? For a 24-month loan, the denominator is 300. The FHA went one step further, and set restrictions on the terms and interest rates of qualifying mortgages, typically requiring fully amortizing mortgages to carry terms to maturity in excess of 15 years, with interest rates exceeding 5% annually in only isolated cases. Some mortgage lenders are known to allow as high as 55%. == Monthly payment formula == The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. For example, $100,000 mortgaged (without fees, since they add into the calculation in a different way) over 15 years costs a total of $193,429.80 (interest is 93.430% of principal), but over 30 years, costs a total of $315,925.20 (interest is 215.925% of principal). The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. S&Ls;, on the other hand, tended to offer 11 to 12 year fully amortizing mortgages, and would generally write mortgages with loan-to-value ratios well in excess of 50%. That is, the APR for a 30-year loan cannot be compared to the APR for a 20-year loan. A simple explanation would be as follows: suppose that the total finance charge for a 12-month loan was $78.00. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. ",131, 9.73,25.6773, 135.36,0.0761,B
-"A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?","Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. :For a 30-year loan with monthly payments, n = 30 \text{ years} \times 12 \text{ months/year} = 360\text{ months} Note that the interest rate is commonly referred to as an annual percentage rate (e.g. 8% APR), but in the above formula, since the payments are monthly, the rate i must be in terms of a monthly percent. thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. For example, $100,000 mortgaged (without fees, since they add into the calculation in a different way) over 15 years costs a total of $193,429.80 (interest is 93.430% of principal), but over 30 years, costs a total of $315,925.20 (interest is 215.925% of principal). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. The expansion P\approx P_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 . ====Example of mortgage payment==== For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find: T=30 I=0.045 which gives X=\frac{1}{2}IT=.675 so that P\approx P_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is P=\$608.02 so the approximation is an overestimate of about a sixth of a percent. ===Investing: monthly deposits=== Given a principal (initial) deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Some mortgage lenders are known to allow as high as 55%. == Monthly payment formula == The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. A money factor of .0030 is equivalent to a monthly interest rate of 0.6% and an APR of 7.2%. For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. For a 24-month loan, the denominator is 300. During the second month the borrower has use of two $1000 (2/3) amounts and so the payment should be $1000 plus two $10 interest fees. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). thumb|279px|Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. At the end of the month, the borrower pays back one $1000 and the $30 interest. If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The second value evaluated within the context of the above formula will provide the required interest rate. In that case the formula becomes: :: S -A = R (1 + \mathrm{APR}/100)^{-t_N} + \sum_{k=1}^N A_k (1 + \mathrm{APR}/100)^{-t_k} :where: :: S is the borrowed amount or principal amount. ",−2,8.87,15.757,27,804.62,E
-"Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \mathrm{~s}$ if the bar is made of silver.","The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While thermal bars can form in both fall and spring, most studies of the thermal bar have investigated aspects of the feature in the spring, when the lake is warming up and the summer thermocline is beginning to form. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. Additional studies have been carried out in Lake Ladoga,S. S. Zilitinkevich, K. D. Kreiman, and A. Yu. Terzhevik, “The Thermal Bar,” Journal of Fluid Mechanics 236, no. 1 (1992): 27-42. In metallurgy, cold forming or cold working is any metalworking process in which metal is shaped below its recrystallization temperature, usually at the ambient temperature. This could be used to model heat conduction in a rod. The distribution approaches equilibrium over time. thumb|The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). Thus, if is the temperature, tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too. ==Specific examples== ===Heat flow in a uniform rod=== For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy . Since the thermal conductivity of ice is so small (1.6 - 2.4 W mK−1) compared with most every other ceramic (ex. Al2O3= 40 W mK−1), the growing ice will have a significant insulative effect on the localized thermal conditions within the slurry. This experiment is possible for ice at −10 °C or cooler, and while essentially valid, the details of the process by which the wire passes through the ice are complex. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. thumb|Classic experiment involving regelation of an ice block as a tensioned wire passes through it. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. Therefore, it is sometimes more economical to cold work a less costly and weaker metal than to hot work a more expensive metal that can be heat treated, especially if precision or a fine surface finish is required as well. In large lakes this condition may persist for weeks, during which a temperature front known as a thermal bar forms between the stratified and unstratified areas of the lake. The thermal resistance of the ceramic is significantly smaller than that of the ice however, so the apparent resistance can be expressed as the lower Rceramic. The name comes from the highly polished appearance of the rods; there is no silver in the alloy. The melting point of ice falls by 0.0072 °C for each additional atm of pressure applied. It has been shown that a linearly decreasing temperature on one side of a freeze-cast will result in near- constant solidification velocity, yielding ice crystals with an almost constant thickness along the SSZ of an entire sample. This approach enables a prediction of the ice-front velocity from the thermal parameters of the suspension. ",2598960,7,35.91,-9.54,310,C
-"Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$.","right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in MATLAB. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Colors vary from blue to yellow with time. thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in Mathematica. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In the end, this result shows the interval of definition of the solution does not depend on the Lipschitz constant of the field, but only on the interval of definition of the field and its maximum absolute value. == Other existence theorems == The Picard–Lindelöf theorem shows that the solution exists and that it is unique. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Notice the drastic change in the solutions with respect to the solution obtained with MATLAB. thumb|A chaotic attractor found with parameter values \alpha=1.1 and \gamma=0.87 and initial conditions x_0=-1, y_0=-0, and z_0=0.5, using the default ODE solver in Mathematica. For non-positive integers, the gamma function is not defined. A simple proof of existence of the solution is obtained by successive approximations. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. ",2.8,2,-6.8,840,30,B
+$$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$","Period 2 is the first period in the periodic table from which periodic trends can be drawn. The constant function , where is independent of , is periodic with any period, but lacks a fundamental period. Since we are calculating a sine series, a_n=0\ \quad \forall n Now, b_n= \frac{2}{\pi} \int_0^\pi \cos(x)\sin(nx)\,\mathrm{d}x = \frac{2n((-1)^n+1)}{\pi(n^2-1)}\quad \forall n\ge 2 When n is odd, b_n=0 When n is even, b_n={4n \over \pi(n^2-1)} thus b_{2k}={8k \over \pi(4k^2-1)} With the special case b_1=0, hence the required Fourier sine series is \cos(x) = {{8 \over \pi} \sum_{n=1}^{\infty} {n \over(4n^2-1)}\sin(2nx)} Category:Fourier series In a quantum mechanical description of atomic structure, this period corresponds to the filling of the second () shell, more specifically its 2s and 2p subshells. The fundamental series is a set of spectral lines in a set caused by transition between d and f orbitals in atoms. Example Calculate the half range Fourier sine series for the function f(x)=\cos(x) where 0. This is a list of some well-known periodic functions. A period 2 element is one of the chemical elements in the second row (or period) of the periodic table of the chemical elements. Period 2 only has two metals (lithium and beryllium) of eight elements, less than for any subsequent period both by number and by proportion. The second period contains the elements lithium, beryllium, boron, carbon, nitrogen, oxygen, fluorine, and neon. Year 186 (CLXXXVI) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. Period 2 elements (carbon, nitrogen, oxygen, fluorine and neon) obey the octet rule in that they need eight electrons to complete their valence shell (lithium and beryllium obey duet rule, boron is electron deficient.), where at most eight electrons can be accommodated: two in the 2s orbital and six in the 2p subshell. ==Periodic trends== thumb|Calculated atomic radii of period 2 elements in picometers. In mathematics, a half range Fourier series is a Fourier series defined on an interval [0,L] instead of the more common [-L,L], with the implication that the analyzed function f(x), x\in[0,L] should be extended to [-L,0] as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). Period 1, which only contains two elements (hydrogen and helium), is too small to draw any conclusive trends from it, especially because the two elements behave nothing like other s-block elements. The fundamental series was described as badly-named. thumb|The first 15,000 partial sums of 0 + 1 − 2 + 3 − 4 + ... Since the function is defined for all , one can still take the limit as x approaches 1, and this is the definition of the Abel sum: \lim_{x\rightarrow 1^{-}}\sum_{n=1}^\infty n(-x)^{n-1} = \lim_{x\rightarrow 1^{-}}\frac{1}{(1+x)^2} = \frac14. ===Euler and Borel=== right|thumb|Euler summation to − . All period 2 elements completely obey the Madelung rule; in period 2, lithium and beryllium fill the 2s subshell, and boron, carbon, nitrogen, oxygen, fluorine, and neon fill the 2p subshell. They became known as the fundamental series. In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. In 1909 W. M. Hicks produced approximate formulas for the various series and noticed that this series had a simpler formula than the others and thus called it the ""fundamental series"" and used the letter F. In the case where an = bn = (−1)n, the terms of the Cauchy product are given by the finite diagonal sums \begin{array}{rcl} c_n & = &\displaystyle \sum_{k=0}^n a_k b_{n-k}=\sum_{k=0}^n (-1)^k (-1)^{n-k} \\\\[1em] & = &\displaystyle \sum_{k=0}^n (-1)^n = (-1)^n(n+1). \end{array} The product series is then \sum_{n=0}^\infty(-1)^n(n+1) = 1-2+3-4+\cdots. ",0.6321205588,4,"""7.136""", -194,-0.10,B
+"A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900?","thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? For a 24-month loan, the denominator is 300. The FHA went one step further, and set restrictions on the terms and interest rates of qualifying mortgages, typically requiring fully amortizing mortgages to carry terms to maturity in excess of 15 years, with interest rates exceeding 5% annually in only isolated cases. Some mortgage lenders are known to allow as high as 55%. == Monthly payment formula == The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. For example, $100,000 mortgaged (without fees, since they add into the calculation in a different way) over 15 years costs a total of $193,429.80 (interest is 93.430% of principal), but over 30 years, costs a total of $315,925.20 (interest is 215.925% of principal). The minimum possible payment rate is that which just covers the loan interest – a borrower would in theory pay this amount forever because there is never any decrease in loan capital. Finally the model demonstrates that it is to the modest advantage of the mortgage holder to increase frequency of payment where practically possible. ==Summary of formulae and online calculators== Annual payment rate (mortgage loan): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_0\cdot r}{1 - e^{-rT}} Annual payment rate (sinking fund): M_a=\lim_{N\to\infty}N\cdot x(N)=\frac{P_T\cdot r}{e^{rT}-1} Future value: F_v(t) = \frac{M_a}{r}(e^{rt}-1) Present value: P_v(t) = \frac{M_a}{r}(1 - e^{-rt}) Loan balance: P(t) = \frac{M_a}{r}(1 - e^{-r(T-t)}) Loan period: T=-\frac{1}{r}\ln\left(1-\frac{P_0 r}{M_a}\right) Half- life of loan: t_{\frac{1}{2}}=\frac{1}{r}\ln\left(\frac{1+e^{rT}}{2}\right) Interest rate: r\approx\frac{2}{T}\ln{\frac{M_aT}{P_0}} r=\frac{1}{T}\left (W(-se^{-s})+s\right )\text{ with }s=\frac{M_at}{P_0} Universal mortgage calculator. If the $10 fee were considered, the monthly interest increases by 10% ($10/$100), and the effective APR becomes approximately 435% (1.1512 = 5.3503, which equals a 435% increase). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. If n were 24, the sum of the numbers from 1 to 24 is 24 * (24+1) / 2 = (24 * 25) / 2 = 300, which is the loan's denominator, D. S&Ls;, on the other hand, tended to offer 11 to 12 year fully amortizing mortgages, and would generally write mortgages with loan-to-value ratios well in excess of 50%. That is, the APR for a 30-year loan cannot be compared to the APR for a 20-year loan. A simple explanation would be as follows: suppose that the total finance charge for a 12-month loan was $78.00. thumb|right|350px|The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies Analogous to continuous compounding, a continuous annuity \- Entry on continuous annuity, p. 86 is an ordinary annuity in which the payment interval is narrowed indefinitely. Given an annual interest rate r and a borrower with an annual payment capability MN (divided into N equal payments made at time intervals Δt where Δt = 1/N years), we may write: :\begin{align} P_{t+\Delta t} & = P_t+(rP_t- M_N)\Delta t\\\\[12pt] \dfrac{P_{t+\Delta t}-P_t}{\Delta t} & = rP_t-M_N \end{align} If N is increased indefinitely so that Δt → 0, we obtain the continuous time differential equation: :{\operatorname{d}P(t)\over\operatorname{d}t}=rP(t)-M_a Beckwith: Equation (25) p. 123Hackman: Equation (2) p.1 Note that for there to be a continually diminishing mortgage balance, the following inequality must hold: :P_0 \leqslant \frac{M_a}{r} Where equality holds, the mortgage becomes a perpetuity. If the fee is not considered, this loan has an effective APR of approximately 80% (1.0512 = 1.7959, which is approximately an 80% increase). Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. ",131, 9.73,"""25.6773""", 135.36,0.0761,B
+"A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?","Mortgage calculators can be used to answer such questions as: If one borrows $250,000 at a 7% annual interest rate and pays the loan back over thirty years, with $3,000 annual property tax payment, $1,500 annual property insurance cost and 0.5% annual private mortgage insurance payment, what will the monthly payment be? If in practice the loan is being repaid via monthly instalments, the decimal portion can be converted to months and rounded so this answer would equate to 172 months. ==Calculation of interest rate== In the discrete time interval model, calculation of a mortgage based interest rate given the remaining parameters has not been possible using analytic methods. :For a 30-year loan with monthly payments, n = 30 \text{ years} \times 12 \text{ months/year} = 360\text{ months} Note that the interest rate is commonly referred to as an annual percentage rate (e.g. 8% APR), but in the above formula, since the payments are monthly, the rate i must be in terms of a monthly percent. thumb|385px|30 year mortgage on a $250,000 loan thumb|30 year mortgage of $250,000 at different interest rates Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. For example, $100,000 mortgaged (without fees, since they add into the calculation in a different way) over 15 years costs a total of $193,429.80 (interest is 93.430% of principal), but over 30 years, costs a total of $315,925.20 (interest is 215.925% of principal). This can be expressed mathematically by : p = \frac{P_0\cdot r\cdot (1+r)^n}{(1+r)^n-1} :where: ::p is the payment made each period :: P0 is the initial principal :: r is the percentage rate used each payment :: n is the number of payments This also explains why a 15-year mortgage and a 30-year mortgage with the same APR would have different monthly payments and a different total amount of interest paid. The expansion P\approx P_0 \left(1 + X + \frac{X^2}{3}\right) is valid to better than 1% provided X\le 1 . ====Example of mortgage payment==== For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find: T=30 I=0.045 which gives X=\frac{1}{2}IT=.675 so that P\approx P_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96 The exact payment amount is P=\$608.02 so the approximation is an overestimate of about a sixth of a percent. ===Investing: monthly deposits=== Given a principal (initial) deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. Figure 1 Month Numerator Denominator Percentage of total interest Monthly interest 1 12 78 15.4% $77.00 2 11 78 14.1% $70.50 3 10 78 12.8% $64.00 4 9 78 11.5% $57.50 5 8 78 10.3% $51.50 6 7 78 9.0% $45.00 7 6 78 7.7% $38.50 8 5 78 6.4% $32.00 9 4 78 5.1% $25.50 10 3 78 3.8% $19.00 11 2 78 2.6% $13.00 12 1 78 1.3% $6.50 ==History== Prior to 1935, a borrower might have entered a contract with the lender to repay off a principal plus the pre-calculated total interest divided equally into the monthly repayments. Some mortgage lenders are known to allow as high as 55%. == Monthly payment formula == The fixed monthly payment for a fixed rate mortgage is the amount paid by the borrower every month that ensures that the loan is paid off in full with interest at the end of its term. ""A capital budgeting model of the supply and demand of loanable funds"", Journal of Macroeconomics 12, Summer 1990, pp. 427-436 (specifically p. 430). = \frac {rP(1+r)^N}{(1+r)^N-1}, & r e 0; \\\ \frac{P}{N}, & r = 0. \end{cases} }} For example, for a home loan of $200,000 with a fixed yearly interest rate of 6.5% for 30 years, the principal is P=200000, the monthly interest rate is r=0.065/12, the number of monthly payments is N=30\cdot 12=360, the fixed monthly payment equals $1,264.14. Over the life of a 30-year loan, this amounts to $23,070.86, which is over 11% of the original loan amount. ===Certain fees are not considered=== Some classes of fees are deliberately not included in the calculation of APR. A money factor of .0030 is equivalent to a monthly interest rate of 0.6% and an APR of 7.2%. For home owners with mortgages the important parameter to keep in mind is the time constant of the equation which is simply the reciprocal of the annual interest rate r. For a 24-month loan, the denominator is 300. During the second month the borrower has use of two $1000 (2/3) amounts and so the payment should be $1000 plus two $10 interest fees. Although this article focuses primarily on mortgages, the methods employed are relevant to any situation in which payment or saving is effected by a regular stream of fixed interval payments (annuity). ==Derivation of time-continuous equation== The classical formula for the present value of a series of n fixed monthly payments amount x invested at a monthly interest rate i% is: :P_v(n) = \frac{x(1 - (1 + i)^{-n})}{i} The formula may be re-arranged to determine the monthly payment x on a loan of amount P0 taken out for a period of n months at a monthly interest rate of i%: :x = \frac{P_0\cdot i}{1 - (1 + i)^{-n}} We begin with a small adjustment of the formula: replace i with r/N where r is the annual interest rate and N is the annual frequency of compounding periods (N = 12 for monthly payments). thumb|279px|Parts of total cost and effective APR for a 12-month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. At the end of the month, the borrower pays back one $1000 and the $30 interest. If, in the second case, equal monthly payments are made of $946.01 against 9.569% compounded monthly then it takes 240 months to pay the loan back. Subtracting the original principal from this amount gives the amount of interest received: 1938.84-1500=438.84 ====Example 2==== Suppose the same amount of $1,500 is compounded biennially (every 2 years). The second value evaluated within the context of the above formula will provide the required interest rate. In that case the formula becomes: :: S -A = R (1 + \mathrm{APR}/100)^{-t_N} + \sum_{k=1}^N A_k (1 + \mathrm{APR}/100)^{-t_k} :where: :: S is the borrowed amount or principal amount. ",−2,8.87,"""15.757""",27,804.62,E
+"Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \mathrm{~s}$ if the bar is made of silver.","The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While thermal bars can form in both fall and spring, most studies of the thermal bar have investigated aspects of the feature in the spring, when the lake is warming up and the summer thermocline is beginning to form. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. Additional studies have been carried out in Lake Ladoga,S. S. Zilitinkevich, K. D. Kreiman, and A. Yu. Terzhevik, “The Thermal Bar,” Journal of Fluid Mechanics 236, no. 1 (1992): 27-42. In metallurgy, cold forming or cold working is any metalworking process in which metal is shaped below its recrystallization temperature, usually at the ambient temperature. This could be used to model heat conduction in a rod. The distribution approaches equilibrium over time. thumb|The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). Thus, if is the temperature, tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point. Unlike the elastic and electromagnetic waves, the speed of a diffusion wave drops with time: as it spreads over a larger region, the temperature gradient decreases, and therefore the heat flow decreases too. ==Specific examples== ===Heat flow in a uniform rod=== For heat flow, the heat equation follows from the physical laws of conduction of heat and conservation of energy . Since the thermal conductivity of ice is so small (1.6 - 2.4 W mK−1) compared with most every other ceramic (ex. Al2O3= 40 W mK−1), the growing ice will have a significant insulative effect on the localized thermal conditions within the slurry. This experiment is possible for ice at −10 °C or cooler, and while essentially valid, the details of the process by which the wire passes through the ice are complex. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. thumb|Classic experiment involving regelation of an ice block as a tensioned wire passes through it. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. Therefore, it is sometimes more economical to cold work a less costly and weaker metal than to hot work a more expensive metal that can be heat treated, especially if precision or a fine surface finish is required as well. In large lakes this condition may persist for weeks, during which a temperature front known as a thermal bar forms between the stratified and unstratified areas of the lake. The thermal resistance of the ceramic is significantly smaller than that of the ice however, so the apparent resistance can be expressed as the lower Rceramic. The name comes from the highly polished appearance of the rods; there is no silver in the alloy. The melting point of ice falls by 0.0072 °C for each additional atm of pressure applied. It has been shown that a linearly decreasing temperature on one side of a freeze-cast will result in near- constant solidification velocity, yielding ice crystals with an almost constant thickness along the SSZ of an entire sample. This approach enables a prediction of the ice-front velocity from the thermal parameters of the suspension. ",2598960,7,"""35.91""",-9.54,310,C
+"Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$.","right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in MATLAB. A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. Colors vary from blue to yellow with time. thumb|Trajectory of a solution with parameter values \alpha=0.05 and \gamma=0.1 and initial conditions x_0=0.1, y_0=-0.1, and z_0=0.1, using the default ODE solver in Mathematica. Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In the end, this result shows the interval of definition of the solution does not depend on the Lipschitz constant of the field, but only on the interval of definition of the field and its maximum absolute value. == Other existence theorems == The Picard–Lindelöf theorem shows that the solution exists and that it is unique. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Notice the drastic change in the solutions with respect to the solution obtained with MATLAB. thumb|A chaotic attractor found with parameter values \alpha=1.1 and \gamma=0.87 and initial conditions x_0=-1, y_0=-0, and z_0=0.5, using the default ODE solver in Mathematica. For non-positive integers, the gamma function is not defined. A simple proof of existence of the solution is obtained by successive approximations. Evidently, the functions are computing the Taylor series expansion of our known solution y=\tan(t). An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. ",2.8,2,"""-6.8""",840,30,B
 "A tank contains 100 gal of water and $50 \mathrm{oz}$ of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2} \sin t\right) \mathrm{oz} / \mathrm{gal}$ flows into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture in the tank flows out at the same rate.
-The long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?","The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. If used correctly, the Leeson equation gives a useful prediction of oscillator performance in this range. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. ==Formula for phase of an oscillation or a periodic signal== The phase of an oscillation or signal refers to a sinusoidal function such as the following: :\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \tfrac{\pi}{2}\right) \end{align} where \textstyle A, \textstyle f, and \textstyle \varphi are constant parameters called the amplitude, frequency, and phase of the sinusoid. However it is possible that the concentrations of some reaction intermediates oscillate, and also that the rate of formation of products oscillates.Espenson, J.H. Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002) p.190 Theoretical models of oscillating reactions have been studied by chemists, physicists, and mathematicians. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z at a given excitation frequency. For waves on a string, or in a medium such as water, the amplitude is a displacement. Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum. Nonlinear Oscillations is a quarterly peer-reviewed mathematical journal that was established in 1998. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). thumb|Duffing oscillator plot, containing phase plot, trajectory, strange attractor, Poincare section, and double well potential plot. At values of t when the difference is zero, the two signals are said to be in phase, otherwise they are out of phase with each other. thumb|right|A stirred BZ reaction mixture showing changes in color over time A chemical oscillator is a complex mixture of reacting chemical compounds in which the concentration of one or more components exhibits periodic changes. The square of the amplitude is proportional to the intensity of the wave. For arguments t when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. Nonlinear Oscillations is a translation of the Ukrainian journal Neliniyni Kolyvannya (). The common misunderstanding, that is the oscillator output level, may result from derivations that are not completely general. The parameters are (0.5, 0.0625, 0.1, 2.5, 2.0) thumb|The strange attractor of the Duffing oscillator, through 4 periods (8pi time). In the above equation, if is set to zero the equation represents a linear analysis of a feedback oscillator in the general case (and flicker noise is not included), it is for this that Leeson is most recognised, showing a -20 dB/ decade of offset frequency slope. An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: :x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi), with z^2=a^2+b^2 and \tan\phi = \frac{b}{a}. As a result, : \begin{align} & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma \qquad \text{and} \\\ & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0. \end{align} Squaring both equations and adding leads to the amplitude frequency response: :\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^{2}+\left(\delta\omega\right)^2\right]\,z^2=\gamma^{2}, as stated above. The amplitude of a non-periodic signal is its magnitude compared with a reference value. ",1590,0.24995,-1.46,10.4,9,B
-A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$,"The graph shows the effect of a tuned mass damper on a simple spring–mass–damper system, excited by vibrations with an amplitude of one unit of force applied to the main mass, m1. Now considering m2 = , the blue line shows the motion of the damping mass and the red line shows the motion of the primary mass. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. The natural frequency of the tuned mass damper is basically defined by its spring constant and the damping ratio determined by the dashpot. As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). The effective mass of the spring can be determined by finding its kinetic energy. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. As the frequency increases m2 moves out of phase with m1 until at around 9.5 Hz it is 180° out of phase with m1, maximizing the damping effect by maximizing the amplitude of x2 − x1, this maximizes the energy dissipated into c2 and simultaneously pulls on the primary mass in the same direction as the motor mounts. ==Mass dampers in automobiles== ===Motorsport=== The tuned mass damper was introduced as part of the suspension system by Renault on its 2005 F1 car (the Renault R25), at the 2005 Brazilian Grand Prix. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations The split between the two peaks can be changed by altering the mass of the damper (m2). This energy represents the amount of mechanical energy being converted to heat in a volume of material resulting in damping. thumb|Damping capacity ==References== Category:Materials science alt=mass connected to the ground with a spring and damper in parallel|thumb|Classic model used for deriving the equations of a mass spring damper model The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Critically damped means the machine has the right amount of damping torque and is ready to be used for experiments. ==References== 1\. The tuned parameter of the tuned mass damper enables the auxiliary mass to oscillate with a phase shift with respect to the motion of the structure. They are frequently used in power transmission, automobiles and buildings. ==Principle== thumb|A schematic of a simple spring–mass–damper system used to demonstrate the tuned mass damper system Tuned mass dampers stabilize against violent motion caused by harmonic vibration. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass Dampers were fitted in response. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Damping torque is provided by indicating instrument. The heights of the two peaks can be adjusted by changing the stiffness of the spring in the tuned mass damper. Finally, if it is critically damped, it has an equal amount of deflection and controlling torque, thus allowing the pointer to quickly find the correct value, without the system oscillating past that value. ",8,1000,-1.78,773,0.366,A
-"Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \mathrm{~g} / \mathrm{gal}$.","500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. This depends on: the microorganism, the disinfectant being used, the concentration of the disinfectant, the contact time, and the temperature and pH of the water. ==Kinetics== The disinfection kinetics are conventionally calculated via the Chick-Watson model, named for the work of Harriette Chick and H. E. Watson. A portion of such a table is reproduced below. ==Example CT Table== CT Values for the Inactivation of Giardia Cysts by Free Chlorine at 5 °C and pH ≈ 7.0: Chlorine Concentration (mg/L) 1 log inactivation (mg·L−1·min) 2 log inactivation (mg·L−1·min) 3 log inactivation (mg·L−1·min) 0.6 48 95 143 1.2 51 101 152 1.8 54 108 162 2.4 57 115 172 Full tables are much larger than this example and should be obtained from the regulatory agency for a particular jurisdiction. ==See also== * Chlorination * Disinfectant ==References== ==External links== * Category:Water treatment Category:Chlorine A CT value is the product of the concentration of a disinfectant (e.g. free chlorine) and the contact time with the water being disinfected. John Martin-Dye (21 May 1940 – 31 December 2022John Martin-Dye) was a retired British swimmer. ==Swimming career== He won a silver medal in the 4 × 100 m freestyle relay at the 1962 European Aquatics Championships. CT Values are an important part of calculating disinfectant dosage for the chlorination of drinking water. The calculated CT value is the product of the disinfectant residual (in mg/L) and the detention time (in minutes), through the section at peak hourly flow. Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. For example if Lake Michigan was emptied, it would take 99 years for its tributaries to completely refill the lake. ==List of residence times of lake water== The residence time listed is taken from the infobox in the associated article unless otherwise specified. In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. These tables express the required CT values to achieve a desired removal of microorganisms of interest in drinking water (e.g. Giardia lamblia cysts) for a given disinfectant under constant temperature and pH conditions. Where: * (\frac{N}{N_0}) \\! is the survival ratio for the microorganisms being killed * \Lambda_{CW} \\! is the Chick-Watson coefficient of specific lethality * C \\! is the concentration of the disinfectant (typically in mg/L) * n \\! is the coefficient of dilution, frequently assumed to be 1 * t \\! is the contact time (typically in minutes or seconds) The survival ratio is commonly expressed as an inactivation ratio (in %) or as the number of reductions in the order of magnitude of the microorganism concentration. Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. In theory it would be possible to integrate a system of hydrodynamic equations with variable boundary conditions over a very long period sufficient for inflowing water particles to exit the lake. Lake retention time (also called the residence time of lake water, or the water age or flushing time) is a calculated quantity expressing the mean time that water (or some dissolved substance) spends in a particular lake. The concentration of this admixture should be small and the gradient of this concentration should be also small. ",-1368,7.136,3.0,9.8,0.36,B
-"For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.  If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.","The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. The most appropriate are the Reynolds number, given by \mathrm{Re} = \frac{u\sqrt{A}}{ u} and the drag coefficient, given by c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}. The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. An object moving downward faster than the terminal velocity (for example because it was thrown downwards, it fell from a thinner part of the atmosphere, or it changed shape) will slow down until it reaches the terminal velocity. As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). The terminal speed of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional surface area. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. Air is 1.293 kg/m3 at 0°C and 1 atmosphere *u is the flow velocity relative to the object, *A is the reference area, and *c_{\rm d} is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass m has to be reduced by the displaced fluid mass \rho V, with V the volume of the object. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. Other than fixed properties of the system, this form of the equation shows that the impact force depends only on the fall factor. For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000.Drag Force For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).See Batchelor (1967), p. 341. == Discussion == The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. The mass m0 used in the fall is 80 kg. Based on air resistance, for example, the terminal speed of a skydiver in a belly-to-earth (i.e., face down) free fall position is about . The expression for the drag force given by equation () is called Stokes' law. ", -6.04697,0.5,273.0,0.0408,817.90,D
+The long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?","The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. If used correctly, the Leeson equation gives a useful prediction of oscillator performance in this range. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. ==Formula for phase of an oscillation or a periodic signal== The phase of an oscillation or signal refers to a sinusoidal function such as the following: :\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \tfrac{\pi}{2}\right) \end{align} where \textstyle A, \textstyle f, and \textstyle \varphi are constant parameters called the amplitude, frequency, and phase of the sinusoid. However it is possible that the concentrations of some reaction intermediates oscillate, and also that the rate of formation of products oscillates.Espenson, J.H. Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002) p.190 Theoretical models of oscillating reactions have been studied by chemists, physicists, and mathematicians. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z at a given excitation frequency. For waves on a string, or in a medium such as water, the amplitude is a displacement. Leeson's equation is an empirical expression that describes an oscillator's phase noise spectrum. Nonlinear Oscillations is a quarterly peer-reviewed mathematical journal that was established in 1998. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). thumb|Duffing oscillator plot, containing phase plot, trajectory, strange attractor, Poincare section, and double well potential plot. At values of t when the difference is zero, the two signals are said to be in phase, otherwise they are out of phase with each other. thumb|right|A stirred BZ reaction mixture showing changes in color over time A chemical oscillator is a complex mixture of reacting chemical compounds in which the concentration of one or more components exhibits periodic changes. The square of the amplitude is proportional to the intensity of the wave. For arguments t when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. Nonlinear Oscillations is a translation of the Ukrainian journal Neliniyni Kolyvannya (). The common misunderstanding, that is the oscillator output level, may result from derivations that are not completely general. The parameters are (0.5, 0.0625, 0.1, 2.5, 2.0) thumb|The strange attractor of the Duffing oscillator, through 4 periods (8pi time). In the above equation, if is set to zero the equation represents a linear analysis of a feedback oscillator in the general case (and flicker noise is not included), it is for this that Leeson is most recognised, showing a -20 dB/ decade of offset frequency slope. An introduction to nonlinear chemical dynamics: oscillations, waves, patterns, and chaos. Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: :x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi), with z^2=a^2+b^2 and \tan\phi = \frac{b}{a}. As a result, : \begin{align} & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma \qquad \text{and} \\\ & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0. \end{align} Squaring both equations and adding leads to the amplitude frequency response: :\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^{2}+\left(\delta\omega\right)^2\right]\,z^2=\gamma^{2}, as stated above. The amplitude of a non-periodic signal is its magnitude compared with a reference value. ",1590,0.24995,"""-1.46""",10.4,9,B
+A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$,"The graph shows the effect of a tuned mass damper on a simple spring–mass–damper system, excited by vibrations with an amplitude of one unit of force applied to the main mass, m1. Now considering m2 = , the blue line shows the motion of the damping mass and the red line shows the motion of the primary mass. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. The natural frequency of the tuned mass damper is basically defined by its spring constant and the damping ratio determined by the dashpot. As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). The effective mass of the spring can be determined by finding its kinetic energy. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. As the frequency increases m2 moves out of phase with m1 until at around 9.5 Hz it is 180° out of phase with m1, maximizing the damping effect by maximizing the amplitude of x2 − x1, this maximizes the energy dissipated into c2 and simultaneously pulls on the primary mass in the same direction as the motor mounts. ==Mass dampers in automobiles== ===Motorsport=== The tuned mass damper was introduced as part of the suspension system by Renault on its 2005 F1 car (the Renault R25), at the 2005 Brazilian Grand Prix. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = 0 This has the solution: : x = A e^{-\omega_n t \left(\zeta + \sqrt{\zeta^2-1}\right)} + B e^{-\omega_n t \left(\zeta - \sqrt{\zeta^2-1}\right)} If \zeta < 1 then \zeta^2-1 is negative, meaning the square root will be negative the solution will have an oscillatory component. ==See also== * Numerical methods * Soft body dynamics#Spring/mass models * Finite element analysis ==References== Category:Classical mechanics Category:Mechanical vibrations The split between the two peaks can be changed by altering the mass of the damper (m2). This energy represents the amount of mechanical energy being converted to heat in a volume of material resulting in damping. thumb|Damping capacity ==References== Category:Materials science alt=mass connected to the ground with a spring and damper in parallel|thumb|Classic model used for deriving the equations of a mass spring damper model The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Critically damped means the machine has the right amount of damping torque and is ready to be used for experiments. ==References== 1\. The tuned parameter of the tuned mass damper enables the auxiliary mass to oscillate with a phase shift with respect to the motion of the structure. They are frequently used in power transmission, automobiles and buildings. ==Principle== thumb|A schematic of a simple spring–mass–damper system used to demonstrate the tuned mass damper system Tuned mass dampers stabilize against violent motion caused by harmonic vibration. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass Dampers were fitted in response. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Damping torque is provided by indicating instrument. The heights of the two peaks can be adjusted by changing the stiffness of the spring in the tuned mass damper. Finally, if it is critically damped, it has an equal amount of deflection and controlling torque, thus allowing the pointer to quickly find the correct value, without the system oscillating past that value. ",8,1000,"""-1.78""",773,0.366,A
+"Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \mathrm{~g} / \mathrm{gal}$.","500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. This depends on: the microorganism, the disinfectant being used, the concentration of the disinfectant, the contact time, and the temperature and pH of the water. ==Kinetics== The disinfection kinetics are conventionally calculated via the Chick-Watson model, named for the work of Harriette Chick and H. E. Watson. A portion of such a table is reproduced below. ==Example CT Table== CT Values for the Inactivation of Giardia Cysts by Free Chlorine at 5 °C and pH ≈ 7.0: Chlorine Concentration (mg/L) 1 log inactivation (mg·L−1·min) 2 log inactivation (mg·L−1·min) 3 log inactivation (mg·L−1·min) 0.6 48 95 143 1.2 51 101 152 1.8 54 108 162 2.4 57 115 172 Full tables are much larger than this example and should be obtained from the regulatory agency for a particular jurisdiction. ==See also== * Chlorination * Disinfectant ==References== ==External links== * Category:Water treatment Category:Chlorine A CT value is the product of the concentration of a disinfectant (e.g. free chlorine) and the contact time with the water being disinfected. John Martin-Dye (21 May 1940 – 31 December 2022John Martin-Dye) was a retired British swimmer. ==Swimming career== He won a silver medal in the 4 × 100 m freestyle relay at the 1962 European Aquatics Championships. CT Values are an important part of calculating disinfectant dosage for the chlorination of drinking water. The calculated CT value is the product of the disinfectant residual (in mg/L) and the detention time (in minutes), through the section at peak hourly flow. Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. For example if Lake Michigan was emptied, it would take 99 years for its tributaries to completely refill the lake. ==List of residence times of lake water== The residence time listed is taken from the infobox in the associated article unless otherwise specified. In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. These tables express the required CT values to achieve a desired removal of microorganisms of interest in drinking water (e.g. Giardia lamblia cysts) for a given disinfectant under constant temperature and pH conditions. Where: * (\frac{N}{N_0}) \\! is the survival ratio for the microorganisms being killed * \Lambda_{CW} \\! is the Chick-Watson coefficient of specific lethality * C \\! is the concentration of the disinfectant (typically in mg/L) * n \\! is the coefficient of dilution, frequently assumed to be 1 * t \\! is the contact time (typically in minutes or seconds) The survival ratio is commonly expressed as an inactivation ratio (in %) or as the number of reductions in the order of magnitude of the microorganism concentration. Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. In theory it would be possible to integrate a system of hydrodynamic equations with variable boundary conditions over a very long period sufficient for inflowing water particles to exit the lake. Lake retention time (also called the residence time of lake water, or the water age or flushing time) is a calculated quantity expressing the mean time that water (or some dissolved substance) spends in a particular lake. The concentration of this admixture should be small and the gradient of this concentration should be also small. ",-1368,7.136,"""3.0""",9.8,0.36,B
+"For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity.  If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.","The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. The most appropriate are the Reynolds number, given by \mathrm{Re} = \frac{u\sqrt{A}}{ u} and the drag coefficient, given by c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}. The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. An object moving downward faster than the terminal velocity (for example because it was thrown downwards, it fell from a thinner part of the atmosphere, or it changed shape) will slow down until it reaches the terminal velocity. As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). The terminal speed of an object changes due to the properties of the fluid, the mass of the object and its projected cross-sectional surface area. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. Air is 1.293 kg/m3 at 0°C and 1 atmosphere *u is the flow velocity relative to the object, *A is the reference area, and *c_{\rm d} is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass m has to be reduced by the displaced fluid mass \rho V, with V the volume of the object. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. Other than fixed properties of the system, this form of the equation shows that the impact force depends only on the fall factor. For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000.Drag Force For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).See Batchelor (1967), p. 341. == Discussion == The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. The mass m0 used in the fall is 80 kg. Based on air resistance, for example, the terminal speed of a skydiver in a belly-to-earth (i.e., face down) free fall position is about . The expression for the drag force given by equation () is called Stokes' law. ", -6.04697,0.5,"""273.0""",0.0408,817.90,D
 "Consider the initial value problem
 $$
 3 u^{\prime \prime}-u^{\prime}+2 u=0, \quad u(0)=2, \quad u^{\prime}(0)=0
 $$
-For $t>0$ find the first time at which $|u(t)|=10$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad u(t_0)=u_0, after linearizing locally about a fixed or local state u^*: : L = \frac{\partial f}{\partial u}(u^*); \qquad N = f(u) - L u. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. For example, for the equation (a<0), the stationary solution is , which is obtained for the initial condition . We use their notation, and assume that the unknown function is u, and that we have a known solution u_n at time t_n. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Originally developed for solving stiff differential equations, the methods have been used to solve partial differential equations including hyperbolic as well as parabolic problems, such as the heat equation. ==Introduction== We consider initial value problems of the form, :u'(t) = L u(t) + N(u(t) ), \qquad u(t_0)=u_0, \qquad \qquad (1) where L is composed of linear terms, and N is composed of the non-linear terms. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. ", 10.7598,0.082,25.6773,1.5377,2,A
+For $t>0$ find the first time at which $|u(t)|=10$.","A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. These problems can come from a more typical initial value problem :u'(t) = f(u(t)), \qquad u(t_0)=u_0, after linearizing locally about a fixed or local state u^*: : L = \frac{\partial f}{\partial u}(u^*); \qquad N = f(u) - L u. This happens for example for the equation , which has at least two solutions corresponding to the initial condition such as: or :y(t)=\begin{cases} \left (\tfrac{at}{3} \right )^{3} & t<0\\\ \ \ \ \ 0 & t \ge 0, \end{cases} so the previous state of the system is not uniquely determined by its state after t = 0. Indeed, rather than being unique, this equation has three solutions:, p. 7 :y(t) = 0, \qquad y(t) = \pm\left (\tfrac23 t\right)^{\frac{3}{2}}. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. Beginning with another initial condition , the solution y(t) tends toward the stationary point, but reaches it only at the limit of infinite time, so the uniqueness of solutions (over all finite times) is guaranteed. In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. For example, for the equation (a<0), the stationary solution is , which is obtained for the initial condition . We use their notation, and assume that the unknown function is u, and that we have a known solution u_n at time t_n. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value problem. Originally developed for solving stiff differential equations, the methods have been used to solve partial differential equations including hyperbolic as well as parabolic problems, such as the heat equation. ==Introduction== We consider initial value problems of the form, :u'(t) = L u(t) + N(u(t) ), \qquad u(t_0)=u_0, \qquad \qquad (1) where L is composed of linear terms, and N is composed of the non-linear terms. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval [t_0-\varepsilon, t_0+\varepsilon]., Theorem I.3.1 == Proof sketch == The proof relies on transforming the differential equation, and applying the Banach fixed-point theorem. See Newton's method of successive approximation for instruction. == Example of Picard iteration == thumb|Four Picard iteration steps and their limit Let y(t)=\tan(t), the solution to the equation y'(t)=1+y(t)^2 with initial condition y(t_0)=y_0=0,t_0=0. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. == Definition == An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. Its behavior through time can be traced with a closed form solution conditional on an initial condition vector X_0. ", 10.7598,0.082,"""25.6773""",1.5377,2,A
 "Consider the initial value problem
 $$
 9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=a>0, \quad y^{\prime}(0)=-1
 $$
-Find the critical value of $a$ that separates solutions that become negative from those that are always positive.","By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. More generally one might expect :f(\tau)=A \tau^k \left(1+b\tau ^{k_1} + \cdots\right) ==The most important critical exponents== Let us assume that the system has two different phases characterized by an order parameter , which vanishes at and above . A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Now we shall check when these solutions are singular solutions. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Critical exponents describe the behavior of physical quantities near continuous phase transitions. Another solution is given by : y_s(x) = 0 . Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Consider the disordered phase (), ordered phase () and critical temperature () phases separately. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Furthermore, for a given x ot=0, this is the unique solution going through (x,y(x)). ==Failure of uniqueness== Consider the differential equation : y'(x)^2 = 4y(x) . If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Following the standard convention, the critical exponents related to the ordered phase are primed. The set on which a solution is singular may be as small as a single point or as large as the full real line. These can be found by taking two constant c_1 < c_2 and defining a solution y(x) to be (x-c_1)^2 when x < c_1, to be 0 when c_1\leq x\leq c_2, and to be (x-c_2)^2 when x > c_2. Uniqueness fails for these solutions on the interval c_1\leq x\leq c_2, and the solutions are singular, in the sense that the second derivative fails to exist, at x=c_1 and x=c_2. ==Further example of failure of uniqueness== The previous example might give the erroneous impression that failure of uniqueness is directly related to y(x)=0. ",5.1,10,1.5,2688,0.0384,C
-"In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\vec{a}, 2.0 \mathrm{~km}$ due east (directly toward the east); (b) $\vec{b}, 2.0 \mathrm{~km} 30^{\circ}$ north of east (at an angle of $30^{\circ}$ toward the north from due east); (c) $\vec{c}, 1.0 \mathrm{~km}$ due west. Alternatively, you may substitute either $-\vec{b}$ for $\vec{b}$ or $-\vec{c}$ for $\vec{c}$. What is the greatest distance you can be from base camp at the end of the third displacement? (We are not concerned about the direction.)","The distance from the camp post to the market is (1/4-1/15), which will be repeated 3 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/15)*3=1.45 units; *# For m=1/2>(1/3+1/15), another midway camp post is necessary at a distance of 1/3 units from the first camp post, where a total of 1 unit of bananas will accrue. The distance from the second camp post to the market is [1/2-(1/4+1/18)], which will be repeated twice by the camel, and the maximum amount of bananas tranported to the market is 1-[1/2-(1/4+1/18)]*2=0.61 units. The distance from the second camp post to the market is [1/2-(1/3+1/15)], which will be traveled only once by the camel, and the maximum amount of bananas tranported to the market is 1-[1/2-(1/3+1/15)]*1=0.9 units. The other base is at m units of distance away. The other base is at m units of distance away. The distance from the camp post to the market is (1/4-1/18), which will be repeated 4 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/18)*4=1.22 units; *# For m=1/2>(1/4+1/18), another midway camp post is necessary at a distance of 1/4 units from the first camp post, where a total of 1 unit of bananas (plus 1/4 units for the camel's final return) will accrue. Here \\{n\\}=n-\lfloor n \rfloor is the fractional part of . === Other variants of the problem === In the camel and bananas problem, assuming the merchant has a total of n=7/3 units of bananas at the starting base and the market is at m units of distance away: * If m> \mathrm{cross}(n)=1/15+1/3+1, no solution exists for this problem; * If m\leq \mathrm{cross}(n)-\mathrm{cross}(\lfloor n\rfloor)=1/15, no midway camp post is necessary for the transport, and the distance m will be repeated for 2 \lceil n \rceil-1=5 times by the camel, so the maximum amount of bananas tranported to the market is n-m\times (2 \lceil n \rceil-1)=7/3-5m; * If \mathrm{cross}(n)-\mathrm{cross}(\lfloor n\rfloor), the optimal solution to transport that maximum amount of bananas requires some midway camp posts: *# For m=1/4<(1/3+1/15), only one midway camp post is necessary at a distance of 1/15 units away from the starting base, where a total of 2 units of bananas will accrue. *At the vertex (0,0), the degree is one: the only possible direction for both climbers to go is onto the mountain. If the camel is required to eventually return to the starting base, then the \mathrm{explore}(n) function applies (still assuming n=7/3): *If m> \mathrm{explore}(n)=1/18+1/4+1/2, no solution exists for this problem; * If m\leq \mathrm{explore}(n)-\mathrm{explore}(\lfloor n\rfloor)=1/18, no midway camp post is necessary for the transport, and the distance m will be repeated for 2 \lceil n \rceil=6 times by the camel, so the maximum amount of bananas tranported to the market is n-m\times (2 \lceil n \rceil)=7/3-6m; * If \mathrm{explore}(n)-\mathrm{explore}(\lfloor n\rfloor), the optimal solution to transport that maximum amount of bananas requires some midway camp posts: *# For m=1/4<(1/4+1/18), only one midway camp post is necessary at a distance of 1/18 units away from the starting base, where a total of 2 units of bananas (plus 1/18 units for the camel's final return) will accrue. A route summit is the highest point on a transportation route crossing higher ground. In summary, the maximum distance reachable by the jeep (with a fuel capacity for 1 unit of distance at any time) in n trips (with n-1 midway fuel dumps and consuming a total of n units of fuel) is * \mathrm{explore}(n)= \frac{1}{2}H_{n}=\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots + \frac{1}{2n}, for exploring the desert where the jeep must return to the base at the end of every trip; * \mathrm{cross}(n)=H_{2n-1}-\frac{1}{2}H_{n-1}=1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} , for crossing the desert where the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel. The maximum distance achievable for the ""cross the desert"" problem with units of fuel is :\mathrm{cross}(n) = \int_0^n \frac{\mathrm{d}f}{2 \lceil n-f \rceil - 1} with the first fuel dump located at \\{n\\}/(2 \lceil n \rceil-1) units of distance away from the starting base, the second one at 1/(2 \lceil n \rceil-3) units of distance away from the first fuel dump, the third one at 1/(2 \lceil n \rceil-5) units of distance away from the second fuel dump, and so on. thumb|250px|Plot of amount of fuel f vs distance from origin d for exploring (1–3) and crossing (I–III) versions of the jeep problem for three units of fuel - coloured arrows denote depots, diagonal segments denote travel and vertical segments denote fuel transfer The jeep problem, desert crossing problem or exploration problem""Exploration problems. In the general case, the maximum distance achievable for the ""explore the desert"" problem with units of fuel is :\mathrm{explore}(n) = \int_0^n \frac{\mathrm{d}f}{2 \lceil n-f \rceil} with the first fuel dump located at \\{n\\}/(2 \lceil n \rceil) units of distance away from the starting base, the second one at 1/(2 \lceil n \rceil-2) units of distance away from the first fuel dump, the third one at 1/(2 \lceil n \rceil-4) units of distance away from the second fuel dump, and so on. In either case the objective is to maximize the distance traveled by the jeep on its final trip. *At a vertex where both climbers are at peaks or both climbers are at valleys, the degree is four: both climbers may choose independently of each other which direction to go. If the last and final traveler also needs to return to the starting base, then he would only travel 1/(n+1) unit alone so that he has n/(n+1) units of supply to return, so the longest distance n travelers can reach is :\mathrm{travel_{return}}(n)=\frac{n}{n+1}=1-\frac{1}{n+1} Equating this to m, one may solve for the minimum number of travelers needed to travel m units of distance. A strategy that maximizes the distance traveled on the final trip for the ""exploring the desert"" variant is as follows: *The jeep makes n trips. A variant of this problem gives the total number of cars available, and asks for the maximum distance that can be reached. ==Solution== thumb|250px|Solution to ""exploring the desert"" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip. Mathematical Recreations and Essays, Thirteenth Edition, Dover, p32. . is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The problem asks for the minimum number of accompanying travelers needed to reach the other base. thumb|Approach (α) and departure angle (β) of a vehicle Approach angle is the maximum angle of a ramp onto which a vehicle can climb from a horizontal plane without interference. ",14.34457,13,4.8,140,0.18,C
+Find the critical value of $a$ that separates solutions that become negative from those that are always positive.","By considering the two sets of solutions above, one can see that the solution fails to be unique when y=0. More generally one might expect :f(\tau)=A \tau^k \left(1+b\tau ^{k_1} + \cdots\right) ==The most important critical exponents== Let us assume that the system has two different phases characterized by an order parameter , which vanishes at and above . A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies :y(t_0) = y_0. ;Second example The solution of : y'+3y=6t+5,\qquad y(0)=3 can be found to be : y(t)=2e^{-3t}+2t+1. In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Since the equation being studied is a first-order equation, the initial conditions are the initial x and y values. \, Indeed, : \begin{align} y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\\ &= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\\ &= 6t+5. \end{align} ==Notes== ==See also== * Boundary value problem * Constant of integration * Integral curve == References == * * * * * * Category:Boundary conditions el:Αρχική τιμή it:Problema ai valori iniziali sv:Begynnelsevärdesproblem Now we shall check when these solutions are singular solutions. Solutions which are singular in the sense that the initial value problem fails to have a unique solution need not be singular functions. right|thumb|390px|Solutions to the differential equation \frac{dy}{dx} = \frac{1}{2y} subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). Critical exponents describe the behavior of physical quantities near continuous phase transitions. Another solution is given by : y_s(x) = 0 . Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. y(t)=f(t,y(t),y'(t)). == Existence and uniqueness of solutions == The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t0 if f is continuous on a region containing t0 and y0 and satisfies the Lipschitz condition on the variable y. Consider the disordered phase (), ordered phase () and critical temperature () phases separately. Rearrange the equation so that y is on the left hand side : \frac{y'(t)}{y(t)} = 0.85 Now integrate both sides with respect to t (this introduces an unknown constant B). : \int \frac{y'(t)}{y(t)}\,dt = \int 0.85\,dt : \ln |y(t)| = 0.85t + B Eliminate the logarithm with exponentiation on both sides : | y(t) | = e^Be^{0.85t} Let C be a new unknown constant, C = \pm e^B, so : y(t) = Ce^{0.85t} Now we need to find a value for C. Use y(0) = 19 as given at the start and substitute 0 for t and 19 for y : 19 = C e^{0.85 \cdot 0} : C = 19 this gives the final solution of y(t) = 19e^{0.85t}. A singular solution ys(x) of an ordinary differential equation is a solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution at some point on the solution. Furthermore, for a given x ot=0, this is the unique solution going through (x,y(x)). ==Failure of uniqueness== Consider the differential equation : y'(x)^2 = 4y(x) . If x + 2p = 0 then we get that p = −½x and substituting in the ODE gives : y_s(x) = -\tfrac{1}{2}x^2 + (-\tfrac{1}{2}x)^2 = -\tfrac{1}{4} x^2. Following the standard convention, the critical exponents related to the ordered phase are primed. The set on which a solution is singular may be as small as a single point or as large as the full real line. These can be found by taking two constant c_1 < c_2 and defining a solution y(x) to be (x-c_1)^2 when x < c_1, to be 0 when c_1\leq x\leq c_2, and to be (x-c_2)^2 when x > c_2. Uniqueness fails for these solutions on the interval c_1\leq x\leq c_2, and the solutions are singular, in the sense that the second derivative fails to exist, at x=c_1 and x=c_2. ==Further example of failure of uniqueness== The previous example might give the erroneous impression that failure of uniqueness is directly related to y(x)=0. ",5.1,10,"""1.5""",2688,0.0384,C
+"In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) $\vec{a}, 2.0 \mathrm{~km}$ due east (directly toward the east); (b) $\vec{b}, 2.0 \mathrm{~km} 30^{\circ}$ north of east (at an angle of $30^{\circ}$ toward the north from due east); (c) $\vec{c}, 1.0 \mathrm{~km}$ due west. Alternatively, you may substitute either $-\vec{b}$ for $\vec{b}$ or $-\vec{c}$ for $\vec{c}$. What is the greatest distance you can be from base camp at the end of the third displacement? (We are not concerned about the direction.)","The distance from the camp post to the market is (1/4-1/15), which will be repeated 3 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/15)*3=1.45 units; *# For m=1/2>(1/3+1/15), another midway camp post is necessary at a distance of 1/3 units from the first camp post, where a total of 1 unit of bananas will accrue. The distance from the second camp post to the market is [1/2-(1/4+1/18)], which will be repeated twice by the camel, and the maximum amount of bananas tranported to the market is 1-[1/2-(1/4+1/18)]*2=0.61 units. The distance from the second camp post to the market is [1/2-(1/3+1/15)], which will be traveled only once by the camel, and the maximum amount of bananas tranported to the market is 1-[1/2-(1/3+1/15)]*1=0.9 units. The other base is at m units of distance away. The other base is at m units of distance away. The distance from the camp post to the market is (1/4-1/18), which will be repeated 4 times by the camel, and the maximum amount of bananas tranported to the market is 2-(1/4-1/18)*4=1.22 units; *# For m=1/2>(1/4+1/18), another midway camp post is necessary at a distance of 1/4 units from the first camp post, where a total of 1 unit of bananas (plus 1/4 units for the camel's final return) will accrue. Here \\{n\\}=n-\lfloor n \rfloor is the fractional part of . === Other variants of the problem === In the camel and bananas problem, assuming the merchant has a total of n=7/3 units of bananas at the starting base and the market is at m units of distance away: * If m> \mathrm{cross}(n)=1/15+1/3+1, no solution exists for this problem; * If m\leq \mathrm{cross}(n)-\mathrm{cross}(\lfloor n\rfloor)=1/15, no midway camp post is necessary for the transport, and the distance m will be repeated for 2 \lceil n \rceil-1=5 times by the camel, so the maximum amount of bananas tranported to the market is n-m\times (2 \lceil n \rceil-1)=7/3-5m; * If \mathrm{cross}(n)-\mathrm{cross}(\lfloor n\rfloor), the optimal solution to transport that maximum amount of bananas requires some midway camp posts: *# For m=1/4<(1/3+1/15), only one midway camp post is necessary at a distance of 1/15 units away from the starting base, where a total of 2 units of bananas will accrue. *At the vertex (0,0), the degree is one: the only possible direction for both climbers to go is onto the mountain. If the camel is required to eventually return to the starting base, then the \mathrm{explore}(n) function applies (still assuming n=7/3): *If m> \mathrm{explore}(n)=1/18+1/4+1/2, no solution exists for this problem; * If m\leq \mathrm{explore}(n)-\mathrm{explore}(\lfloor n\rfloor)=1/18, no midway camp post is necessary for the transport, and the distance m will be repeated for 2 \lceil n \rceil=6 times by the camel, so the maximum amount of bananas tranported to the market is n-m\times (2 \lceil n \rceil)=7/3-6m; * If \mathrm{explore}(n)-\mathrm{explore}(\lfloor n\rfloor), the optimal solution to transport that maximum amount of bananas requires some midway camp posts: *# For m=1/4<(1/4+1/18), only one midway camp post is necessary at a distance of 1/18 units away from the starting base, where a total of 2 units of bananas (plus 1/18 units for the camel's final return) will accrue. A route summit is the highest point on a transportation route crossing higher ground. In summary, the maximum distance reachable by the jeep (with a fuel capacity for 1 unit of distance at any time) in n trips (with n-1 midway fuel dumps and consuming a total of n units of fuel) is * \mathrm{explore}(n)= \frac{1}{2}H_{n}=\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \cdots + \frac{1}{2n}, for exploring the desert where the jeep must return to the base at the end of every trip; * \mathrm{cross}(n)=H_{2n-1}-\frac{1}{2}H_{n-1}=1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} , for crossing the desert where the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel. The maximum distance achievable for the ""cross the desert"" problem with units of fuel is :\mathrm{cross}(n) = \int_0^n \frac{\mathrm{d}f}{2 \lceil n-f \rceil - 1} with the first fuel dump located at \\{n\\}/(2 \lceil n \rceil-1) units of distance away from the starting base, the second one at 1/(2 \lceil n \rceil-3) units of distance away from the first fuel dump, the third one at 1/(2 \lceil n \rceil-5) units of distance away from the second fuel dump, and so on. thumb|250px|Plot of amount of fuel f vs distance from origin d for exploring (1–3) and crossing (I–III) versions of the jeep problem for three units of fuel - coloured arrows denote depots, diagonal segments denote travel and vertical segments denote fuel transfer The jeep problem, desert crossing problem or exploration problem""Exploration problems. In the general case, the maximum distance achievable for the ""explore the desert"" problem with units of fuel is :\mathrm{explore}(n) = \int_0^n \frac{\mathrm{d}f}{2 \lceil n-f \rceil} with the first fuel dump located at \\{n\\}/(2 \lceil n \rceil) units of distance away from the starting base, the second one at 1/(2 \lceil n \rceil-2) units of distance away from the first fuel dump, the third one at 1/(2 \lceil n \rceil-4) units of distance away from the second fuel dump, and so on. In either case the objective is to maximize the distance traveled by the jeep on its final trip. *At a vertex where both climbers are at peaks or both climbers are at valleys, the degree is four: both climbers may choose independently of each other which direction to go. If the last and final traveler also needs to return to the starting base, then he would only travel 1/(n+1) unit alone so that he has n/(n+1) units of supply to return, so the longest distance n travelers can reach is :\mathrm{travel_{return}}(n)=\frac{n}{n+1}=1-\frac{1}{n+1} Equating this to m, one may solve for the minimum number of travelers needed to travel m units of distance. A strategy that maximizes the distance traveled on the final trip for the ""exploring the desert"" variant is as follows: *The jeep makes n trips. A variant of this problem gives the total number of cars available, and asks for the maximum distance that can be reached. ==Solution== thumb|250px|Solution to ""exploring the desert"" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip. Mathematical Recreations and Essays, Thirteenth Edition, Dover, p32. . is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The problem asks for the minimum number of accompanying travelers needed to reach the other base. thumb|Approach (α) and departure angle (β) of a vehicle Approach angle is the maximum angle of a ramp onto which a vehicle can climb from a horizontal plane without interference. ",14.34457,13,"""4.8""",140,0.18,C
 """Top gun"" pilots have long worried about taking a turn too tightly. As a pilot's body undergoes centripetal acceleration, with the head toward the center of curvature, the blood pressure in the brain decreases, leading to loss of brain function.
 There are several warning signs. When the centripetal acceleration is $2 g$ or $3 g$, the pilot feels heavy. At about $4 g$, the pilot's vision switches to black and white and narrows to ""tunnel vision."" If that acceleration is sustained or increased, vision ceases and, soon after, the pilot is unconscious - a condition known as $g$-LOC for "" $g$-induced loss of consciousness.""
 
-What is the magnitude of the acceleration, in $g$ units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of $\vec{v}_i=(400 \hat{\mathrm{i}}+500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and $24.0 \mathrm{~s}$ later leaves the turn with a velocity of $\vec{v}_f=(-400 \hat{\mathrm{i}}-500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ ?","Incidents of acceleration-induced loss of consciousness have caused fatal accidents in aircraft capable of sustaining high-g for considerable periods. Skilled pilots can use this loss of vision as their indicator that they are at maximum turn performance without losing consciousness. The danger of g-LOC to aircraft pilots is magnified because on relaxation of g-force there is a period of disorientation before full sensation is re-gained. If G-LOC occurs at low altitude, this momentary lapse can prove fatal and even highly experienced pilots can pull straight to a G-LOC condition without first perceiving the visual onset warnings that would normally be used as the sign to back off from pulling any more gs. A further increase in g-forces will cause g-LOC where consciousness is lost. g-force induced loss of consciousness (abbreviated as G-LOC, pronounced ""JEE- lock"") is a term generally used in aerospace physiology to describe a loss of consciousness occurring from excessive and sustained g-forces draining blood away from the brain causing cerebral hypoxia. If g-forces increase further, complete loss of vision will occur, while consciousness remains. High-g training for pilots of high performance aircraft or spacecraft often includes ground training for G-LOC in special centrifuges, with some profiles exposing pilots to 9 gs for a sustained period. ==Effects of g-forces== Under increasing positive g-force, blood in the body will tend to move from the head toward the feet. In general, when the g-force pushes the body forwards (colloquially known as 'eyeballs in'NASA Physiological Acceleration Systems ) a much higher tolerance is shown than when g-force is pushing the body backwards ('eyeballs out') since blood vessels in the retina appear more sensitive to that direction.NASA Technical note D-337, Centrifuge Study of Pilot Tolerance to Acceleration and the Effects of Acceleration on Pilot Performance, by Brent Y. Creer, Captain Harald A. Smedal, USN (MC), and Rodney C. Wingrove ==G-suits== A g-suit is worn by aviators and astronauts who are subject to high levels of acceleration and, hence, increasing positive g. thumb|upright=1.2|If the cockpit lost pressure while the aircraft was above the Armstrong limit, even a positive pressure oxygen mask could not sustain pilot consciousness. This is not true in 0 g when you strafe up; that is an eyeballs-down maneuver, which is the same force as a blackout where blood rushes to the feet, and this force is parallel to the spine. A pilot aiming for a 500-foot per minute descent, for example, may find themselves descending more rapidly than intended. This is doubly dangerous because, on recovery as g is reduced, a period of several seconds of disorientation occurs, during which the aircraft can dive into the ground. thumb|A pilot in a Cessna 152 performing a steep turn as seen from the cockpit. The reverse effect is experienced in advanced aerobatic maneuvers under negative g-forces, where excess blood moves towards the brain and eyes (""redout""). Black-out and g-LOC have caused a number of fatal aircraft accidents.Amos, Smith, ""Report: Blue Angels pilot became disoriented"" , Military Times, January 16, 2008. == Operation == If blood is allowed to pool in the lower areas of the body, the brain will be deprived of blood. By using a modern g-suit in combination with anti-g strain techniques, a trained pilot is now expected to endure accelerations of up to nine g without blacking out. A trained, fit individual wearing a g suit and practising the straining maneuver can, with some difficulty, sustain up to 9 g without loss of consciousness. A trained, fit individual wearing a g suit and practicing the straining manoeuvre can, with some difficulty, sustain up to 12-14g without loss of consciousness. Upon regaining cerebral blood flow, the G-LOC victim usually experiences myoclonic convulsions (often called the ‘funky chicken’) and often full amnesia of the event is experienced. The most dangerous pilot-induced oscillations can occur during landing. * G-LOC – where consciousness is lost. ",0.132,1000,83.81,0.00017,5654.86677646,C
-"The world’s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball?","If the string should be raised off the ground, all the way along the equator, how much longer would the string be? Even more surprising is that the size of the sphere or circle around which the string is spanned is irrelevant, and may be anything from the size of an atom to the Milky Way -- the result depends only on the amount it is raised. The scale length of a string instrument is the maximum vibrating length of the strings that produce sound, and determines the range of tones that string can produce at a given tension. The question that is then posed is whether the gap between string and Earth will allow the passage of a car, a cat or a thin knife blade. ==Solution== thumb|Visual analogue using a square As the string must be raised all along the entire circumference, one might expect several kilometres of additional string. 300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1). 275px|thumb|Stick figure of 1.75 meters standing next to a violin string of .33 meters and a long string instrument string of 10 meters. On instruments in which strings are not ""stopped"" (typically by frets or the player's fingers) or divided in length (such as in the piano), it is the actual length of string between the nut and the bridge. More formally, let c be the Earth's circumference, r its radius, Δc the added string length and Δr the added radius. has a circumference of 2R , : \begin{align} c + \varDelta c & = 2 \pi (r + \varDelta r) \\\ 2 \pi r + \varDelta c & = 2 \pi r + 2 \pi \varDelta r \\\ \varDelta c & = 2 \pi \varDelta r \\\ \therefore\; \varDelta r & = \frac{\varDelta c}{2 \pi} \end{align} regardless of the value of c . In a version of this puzzle, string is tightly wrapped around the equator of a perfectly spherical Earth. For example if a string 0.33 meters long, of given mass and tension, produces A440, a string with identical mass and tension but eight times as long, 2.64 meters, produces 55 hertz. thumb|300px|Visualisation showing that the length added to the circumference (blue) is dependent only on the additional radius (red) and not the original circumference (grey) String girdling Earth is a mathematical puzzle with a counterintuitive solution. Elevenstring may refer to: * The eleven-string alto guitar * A fictional musical instrument in The Hydrogen Sonata by Iain M. Banks It is also called string length. Alternatively, of string is spliced into the original string, and the extended string rearranged so that it is at a uniform height above the equator. The long-string instrument is a musical instrument in which the string is of such a length that the fundamental transverse wave is below what a person can hear as a tone (±20 Hz). In the second phrasing, considering that is almost negligible compared with the circumference, the first response may be that the new position of the string will be no different from the original surface-hugging position. This long-string instrument's range is centered on the octave of middle C and extends above and below this by an octave. The string spelled by the edges from the root to such a node is a longest repeated substring. If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher. Springer. p.10. . ==Equations== The natural frequency is: *a) Inversely proportional to the length of the string (the law of Pythagoras), *b) Proportional to the square root of the stretching force, and *c) Inversely proportional to the square root of the mass per length. : f_0 \propto \tfrac{1}{L}. (equation 26) : f_0 \propto \sqrt{F}. (equation 27) : f_0 \propto \frac{1}{\sqrt{\mu}}. (equation 28) Thus, for example, all other properties of the string being equal, to make the note one octave higher (2/1) one would need either to decrease its length by half (1/2), to increase the tension to the square (4), or to decrease its mass per length by the inverse square (1/4). The Long- String Instrument. The strings of the bass octave extend the instrument's full 90 feet. ",-11.2,0.4207,2.0,479,0.245,C
+What is the magnitude of the acceleration, in $g$ units, of a pilot whose aircraft enters a horizontal circular turn with a velocity of $\vec{v}_i=(400 \hat{\mathrm{i}}+500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ and $24.0 \mathrm{~s}$ later leaves the turn with a velocity of $\vec{v}_f=(-400 \hat{\mathrm{i}}-500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}$ ?","Incidents of acceleration-induced loss of consciousness have caused fatal accidents in aircraft capable of sustaining high-g for considerable periods. Skilled pilots can use this loss of vision as their indicator that they are at maximum turn performance without losing consciousness. The danger of g-LOC to aircraft pilots is magnified because on relaxation of g-force there is a period of disorientation before full sensation is re-gained. If G-LOC occurs at low altitude, this momentary lapse can prove fatal and even highly experienced pilots can pull straight to a G-LOC condition without first perceiving the visual onset warnings that would normally be used as the sign to back off from pulling any more gs. A further increase in g-forces will cause g-LOC where consciousness is lost. g-force induced loss of consciousness (abbreviated as G-LOC, pronounced ""JEE- lock"") is a term generally used in aerospace physiology to describe a loss of consciousness occurring from excessive and sustained g-forces draining blood away from the brain causing cerebral hypoxia. If g-forces increase further, complete loss of vision will occur, while consciousness remains. High-g training for pilots of high performance aircraft or spacecraft often includes ground training for G-LOC in special centrifuges, with some profiles exposing pilots to 9 gs for a sustained period. ==Effects of g-forces== Under increasing positive g-force, blood in the body will tend to move from the head toward the feet. In general, when the g-force pushes the body forwards (colloquially known as 'eyeballs in'NASA Physiological Acceleration Systems ) a much higher tolerance is shown than when g-force is pushing the body backwards ('eyeballs out') since blood vessels in the retina appear more sensitive to that direction.NASA Technical note D-337, Centrifuge Study of Pilot Tolerance to Acceleration and the Effects of Acceleration on Pilot Performance, by Brent Y. Creer, Captain Harald A. Smedal, USN (MC), and Rodney C. Wingrove ==G-suits== A g-suit is worn by aviators and astronauts who are subject to high levels of acceleration and, hence, increasing positive g. thumb|upright=1.2|If the cockpit lost pressure while the aircraft was above the Armstrong limit, even a positive pressure oxygen mask could not sustain pilot consciousness. This is not true in 0 g when you strafe up; that is an eyeballs-down maneuver, which is the same force as a blackout where blood rushes to the feet, and this force is parallel to the spine. A pilot aiming for a 500-foot per minute descent, for example, may find themselves descending more rapidly than intended. This is doubly dangerous because, on recovery as g is reduced, a period of several seconds of disorientation occurs, during which the aircraft can dive into the ground. thumb|A pilot in a Cessna 152 performing a steep turn as seen from the cockpit. The reverse effect is experienced in advanced aerobatic maneuvers under negative g-forces, where excess blood moves towards the brain and eyes (""redout""). Black-out and g-LOC have caused a number of fatal aircraft accidents.Amos, Smith, ""Report: Blue Angels pilot became disoriented"" , Military Times, January 16, 2008. == Operation == If blood is allowed to pool in the lower areas of the body, the brain will be deprived of blood. By using a modern g-suit in combination with anti-g strain techniques, a trained pilot is now expected to endure accelerations of up to nine g without blacking out. A trained, fit individual wearing a g suit and practising the straining maneuver can, with some difficulty, sustain up to 9 g without loss of consciousness. A trained, fit individual wearing a g suit and practicing the straining manoeuvre can, with some difficulty, sustain up to 12-14g without loss of consciousness. Upon regaining cerebral blood flow, the G-LOC victim usually experiences myoclonic convulsions (often called the ‘funky chicken’) and often full amnesia of the event is experienced. The most dangerous pilot-induced oscillations can occur during landing. * G-LOC – where consciousness is lost. ",0.132,1000,"""83.81""",0.00017,5654.86677646,C
+"The world’s largest ball of string is about 2 m in radius. To the nearest order of magnitude, what is the total length L of the string in the ball?","If the string should be raised off the ground, all the way along the equator, how much longer would the string be? Even more surprising is that the size of the sphere or circle around which the string is spanned is irrelevant, and may be anything from the size of an atom to the Milky Way -- the result depends only on the amount it is raised. The scale length of a string instrument is the maximum vibrating length of the strings that produce sound, and determines the range of tones that string can produce at a given tension. The question that is then posed is whether the gap between string and Earth will allow the passage of a car, a cat or a thin knife blade. ==Solution== thumb|Visual analogue using a square As the string must be raised all along the entire circumference, one might expect several kilometres of additional string. 300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1). 275px|thumb|Stick figure of 1.75 meters standing next to a violin string of .33 meters and a long string instrument string of 10 meters. On instruments in which strings are not ""stopped"" (typically by frets or the player's fingers) or divided in length (such as in the piano), it is the actual length of string between the nut and the bridge. More formally, let c be the Earth's circumference, r its radius, Δc the added string length and Δr the added radius. has a circumference of 2R , : \begin{align} c + \varDelta c & = 2 \pi (r + \varDelta r) \\\ 2 \pi r + \varDelta c & = 2 \pi r + 2 \pi \varDelta r \\\ \varDelta c & = 2 \pi \varDelta r \\\ \therefore\; \varDelta r & = \frac{\varDelta c}{2 \pi} \end{align} regardless of the value of c . In a version of this puzzle, string is tightly wrapped around the equator of a perfectly spherical Earth. For example if a string 0.33 meters long, of given mass and tension, produces A440, a string with identical mass and tension but eight times as long, 2.64 meters, produces 55 hertz. thumb|300px|Visualisation showing that the length added to the circumference (blue) is dependent only on the additional radius (red) and not the original circumference (grey) String girdling Earth is a mathematical puzzle with a counterintuitive solution. Elevenstring may refer to: * The eleven-string alto guitar * A fictional musical instrument in The Hydrogen Sonata by Iain M. Banks It is also called string length. Alternatively, of string is spliced into the original string, and the extended string rearranged so that it is at a uniform height above the equator. The long-string instrument is a musical instrument in which the string is of such a length that the fundamental transverse wave is below what a person can hear as a tone (±20 Hz). In the second phrasing, considering that is almost negligible compared with the circumference, the first response may be that the new position of the string will be no different from the original surface-hugging position. This long-string instrument's range is centered on the octave of middle C and extends above and below this by an octave. The string spelled by the edges from the root to such a node is a longest repeated substring. If the tension on a string is ten lbs., it must be increased to 40 lbs. for a pitch an octave higher. Springer. p.10. . ==Equations== The natural frequency is: *a) Inversely proportional to the length of the string (the law of Pythagoras), *b) Proportional to the square root of the stretching force, and *c) Inversely proportional to the square root of the mass per length. : f_0 \propto \tfrac{1}{L}. (equation 26) : f_0 \propto \sqrt{F}. (equation 27) : f_0 \propto \frac{1}{\sqrt{\mu}}. (equation 28) Thus, for example, all other properties of the string being equal, to make the note one octave higher (2/1) one would need either to decrease its length by half (1/2), to increase the tension to the square (4), or to decrease its mass per length by the inverse square (1/4). The Long- String Instrument. The strings of the bass octave extend the instrument's full 90 feet. ",-11.2,0.4207,"""2.0""",479,0.245,C
 "You drive a beat-up pickup truck along a straight road for $8.4 \mathrm{~km}$ at $70 \mathrm{~km} / \mathrm{h}$, at which point the truck runs out of gasoline and stops. Over the next $30 \mathrm{~min}$, you walk another $2.0 \mathrm{~km}$ farther along the road to a gasoline station.
-What is your overall displacement from the beginning of your drive to your arrival at the station?","A displacement may be also described as a relative position (resulting from the motion), that is, as the final position of a point relative to its initial position . Displacement is usually measured in units of tonnes or long tons. ==Definitions== There are terms for the displacement of a vessel under specified conditions: ===Loaded displacement=== *Loaded displacement is the weight of the ship including cargo, passengers, fuel, water, stores, dunnage and such other items necessary for use on a voyage. thumb|Displacement versus distance travelled along a path In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. Engine displacement is the measure of the cylinder volume swept by all of the pistons of a piston engine, excluding the combustion chambers. right|thumb|upright=1.3|Load lines, by showing how low a ship is sitting in the water, make it possible to determine displacement. In fluid mechanics, displacement occurs when an object is largely immersed in a fluid, pushing it out of the way and taking its place. The displacement or displacement tonnage of a ship is its weight. The process of determining a vessel's displacement begins with measuring its draft.George, 2005. p.5. Ship displacement varies by a vessel's degree of load, from its empty weight as designed (known as ""lightweight tonnage"") to its maximum load. A displacement may be identified with the translation that maps the initial position to the final position. For this reason displacement is one of the measures often used in advertising, as well as regulating, motor vehicles. The corresponding displacement vector can be defined as the difference between the final and initial positions: s = x_\textrm{f} - x_\textrm{i} = \Delta{x} In considering motions of objects over time, the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The weight of the displaced fluid can be found mathematically. Therefore, the weight of the displaced fluid can be expressed as . The ship's hydrostatic tables show the corresponding volume displaced.George, 2005. p. 465. The fluid displaced has a weight , where is acceleration due to gravity. Ship displacement should not be confused with measurements of volume or capacity typically used for commercial vessels and measured by tonnage: net tonnage and gross tonnage. ==Calculation== thumb|Shipboard stability computer programs can be used to calculate a vessel's displacement. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity, which is a vector, and differs thus from the average speed, which is a scalar quantity. == Rigid body == In dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. Therefore, they are generally taxed as 1.5 times their stated physical displacement (1.3 litres becomes effectively 2.0, 2.0 becomes effectively 3.0), although actual power outputs can be higher than suggested by this conversion factor. == Automotive model names == Historically, many car model names have included their engine displacement. The formula is: : \text{Displacement} = \text{stroke length} \times \pi \left(\frac{\text{bore}}{2}\right)^2 \times \text{number of cylinders} Using this formula for non-typical types of engine, such as the Wankel design and the oval-piston type used in Honda NR motorcycles, can sometimes yield misleading results when attempting to compare engines. ",10.4, 258.14,3.0,273,-0.347,A
+What is your overall displacement from the beginning of your drive to your arrival at the station?","A displacement may be also described as a relative position (resulting from the motion), that is, as the final position of a point relative to its initial position . Displacement is usually measured in units of tonnes or long tons. ==Definitions== There are terms for the displacement of a vessel under specified conditions: ===Loaded displacement=== *Loaded displacement is the weight of the ship including cargo, passengers, fuel, water, stores, dunnage and such other items necessary for use on a voyage. thumb|Displacement versus distance travelled along a path In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. Engine displacement is the measure of the cylinder volume swept by all of the pistons of a piston engine, excluding the combustion chambers. right|thumb|upright=1.3|Load lines, by showing how low a ship is sitting in the water, make it possible to determine displacement. In fluid mechanics, displacement occurs when an object is largely immersed in a fluid, pushing it out of the way and taking its place. The displacement or displacement tonnage of a ship is its weight. The process of determining a vessel's displacement begins with measuring its draft.George, 2005. p.5. Ship displacement varies by a vessel's degree of load, from its empty weight as designed (known as ""lightweight tonnage"") to its maximum load. A displacement may be identified with the translation that maps the initial position to the final position. For this reason displacement is one of the measures often used in advertising, as well as regulating, motor vehicles. The corresponding displacement vector can be defined as the difference between the final and initial positions: s = x_\textrm{f} - x_\textrm{i} = \Delta{x} In considering motions of objects over time, the instantaneous velocity of the object is the rate of change of the displacement as a function of time. The weight of the displaced fluid can be found mathematically. Therefore, the weight of the displaced fluid can be expressed as . The ship's hydrostatic tables show the corresponding volume displaced.George, 2005. p. 465. The fluid displaced has a weight , where is acceleration due to gravity. Ship displacement should not be confused with measurements of volume or capacity typically used for commercial vessels and measured by tonnage: net tonnage and gross tonnage. ==Calculation== thumb|Shipboard stability computer programs can be used to calculate a vessel's displacement. Study of these higher order derivatives can improve approximations of the original displacement function. Such higher-order terms are required in order to accurately represent the displacement function as a sum of an infinite series, enabling several analytical techniques in engineering and physics. For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity, which is a vector, and differs thus from the average speed, which is a scalar quantity. == Rigid body == In dealing with the motion of a rigid body, the term displacement may also include the rotations of the body. Therefore, they are generally taxed as 1.5 times their stated physical displacement (1.3 litres becomes effectively 2.0, 2.0 becomes effectively 3.0), although actual power outputs can be higher than suggested by this conversion factor. == Automotive model names == Historically, many car model names have included their engine displacement. The formula is: : \text{Displacement} = \text{stroke length} \times \pi \left(\frac{\text{bore}}{2}\right)^2 \times \text{number of cylinders} Using this formula for non-typical types of engine, such as the Wankel design and the oval-piston type used in Honda NR motorcycles, can sometimes yield misleading results when attempting to compare engines. ",10.4, 258.14,"""3.0""",273,-0.347,A
 "A heavy object can sink into the ground during an earthquake if the shaking causes the ground to undergo liquefaction, in which the soil grains experience little friction as they slide over one another. The ground is then effectively quicksand. The possibility of liquefaction in sandy ground can be predicted in terms of the void ratio $e$ for a sample of the ground:
 $$
 e=\frac{V_{\text {voids }}}{V_{\text {grains }}} .
 $$
-Here, $V_{\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\rho_{\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\rho_{\mathrm{SiO}_2}=2.600 \times 10^3 \mathrm{~kg} / \mathrm{m}^3$.","Eng., 139(3), 407–419. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000743 ] #The earthquake load, measured as cyclic stress ratio CSR=\frac{\tau_{av}}{\sigma'_{v}}=0,65\frac{a_{max}}{g}\frac{\sigma_{v}}{\sigma'_{v}}r_d Evaluation of soil liquefaction from surface analysis #the capacity of the soil to resist liquefaction, expressed in terms of the cyclic resistance ratio (CRR) ==Earthquake liquefaction== Pressures generated during large earthquakes can force underground water and liquefied sand to the surface. Soil liquefaction may occur in partially saturated soil when it is shaken by an earthquake or similar forces. Aluminium, for example, has a density of about 2.7 grams per cubic centimeter, but a piece of aluminium will float on top of quicksand until motion causes the sand to liquefy. The saturated sediment may appear quite solid until a change in pressure or a shock initiates the liquefaction, causing the sand to form a suspension with each grain surrounded by a thin film of water. ""Liquefaction of sands and its evaluation."", Proceedings of the 1st International Conference on Earthquake Geotechnical Engineering, Tokyo The resistance of the cohesionless soil to liquefaction will depend on the density of the soil, confining stresses, soil structure (fabric, age and cementation), the magnitude and duration of the cyclic loading, and the extent to which shear stress reversal occurs. ==Liquefaction potential: simplified empirical analysis== Three parameters are needed to assess liquefaction potential using the simplified empirical method: #A measure of soil resistance to liquefaction: Standard Penetration Resistance (SPT), [Cetin, K.O., Seed, R.B., Armen Der Kiureghian, M., Tokimatsu, K., Harder, L.F. Jr., Kayen, R.E., Moss, R.E.S. (2004) SPT-Based Probabilistic and Deterministic Assessment of Seismic Soil Liquefaction Potential, Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 130, No. 12, December 2004, pp. 1314-1340. http://ascelibrary.org/doi/abs/10.1061/%28ASCE%291090-0241%282004%29130%3A12%281314%29 ] [I.M. Idriss, Ross W. Boulanger, 2nd Ishihara Lecture: SPT- and CPT-based relationships for the residual shear strength of liquefied soils, Soil Dynamics and Earthquake Engineering, Volume 68, 2015, Pages 57 68, ISSN 0267-7261, https://doi.org/10.1016/j.soildyn.2014.09.010.] Thus, shear strains are significantly less than a true state of soil liquefaction. ==Occurrence== Liquefaction is more likely to occur in loose to moderately saturated granular soils with poor drainage, such as silty sands or sands and gravels containing impermeable sediments. Soil liquefaction occurs when a cohesionless saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, in which material that is ordinarily a solid behaves like a liquid. One positive aspect of soil liquefaction is the tendency for the effects of earthquake shaking to be significantly damped (reduced) for the remainder of the earthquake. Liquefaction is more likely to occur in sandy or non-plastic silty soils, but may in rare cases occur in gravels and clays (see quick clay). In the case of earthquakes, the shaking force can increase the pressure of shallow groundwater, liquefying sand and silt deposits. If the soil strain-hardens, e.g. moderately dense to dense sand, flow liquefaction will generally not occur. Soil liquefaction induced by earthquake shaking is a major contributor to urban seismic risk. ==Effects== The effects of soil liquefaction on the built environment can be extremely damaging. Objects in liquefied sand sink to the level at which the weight of the object is equal to the weight of the displaced soil/water mix and the submerged object floats due to its buoyancy. Deposits most susceptible to liquefaction are young (Holocene-age, deposited within the last 10,000 years) sands and silts of similar grain size (well- sorted), in beds at least metres thick, and saturated with water. When water in the sand cannot escape, it creates a liquefied soil that loses strength and cannot support weight. When the water trapped in the batch of sand cannot escape, it creates liquefied soil that can no longer resist force. The saturated sediment may appear quite solid until a sudden change in pressure or shock initiates liquefaction. Recent Advances in Soil Liquefaction Engineering: A Unified and Consistent Framework, 26th Annual ASCE Los Angeles Geotechnical Spring Seminar, Long Beach, California, April 30, 2003, Earthquake Engineering Research Center ==External links== * Soil Liquefaction * Liquefaction – Pacific Northwest Seismic Network * Liquefaction in Chiba, Japan on YouTube recorded during the 2011 Tohoku earthquake Category:Earthquake engineering Liquifaction, soil Category:Sedimentology Category:Seismology Category:Soil mechanics Category:Natural disasters By weight, its mineral composition is about 40–40–20% concentration of sand–silt–clay, respectively. The phenomenon of dilatancy can be observed in a drained simple shear test on a sample of dense sand. Examples of soil liquefaction include quicksand, quick clay, turbidity currents and earthquake- induced liquefaction. To move within the quicksand, a person or object must apply sufficient pressure on the compacted sand to re-introduce enough water to liquefy it. ",0.70710678,2598960,-233.0,0.9966,1.4,E
-"What is the angle $\phi$ between $\vec{a}=3.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}$ and $\vec{b}=$ $-2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{k}}$ ? (Caution: Although many of the following steps can be bypassed with a vector-capable calculator, you will learn more about scalar products if, at least here, you use these steps.)","===Vector rejection=== By definition, \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}}. ==Properties== ===Scalar projection=== The scalar projection on is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. Hence, a_2 = \left\|\mathbf{a}\right\| \sin \theta = \frac {\mathbf{a} \cdot \mathbf{b}^\perp} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a}_y \mathbf{b}_x - \mathbf{a}_x \mathbf{b}_y} {\left\|\mathbf{b}\right\| }. Which finally gives: \mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ . In mathematics, the scalar projection of a vector \mathbf{a} on (or onto) a vector \mathbf{b}, also known as the scalar resolute of \mathbf{a} in the direction of \mathbf{b}, is given by: :s = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b}, where the operator \cdot denotes a dot product, \hat{\mathbf{b}} is the unit vector in the direction of \mathbf{b}, \left\|\mathbf{a}\right\| is the length of \mathbf{a}, and \theta is the angle between \mathbf{a} and \mathbf{b}. Namely, it is defined as \mathbf{a}_1 = a_1 \mathbf{\hat b} = (\left\|\mathbf{a}\right\| \cos \theta) \mathbf{\hat b} where a_1 is the corresponding scalar projection, as defined above, and \mathbf{\hat b} is the unit vector with the same direction as : \mathbf{\hat b} = \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} ===Vector rejection=== By definition, the vector rejection of on is: \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \left(\left\|\mathbf{a}\right\| \cos \theta\right) \mathbf{\hat b} ==Definitions in terms of a and b== When is not known, the cosine of can be computed in terms of and , by the following property of the dot product \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta ===Scalar projection=== By the above- mentioned property of the dot product, the definition of the scalar projection becomes: a_1 = \left\|\mathbf{a}\right\| \cos \theta = \frac {\mathbf{a} \cdot \mathbf{b}} { \left\|\mathbf{b}\right\|}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. thumb|300px|Geometric interpretation of the angle between two vectors defined using an inner product alt=Scalar product spaces, inner product spaces, Hermitian product spaces.|thumb|300px|Scalar product spaces, over any field, have ""scalar products"" that are symmetrical and linear in the first argument. Multiplying the scalar projection of \mathbf{a} on \mathbf{b} by \mathbf{\hat b} converts it into the above-mentioned orthogonal projection, also called vector projection of \mathbf{a} on \mathbf{b}. ==Definition based on angle θ== If the angle \theta between \mathbf{a} and \mathbf{b} is known, the scalar projection of \mathbf{a} on \mathbf{b} can be computed using :s = \left\|\mathbf{a}\right\| \cos \theta . (s = \left\|\mathbf{a}_1\right\| in the figure) ==Definition in terms of a and b== When \theta is not known, the cosine of \theta can be computed in terms of \mathbf{a} and \mathbf{b}, by the following property of the dot product \mathbf{a} \cdot \mathbf{b}: : \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta By this property, the definition of the scalar projection s becomes: : s = \left\|\mathbf{a}_1\right\| = \left\|\mathbf{a}\right\| \cos \theta = \left\|\mathbf{a}\right\| \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| }\, ==Properties== The scalar projection has a negative sign if 90^\circ < \theta \le 180^\circ. The scalar projection is a scalar, equal to the length of the orthogonal projection of \mathbf{a} on \mathbf{b}, with a negative sign if the projection has an opposite direction with respect to \mathbf{b}. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. More exactly, if the vector projection is denoted \mathbf{a}_1 and its length \left\|\mathbf{a}_1\right\|: : s = \left\|\mathbf{a}_1\right\| if 0^\circ \le \theta \le 90^\circ, : s = -\left\|\mathbf{a}_1\right\| if 90^\circ < \theta \le 180^\circ. ==See also== * Scalar product * Cross product * Vector projection ==Sources== *Dot products - www.mit.org *Scalar projection - Flexbooks.ck12.org *Scalar Projection & Vector Projection - medium.com *Lesson Explainer: Scalar Projection | Nagwa Category:Operations on vectors In turn, the scalar projection is defined as a_1 = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b} where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and θ is the angle between and . It coincides with the length of the vector projection if the angle is smaller than 90°. Any vector field can be written in terms of the unit vectors as: \mathbf A = A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z} = A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi} The spherical unit vectors are related to the Cartesian unit vectors by: \begin{bmatrix}\boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\\ -\sin\phi & \cos\phi & 0 \end{bmatrix} \begin{bmatrix} \mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. It is a vector parallel to , defined as \mathbf{a}_1 = a_1\mathbf{\hat b} where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. In two dimensions, this becomes a_1 = \frac {\mathbf{a}_x \mathbf{b}_x + \mathbf{a}_y \mathbf{b}_y} {\left\|\mathbf{b}\right\|}. ===Vector projection=== Similarly, the definition of the vector projection of onto becomes: \mathbf{a}_1 = a_1 \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|}, which is equivalent to either \mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b}, or \mathbf{a}_1 = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ . ===Scalar rejection=== In two dimensions, the scalar rejection is equivalent to the projection of onto \mathbf{b}^\perp = \begin{pmatrix}-\mathbf{b}_y & \mathbf{b}_x\end{pmatrix}, which is \mathbf{b} = \begin{pmatrix}\mathbf{b}_x & \mathbf{b}_y\end{pmatrix} rotated 90° to the left. The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . This means that \mathbf{A} = \mathbf{P} = \rho \mathbf{\hat \rho} + z \mathbf{\hat z}. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. The vector projection of on and the corresponding rejection are sometimes denoted by and , respectively. ==Definitions based on angle θ== ===Scalar projection=== The scalar projection of on is a scalar equal to a_1 = \left\|\mathbf{a}\right\| \cos \theta , where θ is the angle between and . ",0.03,804.62,109.0,2.7,8,C
-"During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\vec{d}=(-3.0 \mathrm{~m}) \hat{\mathrm{i}}$ while a steady wind pushes against the crate with a force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(-6.0 \mathrm{~N}) \hat{\mathrm{j}}$. The situation and coordinate axes are shown in Figure. If the crate has a kinetic energy of $10 \mathrm{~J}$ at the beginning of displacement $\vec{d}$, what is its kinetic energy at the end of $\vec{d}$ ?","Kinetic energy is the movement energy of an object. The velocity v of the car can be determined from the length of the skid using the work–energy principle, kWs = \frac{W}{2g} v^2,\quad\text{or}\quad v = \sqrt{2ksg}. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. Where e_k is the specific kinetic energy and v is velocity. The work of the net force is calculated as the product of its magnitude and the particle displacement. The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). It is known as the work–energy principle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = m \int_{t_1}^{t_2} \mathbf{a} \cdot \mathbf{v}dt = \frac{m}{2} \int_{t_1}^{t_2} \frac{d v^2}{dt}\,dt = \frac{m}{2} \int_{v^2_1}^{v^2_2} d v^2 = \frac{mv_2^2}{2} - \frac{mv_1^2}{2} = \Delta E_\text{k} The identity \mathbf{a} \cdot \mathbf{v} = \frac{1}{2} \frac{d v^2}{dt} requires some algebra. John Wiley & Sons :E_\text{k} = \frac{1}{2} mv^2 Dividing by V, the unit of volume: :\begin{align} \frac{E_\text{k}}{V} &= \frac{1}{2} \frac{m}{V}v^2 \\\ q &= \frac{1}{2} \rho v^2 \end{align} where q is the dynamic pressure, and ρ is the density of the incompressible fluid. ===Frame of reference=== The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. Specific kinetic energy is kinetic energy of an object per unit of mass. For example: the energy of TNT is 4.6 MJ/kg, and the energy of a kinetic kill vehicle with a closing speed of is 50 MJ/kg. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. The work done is given by the dot product of the two vectors. Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by the force to bring the object from rest to velocity v is equal to the work necessary to do the reverse: :E_\text{k} = \int_0^t \mathbf{F} \cdot d \mathbf{x} = \int_0^t \mathbf{v} \cdot d (m \mathbf{v}) = \int_0^t d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}. ",-383,11,2.0,4.0,1.602,D
+Here, $V_{\text {grains }}$ is the total volume of the sand grains in the sample and $V_{\text {voids }}$ is the total volume between the grains (in the voids). If $e$ exceeds a critical value of 0.80 , liquefaction can occur during an earthquake. What is the corresponding sand density $\rho_{\text {sand }}$ ? Solid silicon dioxide (the primary component of sand) has a density of $\rho_{\mathrm{SiO}_2}=2.600 \times 10^3 \mathrm{~kg} / \mathrm{m}^3$.","Eng., 139(3), 407–419. http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000743 ] #The earthquake load, measured as cyclic stress ratio CSR=\frac{\tau_{av}}{\sigma'_{v}}=0,65\frac{a_{max}}{g}\frac{\sigma_{v}}{\sigma'_{v}}r_d Evaluation of soil liquefaction from surface analysis #the capacity of the soil to resist liquefaction, expressed in terms of the cyclic resistance ratio (CRR) ==Earthquake liquefaction== Pressures generated during large earthquakes can force underground water and liquefied sand to the surface. Soil liquefaction may occur in partially saturated soil when it is shaken by an earthquake or similar forces. Aluminium, for example, has a density of about 2.7 grams per cubic centimeter, but a piece of aluminium will float on top of quicksand until motion causes the sand to liquefy. The saturated sediment may appear quite solid until a change in pressure or a shock initiates the liquefaction, causing the sand to form a suspension with each grain surrounded by a thin film of water. ""Liquefaction of sands and its evaluation."", Proceedings of the 1st International Conference on Earthquake Geotechnical Engineering, Tokyo The resistance of the cohesionless soil to liquefaction will depend on the density of the soil, confining stresses, soil structure (fabric, age and cementation), the magnitude and duration of the cyclic loading, and the extent to which shear stress reversal occurs. ==Liquefaction potential: simplified empirical analysis== Three parameters are needed to assess liquefaction potential using the simplified empirical method: #A measure of soil resistance to liquefaction: Standard Penetration Resistance (SPT), [Cetin, K.O., Seed, R.B., Armen Der Kiureghian, M., Tokimatsu, K., Harder, L.F. Jr., Kayen, R.E., Moss, R.E.S. (2004) SPT-Based Probabilistic and Deterministic Assessment of Seismic Soil Liquefaction Potential, Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 130, No. 12, December 2004, pp. 1314-1340. http://ascelibrary.org/doi/abs/10.1061/%28ASCE%291090-0241%282004%29130%3A12%281314%29 ] [I.M. Idriss, Ross W. Boulanger, 2nd Ishihara Lecture: SPT- and CPT-based relationships for the residual shear strength of liquefied soils, Soil Dynamics and Earthquake Engineering, Volume 68, 2015, Pages 57 68, ISSN 0267-7261, https://doi.org/10.1016/j.soildyn.2014.09.010.] Thus, shear strains are significantly less than a true state of soil liquefaction. ==Occurrence== Liquefaction is more likely to occur in loose to moderately saturated granular soils with poor drainage, such as silty sands or sands and gravels containing impermeable sediments. Soil liquefaction occurs when a cohesionless saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, in which material that is ordinarily a solid behaves like a liquid. One positive aspect of soil liquefaction is the tendency for the effects of earthquake shaking to be significantly damped (reduced) for the remainder of the earthquake. Liquefaction is more likely to occur in sandy or non-plastic silty soils, but may in rare cases occur in gravels and clays (see quick clay). In the case of earthquakes, the shaking force can increase the pressure of shallow groundwater, liquefying sand and silt deposits. If the soil strain-hardens, e.g. moderately dense to dense sand, flow liquefaction will generally not occur. Soil liquefaction induced by earthquake shaking is a major contributor to urban seismic risk. ==Effects== The effects of soil liquefaction on the built environment can be extremely damaging. Objects in liquefied sand sink to the level at which the weight of the object is equal to the weight of the displaced soil/water mix and the submerged object floats due to its buoyancy. Deposits most susceptible to liquefaction are young (Holocene-age, deposited within the last 10,000 years) sands and silts of similar grain size (well- sorted), in beds at least metres thick, and saturated with water. When water in the sand cannot escape, it creates a liquefied soil that loses strength and cannot support weight. When the water trapped in the batch of sand cannot escape, it creates liquefied soil that can no longer resist force. The saturated sediment may appear quite solid until a sudden change in pressure or shock initiates liquefaction. Recent Advances in Soil Liquefaction Engineering: A Unified and Consistent Framework, 26th Annual ASCE Los Angeles Geotechnical Spring Seminar, Long Beach, California, April 30, 2003, Earthquake Engineering Research Center ==External links== * Soil Liquefaction * Liquefaction – Pacific Northwest Seismic Network * Liquefaction in Chiba, Japan on YouTube recorded during the 2011 Tohoku earthquake Category:Earthquake engineering Liquifaction, soil Category:Sedimentology Category:Seismology Category:Soil mechanics Category:Natural disasters By weight, its mineral composition is about 40–40–20% concentration of sand–silt–clay, respectively. The phenomenon of dilatancy can be observed in a drained simple shear test on a sample of dense sand. Examples of soil liquefaction include quicksand, quick clay, turbidity currents and earthquake- induced liquefaction. To move within the quicksand, a person or object must apply sufficient pressure on the compacted sand to re-introduce enough water to liquefy it. ",0.70710678,2598960,"""-233.0""",0.9966,1.4,E
+"What is the angle $\phi$ between $\vec{a}=3.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}$ and $\vec{b}=$ $-2.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{k}}$ ? (Caution: Although many of the following steps can be bypassed with a vector-capable calculator, you will learn more about scalar products if, at least here, you use these steps.)","===Vector rejection=== By definition, \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}}. ==Properties== ===Scalar projection=== The scalar projection on is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees. Hence, a_2 = \left\|\mathbf{a}\right\| \sin \theta = \frac {\mathbf{a} \cdot \mathbf{b}^\perp} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a}_y \mathbf{b}_x - \mathbf{a}_x \mathbf{b}_y} {\left\|\mathbf{b}\right\| }. Which finally gives: \mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ . In mathematics, the scalar projection of a vector \mathbf{a} on (or onto) a vector \mathbf{b}, also known as the scalar resolute of \mathbf{a} in the direction of \mathbf{b}, is given by: :s = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b}, where the operator \cdot denotes a dot product, \hat{\mathbf{b}} is the unit vector in the direction of \mathbf{b}, \left\|\mathbf{a}\right\| is the length of \mathbf{a}, and \theta is the angle between \mathbf{a} and \mathbf{b}. Namely, it is defined as \mathbf{a}_1 = a_1 \mathbf{\hat b} = (\left\|\mathbf{a}\right\| \cos \theta) \mathbf{\hat b} where a_1 is the corresponding scalar projection, as defined above, and \mathbf{\hat b} is the unit vector with the same direction as : \mathbf{\hat b} = \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} ===Vector rejection=== By definition, the vector rejection of on is: \mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 Hence, \mathbf{a}_2 = \mathbf{a} - \left(\left\|\mathbf{a}\right\| \cos \theta\right) \mathbf{\hat b} ==Definitions in terms of a and b== When is not known, the cosine of can be computed in terms of and , by the following property of the dot product \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta ===Scalar projection=== By the above- mentioned property of the dot product, the definition of the scalar projection becomes: a_1 = \left\|\mathbf{a}\right\| \cos \theta = \frac {\mathbf{a} \cdot \mathbf{b}} { \left\|\mathbf{b}\right\|}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. thumb|300px|Geometric interpretation of the angle between two vectors defined using an inner product alt=Scalar product spaces, inner product spaces, Hermitian product spaces.|thumb|300px|Scalar product spaces, over any field, have ""scalar products"" that are symmetrical and linear in the first argument. Multiplying the scalar projection of \mathbf{a} on \mathbf{b} by \mathbf{\hat b} converts it into the above-mentioned orthogonal projection, also called vector projection of \mathbf{a} on \mathbf{b}. ==Definition based on angle θ== If the angle \theta between \mathbf{a} and \mathbf{b} is known, the scalar projection of \mathbf{a} on \mathbf{b} can be computed using :s = \left\|\mathbf{a}\right\| \cos \theta . (s = \left\|\mathbf{a}_1\right\| in the figure) ==Definition in terms of a and b== When \theta is not known, the cosine of \theta can be computed in terms of \mathbf{a} and \mathbf{b}, by the following property of the dot product \mathbf{a} \cdot \mathbf{b}: : \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta By this property, the definition of the scalar projection s becomes: : s = \left\|\mathbf{a}_1\right\| = \left\|\mathbf{a}\right\| \cos \theta = \left\|\mathbf{a}\right\| \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| }\, ==Properties== The scalar projection has a negative sign if 90^\circ < \theta \le 180^\circ. The scalar projection is a scalar, equal to the length of the orthogonal projection of \mathbf{a} on \mathbf{b}, with a negative sign if the projection has an opposite direction with respect to \mathbf{b}. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. More exactly, if the vector projection is denoted \mathbf{a}_1 and its length \left\|\mathbf{a}_1\right\|: : s = \left\|\mathbf{a}_1\right\| if 0^\circ \le \theta \le 90^\circ, : s = -\left\|\mathbf{a}_1\right\| if 90^\circ < \theta \le 180^\circ. ==See also== * Scalar product * Cross product * Vector projection ==Sources== *Dot products - www.mit.org *Scalar projection - Flexbooks.ck12.org *Scalar Projection & Vector Projection - medium.com *Lesson Explainer: Scalar Projection | Nagwa Category:Operations on vectors In turn, the scalar projection is defined as a_1 = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b} where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and θ is the angle between and . It coincides with the length of the vector projection if the angle is smaller than 90°. Any vector field can be written in terms of the unit vectors as: \mathbf A = A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z} = A_r\boldsymbol{\hat r} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi} The spherical unit vectors are related to the Cartesian unit vectors by: \begin{bmatrix}\boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\\ -\sin\phi & \cos\phi & 0 \end{bmatrix} \begin{bmatrix} \mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. It is a vector parallel to , defined as \mathbf{a}_1 = a_1\mathbf{\hat b} where a_1 is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. In two dimensions, this becomes a_1 = \frac {\mathbf{a}_x \mathbf{b}_x + \mathbf{a}_y \mathbf{b}_y} {\left\|\mathbf{b}\right\|}. ===Vector projection=== Similarly, the definition of the vector projection of onto becomes: \mathbf{a}_1 = a_1 \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|}, which is equivalent to either \mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b}, or \mathbf{a}_1 = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ . ===Scalar rejection=== In two dimensions, the scalar rejection is equivalent to the projection of onto \mathbf{b}^\perp = \begin{pmatrix}-\mathbf{b}_y & \mathbf{b}_x\end{pmatrix}, which is \mathbf{b} = \begin{pmatrix}\mathbf{b}_x & \mathbf{b}_y\end{pmatrix} rotated 90° to the left. The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . This means that \mathbf{A} = \mathbf{P} = \rho \mathbf{\hat \rho} + z \mathbf{\hat z}. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. The vector projection of on and the corresponding rejection are sometimes denoted by and , respectively. ==Definitions based on angle θ== ===Scalar projection=== The scalar projection of on is a scalar equal to a_1 = \left\|\mathbf{a}\right\| \cos \theta , where θ is the angle between and . ",0.03,804.62,"""109.0""",2.7,8,C
+"During a storm, a crate of crepe is sliding across a slick, oily parking lot through a displacement $\vec{d}=(-3.0 \mathrm{~m}) \hat{\mathrm{i}}$ while a steady wind pushes against the crate with a force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}+(-6.0 \mathrm{~N}) \hat{\mathrm{j}}$. The situation and coordinate axes are shown in Figure. If the crate has a kinetic energy of $10 \mathrm{~J}$ at the beginning of displacement $\vec{d}$, what is its kinetic energy at the end of $\vec{d}$ ?","Kinetic energy is the movement energy of an object. The velocity v of the car can be determined from the length of the skid using the work–energy principle, kWs = \frac{W}{2g} v^2,\quad\text{or}\quad v = \sqrt{2ksg}. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. Where e_k is the specific kinetic energy and v is velocity. The work of the net force is calculated as the product of its magnitude and the particle displacement. The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). It is known as the work–energy principle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = m \int_{t_1}^{t_2} \mathbf{a} \cdot \mathbf{v}dt = \frac{m}{2} \int_{t_1}^{t_2} \frac{d v^2}{dt}\,dt = \frac{m}{2} \int_{v^2_1}^{v^2_2} d v^2 = \frac{mv_2^2}{2} - \frac{mv_1^2}{2} = \Delta E_\text{k} The identity \mathbf{a} \cdot \mathbf{v} = \frac{1}{2} \frac{d v^2}{dt} requires some algebra. John Wiley & Sons :E_\text{k} = \frac{1}{2} mv^2 Dividing by V, the unit of volume: :\begin{align} \frac{E_\text{k}}{V} &= \frac{1}{2} \frac{m}{V}v^2 \\\ q &= \frac{1}{2} \rho v^2 \end{align} where q is the dynamic pressure, and ρ is the density of the incompressible fluid. ===Frame of reference=== The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. Specific kinetic energy is kinetic energy of an object per unit of mass. For example: the energy of TNT is 4.6 MJ/kg, and the energy of a kinetic kill vehicle with a closing speed of is 50 MJ/kg. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. The work done is given by the dot product of the two vectors. Assuming the object was at rest at time 0, we integrate from time 0 to time t because the work done by the force to bring the object from rest to velocity v is equal to the work necessary to do the reverse: :E_\text{k} = \int_0^t \mathbf{F} \cdot d \mathbf{x} = \int_0^t \mathbf{v} \cdot d (m \mathbf{v}) = \int_0^t d \left(\frac{m v^2}{2}\right) = \frac{m v^2}{2}. ",-383,11,"""2.0""",4.0,1.602,D
 "When the force on an object depends on the position of the object, we cannot find the work done by it on the object by simply multiplying the force by the displacement. The reason is that there is no one value for the force-it changes. So, we must find the work in tiny little displacements and then add up all the work results. We effectively say, ""Yes, the force varies over any given tiny little displacement, but the variation is so small we can approximate the force as being constant during the displacement."" Sure, it is not precise, but if we make the displacements infinitesimal, then our error becomes infinitesimal and the result becomes precise. But, to add an infinite number of work contributions by hand would take us forever, longer than a semester. So, we add them up via an integration, which allows us to do all this in minutes (much less than a semester).
 
-Force $\vec{F}=\left(3 x^2 \mathrm{~N}\right) \hat{\mathrm{i}}+(4 \mathrm{~N}) \hat{\mathrm{j}}$, with $x$ in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates $(2 \mathrm{~m}, 3 \mathrm{~m})$ to $(3 \mathrm{~m}, 0 \mathrm{~m})$ ? Does the speed of the particle increase, decrease, or remain the same?","The work of the net force is calculated as the product of its magnitude and the particle displacement. The small amount of work by the forces over the small displacements can be determined by approximating the displacement by so \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \ldots + \mathbf{F}_n\cdot\mathbf{V}_n\delta t or \delta W = \sum_{i=1}^n \mathbf{F}_i\cdot (\boldsymbol{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \dot{\mathbf{d}})\delta t. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. The small amount of work that occurs over an instant of time is calculated as \delta W = \mathbf{F} \cdot d\mathbf{s} = \mathbf{F} \cdot \mathbf{v}dt where the is the power over the instant . Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector), and the velocity vector of the point of application. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The time derivative of the integral for work yields the instantaneous power, \frac{dW}{dt} = P(t) = \mathbf{F}\cdot \mathbf{v} . ===Path independence=== If the work for an applied force is independent of the path, then the work done by the force, by the gradient theorem, defines a potential function which is evaluated at the start and end of the trajectory of the point of application. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. Therefore, work need only be computed for the gravitational forces acting on the bodies. Notice that the work done by gravity depends only on the vertical movement of the object. The work done by a conservative force is equal to the negative of change in potential energy during that process. ",0.14,210,5.9,1855, 7.0,E
+Force $\vec{F}=\left(3 x^2 \mathrm{~N}\right) \hat{\mathrm{i}}+(4 \mathrm{~N}) \hat{\mathrm{j}}$, with $x$ in meters, acts on a particle, changing only the kinetic energy of the particle. How much work is done on the particle as it moves from coordinates $(2 \mathrm{~m}, 3 \mathrm{~m})$ to $(3 \mathrm{~m}, 0 \mathrm{~m})$ ? Does the speed of the particle increase, decrease, or remain the same?","The work of the net force is calculated as the product of its magnitude and the particle displacement. The small amount of work by the forces over the small displacements can be determined by approximating the displacement by so \delta W = \mathbf{F}_1\cdot\mathbf{V}_1\delta t+\mathbf{F}_2\cdot\mathbf{V}_2\delta t + \ldots + \mathbf{F}_n\cdot\mathbf{V}_n\delta t or \delta W = \sum_{i=1}^n \mathbf{F}_i\cdot (\boldsymbol{\omega}\times(\mathbf{X}_i-\mathbf{d}) + \dot{\mathbf{d}})\delta t. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. The small amount of work that occurs over an instant of time is calculated as \delta W = \mathbf{F} \cdot d\mathbf{s} = \mathbf{F} \cdot \mathbf{v}dt where the is the power over the instant . Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector), and the velocity vector of the point of application. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The time derivative of the integral for work yields the instantaneous power, \frac{dW}{dt} = P(t) = \mathbf{F}\cdot \mathbf{v} . ===Path independence=== If the work for an applied force is independent of the path, then the work done by the force, by the gradient theorem, defines a potential function which is evaluated at the start and end of the trajectory of the point of application. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. Therefore, work need only be computed for the gravitational forces acting on the bodies. Notice that the work done by gravity depends only on the vertical movement of the object. The work done by a conservative force is equal to the negative of change in potential energy during that process. ",0.14,210,"""5.9""",1855, 7.0,E
 " The charges of an electron and a positron are $-e$ and $+e$. The mass of each is $9.11 \times 10^{-31} \mathrm{~kg}$. What is the ratio of the electrical force to the gravitational force between an electron and a positron?
-","The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. It turns out that gravitational forces are negligible on the subatomic level, due to the extremely small masses of subatomic particles. === Electron === The electron charge-to-mass quotient, -e/m_{e}, is a quantity that may be measured in experimental physics. Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. In this approximation, the energy levels are different because of a different effective mass, μ, in the energy equation (see electron energy levels for a derivation): E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2}, where: * is the charge magnitude of the electron (same as the positron), * is the Planck constant, * is the electric constant (otherwise known as the permittivity of free space), * is the reduced mass: \mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2}, where and are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles). Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. So finally, the energy levels of positronium are given by E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}. Approximately: * ~60% of positrons will directly annihilate with an electron without forming positronium. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Positron paths in a cloud- chamber trace the same helical path as an electron but rotate in the opposite direction with respect to the magnetic field direction due to their having the same magnitude of charge-to-mass ratio but with opposite charge and, therefore, opposite signed charge-to-mass ratios. As with any two charged objects, electrons and positrons may also interact with each other without annihilating, in general by elastic scattering. ==Low-energy case== There are only a very limited set of possibilities for the final state. Rearranging, it is possible to solve for the charge-to-mass ratio of an electron as \frac{e}{m} = \frac{4 \pi c}{B(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})}\frac{\delta D}{D\Delta D} \, . ==See also== *Gyromagnetic ratio *Thomson (unit) == References== == Bibliography == * * * CC. However, because of the reduced mass, the frequencies of the spectral lines are less than half of those for the corresponding hydrogen lines. ==States== The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. Corrections that involved higher orders were then calculated in a non-relativistic quantum electrodynamics. == Exotic compounds == Molecular bonding was predicted for positronium. Positronium can also be considered by a particular form of the two-body Dirac equation; Two particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of Janine Shertzer. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to- charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Another difficulty was the difference in masses of the electron and the proton. This gives : = , which is much larger than the length scale associated with the electron's charge. By doing this, he showed that the electron was in fact a particle with a mass and a charge, and that its mass-to-charge ratio was much smaller than that of the hydrogen ion H+. thumb|350px|Naturally occurring electron-positron annihilation as a result of beta plus decay Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. Positrons, because of the direction that their paths curled, were at first mistaken for electrons travelling in the opposite direction. In 1897, the mass-to-charge ratio of the electron was first measured by J. J. Thomson.J. J. Thomson (1856–1940) Philosophical Magazine, 44, 293 (1897). ",4.16,5.51,27.211,-36.5,7.136,A
+","The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. It turns out that gravitational forces are negligible on the subatomic level, due to the extremely small masses of subatomic particles. === Electron === The electron charge-to-mass quotient, -e/m_{e}, is a quantity that may be measured in experimental physics. Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. In this approximation, the energy levels are different because of a different effective mass, μ, in the energy equation (see electron energy levels for a derivation): E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2}, where: * is the charge magnitude of the electron (same as the positron), * is the Planck constant, * is the electric constant (otherwise known as the permittivity of free space), * is the reduced mass: \mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2}, where and are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles). Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. So finally, the energy levels of positronium are given by E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}. Approximately: * ~60% of positrons will directly annihilate with an electron without forming positronium. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Positron paths in a cloud- chamber trace the same helical path as an electron but rotate in the opposite direction with respect to the magnetic field direction due to their having the same magnitude of charge-to-mass ratio but with opposite charge and, therefore, opposite signed charge-to-mass ratios. As with any two charged objects, electrons and positrons may also interact with each other without annihilating, in general by elastic scattering. ==Low-energy case== There are only a very limited set of possibilities for the final state. Rearranging, it is possible to solve for the charge-to-mass ratio of an electron as \frac{e}{m} = \frac{4 \pi c}{B(m_{j,f}g_{J,f}-m_{j,i}g_{J,i})}\frac{\delta D}{D\Delta D} \, . ==See also== *Gyromagnetic ratio *Thomson (unit) == References== == Bibliography == * * * CC. However, because of the reduced mass, the frequencies of the spectral lines are less than half of those for the corresponding hydrogen lines. ==States== The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. Corrections that involved higher orders were then calculated in a non-relativistic quantum electrodynamics. == Exotic compounds == Molecular bonding was predicted for positronium. Positronium can also be considered by a particular form of the two-body Dirac equation; Two particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of Janine Shertzer. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to- charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Another difficulty was the difference in masses of the electron and the proton. This gives : = , which is much larger than the length scale associated with the electron's charge. By doing this, he showed that the electron was in fact a particle with a mass and a charge, and that its mass-to-charge ratio was much smaller than that of the hydrogen ion H+. thumb|350px|Naturally occurring electron-positron annihilation as a result of beta plus decay Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. Positrons, because of the direction that their paths curled, were at first mistaken for electrons travelling in the opposite direction. In 1897, the mass-to-charge ratio of the electron was first measured by J. J. Thomson.J. J. Thomson (1856–1940) Philosophical Magazine, 44, 293 (1897). ",4.16,5.51,"""27.211""",-36.5,7.136,A
 "In Fig. 21-26, particle 1 of charge $+q$ and particle 2 of charge $+4.00 q$ are held at separation $L=9.00 \mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_3$ is to be located such that the three particles remain in place when released, what must be the $x$ coordinate of particle 3?
-","The position coordinates xj and xk are replaced > by their relative position rjk = xj − xk and by the vector to their center > of mass Rjk = (mj qj \+ mkqk)/(mj \+ mk). thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. A trion is a localized excitation which consists of three charged particles. Electrostatic separation is a process that uses electrostatic charges to separate crushed particles of material. An electrostatic separator is a device for separating particles by mass in a low energy charged beam. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. thumb|540x540px|Oppositely charged particles interact as they are moved through a column. This way, when a mixture of particles falls past a repelling object, the particles with the correct charge fall away from the other particles when they are repelled by the similarly charged object. The vector \boldsymbol{r}_N is the center of mass of all the bodies and \boldsymbol{r}_1 is the relative coordinate between the particles 1 and 2: The result one is left with is thus a system of N-1 translationally invariant coordinates \boldsymbol{r}_1, \dots, \boldsymbol{r}_{N-1} and a center of mass coordinate \boldsymbol{r}_N, from iteratively reducing two-body systems within the many- body system. In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. Electrostatic separation is a preferred sorting method when dealing with separating conductors from electrostatic separation non- conductors. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. Generally, electrostatic charges are used to attract or repel differently charged material.Science Direct.com When electrostatic separation uses the force of attraction to sort particles, conducting particles stick to an oppositely charged object, such as a metal drum, thereby separating them from the particle mixture. The order of children indicates the relative coordinate points > from xk to xj. For example, the surface charge of adsorbent is described by the ion that lies on the surface of the particle (adsorbent) structure like image. In a similar way to that in which electrostatic separation sorts particles with different electrostatic charges magnetic beneficiation sorts particles that respond to a magnetic field. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. A negative trion consists of two electrons and one hole and a positive trion consists of two holes and one electron. Surface Charging and Points of Zero Charge. The charged particle beams that can be manipulated in particle accelerators can be subdivided into electron beams, ion beams and proton beams. ==Common types== * Electron beam, or cathode ray, such as in a scanning electron microscope or in accelerators such as the Large Electron–Positron Collider or synchrotron light sources. An industrial process used to separate large amounts of material particles, electrostatic separating is most often used in the process of sorting mineral ore. Experiments showing electrostatic sorting in action can help make the process more clear. ",0.9731,313,3.0,0.14,12,C
+","The position coordinates xj and xk are replaced > by their relative position rjk = xj − xk and by the vector to their center > of mass Rjk = (mj qj \+ mkqk)/(mj \+ mk). thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. A trion is a localized excitation which consists of three charged particles. Electrostatic separation is a process that uses electrostatic charges to separate crushed particles of material. An electrostatic separator is a device for separating particles by mass in a low energy charged beam. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. thumb|540x540px|Oppositely charged particles interact as they are moved through a column. This way, when a mixture of particles falls past a repelling object, the particles with the correct charge fall away from the other particles when they are repelled by the similarly charged object. The vector \boldsymbol{r}_N is the center of mass of all the bodies and \boldsymbol{r}_1 is the relative coordinate between the particles 1 and 2: The result one is left with is thus a system of N-1 translationally invariant coordinates \boldsymbol{r}_1, \dots, \boldsymbol{r}_{N-1} and a center of mass coordinate \boldsymbol{r}_N, from iteratively reducing two-body systems within the many- body system. In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. Electrostatic separation is a preferred sorting method when dealing with separating conductors from electrostatic separation non- conductors. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. Generally, electrostatic charges are used to attract or repel differently charged material.Science Direct.com When electrostatic separation uses the force of attraction to sort particles, conducting particles stick to an oppositely charged object, such as a metal drum, thereby separating them from the particle mixture. The order of children indicates the relative coordinate points > from xk to xj. For example, the surface charge of adsorbent is described by the ion that lies on the surface of the particle (adsorbent) structure like image. In a similar way to that in which electrostatic separation sorts particles with different electrostatic charges magnetic beneficiation sorts particles that respond to a magnetic field. These coordinates are particularly common in treating polyatomic molecules and chemical reactions, and in celestial mechanics. A negative trion consists of two electrons and one hole and a positive trion consists of two holes and one electron. Surface Charging and Points of Zero Charge. The charged particle beams that can be manipulated in particle accelerators can be subdivided into electron beams, ion beams and proton beams. ==Common types== * Electron beam, or cathode ray, such as in a scanning electron microscope or in accelerators such as the Large Electron–Positron Collider or synchrotron light sources. An industrial process used to separate large amounts of material particles, electrostatic separating is most often used in the process of sorting mineral ore. Experiments showing electrostatic sorting in action can help make the process more clear. ",0.9731,313,"""3.0""",0.14,12,C
 "Two charged particles are fixed to an $x$ axis: Particle 1 of charge $q_1=2.1 \times 10^{-8} \mathrm{C}$ is at position $x=20 \mathrm{~cm}$ and particle 2 of charge $q_2=-4.00 q_1$ is at position $x=70 \mathrm{~cm}$. At what coordinate on the axis (other than at infinity) is the net electric field produced by the two particles equal to zero?
-","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The particle located experiences an interaction with the electric field. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . Surface Charging and Points of Zero Charge. Electric field work is the work performed by an electric field on a charged particle in its vicinity. In the example both charges are positive; this equation is applicable to any charge configuration (as the product of the charges will be either positive or negative according to their (dis)similarity). 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. To move q+ closer to Q+ (starting from r_0 = \infty , where the potential energy=0, for convenience), we would have to apply an external force against the Coulomb field and positive work would be performed. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. : W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{Q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r} where :Q is the electric charge of the particle :E is the electric field, which at a location is the force at that location divided by a unit ('test') charge :FE is the Coulomb (electric) force :r is the displacement :\cdot is the dot product operator ==Mathematical description== Given a charged object in empty space, Q+. In physics, a charged particle is a particle with an electric charge. thumb|Circuit diagram of a charge qubit circuit. As the field lines are pulled together tightly by gluons, they do not ""bow"" outwards as much as an electric field between electric charges. The work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in electric potential at those points. The modern version of these equations is called Maxwell's equations. ====Electrostatics==== A charged test particle with charge q experiences a force F based solely on its charge. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. So, integrating and using Coulomb's Law for the force: :U(r) = \Delta U = -\int_{r_0}^{r} \mathbf{F}_{ext} \cdot \, d \mathbf{r}= -\int_{r_0}^{r} \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{\mathbf{r^2}} \cdot \, d \mathbf{r}= \frac{q_1q_2}{4\pi\varepsilon_0}\left(\frac{1}{r_0}- \frac{1}{r}\right) = -\frac{q_1q_2}{4\pi\varepsilon_0} \frac{1}{r} Now, use the relationship : W = -\Delta U \\! A related concept in electrochemistry is the electrode potential at the point of zero charge. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta{\left( \mathbf r - \mathbf r_2 \right)}, which has a Fourier transform \mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). thumb|Electrical double layer (EDL) around a negatively charged particle in suspension in water. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. ",200,5040,4.86,-30,0.6321205588,D
-The volume charge density of a solid nonconducting sphere of radius $R=5.60 \mathrm{~cm}$ varies with radial distance $r$ as given by $\rho=$ $\left(14.1 \mathrm{pC} / \mathrm{m}^3\right) r / R$. What is the sphere's total charge?,"Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into ""free"" and ""bound"" charges. The volume can also be expressed in terms of A_n, the area of the unit -sphere. == Formulas == The first volumes are as follows: Dimension Volume of a ball of radius Radius of a ball of volume 0 1 (all 0-balls have volume 1) 1 2R \frac{V}{2}=0.5\times V 2 \pi R^2 \approx 3.142\times R^2 \frac{V^{1/2}}{\sqrt{\pi}}\approx 0.564\times V^{\frac{1}{2}} 3 \frac{4\pi}{3} R^3 \approx 4.189\times R^3 \left(\frac{3V}{4\pi}\right)^{1/3}\approx 0.620\times V^{1/3} 4 \frac{\pi^2}{2} R^4 \approx 4.935\times R^4 \frac{(2V)^{1/4}}{\sqrt{\pi}}\approx 0.671\times V^{1/4} 5 \frac{8\pi^2}{15} R^5\approx 5.264\times R^5 \left(\frac{15V}{8\pi^2}\right)^{1/5}\approx 0.717\times V^{1/5} 6 \frac{\pi^3}{6} R^6 \approx 5.168 \times R^6 \frac{(6V)^{1/6}}{\sqrt{\pi}}\approx 0.761\times V^{1/6} 7 \frac{16\pi^3}{105} R^7 \approx 4.725\times R^7 \left(\frac{105V}{16\pi^3}\right)^{1/7}\approx 0.801\times V^{1/7} 8 \frac{\pi^4}{24} R^8 \approx 4.059\times R^8 \frac{(24V)^{1/8}}{\sqrt{\pi}}\approx 0.839\times V^{1/8} 9 \frac{32\pi^4}{945} R^9 \approx 3.299\times R^9 \left(\frac{945V}{32\pi^4}\right)^{1/9}\approx 0.876\times V^{1/9} 10 \frac{\pi^5}{120} R^{10} \approx 2.550\times R^{10} \frac{(120V)^{1/10}}{\sqrt{\pi}}\approx 0.911\times V^{1/10} 11 \frac{64\pi^5}{10395} R^{11} \approx 1.884\times R^{11} \left(\frac{10395V}{64\pi^5}\right)^{1/11}\approx 0.944\times V^{1/11} 12 \frac{\pi^6}{720} R^{12} \approx 1.335\times R^{12} \frac{(720V)^{1/12}}{\sqrt{\pi}}\approx 0.976\times V^{1/12} 13 \frac{128\pi^6}{135135} R^{13} \approx 0.911\times R^{13} \left(\frac{135135V}{128\pi^6}\right)^{1/13}\approx 1.007\times V^{1/13} 14 \frac{\pi^7}{5040} R^{14} \approx 0.599\times R^{14} \frac{(5040V)^{1/14}}{\sqrt{\pi}}\approx 1.037\times V^{1/14} 15 \frac{256\pi^7}{2027025} R^{15} \approx 0.381\times R^{15} \left(\frac{2027025V}{256\pi^7}\right)^{1/15}\approx 1.066\times V^{1/15} n Vn(R) Rn(V) === Two-dimension recurrence relation === As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation: : V_n(R) = \begin{cases} 1 &\text{if } n=0,\\\\[0.5ex] 2R &\text{if } n=1,\\\\[0.5ex] \dfrac{2\pi}{n}R^2 \times V_{n-2}(R) &\text{otherwise}. \end{cases} This allows computation of in approximately steps. === Closed form === The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06. The volume of a -ball of radius is R^nV_n, where V_n is the volume of the unit -ball, the -ball of radius . The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then: :A_{n-1}(r) = r^{n-1} A_{n-1}(1). In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. However, if then the correction factor is : the surface area of an sphere of radius in is times the derivative of the volume of an ball. The radius can also be calculated as :r_{\rm s}= \left(\frac{3M}{4\pi Z \rho N_{\rm A}}\right)^\frac{1}{3}\,, where M is molar mass, Z is amount of free electrons per atom, \rho is mass density, and N_{\rm A} is the Avogadro constant. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. The volume of an odd-dimensional ball is :V_{2k+1}(R) = \frac{2(2\pi)^k}{(2k + 1)!!}R^{2k+1}. The volume of an ball of radius is :V^p_n(R) = \frac{\Bigl(2\,\Gamma\bigl(\tfrac1p + 1\bigr)\Bigr)^n}{\Gamma\bigl(\tfrac np + 1\bigr)}R^n. To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation : :V_n(R) = \int_0^R \frac{2\pi^{n/2}}{\Gamma\bigl(\tfrac n2\bigr)} \,r^{n-1}\,dr = \frac{2\pi^{n/2}}{n\,\Gamma\bigl(\tfrac n2\bigr)}R^n = \frac{\pi^{n/2}}{\Gamma\bigl(\tfrac n2 + 1\bigr)}R^n. === Geometric proof === The relations V_{n+1}(R) = \frac{R}{n+1}A_n(R) and A_{n+1}(R) = (2\pi R)V_n(R) and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. ""How Small Is a Unit Ball?"", Mathematics Magazine, Volume 62, Issue 2, 1989, pp. 101–107, https://doi.org/10.1080/0025570X.1989.11977419. === Relation with surface area === Let denote the hypervolume of the -sphere of radius . The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by: :A_{n-1}(R) = \frac{d}{dR} V_{n}(R) = \frac{n}{R}V_{n}(R). The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. As always, the integral of the charge density over a region of space is the charge contained in that region. Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume: :\begin{align} R_n(V) &= \frac{\Gamma\bigl(\tfrac n2 + 1\bigr)^{1/n}}{\sqrt{\pi}}V^{1/n} \\\ &= \left(\frac{n!! This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... Also, A_{n-1}(R) = \frac{dV_n(R)}{dR} because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. ",0.332,170,0.6,7.78,292,D
+","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The particle located experiences an interaction with the electric field. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . Surface Charging and Points of Zero Charge. Electric field work is the work performed by an electric field on a charged particle in its vicinity. In the example both charges are positive; this equation is applicable to any charge configuration (as the product of the charges will be either positive or negative according to their (dis)similarity). 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. To move q+ closer to Q+ (starting from r_0 = \infty , where the potential energy=0, for convenience), we would have to apply an external force against the Coulomb field and positive work would be performed. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. : W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{Q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r} where :Q is the electric charge of the particle :E is the electric field, which at a location is the force at that location divided by a unit ('test') charge :FE is the Coulomb (electric) force :r is the displacement :\cdot is the dot product operator ==Mathematical description== Given a charged object in empty space, Q+. In physics, a charged particle is a particle with an electric charge. thumb|Circuit diagram of a charge qubit circuit. As the field lines are pulled together tightly by gluons, they do not ""bow"" outwards as much as an electric field between electric charges. The work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in electric potential at those points. The modern version of these equations is called Maxwell's equations. ====Electrostatics==== A charged test particle with charge q experiences a force F based solely on its charge. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. So, integrating and using Coulomb's Law for the force: :U(r) = \Delta U = -\int_{r_0}^{r} \mathbf{F}_{ext} \cdot \, d \mathbf{r}= -\int_{r_0}^{r} \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{\mathbf{r^2}} \cdot \, d \mathbf{r}= \frac{q_1q_2}{4\pi\varepsilon_0}\left(\frac{1}{r_0}- \frac{1}{r}\right) = -\frac{q_1q_2}{4\pi\varepsilon_0} \frac{1}{r} Now, use the relationship : W = -\Delta U \\! A related concept in electrochemistry is the electrode potential at the point of zero charge. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta{\left( \mathbf r - \mathbf r_2 \right)}, which has a Fourier transform \mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). thumb|Electrical double layer (EDL) around a negatively charged particle in suspension in water. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. ",200,5040,"""4.86""",-30,0.6321205588,D
+The volume charge density of a solid nonconducting sphere of radius $R=5.60 \mathrm{~cm}$ varies with radial distance $r$ as given by $\rho=$ $\left(14.1 \mathrm{pC} / \mathrm{m}^3\right) r / R$. What is the sphere's total charge?,"Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into ""free"" and ""bound"" charges. The volume can also be expressed in terms of A_n, the area of the unit -sphere. == Formulas == The first volumes are as follows: Dimension Volume of a ball of radius Radius of a ball of volume 0 1 (all 0-balls have volume 1) 1 2R \frac{V}{2}=0.5\times V 2 \pi R^2 \approx 3.142\times R^2 \frac{V^{1/2}}{\sqrt{\pi}}\approx 0.564\times V^{\frac{1}{2}} 3 \frac{4\pi}{3} R^3 \approx 4.189\times R^3 \left(\frac{3V}{4\pi}\right)^{1/3}\approx 0.620\times V^{1/3} 4 \frac{\pi^2}{2} R^4 \approx 4.935\times R^4 \frac{(2V)^{1/4}}{\sqrt{\pi}}\approx 0.671\times V^{1/4} 5 \frac{8\pi^2}{15} R^5\approx 5.264\times R^5 \left(\frac{15V}{8\pi^2}\right)^{1/5}\approx 0.717\times V^{1/5} 6 \frac{\pi^3}{6} R^6 \approx 5.168 \times R^6 \frac{(6V)^{1/6}}{\sqrt{\pi}}\approx 0.761\times V^{1/6} 7 \frac{16\pi^3}{105} R^7 \approx 4.725\times R^7 \left(\frac{105V}{16\pi^3}\right)^{1/7}\approx 0.801\times V^{1/7} 8 \frac{\pi^4}{24} R^8 \approx 4.059\times R^8 \frac{(24V)^{1/8}}{\sqrt{\pi}}\approx 0.839\times V^{1/8} 9 \frac{32\pi^4}{945} R^9 \approx 3.299\times R^9 \left(\frac{945V}{32\pi^4}\right)^{1/9}\approx 0.876\times V^{1/9} 10 \frac{\pi^5}{120} R^{10} \approx 2.550\times R^{10} \frac{(120V)^{1/10}}{\sqrt{\pi}}\approx 0.911\times V^{1/10} 11 \frac{64\pi^5}{10395} R^{11} \approx 1.884\times R^{11} \left(\frac{10395V}{64\pi^5}\right)^{1/11}\approx 0.944\times V^{1/11} 12 \frac{\pi^6}{720} R^{12} \approx 1.335\times R^{12} \frac{(720V)^{1/12}}{\sqrt{\pi}}\approx 0.976\times V^{1/12} 13 \frac{128\pi^6}{135135} R^{13} \approx 0.911\times R^{13} \left(\frac{135135V}{128\pi^6}\right)^{1/13}\approx 1.007\times V^{1/13} 14 \frac{\pi^7}{5040} R^{14} \approx 0.599\times R^{14} \frac{(5040V)^{1/14}}{\sqrt{\pi}}\approx 1.037\times V^{1/14} 15 \frac{256\pi^7}{2027025} R^{15} \approx 0.381\times R^{15} \left(\frac{2027025V}{256\pi^7}\right)^{1/15}\approx 1.066\times V^{1/15} n Vn(R) Rn(V) === Two-dimension recurrence relation === As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation: : V_n(R) = \begin{cases} 1 &\text{if } n=0,\\\\[0.5ex] 2R &\text{if } n=1,\\\\[0.5ex] \dfrac{2\pi}{n}R^2 \times V_{n-2}(R) &\text{otherwise}. \end{cases} This allows computation of in approximately steps. === Closed form === The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06. The volume of a -ball of radius is R^nV_n, where V_n is the volume of the unit -ball, the -ball of radius . The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then: :A_{n-1}(r) = r^{n-1} A_{n-1}(1). In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. However, if then the correction factor is : the surface area of an sphere of radius in is times the derivative of the volume of an ball. The radius can also be calculated as :r_{\rm s}= \left(\frac{3M}{4\pi Z \rho N_{\rm A}}\right)^\frac{1}{3}\,, where M is molar mass, Z is amount of free electrons per atom, \rho is mass density, and N_{\rm A} is the Avogadro constant. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. The volume of an odd-dimensional ball is :V_{2k+1}(R) = \frac{2(2\pi)^k}{(2k + 1)!!}R^{2k+1}. The volume of an ball of radius is :V^p_n(R) = \frac{\Bigl(2\,\Gamma\bigl(\tfrac1p + 1\bigr)\Bigr)^n}{\Gamma\bigl(\tfrac np + 1\bigr)}R^n. To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation : :V_n(R) = \int_0^R \frac{2\pi^{n/2}}{\Gamma\bigl(\tfrac n2\bigr)} \,r^{n-1}\,dr = \frac{2\pi^{n/2}}{n\,\Gamma\bigl(\tfrac n2\bigr)}R^n = \frac{\pi^{n/2}}{\Gamma\bigl(\tfrac n2 + 1\bigr)}R^n. === Geometric proof === The relations V_{n+1}(R) = \frac{R}{n+1}A_n(R) and A_{n+1}(R) = (2\pi R)V_n(R) and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. ""How Small Is a Unit Ball?"", Mathematics Magazine, Volume 62, Issue 2, 1989, pp. 101–107, https://doi.org/10.1080/0025570X.1989.11977419. === Relation with surface area === Let denote the hypervolume of the -sphere of radius . The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by: :A_{n-1}(R) = \frac{d}{dR} V_{n}(R) = \frac{n}{R}V_{n}(R). The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. As always, the integral of the charge density over a region of space is the charge contained in that region. Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume: :\begin{align} R_n(V) &= \frac{\Gamma\bigl(\tfrac n2 + 1\bigr)^{1/n}}{\sqrt{\pi}}V^{1/n} \\\ &= \left(\frac{n!! This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... Also, A_{n-1}(R) = \frac{dV_n(R)}{dR} because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. ",0.332,170,"""0.6""",7.78,292,D
 "Two charged concentric spher-
-ical shells have radii $10.0 \mathrm{~cm}$ and $15.0 \mathrm{~cm}$. The charge on the inner shell is $4.00 \times 10^{-8} \mathrm{C}$, and that on the outer shell is $2.00 \times 10^{-8} \mathrm{C}$. Find the electric field at $r=12.0 \mathrm{~cm}$.","The ""spherium"" model consists of two electrons trapped on the surface of a sphere of radius R. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. As before, the Rabi problem is solved by assuming that the electric field E is oscillatory with constant magnitude E0: E = E_0 (e^{i\omega t} + \text{c.c.}). thumb|right|FitzHugh-Nagumo model in phase space, with a = 0.7, b = 0.8, \tau = 12.5, R = 0.1, I_{ext} = 0.5. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It follows that the exact correlation energy for R = \sqrt{3}/2 is E_{\rm corr} = 1-2/\sqrt{3} \approx -0.1547 which is much larger than the limiting correlation energies of the helium-like ions (-0.0467) or Hooke's atoms (-0.0497). The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Values from other sources may differ significantly (see text) The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. To apply this to the Rabi problem, one assumes that the electric field E is oscillatory in time and constant in space: : E = E_0[e^{i\omega t} + e^{-i\omega t}], and xa is decomposed into a part ua that is in-phase with the driving E field (corresponding to dispersion) and a part va that is out of phase (corresponding to absorption): : x_a = x_0 (u_a \cos \omega t + v_a \sin \omega t). Gallery figures: FitzHugh-Nagumo model, with a = 0.7, \tau = 12.5, R = 0.1, and varying b, I_{ext}. This confirms the view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space. ==Spherium on a 3-sphere== Loos and Gill considered the case of two electrons confined to a 3-sphere repelling Coulombically. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. * For N = 12, electrons reside at the vertices of a regular icosahedron. , Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles. Dividing by the Avogadro constant gives V = = 0.8685 Å, corresponding to r = 0.59 Å. === Polarizability === The polarizability α of a gas is related to its electric susceptibility χ by the relation \alpha = {\varepsilon_0 k_{\rm B}T\over p}\chi_{\rm e} and the electric susceptibility may be calculated from tabulated values of the relative permittivity ε using the relation χ = ε − 1\. The single electron may reside at any point on the surface of the unit sphere. Configurations reproduced in * * * This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Category:Electrostatics Category:Electron Category:Circle packing Category:Unsolved problems in mathematics ",2.50,0.396,0.0182,0.00539,0.686,A
-Assume that a honeybee is a sphere of diameter 1.000 $\mathrm{cm}$ with a charge of $+45.0 \mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \mu \mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \mathrm{pC}$ on the far side. What is the magnitude of the net electrostatic force on the grain due to the bee? ,"thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. If r is the distance (in meters) between two charges, then the force (in newtons) between two point charges q and Q (in coulombs) is: :F = \frac{1}{4\pi \varepsilon_0}\frac{qQ}{r^2}= k_\text{e}\frac{qQ}{r^2}\, , where ε0 is the vacuum permittivity, or permittivity of free space: :\varepsilon_0 \approx \mathrm{8.854\ 187\ 817 \times 10^{-12} ~C^2{\cdot}N^{-1}{\cdot}m^{-2}}. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This force is the average of the discontinuous electric field at the surface charge. Adult worker honey bees consume 3.4–4.3 mg of pollen per day to meet a dry matter requirement of 66–74% protein. A drone is a male honey bee. A honey bee (also spelled honeybee) is a eusocial flying insect within the genus Apis of the bee clade, all native to mainland Afro-Eurasia. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. Defense can vary based on the habitat of the bee. According to Nikolaides, the electrostatic force engenders a long range capillary attraction. Cambridge, Massachusetts and London, England: Harvard University Press.. ==Pollination== thumb|Hind leg of a honey bee with pollen pellet stuck on the pollen basket or corbicula. Adult worker honey bees require 4 mg of utilizable sugars per day and larvae require about 59.4 mg of carbohydrates for proper development. Electrostatic phenomena arise from the forces that electric charges exert on each other. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The familiar phenomenon of a static ""shock"" is caused by the neutralization of charge built up in the body from contact with insulated surfaces. ==Coulomb's law== Coulomb's law states that: 'The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.' In this case, the surface charge density decreases upon approach. Drones depend on worker bees to feed them. ",4,1.8763,-1.49,2.6,7,D
+ical shells have radii $10.0 \mathrm{~cm}$ and $15.0 \mathrm{~cm}$. The charge on the inner shell is $4.00 \times 10^{-8} \mathrm{C}$, and that on the outer shell is $2.00 \times 10^{-8} \mathrm{C}$. Find the electric field at $r=12.0 \mathrm{~cm}$.","The ""spherium"" model consists of two electrons trapped on the surface of a sphere of radius R. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. As before, the Rabi problem is solved by assuming that the electric field E is oscillatory with constant magnitude E0: E = E_0 (e^{i\omega t} + \text{c.c.}). thumb|right|FitzHugh-Nagumo model in phase space, with a = 0.7, b = 0.8, \tau = 12.5, R = 0.1, I_{ext} = 0.5. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. The Rabi problem concerns the response of an atom to an applied harmonic electric field, with an applied frequency very close to the atom's natural frequency. It follows that the exact correlation energy for R = \sqrt{3}/2 is E_{\rm corr} = 1-2/\sqrt{3} \approx -0.1547 which is much larger than the limiting correlation energies of the helium-like ions (-0.0467) or Hooke's atoms (-0.0497). The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Values from other sources may differ significantly (see text) The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. To apply this to the Rabi problem, one assumes that the electric field E is oscillatory in time and constant in space: : E = E_0[e^{i\omega t} + e^{-i\omega t}], and xa is decomposed into a part ua that is in-phase with the driving E field (corresponding to dispersion) and a part va that is out of phase (corresponding to absorption): : x_a = x_0 (u_a \cos \omega t + v_a \sin \omega t). Gallery figures: FitzHugh-Nagumo model, with a = 0.7, \tau = 12.5, R = 0.1, and varying b, I_{ext}. This confirms the view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space. ==Spherium on a 3-sphere== Loos and Gill considered the case of two electrons confined to a 3-sphere repelling Coulombically. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. * For N = 12, electrons reside at the vertices of a regular icosahedron. , Geometric solutions of the Thomson problem for N = 4, 6, and 12 electrons are Platonic solids whose faces are all congruent equilateral triangles. Dividing by the Avogadro constant gives V = = 0.8685 Å, corresponding to r = 0.59 Å. === Polarizability === The polarizability α of a gas is related to its electric susceptibility χ by the relation \alpha = {\varepsilon_0 k_{\rm B}T\over p}\chi_{\rm e} and the electric susceptibility may be calculated from tabulated values of the relative permittivity ε using the relation χ = ε − 1\. The single electron may reside at any point on the surface of the unit sphere. Configurations reproduced in * * * This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Category:Electrostatics Category:Electron Category:Circle packing Category:Unsolved problems in mathematics ",2.50,0.396,"""0.0182""",0.00539,0.686,A
+Assume that a honeybee is a sphere of diameter 1.000 $\mathrm{cm}$ with a charge of $+45.0 \mathrm{pC}$ uniformly spread over its surface. Assume also that a spherical pollen grain of diameter $40.0 \mu \mathrm{m}$ is electrically held on the surface of the bee because the bee's charge induces a charge of $-1.00 \mathrm{pC}$ on the near side of the grain and a charge of $+1.00 \mathrm{pC}$ on the far side. What is the magnitude of the net electrostatic force on the grain due to the bee? ,"thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). Electrostatics involves the buildup of charge on the surface of objects due to contact with other surfaces. If r is the distance (in meters) between two charges, then the force (in newtons) between two point charges q and Q (in coulombs) is: :F = \frac{1}{4\pi \varepsilon_0}\frac{qQ}{r^2}= k_\text{e}\frac{qQ}{r^2}\, , where ε0 is the vacuum permittivity, or permittivity of free space: :\varepsilon_0 \approx \mathrm{8.854\ 187\ 817 \times 10^{-12} ~C^2{\cdot}N^{-1}{\cdot}m^{-2}}. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This force is the average of the discontinuous electric field at the surface charge. Adult worker honey bees consume 3.4–4.3 mg of pollen per day to meet a dry matter requirement of 66–74% protein. A drone is a male honey bee. A honey bee (also spelled honeybee) is a eusocial flying insect within the genus Apis of the bee clade, all native to mainland Afro-Eurasia. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. Defense can vary based on the habitat of the bee. According to Nikolaides, the electrostatic force engenders a long range capillary attraction. Cambridge, Massachusetts and London, England: Harvard University Press.. ==Pollination== thumb|Hind leg of a honey bee with pollen pellet stuck on the pollen basket or corbicula. Adult worker honey bees require 4 mg of utilizable sugars per day and larvae require about 59.4 mg of carbohydrates for proper development. Electrostatic phenomena arise from the forces that electric charges exert on each other. Even though electrostatically induced forces seem to be rather weak, some electrostatic forces are relatively large. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The familiar phenomenon of a static ""shock"" is caused by the neutralization of charge built up in the body from contact with insulated surfaces. ==Coulomb's law== Coulomb's law states that: 'The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.' In this case, the surface charge density decreases upon approach. Drones depend on worker bees to feed them. ",4,1.8763,"""-1.49""",2.6,7,D
 "In the radioactive decay of Eq. 21-13, $\mathrm{a}^{238} \mathrm{U}$ nucleus transforms to ${ }^{234} \mathrm{Th}$ and an ejected ${ }^4 \mathrm{He}$. (These are nuclei, not atoms, and thus electrons are not involved.) When the separation between ${ }^{234} \mathrm{Th}$ and ${ }^4 \mathrm{He}$ is $9.0 \times 10^{-15} \mathrm{~m}$, what are the magnitudes of the electrostatic force between them?
-","See also: H. Geiger and J.M. Nuttall (1912) ""The ranges of α particles from uranium,"" Philosophical Magazine, Series 6, vol. 23, no. 135, pages 439-445. in its modern form the Geiger–Nuttall law is :\log_{10}T_{1/2}=\frac{A(Z)}{\sqrt{E}}+B(Z) where T_{1/2} is the half-life, E the total kinetic energy (of the alpha particle and the daughter nucleus), and A and B are coefficients that depend on the isotope's atomic number Z. The nuclear force is powerfully attractive between nucleons at distances of about 0.8 femtometre (fm, or 0.8×10−15 metre), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge. At distances less than 0.7 fm, the nuclear force becomes repulsive. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). In this case, the effective nuclear charge can be calculated by Coulomb's law. In atomic physics, the effective nuclear charge is the actual amount of positive (nuclear) charge experienced by an electron in a multi-electron atom. thumb|right|300px|Woods–Saxon potential for , relative to V0 with a and R=4.6 fm The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. In nuclear physics, the Geiger–Nuttall law or Geiger–Nuttall rule relates the decay constant of a radioactive isotope with the energy of the alpha particles emitted. In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase-shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. ==The nuclear force as a residual of the strong force== The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. * Nucleons near the surface of the nucleus (i.e. having within a distance of order a) experience a large force towards the center. There are only strong attractions when the total isospin of the set of interacting particles is 0, which is confirmed by experiment.Griffiths, David, Introduction to Elementary Particles Our understanding of the nuclear force is obtained by scattering experiments and the binding energy of light nuclei. At short distances (less than 1.7 fm or so), the attractive nuclear force is stronger than the repulsive Coulomb force between protons; it thus overcomes the repulsion of protons within the nucleus. In contrast, the effective nuclear charge is the attractive positive charge of nuclear protons acting on valence electrons, which is always less than the total number of protons present in a nucleus due to the shielding effect. ==See also== * Atomic orbitals * Core charge * d-block contraction (or scandide contraction) * Electronegativity * Lanthanide contraction * Shielding effect * Slater-type orbitals * Valence electrons * Weak charge ==References== ==Resources== *Brown, Theodore; intekhab khan, H.E.; & Bursten, Bruce (2002). Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. thumb|A conjectural example of an interaction between two neutrons and a proton, the triton or hydrogen-3, which is beta unstable. However, electrons further away are screened from the nucleus by other electrons in between, and feel less electrostatic interaction as a result. Nuclear interaction length is the mean distance travelled by a hadronic particle before undergoing an inelastic nuclear interaction. ==See also== *Nuclear collision length *Radiation length ==External links== *Particle Data Group site Category:Experimental particle physics These nuclear forces are very weak compared to direct gluon forces (""color forces"" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. ",-2,449,5.1,1.8,228,C
-"The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$ at the point $(3.0,3.0) \mathrm{cm}$ and $100 \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$ at the point $(2.0,0) \mathrm{cm}$. What is the $x$ coordinate of the particle?","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. In Cartesian coordinate system \vec{r}=x\,\hat{e}_x+y\,\hat{e}_y+z\,\hat{e}_z. Several other definitions are in use, and so care must be taken in comparing different sources.Wolfram Mathworld, spherical coordinates == Cylindrical coordinate system == === Vector fields === Vectors are defined in cylindrical coordinates by (ρ, φ, z), where * ρ is the length of the vector projected onto the xy- plane, * φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), * z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: \begin{bmatrix} \rho \\\ \phi \\\ z \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\\ \operatorname{arctan}(y / x) \\\ z \end{bmatrix},\ \ \ 0 \le \phi < 2\pi, thumb or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} \rho\cos\phi \\\ \rho\sin\phi \\\ z \end{bmatrix}. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. This is the intersection between the reference plane and the axis. The dot is the point with radial distance , angular coordinate , and height . * The azimuth is the angle between the reference direction on the chosen plane and the line from the origin to the projection of on the plane. The coordinates of the vector r with respect to the basis vectors ei are xi. The relative direction between two points is their relative position normalized as a unit vector: :\Delta \mathbf{\hat{r}}=\Delta \mathbf{r} / \|\Delta \mathbf{r}\| where the denominator is the distance between the two points, \| \Delta \mathbf{r} \|. ==Definition== ===Three dimensions=== In three dimensions, any set of three- dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used. Any vector field can be written in terms of the unit vectors as: \mathbf A = A_x \mathbf{\hat x} + A_y \mathbf{\hat y} + A_z \mathbf{\hat z} = A_\rho \mathbf{\hat \rho} + A_\phi \boldsymbol{\hat \phi} + A_z \mathbf{\hat z} The cylindrical unit vectors are related to the Cartesian unit vectors by: \begin{bmatrix}\mathbf{\hat \rho} \\\ \boldsymbol{\hat\phi} \\\ \mathbf{\hat z}\end{bmatrix} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \\\ -\sin\phi & \cos\phi & 0 \\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The line element is \mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}. Then the -coordinate is the same in both systems, and the correspondence between cylindrical and Cartesian are the same as for polar coordinates, namely \begin{align} x &= \rho \cos \varphi \\\ y &= \rho \sin \varphi \\\ z &= z \end{align} in one direction, and \begin{align} \rho &= \sqrt{x^2+y^2} \\\ \varphi &= \begin{cases} \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\\ \arcsin\left(\frac{y}{\rho}\right) & \text{if } x \geq 0 \\\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y \ge 0\\\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y < 0 \end{cases} \end{align} in the other. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position, or axial position. A cylindrical coordinate system is a three- dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section). Its acceleration is \boldsymbol{a} = \frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \left( \ddot{\rho} - \rho\,\dot\varphi^2 \right)\boldsymbol{\hat \rho} + \left( 2\dot{\rho}\,\dot\varphi + \rho\,\ddot\varphi \right) \hat{\boldsymbol\varphi } + \ddot{z}\,\hat{\boldsymbol{z}} ==See also== *List of canonical coordinate transformations *Vector fields in cylindrical and spherical coordinates *Del in cylindrical and spherical coordinates ==References== ==Further reading== * * * * * * ==External links== * *MathWorld description of cylindrical coordinates *Cylindrical Coordinates Animations illustrating cylindrical coordinates by Frank Wattenberg Category:Three- dimensional coordinate systems Category:Orthogonal coordinate systems de:Polarkoordinaten#Zylinderkoordinaten ro:Coordonate polare#Coordonate cilindrice fi:Koordinaatisto#Sylinterikoordinaatisto The inverse projection is then given by: :\begin{align}\varphi &= \frac{\pi}{2} - \rho \\\ \lambda &=\frac{E \rho} {\sin \varphi_1 \sin \rho} \end{align} where :\rho = \sqrt{ (x\sin \varphi_1)^2 + \left(\varphi_1 - y + \cot \varphi_1\right)^2 }, \qquad E= \tan^{-1}\left(\frac{x\sin \varphi_1}{\varphi_1 - y + \cot \varphi_1}\right). thumb|Radius vector \vec{r} represents the position of a point \mathrm{P}(x,y,z) with respect to origin O. thumb|240px|A cylindrical coordinate system with origin , polar axis , and longitudinal axis . ",0.0625,35,-1.0,344,54.394,C
-"A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude $E=K r^4$, directed radially outward from the center of the sphere. Here $r$ is the radial distance from that center, and $K$ is a constant. What is the volume density $\rho$ of the charge distribution?","Let the first charge distribution \rho_1(\mathbf{r}') be centered on the origin and lie entirely within the second charge distribution \rho_2(\mathbf{r}'). In the interior case, where r' > r, the result is: \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^\ell I_{\ell m} r^\ell \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell m}(\theta, \phi) , with the interior multipole moments defined as I_{\ell m} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \frac{\rho(\mathbf{r}')}{\left( r' \right)^{\ell+1}} \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell m}^{*}(\theta', \phi'). ==Interaction energies of spherical multipoles== A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. As always, the integral of the charge density over a region of space is the charge contained in that region. We also use spherical coordinates throughout, e.g., the vector \mathbf r' has coordinates ( r', \theta', \phi') where r' is the radius, \theta' is the colatitude and \phi' is the azimuthal angle. ==Spherical multipole moments of a point charge== thumb|right|Figure 1: Definitions for the spherical multipole expansion The electric potential due to a point charge located at \mathbf{r'} is given by \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^2 + r^{\prime 2} - 2 r' r \cos \gamma}}. where R \ \stackrel{\mathrm{def}}{=}\ \left|\mathbf{r} - \mathbf{r'} \right| is the distance between the charge position and the observation point and \gamma is the angle between the vectors \mathbf{r} and \mathbf{r'}. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. For three dimensions, this normalization is the number density of the system ( \rho ) multiplied by the volume of the spherical shell, which symbolically can be expressed as \rho \, 4\pi r^2 dr. Using the mathematical identity P_\ell(\cos \theta) \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell 0}(\theta, \phi) the exterior multipole expansion becomes \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \left( \frac{Q_\ell}{r^{\ell+1}} \right) P_\ell(\cos \theta) where the axially symmetric multipole moments are defined Q_\ell \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \rho(\mathbf{r}') \left( r' \right)^\ell P_\ell(\cos \theta') In the limit that the charge is confined to the z-axis, we recover the exterior axial multipole moments. For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density \rho(\mathbf r'). Similarly the interior multipole expansion becomes \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} I_\ell r^\ell P_\ell(\cos \theta) where the axially symmetric interior multipole moments are defined I_\ell \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \frac{\rho(\mathbf{r}')}{\left( r' \right)^{\ell+1}} P_\ell(\cos \theta') In the limit that the charge is confined to the z-axis, we recover the interior axial multipole moments. ==See also== * Solid harmonics * Laplace expansion * Multipole expansion * Legendre polynomials * Axial multipole moments * Cylindrical multipole moments Category:Electromagnetism Category:Potential theory Category:Moment (physics) In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. Taking particle 0 as fixed at the origin of the coordinates, \textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r}) is the average number of particles (among the remaining N-1) to be found in the volume \textstyle d^3r around the position \textstyle \mathbf{r}. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein-Zernike equation or structure-optimized potential refinement. ==Definition== Consider a system of N particles in a volume V (for an average number density \rho =N/V) and at a temperature T (let us also define \textstyle \beta = \frac{1}{kT}). This is expressed by a continuity equation which links the rate of change of charge density \rho(\boldsymbol{x}) and the current density \boldsymbol{J}(\boldsymbol{x}). Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. The delta function has the sifting property for any function f: \int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0) so the delta function ensures that when the charge density is integrated over R, the total charge in R is q: Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q This can be extended to N discrete point-like charge carriers. ",0.71,6,0.36,52,157.875,B
-"Two particles, each with a charge of magnitude $12 \mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive?","thumb|upright=1.25|triangle ABC exsymmedians (red): e_a, e_b, e_c symmedians (green): s_a, s_b, s_c exsymmedian points (red): E_a, E_b, E_c The exsymmedians are three lines associated with a triangle. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). thumb|This triangle diagram is forbidden by Furry's theorem in quantum electrodynamics. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|upright=1.0|Distance from the origin O to the line E calculated with the Hesse normal form. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. Specifically the following formulas apply: :\begin{align} k_a&=a\cdot \frac{2\triangle}{c^2+b^2-a^2} \\\\[6pt] k_b&=b\cdot \frac{2\triangle}{c^2+a^2-b^2} \\\\[6pt] k_c&=c\cdot \frac{2\triangle}{a^2+b^2-c^2} \end{align} Here \triangle denotes the area of the triangle ABC and k_a, k_b, k_c the perpendicular line segments connecting the triangle sides a, b, c with the exsymmedian points E_a, E_b, E_c . == References == * Roger A. Johnson: Advanced Euclidean Geometry. For a triangle ABC with e_a, e_b, e_c being the exsymmedians and s_a, s_b, s_c being the symmedians through the vertices A, B, C two exsymmedians and one symmedian intersect in a common point, that is: :\begin{align} E_a&=e_b \cap e_c \cap s_a \\\ E_b&=e_a \cap e_c \cap s_b \\\ E_c&=e_a \cap e_b \cap s_c \end{align} The length of the perpendicular line segment connecting a triangle side with its associated exsymmedian point is proportional to that triangle side. The triangle formed by the three exsymmedians is the tangential triangle and its vertices, that is the three intersections of the exsymmedians are called exsymmedian points. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. More precisely for a given triangle the exsymmedians are the tangent lines on the triangle's circumcircle through the three vertices of the triangle. The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse, so one of its angles is obtuse and hence has a negative cosine. ==Properties== Any two polar circles of two triangles in an orthocentric system are orthogonal. As the field lines are pulled together tightly by gluons, they do not ""bow"" outwards as much as an electric field between electric charges. It has an eccentricity equal to \sqrt{2}. ==In polygons== ===Triangles=== The legs of a right triangle are perpendicular to each other. To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. We can similarly describe the electric field E so that . * All four angles are equal. ==In computing distances== == Graph of functions == In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. The altitudes of a triangle are perpendicular to their respective bases. Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle. ===Quadrilaterals=== In a square or other rectangle, all pairs of adjacent sides are perpendicular. The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r2 – 4p2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).College Mathematics Journal 29(4), September 1998, p. 331, problem 635. The magnitude |\vec r_s| of {\vec r_s} is the shortest distance from the origin to the plane. ==References== == External links == * Category:Analytic geometry ",0.0000092,9,0.8185,47,-11.875,D
+","See also: H. Geiger and J.M. Nuttall (1912) ""The ranges of α particles from uranium,"" Philosophical Magazine, Series 6, vol. 23, no. 135, pages 439-445. in its modern form the Geiger–Nuttall law is :\log_{10}T_{1/2}=\frac{A(Z)}{\sqrt{E}}+B(Z) where T_{1/2} is the half-life, E the total kinetic energy (of the alpha particle and the daughter nucleus), and A and B are coefficients that depend on the isotope's atomic number Z. The nuclear force is powerfully attractive between nucleons at distances of about 0.8 femtometre (fm, or 0.8×10−15 metre), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge. At distances less than 0.7 fm, the nuclear force becomes repulsive. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). In this case, the effective nuclear charge can be calculated by Coulomb's law. In atomic physics, the effective nuclear charge is the actual amount of positive (nuclear) charge experienced by an electron in a multi-electron atom. thumb|right|300px|Woods–Saxon potential for , relative to V0 with a and R=4.6 fm The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. In nuclear physics, the Geiger–Nuttall law or Geiger–Nuttall rule relates the decay constant of a radioactive isotope with the energy of the alpha particles emitted. In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase-shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. ==The nuclear force as a residual of the strong force== The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. * Nucleons near the surface of the nucleus (i.e. having within a distance of order a) experience a large force towards the center. There are only strong attractions when the total isospin of the set of interacting particles is 0, which is confirmed by experiment.Griffiths, David, Introduction to Elementary Particles Our understanding of the nuclear force is obtained by scattering experiments and the binding energy of light nuclei. At short distances (less than 1.7 fm or so), the attractive nuclear force is stronger than the repulsive Coulomb force between protons; it thus overcomes the repulsion of protons within the nucleus. In contrast, the effective nuclear charge is the attractive positive charge of nuclear protons acting on valence electrons, which is always less than the total number of protons present in a nucleus due to the shielding effect. ==See also== * Atomic orbitals * Core charge * d-block contraction (or scandide contraction) * Electronegativity * Lanthanide contraction * Shielding effect * Slater-type orbitals * Valence electrons * Weak charge ==References== ==Resources== *Brown, Theodore; intekhab khan, H.E.; & Bursten, Bruce (2002). Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. thumb|A conjectural example of an interaction between two neutrons and a proton, the triton or hydrogen-3, which is beta unstable. However, electrons further away are screened from the nucleus by other electrons in between, and feel less electrostatic interaction as a result. Nuclear interaction length is the mean distance travelled by a hadronic particle before undergoing an inelastic nuclear interaction. ==See also== *Nuclear collision length *Radiation length ==External links== *Particle Data Group site Category:Experimental particle physics These nuclear forces are very weak compared to direct gluon forces (""color forces"" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. ",-2,449,"""5.1""",1.8,228,C
+"The electric field in an $x y$ plane produced by a positively charged particle is $7.2(4.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$ at the point $(3.0,3.0) \mathrm{cm}$ and $100 \hat{\mathrm{i}} \mathrm{N} / \mathrm{C}$ at the point $(2.0,0) \mathrm{cm}$. What is the $x$ coordinate of the particle?","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. In Cartesian coordinate system \vec{r}=x\,\hat{e}_x+y\,\hat{e}_y+z\,\hat{e}_z. Several other definitions are in use, and so care must be taken in comparing different sources.Wolfram Mathworld, spherical coordinates == Cylindrical coordinate system == === Vector fields === Vectors are defined in cylindrical coordinates by (ρ, φ, z), where * ρ is the length of the vector projected onto the xy- plane, * φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), * z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: \begin{bmatrix} \rho \\\ \phi \\\ z \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\\ \operatorname{arctan}(y / x) \\\ z \end{bmatrix},\ \ \ 0 \le \phi < 2\pi, thumb or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} \rho\cos\phi \\\ \rho\sin\phi \\\ z \end{bmatrix}. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. This is the intersection between the reference plane and the axis. The dot is the point with radial distance , angular coordinate , and height . * The azimuth is the angle between the reference direction on the chosen plane and the line from the origin to the projection of on the plane. The coordinates of the vector r with respect to the basis vectors ei are xi. The relative direction between two points is their relative position normalized as a unit vector: :\Delta \mathbf{\hat{r}}=\Delta \mathbf{r} / \|\Delta \mathbf{r}\| where the denominator is the distance between the two points, \| \Delta \mathbf{r} \|. ==Definition== ===Three dimensions=== In three dimensions, any set of three- dimensional coordinates and their corresponding basis vectors can be used to define the location of a point in space—whichever is the simplest for the task at hand may be used. Any vector field can be written in terms of the unit vectors as: \mathbf A = A_x \mathbf{\hat x} + A_y \mathbf{\hat y} + A_z \mathbf{\hat z} = A_\rho \mathbf{\hat \rho} + A_\phi \boldsymbol{\hat \phi} + A_z \mathbf{\hat z} The cylindrical unit vectors are related to the Cartesian unit vectors by: \begin{bmatrix}\mathbf{\hat \rho} \\\ \boldsymbol{\hat\phi} \\\ \mathbf{\hat z}\end{bmatrix} = \begin{bmatrix} \cos\phi & \sin\phi & 0 \\\ -\sin\phi & \cos\phi & 0 \\\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The line element is \mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}. Then the -coordinate is the same in both systems, and the correspondence between cylindrical and Cartesian are the same as for polar coordinates, namely \begin{align} x &= \rho \cos \varphi \\\ y &= \rho \sin \varphi \\\ z &= z \end{align} in one direction, and \begin{align} \rho &= \sqrt{x^2+y^2} \\\ \varphi &= \begin{cases} \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\\ \arcsin\left(\frac{y}{\rho}\right) & \text{if } x \geq 0 \\\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y \ge 0\\\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y < 0 \end{cases} \end{align} in the other. The third coordinate may be called the height or altitude (if the reference plane is considered horizontal), longitudinal position, or axial position. A cylindrical coordinate system is a three- dimensional coordinate system that specifies point positions by the distance from a chosen reference axis (axis L in the image opposite), the direction from the axis relative to a chosen reference direction (axis A), and the distance from a chosen reference plane perpendicular to the axis (plane containing the purple section). Its acceleration is \boldsymbol{a} = \frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \left( \ddot{\rho} - \rho\,\dot\varphi^2 \right)\boldsymbol{\hat \rho} + \left( 2\dot{\rho}\,\dot\varphi + \rho\,\ddot\varphi \right) \hat{\boldsymbol\varphi } + \ddot{z}\,\hat{\boldsymbol{z}} ==See also== *List of canonical coordinate transformations *Vector fields in cylindrical and spherical coordinates *Del in cylindrical and spherical coordinates ==References== ==Further reading== * * * * * * ==External links== * *MathWorld description of cylindrical coordinates *Cylindrical Coordinates Animations illustrating cylindrical coordinates by Frank Wattenberg Category:Three- dimensional coordinate systems Category:Orthogonal coordinate systems de:Polarkoordinaten#Zylinderkoordinaten ro:Coordonate polare#Coordonate cilindrice fi:Koordinaatisto#Sylinterikoordinaatisto The inverse projection is then given by: :\begin{align}\varphi &= \frac{\pi}{2} - \rho \\\ \lambda &=\frac{E \rho} {\sin \varphi_1 \sin \rho} \end{align} where :\rho = \sqrt{ (x\sin \varphi_1)^2 + \left(\varphi_1 - y + \cot \varphi_1\right)^2 }, \qquad E= \tan^{-1}\left(\frac{x\sin \varphi_1}{\varphi_1 - y + \cot \varphi_1}\right). thumb|Radius vector \vec{r} represents the position of a point \mathrm{P}(x,y,z) with respect to origin O. thumb|240px|A cylindrical coordinate system with origin , polar axis , and longitudinal axis . ",0.0625,35,"""-1.0""",344,54.394,C
+"A charge distribution that is spherically symmetric but not uniform radially produces an electric field of magnitude $E=K r^4$, directed radially outward from the center of the sphere. Here $r$ is the radial distance from that center, and $K$ is a constant. What is the volume density $\rho$ of the charge distribution?","Let the first charge distribution \rho_1(\mathbf{r}') be centered on the origin and lie entirely within the second charge distribution \rho_2(\mathbf{r}'). In the interior case, where r' > r, the result is: \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^\ell I_{\ell m} r^\ell \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell m}(\theta, \phi) , with the interior multipole moments defined as I_{\ell m} \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \frac{\rho(\mathbf{r}')}{\left( r' \right)^{\ell+1}} \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell m}^{*}(\theta', \phi'). ==Interaction energies of spherical multipoles== A simple formula for the interaction energy of two non-overlapping but concentric charge distributions can be derived. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. As always, the integral of the charge density over a region of space is the charge contained in that region. We also use spherical coordinates throughout, e.g., the vector \mathbf r' has coordinates ( r', \theta', \phi') where r' is the radius, \theta' is the colatitude and \phi' is the azimuthal angle. ==Spherical multipole moments of a point charge== thumb|right|Figure 1: Definitions for the spherical multipole expansion The electric potential due to a point charge located at \mathbf{r'} is given by \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^2 + r^{\prime 2} - 2 r' r \cos \gamma}}. where R \ \stackrel{\mathrm{def}}{=}\ \left|\mathbf{r} - \mathbf{r'} \right| is the distance between the charge position and the observation point and \gamma is the angle between the vectors \mathbf{r} and \mathbf{r'}. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. For three dimensions, this normalization is the number density of the system ( \rho ) multiplied by the volume of the spherical shell, which symbolically can be expressed as \rho \, 4\pi r^2 dr. Using the mathematical identity P_\ell(\cos \theta) \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell 0}(\theta, \phi) the exterior multipole expansion becomes \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} \left( \frac{Q_\ell}{r^{\ell+1}} \right) P_\ell(\cos \theta) where the axially symmetric multipole moments are defined Q_\ell \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \rho(\mathbf{r}') \left( r' \right)^\ell P_\ell(\cos \theta') In the limit that the charge is confined to the z-axis, we recover the exterior axial multipole moments. For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density \rho(\mathbf r'). Similarly the interior multipole expansion becomes \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{\ell=0}^{\infty} I_\ell r^\ell P_\ell(\cos \theta) where the axially symmetric interior multipole moments are defined I_\ell \ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}' \frac{\rho(\mathbf{r}')}{\left( r' \right)^{\ell+1}} P_\ell(\cos \theta') In the limit that the charge is confined to the z-axis, we recover the interior axial multipole moments. ==See also== * Solid harmonics * Laplace expansion * Multipole expansion * Legendre polynomials * Axial multipole moments * Cylindrical multipole moments Category:Electromagnetism Category:Potential theory Category:Moment (physics) In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. Taking particle 0 as fixed at the origin of the coordinates, \textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r}) is the average number of particles (among the remaining N-1) to be found in the volume \textstyle d^3r around the position \textstyle \mathbf{r}. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein-Zernike equation or structure-optimized potential refinement. ==Definition== Consider a system of N particles in a volume V (for an average number density \rho =N/V) and at a temperature T (let us also define \textstyle \beta = \frac{1}{kT}). This is expressed by a continuity equation which links the rate of change of charge density \rho(\boldsymbol{x}) and the current density \boldsymbol{J}(\boldsymbol{x}). Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. The delta function has the sifting property for any function f: \int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0) so the delta function ensures that when the charge density is integrated over R, the total charge in R is q: Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q This can be extended to N discrete point-like charge carriers. ",0.71,6,"""0.36""",52,157.875,B
+"Two particles, each with a charge of magnitude $12 \mathrm{nC}$, are at two of the vertices of an equilateral triangle with edge length $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the third vertex if both charges are positive?","thumb|upright=1.25|triangle ABC exsymmedians (red): e_a, e_b, e_c symmedians (green): s_a, s_b, s_c exsymmedian points (red): E_a, E_b, E_c The exsymmedians are three lines associated with a triangle. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). thumb|This triangle diagram is forbidden by Furry's theorem in quantum electrodynamics. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|upright=1.0|Distance from the origin O to the line E calculated with the Hesse normal form. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. Specifically the following formulas apply: :\begin{align} k_a&=a\cdot \frac{2\triangle}{c^2+b^2-a^2} \\\\[6pt] k_b&=b\cdot \frac{2\triangle}{c^2+a^2-b^2} \\\\[6pt] k_c&=c\cdot \frac{2\triangle}{a^2+b^2-c^2} \end{align} Here \triangle denotes the area of the triangle ABC and k_a, k_b, k_c the perpendicular line segments connecting the triangle sides a, b, c with the exsymmedian points E_a, E_b, E_c . == References == * Roger A. Johnson: Advanced Euclidean Geometry. For a triangle ABC with e_a, e_b, e_c being the exsymmedians and s_a, s_b, s_c being the symmedians through the vertices A, B, C two exsymmedians and one symmedian intersect in a common point, that is: :\begin{align} E_a&=e_b \cap e_c \cap s_a \\\ E_b&=e_a \cap e_c \cap s_b \\\ E_c&=e_a \cap e_b \cap s_c \end{align} The length of the perpendicular line segment connecting a triangle side with its associated exsymmedian point is proportional to that triangle side. The triangle formed by the three exsymmedians is the tangential triangle and its vertices, that is the three intersections of the exsymmedians are called exsymmedian points. Then use the SAS congruence theorem for triangles OPA' and OPB' to conclude that angles POA and POB are equal. More precisely for a given triangle the exsymmedians are the tangent lines on the triangle's circumcircle through the three vertices of the triangle. The trigonometric formula for the radius shows that the polar circle has a real existence only if the triangle is obtuse, so one of its angles is obtuse and hence has a negative cosine. ==Properties== Any two polar circles of two triangles in an orthocentric system are orthogonal. As the field lines are pulled together tightly by gluons, they do not ""bow"" outwards as much as an electric field between electric charges. It has an eccentricity equal to \sqrt{2}. ==In polygons== ===Triangles=== The legs of a right triangle are perpendicular to each other. To prove that the PQ is perpendicular to AB, use the SSS congruence theorem for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. We can similarly describe the electric field E so that . * All four angles are equal. ==In computing distances== == Graph of functions == In the two-dimensional plane, right angles can be formed by two intersected lines if the product of their slopes equals −1. The altitudes of a triangle are perpendicular to their respective bases. Harcourt's theorem concerns the relationship of line segments through a vertex and perpendicular to any line tangent to the triangle's incircle. ===Quadrilaterals=== In a square or other rectangle, all pairs of adjacent sides are perpendicular. The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r2 – 4p2 (where r is the circle's radius and p is the distance from the center point to the point of intersection).College Mathematics Journal 29(4), September 1998, p. 331, problem 635. The magnitude |\vec r_s| of {\vec r_s} is the shortest distance from the origin to the plane. ==References== == External links == * Category:Analytic geometry ",0.0000092,9,"""0.8185""",47,-11.875,D
 "How much work is required to turn an electric dipole $180^{\circ}$ in a uniform electric field of magnitude $E=46.0 \mathrm{~N} / \mathrm{C}$ if the dipole moment has a magnitude of $p=3.02 \times$ $10^{-25} \mathrm{C} \cdot \mathrm{m}$ and the initial angle is $64^{\circ} ?$
-","This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{p} \cdot \mathbf{E},\qquad\ \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, The scalar dot """" product and the negative sign shows the potential energy minimises when the dipole is parallel with field and is maximum when antiparallel while zero when perpendicular. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . thumb|upright=1.2|In the discrete dipole approximation, a larger object is approximated in terms of discrete radiating electric dipoles. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some angle with it. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. However in a non- uniform electric field a dipole may indeed receive a net force since the force on one end of the dipole no longer balances that on the other end. Note that a dipole in such a uniform field may twist and oscillate but receives no overall net force with no linear acceleration of the dipole. Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The torque tends to align the dipole with the field. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin). (See electron electric dipole moment). In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample, \mathcal{M}_\text{Tot} = \mathcal{M}_\text{Trans} + \mathcal{M}_\text{Rot}. It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. Further improvements, or a positive result, would place further limits on which theory takes precedence. == Formal definition == As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that : \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} depends on the point {\mathbf r}_0 about which the moment of the charge distribution \rho({\mathbf r}) is taken. ", 1.16,1.22,0.7812,3.54,-32,B
-"We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \%$. With what force would two copper coins, placed $1.0 \mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \times 10^{22}$ copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) What do you conclude?","This is not the case with copper. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. thumb|upright=1.35|Coin of Tennes. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. And although we would all be walking around with a few $1 coins, they would be replacing several quarters"".Barro, Robert J. and Stevenson, Betsey: Do You Want That In Paper, or Metal? , The Wall Street Journal, Nov. 6, 1997 In advocating abolition of the penny, the Coin Coalition cites three penny-related costs that are passed on to consumers:Should the penny go?, Annelena Lobb, CNN Money, Apr. 11, 2002 *Wrapping charges (stores pay about 60 cents for each roll of 50 pennies) *Lost store productivity from penny users slowing the checkout line *Lost wages paid to clerks counting pennies in the register on each shift James C. Benfield, a partner with Bracy Williams and Company (Washington, D.C.), led the Coalition from 1987 until his death in 2002. The Coin Coalition is an organization supporting the elimination of pennies and dollar bills from U.S. currency. thumb|upright=1.5|Coin of Amyntas. Copper has the highest electrical conductivity rating of all non-precious metals: the electrical resistivity of copper = 16.78 nΩ•m at 20 °C. What is unique about copper is its long mean free path (approximately 100 atomic spacings at room temperature). A dollar coin is a coin valued at one dollar in a given currency. Roughly half of all copper mined is used to manufacture electrical wire and cable conductors. == Properties of copper == ===Electrical conductivity=== Electrical conductivity is a measure of how well a material transports an electric charge. Aluminium has 61% of the conductivity of copper. Manufacturers converted machines to accept the dollar coin at great expense, but the unwillingness of the U.S. government to phase out the dollar bill prevented the coin from becoming popular.$1 Coin Makes No ""Cents"" If $1 Bills Remain , Randy Chilton, Replay Magazine, March 2001 Although copper miners and other interest groups backed the Coin Coalition on this issue, they were unable to match the influence of Save the Greenback, a rival organization supporting continued dollar-bill production. Although 6-nines copper (99.9999% pure) has been produced in small quantities, it is extremely expensive and probably unnecessary for most commercial applications such as magnet, telecommunications, and building wire. Silver, a precious metal, is the only metal with a higher electrical conductivity than copper. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. If the two metals are joined, a galvanic reaction can occur. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Copper has a higher ductility than alternate metal conductors with the exception of gold and silver.Rich, Jack C., 1988, The Materials and Methods of Sculpture. This copper is at least 99.90% pure and has an electrical conductivity of at least 101% IACS. ",2600,1.7,12.0,71,21,B
+","This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{p} \cdot \mathbf{E},\qquad\ \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, The scalar dot """" product and the negative sign shows the potential energy minimises when the dipole is parallel with field and is maximum when antiparallel while zero when perpendicular. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . thumb|upright=1.2|In the discrete dipole approximation, a larger object is approximated in terms of discrete radiating electric dipoles. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some angle with it. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. However in a non- uniform electric field a dipole may indeed receive a net force since the force on one end of the dipole no longer balances that on the other end. Note that a dipole in such a uniform field may twist and oscillate but receives no overall net force with no linear acceleration of the dipole. Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The torque tends to align the dipole with the field. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin). (See electron electric dipole moment). In general the total dipole moment have contributions coming from translations and rotations of the molecules in the sample, \mathcal{M}_\text{Tot} = \mathcal{M}_\text{Trans} + \mathcal{M}_\text{Rot}. It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. Further improvements, or a positive result, would place further limits on which theory takes precedence. == Formal definition == As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that : \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} depends on the point {\mathbf r}_0 about which the moment of the charge distribution \rho({\mathbf r}) is taken. ", 1.16,1.22,"""0.7812""",3.54,-32,B
+"We know that the negative charge on the electron and the positive charge on the proton are equal. Suppose, however, that these magnitudes differ from each other by $0.00010 \%$. With what force would two copper coins, placed $1.0 \mathrm{~m}$ apart, repel each other? Assume that each coin contains $3 \times 10^{22}$ copper atoms. (Hint: A neutral copper atom contains 29 protons and 29 electrons.) What do you conclude?","This is not the case with copper. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. thumb|upright=1.35|Coin of Tennes. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. And although we would all be walking around with a few $1 coins, they would be replacing several quarters"".Barro, Robert J. and Stevenson, Betsey: Do You Want That In Paper, or Metal? , The Wall Street Journal, Nov. 6, 1997 In advocating abolition of the penny, the Coin Coalition cites three penny-related costs that are passed on to consumers:Should the penny go?, Annelena Lobb, CNN Money, Apr. 11, 2002 *Wrapping charges (stores pay about 60 cents for each roll of 50 pennies) *Lost store productivity from penny users slowing the checkout line *Lost wages paid to clerks counting pennies in the register on each shift James C. Benfield, a partner with Bracy Williams and Company (Washington, D.C.), led the Coalition from 1987 until his death in 2002. The Coin Coalition is an organization supporting the elimination of pennies and dollar bills from U.S. currency. thumb|upright=1.5|Coin of Amyntas. Copper has the highest electrical conductivity rating of all non-precious metals: the electrical resistivity of copper = 16.78 nΩ•m at 20 °C. What is unique about copper is its long mean free path (approximately 100 atomic spacings at room temperature). A dollar coin is a coin valued at one dollar in a given currency. Roughly half of all copper mined is used to manufacture electrical wire and cable conductors. == Properties of copper == ===Electrical conductivity=== Electrical conductivity is a measure of how well a material transports an electric charge. Aluminium has 61% of the conductivity of copper. Manufacturers converted machines to accept the dollar coin at great expense, but the unwillingness of the U.S. government to phase out the dollar bill prevented the coin from becoming popular.$1 Coin Makes No ""Cents"" If $1 Bills Remain , Randy Chilton, Replay Magazine, March 2001 Although copper miners and other interest groups backed the Coin Coalition on this issue, they were unable to match the influence of Save the Greenback, a rival organization supporting continued dollar-bill production. Although 6-nines copper (99.9999% pure) has been produced in small quantities, it is extremely expensive and probably unnecessary for most commercial applications such as magnet, telecommunications, and building wire. Silver, a precious metal, is the only metal with a higher electrical conductivity than copper. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. If the two metals are joined, a galvanic reaction can occur. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Copper has a higher ductility than alternate metal conductors with the exception of gold and silver.Rich, Jack C., 1988, The Materials and Methods of Sculpture. This copper is at least 99.90% pure and has an electrical conductivity of at least 101% IACS. ",2600,1.7,"""12.0""",71,21,B
 "What must be the distance between point charge $q_1=$ $26.0 \mu \mathrm{C}$ and point charge $q_2=-47.0 \mu \mathrm{C}$ for the electrostatic force between them to have a magnitude of $5.70 \mathrm{~N}$ ?
-","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). Then the distance between p and q is given by: d(p,q) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2}. Thus if p and q are two points on the real line, then the distance between them is given by: d(p,q) = |p-q|. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. If the polar coordinates of p are (r,\theta) and the polar coordinates of q are (s,\psi), then their distance is given by the law of cosines: d(p,q)=\sqrt{r^2 + s^2 - 2rs\cos(\theta-\psi)}. thumb|upright=1.35|Using the Pythagorean theorem to compute two-dimensional Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. In general, for points given by Cartesian coordinates in n-dimensional Euclidean space, the distance is d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2}. When p and q are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used, although here the absolute value sign indicates the complex norm: d(p,q)=|p-q|. === Higher dimensions === thumb|upright=1.2|Deriving the n-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem In three dimensions, for points given by their Cartesian coordinates, the distance is d(p,q)=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. As an equation, the squared distance can be expressed as a sum of squares: d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2. Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. ",-45,7.27,0.000226,1.61,1.39,E
+","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). Then the distance between p and q is given by: d(p,q) = \sqrt{(q_1-p_1)^2 + (q_2-p_2)^2}. Thus if p and q are two points on the real line, then the distance between them is given by: d(p,q) = |p-q|. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. If the polar coordinates of p are (r,\theta) and the polar coordinates of q are (s,\psi), then their distance is given by the law of cosines: d(p,q)=\sqrt{r^2 + s^2 - 2rs\cos(\theta-\psi)}. thumb|upright=1.35|Using the Pythagorean theorem to compute two-dimensional Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. In general, for points given by Cartesian coordinates in n-dimensional Euclidean space, the distance is d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2}. When p and q are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used, although here the absolute value sign indicates the complex norm: d(p,q)=|p-q|. === Higher dimensions === thumb|upright=1.2|Deriving the n-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem In three dimensions, for points given by their Cartesian coordinates, the distance is d(p,q)=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. As an equation, the squared distance can be expressed as a sum of squares: d^2(p,q) = (p_1 - q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2. Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. ",-45,7.27,"""0.000226""",1.61,1.39,E
 "Three charged particles form a triangle: particle 1 with charge $Q_1=80.0 \mathrm{nC}$ is at $x y$ coordinates $(0,3.00 \mathrm{~mm})$, particle 2 with charge $Q_2$ is at $(0,-3.00 \mathrm{~mm})$, and particle 3 with charge $q=18.0$ $\mathrm{nC}$ is at $(4.00 \mathrm{~mm}, 0)$. In unit-vector notation, what is the electrostatic force on particle 3 due to the other two particles if $Q_2$ is equal to $80.0 \mathrm{nC}$?
-","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. #Lines of application of the actual forces {\scriptstyle \vec{F}_{1}} and \scriptstyle \vec{F}_{2} in the leftmost illustration intersect. Now let H denote the orthocenter of the triangle, then connection vector \overrightarrow{OH} is equal to the sum of the three vectors:Roger A. Johnson: Advanced Euclidean Geometry. After vector addition ""at the location of \scriptstyle\vec{F}_{2}"", the net force is translated to the appropriate line of application, whereof it becomes the resultant force \scriptstyle \vec{F}_{R}. After vector addition is performed ""at the location of \scriptstyle \vec{F}_{1}"", the net force obtained is translated so that its line of application passes through the common intersection point. #Illustration in the middle of the diagram shows two parallel actual forces. With respect to that point all torques are zero, so the torque of the resultant force \scriptstyle \vec{F}_{R} is equal to the sum of the torques of the actual forces. A trion is a localized excitation which consists of three charged particles. Pressure is force magnitude applied over an area. thumb|500px|Graphical placing of the resultant force In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The term resultant force should be understood to refer to both the forces and torques acting on a rigid body, which is why some use the term resultant force–torque. ==Illustration== The diagram illustrates simple graphical methods for finding the line of application of the resultant force of simple planar systems. If this condition is satisfied then there is a point of application for the resultant which results in a pure force. thumb|upright=1.5|sum of three equal lengthed vectors Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. Then the arguments from the previous case are applied to the forces and their components to demonstrate the torque relationships. This point is defined by the property : \mathbf{R} \times \mathbf{F} = \sum_{i=1}^n \mathbf{R}_i \times \mathbf{F}_i, where F is resultant force and Fi form the system of forces. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The sum of these forces and torques yields the resultant force- torque. ==Associated torque== If a point R is selected as the point of application of the resultant force F of a system of n forces Fi then the associated torque T is determined from the formulas : \mathbf{F} = \sum_{i=1}^n \mathbf{F}_i, and : \mathbf{T} = \sum_{i=1}^n (\mathbf{R}_i-\mathbf{R})\times \mathbf{F}_i. With the electrostatic force being proportional to r^{-2}, individual particle-particle interactions are long- range in nature, presenting a challenging computational problem in the simulation of particulate systems. It is useful to note that the point of application R of the resultant force may be anywhere along the line of action of F without changing the value of the associated torque. Notice that the case of two equal but opposite forces F and -F acting at points A and B respectively, yields the resultant W=(F-F, A×F - B× F) = (0, (A-B)×F). The point of application of the resultant force determines its associated torque. Notice that this equation for R has a solution only if the sum of the individual torques on the right side yield a vector that is perpendicular to F. ",0.829,11,-0.5,-214,0.14,A
-A particle of charge $-q_1$ is at the origin of an $x$ axis. (a) At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \mathrm{~mm}$ on the axis? ,"The figure at right shows the electric field lines of two equal charges of opposite sign. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. There, the field vanishes and the lines coming axially from the charges end. Surface Charging and Points of Zero Charge. thumb|Field lines depicting the electric field created by a positive charge (left), negative charge (center), and uncharged object (right). The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|Circuit diagram of a charge qubit circuit. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. However, if the electric field lines for this setup were just drawn on a two-dimensional plane, their two-dimensional density would be proportional to 1/r, an incorrect result for this situation.A. Wolf, S. J. Van Hook, E. R. Weeks, Electric field line diagrams don't work Am. J. Phys., Vol. 64, No. 6. (1996), pp. 714–724 DOI 10.1119/1.18237 == Construction == thumb|upright=1.3|Construction of a field line Given a vector field \mathbf{F}(\mathbf{x}) and a starting point \mathbf{x}_\text{0} a field line can be constructed iteratively by finding the field vector at that point \mathbf{F}(\mathbf{x}_\text{0}). The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. Related values associated with the soil characteristics exist along with the pzc value, including zero point of charge (zpc), point of zero net charge (pznc), etc. == Term definition of point of zero charge == The point of zero charge is the pH for which the net surface charge of adsorbent is equal to zero. A related concept in electrochemistry is the electrode potential at the point of zero charge. The field line can be extended in the opposite direction from \mathbf{x}_\text{0} by taking each step in the opposite direction by using a negative step -ds. == Examples == thumb|420px|Different ways to depict the field of a magnet. thumb|right|250px|The slope field of \frac{dy}{dx}=x^{2}-x-2, with the blue, red, and turquoise lines being \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x+4, \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x, and \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x-4, respectively. For example, Gauss's law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. Points where the field is zero or infinite have no field line through them, since direction cannot be defined there, but can be the endpoints of field lines. For example, electric field lines begin on positive charges and end on negative charges. The modern version of these equations is called Maxwell's equations. ====Electrostatics==== A charged test particle with charge q experiences a force F based solely on its charge. As the field lines are pulled together tightly by gluons, they do not ""bow"" outwards as much as an electric field between electric charges. The point of zero charge (pzc) is generally described as the pH at which the net charge of total particle surface (i.e. absorbent's surface) is equal to zero, which concept has been introduced in the studies dealt with colloidal flocculation to explain pH affecting the phenomenon. ",3.42,6.0,-114.4,11,4,B
-An electron is released from rest in a uniform electric field of magnitude $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.),"Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . thumb|upright=1.3|right|Launch of Electron in start of the ""Birds of the Feather"" mission. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. Relativistic electron beams are streams of electrons moving at relativistic speeds. The term runaway electrons (RE) is used to denote electrons that undergo free fall acceleration into the realm of relativistic particles. Using Searle's formula (1897) for the electromagnetic energy increase of charged bodies with velocity, he calculated the increase of the electron's electromagnetic mass as a function of velocity: :\phi(\beta)=\frac{3}{4\beta^{2}}\left[\frac{1}{\beta}\lg\frac{1-\beta}{1+\beta}+\frac{2}{1-\beta^{2}}\right],\;\beta=\frac{v}{c}, Kaufmann noticed that the observed increase cannot be explained by this formula, so he separated the measured total mass into a mechanical (true) mass and an electromagnetic (apparent) mass, the mechanical mass being considerably greater than the electromagnetic one. Kaufmann's measurements of 1901 (corrected in 1902) showed that the charge-to-mass ratio diminishes and thus the electron's momentum (or mass) increases with velocity. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). It has been suggested that relativistic electron beams could be used to heat and accelerate the reaction mass in electrical rocket engines that Dr. Robert W. Bussard called quiet electric-discharge engines (QEDs). ==References== ==External links== *PEARL Lab @ UHawaii *Applying REBs for the development of high-powered microwaves (HPM) Category:Electron beam Category:Quantum mechanics Category:Special relativity First flight of Electron with a fully autonomous flight termination system on the rocket. === 2020 === First launch for the National Reconnaissance Office in January 2020. In 1906 and 1907, Planck published his own conclusion on the behavior of the inertial mass of electrons with high speeds. In his theory, the longitudinal mass m_L= {{\gamma}^3}m and the transverse mass m_T= {\gamma}m, where \gamma is the Lorentz factor and m is the rest mass of the electron. (a) This front view of the apparatus illustrates the uniform acceleration imposed by the charged condenser plates on the beta particles. Suborbital Flight-1 was the first suborbital launch of the rocket. == Launch statistics == === Launch outcomes === === Launch sites === === Booster tests and recoveries === === Rocket configurations === == Orbital launches == === 2017–2018 === Electron experienced its first successful launch in January 2018, and launched their first mission for NASA in December 2018. In this notation, the effective acceleration voltage |V_\parallel| is often expressed as V_0 T. == Transverse voltage == In symbolic analogy to the longitudinal voltage, one can define effective voltages in two orthogonal directions x,y that are transversal to the particle trajectory V_{x,y} = \frac{1}{q} \vec e_{x,y} \cdot \int \vec F_L(s) \exp\left(i k_\beta s\right)\,\mathrm d s which describe the integrated forces that deflect the particle from its design path. Accelerationen (Accelerations), op. 234, is a waltz composed by Johann Strauss II in 1860 for the Engineering Students' Ball at the Sofienbad-Saal in Vienna. ",16,-3.141592,-0.029,3.51,0.2,D
-Identify $\mathrm{X}$ in the following nuclear reactions: (a) ${ }^1 \mathrm{H}+$ ${ }^9 \mathrm{Be} \rightarrow \mathrm{X}+\mathrm{n} ;$ (b) ${ }^{12} \mathrm{C}+{ }^1 \mathrm{H} \rightarrow \mathrm{X} ;$ (c) ${ }^{15} \mathrm{~N}+{ }^1 \mathrm{H} \rightarrow{ }^4 \mathrm{He}+\mathrm{X}$. Appendix F will help.,"Jonathan Feng et al. attribute this 6.8-σ anomaly to a 17 MeV protophobic X-boson dubbed the X17 particle. The X17 particle is a hypothetical subatomic particle proposed by Attila Krasznahorkay and his colleagues to explain certain anomalous measurement results. Krasznahorkay (2019) posted a preprint announcing that he and his team at ATOMKI had successfully observed the same anomalies in the decay of stable helium atoms as had been observed in beryllium-8, strengthening the case for the existence of the X17 particle. Hexanitrostilbene (HNS), also called JD-X, is an organic compound with the formula [(O2N)3C6H2CH]2. The molecular formula C8H11N (molar mass: 121.18 g/mol) may refer to: * Bicyclo(2.2.1)heptane-2-carbonitrile * Collidines (trimethylpyridines) ** 2,3,4-Trimethylpyridine ** 2,3,5-Trimethylpyridine ** 2,3,6-Trimethylpyridine ** 2,4,5-Trimethylpyridine ** 2,4,6-Trimethylpyridine ** 3,4,5-Trimethylpyridine * Dimethylaniline * Phenethylamine * 1-Phenylethylamine * Xylidines ** 2,3-Xylidine ** 2,4-Xylidine ** 2,5-Xylidine ** 2,6-Xylidine ** 3,4-Xylidine ** 3,5-Xylidine Beryllium-8 (8Be, Be-8) is a radionuclide with 4 neutrons and 4 protons. HNC may refer to: *Hydrogen isocyanide, a molecule with the formula HNC that is important to the field of astrochemistry *Heptanitrocubane, an experimental high explosive *Higher National Certificate, a higher education qualification in the United Kingdom *High Negotiations Committee, a Syrian political- military opposition bloc headquartered in Riyadh *Classical-map Hyper-Netted- Chain equation, a method in many-body theoretical physics for interacting uniform electron liquids in two and three dimensions *Hypernetted-chain equation, a closure relation to solve the Ornstein-Zernike equation commonly applied in statistical mechanics and fluid theory *Hopkins-Nanjing Center, a joint educational venture between Nanjing University and Johns Hopkins University located in Nanjing, China *Habits & Contradictions, album by Schoolboy Q *Huddersfield Narrow Canal, Northern England Feng et al. (2016) proposed that a ""protophobic"" X boson, with a mass of , suppressed couplings to protons relative to neutrons and electrons at femtometer range, could explain the data. The X17 particle could be the force carrier for a postulated fifth force, possibly connected with dark matter, and has been described as a protophobic (i.e., ignoring protons) vector boson with a mass near . The NA64 experiment at CERN looks for the proposed X17 particle by striking the electron beams from the Super Proton Synchrotron on fixed target nuclei. ==History== In 2015, Krasznahorkay and his colleagues at ATOMKI, the Hungarian Institute for Nuclear Research, posited the existence of a new, light boson with a mass of about (i.e., 34 times heavier than the electron). However, stable 8Be would enable alternative reaction pathways in helium burning (such as 8Be + 4He and 8Be + 8Be; constituting a ""beryllium burning"" phase) and possibly affect the abundance of the resultant 12C, 16O, and heavier nuclei, though 1H and 4He would remain the most abundant nuclides. CXOU J061705.3+222127 is a neutron star. While further experiments are needed to corroborate these observations, the influence of a fifth boson has been proposed as ""the most straightforward possibility"". == Role in stellar nucleosynthesis == In stellar nucleosynthesis, two helium-4 nuclei may collide and fuse into a single beryllium-8 nucleus. Furthermore, while other alpha nuclides have similar short-lived resonances, 8Be is exceptionally already in the ground state. The nucleus of helium-4 is particularly stable, having a doubly magic configuration and larger binding energy per nucleon than 8Be. Owing to the instability of 8Be, the triple-alpha process is the only reaction in which 12C and heavier elements may be produced in observed quantities. If the beryllium-8 collides with a helium-4 nucleus before decaying, they can fuse into a carbon-12 nucleus. Beryllium-8 has an extremely short half-life (8.19 seconds), and decays back into two helium-4 nuclei. They reported that this populated a nucleus with A = 8 that near- instantaneously decays into two alpha particles. In an effort to find a dark photon, the team fired protons at thin targets of lithium-7, which created unstable beryllium-8 nuclei that then decayed and produced pairs of electrons and positrons. The X‑17 particle is not consistent with the Standard Model, so its existence would need to be explained by another theory. ==See also== * Axion * List of particles * 750 GeV diphoton excess ==References== Category:Bosons Category:Dark matter Category:Hypothetical elementary particles Category:Force carriers This activity was observed again several months later, and was inferred to originate from 8Be. == Properties == left|thumb|300px|Triple-alpha process Beryllium-8 is unbound with respect to alpha emission by 92 keV; it is a resonance having a width of 6 eV. ",0.366,0.020,nan,-87.8,48,C
-"The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons?","This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The proton radius is approximately one femtometre = . It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton Plutonium is a radioactive chemical element with the symbol Pu and atomic number 94. When one of these neutrons strikes the nucleus of another 238U atom, it is absorbed by the atom, which becomes 239U. Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). Later studies found an empirical relation between the charge radius and the mass number, A, for heavier nuclei (A > 20): :R ≈ r0A where the empirical constant r0 of 1.2–1.5 fm can be interpreted as the Compton wavelength of the proton. The proton radius puzzle is an unanswered problem in physics relating to the size of the proton. Plutonium-239 (239Pu or Pu-239) is an isotope of plutonium. The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. The known isotopes of plutonium range in mass number from 228 to 247. The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. Occasionally, when an atom of 238U is exposed to neutron radiation, its nucleus will capture a neutron, changing it to 239U. [It] was stocked with 24 pounds of plutonium that produced about 240 watts of electricity when it left Earth in 2006, according to Ryan Bechtel, an engineer from the Department of Energy who works on space nuclear power. For the proton, the two radii are the same. ==History== The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909,. under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester, UK. The plutonium grade is usually listed as percentage of 240Pu. The result is again ~5% smaller than the previously-accepted proton radius. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. The fission of one atom of 239Pu generates 207.1 MeV = 3.318 × 10−11 J, i.e. 19.98 TJ/mol = 83.61 TJ/kg, or about 23 gigawatt hours/kg. radiation source (thermal fission of 239Pu) average energy released [MeV] Kinetic energy of fission fragments 175.8 Kinetic energy of prompt neutrons 5.9 Energy carried by prompt γ-rays 7.8 Total instantaneous energy 189.5 Energy of β− particles 5.3 Energy of antineutrinos 7.1 Energy of delayed γ-rays 5.2 Total from decaying fission products 17.6 Energy released by radiative capture of prompt neutrons 11.5 Total heat released in a thermal-spectrum reactor (anti-neutrinos do not contribute) 211.5 == Production == Plutonium is made from uranium-238. 239Pu is normally created in nuclear reactors by transmutation of individual atoms of one of the isotopes of uranium present in the fuel rods. A 5 kg mass of 239Pu contains about atoms. Nucleons have a radius of about 0.8 fm. Weapons- grade plutonium is defined as containing no more than 7% 240Pu; this is achieved by only exposing 238U to neutron sources for short periods of time to minimize the 240Pu produced. ",54.394,6.3,25.6773,1000,3.07,E
+","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. #Lines of application of the actual forces {\scriptstyle \vec{F}_{1}} and \scriptstyle \vec{F}_{2} in the leftmost illustration intersect. Now let H denote the orthocenter of the triangle, then connection vector \overrightarrow{OH} is equal to the sum of the three vectors:Roger A. Johnson: Advanced Euclidean Geometry. After vector addition ""at the location of \scriptstyle\vec{F}_{2}"", the net force is translated to the appropriate line of application, whereof it becomes the resultant force \scriptstyle \vec{F}_{R}. After vector addition is performed ""at the location of \scriptstyle \vec{F}_{1}"", the net force obtained is translated so that its line of application passes through the common intersection point. #Illustration in the middle of the diagram shows two parallel actual forces. With respect to that point all torques are zero, so the torque of the resultant force \scriptstyle \vec{F}_{R} is equal to the sum of the torques of the actual forces. A trion is a localized excitation which consists of three charged particles. Pressure is force magnitude applied over an area. thumb|500px|Graphical placing of the resultant force In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The term resultant force should be understood to refer to both the forces and torques acting on a rigid body, which is why some use the term resultant force–torque. ==Illustration== The diagram illustrates simple graphical methods for finding the line of application of the resultant force of simple planar systems. If this condition is satisfied then there is a point of application for the resultant which results in a pure force. thumb|upright=1.5|sum of three equal lengthed vectors Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. Then the arguments from the previous case are applied to the forces and their components to demonstrate the torque relationships. This point is defined by the property : \mathbf{R} \times \mathbf{F} = \sum_{i=1}^n \mathbf{R}_i \times \mathbf{F}_i, where F is resultant force and Fi form the system of forces. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The sum of these forces and torques yields the resultant force- torque. ==Associated torque== If a point R is selected as the point of application of the resultant force F of a system of n forces Fi then the associated torque T is determined from the formulas : \mathbf{F} = \sum_{i=1}^n \mathbf{F}_i, and : \mathbf{T} = \sum_{i=1}^n (\mathbf{R}_i-\mathbf{R})\times \mathbf{F}_i. With the electrostatic force being proportional to r^{-2}, individual particle-particle interactions are long- range in nature, presenting a challenging computational problem in the simulation of particulate systems. It is useful to note that the point of application R of the resultant force may be anywhere along the line of action of F without changing the value of the associated torque. Notice that the case of two equal but opposite forces F and -F acting at points A and B respectively, yields the resultant W=(F-F, A×F - B× F) = (0, (A-B)×F). The point of application of the resultant force determines its associated torque. Notice that this equation for R has a solution only if the sum of the individual torques on the right side yield a vector that is perpendicular to F. ",0.829,11,"""-0.5""",-214,0.14,A
+A particle of charge $-q_1$ is at the origin of an $x$ axis. (a) At what location on the axis should a particle of charge $-4 q_1$ be placed so that the net electric field is zero at $x=2.0 \mathrm{~mm}$ on the axis? ,"The figure at right shows the electric field lines of two equal charges of opposite sign. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. There, the field vanishes and the lines coming axially from the charges end. Surface Charging and Points of Zero Charge. thumb|Field lines depicting the electric field created by a positive charge (left), negative charge (center), and uncharged object (right). The electric field lines in this case are straight lines that emanate from the charge uniformly in all directions in three-dimensional space. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|Circuit diagram of a charge qubit circuit. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. However, if the electric field lines for this setup were just drawn on a two-dimensional plane, their two-dimensional density would be proportional to 1/r, an incorrect result for this situation.A. Wolf, S. J. Van Hook, E. R. Weeks, Electric field line diagrams don't work Am. J. Phys., Vol. 64, No. 6. (1996), pp. 714–724 DOI 10.1119/1.18237 == Construction == thumb|upright=1.3|Construction of a field line Given a vector field \mathbf{F}(\mathbf{x}) and a starting point \mathbf{x}_\text{0} a field line can be constructed iteratively by finding the field vector at that point \mathbf{F}(\mathbf{x}_\text{0}). The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. Related values associated with the soil characteristics exist along with the pzc value, including zero point of charge (zpc), point of zero net charge (pznc), etc. == Term definition of point of zero charge == The point of zero charge is the pH for which the net surface charge of adsorbent is equal to zero. A related concept in electrochemistry is the electrode potential at the point of zero charge. The field line can be extended in the opposite direction from \mathbf{x}_\text{0} by taking each step in the opposite direction by using a negative step -ds. == Examples == thumb|420px|Different ways to depict the field of a magnet. thumb|right|250px|The slope field of \frac{dy}{dx}=x^{2}-x-2, with the blue, red, and turquoise lines being \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x+4, \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x, and \frac{x^{3}}{3}-\frac{x^{2}}{2}-2x-4, respectively. For example, Gauss's law states that an electric field has sources at positive charges, sinks at negative charges, and neither elsewhere, so electric field lines start at positive charges and end at negative charges. Points where the field is zero or infinite have no field line through them, since direction cannot be defined there, but can be the endpoints of field lines. For example, electric field lines begin on positive charges and end on negative charges. The modern version of these equations is called Maxwell's equations. ====Electrostatics==== A charged test particle with charge q experiences a force F based solely on its charge. As the field lines are pulled together tightly by gluons, they do not ""bow"" outwards as much as an electric field between electric charges. The point of zero charge (pzc) is generally described as the pH at which the net charge of total particle surface (i.e. absorbent's surface) is equal to zero, which concept has been introduced in the studies dealt with colloidal flocculation to explain pH affecting the phenomenon. ",3.42,6.0,"""-114.4""",11,4,B
+An electron is released from rest in a uniform electric field of magnitude $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$. Calculate the acceleration of the electron. (Ignore gravitation.),"Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . thumb|upright=1.3|right|Launch of Electron in start of the ""Birds of the Feather"" mission. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. Relativistic electron beams are streams of electrons moving at relativistic speeds. The term runaway electrons (RE) is used to denote electrons that undergo free fall acceleration into the realm of relativistic particles. Using Searle's formula (1897) for the electromagnetic energy increase of charged bodies with velocity, he calculated the increase of the electron's electromagnetic mass as a function of velocity: :\phi(\beta)=\frac{3}{4\beta^{2}}\left[\frac{1}{\beta}\lg\frac{1-\beta}{1+\beta}+\frac{2}{1-\beta^{2}}\right],\;\beta=\frac{v}{c}, Kaufmann noticed that the observed increase cannot be explained by this formula, so he separated the measured total mass into a mechanical (true) mass and an electromagnetic (apparent) mass, the mechanical mass being considerably greater than the electromagnetic one. Kaufmann's measurements of 1901 (corrected in 1902) showed that the charge-to-mass ratio diminishes and thus the electron's momentum (or mass) increases with velocity. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). It has been suggested that relativistic electron beams could be used to heat and accelerate the reaction mass in electrical rocket engines that Dr. Robert W. Bussard called quiet electric-discharge engines (QEDs). ==References== ==External links== *PEARL Lab @ UHawaii *Applying REBs for the development of high-powered microwaves (HPM) Category:Electron beam Category:Quantum mechanics Category:Special relativity First flight of Electron with a fully autonomous flight termination system on the rocket. === 2020 === First launch for the National Reconnaissance Office in January 2020. In 1906 and 1907, Planck published his own conclusion on the behavior of the inertial mass of electrons with high speeds. In his theory, the longitudinal mass m_L= {{\gamma}^3}m and the transverse mass m_T= {\gamma}m, where \gamma is the Lorentz factor and m is the rest mass of the electron. (a) This front view of the apparatus illustrates the uniform acceleration imposed by the charged condenser plates on the beta particles. Suborbital Flight-1 was the first suborbital launch of the rocket. == Launch statistics == === Launch outcomes === === Launch sites === === Booster tests and recoveries === === Rocket configurations === == Orbital launches == === 2017–2018 === Electron experienced its first successful launch in January 2018, and launched their first mission for NASA in December 2018. In this notation, the effective acceleration voltage |V_\parallel| is often expressed as V_0 T. == Transverse voltage == In symbolic analogy to the longitudinal voltage, one can define effective voltages in two orthogonal directions x,y that are transversal to the particle trajectory V_{x,y} = \frac{1}{q} \vec e_{x,y} \cdot \int \vec F_L(s) \exp\left(i k_\beta s\right)\,\mathrm d s which describe the integrated forces that deflect the particle from its design path. Accelerationen (Accelerations), op. 234, is a waltz composed by Johann Strauss II in 1860 for the Engineering Students' Ball at the Sofienbad-Saal in Vienna. ",16,-3.141592,"""-0.029""",3.51,0.2,D
+Identify $\mathrm{X}$ in the following nuclear reactions: (a) ${ }^1 \mathrm{H}+$ ${ }^9 \mathrm{Be} \rightarrow \mathrm{X}+\mathrm{n} ;$ (b) ${ }^{12} \mathrm{C}+{ }^1 \mathrm{H} \rightarrow \mathrm{X} ;$ (c) ${ }^{15} \mathrm{~N}+{ }^1 \mathrm{H} \rightarrow{ }^4 \mathrm{He}+\mathrm{X}$. Appendix F will help.,"Jonathan Feng et al. attribute this 6.8-σ anomaly to a 17 MeV protophobic X-boson dubbed the X17 particle. The X17 particle is a hypothetical subatomic particle proposed by Attila Krasznahorkay and his colleagues to explain certain anomalous measurement results. Krasznahorkay (2019) posted a preprint announcing that he and his team at ATOMKI had successfully observed the same anomalies in the decay of stable helium atoms as had been observed in beryllium-8, strengthening the case for the existence of the X17 particle. Hexanitrostilbene (HNS), also called JD-X, is an organic compound with the formula [(O2N)3C6H2CH]2. The molecular formula C8H11N (molar mass: 121.18 g/mol) may refer to: * Bicyclo(2.2.1)heptane-2-carbonitrile * Collidines (trimethylpyridines) ** 2,3,4-Trimethylpyridine ** 2,3,5-Trimethylpyridine ** 2,3,6-Trimethylpyridine ** 2,4,5-Trimethylpyridine ** 2,4,6-Trimethylpyridine ** 3,4,5-Trimethylpyridine * Dimethylaniline * Phenethylamine * 1-Phenylethylamine * Xylidines ** 2,3-Xylidine ** 2,4-Xylidine ** 2,5-Xylidine ** 2,6-Xylidine ** 3,4-Xylidine ** 3,5-Xylidine Beryllium-8 (8Be, Be-8) is a radionuclide with 4 neutrons and 4 protons. HNC may refer to: *Hydrogen isocyanide, a molecule with the formula HNC that is important to the field of astrochemistry *Heptanitrocubane, an experimental high explosive *Higher National Certificate, a higher education qualification in the United Kingdom *High Negotiations Committee, a Syrian political- military opposition bloc headquartered in Riyadh *Classical-map Hyper-Netted- Chain equation, a method in many-body theoretical physics for interacting uniform electron liquids in two and three dimensions *Hypernetted-chain equation, a closure relation to solve the Ornstein-Zernike equation commonly applied in statistical mechanics and fluid theory *Hopkins-Nanjing Center, a joint educational venture between Nanjing University and Johns Hopkins University located in Nanjing, China *Habits & Contradictions, album by Schoolboy Q *Huddersfield Narrow Canal, Northern England Feng et al. (2016) proposed that a ""protophobic"" X boson, with a mass of , suppressed couplings to protons relative to neutrons and electrons at femtometer range, could explain the data. The X17 particle could be the force carrier for a postulated fifth force, possibly connected with dark matter, and has been described as a protophobic (i.e., ignoring protons) vector boson with a mass near . The NA64 experiment at CERN looks for the proposed X17 particle by striking the electron beams from the Super Proton Synchrotron on fixed target nuclei. ==History== In 2015, Krasznahorkay and his colleagues at ATOMKI, the Hungarian Institute for Nuclear Research, posited the existence of a new, light boson with a mass of about (i.e., 34 times heavier than the electron). However, stable 8Be would enable alternative reaction pathways in helium burning (such as 8Be + 4He and 8Be + 8Be; constituting a ""beryllium burning"" phase) and possibly affect the abundance of the resultant 12C, 16O, and heavier nuclei, though 1H and 4He would remain the most abundant nuclides. CXOU J061705.3+222127 is a neutron star. While further experiments are needed to corroborate these observations, the influence of a fifth boson has been proposed as ""the most straightforward possibility"". == Role in stellar nucleosynthesis == In stellar nucleosynthesis, two helium-4 nuclei may collide and fuse into a single beryllium-8 nucleus. Furthermore, while other alpha nuclides have similar short-lived resonances, 8Be is exceptionally already in the ground state. The nucleus of helium-4 is particularly stable, having a doubly magic configuration and larger binding energy per nucleon than 8Be. Owing to the instability of 8Be, the triple-alpha process is the only reaction in which 12C and heavier elements may be produced in observed quantities. If the beryllium-8 collides with a helium-4 nucleus before decaying, they can fuse into a carbon-12 nucleus. Beryllium-8 has an extremely short half-life (8.19 seconds), and decays back into two helium-4 nuclei. They reported that this populated a nucleus with A = 8 that near- instantaneously decays into two alpha particles. In an effort to find a dark photon, the team fired protons at thin targets of lithium-7, which created unstable beryllium-8 nuclei that then decayed and produced pairs of electrons and positrons. The X‑17 particle is not consistent with the Standard Model, so its existence would need to be explained by another theory. ==See also== * Axion * List of particles * 750 GeV diphoton excess ==References== Category:Bosons Category:Dark matter Category:Hypothetical elementary particles Category:Force carriers This activity was observed again several months later, and was inferred to originate from 8Be. == Properties == left|thumb|300px|Triple-alpha process Beryllium-8 is unbound with respect to alpha emission by 92 keV; it is a resonance having a width of 6 eV. ",0.366,0.020,"""nan""",-87.8,48,C
+"The nucleus of a plutonium-239 atom contains 94 protons. Assume that the nucleus is a sphere with radius $6.64 \mathrm{fm}$ and with the charge of the protons uniformly spread through the sphere. At the surface of the nucleus, what are the magnitude of the electric field produced by the protons?","This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The proton radius is approximately one femtometre = . It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton Plutonium is a radioactive chemical element with the symbol Pu and atomic number 94. When one of these neutrons strikes the nucleus of another 238U atom, it is absorbed by the atom, which becomes 239U. Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). Later studies found an empirical relation between the charge radius and the mass number, A, for heavier nuclei (A > 20): :R ≈ r0A where the empirical constant r0 of 1.2–1.5 fm can be interpreted as the Compton wavelength of the proton. The proton radius puzzle is an unanswered problem in physics relating to the size of the proton. Plutonium-239 (239Pu or Pu-239) is an isotope of plutonium. The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. The known isotopes of plutonium range in mass number from 228 to 247. The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. Occasionally, when an atom of 238U is exposed to neutron radiation, its nucleus will capture a neutron, changing it to 239U. [It] was stocked with 24 pounds of plutonium that produced about 240 watts of electricity when it left Earth in 2006, according to Ryan Bechtel, an engineer from the Department of Energy who works on space nuclear power. For the proton, the two radii are the same. ==History== The first estimate of a nuclear charge radius was made by Hans Geiger and Ernest Marsden in 1909,. under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester, UK. The plutonium grade is usually listed as percentage of 240Pu. The result is again ~5% smaller than the previously-accepted proton radius. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. The fission of one atom of 239Pu generates 207.1 MeV = 3.318 × 10−11 J, i.e. 19.98 TJ/mol = 83.61 TJ/kg, or about 23 gigawatt hours/kg. radiation source (thermal fission of 239Pu) average energy released [MeV] Kinetic energy of fission fragments 175.8 Kinetic energy of prompt neutrons 5.9 Energy carried by prompt γ-rays 7.8 Total instantaneous energy 189.5 Energy of β− particles 5.3 Energy of antineutrinos 7.1 Energy of delayed γ-rays 5.2 Total from decaying fission products 17.6 Energy released by radiative capture of prompt neutrons 11.5 Total heat released in a thermal-spectrum reactor (anti-neutrinos do not contribute) 211.5 == Production == Plutonium is made from uranium-238. 239Pu is normally created in nuclear reactors by transmutation of individual atoms of one of the isotopes of uranium present in the fuel rods. A 5 kg mass of 239Pu contains about atoms. Nucleons have a radius of about 0.8 fm. Weapons- grade plutonium is defined as containing no more than 7% 240Pu; this is achieved by only exposing 238U to neutron sources for short periods of time to minimize the 240Pu produced. ",54.394,6.3,"""25.6773""",1000,3.07,E
 "A nonconducting spherical shell, with an inner radius of $4.0 \mathrm{~cm}$ and an outer radius of $6.0 \mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell $\rho=b / r$, where $r$ is the distance in meters from the center of the shell and $b=3.0 \mu \mathrm{C} / \mathrm{m}^2$. What is the net charge in the shell?
-","It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry The key to the effectiveness of the hollow charge is its diameter. A typical modern shaped charge, with a metal liner on the charge cavity, can penetrate armor steel to a depth of seven or more times the diameter of the charge (charge diameters, CD), though depths of 10 CD and above have been achieved. A shell is a three-dimensional solid structural element whose thickness is very small compared to its other dimensions, and characterized in structural terms by mid-plane stress which is both coplanar and normal to the surface. Octol-loaded charges with a rounded cone apex generally had higher surface temperatures with an average of 810 K, and the temperature of a tin- lead liner with Comp-B fill averaged 842 K. A 66-pound shaped charge accelerated the gas in a 3-cm glass-walled tube 2 meters in length. Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. A shaped charge is an explosive charge shaped to focus the effect of the explosive's energy. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. The History of Shaped Charges"", Technical Report BRL-TR-3158, U.S. Army Laboratory Command, Ballistic Research Laboratory (Aberdeen Proving Ground, Maryland), p. Available on-line at: Defense Technical Information Center During World War II, shaped-charge munitions were developed by Germany (Panzerschreck, Panzerfaust, Panzerwurfmine, Mistel), Britain (PIAT, Beehive cratering charge), the Soviet Union (RPG-43, RPG-6), the U.S. (bazooka),Donald R. Kennedy, ""History of the Shaped Charge Effect: The First 100 Years "", D.R. Kennedy and Associates, Inc., Mountain View, California, 1983 and Italy (Effetto Pronto Speciale shells for various artillery pieces).https://comandosupremo.com/forums/index.php?attachments/ep_eps-jpg.59/ The development of shaped charges revolutionized anti-tank warfare. In probability and statistics, a spherical contact distribution function, first contact distribution function,D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. *William P. Walters (September 1990) ""The Shaped Charge Concept. A description of Munroe's first shaped-charge experiment appears on p. 453. > Among the experiments made ... was one upon a safe twenty-nine inches cube, > with walls four inches and three quarters thick, made up of plates of iron > and steel ... Shaped charges are used most extensively in the petroleum and natural gas industries, in particular in the completion of oil and gas wells, in which they are detonated to perforate the metal casing of the well at intervals to admit the influx of oil and gas. The location of the charge relative to its target is critical for optimum penetration for two reasons. Contrary to a misconception (possibly resulting from the acronym for high- explosive anti-tank, HEAT) the shaped charge does not depend in any way on heating or melting for its effectiveness; that is, the jet from a shaped charge does not melt its way through armor, as its effect is purely kinetic in nature – however the process does create significant heat and often has a significant secondary incendiary effect after penetration. ==Munroe effect== The Munroe or Neumann effect is the focusing of blast energy by a hollow or void cut on a surface of an explosive. Although Munroe's experiment with the shaped charge was widely publicized in 1900 in Popular Science Monthly, the importance of the tin can ""liner"" of the hollow charge remained unrecognized for another 44 years.Kennedy (1990), p. The limit of this approximation is the shell integral. In general, shaped charges can penetrate a steel plate as thick as 150% to 700%Jane's Ammunition Handbook 1994, pp. 140–141, addresses the reported ≈700 mm penetration of the Swedish 106 3A-HEAT-T and Austrian RAT 700 HEAT projectiles for the 106 mm M40A1 recoilless rifle. of their diameter, depending on the charge quality. ",0.22222222,0.32,1838.50666349,+37,3.8,E
+","It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry The key to the effectiveness of the hollow charge is its diameter. A typical modern shaped charge, with a metal liner on the charge cavity, can penetrate armor steel to a depth of seven or more times the diameter of the charge (charge diameters, CD), though depths of 10 CD and above have been achieved. A shell is a three-dimensional solid structural element whose thickness is very small compared to its other dimensions, and characterized in structural terms by mid-plane stress which is both coplanar and normal to the surface. Octol-loaded charges with a rounded cone apex generally had higher surface temperatures with an average of 810 K, and the temperature of a tin- lead liner with Comp-B fill averaged 842 K. A 66-pound shaped charge accelerated the gas in a 3-cm glass-walled tube 2 meters in length. Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. A shaped charge is an explosive charge shaped to focus the effect of the explosive's energy. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. The History of Shaped Charges"", Technical Report BRL-TR-3158, U.S. Army Laboratory Command, Ballistic Research Laboratory (Aberdeen Proving Ground, Maryland), p. Available on-line at: Defense Technical Information Center During World War II, shaped-charge munitions were developed by Germany (Panzerschreck, Panzerfaust, Panzerwurfmine, Mistel), Britain (PIAT, Beehive cratering charge), the Soviet Union (RPG-43, RPG-6), the U.S. (bazooka),Donald R. Kennedy, ""History of the Shaped Charge Effect: The First 100 Years "", D.R. Kennedy and Associates, Inc., Mountain View, California, 1983 and Italy (Effetto Pronto Speciale shells for various artillery pieces).https://comandosupremo.com/forums/index.php?attachments/ep_eps-jpg.59/ The development of shaped charges revolutionized anti-tank warfare. In probability and statistics, a spherical contact distribution function, first contact distribution function,D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. *William P. Walters (September 1990) ""The Shaped Charge Concept. A description of Munroe's first shaped-charge experiment appears on p. 453. > Among the experiments made ... was one upon a safe twenty-nine inches cube, > with walls four inches and three quarters thick, made up of plates of iron > and steel ... Shaped charges are used most extensively in the petroleum and natural gas industries, in particular in the completion of oil and gas wells, in which they are detonated to perforate the metal casing of the well at intervals to admit the influx of oil and gas. The location of the charge relative to its target is critical for optimum penetration for two reasons. Contrary to a misconception (possibly resulting from the acronym for high- explosive anti-tank, HEAT) the shaped charge does not depend in any way on heating or melting for its effectiveness; that is, the jet from a shaped charge does not melt its way through armor, as its effect is purely kinetic in nature – however the process does create significant heat and often has a significant secondary incendiary effect after penetration. ==Munroe effect== The Munroe or Neumann effect is the focusing of blast energy by a hollow or void cut on a surface of an explosive. Although Munroe's experiment with the shaped charge was widely publicized in 1900 in Popular Science Monthly, the importance of the tin can ""liner"" of the hollow charge remained unrecognized for another 44 years.Kennedy (1990), p. The limit of this approximation is the shell integral. In general, shaped charges can penetrate a steel plate as thick as 150% to 700%Jane's Ammunition Handbook 1994, pp. 140–141, addresses the reported ≈700 mm penetration of the Swedish 106 3A-HEAT-T and Austrian RAT 700 HEAT projectiles for the 106 mm M40A1 recoilless rifle. of their diameter, depending on the charge quality. ",0.22222222,0.32,"""1838.50666349""",+37,3.8,E
 "An electron is shot directly
-Figure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?","thumb|upright=1.3|right|Launch of Electron in start of the ""Birds of the Feather"" mission. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. First mid-air helicopter capture attempt of an Electron first stage following launch. First flight of Electron with a fully autonomous flight termination system on the rocket. === 2020 === First launch for the National Reconnaissance Office in January 2020. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. In physics, a projectile launched with specific initial conditions will have a range. thumb|Scheme of two types of electron capture. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Second stage engine shut down early causing the mission to be lost, but Electron's first stage safely completed a successful splashdown under parachute. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. First, as previously noted, the electron must absorb an amount of energy equivalent to the energy difference between the electron's current energy level and an unoccupied, higher energy level in order to be promoted to that energy level. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The second solution is the useful one for determining the range of the projectile. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: * constrained global optimization (Altschuler et al. 1994), * steepest descent (Claxton and Benson 1966, Erber and Hockney 1991), * random walk (Weinrach et al. 1990), * genetic algorithm (Morris et al. 1996) While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest. ===Continuous spherical shell charge=== thumb|The extreme upper energy limit of the Thomson Problem is given by N^2/2 for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Flight 26 was the first Electron flight to attempt a full catch recovery using a mid-air helicopter catch. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge. Surface Charging and Points of Zero Charge. The energy released is equal to the difference in energy levels between the electron energy states. Suborbital Flight-1 was the first suborbital launch of the rocket. == Launch statistics == === Launch outcomes === === Launch sites === === Booster tests and recoveries === === Rocket configurations === == Orbital launches == === 2017–2018 === Electron experienced its first successful launch in January 2018, and launched their first mission for NASA in December 2018. ",1.5,0.42,0.44,-0.5,4152,C
-"A square metal plate of edge length $8.0 \mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \mathrm{~mm}$ from the center) by assuming that the charge is spread uniformly over the two faces of the plate. ","We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. As an example, consider a charged spherical shell of negligible thickness, with a uniformly distributed charge and radius . We can similarly describe the electric field E so that . Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. Because the field close to the sheet can be approximated as constant, the pillbox is oriented in a way so that the field lines penetrate the disks at the ends of the field at a perpendicular angle and the side of the cylinder are parallel to the field lines. ==See also== * Area * Surface area * Vector calculus * Integration * Divergence theorem * Faraday cage * Field theory * Field line == References == * * ==Further reading== * Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ==External links== * Fields - a chapter from an online textbook Category:Surfaces Category:Electrostatics Category:Carl Friedrich Gauss Thereby is the electrical charge enclosed by the Gaussian surface. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example ""field near infinite line charge"" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. With the same example, using a larger Gaussian surface outside the shell where , Gauss's law will produce a non-zero electric field. ",10.8,0.70710678,2.3613,-0.16,5.4,E
+Figure 23-50 Problem 40. toward the center of a large metal plate that has surface charge density $-2.0 \times 10^{-6} \mathrm{C} / \mathrm{m}^2$. If the initial kinetic energy of the electron is $1.60 \times 10^{-17} \mathrm{~J}$ and if the electron is to stop (due to electrostatic repulsion from the plate) just as it reaches the plate, how far from the plate must the launch point be?","thumb|upright=1.3|right|Launch of Electron in start of the ""Birds of the Feather"" mission. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Electron is a two-stage small-lift launch vehicle built and operated by Rocket Lab. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. First mid-air helicopter capture attempt of an Electron first stage following launch. First flight of Electron with a fully autonomous flight termination system on the rocket. === 2020 === First launch for the National Reconnaissance Office in January 2020. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. In physics, a projectile launched with specific initial conditions will have a range. thumb|Scheme of two types of electron capture. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Second stage engine shut down early causing the mission to be lost, but Electron's first stage safely completed a successful splashdown under parachute. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. First, as previously noted, the electron must absorb an amount of energy equivalent to the energy difference between the electron's current energy level and an unoccupied, higher energy level in order to be promoted to that energy level. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The second solution is the useful one for determining the range of the projectile. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: * constrained global optimization (Altschuler et al. 1994), * steepest descent (Claxton and Benson 1966, Erber and Hockney 1991), * random walk (Weinrach et al. 1990), * genetic algorithm (Morris et al. 1996) While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest. ===Continuous spherical shell charge=== thumb|The extreme upper energy limit of the Thomson Problem is given by N^2/2 for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Flight 26 was the first Electron flight to attempt a full catch recovery using a mid-air helicopter catch. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge. Surface Charging and Points of Zero Charge. The energy released is equal to the difference in energy levels between the electron energy states. Suborbital Flight-1 was the first suborbital launch of the rocket. == Launch statistics == === Launch outcomes === === Launch sites === === Booster tests and recoveries === === Rocket configurations === == Orbital launches == === 2017–2018 === Electron experienced its first successful launch in January 2018, and launched their first mission for NASA in December 2018. ",1.5,0.42,"""0.44""",-0.5,4152,C
+"A square metal plate of edge length $8.0 \mathrm{~cm}$ and negligible thickness has a total charge of $6.0 \times 10^{-6} \mathrm{C}$. Estimate the magnitude $E$ of the electric field just off the center of the plate (at, say, a distance of $0.50 \mathrm{~mm}$ from the center) by assuming that the charge is spread uniformly over the two faces of the plate. ","We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. As an example, consider a charged spherical shell of negligible thickness, with a uniformly distributed charge and radius . We can similarly describe the electric field E so that . Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. Because the field close to the sheet can be approximated as constant, the pillbox is oriented in a way so that the field lines penetrate the disks at the ends of the field at a perpendicular angle and the side of the cylinder are parallel to the field lines. ==See also== * Area * Surface area * Vector calculus * Integration * Divergence theorem * Faraday cage * Field theory * Field line == References == * * ==Further reading== * Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ==External links== * Fields - a chapter from an online textbook Category:Surfaces Category:Electrostatics Category:Carl Friedrich Gauss Thereby is the electrical charge enclosed by the Gaussian surface. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example ""field near infinite line charge"" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. With the same example, using a larger Gaussian surface outside the shell where , Gauss's law will produce a non-zero electric field. ",10.8,0.70710678,"""2.3613""",-0.16,5.4,E
 "A neutron consists of one ""up"" quark of charge $+2 e / 3$ and two ""down"" quarks each having charge $-e / 3$. If we assume that the down quarks are $2.6 \times 10^{-15} \mathrm{~m}$ apart inside the neutron, what is the magnitude of the electrostatic force between them?
-","Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. This same force is much weaker between neutrons and protons, because it is mostly neutralized within them, in the same way that electromagnetic forces between neutral atoms (van der Waals forces) are much weaker than the electromagnetic forces that hold electrons in association with the nucleus, forming the atoms. A Pauli repulsion also occurs between quarks of the same flavour from different nucleons (a proton and a neutron). ===Field strength=== At distances larger than 0.7 fm the force becomes attractive between spin-aligned nucleons, becoming maximal at a center–center distance of about 0.9 fm. Beyond this distance the force drops exponentially, until beyond about 2.0 fm separation, the force is negligible. The strong attraction between nucleons was the side-effect of a more fundamental force that bound the quarks together into protons and neutrons. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). In the context of atomic nuclei, the same strong interaction force (that binds quarks within a nucleon) also binds protons and neutrons together to form a nucleus. The strong interaction is the attractive force that binds the elementary particles called quarks together to form the nucleons (protons and neutrons) themselves. On the smaller scale (less than about 0.8 fm, the radius of a nucleon), it is the force (carried by gluons) that holds quarks together to form protons, neutrons, and other hadron particles. The strong force acts between quarks. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Unlike all other forces (electromagnetic, weak, and gravitational), the strong force does not diminish in strength with increasing distance between pairs of quarks. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. On a larger scale (of about 1 to 3 fm), it is the force (carried by mesons) that binds protons and neutrons (nucleons) together to form the nucleus of an atom. For identical nucleons (such as two neutrons or two protons) this repulsion arises from the Pauli exclusion force. It was known that the nucleus was composed of protons and neutrons and that protons possessed positive electric charge, while neutrons were electrically neutral. The residual strong force is thus a minor residuum of the strong force that binds quarks together into protons and neutrons. The larger the neutron cross section, the more likely a neutron will react with the nucleus. thumb|A conjectural example of an interaction between two neutrons and a proton, the triton or hydrogen-3, which is beta unstable. The nuclear force is nearly independent of whether the nucleons are neutrons or protons. After the verification of the quark model, strong interaction has come to mean QCD. ==Nucleon–nucleon potentials== Two- nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. In nuclear physics, the concept of a neutron cross section is used to express the likelihood of interaction between an incident neutron and a target nucleus. Most of the mass of a common proton or neutron is the result of the strong interaction energy; the individual quarks provide only about 1% of the mass of a proton. ",0,47,635013559600.0,0.5768,3.8,E
-"In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit?","In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. This means that the innermost electrons orbit at approximately 1/2 the Bohr radius. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. In atomic physics, the Bohr model or Rutherford–Bohr model of the atom, presented by Niels Bohr and Ernest Rutherford in 1913, consists of a small, dense nucleus surrounded by orbiting electrons. :The smallest possible value of r in the hydrogen atom () is called the Bohr radius and is equal to: :: r_1 = \frac{\hbar^2}{k_\mathrm{e} e^2 m_\mathrm{e}} \approx 5.29 \times 10^{-11}~\mathrm{m}. Bohr considered circular orbits. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. Once an electron is in this lowest orbit, it can get no closer to the nucleus. This outer electron should be at nearly one Bohr radius from the nucleus. Bohr described angular momentum of the electron orbit as 1/2h while de Broglie's wavelength of described h divided by the electron momentum. This equation determines the electron's speed at any radius: :: v = \sqrt{\frac{Zk_\mathrm{e} e^2}{m_\mathrm{e} r}}. For a hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r3/2. In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability that grows denser near the nucleus. The de Broglie wavelength of an electron is : \lambda = \frac{h}{mv}, which implies that : \frac{nh}{mv} = 2 \pi r, or : \frac{nh}{2 \pi} = mvr, where mvr is the angular momentum of the orbiting electron. It does not work for (neutral) helium. == Refinements == thumb|Elliptical orbits with the same energy and quantized angular momentum Several enhancements to the Bohr model were proposed, most notably the Sommerfeld or Bohr–Sommerfeld models, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. The lowest value of n is 1; this gives the smallest possible orbital radius of 0.0529 nm known as the Bohr radius. An electron in the lowest energy level of hydrogen () therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a ""coincidence"". ", 7.0,3.23,4.0,2.19,0.064,D
-At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?,"thumb|upright=1.35|An ellipse, its minimum bounding box, and its director circle. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). This measure is strictly a ratio of diameters and should not be confused with the covered fraction of the apparent area (disk) of the eclipsed body. A line segment through a circle's center bisecting a chord is perpendicular to the chord. thumb|right|170px Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis ""parallel"" to the axis of revolution. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. thumb|An annular solar eclipse has a magnitude of less than 1.0 The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. In an annular solar eclipse, the magnitude of the eclipse is the ratio between the apparent angular diameters of the Moon and that of the Sun during the maximum eclipse, yielding a ratio less than 1.0. It has the same center as the ellipse, with radius \sqrt{a^2+b^2}, where a and b are the semi- major axis and semi-minor axis of the ellipse. This can be calculated in a single integral similar to the following: :\pi\int_a^b\left(R_\mathrm{O}(x)^2 - R_\mathrm{I}(x)^2\right)\,dx where is the function that is farthest from the axis of rotation and is the function that is closest to the axis of rotation. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. It has an eccentricity equal to \sqrt{2}. ==In polygons== ===Triangles=== The legs of a right triangle are perpendicular to each other. In many cases, it makes more sense to take the distance between points where the intensity falls to 1/e2 = 0.135 times the maximum value. We can similarly describe the electric field E so that . * Line c is perpendicular to line a. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. Note that when integrating along an axis other than the , the graph of the function that is farthest from the axis of rotation may not be that obvious. The magnitude of a partial or annular solar eclipse is always between 0.0 and 1.0, while the magnitude of a total solar eclipse is always greater than or equal to 1.0. ",15.757,0.346,0.166666666,420,0,B
-"Of the charge $Q$ on a tiny sphere, a fraction $\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles.  What value of $\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres? ","The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). Minimum bounding circle may refer to: * Bounding sphere * Smallest circle problem For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). The interaction free energy involving two spherical particles within the DH approximation follows the Yukawa or screened Coulomb potential : U = \frac{Q^2}{4 \pi \epsilon \epsilon_0} \left( \frac{e^{\kappa a}}{1+\kappa a} \right)^2 \frac{e^{-\kappa r}}{r} where r is the center-to- center distance, Q is the particle charge, and a the particle radius. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to- charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. ""Min-Energy Configurations of Electrons On A Sphere"". In this case, the surface charge density decreases upon approach. At larger distances, oppositely charged surfaces repel and equally charged ones attract. ==Charge regulating surfaces== While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations. ",12,3857,-4564.7,0.5,6760000,D
-"In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \times 10^{-13} \mathrm{C} / \mathrm{m}^2$. What is the magnitude of the electron's acceleration when it reaches radial distances $r=0.500 R$?","The radius r is then defined to be the classical electron radius, r_\text{e}, and one arrives at the expression given above. This asymmetric distribution of charge within the particle gives rise to a small negative squared charge radius for the particle as a whole. The electrostatic potential at a distance r from a charge q is :V(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Note that this derivation does not say that r_\text{e} is the actual radius of an electron. It is customary when charge radius takes an imaginary numbered value to report the negative valued square of the charge radius, rather than the charge radius itself, for a particle. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The classical electron radius is given as :r_\text{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\text{e}} c^2} = 2.817 940 3227(19) \times 10^{-15} \text{ m} = 2.817 940 3227(19) \text{ fm} , where e is the elementary charge, m_{\text{e}} is the electron mass, c is the speed of light, and \varepsilon_0 is the permittivity of free space.David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 155. If the sphere is assumed to have constant charge density, \rho, then :q = \rho \frac{4}{3} \pi r^3 and dq = \rho 4 \pi r^2 dr. Integrating for r from zero to the final radius r yields the expression for the total energy U, necessary to assemble the total charge q into a uniform sphere of radius r: :U = \frac{1}{4\pi\varepsilon_0} \frac{3}{5} \frac{q^2}{r}. This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The single electron may reside at any point on the surface of the unit sphere. The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. Although it is sometimes stated that all the electrons in a shell have the same energy, this is an approximation. This numerical value is several times larger than the radius of the proton. The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. The best known particle with a negative squared charge radius is the neutron. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length. Surface Charging and Points of Zero Charge. ",0,49,2.0,96.4365076099,243,A
-A particle of charge $+3.00 \times 10^{-6} \mathrm{C}$ is $12.0 \mathrm{~cm}$ distant from a second particle of charge $-1.50 \times 10^{-6} \mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles.,"If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. Coulomb's law in vector form states that the electrostatic force \mathbf{F}_1 experienced by a charge, q_1 at position \mathbf{r}_1, in the vicinity of another charge, q_2 at position \mathbf{r}_2, in a vacuum is equal to \mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2} where \boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2 is the vectorial distance between the charges, \widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|} a unit vector pointing from q_2 to and \varepsilon_0 the electric constant. The force is along the straight line joining the two charges. If both charges have the same sign (like charges) then the product q_1q_2 is positive and the direction of the force on q_1 is given by \widehat{\mathbf{r}}_{12}; the charges repel each other. If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. == Vector form == thumb|right|350px|In the image, the vector is the force experienced by , and the vector is the force experienced by . The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. If the charges have opposite signs then the product q_1q_2 is negative and the direction of the force on q_1 is _{12};}} the charges attract each other. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In 1767, he conjectured that the force between charges varied as the inverse square of the distance. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. The magnitude of the electric field can be derived from Coulomb's law. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. Value Item 10−21 zepto- (zC) 10−20 (−1/3 e) – Charge of down, strange and bottom quarks 10−19 (2/3 e)—Charge of up, charm and top quarks 10−19 The elementary charge e, i.e. the negative charge on a single electron or the positive charge on a single proton 10−18 atto- (aC) ~ Planck chargePlanck Units 10−17 (92 e) – Positive charge on a uranium nucleus (derived: 92 x ) 10−16 Charge on a dust particle in a plasma 10−15 femto- (fC) Charge on a typical dust particle 10−12 pico- (pC) Charge in typical microwave frequency capacitors 10−9 nano- (nC) Charge in typical radio frequency capacitors 10−6 micro- (μC) Charge in typical audio frequency capacitors 10−6 micro- (μC) ~ Static electricity from rubbing materials together 10−3 milli- (mC) Charge in typical power supply capacitors 10−3 milli- (mC) Charge in CH85-2100-105 high voltage capacitor for microwaves 100 C Two like charges, each of , placed one meter apart, would experience a repulsive force of approximately 100 C Supercapacitor for real-time clock (RTC) (1F x 3.6V) 101 deca- (daC) Charge in a typical thundercloud Hasbrouck, Richard. When the electromagnetic theory is expressed in the International System of Units, force is measured in newtons, charge in coulombs and distance in meters. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. Electrostatic charge on an object can be measured by placing it into the Faraday Cup. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field created by a single source point charge Q at a certain distance from it r in vacuum is given by |\mathbf{E}| = k_\text{e} \frac{|q|}{r^2} A system N of charges q_i stationed at \mathbf{r}_i produces an electric field whose magnitude and direction is, by superposition \mathbf{E}(\mathbf{r}) = {1\over4\pi\varepsilon_0} \sum_{i=1}^N q_i \frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3} == Atomic forces == Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons. The force on a small test charge q at position \boldsymbol{r} in vacuum is given by the integral over the distribution of charge \mathbf{F}(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0}\int dq' \frac{\mathbf{r} - \mathbf{r'}}{|\mathbf{r} - \mathbf{r'}|^3}. where it the ""continuous charge"" version of Coulomb's law is never supposed to be applied to locations for which |\mathbf{r} - \mathbf{r'}| = 0 because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England. This electric force is conventionally called electrostatic force or Coulomb force. ",1.39,0.1591549431,3.8,5.7,2.81,E
+","Neutrons and protons, both nucleons, are affected by the nuclear force almost identically. This same force is much weaker between neutrons and protons, because it is mostly neutralized within them, in the same way that electromagnetic forces between neutral atoms (van der Waals forces) are much weaker than the electromagnetic forces that hold electrons in association with the nucleus, forming the atoms. A Pauli repulsion also occurs between quarks of the same flavour from different nucleons (a proton and a neutron). ===Field strength=== At distances larger than 0.7 fm the force becomes attractive between spin-aligned nucleons, becoming maximal at a center–center distance of about 0.9 fm. Beyond this distance the force drops exponentially, until beyond about 2.0 fm separation, the force is negligible. The strong attraction between nucleons was the side-effect of a more fundamental force that bound the quarks together into protons and neutrons. By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). In the context of atomic nuclei, the same strong interaction force (that binds quarks within a nucleon) also binds protons and neutrons together to form a nucleus. The strong interaction is the attractive force that binds the elementary particles called quarks together to form the nucleons (protons and neutrons) themselves. On the smaller scale (less than about 0.8 fm, the radius of a nucleon), it is the force (carried by gluons) that holds quarks together to form protons, neutrons, and other hadron particles. The strong force acts between quarks. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Unlike all other forces (electromagnetic, weak, and gravitational), the strong force does not diminish in strength with increasing distance between pairs of quarks. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. On a larger scale (of about 1 to 3 fm), it is the force (carried by mesons) that binds protons and neutrons (nucleons) together to form the nucleus of an atom. For identical nucleons (such as two neutrons or two protons) this repulsion arises from the Pauli exclusion force. It was known that the nucleus was composed of protons and neutrons and that protons possessed positive electric charge, while neutrons were electrically neutral. The residual strong force is thus a minor residuum of the strong force that binds quarks together into protons and neutrons. The larger the neutron cross section, the more likely a neutron will react with the nucleus. thumb|A conjectural example of an interaction between two neutrons and a proton, the triton or hydrogen-3, which is beta unstable. The nuclear force is nearly independent of whether the nucleons are neutrons or protons. After the verification of the quark model, strong interaction has come to mean QCD. ==Nucleon–nucleon potentials== Two- nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. In nuclear physics, the concept of a neutron cross section is used to express the likelihood of interaction between an incident neutron and a target nucleus. Most of the mass of a common proton or neutron is the result of the strong interaction energy; the individual quarks provide only about 1% of the mass of a proton. ",0,47,"""635013559600.0""",0.5768,3.8,E
+"In an early model of the hydrogen atom (the Bohr model), the electron orbits the proton in uniformly circular motion. The radius of the circle is restricted (quantized) to certain values given by where $a_0=52.92 \mathrm{pm}$. What is the speed of the electron if it orbits in the smallest allowed orbit?","In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. This means that the innermost electrons orbit at approximately 1/2 the Bohr radius. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. In atomic physics, the Bohr model or Rutherford–Bohr model of the atom, presented by Niels Bohr and Ernest Rutherford in 1913, consists of a small, dense nucleus surrounded by orbiting electrons. :The smallest possible value of r in the hydrogen atom () is called the Bohr radius and is equal to: :: r_1 = \frac{\hbar^2}{k_\mathrm{e} e^2 m_\mathrm{e}} \approx 5.29 \times 10^{-11}~\mathrm{m}. Bohr considered circular orbits. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. Once an electron is in this lowest orbit, it can get no closer to the nucleus. This outer electron should be at nearly one Bohr radius from the nucleus. Bohr described angular momentum of the electron orbit as 1/2h while de Broglie's wavelength of described h divided by the electron momentum. This equation determines the electron's speed at any radius: :: v = \sqrt{\frac{Zk_\mathrm{e} e^2}{m_\mathrm{e} r}}. For a hydrogen atom, the classical orbits have a period T determined by Kepler's third law to scale as r3/2. In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability that grows denser near the nucleus. The de Broglie wavelength of an electron is : \lambda = \frac{h}{mv}, which implies that : \frac{nh}{mv} = 2 \pi r, or : \frac{nh}{2 \pi} = mvr, where mvr is the angular momentum of the orbiting electron. It does not work for (neutral) helium. == Refinements == thumb|Elliptical orbits with the same energy and quantized angular momentum Several enhancements to the Bohr model were proposed, most notably the Sommerfeld or Bohr–Sommerfeld models, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. The lowest value of n is 1; this gives the smallest possible orbital radius of 0.0529 nm known as the Bohr radius. An electron in the lowest energy level of hydrogen () therefore has about 13.6 eV less energy than a motionless electron infinitely far from the nucleus. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree is considered a ""coincidence"". ", 7.0,3.23,"""4.0""",2.19,0.064,D
+At what distance along the central perpendicular axis of a uniformly charged plastic disk of radius $0.600 \mathrm{~m}$ is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk?,"thumb|upright=1.35|An ellipse, its minimum bounding box, and its director circle. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. An obvious choice for this fraction is ½ (−3 dB), in which case the diameter obtained is the full width of the beam at half its maximum intensity (FWHM). This measure is strictly a ratio of diameters and should not be confused with the covered fraction of the apparent area (disk) of the eclipsed body. A line segment through a circle's center bisecting a chord is perpendicular to the chord. thumb|right|170px Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis ""parallel"" to the axis of revolution. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. thumb|An annular solar eclipse has a magnitude of less than 1.0 The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The beam diameter or beam width of an electromagnetic beam is the diameter along any specified line that is perpendicular to the beam axis and intersects it. In an annular solar eclipse, the magnitude of the eclipse is the ratio between the apparent angular diameters of the Moon and that of the Sun during the maximum eclipse, yielding a ratio less than 1.0. It has the same center as the ellipse, with radius \sqrt{a^2+b^2}, where a and b are the semi- major axis and semi-minor axis of the ellipse. This can be calculated in a single integral similar to the following: :\pi\int_a^b\left(R_\mathrm{O}(x)^2 - R_\mathrm{I}(x)^2\right)\,dx where is the function that is farthest from the axis of rotation and is the function that is closest to the axis of rotation. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. It has an eccentricity equal to \sqrt{2}. ==In polygons== ===Triangles=== The legs of a right triangle are perpendicular to each other. In many cases, it makes more sense to take the distance between points where the intensity falls to 1/e2 = 0.135 times the maximum value. We can similarly describe the electric field E so that . * Line c is perpendicular to line a. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. Note that when integrating along an axis other than the , the graph of the function that is farthest from the axis of rotation may not be that obvious. The magnitude of a partial or annular solar eclipse is always between 0.0 and 1.0, while the magnitude of a total solar eclipse is always greater than or equal to 1.0. ",15.757,0.346,"""0.166666666""",420,0,B
+"Of the charge $Q$ on a tiny sphere, a fraction $\alpha$ is to be transferred to a second, nearby sphere. The spheres can be treated as particles.  What value of $\alpha$ maximizes the magnitude $F$ of the electrostatic force between the two spheres? ","The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). Minimum bounding circle may refer to: * Bounding sphere * Smallest circle problem For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). The interaction free energy involving two spherical particles within the DH approximation follows the Yukawa or screened Coulomb potential : U = \frac{Q^2}{4 \pi \epsilon \epsilon_0} \left( \frac{e^{\kappa a}}{1+\kappa a} \right)^2 \frac{e^{-\kappa r}}{r} where r is the center-to- center distance, Q is the particle charge, and a the particle radius. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to- charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. ""Min-Energy Configurations of Electrons On A Sphere"". In this case, the surface charge density decreases upon approach. At larger distances, oppositely charged surfaces repel and equally charged ones attract. ==Charge regulating surfaces== While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations. ",12,3857,"""-4564.7""",0.5,6760000,D
+"In a spherical metal shell of radius $R$, an electron is shot from the center directly toward a tiny hole in the shell, through which it escapes. The shell is negatively charged with a surface charge density (charge per unit area) of $6.90 \times 10^{-13} \mathrm{C} / \mathrm{m}^2$. What is the magnitude of the electron's acceleration when it reaches radial distances $r=0.500 R$?","The radius r is then defined to be the classical electron radius, r_\text{e}, and one arrives at the expression given above. This asymmetric distribution of charge within the particle gives rise to a small negative squared charge radius for the particle as a whole. The electrostatic potential at a distance r from a charge q is :V(r) = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. Note that this derivation does not say that r_\text{e} is the actual radius of an electron. It is customary when charge radius takes an imaginary numbered value to report the negative valued square of the charge radius, rather than the charge radius itself, for a particle. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The classical electron radius is given as :r_\text{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\text{e}} c^2} = 2.817 940 3227(19) \times 10^{-15} \text{ m} = 2.817 940 3227(19) \text{ fm} , where e is the elementary charge, m_{\text{e}} is the electron mass, c is the speed of light, and \varepsilon_0 is the permittivity of free space.David J. Griffiths, Introduction to Quantum Mechanics, Prentice-Hall, 1995, p. 155. If the sphere is assumed to have constant charge density, \rho, then :q = \rho \frac{4}{3} \pi r^3 and dq = \rho 4 \pi r^2 dr. Integrating for r from zero to the final radius r yields the expression for the total energy U, necessary to assemble the total charge q into a uniform sphere of radius r: :U = \frac{1}{4\pi\varepsilon_0} \frac{3}{5} \frac{q^2}{r}. This gives a charge radius for the gold nucleus (A = 197) of about 7.69 fm.. ==Modern measurements== Modern direct measurements are based on precision measurements of the atomic energy levels in hydrogen and deuterium, and measurements of scattering of electrons by nuclei... The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The single electron may reside at any point on the surface of the unit sphere. The rms charge radius is a measure of the size of an atomic nucleus, particularly the proton distribution. Although it is sometimes stated that all the electrons in a shell have the same energy, this is an approximation. This numerical value is several times larger than the radius of the proton. The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. The best known particle with a negative squared charge radius is the neutron. According to modern understanding, the electron is a point particle with a point charge and no spatial extent. The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length. Surface Charging and Points of Zero Charge. ",0,49,"""2.0""",96.4365076099,243,A
+A particle of charge $+3.00 \times 10^{-6} \mathrm{C}$ is $12.0 \mathrm{~cm}$ distant from a second particle of charge $-1.50 \times 10^{-6} \mathrm{C}$. Calculate the magnitude of the electrostatic force between the particles.,"If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. Coulomb's law in vector form states that the electrostatic force \mathbf{F}_1 experienced by a charge, q_1 at position \mathbf{r}_1, in the vicinity of another charge, q_2 at position \mathbf{r}_2, in a vacuum is equal to \mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2} where \boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2 is the vectorial distance between the charges, \widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|} a unit vector pointing from q_2 to and \varepsilon_0 the electric constant. The force is along the straight line joining the two charges. If both charges have the same sign (like charges) then the product q_1q_2 is positive and the direction of the force on q_1 is given by \widehat{\mathbf{r}}_{12}; the charges repel each other. If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. == Vector form == thumb|right|350px|In the image, the vector is the force experienced by , and the vector is the force experienced by . The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. If the charges have opposite signs then the product q_1q_2 is negative and the direction of the force on q_1 is _{12};}} the charges attract each other. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In 1767, he conjectured that the force between charges varied as the inverse square of the distance. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. The magnitude of the electric field can be derived from Coulomb's law. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. Value Item 10−21 zepto- (zC) 10−20 (−1/3 e) – Charge of down, strange and bottom quarks 10−19 (2/3 e)—Charge of up, charm and top quarks 10−19 The elementary charge e, i.e. the negative charge on a single electron or the positive charge on a single proton 10−18 atto- (aC) ~ Planck chargePlanck Units 10−17 (92 e) – Positive charge on a uranium nucleus (derived: 92 x ) 10−16 Charge on a dust particle in a plasma 10−15 femto- (fC) Charge on a typical dust particle 10−12 pico- (pC) Charge in typical microwave frequency capacitors 10−9 nano- (nC) Charge in typical radio frequency capacitors 10−6 micro- (μC) Charge in typical audio frequency capacitors 10−6 micro- (μC) ~ Static electricity from rubbing materials together 10−3 milli- (mC) Charge in typical power supply capacitors 10−3 milli- (mC) Charge in CH85-2100-105 high voltage capacitor for microwaves 100 C Two like charges, each of , placed one meter apart, would experience a repulsive force of approximately 100 C Supercapacitor for real-time clock (RTC) (1F x 3.6V) 101 deca- (daC) Charge in a typical thundercloud Hasbrouck, Richard. When the electromagnetic theory is expressed in the International System of Units, force is measured in newtons, charge in coulombs and distance in meters. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. Electrostatic charge on an object can be measured by placing it into the Faraday Cup. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field created by a single source point charge Q at a certain distance from it r in vacuum is given by |\mathbf{E}| = k_\text{e} \frac{|q|}{r^2} A system N of charges q_i stationed at \mathbf{r}_i produces an electric field whose magnitude and direction is, by superposition \mathbf{E}(\mathbf{r}) = {1\over4\pi\varepsilon_0} \sum_{i=1}^N q_i \frac{\mathbf{r}-\mathbf{r}_i}{|\mathbf{r}-\mathbf{r}_i|^3} == Atomic forces == Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons. The force on a small test charge q at position \boldsymbol{r} in vacuum is given by the integral over the distribution of charge \mathbf{F}(\mathbf{r}) = \frac{q}{4\pi\varepsilon_0}\int dq' \frac{\mathbf{r} - \mathbf{r'}}{|\mathbf{r} - \mathbf{r'}|^3}. where it the ""continuous charge"" version of Coulomb's law is never supposed to be applied to locations for which |\mathbf{r} - \mathbf{r'}| = 0 because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England. This electric force is conventionally called electrostatic force or Coulomb force. ",1.39,0.1591549431,"""3.8""",5.7,2.81,E
 "A charged particle produces an electric field with a magnitude of $2.0 \mathrm{~N} / \mathrm{C}$ at a point that is $50 \mathrm{~cm}$ away from the particle. What is the magnitude of the particle's charge?
-","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The voltage between two points is defined as:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 685–686 {\Delta V} = -\int {\mathbf E \cdot d \boldsymbol \ell} with d \boldsymbol \ell the element of path along the integration of electric field vector E. Electric field work is the work performed by an electric field on a charged particle in its vicinity. List of orders of magnitude for electric charge Factor [Coulomb] SI prefix8th edition of the official brochure of the BIPM (SI units and prefixes). 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. : W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{Q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r} where :Q is the electric charge of the particle :E is the electric field, which at a location is the force at that location divided by a unit ('test') charge :FE is the Coulomb (electric) force :r is the displacement :\cdot is the dot product operator ==Mathematical description== Given a charged object in empty space, Q+. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. In physics, a charged particle is a particle with an electric charge. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in electric potential at those points. From these two constants, the elementary charge can be deduced: e = \frac{2}{R_\text{K} K_\text{J}}. ===CODATA method=== The relation used by CODATA to determine elementary charge was: e^2 = \frac{2h \alpha}{\mu_0 c} = 2h \alpha \varepsilon_0 c, where h is the Planck constant, α is the fine-structure constant, μ0 is the magnetic constant, ε0 is the electric constant, and c is the speed of light. The work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points. Prior to this change, the elementary charge was a measured quantity whose magnitude was determined experimentally. right|thumb|150px|V, I, and R, the parameters of Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation: V = {E}{\ell} \ \ \text{or} \ \ E = \frac{V}{\ell}. The particle located experiences an interaction with the electric field. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). This article is a progressive and labeled list of the SI charge orders of magnitude, with certain examples appended to some list objects. Later, the name electron was assigned to the particle and the unit of charge e lost its name. In some other natural unit systems the unit of charge is defined as \sqrt{\varepsilon_0\hbar c}, with the result that e = \sqrt{4\pi\alpha}\sqrt{\varepsilon_0 \hbar c} \approx 0.30282212088 \sqrt{\varepsilon_0 \hbar c}, where is the fine-structure constant, is the speed of light, is the electric constant, and is the reduced Planck constant. == Quantization == Charge quantization is the principle that the charge of any object is an integer multiple of the elementary charge. 104 ~ Charge on one mole of electrons (Faraday constant) 105 Automotive battery charge. 50Ah = 106 mega- (MC) Charge needed to produce 1 kg of aluminium from bauxite in an electrolytic cell 107 108 Charge in world's largest battery bank (36 MWh), assuming 220 VAC outputhttp://www.popsci.com/science/article/2012-01/china-builds-worlds- largest-battery-36-megawatt-hour-behemoth - China Builds the World's Largest Battery – 01.04.2012 == References == Charge ",8, 1.16,56.0,30,2.81,C
-"In Millikan's experiment, an oil drop of radius $1.64 \mu \mathrm{m}$ and density $0.851 \mathrm{~g} / \mathrm{cm}^3$ is suspended in chamber C (Fig. 22-16) when a downward electric field of $1.92 \times 10^5 \mathrm{~N} / \mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$.","Using the known electric field, Millikan and Fletcher could determine the charge on the oil droplet. right|thumb|Millikan's setup for the oil drop experiment|300x300px The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron). Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. Robert A. Millikan and Harvey Fletcher's oil drop experiment first directly measured the magnitude of the elementary charge in 1909, differing from the modern accepted value by just 0.6%. As of 2015, no evidence for fractional charge particles has been found after measuring over 100 million drops. ==Experimental procedure== === Apparatus=== right|thumb|Simplified scheme of Millikan's oil drop experiment|576x576px right|thumb|Oil drop experiment apparatus|335x335px Millikan's and Fletcher's apparatus incorporated a parallel pair of horizontal metal plates. Now the field is turned back on, and the electric force on the drop is :F_E = q E \, where q is the charge on the oil drop and E is the electric field between the plates. By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. The limit to the precision of the method is the measurement of F: the best experimental value has a relative uncertainty of 1.6 ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge. === Oil-drop experiment === A famous method for measuring e is Millikan's oil-drop experiment. With the electrical field calculated, they could measure the droplet's charge, the charge on a single electron being (). So the mass discharged is m = \frac{x M}{v N_{\rm A}} = \frac{Q M}{e N_{\rm A} v} = \frac{Q M}{vF} where * is the Avogadro constant; * is the total charge, equal to the number of electrons () times the elementary charge ; * is the Faraday constant. ==Mathematical form== Faraday's laws can be summarized by :Z = \frac{m}{Q} = \frac{1}{F}\left(\frac{M}{v}\right) = \frac{E}{F} where is the molar mass of the substance (usually given in SI units of grams per mole) and is the valency of the ions . The droplets entered the space between the plates and, because they were charged, could be made to rise and fall by changing the voltage across the plates. ===Method=== thumb|372x372px Initially the oil drops are allowed to fall between the plates with the electric field turned off. Millikan and Fletcher's experiment involved measuring the force on oil droplets in a glass chamber sandwiched between two electrodes, one above and one below. A small drop of oil in an electric field would move at a rate that balanced the forces of gravity, viscosity (of traveling through the air), and electric force. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. By 1937 it was ""quite obvious"" that Millikan's value could not be maintained any longer, and the established value became or . == References == ==Further reading== * * * ==External links== * Simulation of the oil drop experiment (requires JavaScript) * Thomsen, Marshall, ""Good to the Last Drop"". The drop is allowed to fall and its terminal velocity v1 in the absence of an electric field is calculated. The experiment entailed observing tiny electrically charged droplets of oil located between two parallel metal surfaces, forming the plates of a capacitor. First, with zero applied electric field, the velocity of a falling droplet was measured. Reasons for a failure to generate a complete observation include annotations regarding the apparatus setup, oil drop production, and atmospheric effects which invalidated, in Millikan's opinion (borne out by the reduced error in this set), a given particular measurement. ==Millikan's experiment as an example of psychological effects in scientific methodology== thumb|A scatter plot of electron charge measurements as suggested by Feynman, using papers published from 1913-1951 In a commencement address given at the California Institute of Technology (Caltech) in 1974 (and reprinted in Surely You're Joking, Mr. Feynman! in 1985 as well as in The Pleasure of Finding Things Out in 1999), physicist Richard Feynman noted: (adapted from the 1974 California Institute of Technology commencement address), Donald Simanek's Pages, Lock Haven University, rev. December 2017. the value of the elementary charge is defined to be exactly . A mist of atomized oil drops was introduced through a small hole in the top plate and was ionized by an x-ray, making them negatively charged. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. ",2.00,12,0.0,62.8318530718,-5,E
-"The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \mu \mathrm{C}, x_1=3.5 \mathrm{~cm}, y_1=0.50 \mathrm{~cm}$, and $q_2=-4.0 \mu \mathrm{C}, x_2=-2.0 \mathrm{~cm}, y_2=1.5 \mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1.","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. Then, by Newton's second law, \ddot{\mathbf{r}} = \ddot{\mathbf{x}}_{1} - \ddot{\mathbf{x}}_{2} = \left( \frac{\mathbf{F}_{21}}{m_{1}} - \frac{\mathbf{F}_{12}}{m_{2}} \right) = \left(\frac{1}{m_{1}} + \frac{1}{m_{2}} \right)\mathbf{F}_{21} The final equation derives from Newton's third law; the force of the second body on the first body (F21) is equal and opposite to the force of the first body on the second (F12). The Darwin interaction term in the Lagrangian is then L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 where again we kept only the lowest order term in . ==Lagrangian equations of motion== The equation of motion for one of the particles is \frac{d}{dt} \frac{\partial}{\partial \mathbf v_1} L\left( \mathbf r_1 , \mathbf v_1 \right) = abla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) \frac{d \mathbf p_1}{dt} = abla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) where is the momentum of the particle. ===Free particle=== The equation of motion for a free particle neglecting interactions between the two particles is \frac{d}{dt} \left[ \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 \right] = 0 \mathbf p_1 = \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 ===Interacting particles=== For interacting particles, the equation of motion becomes \frac{d}{dt} \left[ \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 + \frac{q_1}{c} \mathbf A\left( \mathbf r_1 \right) \right] = \- abla \frac{q_1 q_2}{r} \+ abla \left[ \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 \right] \frac{d \mathbf p_1}{dt} = \frac{q_1 q_2}{r^2} \hat{\mathbf r} \+ \frac{q_1 q_2}{r^2} \frac{1}{2c^2} \left\\{ \mathbf v_1 \left( { \hat\mathbf{r} \cdot \mathbf v_2} \right) \+ \mathbf v_2 \left( { \hat\mathbf{r} \cdot \mathbf v_1}\right) \- \hat\mathbf{r} \left[ \mathbf v_1 \cdot \left( \mathbf 1 +3 \hat\mathbf{r} \hat\mathbf{r}\right)\cdot \mathbf v_2\right] \right\\} \mathbf p_1 = \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1\mathbf v_1 \+ \frac{q_1}{c} \mathbf A\left( \mathbf r_1 \right) \mathbf A \left( \mathbf r_1 \right) = \frac{q_2}{2c} \frac{1}{r} \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 \mathbf r = \mathbf r_1 - \mathbf r_2 ==Hamiltonian for two particles in a vacuum== The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation H = \mathbf p_1 \cdot \mathbf v_1 + \mathbf p_2 \cdot \mathbf v_2 - L. The difference vector Δr = rBC − rAB equals ΔvΔt (green line), where Δv = vBC − vAB is the change in velocity resulting from the force at point B. The Hamiltonian becomes H\left( \mathbf r_1 , \mathbf p_1 ,\mathbf r_2 , \mathbf p_2 \right)= \left( 1 - \frac{1}{4} \frac{p_1^2}{m_1^2 c^2} \right) \frac{p_1^2}{2 m_1} \; + \; \left( 1 - \frac{1}{4} \frac{p_2^2}{m_2^2 c^2} \right) \frac{p_2^2}{2 m_2} \; + \; \frac{q_1 q_2}{r} \; - \; \frac{q_1 q_2}{r} \frac{1}{2m_1 m_2 c^2} \mathbf p_1\cdot \left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right] \cdot\mathbf p_2 . ==Hamiltonian equations of motion== The Hamiltonian equations of motion are \mathbf v_1 = \frac{\partial H}{\partial \mathbf p_1} and \frac{d \mathbf p_1}{dt} = - abla_1 H which yield \mathbf v_1 = \left( 1- \frac{1}{2} \frac{p_1^2}{m_1^2 c^2} \right) \frac{\mathbf p_1}{m_1} \- \frac{q_1 q_2}{2m_1m_2 c^2} \frac{1}{r} \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf p_2 and \frac{d \mathbf p_1}{dt} = \frac{q_1 q_2}{r^2}{\hat{\mathbf r}} \; + \; \frac{q_1 q_2}{r^2} \frac{1}{2m_1 m_2 c^2} \left\\{ \mathbf p_1 \left( { {\hat{\mathbf r}}\cdot \mathbf p_2} \right) \+ \mathbf p_2 \left( { {\hat{\mathbf r}}\cdot \mathbf p_1}\right) \- {\hat{\mathbf r}} \left[ \mathbf p_1 \cdot \left( \mathbf 1 +3 {\hat{\mathbf r}}{\hat{\mathbf r}}\right)\cdot \mathbf p_2\right] \right\\} Note that the quantum mechanical Breit equation originally used the Darwin Lagrangian with the Darwin Hamiltonian as its classical starting point though the Breit equation would be better vindicated by the Wheeler–Feynman absorber theory and better yet quantum electrodynamics. ==See also== * Static forces and virtual-particle exchange * Breit equation * Wheeler–Feynman absorber theory ==References== Category:Magnetostatics Category:Equations of physics Thus, the equation of motion for r can be written in the form \mu \ddot{\mathbf{r}} = \mathbf{F} where \mu is the reduced mass \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2} As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. ==Qualitative properties== ===Planar motion=== thumb|right|alt=The image shows a yellow disc with three vectors. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta{\left( \mathbf r - \mathbf r_2 \right)}, which has a Fourier transform \mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). Since the potential profile passes through a minimum at the mid-plane, it is easiest to evaluate the disjoining pressure at the midplane. To demonstrate this, let x1 and x2 be the positions of the two particles, and let r = x1 − x2 be their relative position. A way to correct this problem is to shift the force to zero at r_c, thus removing the discontinuity. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The potential at the midplane is thus given by twice the value of this potential at a distance z = h/2. In other words, the azimuthal angles of the two particles are related by the equation φ2(t) = k φ1(t). Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows. The perpendicular force is Ze^2/4\pi\epsilon_0 b^2 at the closest approach and the duration of the encounter is about b/v. If F is a central force, it must be parallel to the vector rB from the center O to the point B (dashed green line); in that case, Δr is also parallel to rB. ",35,-3.141592,-1.0,2,0.15,A
-An electron on the axis of an electric dipole is $25 \mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \times 10^{-29} \mathrm{C} \cdot \mathrm{m}$ ? Assume that $25 \mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole.,"The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. (See electron electric dipole moment). The SI unit for electric dipole moment is the coulomb-meter (C⋅m). It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum , its magnetic dipole moment is given by: \boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,, where e is the electron rest mass. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. Further improvements, or a positive result, would place further limits on which theory takes precedence. == Formal definition == As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that : \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} depends on the point {\mathbf r}_0 about which the moment of the charge distribution \rho({\mathbf r}) is taken. The electric dipole moment vector also points from the negative charge to the positive charge. Experiments have been performed to measure the electric dipole moment of various particles like the electron and the neutron. However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The dipole moment is particularly useful in the context of an overall neutral system of charges, for example a pair of opposite charges, or a neutral conductor in a uniform electric field. A nonzero electric dipole moment can only exist if the centers of the negative and positive charge distribution inside the particle do not coincide. See Landé g-factor for details. ==Example: hydrogen atom== For a hydrogen atom, an electron occupying the atomic orbital , the magnetic dipole moment is given by :\mu_\text{L} = -g_\text{L} \frac{\mu_\text{B}}{\hbar}\langle\Psi_{n,\ell,m}|L|\Psi_{n,\ell,m}\rangle = -\mu_\text{B}\sqrt{\ell(\ell + 1)}. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The q^2 = 0 form factor F_1(0) = -e is the electron's charge, \mu = [\,F_1(0)+F_2(0)\,]/[\,2\,m_{\rm e}\,] is its static magnetic dipole moment, and -F_3(0)/[\,2\,m_{\rm e}\,] provides the formal definion of the electron's electric dipole moment. Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.Many theorists predict elementary particles can have very tiny electric dipole moments, possibly without separated charge. The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin). ",6.6,0.5,0.241,1.8,30,A
-"Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?","Consequently, there is always a small current of approximately 2pA per square metre transporting charged particles in the form of atmospheric ions between the ionosphere and the surface. === Fair weather=== This current is carried by ions present in the atmosphere (generated mainly by cosmic rays in the free troposphere and above, and by radioactivity in the lowest 1km or so). The movement of charge between the Earth's surface, the atmosphere, and the ionosphere is known as the global atmospheric electrical circuit. Near the surface of the Earth, the magnitude of the field is on average around 100 V/m. Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). There is a weak conduction current of atmospheric ions moving in the atmospheric electric field, about 2 picoamperes per square meter, and the air is weakly conductive due to the presence of these atmospheric ions. ===Variations=== Global daily cycles in the atmospheric electric field, with a minimum around 03 UT and peaking roughly 16 hours later, were researched by the Carnegie Institution of Washington in the 20th century. The protonosphere is a layer of the Earth's atmosphere (or any planet with a similar atmosphere) where the dominant components are atomic hydrogen and ionic hydrogen (protons). Atmospheric ions created by cosmic rays and natural radioactivity move in the electric field, so a very small current flows through the atmosphere, even away from thunderstorms. A global atmospheric electrical circuit is the continuous movement of atmospheric charge carriers, such as ions, between an upper conductive layer (often an ionosphere) and surface. The Earth's electrical environment. A ring current is an electric current carried by charged particles trapped in a planet's magnetosphere. thumb|400px|A snapshot of the variation of the Earth's magnetic field from its intrinsic field at 400 km altitude, due to the ionospheric current systems. Because of this, the ionosphere is positively charged relative to the earth. * R Reiter, Relationships Between Atmospheric Electric Phenomena and Simultaneous Meteorological Conditions. 1960 * J. Law, The ionisation of the atmosphere near the ground in fair weather. Global-scale ionospheric circulation establishes a Sq (solar quiet) current system in the E region of the Earth's ionosphere (100-130 km altitude), and a primary eastwards electric field near day-side magnetic equator, where the magnetic field is horizontal and northwards. * R Markson, Modulation of the earth's electric field by cosmic radiation. Over a flat field on a day with clear skies, the atmospheric potential gradient is approximately 120 V/m. This Carnegie curve variation has been described as ""the fundamental electrical heartbeat of the planet"".Liz Kalaugher, Atmospheric electricity affects cloud height 3 March 2013, physicsworld.com accessed 15 April 2021 Even away from thunderstorms, atmospheric electricity can be highly variable, but, generally, the electric field is enhanced in fogs and dust whereas the atmospheric electrical conductivity is diminished. === Links with biology === The atmospheric potential gradient leads to an ion flow from the positively charged atmosphere to the negatively charged earth surface. The global electrical circuit is also relevant to the study of human health and air pollution, due to the interaction of ions and aerosols. * Bespalov P.A., Chugunov Yu. V. and Davydenko S.S., Planetary electric generator under fair- weather condition with altitude-dependent atmospheric conductivity, Journal of Atmospheric and Terrestrial Physics, v.58, #5,pp. 605–611,1996 * DG Yerg, KR Johnson, Short-period fluctuations in the fair weather electric field. Physics – Doklady, Volume 39, Issue 8, August 1994, pp. 553–555 * Bespalov, P. A.; Chugunov, Yu. V.; Davydenko, S. S. Planetary electric generator under fair-weather conditions with altitude-dependent atmospheric conductivity. Discoveries about the electrification of the atmosphere via sensitive electrical instruments and ideas on how the Earth's negative charge is maintained were developed mainly in the 20th century, with CTR Wilson playing an important part.Encyclopedia of Geomagnetism and Paleomagnetism - Page 359 Current research on atmospheric electricity focuses mainly on lightning, particularly high-energy particles and transient luminous events, and the role of non-thunderstorm electrical processes in weather and climate. ==Description== Atmospheric electricity is always present, and during fine weather away from thunderstorms, the air above the surface of Earth is positively charged, while the Earth's surface charge is negative. Atmospheric electricity is an interdisciplinary topic with a long history, involving concepts from electrostatics, atmospheric physics, meteorology and Earth science. ",0.318, 11.58,3.03,122,460.5,D
-"An electric field $\vec{E}$ with an average magnitude of about $150 \mathrm{~N} / \mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to ""float"" a sulfur sphere weighing $4.4 \mathrm{~N}$ in this field by charging the sphere. What charge (both sign and magnitude) must be used?","This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. A surface charge is an electric charge present on a two-dimensional surface. Charges are arbitrarily labeled as positive(+) or negative(-). At a certain pH, the average surface charge will be equal to zero; this is known as the point of zero charge (PZC). It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). Which leads to the simple expression: \sigma=\frac{\varepsilon \varepsilon_0 \psi_0}{\lambda_D} thumb|right|upright=1.25|alt=A bulk solid, containing positive charge, borders a bulk liquid, containing negative charge. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. Surface charge practically always appears on the particle surface when it is placed into a fluid. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. If the number of adsorbed cations exceeds the number of adsorbed anions, the surface would have a net positive electric charge. When the cross does not reach the edges of the field, it becomes a mobile charge. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Thereby is the electrical charge enclosed by the Gaussian surface. The term charge can also be used as a verb; for example, if an escutcheon depicts three lions, it is said to be charged with three lions; similarly, a crest or even a charge itself may be ""charged"", such as a pair of eagle wings charged with trefoils (as on the coat of arms of Brandenburg). By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. In chemistry, there are many different processes which can lead to a surface being charged, including adsorption of ions, protonation or deprotonation, and, as discussed above, the application of an external electric field. ",0.0625,-0.029,0.118,0.064,24.4,B
-"Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \mathrm{~N}$ when their center-to-center separation is $50.0 \mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.0360 \mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them?","thumb|300px|right|visualized induced-charge electrokinetic flow pattern around a carbon-steel sphere (diameter=1.2mm). For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Surface Charging and Points of Zero Charge. Double layer forces occur between charged objects across liquids, typically water. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. An electrical double layer develops near charged surfaces (or another charged objects) in aqueous solutions. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. Chenhui Peng et al.C. Peng, I. Lazo, S. V. Shiyanovskii, O. D. Lavrentovich , Induced-charge electro-osmosis around metal and Janus spheres in water: Patterns of flow and breaking symmetries, arXiv preprint , (2014) also experimentally showed the patterns of electro-osmotic flow around an Au sphere when alternating current (AC) is involved (E=10mV/μm, f=1 kHz). For unequally charged objects and eventually at shorted distances, these forces may also be attractive. Within this double layer, the first layer corresponds to the charged surface. As a result of this migration, the negative charges move to the side which is close to the positive (or higher) voltage while the positive charges move to the opposite side of the particle. :*If abla = 0, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2). Aspects, 73, (1993) 29-48.Electrokinetics and Electrohydrodynamics in Microsystems CISM Courses and Lectures Volume 530, 2011, pp 221-297 Induced-Charge Electrokinetic Phenomena Martin Z. BazantY. At larger distances, oppositely charged surfaces repel and equally charged ones attract. ==Charge regulating surfaces== While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations. In this case, the surface charge density decreases upon approach. Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere. Induced-charge electrokinetics in physics is the electrically driven fluid flow and particle motion in a liquid electrolyte.V. G. Levich, Physicochemical Hydrodynamics. Englewood Cliffs, N.J., Prentice-Hall, (1962) Consider a metal particle (which is neutrally charged but electrically conducting) in contact with an aqueous solution in a chamber/channel. The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the ""sticky-sphere problem"". ",0.2115,0.11,30.0,-1.00, -31.95,D
-"A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \mathrm{~cm}$ away, in a time $1.5 \times 10^{-8} \mathrm{~s}$. What is the speed of the electron as it strikes the second plate? ","This velocity is the speed with which electromagnetic waves penetrate into the conductor and is not the drift velocity of the conduction electrons. Without the presence of an electric field, the electrons have no net velocity. * ""Velocity of Propagation of Electric Field"", Theory and Calculation of Transient Electric Phenomena and Oscillations by Charles Proteus Steinmetz, Chapter VIII, p. 394-, McGraw-Hill, 1920. Relativistic electron beams are streams of electrons moving at relativistic speeds. When a DC voltage is applied, the electron drift velocity will increase in speed proportionally to the strength of the electric field. The inch per second is a unit of speed or velocity. That is, the velocity of propagation has no appreciable effect unless the return conductor is very distant, or entirely absent, or the frequency is so high that the distance to the return conductor is an appreciable portion of the wavelength.Theory and calculation of transient electric phenomena and oscillations By Charles Proteus Steinmetz ==Electric drift== The drift velocity deals with the average velocity of a particle, such as an electron, due to an electric field. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. AC voltages cause no net movement; the electrons oscillate back and forth in response to the alternating electric field (over a distance of a few micrometers – see example calculation). ==See also== *Speed of light *Speed of gravity *Speed of sound *Telegrapher's equations *Reflections of signals on conducting lines ==References== ==Further reading== * Alfvén, H. (1950). The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant. The word electricity refers generally to the movement of electrons (or other charge carriers) through a conductor in the presence of a potential difference or an electric field. In everyday electrical and electronic devices, the signals travel as electromagnetic waves typically at 50%–99% of the speed of light in vacuum, while the electrons themselves move much more slowly; see drift velocity and electron mobility. ==Electromagnetic waves== The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave traveling along (guided by) the cable. As a consequence of Snell's Law and the extremely low speed, electromagnetic waves always enter good conductors in a direction that is within a milliradian of normal to the surface, regardless of the angle of incidence. ===Electromagnetic waves in circuits=== In the theoretical investigation of electric circuits, the velocity of propagation of the electromagnetic field through space is usually not considered; the field is assumed, as a precondition, to be present throughout space. It has been suggested that relativistic electron beams could be used to heat and accelerate the reaction mass in electrical rocket engines that Dr. Robert W. Bussard called quiet electric-discharge engines (QEDs). ==References== ==External links== *PEARL Lab @ UHawaii *Applying REBs for the development of high-powered microwaves (HPM) Category:Electron beam Category:Quantum mechanics Category:Special relativity In copper at 60Hz, v \approx 3.2m/s. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. In other words, the greater the distance from the conductor, the more the electric field lags. In general, an electron will propagate randomly in a conductor at the Fermi velocity.Academic Press dictionary of science and technology By Christopher G. Morris, Academic Press. The electric field starts at the conductor, and propagates through space at the velocity of light (which depends on the material it is traveling through). Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. The drift velocity in a 2 mm diameter copper wire in 1 ampere current is approximately 8 cm per hour. ",0.0547,-11.875,6.0,2.7,22,D
-" Two point charges of $30 \mathrm{nC}$ and $-40 \mathrm{nC}$ are held fixed on an $x$ axis, at the origin and at $x=72 \mathrm{~cm}$, respectively. A particle with a charge of $42 \mu \mathrm{C}$ is released from rest at $x=28 \mathrm{~cm}$. If the initial acceleration of the particle has a magnitude of $100 \mathrm{~km} / \mathrm{s}^2$, what is the particle's mass?","The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. This differential equation is the classic equation of motion for charged particles. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This differential equation is the classic equation of motion of a charged particle in a vacuum. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. In this case, one must solve the PB equation together with an appropriate model of the surface charging process. The CODATA recommended value for an electron is ==Origin== When charged particles move in electric and magnetic fields the following two laws apply: *Lorentz force law: \mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), *Newton's second law of motion:\mathbf{F}=m\mathbf{a} = m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} where F is the force applied to the ion, m is the mass of the particle, a is the acceleration, Q is the electric charge, E is the electric field, and v × B is the cross product of the ion's velocity and the magnetic flux density. Thus, the m/z of an ion alone neither infers mass nor the number of charges. Often, the charge can be inferred from theoretical considerations, so the charge-to-mass ratio provides a way to calculate the mass of a particle. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. At very low energies, space charge has a large effect on a particle beam and thus becomes hard to calculate. Often, the charge-to-mass ratio can be determined by observing the deflection of a charged particle in an external magnetic field. A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. When performing a modeling task for any accelerator operation, the results of charged particle beam dynamics simulations must feed into the associated application. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. Codes for this computation include *ABCI ABCI home page at kek.jp *ACE3P ACE3P at slac.stanford.gov *CST Studio Suite CST, Computer Simulation Technology at cst.com *GdfidLGdfidL, Gitter drueber, fertig ist die Laube at gdfidl.de *TBCI T. Weiland, DESY * VSim ==Magnet and other hardware-modeling codes== To control the charged particle beam, appropriate electric and magnetic fields must be created. ",41.40,2.2,-1.0,0.9830,7.25,B
-"In Fig. 21-26, particle 1 of charge $-5.00 q$ and particle 2 of charge $+2.00 q$ are held at separation $L$ on an $x$ axis. If particle 3 of unknown charge $q_3$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the $x$ coordinate of particle 3?","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . In physics, a charged particle is a particle with an electric charge. Surface Charging and Points of Zero Charge. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. Related values associated with the soil characteristics exist along with the pzc value, including zero point of charge (zpc), point of zero net charge (pznc), etc. == Term definition of point of zero charge == The point of zero charge is the pH for which the net surface charge of adsorbent is equal to zero. With the electrostatic force being proportional to r^{-2}, individual particle-particle interactions are long- range in nature, presenting a challenging computational problem in the simulation of particulate systems. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. thumb|Electrical double layer (EDL) around a negatively charged particle in suspension in water. For example, the surface charge of adsorbent is described by the ion that lies on the surface of the particle (adsorbent) structure like image. Of interest, this represents the first three-dimensional solution. Charges are arbitrarily labeled as positive(+) or negative(-). A related concept in electrochemistry is the electrode potential at the point of zero charge. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. Also, this represents the first two-dimensional solution. Simply applying this cutoff method introduces a discontinuity in the force at r_c that results in particles experiencing sudden impulses when other particles cross the boundary of their respective interaction spheres. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. The single electron may reside at any point on the surface of the unit sphere. Only the existence of two 'types' of charges is known, there isn't anything inherent about positive charges that makes them positive, and the same goes for the negative charge. == Examples == === Positively charged particles === * protons and atomic nuclei * positrons (antielectrons) * alpha particles * positive charged pions * cations === Negatively charged particles === * electrons * antiprotons * muons * tauons * negative charged pions * anions === Particles without an electric charge=== * neutrons * photons * neutrinos * neutral pions *z boson *higgs boson *atoms ==References== * * * * * ==External links== * Charged particle motion in E/B Field Category:Charge carriers Category:Particle physics ", 135.36,1.5,3.03,2.72,0.6749,D
-An isolated conductor has net charge $+10 \times 10^{-6} \mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \times 10^{-6} \mathrm{C}$. What is the charge on the cavity wall?,"Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. In modern cavity wall construction, cavity insulation is typically added. thumb|Circuit diagram of a charge qubit circuit. A cavity wall is a type of wall that has a hollow center. He shares half of the prize for developing a new field called cavity quantum electrodynamics (CQED) – whereby the properties of an atom are controlled by placing it in an optical or microwave cavity. This results in two overlapping layers of the superconducting metal, in between which a thin layer of insulator (normally aluminum oxide) is deposited. == Hamiltonian == If the Josephson junction has a junction capacitance C_{\rm J}, and the gate capacitor C_{\rm g}, then the charging (Coulomb) energy of one Cooper pair is: :E_{\rm C}=(2e)^2/2(C_{\rm g}+C_{\rm J}). In superconducting quantum computing, a charge qubit is formed by a tiny superconducting island coupled by a Josephson junction (or practically, superconducting tunnel junction) to a superconducting reservoir (see figure). In quantum computing, a charge qubit (also known as Cooper-pair box) is a qubit whose basis states are charge states (i.e. states which represent the presence or absence of excess Cooper pairs in the island). QIT may refer to: * QIT-Fer et Titane, a Canadian mining company * Quadrupole ion trap * Quantum information theory * Queensland University of Technology * Q = It, the formula describing charge in terms of current and time The conductance quantum, denoted by the symbol , is the quantized unit of electrical conductance. The resonance frequencies are given by \lambda/2: \quad u_n=\frac{c}{\sqrt{\varepsilon_{\text{eff}}}}\frac{n}{2 \ell} \quad (n=1,2,3,\ldots) \qquad \lambda/4:\quad u_n=\frac{c}{\sqrt{\varepsilon_{\text{eff}}}}\frac{2n+1}{4 \ell} \quad (n=0,1,2,\ldots) with \varepsilon_{\text{eff}} being the effective dielectric permittivity of the device. == Artificial atoms, Qubits == The first realized artificial atom in circuit QED was the so-called Cooper-pair box, also known as the charge qubit. Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter (quantum optics). The voltage is: V = -\frac{(\mu_1 - \mu_2)}{e} , where e is the electron charge. If the detuning is significantly larger than the combined cavity and atomic linewidth the cavity states are merely shifted by \pm g^2/\Delta (with the detuning \Delta=\omega_a-\omega_r) depending on the atomic state. *Cavity wall, Energy Saving Trust *Cavity wall insulation (CWI): consumer guide to issues arising from installations, 14 October 2019, Department for Business, Energy & Industrial Strategy *Cavity Wall Insulation Victims Alliance Category:Masonry Category:Construction Category:Types of wall Structural properties of a concrete-block cavity- wall construction sponsored by the National Concrete Masonry Association. This immobilises the air within the cavity (air is still the actual insulator), preventing convection, and can substantially reduce space heating costs. thumb|A wall that has had cavity wall insulation installed (after construction), with refilled holes highlighted with arrows During construction of new buildings, cavities are often filled with glass fiber wool or mineral wool panels placed between the two leaves (sides) of the wall, but many other building insulation materials offer various advantages and many others are also widely used. Recent work has shown T2 times approaching 100 μs using a type of charge qubit known as a transmon inside a three-dimensional superconducting cavity.C. Rigetti et al., ""Superconducting qubit in waveguide cavity with coherence time approaching 0.1 ms,"" arXiv:1202.5533 (2012) Understanding the limits of T2 is an active area of research in the field of superconducting quantum computing. == Fabrication == Charge qubits are fabricated using techniques similar to those used for microelectronics. Cavity wall insulation also helps to prevent convection and can keep a house warm by making sure that less heat is lost through walls; this can also thus be a more cost-efficient way of heating a house. One function of the cavity is to drain water through weep holes at the base of the wall system or above windows. In 1999, coherent oscillations in the charge Qubit were first observed by Nakamura et al. Manipulation of the quantum states and full realization of the charge qubit was observed 2 years later. Thermal mass cavity walls are thick walls. ",4.86,0.9,1.11,1,-3.0,E
+","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The voltage between two points is defined as:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 685–686 {\Delta V} = -\int {\mathbf E \cdot d \boldsymbol \ell} with d \boldsymbol \ell the element of path along the integration of electric field vector E. Electric field work is the work performed by an electric field on a charged particle in its vicinity. List of orders of magnitude for electric charge Factor [Coulomb] SI prefix8th edition of the official brochure of the BIPM (SI units and prefixes). 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. : W = Q \int_{a}^{b} \mathbf{E} \cdot \, d \mathbf{r} = Q \int_{a}^{b} \frac{\mathbf{F_E}}{Q} \cdot \, d \mathbf{r}= \int_{a}^{b} \mathbf{F_E} \cdot \, d \mathbf{r} where :Q is the electric charge of the particle :E is the electric field, which at a location is the force at that location divided by a unit ('test') charge :FE is the Coulomb (electric) force :r is the displacement :\cdot is the dot product operator ==Mathematical description== Given a charged object in empty space, Q+. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. In physics, a charged particle is a particle with an electric charge. The work per unit of charge is defined by moving a negligible test charge between two points, and is expressed as the difference in electric potential at those points. From these two constants, the elementary charge can be deduced: e = \frac{2}{R_\text{K} K_\text{J}}. ===CODATA method=== The relation used by CODATA to determine elementary charge was: e^2 = \frac{2h \alpha}{\mu_0 c} = 2h \alpha \varepsilon_0 c, where h is the Planck constant, α is the fine-structure constant, μ0 is the magnetic constant, ε0 is the electric constant, and c is the speed of light. The work per unit of charge, when moving a negligible test charge between two points, is defined as the voltage between those points. Prior to this change, the elementary charge was a measured quantity whose magnitude was determined experimentally. right|thumb|150px|V, I, and R, the parameters of Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. If the applied E field is uniform and oriented along the length of the conductor as shown in the figure, then defining the voltage V in the usual convention of being opposite in direction to the field (see figure), and with the understanding that the voltage V is measured differentially across the length of the conductor allowing us to drop the Δ symbol, the above vector equation reduces to the scalar equation: V = {E}{\ell} \ \ \text{or} \ \ E = \frac{V}{\ell}. The particle located experiences an interaction with the electric field. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). This article is a progressive and labeled list of the SI charge orders of magnitude, with certain examples appended to some list objects. Later, the name electron was assigned to the particle and the unit of charge e lost its name. In some other natural unit systems the unit of charge is defined as \sqrt{\varepsilon_0\hbar c}, with the result that e = \sqrt{4\pi\alpha}\sqrt{\varepsilon_0 \hbar c} \approx 0.30282212088 \sqrt{\varepsilon_0 \hbar c}, where is the fine-structure constant, is the speed of light, is the electric constant, and is the reduced Planck constant. == Quantization == Charge quantization is the principle that the charge of any object is an integer multiple of the elementary charge. 104 ~ Charge on one mole of electrons (Faraday constant) 105 Automotive battery charge. 50Ah = 106 mega- (MC) Charge needed to produce 1 kg of aluminium from bauxite in an electrolytic cell 107 108 Charge in world's largest battery bank (36 MWh), assuming 220 VAC outputhttp://www.popsci.com/science/article/2012-01/china-builds-worlds- largest-battery-36-megawatt-hour-behemoth - China Builds the World's Largest Battery – 01.04.2012 == References == Charge ",8, 1.16,"""56.0""",30,2.81,C
+"In Millikan's experiment, an oil drop of radius $1.64 \mu \mathrm{m}$ and density $0.851 \mathrm{~g} / \mathrm{cm}^3$ is suspended in chamber C (Fig. 22-16) when a downward electric field of $1.92 \times 10^5 \mathrm{~N} / \mathrm{C}$ is applied. Find the charge on the drop, in terms of $e$.","Using the known electric field, Millikan and Fletcher could determine the charge on the oil droplet. right|thumb|Millikan's setup for the oil drop experiment|300x300px The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron). Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. Robert A. Millikan and Harvey Fletcher's oil drop experiment first directly measured the magnitude of the elementary charge in 1909, differing from the modern accepted value by just 0.6%. As of 2015, no evidence for fractional charge particles has been found after measuring over 100 million drops. ==Experimental procedure== === Apparatus=== right|thumb|Simplified scheme of Millikan's oil drop experiment|576x576px right|thumb|Oil drop experiment apparatus|335x335px Millikan's and Fletcher's apparatus incorporated a parallel pair of horizontal metal plates. Now the field is turned back on, and the electric force on the drop is :F_E = q E \, where q is the charge on the oil drop and E is the electric field between the plates. By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. The limit to the precision of the method is the measurement of F: the best experimental value has a relative uncertainty of 1.6 ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge. === Oil-drop experiment === A famous method for measuring e is Millikan's oil-drop experiment. With the electrical field calculated, they could measure the droplet's charge, the charge on a single electron being (). So the mass discharged is m = \frac{x M}{v N_{\rm A}} = \frac{Q M}{e N_{\rm A} v} = \frac{Q M}{vF} where * is the Avogadro constant; * is the total charge, equal to the number of electrons () times the elementary charge ; * is the Faraday constant. ==Mathematical form== Faraday's laws can be summarized by :Z = \frac{m}{Q} = \frac{1}{F}\left(\frac{M}{v}\right) = \frac{E}{F} where is the molar mass of the substance (usually given in SI units of grams per mole) and is the valency of the ions . The droplets entered the space between the plates and, because they were charged, could be made to rise and fall by changing the voltage across the plates. ===Method=== thumb|372x372px Initially the oil drops are allowed to fall between the plates with the electric field turned off. Millikan and Fletcher's experiment involved measuring the force on oil droplets in a glass chamber sandwiched between two electrodes, one above and one below. A small drop of oil in an electric field would move at a rate that balanced the forces of gravity, viscosity (of traveling through the air), and electric force. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. By 1937 it was ""quite obvious"" that Millikan's value could not be maintained any longer, and the established value became or . == References == ==Further reading== * * * ==External links== * Simulation of the oil drop experiment (requires JavaScript) * Thomsen, Marshall, ""Good to the Last Drop"". The drop is allowed to fall and its terminal velocity v1 in the absence of an electric field is calculated. The experiment entailed observing tiny electrically charged droplets of oil located between two parallel metal surfaces, forming the plates of a capacitor. First, with zero applied electric field, the velocity of a falling droplet was measured. Reasons for a failure to generate a complete observation include annotations regarding the apparatus setup, oil drop production, and atmospheric effects which invalidated, in Millikan's opinion (borne out by the reduced error in this set), a given particular measurement. ==Millikan's experiment as an example of psychological effects in scientific methodology== thumb|A scatter plot of electron charge measurements as suggested by Feynman, using papers published from 1913-1951 In a commencement address given at the California Institute of Technology (Caltech) in 1974 (and reprinted in Surely You're Joking, Mr. Feynman! in 1985 as well as in The Pleasure of Finding Things Out in 1999), physicist Richard Feynman noted: (adapted from the 1974 California Institute of Technology commencement address), Donald Simanek's Pages, Lock Haven University, rev. December 2017. the value of the elementary charge is defined to be exactly . A mist of atomized oil drops was introduced through a small hole in the top plate and was ionized by an x-ray, making them negatively charged. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. ",2.00,12,"""0.0""",62.8318530718,-5,E
+"The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_1=+3.0 \mu \mathrm{C}, x_1=3.5 \mathrm{~cm}, y_1=0.50 \mathrm{~cm}$, and $q_2=-4.0 \mu \mathrm{C}, x_2=-2.0 \mathrm{~cm}, y_2=1.5 \mathrm{~cm}$. Find the magnitude of the electrostatic force on particle 2 due to particle 1.","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. Then, by Newton's second law, \ddot{\mathbf{r}} = \ddot{\mathbf{x}}_{1} - \ddot{\mathbf{x}}_{2} = \left( \frac{\mathbf{F}_{21}}{m_{1}} - \frac{\mathbf{F}_{12}}{m_{2}} \right) = \left(\frac{1}{m_{1}} + \frac{1}{m_{2}} \right)\mathbf{F}_{21} The final equation derives from Newton's third law; the force of the second body on the first body (F21) is equal and opposite to the force of the first body on the second (F12). The Darwin interaction term in the Lagrangian is then L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 where again we kept only the lowest order term in . ==Lagrangian equations of motion== The equation of motion for one of the particles is \frac{d}{dt} \frac{\partial}{\partial \mathbf v_1} L\left( \mathbf r_1 , \mathbf v_1 \right) = abla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) \frac{d \mathbf p_1}{dt} = abla_1 L\left( \mathbf r_1 , \mathbf v_1 \right) where is the momentum of the particle. ===Free particle=== The equation of motion for a free particle neglecting interactions between the two particles is \frac{d}{dt} \left[ \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 \right] = 0 \mathbf p_1 = \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 ===Interacting particles=== For interacting particles, the equation of motion becomes \frac{d}{dt} \left[ \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1 \mathbf v_1 + \frac{q_1}{c} \mathbf A\left( \mathbf r_1 \right) \right] = \- abla \frac{q_1 q_2}{r} \+ abla \left[ \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 \right] \frac{d \mathbf p_1}{dt} = \frac{q_1 q_2}{r^2} \hat{\mathbf r} \+ \frac{q_1 q_2}{r^2} \frac{1}{2c^2} \left\\{ \mathbf v_1 \left( { \hat\mathbf{r} \cdot \mathbf v_2} \right) \+ \mathbf v_2 \left( { \hat\mathbf{r} \cdot \mathbf v_1}\right) \- \hat\mathbf{r} \left[ \mathbf v_1 \cdot \left( \mathbf 1 +3 \hat\mathbf{r} \hat\mathbf{r}\right)\cdot \mathbf v_2\right] \right\\} \mathbf p_1 = \left( 1 + \frac{1}{2} \frac{v_1^2}{c^2} \right)m_1\mathbf v_1 \+ \frac{q_1}{c} \mathbf A\left( \mathbf r_1 \right) \mathbf A \left( \mathbf r_1 \right) = \frac{q_2}{2c} \frac{1}{r} \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf v_2 \mathbf r = \mathbf r_1 - \mathbf r_2 ==Hamiltonian for two particles in a vacuum== The Darwin Hamiltonian for two particles in a vacuum is related to the Lagrangian by a Legendre transformation H = \mathbf p_1 \cdot \mathbf v_1 + \mathbf p_2 \cdot \mathbf v_2 - L. The difference vector Δr = rBC − rAB equals ΔvΔt (green line), where Δv = vBC − vAB is the change in velocity resulting from the force at point B. The Hamiltonian becomes H\left( \mathbf r_1 , \mathbf p_1 ,\mathbf r_2 , \mathbf p_2 \right)= \left( 1 - \frac{1}{4} \frac{p_1^2}{m_1^2 c^2} \right) \frac{p_1^2}{2 m_1} \; + \; \left( 1 - \frac{1}{4} \frac{p_2^2}{m_2^2 c^2} \right) \frac{p_2^2}{2 m_2} \; + \; \frac{q_1 q_2}{r} \; - \; \frac{q_1 q_2}{r} \frac{1}{2m_1 m_2 c^2} \mathbf p_1\cdot \left[\mathbf 1 + \mathbf{\hat r} \mathbf{\hat r}\right] \cdot\mathbf p_2 . ==Hamiltonian equations of motion== The Hamiltonian equations of motion are \mathbf v_1 = \frac{\partial H}{\partial \mathbf p_1} and \frac{d \mathbf p_1}{dt} = - abla_1 H which yield \mathbf v_1 = \left( 1- \frac{1}{2} \frac{p_1^2}{m_1^2 c^2} \right) \frac{\mathbf p_1}{m_1} \- \frac{q_1 q_2}{2m_1m_2 c^2} \frac{1}{r} \left[\mathbf 1 + \hat\mathbf r \hat\mathbf r\right] \cdot \mathbf p_2 and \frac{d \mathbf p_1}{dt} = \frac{q_1 q_2}{r^2}{\hat{\mathbf r}} \; + \; \frac{q_1 q_2}{r^2} \frac{1}{2m_1 m_2 c^2} \left\\{ \mathbf p_1 \left( { {\hat{\mathbf r}}\cdot \mathbf p_2} \right) \+ \mathbf p_2 \left( { {\hat{\mathbf r}}\cdot \mathbf p_1}\right) \- {\hat{\mathbf r}} \left[ \mathbf p_1 \cdot \left( \mathbf 1 +3 {\hat{\mathbf r}}{\hat{\mathbf r}}\right)\cdot \mathbf p_2\right] \right\\} Note that the quantum mechanical Breit equation originally used the Darwin Lagrangian with the Darwin Hamiltonian as its classical starting point though the Breit equation would be better vindicated by the Wheeler–Feynman absorber theory and better yet quantum electrodynamics. ==See also== * Static forces and virtual-particle exchange * Breit equation * Wheeler–Feynman absorber theory ==References== Category:Magnetostatics Category:Equations of physics Thus, the equation of motion for r can be written in the form \mu \ddot{\mathbf{r}} = \mathbf{F} where \mu is the reduced mass \mu = \frac{1}{\frac{1}{m_1} + \frac{1}{m_2}} = \frac{m_1 m_2}{m_1 + m_2} As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. ==Qualitative properties== ===Planar motion=== thumb|right|alt=The image shows a yellow disc with three vectors. The current generated by the second particle is \mathbf J = q_2 \mathbf v_2 \delta{\left( \mathbf r - \mathbf r_2 \right)}, which has a Fourier transform \mathbf J\left( \mathbf k \right) \equiv \int d^3r \exp\left( -i\mathbf k \cdot \mathbf r \right) \mathbf J\left( \mathbf r \right) = q_2 \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right). Since the potential profile passes through a minimum at the mid-plane, it is easiest to evaluate the disjoining pressure at the midplane. To demonstrate this, let x1 and x2 be the positions of the two particles, and let r = x1 − x2 be their relative position. A way to correct this problem is to shift the force to zero at r_c, thus removing the discontinuity. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The potential at the midplane is thus given by twice the value of this potential at a distance z = h/2. In other words, the azimuthal angles of the two particles are related by the equation φ2(t) = k φ1(t). Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows. The perpendicular force is Ze^2/4\pi\epsilon_0 b^2 at the closest approach and the duration of the encounter is about b/v. If F is a central force, it must be parallel to the vector rB from the center O to the point B (dashed green line); in that case, Δr is also parallel to rB. ",35,-3.141592,"""-1.0""",2,0.15,A
+An electron on the axis of an electric dipole is $25 \mathrm{~nm}$ from the center of the dipole. What is the magnitude of the electrostatic force on the electron if the dipole moment is $3.6 \times 10^{-29} \mathrm{C} \cdot \mathrm{m}$ ? Assume that $25 \mathrm{~nm}$ is much larger than the separation of the charged particles that form the dipole.,"The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. (See electron electric dipole moment). The SI unit for electric dipole moment is the coulomb-meter (C⋅m). It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum , its magnetic dipole moment is given by: \boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,, where e is the electron rest mass. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. Further improvements, or a positive result, would place further limits on which theory takes precedence. == Formal definition == As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that : \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} depends on the point {\mathbf r}_0 about which the moment of the charge distribution \rho({\mathbf r}) is taken. The electric dipole moment vector also points from the negative charge to the positive charge. Experiments have been performed to measure the electric dipole moment of various particles like the electron and the neutron. However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The dipole moment is particularly useful in the context of an overall neutral system of charges, for example a pair of opposite charges, or a neutral conductor in a uniform electric field. A nonzero electric dipole moment can only exist if the centers of the negative and positive charge distribution inside the particle do not coincide. See Landé g-factor for details. ==Example: hydrogen atom== For a hydrogen atom, an electron occupying the atomic orbital , the magnetic dipole moment is given by :\mu_\text{L} = -g_\text{L} \frac{\mu_\text{B}}{\hbar}\langle\Psi_{n,\ell,m}|L|\Psi_{n,\ell,m}\rangle = -\mu_\text{B}\sqrt{\ell(\ell + 1)}. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The q^2 = 0 form factor F_1(0) = -e is the electron's charge, \mu = [\,F_1(0)+F_2(0)\,]/[\,2\,m_{\rm e}\,] is its static magnetic dipole moment, and -F_3(0)/[\,2\,m_{\rm e}\,] provides the formal definion of the electron's electric dipole moment. Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.Many theorists predict elementary particles can have very tiny electric dipole moments, possibly without separated charge. The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin). ",6.6,0.5,"""0.241""",1.8,30,A
+"Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. What would be the electric current intercepted by the total surface area of the planet?","Consequently, there is always a small current of approximately 2pA per square metre transporting charged particles in the form of atmospheric ions between the ionosphere and the surface. === Fair weather=== This current is carried by ions present in the atmosphere (generated mainly by cosmic rays in the free troposphere and above, and by radioactivity in the lowest 1km or so). The movement of charge between the Earth's surface, the atmosphere, and the ionosphere is known as the global atmospheric electrical circuit. Near the surface of the Earth, the magnitude of the field is on average around 100 V/m. Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). There is a weak conduction current of atmospheric ions moving in the atmospheric electric field, about 2 picoamperes per square meter, and the air is weakly conductive due to the presence of these atmospheric ions. ===Variations=== Global daily cycles in the atmospheric electric field, with a minimum around 03 UT and peaking roughly 16 hours later, were researched by the Carnegie Institution of Washington in the 20th century. The protonosphere is a layer of the Earth's atmosphere (or any planet with a similar atmosphere) where the dominant components are atomic hydrogen and ionic hydrogen (protons). Atmospheric ions created by cosmic rays and natural radioactivity move in the electric field, so a very small current flows through the atmosphere, even away from thunderstorms. A global atmospheric electrical circuit is the continuous movement of atmospheric charge carriers, such as ions, between an upper conductive layer (often an ionosphere) and surface. The Earth's electrical environment. A ring current is an electric current carried by charged particles trapped in a planet's magnetosphere. thumb|400px|A snapshot of the variation of the Earth's magnetic field from its intrinsic field at 400 km altitude, due to the ionospheric current systems. Because of this, the ionosphere is positively charged relative to the earth. * R Reiter, Relationships Between Atmospheric Electric Phenomena and Simultaneous Meteorological Conditions. 1960 * J. Law, The ionisation of the atmosphere near the ground in fair weather. Global-scale ionospheric circulation establishes a Sq (solar quiet) current system in the E region of the Earth's ionosphere (100-130 km altitude), and a primary eastwards electric field near day-side magnetic equator, where the magnetic field is horizontal and northwards. * R Markson, Modulation of the earth's electric field by cosmic radiation. Over a flat field on a day with clear skies, the atmospheric potential gradient is approximately 120 V/m. This Carnegie curve variation has been described as ""the fundamental electrical heartbeat of the planet"".Liz Kalaugher, Atmospheric electricity affects cloud height 3 March 2013, physicsworld.com accessed 15 April 2021 Even away from thunderstorms, atmospheric electricity can be highly variable, but, generally, the electric field is enhanced in fogs and dust whereas the atmospheric electrical conductivity is diminished. === Links with biology === The atmospheric potential gradient leads to an ion flow from the positively charged atmosphere to the negatively charged earth surface. The global electrical circuit is also relevant to the study of human health and air pollution, due to the interaction of ions and aerosols. * Bespalov P.A., Chugunov Yu. V. and Davydenko S.S., Planetary electric generator under fair- weather condition with altitude-dependent atmospheric conductivity, Journal of Atmospheric and Terrestrial Physics, v.58, #5,pp. 605–611,1996 * DG Yerg, KR Johnson, Short-period fluctuations in the fair weather electric field. Physics – Doklady, Volume 39, Issue 8, August 1994, pp. 553–555 * Bespalov, P. A.; Chugunov, Yu. V.; Davydenko, S. S. Planetary electric generator under fair-weather conditions with altitude-dependent atmospheric conductivity. Discoveries about the electrification of the atmosphere via sensitive electrical instruments and ideas on how the Earth's negative charge is maintained were developed mainly in the 20th century, with CTR Wilson playing an important part.Encyclopedia of Geomagnetism and Paleomagnetism - Page 359 Current research on atmospheric electricity focuses mainly on lightning, particularly high-energy particles and transient luminous events, and the role of non-thunderstorm electrical processes in weather and climate. ==Description== Atmospheric electricity is always present, and during fine weather away from thunderstorms, the air above the surface of Earth is positively charged, while the Earth's surface charge is negative. Atmospheric electricity is an interdisciplinary topic with a long history, involving concepts from electrostatics, atmospheric physics, meteorology and Earth science. ",0.318, 11.58,"""3.03""",122,460.5,D
+"An electric field $\vec{E}$ with an average magnitude of about $150 \mathrm{~N} / \mathrm{C}$ points downward in the atmosphere near Earth's surface. We wish to ""float"" a sulfur sphere weighing $4.4 \mathrm{~N}$ in this field by charging the sphere. What charge (both sign and magnitude) must be used?","This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. A surface charge is an electric charge present on a two-dimensional surface. Charges are arbitrarily labeled as positive(+) or negative(-). At a certain pH, the average surface charge will be equal to zero; this is known as the point of zero charge (PZC). It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). Which leads to the simple expression: \sigma=\frac{\varepsilon \varepsilon_0 \psi_0}{\lambda_D} thumb|right|upright=1.25|alt=A bulk solid, containing positive charge, borders a bulk liquid, containing negative charge. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. Surface charge practically always appears on the particle surface when it is placed into a fluid. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. If the number of adsorbed cations exceeds the number of adsorbed anions, the surface would have a net positive electric charge. When the cross does not reach the edges of the field, it becomes a mobile charge. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Thereby is the electrical charge enclosed by the Gaussian surface. The term charge can also be used as a verb; for example, if an escutcheon depicts three lions, it is said to be charged with three lions; similarly, a crest or even a charge itself may be ""charged"", such as a pair of eagle wings charged with trefoils (as on the coat of arms of Brandenburg). By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. In chemistry, there are many different processes which can lead to a surface being charged, including adsorption of ions, protonation or deprotonation, and, as discussed above, the application of an external electric field. ",0.0625,-0.029,"""0.118""",0.064,24.4,B
+"Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \mathrm{~N}$ when their center-to-center separation is $50.0 \mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.0360 \mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them?","thumb|300px|right|visualized induced-charge electrokinetic flow pattern around a carbon-steel sphere (diameter=1.2mm). For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Surface Charging and Points of Zero Charge. Double layer forces occur between charged objects across liquids, typically water. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. An electrical double layer develops near charged surfaces (or another charged objects) in aqueous solutions. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. Chenhui Peng et al.C. Peng, I. Lazo, S. V. Shiyanovskii, O. D. Lavrentovich , Induced-charge electro-osmosis around metal and Janus spheres in water: Patterns of flow and breaking symmetries, arXiv preprint , (2014) also experimentally showed the patterns of electro-osmotic flow around an Au sphere when alternating current (AC) is involved (E=10mV/μm, f=1 kHz). For unequally charged objects and eventually at shorted distances, these forces may also be attractive. Within this double layer, the first layer corresponds to the charged surface. As a result of this migration, the negative charges move to the side which is close to the positive (or higher) voltage while the positive charges move to the opposite side of the particle. :*If abla = 0, then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2). Aspects, 73, (1993) 29-48.Electrokinetics and Electrohydrodynamics in Microsystems CISM Courses and Lectures Volume 530, 2011, pp 221-297 Induced-Charge Electrokinetic Phenomena Martin Z. BazantY. At larger distances, oppositely charged surfaces repel and equally charged ones attract. ==Charge regulating surfaces== While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations. In this case, the surface charge density decreases upon approach. Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere. Induced-charge electrokinetics in physics is the electrically driven fluid flow and particle motion in a liquid electrolyte.V. G. Levich, Physicochemical Hydrodynamics. Englewood Cliffs, N.J., Prentice-Hall, (1962) Consider a metal particle (which is neutrally charged but electrically conducting) in contact with an aqueous solution in a chamber/channel. The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the ""sticky-sphere problem"". ",0.2115,0.11,"""30.0""",-1.00, -31.95,D
+"A uniform electric field exists in a region between two oppositely charged plates. An electron is released from rest at the surface of the negatively charged plate and strikes the surface of the opposite plate, $2.0 \mathrm{~cm}$ away, in a time $1.5 \times 10^{-8} \mathrm{~s}$. What is the speed of the electron as it strikes the second plate? ","This velocity is the speed with which electromagnetic waves penetrate into the conductor and is not the drift velocity of the conduction electrons. Without the presence of an electric field, the electrons have no net velocity. * ""Velocity of Propagation of Electric Field"", Theory and Calculation of Transient Electric Phenomena and Oscillations by Charles Proteus Steinmetz, Chapter VIII, p. 394-, McGraw-Hill, 1920. Relativistic electron beams are streams of electrons moving at relativistic speeds. When a DC voltage is applied, the electron drift velocity will increase in speed proportionally to the strength of the electric field. The inch per second is a unit of speed or velocity. That is, the velocity of propagation has no appreciable effect unless the return conductor is very distant, or entirely absent, or the frequency is so high that the distance to the return conductor is an appreciable portion of the wavelength.Theory and calculation of transient electric phenomena and oscillations By Charles Proteus Steinmetz ==Electric drift== The drift velocity deals with the average velocity of a particle, such as an electron, due to an electric field. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. AC voltages cause no net movement; the electrons oscillate back and forth in response to the alternating electric field (over a distance of a few micrometers – see example calculation). ==See also== *Speed of light *Speed of gravity *Speed of sound *Telegrapher's equations *Reflections of signals on conducting lines ==References== ==Further reading== * Alfvén, H. (1950). The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant. The word electricity refers generally to the movement of electrons (or other charge carriers) through a conductor in the presence of a potential difference or an electric field. In everyday electrical and electronic devices, the signals travel as electromagnetic waves typically at 50%–99% of the speed of light in vacuum, while the electrons themselves move much more slowly; see drift velocity and electron mobility. ==Electromagnetic waves== The speed at which energy or signals travel down a cable is actually the speed of the electromagnetic wave traveling along (guided by) the cable. As a consequence of Snell's Law and the extremely low speed, electromagnetic waves always enter good conductors in a direction that is within a milliradian of normal to the surface, regardless of the angle of incidence. ===Electromagnetic waves in circuits=== In the theoretical investigation of electric circuits, the velocity of propagation of the electromagnetic field through space is usually not considered; the field is assumed, as a precondition, to be present throughout space. It has been suggested that relativistic electron beams could be used to heat and accelerate the reaction mass in electrical rocket engines that Dr. Robert W. Bussard called quiet electric-discharge engines (QEDs). ==References== ==External links== *PEARL Lab @ UHawaii *Applying REBs for the development of high-powered microwaves (HPM) Category:Electron beam Category:Quantum mechanics Category:Special relativity In copper at 60Hz, v \approx 3.2m/s. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. In other words, the greater the distance from the conductor, the more the electric field lags. In general, an electron will propagate randomly in a conductor at the Fermi velocity.Academic Press dictionary of science and technology By Christopher G. Morris, Academic Press. The electric field starts at the conductor, and propagates through space at the velocity of light (which depends on the material it is traveling through). Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. The drift velocity in a 2 mm diameter copper wire in 1 ampere current is approximately 8 cm per hour. ",0.0547,-11.875,"""6.0""",2.7,22,D
+" Two point charges of $30 \mathrm{nC}$ and $-40 \mathrm{nC}$ are held fixed on an $x$ axis, at the origin and at $x=72 \mathrm{~cm}$, respectively. A particle with a charge of $42 \mu \mathrm{C}$ is released from rest at $x=28 \mathrm{~cm}$. If the initial acceleration of the particle has a magnitude of $100 \mathrm{~km} / \mathrm{s}^2$, what is the particle's mass?","The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. This differential equation is the classic equation of motion for charged particles. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. This differential equation is the classic equation of motion of a charged particle in a vacuum. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. In this case, one must solve the PB equation together with an appropriate model of the surface charging process. The CODATA recommended value for an electron is ==Origin== When charged particles move in electric and magnetic fields the following two laws apply: *Lorentz force law: \mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), *Newton's second law of motion:\mathbf{F}=m\mathbf{a} = m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} where F is the force applied to the ion, m is the mass of the particle, a is the acceleration, Q is the electric charge, E is the electric field, and v × B is the cross product of the ion's velocity and the magnetic flux density. Thus, the m/z of an ion alone neither infers mass nor the number of charges. Often, the charge can be inferred from theoretical considerations, so the charge-to-mass ratio provides a way to calculate the mass of a particle. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. At very low energies, space charge has a large effect on a particle beam and thus becomes hard to calculate. Often, the charge-to-mass ratio can be determined by observing the deflection of a charged particle in an external magnetic field. A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. When performing a modeling task for any accelerator operation, the results of charged particle beam dynamics simulations must feed into the associated application. This equation can be extended to more highly charged particles by reinterpreting the charge Q as an effective charge. Codes for this computation include *ABCI ABCI home page at kek.jp *ACE3P ACE3P at slac.stanford.gov *CST Studio Suite CST, Computer Simulation Technology at cst.com *GdfidLGdfidL, Gitter drueber, fertig ist die Laube at gdfidl.de *TBCI T. Weiland, DESY * VSim ==Magnet and other hardware-modeling codes== To control the charged particle beam, appropriate electric and magnetic fields must be created. ",41.40,2.2,"""-1.0""",0.9830,7.25,B
+"In Fig. 21-26, particle 1 of charge $-5.00 q$ and particle 2 of charge $+2.00 q$ are held at separation $L$ on an $x$ axis. If particle 3 of unknown charge $q_3$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the $x$ coordinate of particle 3?","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. Here and are the charges on particles 1 and 2 respectively, and are the masses of the particles, and are the velocities of the particles, is the speed of light, is the vector between the two particles, and \hat\mathbf r is the unit vector in the direction of . In physics, a charged particle is a particle with an electric charge. Surface Charging and Points of Zero Charge. A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order {v^2}/{c^2} between two charged particles in a vacuum and is given by L = L_\text{f} + L_\text{int}, where the free particle Lagrangian is L_\text{f} = \frac{1}{2} m_1 v_1^2 + \frac{1}{8c^2} m_1 v_1^4 + \frac{1}{2} m_2 v_2^2 + \frac{1}{8c^2} m_2 v_2^4, and the interaction Lagrangian is L_\text{int} = L_\text{C} + L_\text{D}, where the Coulomb interaction is L_\text{C} = -\frac{q_1 q_2}{r}, and the Darwin interaction is L_\text{D} = \frac{q_1 q_2}{r} \frac{1}{2c^2} \mathbf v_1 \cdot \left[\mathbf 1 + \hat\mathbf{r} \hat\mathbf{r}\right] \cdot \mathbf v_2. Related values associated with the soil characteristics exist along with the pzc value, including zero point of charge (zpc), point of zero net charge (pznc), etc. == Term definition of point of zero charge == The point of zero charge is the pH for which the net surface charge of adsorbent is equal to zero. With the electrostatic force being proportional to r^{-2}, individual particle-particle interactions are long- range in nature, presenting a challenging computational problem in the simulation of particulate systems. IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. thumb|Electrical double layer (EDL) around a negatively charged particle in suspension in water. For example, the surface charge of adsorbent is described by the ion that lies on the surface of the particle (adsorbent) structure like image. Of interest, this represents the first three-dimensional solution. Charges are arbitrarily labeled as positive(+) or negative(-). A related concept in electrochemistry is the electrode potential at the point of zero charge. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. Also, this represents the first two-dimensional solution. Simply applying this cutoff method introduces a discontinuity in the force at r_c that results in particles experiencing sudden impulses when other particles cross the boundary of their respective interaction spheres. Assuming a normal distribution of particle positions and impulses, a charged particle beam (or a bunch of the beam) is characterized by * the species of particle, e.g. electrons, protons, or atomic nuclei * the mean energy of the particles, often expressed in electronvolts (typically keV to GeV) * the (average) particle current, often expressed in amperes * the particle beam size, often using the so-called β-function * the beam emittance, a measure of the area occupied by the beam in one of several phase spaces. The single electron may reside at any point on the surface of the unit sphere. Only the existence of two 'types' of charges is known, there isn't anything inherent about positive charges that makes them positive, and the same goes for the negative charge. == Examples == === Positively charged particles === * protons and atomic nuclei * positrons (antielectrons) * alpha particles * positive charged pions * cations === Negatively charged particles === * electrons * antiprotons * muons * tauons * negative charged pions * anions === Particles without an electric charge=== * neutrons * photons * neutrinos * neutral pions *z boson *higgs boson *atoms ==References== * * * * * ==External links== * Charged particle motion in E/B Field Category:Charge carriers Category:Particle physics ", 135.36,1.5,"""3.03""",2.72,0.6749,D
+An isolated conductor has net charge $+10 \times 10^{-6} \mathrm{C}$ and a cavity with a particle of charge $q=+3.0 \times 10^{-6} \mathrm{C}$. What is the charge on the cavity wall?,"Cavity quantum electrodynamics (cavity QED) is the study of the interaction between light confined in a reflective cavity and atoms or other particles, under conditions where the quantum nature of photons is significant. In modern cavity wall construction, cavity insulation is typically added. thumb|Circuit diagram of a charge qubit circuit. A cavity wall is a type of wall that has a hollow center. He shares half of the prize for developing a new field called cavity quantum electrodynamics (CQED) – whereby the properties of an atom are controlled by placing it in an optical or microwave cavity. This results in two overlapping layers of the superconducting metal, in between which a thin layer of insulator (normally aluminum oxide) is deposited. == Hamiltonian == If the Josephson junction has a junction capacitance C_{\rm J}, and the gate capacitor C_{\rm g}, then the charging (Coulomb) energy of one Cooper pair is: :E_{\rm C}=(2e)^2/2(C_{\rm g}+C_{\rm J}). In superconducting quantum computing, a charge qubit is formed by a tiny superconducting island coupled by a Josephson junction (or practically, superconducting tunnel junction) to a superconducting reservoir (see figure). In quantum computing, a charge qubit (also known as Cooper-pair box) is a qubit whose basis states are charge states (i.e. states which represent the presence or absence of excess Cooper pairs in the island). QIT may refer to: * QIT-Fer et Titane, a Canadian mining company * Quadrupole ion trap * Quantum information theory * Queensland University of Technology * Q = It, the formula describing charge in terms of current and time The conductance quantum, denoted by the symbol , is the quantized unit of electrical conductance. The resonance frequencies are given by \lambda/2: \quad u_n=\frac{c}{\sqrt{\varepsilon_{\text{eff}}}}\frac{n}{2 \ell} \quad (n=1,2,3,\ldots) \qquad \lambda/4:\quad u_n=\frac{c}{\sqrt{\varepsilon_{\text{eff}}}}\frac{2n+1}{4 \ell} \quad (n=0,1,2,\ldots) with \varepsilon_{\text{eff}} being the effective dielectric permittivity of the device. == Artificial atoms, Qubits == The first realized artificial atom in circuit QED was the so-called Cooper-pair box, also known as the charge qubit. Circuit quantum electrodynamics (circuit QED) provides a means of studying the fundamental interaction between light and matter (quantum optics). The voltage is: V = -\frac{(\mu_1 - \mu_2)}{e} , where e is the electron charge. If the detuning is significantly larger than the combined cavity and atomic linewidth the cavity states are merely shifted by \pm g^2/\Delta (with the detuning \Delta=\omega_a-\omega_r) depending on the atomic state. *Cavity wall, Energy Saving Trust *Cavity wall insulation (CWI): consumer guide to issues arising from installations, 14 October 2019, Department for Business, Energy & Industrial Strategy *Cavity Wall Insulation Victims Alliance Category:Masonry Category:Construction Category:Types of wall Structural properties of a concrete-block cavity- wall construction sponsored by the National Concrete Masonry Association. This immobilises the air within the cavity (air is still the actual insulator), preventing convection, and can substantially reduce space heating costs. thumb|A wall that has had cavity wall insulation installed (after construction), with refilled holes highlighted with arrows During construction of new buildings, cavities are often filled with glass fiber wool or mineral wool panels placed between the two leaves (sides) of the wall, but many other building insulation materials offer various advantages and many others are also widely used. Recent work has shown T2 times approaching 100 μs using a type of charge qubit known as a transmon inside a three-dimensional superconducting cavity.C. Rigetti et al., ""Superconducting qubit in waveguide cavity with coherence time approaching 0.1 ms,"" arXiv:1202.5533 (2012) Understanding the limits of T2 is an active area of research in the field of superconducting quantum computing. == Fabrication == Charge qubits are fabricated using techniques similar to those used for microelectronics. Cavity wall insulation also helps to prevent convection and can keep a house warm by making sure that less heat is lost through walls; this can also thus be a more cost-efficient way of heating a house. One function of the cavity is to drain water through weep holes at the base of the wall system or above windows. In 1999, coherent oscillations in the charge Qubit were first observed by Nakamura et al. Manipulation of the quantum states and full realization of the charge qubit was observed 2 years later. Thermal mass cavity walls are thick walls. ",4.86,0.9,"""1.11""",1,-3.0,E
 "Point charges of $+6.0 \mu \mathrm{C}$ and $-4.0 \mu \mathrm{C}$ are placed on an $x$ axis, at $x=8.0 \mathrm{~m}$ and $x=16 \mathrm{~m}$, respectively. What charge must be placed at $x=24 \mathrm{~m}$ so that any charge placed at the origin would experience no electrostatic force?
-","Surface Charging and Points of Zero Charge. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Electric charge can be positive or negative (commonly carried by protons and electrons respectively, by convention). IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. The sign of the space charge can be either negative or positive. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). ChargePoint (formerly Coulomb Technologies) is an American electric vehicle infrastructure company based in Campbell, California. By convention, the charge of an electron is negative, −e, while that of a proton is positive, +e. Space charge is an interpretation of a collection of electric charges in which excess electric charge is treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges. The proton has a charge of +e, and the electron has a charge of −e. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. No force, either of attraction or of repulsion, can be observed between an electrified body and a body not electrified.James Clerk Maxwell (1891) A Treatise on Electricity and Magnetism, pp. 32–33, Dover Publications ==The role of charge in electric current== Electric current is the flow of electric charge through an object. Electric charges produce electric fields. In ordinary matter, negative charge is carried by electrons, and positive charge is carried by the protons in the nuclei of atoms. Rear..JPG|Ford Focus being charged on a roadside ChargePoint station. In June 2017, ChargePoint took over 9,800 electric vehicle charging spots from GE. In contemporary understanding, positive charge is now defined as the charge of a glass rod after being rubbed with a silk cloth, but it is arbitrary which type of charge is called positive and which is called negative. A related concept in electrochemistry is the electrode potential at the point of zero charge. In this case, the electrical potential profile ψ(z) near a charged interface will only depend on the position z. ",-4.37 ,0.195,0.042,2.534324263,-45,E
-"The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of $300 \mathrm{~m}$ the field has magnitude $60.0 \mathrm{~N} / \mathrm{C}$; at an altitude of $200 \mathrm{~m}$, the magnitude is $100 \mathrm{~N} / \mathrm{C}$. Find the net amount of charge contained in a cube $100 \mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \mathrm{~m}$.","220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. thumb|400px|A snapshot of the variation of the Earth's magnetic field from its intrinsic field at 400 km altitude, due to the ionospheric current systems. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Thus, v_i is the number of vertices where the given number of edges meet, e is the total number of edges, f_3 is the number of triangular faces, f_4 is the number of quadrilateral faces, and \theta_1 is the smallest angle subtended by vectors associated with the nearest charge pair. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. The horizontal intensity of magnetic field peaks at ~12 LT. For conductors, p=3. * For N = 4, electrons reside at the vertices of a regular tetrahedron. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The mechanism that produced the variation in the magnetic field was proposed as a band of current about 300 km in width flowing over the dip equator. Int.,159, 521-547. == External links == * A movie of the magnetic fields generated by the equatorial electrojet, . This electric field gives a primary eastwards Pedersen current. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. Global-scale ionospheric circulation establishes a Sq (solar quiet) current system in the E region of the Earth's ionosphere (100-130 km altitude), and a primary eastwards electric field near day-side magnetic equator, where the magnetic field is horizontal and northwards. * For N = 3, electrons reside at the vertices of an equilateral triangle about any great circle.. Electr. 52, 449 – 451 *Chapman, S. 1951, The equatorial electrojet as detected from the abnormal electric current distribution above Huancayo, Peru, and elsewhere. The equatorial electrojet (EEJ) is a narrow ribbon of current flowing eastward in the day time equatorial region of the Earth's ionosphere. We can similarly describe the electric field E so that . Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. ",3.54,61,6.0,-0.16,-3.8,A
-What would be the magnitude of the electrostatic force between two 1.00 C point charges separated by a distance of $1.00 \mathrm{~m}$  if such point charges existed (they do not) and this configuration could be set up?,"If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). A common question arises concerning the interaction of a point charge with its own electrostatic potential. The force is along the straight line joining the two charges. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. == Vector form == thumb|right|350px|In the image, the vector is the force experienced by , and the vector is the force experienced by . The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). Coulomb's law in vector form states that the electrostatic force \mathbf{F}_1 experienced by a charge, q_1 at position \mathbf{r}_1, in the vicinity of another charge, q_2 at position \mathbf{r}_2, in a vacuum is equal to \mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2} where \boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2 is the vectorial distance between the charges, \widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|} a unit vector pointing from q_2 to and \varepsilon_0 the electric constant. The electrostatic force \mathbf{F}_2 experienced by q_2, according to Newton's third law, is === System of discrete charges === The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. In 1767, he conjectured that the force between charges varied as the inverse square of the distance. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. ",+65.49,0.0408,0.829,10.8,8.99,E
-An electric dipole consisting of charges of magnitude $1.50 \mathrm{nC}$ separated by $6.20 \mu \mathrm{m}$ is in an electric field of strength 1100 $\mathrm{N} / \mathrm{C}$. What is the magnitude of the electric dipole moment?,"Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole: \left|\mathbf{\tau}\right| = \left|q\mathbf{r}\right| \left|\mathbf{E}\right|\sin\theta. (See electron electric dipole moment). For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn. The dipole moment is particularly useful in the context of an overall neutral system of charges, for example a pair of opposite charges, or a neutral conductor in a uniform electric field. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. In the case of two classical point charges, +q and -q, with a displacement vector, \mathbf{r}, pointing from the negative charge to the positive charge, the electric dipole moment is given by \mathbf{p} = q\mathbf{r}. The electric dipole moment vector also points from the negative charge to the positive charge. It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. If the charge, e, is omitted from the electric dipole operator during this calculation, one obtains \mathbf{R}_\alpha as used in oscillator strength. == Applications == The transition dipole moment is useful for determining if transitions are allowed under the electric dipole interaction. This is the vector sum of the individual dipole moments of the neutral charge pairs. The transition dipole moment or transition moment, usually denoted \mathbf{d}_{nm} for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). ",1110,9.30,1.94,0.11,1.41,B
-What equal positive charges would have to be placed on Earth and on the Moon to neutralize their gravitational attraction?,"The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. There is growing evidence that fine particles of moondust might actually float, ejected from the lunar surface by electrostatic repulsion. The charged particle lunar environment experiment was deployed approximately 3 meters northeast of the central station. Volta Latitude Longitude Diameter B 54.6° N 83.5° W 9 km D 52.5° N 83.3° W 20 km ==References== * * * * * * * * * * * * Category:Impact craters on the Moon Category:Alessandro Volta Volta is a lunar impact crater near the northwest limb of the Moon. Lunar Gravity Coefficients nm Jn Cnm Snm 20 203.3 × 10−6 — — 21 — 0 0 22 — 22.4 × 10−6 0 30 8.46 × 10−6 — — 31 — 28.48 × 10−6 5.89 × 10−6 32 — 4.84 × 10−6 1.67 × 10−6 33 — 1.71 × 10−6 −0.25 × 10−6 The J2 coefficient for an oblate shape to the gravity field is affected by rotation and solid-body tides whereas C22 is affected by solid-body tides. Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% (= 1/6) of what they weigh on the Earth. ==Gravitational field== The gravitational field of the Moon has been measured by tracking the radio signals emitted by orbiting spacecraft. Consequently, it is conventional to express the lunar mass M multiplied by the gravitational constant G. The gravitational potential V at an external point is conventionally expressed as positive in astronomy and geophysics, but negative in physics. Lunar Gravity Fields Designation Degree Mission IDs Citation LP165P 165 LO A15 A16 Cl LP GLGM3 150 LO A15 A16 Cl LP CEGM01 50 Ch 1 SGM100h 100 LO A15 A16 Cl LP K/S SGM150J 150 LO A15 A16 Cl LP K/S CEGM02 100 LO A15 A16 Cl LP K/S Ch1 GL0420A 420 G GL0660B 660 G GRGM660PRIM 660 G GL0900D 900 G GRGM900C 900 G GRGM1200A 1200 G CEGM03 100 LO A15 A16 Cl LP Ch1 K/S Ch5T1 A major feature of the Moon's gravitational field is the presence of mascons, which are large positive gravity anomalies associated with some of the giant impact basins. Detailed data collected has shown that for low lunar orbit the only ""stable"" orbits are at inclinations near 27°, 50°, 76°, and 86°. The mass of the Moon is M = 7.3458 × 1022 kg and the mean density is 3346 kg/m3. Much of these details are still speculative, but the Lunar Prospector spacecraft detected changes in the lunar nightside voltage during magnetotail crossings, jumping from -200 V to -1000 V. In physics, a neutral particle is a particle with no electric charge, such as a neutron. One of these (analyzer A) pointed toward local lunar vertical, and the other (analyzer B) to a point 60 deg from vertical toward lunar west. The C31 coefficient is large. ==Simulating lunar gravity== In January 2022 China was reported by the South China Morning Post to have built a small (60 centimeters in diameter) research facility to simulate low lunar gravity with the help of magnets. thumb|right|300px|Total magnetic field strength at the surface of the Moon as derived from the Lunar Prospector electron reflectometer experiment. Voltaire is an impact crater on Mars's moon Deimos and is approximately across. thumb|upright=1.2|A close-up view of the CPLEE on the Moon's surface thumb|upright|The CPLEE with the ALSEP central station in the background The Charged Particle Lunar Environment Experiment (CPLEE), placed on the lunar surface by the Apollo 14 mission as part of the Apollo Lunar Surface Experiments Package (ALSEP), was designed to measure the energy spectra of low-energy charged particles striking the lunar surface. The center of gravity of the Moon does not coincide exactly with its geometric center, but is displaced toward the Earth by about 2 kilometers.Nine Planets == Mass of Moon == The gravitational constant G is less accurate than the product of G and masses for Earth and Moon. The most notable of these are Volta D in the southeast and Volta B to the northeast. ==Satellite craters== By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Volta. ",0.6957,0.84,6.283185307,5.7,0.14,D
+","Surface Charging and Points of Zero Charge. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. Electric charge can be positive or negative (commonly carried by protons and electrons respectively, by convention). IUPAC defines the potential at the point of zero charge as the potential of an electrode (against a defined reference electrode) at which one of the charges defined is zero. The sign of the space charge can be either negative or positive. The potential of zero charge is used for determination of the absolute electrode potential in a given electrolyte. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). ChargePoint (formerly Coulomb Technologies) is an American electric vehicle infrastructure company based in Campbell, California. By convention, the charge of an electron is negative, −e, while that of a proton is positive, +e. Space charge is an interpretation of a collection of electric charges in which excess electric charge is treated as a continuum of charge distributed over a region of space (either a volume or an area) rather than distinct point-like charges. The proton has a charge of +e, and the electron has a charge of −e. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. No force, either of attraction or of repulsion, can be observed between an electrified body and a body not electrified.James Clerk Maxwell (1891) A Treatise on Electricity and Magnetism, pp. 32–33, Dover Publications ==The role of charge in electric current== Electric current is the flow of electric charge through an object. Electric charges produce electric fields. In ordinary matter, negative charge is carried by electrons, and positive charge is carried by the protons in the nuclei of atoms. Rear..JPG|Ford Focus being charged on a roadside ChargePoint station. In June 2017, ChargePoint took over 9,800 electric vehicle charging spots from GE. In contemporary understanding, positive charge is now defined as the charge of a glass rod after being rubbed with a silk cloth, but it is arbitrary which type of charge is called positive and which is called negative. A related concept in electrochemistry is the electrode potential at the point of zero charge. In this case, the electrical potential profile ψ(z) near a charged interface will only depend on the position z. ",-4.37 ,0.195,"""0.042""",2.534324263,-45,E
+"The electric field in a certain region of Earth's atmosphere is directed vertically down. At an altitude of $300 \mathrm{~m}$ the field has magnitude $60.0 \mathrm{~N} / \mathrm{C}$; at an altitude of $200 \mathrm{~m}$, the magnitude is $100 \mathrm{~N} / \mathrm{C}$. Find the net amount of charge contained in a cube $100 \mathrm{~m}$ on edge, with horizontal faces at altitudes of 200 and $300 \mathrm{~m}$.","220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. The Pauthenier equation states K. Adamiak, ""Rate of charging of spherical particles by monopolar ions in electric fields"", IEEE Transactions on Industry Applications 38, 1001-1008 (2002) that the maximum charge accumulated by a particle modelled by a small sphere passing through an electric field is given by: Q_{\mathrm{max}}=4\pi R^2\epsilon_0pE where \epsilon_0 is the permittivity of free space, R is the radius of the sphere, E is the electric field strength, and p is a material dependent constant. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. thumb|400px|A snapshot of the variation of the Earth's magnetic field from its intrinsic field at 400 km altitude, due to the ionospheric current systems. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Thus, v_i is the number of vertices where the given number of edges meet, e is the total number of edges, f_3 is the number of triangular faces, f_4 is the number of quadrilateral faces, and \theta_1 is the smallest angle subtended by vectors associated with the nearest charge pair. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. The horizontal intensity of magnetic field peaks at ~12 LT. For conductors, p=3. * For N = 4, electrons reside at the vertices of a regular tetrahedron. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The mechanism that produced the variation in the magnetic field was proposed as a band of current about 300 km in width flowing over the dip equator. Int.,159, 521-547. == External links == * A movie of the magnetic fields generated by the equatorial electrojet, . This electric field gives a primary eastwards Pedersen current. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. Global-scale ionospheric circulation establishes a Sq (solar quiet) current system in the E region of the Earth's ionosphere (100-130 km altitude), and a primary eastwards electric field near day-side magnetic equator, where the magnetic field is horizontal and northwards. * For N = 3, electrons reside at the vertices of an equilateral triangle about any great circle.. Electr. 52, 449 – 451 *Chapman, S. 1951, The equatorial electrojet as detected from the abnormal electric current distribution above Huancayo, Peru, and elsewhere. The equatorial electrojet (EEJ) is a narrow ribbon of current flowing eastward in the day time equatorial region of the Earth's ionosphere. We can similarly describe the electric field E so that . Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. ",3.54,61,"""6.0""",-0.16,-3.8,A
+What would be the magnitude of the electrostatic force between two 1.00 C point charges separated by a distance of $1.00 \mathrm{~m}$  if such point charges existed (they do not) and this configuration could be set up?,"If is the distance between the charges, the magnitude of the force is |\mathbf{F}|=\frac{|q_1q_2|}{4\pi\varepsilon_0 r^2}, where is the electric constant. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). A common question arises concerning the interaction of a point charge with its own electrostatic potential. The force is along the straight line joining the two charges. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. == Vector form == thumb|right|350px|In the image, the vector is the force experienced by , and the vector is the force experienced by . The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). Coulomb's law in vector form states that the electrostatic force \mathbf{F}_1 experienced by a charge, q_1 at position \mathbf{r}_1, in the vicinity of another charge, q_2 at position \mathbf{r}_2, in a vacuum is equal to \mathbf{F}_1 = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{r}_1-\mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}_{12}}{|\mathbf{r}_{12}|^2} where \boldsymbol{r}_{12} = \boldsymbol{r}_1 - \boldsymbol{r}_2 is the vectorial distance between the charges, \widehat{\mathbf{r}}_{12}=\frac{\mathbf{r}_{12}}{|\mathbf{r}_{12}|} a unit vector pointing from q_2 to and \varepsilon_0 the electric constant. The electrostatic force \mathbf{F}_2 experienced by q_2, according to Newton's third law, is === System of discrete charges === The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. In 1767, he conjectured that the force between charges varied as the inverse square of the distance. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. ",+65.49,0.0408,"""0.829""",10.8,8.99,E
+An electric dipole consisting of charges of magnitude $1.50 \mathrm{nC}$ separated by $6.20 \mu \mathrm{m}$ is in an electric field of strength 1100 $\mathrm{N} / \mathrm{C}$. What is the magnitude of the electric dipole moment?,"Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole: \left|\mathbf{\tau}\right| = \left|q\mathbf{r}\right| \left|\mathbf{E}\right|\sin\theta. (See electron electric dipole moment). For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn. The dipole moment is particularly useful in the context of an overall neutral system of charges, for example a pair of opposite charges, or a neutral conductor in a uniform electric field. The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . Some authors may split in half and use since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. In the case of two classical point charges, +q and -q, with a displacement vector, \mathbf{r}, pointing from the negative charge to the positive charge, the electric dipole moment is given by \mathbf{p} = q\mathbf{r}. The electric dipole moment vector also points from the negative charge to the positive charge. It is possible to calculate dipole moments from electronic structure theory, either as a response to constant electric fields or from the density matrix. If the charge, e, is omitted from the electric dipole operator during this calculation, one obtains \mathbf{R}_\alpha as used in oscillator strength. == Applications == The transition dipole moment is useful for determining if transitions are allowed under the electric dipole interaction. This is the vector sum of the individual dipole moments of the neutral charge pairs. The transition dipole moment or transition moment, usually denoted \mathbf{d}_{nm} for a transition between an initial state, m, and a final state, n, is the electric dipole moment associated with the transition between the two states. This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). ",1110,9.30,"""1.94""",0.11,1.41,B
+What equal positive charges would have to be placed on Earth and on the Moon to neutralize their gravitational attraction?,"The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. There is growing evidence that fine particles of moondust might actually float, ejected from the lunar surface by electrostatic repulsion. The charged particle lunar environment experiment was deployed approximately 3 meters northeast of the central station. Volta Latitude Longitude Diameter B 54.6° N 83.5° W 9 km D 52.5° N 83.3° W 20 km ==References== * * * * * * * * * * * * Category:Impact craters on the Moon Category:Alessandro Volta Volta is a lunar impact crater near the northwest limb of the Moon. Lunar Gravity Coefficients nm Jn Cnm Snm 20 203.3 × 10−6 — — 21 — 0 0 22 — 22.4 × 10−6 0 30 8.46 × 10−6 — — 31 — 28.48 × 10−6 5.89 × 10−6 32 — 4.84 × 10−6 1.67 × 10−6 33 — 1.71 × 10−6 −0.25 × 10−6 The J2 coefficient for an oblate shape to the gravity field is affected by rotation and solid-body tides whereas C22 is affected by solid-body tides. Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% (= 1/6) of what they weigh on the Earth. ==Gravitational field== The gravitational field of the Moon has been measured by tracking the radio signals emitted by orbiting spacecraft. Consequently, it is conventional to express the lunar mass M multiplied by the gravitational constant G. The gravitational potential V at an external point is conventionally expressed as positive in astronomy and geophysics, but negative in physics. Lunar Gravity Fields Designation Degree Mission IDs Citation LP165P 165 LO A15 A16 Cl LP GLGM3 150 LO A15 A16 Cl LP CEGM01 50 Ch 1 SGM100h 100 LO A15 A16 Cl LP K/S SGM150J 150 LO A15 A16 Cl LP K/S CEGM02 100 LO A15 A16 Cl LP K/S Ch1 GL0420A 420 G GL0660B 660 G GRGM660PRIM 660 G GL0900D 900 G GRGM900C 900 G GRGM1200A 1200 G CEGM03 100 LO A15 A16 Cl LP Ch1 K/S Ch5T1 A major feature of the Moon's gravitational field is the presence of mascons, which are large positive gravity anomalies associated with some of the giant impact basins. Detailed data collected has shown that for low lunar orbit the only ""stable"" orbits are at inclinations near 27°, 50°, 76°, and 86°. The mass of the Moon is M = 7.3458 × 1022 kg and the mean density is 3346 kg/m3. Much of these details are still speculative, but the Lunar Prospector spacecraft detected changes in the lunar nightside voltage during magnetotail crossings, jumping from -200 V to -1000 V. In physics, a neutral particle is a particle with no electric charge, such as a neutron. One of these (analyzer A) pointed toward local lunar vertical, and the other (analyzer B) to a point 60 deg from vertical toward lunar west. The C31 coefficient is large. ==Simulating lunar gravity== In January 2022 China was reported by the South China Morning Post to have built a small (60 centimeters in diameter) research facility to simulate low lunar gravity with the help of magnets. thumb|right|300px|Total magnetic field strength at the surface of the Moon as derived from the Lunar Prospector electron reflectometer experiment. Voltaire is an impact crater on Mars's moon Deimos and is approximately across. thumb|upright=1.2|A close-up view of the CPLEE on the Moon's surface thumb|upright|The CPLEE with the ALSEP central station in the background The Charged Particle Lunar Environment Experiment (CPLEE), placed on the lunar surface by the Apollo 14 mission as part of the Apollo Lunar Surface Experiments Package (ALSEP), was designed to measure the energy spectra of low-energy charged particles striking the lunar surface. The center of gravity of the Moon does not coincide exactly with its geometric center, but is displaced toward the Earth by about 2 kilometers.Nine Planets == Mass of Moon == The gravitational constant G is less accurate than the product of G and masses for Earth and Moon. The most notable of these are Volta D in the southeast and Volta B to the northeast. ==Satellite craters== By convention these features are identified on lunar maps by placing the letter on the side of the crater midpoint that is closest to Volta. ",0.6957,0.84,"""6.283185307""",5.7,0.14,D
 "The initial charges on the three identical metal spheres in Fig. 21-24 are the following: sphere $A, Q$; sphere $B,-Q / 4$; and sphere $C, Q / 2$, where $Q=2.00 \times 10^{-14}$ C. Spheres $A$ and $B$ are fixed in place, with a center-to-center separation of $d=1.20 \mathrm{~m}$, which is much larger than the spheres. Sphere $C$ is touched first to sphere $A$ and then to sphere $B$ and is then removed. What then is the magnitude of the electrostatic force between spheres $A$ and $B$ ?
-","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|280px|Scheme of the colloidal probe technique for direct force measurements in the sphere-plane and sphere-sphere geometries. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). For unequally charged objects and eventually at shorted distances, these forces may also be attractive. Simply applying this cutoff method introduces a discontinuity in the force at r_c that results in particles experiencing sudden impulses when other particles cross the boundary of their respective interaction spheres. The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, k_\text{B} T, where k_\text{B} is the Boltzmann constant and T is the absolute temperature in kelvins. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. It is also possible to study forces between colloidal particles by attaching another particle to the substrate and perform the measurement in the sphere-sphere geometry, see figure above. thumb|left|350px|Principle of the force measurements by the colloidal probe technique. According to Nikolaides, the electrostatic force engenders a long range capillary attraction. thumb|Computed electrostatic equipotentials (black contours) between two electrically charged spheres In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential.Weisstein, Eric W. ""Equipotential Curve."" A way to correct this problem is to shift the force to zero at r_c, thus removing the discontinuity. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. In the particular case of electrostatic forces, as the force magnitude is large at the boundary, this unphysical feature can compromise simulation accuracy. The exponential nature of these repulsive forces and the fact that its range is given by the Debye length was confirmed experimentally by direct force measurements, including surface forces apparatus, colloidal probe technique, or optical tweezers. ",0.011,4.68,1.1,62.2,0.686,B
-A $10.0 \mathrm{~g}$ block with a charge of $+8.00 \times 10^{-5} \mathrm{C}$ is placed in an electric field $\vec{E}=(3000 \hat{\mathrm{i}}-600 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$. What is the magnitude of the electrostatic force on the block?,"The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration. {{block indent|em=1.2|text=The electrostatic potential energy, UE, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rrefThe reference zero is usually taken to be a state in which the individual point charges are very well separated (""are at infinite separation"") and are at rest. to that position r.Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 {{Equation box 1 |indent=: |equation=U_\mathrm{E}(\mathbf r) = -W_{r_{\rm ref} \rightarrow r } = -\int_{{\mathbf{r}}_{\rm ref}}^\mathbf{r} q\mathbf{E}(\mathbf{r'}) \cdot \mathrm{d} \mathbf{r'}, |cellpadding |border |border colour = #50C878 |background colour = #ECFCF4}} where E is the electrostatic field and dr' is the displacement vector in a curve from the reference position rref to the final position r.}} Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. Electromotive force in electrostatic units is the statvolt (in the centimeter gram second system of units equal in amount to an erg per electrostatic unit of charge). ==Formal definitions== Inside a source of emf (such as a battery) that is open-circuited, a charge separation occurs between the negative terminal N and the positive terminal P. In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal{E} or {\xi}) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. (A virtual experiment based on the energy transfert between capacitor plates reveals that an additional term must be taken into account when the electrostatic energy is expressed in terms of the electric field and displacement vectors . The electrostatic field does not contribute to the net emf around a circuit because the electrostatic portion of the electric field is conservative (i.e., the work done against the field around a closed path is zero, see Kirchhoff's voltage law, which is valid, as long as the circuit elements remain at rest and radiation is ignored ). This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The magnitude of the emf for the battery (or other source) is the value of this open-circuit voltage. In the particular case of electrostatic forces, as the force magnitude is large at the boundary, this unphysical feature can compromise simulation accuracy. The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula. {{math proof |title=Outline of proof |proof= The electrostatic force F acting on a charge q can be written in terms of the electric field E as \mathbf{F} = q\mathbf{E} , By definition, the change in electrostatic potential energy, UE, of a point charge q that has moved from the reference position rref to position r in the presence of an electric field E is the negative of the work done by the electrostatic force to bring it from the reference position rref to that position r. The total electrostatic potential energy stored in a capacitor is given by U_E = \frac{1}{2}QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C} where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Years earlier, Alessandro Volta, who had measured a contact potential difference at the metal–metal (electrode–electrode) interface of his cells, held the incorrect opinion that contact alone (without taking into account a chemical reaction) was the origin of the emf. == Notation and units of measurement == Electromotive force is often denoted by \mathcal{E} or ℰ. Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. This emf is the work done on a unit charge by the source's nonelectrostatic field \boldsymbol{E}' when the charge moves from N to P. So, E and ds must be parallel: \mathbf{E} \cdot \mathrm{d} \mathbf{s} = |\mathbf{E}| \cdot |\mathrm{d}\mathbf{s}|\cos(0) = E \mathrm{d}s Using Coulomb's law, the electric field is given by |\mathbf{E}| = E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{s^2} and the integral can be easily evaluated: U_E(r) = -\int_\infty^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} = -\int_\infty^r \frac{1}{4\pi\varepsilon_0}\frac{qQ}{s^2}{\rm d}s = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{r} = k_e\frac{qQ}{r} }} ===One point charge q in the presence of n point charges Qi=== thumb|Electrostatic potential energy of q due to Q1 and Q2 charge system:U_E = q\frac{1}{4 \pi \varepsilon_0} \left(\frac{Q_1}{r_1} + \frac{Q_2}{r_2} \right) The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qi, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, ri is the distance between the point charges q and Qi, and q and Qi are the assigned values of the charges. ==Electrostatic potential energy stored in a system of point charges== The electrostatic potential energy UE stored in a system of N charges q1, q2, …, qN at positions r1, r2, …, rN respectively, is: |}} where, for each i value, Φ(ri) is the electrostatic potential due to all point charges except the one at ri,The factor of one half accounts for the 'double counting' of charge pairs. * Both a 1 volt emf and a 1 volt potential difference correspond to 1 joule per coulomb of charge. ",131,2.14,0.0547,-191.2,0.245,E
-"Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \mathrm{~cm}$. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the outer shell. What is the magnitude $E$ of the electric field at radial distance $r=4.0 \mathrm{~cm}$?","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Let f(\mathbf{r}^{\prime}) be the second charge density, and define \lambda(\rho, \theta) as its integral over z \lambda(\rho, \theta) = \int dz \, f(\rho, \theta, z) The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles U = \int d\theta \, d\rho \, \rho \, \lambda(\rho, \theta) \Phi(\rho, \theta) If the cylindrical multipoles are exterior, this equation becomes U = \frac{-Q_1}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \int d\theta \, d\rho \left[ C_{1k} \frac{\cos k\theta}{\rho^{k-1}} + S_{1k} \frac{\sin k\theta}{\rho^{k-1}}\right] \lambda(\rho, \theta) where Q_{1}, C_{1k} and S_{1k} are the cylindrical multipole moments of charge distribution 1. By assumption, the line charges are infinitely long and aligned with the z axis. ==Cylindrical multipole moments of a line charge== frame|right|Figure 1: Definitions for cylindrical multipoles; looking down the z' axis The electric potential of a line charge \lambda located at (\rho', \theta') is given by \Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \ln R = \frac{-\lambda}{4\pi\epsilon} \ln \left| \rho^{2} + \left( \rho' \right)^{2} - 2\rho\rho'\cos (\theta - \theta' ) \right| where R is the shortest distance between the line charge and the observation point. Here, R is the distance from the origin while r is the distance from the central axis of a cylinder as in the (r,\phi,z) cylindrical coordinate system. It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The same equation shows that multipole moments higher than the first non-zero moment do depend on the choice of origin (in general). ==Interior axial multipole moments== Conversely, if the radius r is smaller than the smallest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{min}), the electric potential may be written \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta ) where the interior axial multipole moments I_{k} are defined I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}} Special cases include the interior axial monopole moment ( eq the total charge) M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}, the interior axial dipole moment M_{1} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{2}}, etc. Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as \frac{1}{R}. We can similarly describe the electric field E so that . If the radius r of the observation point P is greater than the largest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{max}), the electric potential may be written \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta ) where the axial multipole moments M_{k} are defined M_{k} \equiv \int d\zeta \ \lambda(\zeta) \zeta^{k} Special cases include the axial monopole moment (=total charge) M_{0} \equiv \int d\zeta \ \lambda(\zeta), the axial dipole moment M_{1} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta, and the axial quadrupole moment M_{2} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta^{2}. This energy formula can be reduced to a remarkably simple form U = \frac{-Q_{1}}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{1k} I_{2k} + S_{1k} J_{2k} \right) where I_{2k} and J_{2k} are the interior cylindrical multipoles of the second charge density. Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves: : \frac{{e^{ik_0 r} }} {r} = i\int\limits_0^\infty {dk_\rho \frac J_0 (k_\rho \rho )e^{ik_z \left| z \right|} } Where : k_z=(k_0^2-k_\rho^2)^{1/2} The notation used here is different form that above: r is now the distance from the origin and \rho is the radial distance in a cylindrical coordinate system defined as (\rho,\phi,z). The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles U = \frac{-Q_1\ln \rho'}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{2k} I_{1k} + S_{2k} J_{1k} \right) where I_{1k} and J_{1k} are the interior cylindrical multipole moments of charge distribution 1, and C_{2k} and S_{2k} are the exterior cylindrical multipoles of the second charge density. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density \lambda(z) localized to the z-axis. frame|right|Figure 1: Point charge on the z axis; Definitions for axial multipole expansion ==Axial multipole moments of a point charge== The electric potential of a point charge q located on the z-axis at z=a (Fig. 1) equals \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + a^{2} - 2 a r \cos \theta}}. At short distances (\frac{r}{\zeta_\text{min}} \ll 1), the potential is well- approximated by the leading nonzero interior multipole term. ==See also== * Potential theory * Multipole expansion * Spherical multipole moments * Cylindrical multipole moments * Solid harmonics * Laplace expansion ==References== Category:Electromagnetism Category:Potential theory Category:Moment (physics) If the radius r of the observation point is greater than a, we may factor out \frac{1}{r} and expand the square root in powers of (a/r)<1 using Legendre polynomials \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta ) where the axial multipole moments M_{k} \equiv q a^{k} contain everything specific to a given charge distribution; the other parts of the electric potential depend only on the coordinates of the observation point P. Special cases include the axial monopole moment M_{0}=q, the axial dipole moment M_{1}=q a and the axial quadrupole moment M_{2} \equiv q a^{2}. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. Thus, at large distances (\frac{\zeta_\text{max}}{r} \ll 1), the potential is well-approximated by the leading nonzero multipole term. Conversely, if the radius r is less than a, we may factor out \frac{1}{a} and expand in powers of (r/a)<1, once again using Legendre polynomials \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon a} \sum_{k=0}^{\infty} \left( \frac{r}{a} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta ) where the interior axial multipole moments I_{k} \equiv \frac{q}{a^{k+1}} contain everything specific to a given charge distribution; the other parts depend only on the coordinates of the observation point P. ==General axial multipole moments== To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimal charge element \lambda(\zeta)\ d\zeta, where \lambda(\zeta) represents the charge density at position z=\zeta on the z-axis. thumb|An annular solar eclipse has a magnitude of less than 1.0 The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry ",57.2,-1.00,-111.92,0.9974,2.3,E
-A particle of charge $1.8 \mu \mathrm{C}$ is at the center of a Gaussian cube $55 \mathrm{~cm}$ on edge. What is the net electric flux through the surface?,"It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). It is defined as the closed surface in three dimensional space by which the flux of vector field be calculated. == Common Gaussian surfaces == Most calculations using Gaussian surfaces begin by implementing Gauss's law (for electricity):Introduction to electrodynamics (4th Edition), D. J. Griffiths, 2012, :}{\varepsilon_0}.}} For a closed Gaussian surface, electric flux is given by: where * is the electric field, * is any closed surface, * is the total electric charge inside the surface , * is the electric constant (a universal constant, also called the ""permittivity of free space"") () This relation is known as Gauss' law for electric fields in its integral form and it is one of Maxwell's equations. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form. thumb|A tiny Gauss's box whose sides are perpendicular to a conductor's surface is used to find the local surface charge once the electric potential and the electric field are calculated by solving Laplace's equation. Thereby is the electrical charge enclosed by the Gaussian surface. Electric flux through its surface is zero. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. Gauss's law may be expressed as: \Phi_E = \frac{Q}{\varepsilon_0} where is the electric flux through a closed surface enclosing any volume , is the total charge enclosed within , and is the electric constant. A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field.Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, It is an arbitrary closed surface (the boundary of a 3-dimensional region ) used in conjunction with Gauss's law for the corresponding field (Gauss's law, Gauss's law for magnetism, or Gauss's law for gravity) by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution. The flux passing consists of the three contributions: : For surfaces a and b, and will be perpendicular. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. In electromagnetism, electric flux is the measure of the electric field through a given surface,Purcell, pp. 22–26 although an electric field in itself cannot flow. The electric flux over a surface is therefore given by the surface integral: \Phi_E = \iint_S \mathbf{E} \cdot \textrm{d}\mathbf{S} where is the electric field and is a differential area on the closed surface with an outward facing surface normal defining its direction. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. While the electric flux is not affected by charges that are not within the closed surface, the net electric field, can be affected by charges that lie outside the closed surface. *MISN-0-133 Gauss's Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET. If the electric field is uniform, the electric flux passing through a surface of vector area is \Phi_E = \mathbf{E} \cdot \mathbf{S} = ES \cos \theta, where is the electric field (having units of ), is its magnitude, is the area of the surface, and is the angle between the electric field lines and the normal (perpendicular) to . The electric field is perpendicular, locally, to the equipotential surface of the conductor, and zero inside; its flux πa2·E, by Gauss's law equals πa2·σ/ε0. Under these circumstances, Gauss's law modifies to \Phi_E = \frac{Q_\mathrm{free}}{\varepsilon} for the integral form, and abla \cdot \mathbf{E} = \frac{\rho_\mathrm{free}}{\varepsilon} for the differential form. ==Interpretations== ===In terms of fields of force=== Gauss's theorem can be interpreted in terms of the lines of force of the field as follows: The flux through a closed surface is dependent upon both the magnitude and direction of the electric field lines penetrating the surface. The electric flux is defined as a surface integral of the electric field: : where is the electric field, is a vector representing an infinitesimal element of area of the surface, and represents the dot product of two vectors. In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. ",2.0,35,5.7,0.2553,+17.7,A
-The drum of a photocopying machine has a length of $42 \mathrm{~cm}$ and a diameter of $12 \mathrm{~cm}$. The electric field just above the drum's surface is $2.3 \times 10^5 \mathrm{~N} / \mathrm{C}$. What is the total charge on the drum? ,"The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into ""free"" and ""bound"" charges. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Unlike the case of the metal, the image charge q' is not exactly opposite to the real charge: q'=\frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_1 + \varepsilon_2}q. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Electrostatic charge on an object can be measured by placing it into the Faraday Cup. If we assume for simplicity (without loss of generality) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the polar angle θ and is given by: : \sigma(\theta) = \varepsilon_0 \left.\frac{\partial V}{\partial r} \right|_{r=R} =\frac{-q\left(R^2-p^2\right)}{4\pi R\left(R^2+p^2-2pR\cos\theta\right)^{3/2}} The total charge on the sphere may be found by integrating over all angles: : Q_t=\int_0^\pi d\theta \int_0^{2\pi} d\phi\,\,\sigma(\theta) R^2\sin\theta = -q Note that the reciprocal problem is also solved by this method. It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. A surface charge is an electric charge present on a two-dimensional surface. Just as in the first case, the image charge will have charge −qR/p and will be located at vector position \left(R^2 / p^2\right) \mathbf{p}. Free charges are the excess charges which can move into electrostatic equilibrium, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute electric currents. === Total charge densities === In terms of volume charge densities, the total charge density is: \rho = \rho_\text{f} + \rho_\text{b}\,. as for surface charge densities: \sigma = \sigma_\text{f} + \sigma_\text{b}\,. where subscripts ""f"" and ""b"" denote ""free"" and ""bound"" respectively. === Bound charge === The bound surface charge is the charge piled up at the surface of the dielectric, given by the dipole moment perpendicular to the surface: q_b = \frac{\mathbf{d} \cdot\mathbf{\hat{n}}}{|\mathbf{s}|} where s is the separation between the point charges constituting the dipole, \mathbf{d} is the electric dipole moment, \mathbf{\hat{n}} is the unit normal vector to the surface. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. ",32,0.32,1000.0,24,1.2,B
-A spherical water drop $1.20 \mu \mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \mathrm{~N} / \mathrm{C}$. What is the magnitude of the gravitational force on the drop?,"If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac{\rho d}{\mu} V. thumb|upright=0.7|The downward force of gravity (Fg) equals the restraining force of drag (Fd) plus the buoyancy. Fluid Dynamics Research 12.2 (1993): 61-93 In the case of floating, a drop will float on the surface for several seconds. A new model for the equilibrium shape of raindrops. The resulting outcome depends on the properties of the drop, the surface, and the surrounding fluid, which is most commonly a gas. == On a dry solid surface == When a liquid drop strikes a dry solid surface, it generally spreads on the surface, and then will retract if the impact is energetic enough to cause the drop to spread out more than it would generally spread due to its static receding contact angle. The Beard and Chuang model is a well known and leading theoretical force balance model used to derive the rotational cross-sections of raindrops in their equilibrium state by employing Chebyshev polynomials in series. thumb|Beard and Chuang model of raindrop The radius-vector of the raindrop's surface r(\theta) in vertical angular direction \theta is equal to r(\theta)=a [ 1 + \sum c_n cos(n \theta) ] , where shape coefficients c_n \cdot 10^4 are defined for the raindrops with different equivolumetric diameter as in following table d(mm) n = 0 1 2 3 4 5 6 7 8 9 10 2.0 -131 -120 -376 -96 -4 15 5 0 -2 0 1 2.5 -201 -172 -567 -137 3 29 8 -2 -4 0 1 3.0 -282 -230 -779 -175 21 46 11 -6 -7 0 3 3.5 -369 -285 -998 -207 48 68 13 -13 -10 0 5 4.0 -458 -335 -1211 -227 83 89 12 -21 -13 1 8 4.5 -549 -377 -1421 -240 126 110 9 -31 -16 4 11 5.0 -644 -416 -1629 -246 176 131 2 -44 -18 9 14 5.5 -742 -454 -1837 -244 234 150 -7 -58 -19 15 19 6.0 -840 -480 -2034 -237 297 166 -21 -72 -19 24 23 == Applications == The description of raindrop shape has some rather practical uses. To find a relationship between drop size and contact time for low Weber number impacts (We << 1) on superhydrophobic surfaces (which experience little deformation), a simple balance between inertia (\rho R / \tau^2) and capillarity (\sigma/R^2) can be used,Richard, Denis, Christophe Clanet, and David Quéré. alt=|thumb|300x300px|Low Force Waterfalls Low Force is an 18-foot (5.5m) high set of falls on the River Tees, England, UK. This outcome is representative of impact of small, low-velocity drops onto smooth wetting surfaces. In fluid dynamics, drop impact occurs when a drop of liquid strikes a solid or liquid surface. Hydrometeor loading is the induced drag effects on the atmosphere from a falling hydrometeor. thumb|A drop striking a liquid surface; in this case, both the drop and the surface are water. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. ""Phenomena of liquid drop impact on solid and liquid surfaces."" When falling at terminal velocity, the value of this drag is equal to grh, where g is the acceleration due to gravity and rh is the mixing ratio of the hydrometeors. If the droplet is split into multiple droplets, the contact time is reduced. thumb|Breakup of a water drop impacting a superhydrophobic surface at a Weber number of approximately 214. Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass m has to be reduced by the displaced fluid mass \rho V, with V the volume of the object. ""Surface phenomena: Contact time of a bouncing drop."" Hydrometeor loading has a net-negative effect on the atmospheric buoyancy equations. * For large We (for which the magnitude depends on the specific surface structure), many satellite drops break off during spreading and/or retraction of the drop. == On a wet solid surface == When a liquid drop strikes a wet solid surface (a surface covered with a thin layer of liquid that exceeds the height of surface roughness), either spreading or splashing will occur. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. ",8.87,0,61.0,0,1.8763,A
-How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \times 10^{-7} \mathrm{C}$ ?,"For an electron, it has a value of . Ten years later, he switched to electron to describe these elementary charges, writing in 1894: ""... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron"". For example, in one instance a Penning trap was used to contain a single electron for a period of 10 months. When there is an excess of electrons, the object is said to be negatively charged. The electron ( or ) is a subatomic particle with a negative one elementary electric charge. The word electron is a combination of the words _electr_ ic and i _on_.""electron, n.2"". The electron, on the other hand, is thought to be stable on theoretical grounds: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation. # Use the previous solution for making r, # excluding coin elif coin > r: m[c][r] = m[c - 1][r] # coin can be used. The version of this problem assumed that the people making change will use the minimum number of coins (from the denominations available). thumb|upright=1.7|Contrasting differences between discrete and continuous electron multipliers. Using the previous solution for making r (without using coin). thumb|upright=1.35|Coin of Tennes. The electron's mass is approximately 1/1836 that of the proton. The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. The upper bound of the electron radius of 10−18 meters can be derived using the uncertainty relation in energy. Within the limits of experimental accuracy, the electron charge is identical to the charge of a proton, but with the opposite sign. In turn, he divided the shells into a number of cells each of which contained one pair of electrons. Hence, about one electron for every billion electron-positron pairs survived. When there are fewer electrons than the number of protons in nuclei, the object is said to be positively charged. On the other hand, a point-like electron (zero radius) generates serious mathematical difficulties due to the self-energy of the electron tending to infinity.Eduard Shpolsky, Atomic physics (Atomnaia fizika), second edition, 1951 Observation of a single electron in a Penning trap suggests the upper limit of the particle's radius to be 10−22 meters. thumb|upright=1.5|Coin of Epander. Using the previous solution for making r - coin (without # using coin) plus this 1 extra coin. else: m[c][r] = min(m[c - 1][r], 1 + m[c][r - coin]) return m[-1][-1] ===Dynamic programming with the probabilistic convolution tree=== The probabilistic convolution tree can also be used as a more efficient dynamic programming approach. ",6.3,1.4,4.738,-167,34,A
+","The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|280px|Scheme of the colloidal probe technique for direct force measurements in the sphere-plane and sphere-sphere geometries. The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). For unequally charged objects and eventually at shorted distances, these forces may also be attractive. Simply applying this cutoff method introduces a discontinuity in the force at r_c that results in particles experiencing sudden impulses when other particles cross the boundary of their respective interaction spheres. The Bjerrum length (after Danish chemist Niels Bjerrum 1879–1958 ) is the separation at which the electrostatic interaction between two elementary charges is comparable in magnitude to the thermal energy scale, k_\text{B} T, where k_\text{B} is the Boltzmann constant and T is the absolute temperature in kelvins. Due to the screening by the electrolyte, the range of the force is given by the Debye length and its strength by the surface potential (or surface charge density). This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. The electrodipping force is a force proposed to explain the observed attraction that arises among small colloidal particles attached to an interface between immiscible liquids. It is also possible to study forces between colloidal particles by attaching another particle to the substrate and perform the measurement in the sphere-sphere geometry, see figure above. thumb|left|350px|Principle of the force measurements by the colloidal probe technique. According to Nikolaides, the electrostatic force engenders a long range capillary attraction. thumb|Computed electrostatic equipotentials (black contours) between two electrically charged spheres In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential.Weisstein, Eric W. ""Equipotential Curve."" A way to correct this problem is to shift the force to zero at r_c, thus removing the discontinuity. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. In the particular case of electrostatic forces, as the force magnitude is large at the boundary, this unphysical feature can compromise simulation accuracy. The exponential nature of these repulsive forces and the fact that its range is given by the Debye length was confirmed experimentally by direct force measurements, including surface forces apparatus, colloidal probe technique, or optical tweezers. ",0.011,4.68,"""1.1""",62.2,0.686,B
+A $10.0 \mathrm{~g}$ block with a charge of $+8.00 \times 10^{-5} \mathrm{C}$ is placed in an electric field $\vec{E}=(3000 \hat{\mathrm{i}}-600 \hat{\mathrm{j}}) \mathrm{N} / \mathrm{C}$. What is the magnitude of the electrostatic force on the block?,"The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. The force exerted by I on a nearby charge q with velocity v is : \mathbf{F}(\mathbf{r}) = q\mathbf{v} \times \mathbf{B}(\mathbf{r}), where B(r) is the magnetic field, which is determined from I by the Biot–Savart law: :\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{I d\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Alternatively, the electric potential energy of any given charge or system of charges is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration. {{block indent|em=1.2|text=The electrostatic potential energy, UE, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position rrefThe reference zero is usually taken to be a state in which the individual point charges are very well separated (""are at infinite separation"") and are at rest. to that position r.Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 {{Equation box 1 |indent=: |equation=U_\mathrm{E}(\mathbf r) = -W_{r_{\rm ref} \rightarrow r } = -\int_{{\mathbf{r}}_{\rm ref}}^\mathbf{r} q\mathbf{E}(\mathbf{r'}) \cdot \mathrm{d} \mathbf{r'}, |cellpadding |border |border colour = #50C878 |background colour = #ECFCF4}} where E is the electrostatic field and dr' is the displacement vector in a curve from the reference position rref to the final position r.}} Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. Electromotive force in electrostatic units is the statvolt (in the centimeter gram second system of units equal in amount to an erg per electrostatic unit of charge). ==Formal definitions== Inside a source of emf (such as a battery) that is open-circuited, a charge separation occurs between the negative terminal N and the positive terminal P. In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted \mathcal{E} or {\xi}) is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. (A virtual experiment based on the energy transfert between capacitor plates reveals that an additional term must be taken into account when the electrostatic energy is expressed in terms of the electric field and displacement vectors . The electrostatic field does not contribute to the net emf around a circuit because the electrostatic portion of the electric field is conservative (i.e., the work done against the field around a closed path is zero, see Kirchhoff's voltage law, which is valid, as long as the circuit elements remain at rest and radiation is ignored ). This can be accomplished with a variety of functions, but the most simple/computationally efficient approach is to simply subtract the value of the electrostatic force magnitude at the cutoff distance as such: \displaystyle F_{SF}(r) = \begin{cases} \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} - \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r_c^{2}} & \text{for } r \le r_c \\\ 0 & \text{for } r > r_c. \end{cases} As mentioned before, the shifted force (SF) method is generally suited for systems that do not have net electrostatic interactions that are long-range in nature. The magnitude of the emf for the battery (or other source) is the value of this open-circuit voltage. In the particular case of electrostatic forces, as the force magnitude is large at the boundary, this unphysical feature can compromise simulation accuracy. The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula. {{math proof |title=Outline of proof |proof= The electrostatic force F acting on a charge q can be written in terms of the electric field E as \mathbf{F} = q\mathbf{E} , By definition, the change in electrostatic potential energy, UE, of a point charge q that has moved from the reference position rref to position r in the presence of an electric field E is the negative of the work done by the electrostatic force to bring it from the reference position rref to that position r. The total electrostatic potential energy stored in a capacitor is given by U_E = \frac{1}{2}QV = \frac{1}{2} CV^2 = \frac{Q^2}{2C} where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. Years earlier, Alessandro Volta, who had measured a contact potential difference at the metal–metal (electrode–electrode) interface of his cells, held the incorrect opinion that contact alone (without taking into account a chemical reaction) was the origin of the emf. == Notation and units of measurement == Electromotive force is often denoted by \mathcal{E} or ℰ. Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. This emf is the work done on a unit charge by the source's nonelectrostatic field \boldsymbol{E}' when the charge moves from N to P. So, E and ds must be parallel: \mathbf{E} \cdot \mathrm{d} \mathbf{s} = |\mathbf{E}| \cdot |\mathrm{d}\mathbf{s}|\cos(0) = E \mathrm{d}s Using Coulomb's law, the electric field is given by |\mathbf{E}| = E = \frac{1}{4\pi\varepsilon_0}\frac{Q}{s^2} and the integral can be easily evaluated: U_E(r) = -\int_\infty^r q\mathbf{E} \cdot \mathrm{d} \mathbf{s} = -\int_\infty^r \frac{1}{4\pi\varepsilon_0}\frac{qQ}{s^2}{\rm d}s = \frac{1}{4\pi\varepsilon_0}\frac{qQ}{r} = k_e\frac{qQ}{r} }} ===One point charge q in the presence of n point charges Qi=== thumb|Electrostatic potential energy of q due to Q1 and Q2 charge system:U_E = q\frac{1}{4 \pi \varepsilon_0} \left(\frac{Q_1}{r_1} + \frac{Q_2}{r_2} \right) The electrostatic potential energy, UE, of one point charge q in the presence of n point charges Qi, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, ri is the distance between the point charges q and Qi, and q and Qi are the assigned values of the charges. ==Electrostatic potential energy stored in a system of point charges== The electrostatic potential energy UE stored in a system of N charges q1, q2, …, qN at positions r1, r2, …, rN respectively, is: |}} where, for each i value, Φ(ri) is the electrostatic potential due to all point charges except the one at ri,The factor of one half accounts for the 'double counting' of charge pairs. * Both a 1 volt emf and a 1 volt potential difference correspond to 1 joule per coulomb of charge. ",131,2.14,"""0.0547""",-191.2,0.245,E
+"Two long, charged, thin-walled, concentric cylindrical shells have radii of 3.0 and $6.0 \mathrm{~cm}$. The charge per unit length is $5.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the inner shell and $-7.0 \times 10^{-6} \mathrm{C} / \mathrm{m}$ on the outer shell. What is the magnitude $E$ of the electric field at radial distance $r=4.0 \mathrm{~cm}$?","Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Let f(\mathbf{r}^{\prime}) be the second charge density, and define \lambda(\rho, \theta) as its integral over z \lambda(\rho, \theta) = \int dz \, f(\rho, \theta, z) The electrostatic energy is given by the integral of the charge multiplied by the potential due to the cylindrical multipoles U = \int d\theta \, d\rho \, \rho \, \lambda(\rho, \theta) \Phi(\rho, \theta) If the cylindrical multipoles are exterior, this equation becomes U = \frac{-Q_1}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} \int d\theta \, d\rho \left[ C_{1k} \frac{\cos k\theta}{\rho^{k-1}} + S_{1k} \frac{\sin k\theta}{\rho^{k-1}}\right] \lambda(\rho, \theta) where Q_{1}, C_{1k} and S_{1k} are the cylindrical multipole moments of charge distribution 1. By assumption, the line charges are infinitely long and aligned with the z axis. ==Cylindrical multipole moments of a line charge== frame|right|Figure 1: Definitions for cylindrical multipoles; looking down the z' axis The electric potential of a line charge \lambda located at (\rho', \theta') is given by \Phi(\rho, \theta) = \frac{-\lambda}{2\pi\epsilon} \ln R = \frac{-\lambda}{4\pi\epsilon} \ln \left| \rho^{2} + \left( \rho' \right)^{2} - 2\rho\rho'\cos (\theta - \theta' ) \right| where R is the shortest distance between the line charge and the observation point. Here, R is the distance from the origin while r is the distance from the central axis of a cylinder as in the (r,\phi,z) cylindrical coordinate system. It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). The same equation shows that multipole moments higher than the first non-zero moment do depend on the choice of origin (in general). ==Interior axial multipole moments== Conversely, if the radius r is smaller than the smallest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{min}), the electric potential may be written \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta ) where the interior axial multipole moments I_{k} are defined I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}} Special cases include the interior axial monopole moment ( eq the total charge) M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}, the interior axial dipole moment M_{1} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{2}}, etc. Cylindrical multipole moments are the coefficients in a series expansion of a potential that varies logarithmically with the distance to a source, i.e., as \ln \ R. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as \frac{1}{R}. We can similarly describe the electric field E so that . If the radius r of the observation point P is greater than the largest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{max}), the electric potential may be written \Phi(\mathbf{r}) = \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta ) where the axial multipole moments M_{k} are defined M_{k} \equiv \int d\zeta \ \lambda(\zeta) \zeta^{k} Special cases include the axial monopole moment (=total charge) M_{0} \equiv \int d\zeta \ \lambda(\zeta), the axial dipole moment M_{1} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta, and the axial quadrupole moment M_{2} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta^{2}. This energy formula can be reduced to a remarkably simple form U = \frac{-Q_{1}}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \ln \rho \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{1k} I_{2k} + S_{1k} J_{2k} \right) where I_{2k} and J_{2k} are the interior cylindrical multipoles of the second charge density. Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. In alternative notation, the Sommerfeld identity can be more easily seen as an expansion of a spherical wave in terms of cylindrically-symmetric waves: : \frac{{e^{ik_0 r} }} {r} = i\int\limits_0^\infty {dk_\rho \frac J_0 (k_\rho \rho )e^{ik_z \left| z \right|} } Where : k_z=(k_0^2-k_\rho^2)^{1/2} The notation used here is different form that above: r is now the distance from the origin and \rho is the radial distance in a cylindrical coordinate system defined as (\rho,\phi,z). The analogous formula holds if charge density 1 is composed of interior cylindrical multipoles U = \frac{-Q_1\ln \rho'}{2\pi\epsilon} \int d\rho \, \rho \, \lambda(\rho, \theta) \+ \frac{1}{2\pi\epsilon} \sum_{k=1}^{\infty} k \left( C_{2k} I_{1k} + S_{2k} J_{1k} \right) where I_{1k} and J_{1k} are the interior cylindrical multipole moments of charge distribution 1, and C_{2k} and S_{2k} are the exterior cylindrical multipoles of the second charge density. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density \lambda(z) localized to the z-axis. frame|right|Figure 1: Point charge on the z axis; Definitions for axial multipole expansion ==Axial multipole moments of a point charge== The electric potential of a point charge q located on the z-axis at z=a (Fig. 1) equals \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon} \frac{1}{R} = \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + a^{2} - 2 a r \cos \theta}}. At short distances (\frac{r}{\zeta_\text{min}} \ll 1), the potential is well- approximated by the leading nonzero interior multipole term. ==See also== * Potential theory * Multipole expansion * Spherical multipole moments * Cylindrical multipole moments * Solid harmonics * Laplace expansion ==References== Category:Electromagnetism Category:Potential theory Category:Moment (physics) If the radius r of the observation point is greater than a, we may factor out \frac{1}{r} and expand the square root in powers of (a/r)<1 using Legendre polynomials \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon r} \sum_{k=0}^{\infty} \left( \frac{a}{r} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k} \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta ) where the axial multipole moments M_{k} \equiv q a^{k} contain everything specific to a given charge distribution; the other parts of the electric potential depend only on the coordinates of the observation point P. Special cases include the axial monopole moment M_{0}=q, the axial dipole moment M_{1}=q a and the axial quadrupole moment M_{2} \equiv q a^{2}. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. Thus, at large distances (\frac{\zeta_\text{max}}{r} \ll 1), the potential is well-approximated by the leading nonzero multipole term. Conversely, if the radius r is less than a, we may factor out \frac{1}{a} and expand in powers of (r/a)<1, once again using Legendre polynomials \Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon a} \sum_{k=0}^{\infty} \left( \frac{r}{a} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta ) where the interior axial multipole moments I_{k} \equiv \frac{q}{a^{k+1}} contain everything specific to a given charge distribution; the other parts depend only on the coordinates of the observation point P. ==General axial multipole moments== To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimal charge element \lambda(\zeta)\ d\zeta, where \lambda(\zeta) represents the charge density at position z=\zeta on the z-axis. thumb|An annular solar eclipse has a magnitude of less than 1.0 The magnitude of eclipse is the fraction of the angular diameter of a celestial body being eclipsed. The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry ",57.2,-1.00,"""-111.92""",0.9974,2.3,E
+A particle of charge $1.8 \mu \mathrm{C}$ is at the center of a Gaussian cube $55 \mathrm{~cm}$ on edge. What is the net electric flux through the surface?,"It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). It is defined as the closed surface in three dimensional space by which the flux of vector field be calculated. == Common Gaussian surfaces == Most calculations using Gaussian surfaces begin by implementing Gauss's law (for electricity):Introduction to electrodynamics (4th Edition), D. J. Griffiths, 2012, :}{\varepsilon_0}.}} For a closed Gaussian surface, electric flux is given by: where * is the electric field, * is any closed surface, * is the total electric charge inside the surface , * is the electric constant (a universal constant, also called the ""permittivity of free space"") () This relation is known as Gauss' law for electric fields in its integral form and it is one of Maxwell's equations. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form. thumb|A tiny Gauss's box whose sides are perpendicular to a conductor's surface is used to find the local surface charge once the electric potential and the electric field are calculated by solving Laplace's equation. Thereby is the electrical charge enclosed by the Gaussian surface. Electric flux through its surface is zero. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. Gauss's law may be expressed as: \Phi_E = \frac{Q}{\varepsilon_0} where is the electric flux through a closed surface enclosing any volume , is the total charge enclosed within , and is the electric constant. A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field.Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, It is an arbitrary closed surface (the boundary of a 3-dimensional region ) used in conjunction with Gauss's law for the corresponding field (Gauss's law, Gauss's law for magnetism, or Gauss's law for gravity) by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution. The flux passing consists of the three contributions: : For surfaces a and b, and will be perpendicular. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. In electromagnetism, electric flux is the measure of the electric field through a given surface,Purcell, pp. 22–26 although an electric field in itself cannot flow. The electric flux over a surface is therefore given by the surface integral: \Phi_E = \iint_S \mathbf{E} \cdot \textrm{d}\mathbf{S} where is the electric field and is a differential area on the closed surface with an outward facing surface normal defining its direction. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. While the electric flux is not affected by charges that are not within the closed surface, the net electric field, can be affected by charges that lie outside the closed surface. *MISN-0-133 Gauss's Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET. If the electric field is uniform, the electric flux passing through a surface of vector area is \Phi_E = \mathbf{E} \cdot \mathbf{S} = ES \cos \theta, where is the electric field (having units of ), is its magnitude, is the area of the surface, and is the angle between the electric field lines and the normal (perpendicular) to . The electric field is perpendicular, locally, to the equipotential surface of the conductor, and zero inside; its flux πa2·E, by Gauss's law equals πa2·σ/ε0. Under these circumstances, Gauss's law modifies to \Phi_E = \frac{Q_\mathrm{free}}{\varepsilon} for the integral form, and abla \cdot \mathbf{E} = \frac{\rho_\mathrm{free}}{\varepsilon} for the differential form. ==Interpretations== ===In terms of fields of force=== Gauss's theorem can be interpreted in terms of the lines of force of the field as follows: The flux through a closed surface is dependent upon both the magnitude and direction of the electric field lines penetrating the surface. The electric flux is defined as a surface integral of the electric field: : where is the electric field, is a vector representing an infinitesimal element of area of the surface, and represents the dot product of two vectors. In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. ",2.0,35,"""5.7""",0.2553,+17.7,A
+The drum of a photocopying machine has a length of $42 \mathrm{~cm}$ and a diameter of $12 \mathrm{~cm}$. The electric field just above the drum's surface is $2.3 \times 10^5 \mathrm{~N} / \mathrm{C}$. What is the total charge on the drum? ,"The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into ""free"" and ""bound"" charges. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Unlike the case of the metal, the image charge q' is not exactly opposite to the real charge: q'=\frac{\varepsilon_1 - \varepsilon_2}{\varepsilon_1 + \varepsilon_2}q. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. Electrostatic charge on an object can be measured by placing it into the Faraday Cup. If we assume for simplicity (without loss of generality) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the polar angle θ and is given by: : \sigma(\theta) = \varepsilon_0 \left.\frac{\partial V}{\partial r} \right|_{r=R} =\frac{-q\left(R^2-p^2\right)}{4\pi R\left(R^2+p^2-2pR\cos\theta\right)^{3/2}} The total charge on the sphere may be found by integrating over all angles: : Q_t=\int_0^\pi d\theta \int_0^{2\pi} d\phi\,\,\sigma(\theta) R^2\sin\theta = -q Note that the reciprocal problem is also solved by this method. It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. A surface charge is an electric charge present on a two-dimensional surface. Just as in the first case, the image charge will have charge −qR/p and will be located at vector position \left(R^2 / p^2\right) \mathbf{p}. Free charges are the excess charges which can move into electrostatic equilibrium, i.e. when the charges are not moving and the resultant electric field is independent of time, or constitute electric currents. === Total charge densities === In terms of volume charge densities, the total charge density is: \rho = \rho_\text{f} + \rho_\text{b}\,. as for surface charge densities: \sigma = \sigma_\text{f} + \sigma_\text{b}\,. where subscripts ""f"" and ""b"" denote ""free"" and ""bound"" respectively. === Bound charge === The bound surface charge is the charge piled up at the surface of the dielectric, given by the dipole moment perpendicular to the surface: q_b = \frac{\mathbf{d} \cdot\mathbf{\hat{n}}}{|\mathbf{s}|} where s is the separation between the point charges constituting the dipole, \mathbf{d} is the electric dipole moment, \mathbf{\hat{n}} is the unit normal vector to the surface. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. ",32,0.32,"""1000.0""",24,1.2,B
+A spherical water drop $1.20 \mu \mathrm{m}$ in diameter is suspended in calm air due to a downward-directed atmospheric electric field of magnitude $E=462 \mathrm{~N} / \mathrm{C}$. What is the magnitude of the gravitational force on the drop?,"If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac{\rho d}{\mu} V. thumb|upright=0.7|The downward force of gravity (Fg) equals the restraining force of drag (Fd) plus the buoyancy. Fluid Dynamics Research 12.2 (1993): 61-93 In the case of floating, a drop will float on the surface for several seconds. A new model for the equilibrium shape of raindrops. The resulting outcome depends on the properties of the drop, the surface, and the surrounding fluid, which is most commonly a gas. == On a dry solid surface == When a liquid drop strikes a dry solid surface, it generally spreads on the surface, and then will retract if the impact is energetic enough to cause the drop to spread out more than it would generally spread due to its static receding contact angle. The Beard and Chuang model is a well known and leading theoretical force balance model used to derive the rotational cross-sections of raindrops in their equilibrium state by employing Chebyshev polynomials in series. thumb|Beard and Chuang model of raindrop The radius-vector of the raindrop's surface r(\theta) in vertical angular direction \theta is equal to r(\theta)=a [ 1 + \sum c_n cos(n \theta) ] , where shape coefficients c_n \cdot 10^4 are defined for the raindrops with different equivolumetric diameter as in following table d(mm) n = 0 1 2 3 4 5 6 7 8 9 10 2.0 -131 -120 -376 -96 -4 15 5 0 -2 0 1 2.5 -201 -172 -567 -137 3 29 8 -2 -4 0 1 3.0 -282 -230 -779 -175 21 46 11 -6 -7 0 3 3.5 -369 -285 -998 -207 48 68 13 -13 -10 0 5 4.0 -458 -335 -1211 -227 83 89 12 -21 -13 1 8 4.5 -549 -377 -1421 -240 126 110 9 -31 -16 4 11 5.0 -644 -416 -1629 -246 176 131 2 -44 -18 9 14 5.5 -742 -454 -1837 -244 234 150 -7 -58 -19 15 19 6.0 -840 -480 -2034 -237 297 166 -21 -72 -19 24 23 == Applications == The description of raindrop shape has some rather practical uses. To find a relationship between drop size and contact time for low Weber number impacts (We << 1) on superhydrophobic surfaces (which experience little deformation), a simple balance between inertia (\rho R / \tau^2) and capillarity (\sigma/R^2) can be used,Richard, Denis, Christophe Clanet, and David Quéré. alt=|thumb|300x300px|Low Force Waterfalls Low Force is an 18-foot (5.5m) high set of falls on the River Tees, England, UK. This outcome is representative of impact of small, low-velocity drops onto smooth wetting surfaces. In fluid dynamics, drop impact occurs when a drop of liquid strikes a solid or liquid surface. Hydrometeor loading is the induced drag effects on the atmosphere from a falling hydrometeor. thumb|A drop striking a liquid surface; in this case, both the drop and the surface are water. After reaching the local terminal velocity, while continuing the fall, speed decreases to change with the local terminal speed. ===Derivation for terminal velocity=== Using mathematical terms, defining down to be positive, the net force acting on an object falling near the surface of Earth is (according to the drag equation): F_\text{net} = m a = m g - \frac{1}{2} \rho v^2 A C_d, with v(t) the velocity of the object as a function of time t. ""Phenomena of liquid drop impact on solid and liquid surfaces."" When falling at terminal velocity, the value of this drag is equal to grh, where g is the acceleration due to gravity and rh is the mixing ratio of the hydrometeors. If the droplet is split into multiple droplets, the contact time is reduced. thumb|Breakup of a water drop impacting a superhydrophobic surface at a Weber number of approximately 214. Buoyancy effects, due to the upward force on the object by the surrounding fluid, can be taken into account using Archimedes' principle: the mass m has to be reduced by the displaced fluid mass \rho V, with V the volume of the object. ""Surface phenomena: Contact time of a bouncing drop."" Hydrometeor loading has a net-negative effect on the atmospheric buoyancy equations. * For large We (for which the magnitude depends on the specific surface structure), many satellite drops break off during spreading and/or retraction of the drop. == On a wet solid surface == When a liquid drop strikes a wet solid surface (a surface covered with a thin layer of liquid that exceeds the height of surface roughness), either spreading or splashing will occur. The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. ",8.87,0,"""61.0""",0,1.8763,A
+How many electrons would have to be removed from a coin to leave it with a charge of $+1.0 \times 10^{-7} \mathrm{C}$ ?,"For an electron, it has a value of . Ten years later, he switched to electron to describe these elementary charges, writing in 1894: ""... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron"". For example, in one instance a Penning trap was used to contain a single electron for a period of 10 months. When there is an excess of electrons, the object is said to be negatively charged. The electron ( or ) is a subatomic particle with a negative one elementary electric charge. The word electron is a combination of the words _electr_ ic and i _on_.""electron, n.2"". The electron, on the other hand, is thought to be stable on theoretical grounds: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation. # Use the previous solution for making r, # excluding coin elif coin > r: m[c][r] = m[c - 1][r] # coin can be used. The version of this problem assumed that the people making change will use the minimum number of coins (from the denominations available). thumb|upright=1.7|Contrasting differences between discrete and continuous electron multipliers. Using the previous solution for making r (without using coin). thumb|upright=1.35|Coin of Tennes. The electron's mass is approximately 1/1836 that of the proton. The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. The upper bound of the electron radius of 10−18 meters can be derived using the uncertainty relation in energy. Within the limits of experimental accuracy, the electron charge is identical to the charge of a proton, but with the opposite sign. In turn, he divided the shells into a number of cells each of which contained one pair of electrons. Hence, about one electron for every billion electron-positron pairs survived. When there are fewer electrons than the number of protons in nuclei, the object is said to be positively charged. On the other hand, a point-like electron (zero radius) generates serious mathematical difficulties due to the self-energy of the electron tending to infinity.Eduard Shpolsky, Atomic physics (Atomnaia fizika), second edition, 1951 Observation of a single electron in a Penning trap suggests the upper limit of the particle's radius to be 10−22 meters. thumb|upright=1.5|Coin of Epander. Using the previous solution for making r - coin (without # using coin) plus this 1 extra coin. else: m[c][r] = min(m[c - 1][r], 1 + m[c][r - coin]) return m[-1][-1] ===Dynamic programming with the probabilistic convolution tree=== The probabilistic convolution tree can also be used as a more efficient dynamic programming approach. ",6.3,1.4,"""4.738""",-167,34,A
 "An unknown charge sits on a conducting solid sphere of radius $10 \mathrm{~cm}$. If the electric field $15 \mathrm{~cm}$ from the center of the sphere has the magnitude $3.0 \times 10^3 \mathrm{~N} / \mathrm{C}$ and is directed radially inward, what is the net charge on the sphere?
-","It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. thumb|300px|right|visualized induced-charge electrokinetic flow pattern around a carbon-steel sphere (diameter=1.2mm). We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. The 15th parallel north is a circle of latitude that is 15 degrees north of the Earth's equatorial plane. The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. Thereby is the electrical charge enclosed by the Gaussian surface. This may also be written as V = \frac{2\pi r^3}{3} (1-\cos\varphi)\,, where is half the cone angle, i.e., is the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center. 400px|thumb|A spherical sector (blue) thumb|A spherical sector In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. It is the three-dimensional analogue of the sector of a circle. ==Volume== If the radius of the sphere is denoted by and the height of the cap by , the volume of the spherical sector is V = \frac{2\pi r^2 h}{3}\,. The volume of the sector is related to the area of the cap by: V = \frac{rA}{3}\,. ==Area== The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2\pi rh\,. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. ",1.6,2,-7.5,48,0.323,C
-Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite $1.3 \mathrm{~m}$ in diameter accumulates $2.4 \mu \mathrm{C}$ of charge in one orbital revolution. Find the resulting surface charge density. ,"A satellite shielded by 3 mm of aluminium in an elliptic orbit () passing the radiation belts will receive about 2,500 rem (25 Sv) per year. Radiation belt electrons are also constantly removed by collisions with Earth's atmosphere, losses to the magnetopause, and their outward radial diffusion. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. Charge carrier densities involve equations concerning the electrical conductivity and related phenomena like the thermal conductivity. ==Calculation== The carrier density is usually obtained theoretically by integrating the density of states over the energy range of charge carriers in the material (e.g. integrating over the conduction band for electrons, integrating over the valence band for holes). This charge is equal to d q = \rho \, v\, dt \, dA, where is the charge density at . To show this mathematically, charge carrier density is a particle density, so integrating it over a volume V gives the number of charge carriers N in that volume N=\int_V n(\mathbf r) \,dV. where n(\mathbf r) is the position-dependent charge carrier density. As always, the integral of the charge density over a region of space is the charge contained in that region. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The outer electron radiation belt is mostly produced by inward radial diffusion and local acceleration due to transfer of energy from whistler-mode plasma waves to radiation belt electrons. If the total number of charge carriers is known, the carrier density can be found by simply dividing by the volume. At position at time , the distribution of charge flowing is described by the current density: \mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) where * is the current density vector; * is the particles' average drift velocity (SI unit: m∙s−1); *\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) is the charge density (SI unit: coulombs per cubic metre), in which ** is the number of particles per unit volume (""number density"") (SI unit: m−3); ** is the charge of the individual particles with density (SI unit: coulombs). The outer Van Allen belt consists mainly of electrons. Spacecraft charging is what happens when charged particles from the surrounding energetic environment stop on either the exterior of a spacecraft or the interior, such as in conductors. ==References== Category:Spaceflight The charge density of the system at a point r is a sum of the charge densities for each charge qi at position ri, where : \rho_q(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R: Q = \int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by \rho_q(\mathbf{r}) = q n(\mathbf{r})\,. Material Number of valence electrons Carrier density (1/cm3) at 300K Copper 1 Silver 1 Gold 1 Beryllium 2 Magnesium 2 Calcium 2 Strontium 2 Barium 2 Niobium 1 Iron 2 Manganese 2 Zinc 2 Cadmium 2 Aluminum 3 Gallium 3 Indium 3 Thallium 3 Tin 4 Lead 4 Bismuth 5 Antimony 5 The values for n among metals inferred for example by the Hall effect are often on the same orders of magnitude, but this simple model cannot predict carrier density to very high accuracy. ==Measurement== The density of charge carriers can be determined in many cases using the Hall effect, the voltage of which depends inversely on the carrier density. ==References== Category:Density Category:Charge carriers thumb|320px|A cross section of Van Allen radiation belts A Van Allen radiation belt is a zone of energetic charged particles, most of which originate from the solar wind, that are captured by and held around a planet by that planet's magnetosphere. As of 2014, it remains uncertain if there are any negative unintended consequences to removing these radiation belts. ==See also== * Dipole model of the Earth's magnetic field * L-shell * List of artificial radiation belts * List of plasma (physics) articles * Space weather == Explanatory notes == == Citations == ==Additional sources== * * * Part I: Radial transport, pp. 1679–1693, ; Part II: Local acceleration and loss, pp. 1694–1713, . == External links == * An explanation of the belts by David P. Stern and Mauricio Peredo * Background: Trapped particle radiation models—Introduction to the trapped radiation belts by SPENVIS * SPENVIS—Space Environment, Effects, and Education System—Gateway to the SPENVIS orbital dose calculation software *The Van Allen Probes Web Site Johns Hopkins University Applied Physics Laboratory Category:1958 in science Category:Articles containing video clips Category:Geomagnetism Category:Space physics Category:Space plasmas The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. == Discrete charges == For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function: \rho_q(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_0) where r is the position to calculate the charge. Charge carrier density, also known as carrier concentration, denotes the number of charge carriers in per volume. Miniaturization and digitization of electronics and logic circuits have made satellites more vulnerable to radiation, as the total electric charge in these circuits is now small enough so as to be comparable with the charge of incoming ions. ",+11,3930,4.5,5.4,2,C
- A charge of $20 \mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc?,"The radius of such a curve is 5729.57795. Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is , where is degree and is radius. The distance from the vertex to the center of curvature is the radius of curvature of the surface. Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose without kinking it, damaging it, or shortening its life. thumb|A concave mirror with light rays thumb|400px|Center of curvature In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. Degree of curve or degree of curvature is a measure of curvature of a circular arc used in civil engineering for its easy use in layout surveying. ==Definition== The degree of curvature is defined as the central angle to the ends of an agreed length of either an arc or a chord; various lengths are commonly used in different areas of practice. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. * If the vertex lies to the right of the center of curvature, the radius of curvature is negative. The is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00, etc. Metric work may use similar notation, such as kilometers plus meters 1+000. ==Formulas for radius of curvature== alt=Degree of Curvature Formula Explanation|thumb|Diagram showing different parts of the curve used in the formula Degree of curvature can be converted to radius of curvature by the following formulae: ===Formula from arc length=== r = \frac{180^\circ A}{\pi D_\text{C}} where A is arc length, r is radius of curvature, and D_\text{C} is degree of curvature, arc definition Substitute deflection angle for degree of curvature or make arc length equal to 100 feet. ===Formula from chord length=== r = \frac{C}{2 \sin \left( \frac{D_\text{C}}{2} \right) } where C is chord length, r is radius of curvature and D_\text{C} is degree of curvature, chord definition ===Formula from radius=== D_\text{C} = 5729.58/r === Example === As an example, a curve with an arc length of 600 units that has an overall sweep of 6 degrees is a 1-degree curve: For every 100 feet of arc, the bearing changes by 1 degree. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve.* The osculating circle to the curve is centered at the centre of curvature. By using degrees of curvature, curve setting can be easily done with the help of a transit or theodolite and a chain, tape, or rope of a prescribed length. === Length selection === The usual distance used to compute degree of curvature in North American road work is of arc. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. For metal tubing, bend radius is to the centerline of tubing, not the exterior. ==References== Category:Cables Category:Fiber optics Category:Plumbing vi:Bán kính cong The sign convention for the optical radius of curvature is as follows: * If the vertex lies to the left of the center of curvature, the radius of curvature is positive. Suppose this arc includes point E within it. The minimum bending radius will vary with different cable designs. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. In an n-degree curve, the forward bearing changes by n degrees over the standard length of arc or chord. ==Usage== Curvature is usually measured in radius of curvature. thumb|upright=1.3|Radius of curvature sign convention for optical design Radius of curvature (ROC) has specific meaning and sign convention in optical design. The smaller the bend radius, the greater the material flexibility (as the radius of curvature decreases, the curvature increases). If the chord definition is used, each 100-unit chord length will sweep 1 degree with a radius of 5729.651 units, and the chord of the whole curve will be slightly shorter than 600 units. == See also == * Geometric design of roads * Highway engineering * Lateral motion device * Minimum railway curve radius * Radius of curvature (applications) * Railway systems engineering * Track geometry * Track transition curve * Transition curve * Turning radius == References == == External links == * * http://www.tpub.com/content/engineering/14071/css/14071_242.htm * * * * * * * Category:Surveying Category:Transportation engineering Category:Track geometry ",38,-214,0.333333,2.19,4.16,A
-Calculate the number of coulombs of positive charge in 250 $\mathrm{cm}^3$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.),"One coulomb is the charge of approximately , where the number is the reciprocal of This is also 160.2176634 zC of charge. In this case, the effective nuclear charge can be calculated by Coulomb's law. For example, a neutral chlorine atom has 17 protons and 17 electrons, whereas a Cl− anion has 17 protons and 18 electrons for a total charge of −1. The internationally accepted value of a proton's charge radius is . In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). It is positive (repulsive) to a radial distance of about 0.6 fm, negative (attractive) at greater distances, and very weak beyond about 2 fm. === Charge radius in solvated proton, hydronium === The radius of the hydrated proton appears in the Born equation for calculating the hydration enthalpy of hydronium. == Interaction of free protons with ordinary matter == Although protons have affinity for oppositely charged electrons, this is a relatively low-energy interaction and so free protons must lose sufficient velocity (and kinetic energy) in order to become closely associated and bound to electrons. It is impossible to realize exactly 1 C of charge, since the number of elementary charges is not an integer. In this case, one says that the ""elementary charge"" is three times as large as the ""quantum of charge"". Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge. In January 2013, an updated value for the charge radius of a proton——was published. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. Protons are composed of two up quarks of charge +e and one down quark of charge −e. The effective nuclear charge on such an electron is given by the following equation: Z_\mathrm{eff} = Z - S where *Z is the number of protons in the nucleus (atomic number), and *S is the shielding constant. A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 e (elementary charge). By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. It is possible to determine the strength of the nuclear charge by the oxidation number of the atom. In atomic physics, the effective nuclear charge is the actual amount of positive (nuclear) charge experienced by an electron in a multi-electron atom. The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). Their measurement of the root-mean-square charge radius of a proton is "", which differs by 5.0 standard deviations from the CODATA value of "". The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. The chemical properties of each atom are determined by the number of (negatively charged) electrons, which for neutral atoms is equal to the number of (positive) protons so that the total charge is zero. ",−1.642876,1.3,5040.0,38,+7.3,B
-A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \times 10^{-9} \mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \times 10^{-6} \mathrm{~N}$ when placed in this field. What is the magnitude of the electric field? ,"Near the surface of the Earth, the magnitude of the field is on average around 100 V/m. Because the atmospheric electric field is negatively directed in fair weather, the convention is to refer to the potential gradient, which has the opposite sign and is about 100 V/m at the surface, away from thunderstorms. From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge. This Carnegie curve variation has been described as ""the fundamental electrical heartbeat of the planet"".Liz Kalaugher, Atmospheric electricity affects cloud height 3 March 2013, physicsworld.com accessed 15 April 2021 Even away from thunderstorms, atmospheric electricity can be highly variable, but, generally, the electric field is enhanced in fogs and dust whereas the atmospheric electrical conductivity is diminished. === Links with biology === The atmospheric potential gradient leads to an ion flow from the positively charged atmosphere to the negatively charged earth surface. The electric field is a vector field... and has a magnitude and direction."" This aura is the electric field due to the charge. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Atmospheric electricity is an interdisciplinary topic with a long history, involving concepts from electrostatics, atmospheric physics, meteorology and Earth science. Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). Thunderstorms act as a giant battery in the atmosphere, charging up the electrosphere to about 400,000 volts with respect to the surface. The movement of charge between the Earth's surface, the atmosphere, and the ionosphere is known as the global atmospheric electrical circuit. thumb|upright=1.25|Convective cloud's thickness, between its base and top, shown on the background scale at different stages of its life The cloud height, more commonly known as cloud thickness or depth, is the distance between the cloud base and the cloud top. The Coulomb force on a charge of magnitude q at any point in space is equal to the product of the charge and the electric field at that point \mathbf{F} = q\mathbf{E} The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1. ===Superposition principle=== Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. Over a flat field on a day with clear skies, the atmospheric potential gradient is approximately 120 V/m. The derived SI unit for the electric field is the volt per meter (V/m), which is equal to the newton per coulomb (N/C)., p. 23 ==Description== The electric field is defined at each point in space as the force per unit charge that would be experienced by a vanishingly small positive test charge if held stationary at that point. This is the electric field at point \mathbf{x}_0 due to the point charge q_1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf{x}_0. There is a weak conduction current of atmospheric ions moving in the atmospheric electric field, about 2 picoamperes per square meter, and the air is weakly conductive due to the presence of these atmospheric ions. ===Variations=== Global daily cycles in the atmospheric electric field, with a minimum around 03 UT and peaking roughly 16 hours later, were researched by the Carnegie Institution of Washington in the 20th century. Discoveries about the electrification of the atmosphere via sensitive electrical instruments and ideas on how the Earth's negative charge is maintained were developed mainly in the 20th century, with CTR Wilson playing an important part.Encyclopedia of Geomagnetism and Paleomagnetism - Page 359 Current research on atmospheric electricity focuses mainly on lightning, particularly high-energy particles and transient luminous events, and the role of non-thunderstorm electrical processes in weather and climate. ==Description== Atmospheric electricity is always present, and during fine weather away from thunderstorms, the air above the surface of Earth is positively charged, while the Earth's surface charge is negative. As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. To make it easy to calculate the Coulomb force on any charge at position \mathbf{x}_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the source charge) \mathbf{E}_{1} (\mathbf{x}_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1 }{4\pi\varepsilon_0} \frac{\hat \mathbf{r}_{01}}{r_{01}^2} Where *\mathbf{E}_{1} (\mathbf{x}_0) is the component of the electric field at q_0 due to q_1 . An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them.Browne, p 225: ""... around every charge there is an aura that fills all space. ",6.2,2.74,2.8108,1.1,1.5,E
+","It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). Note: Here N is used as a continuous variable that represents the infinitely divisible charge, Q, distributed across the spherical shell. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. thumb|300px|right|visualized induced-charge electrokinetic flow pattern around a carbon-steel sphere (diameter=1.2mm). We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. Related problems include the study of the geometry of the minimum energy configuration and the study of the large behavior of the minimum energy. == Mathematical statement == The electrostatic interaction energy occurring between each pair of electrons of equal charges (e_i = e_j = e, with e the elementary charge of an electron) is given by Coulomb's law, : U_{ij}(N)={e_i e_j \over 4\pi\epsilon_0 r_{ij}}, where, \epsilon_0 is the electric constant and r_{ij}=|\mathbf{r}_i - \mathbf{r}_j| is the distance between each pair of electrons located at points on the sphere defined by vectors \mathbf{r}_i and \mathbf{r}_j, respectively. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. The 15th parallel north is a circle of latitude that is 15 degrees north of the Earth's equatorial plane. The great circle is often considered to define an equator about the sphere and the two points perpendicular to the plane are often considered poles to aid in discussions about the electrostatic configurations of many-N electron solutions. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. Thereby is the electrical charge enclosed by the Gaussian surface. This may also be written as V = \frac{2\pi r^3}{3} (1-\cos\varphi)\,, where is half the cone angle, i.e., is the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center. 400px|thumb|A spherical sector (blue) thumb|A spherical sector In geometry, a spherical sector, also known as a spherical cone, is a portion of a sphere or of a ball defined by a conical boundary with apex at the center of the sphere. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. It is the three-dimensional analogue of the sector of a circle. ==Volume== If the radius of the sphere is denoted by and the height of the cap by , the volume of the spherical sector is V = \frac{2\pi r^2 h}{3}\,. The volume of the sector is related to the area of the cap by: V = \frac{rA}{3}\,. ==Area== The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2\pi rh\,. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. * For N = 5, a mathematically rigorous computer-aided solution was reported in 2010 with electrons residing at vertices of a triangular dipyramid. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. ",1.6,2,"""-7.5""",48,0.323,C
+Space vehicles traveling through Earth's radiation belts can intercept a significant number of electrons. The resulting charge buildup can damage electronic components and disrupt operations. Suppose a spherical metal satellite $1.3 \mathrm{~m}$ in diameter accumulates $2.4 \mu \mathrm{C}$ of charge in one orbital revolution. Find the resulting surface charge density. ,"A satellite shielded by 3 mm of aluminium in an elliptic orbit () passing the radiation belts will receive about 2,500 rem (25 Sv) per year. Radiation belt electrons are also constantly removed by collisions with Earth's atmosphere, losses to the magnetopause, and their outward radial diffusion. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. Charge carrier densities involve equations concerning the electrical conductivity and related phenomena like the thermal conductivity. ==Calculation== The carrier density is usually obtained theoretically by integrating the density of states over the energy range of charge carriers in the material (e.g. integrating over the conduction band for electrons, integrating over the valence band for holes). This charge is equal to d q = \rho \, v\, dt \, dA, where is the charge density at . To show this mathematically, charge carrier density is a particle density, so integrating it over a volume V gives the number of charge carriers N in that volume N=\int_V n(\mathbf r) \,dV. where n(\mathbf r) is the position-dependent charge carrier density. As always, the integral of the charge density over a region of space is the charge contained in that region. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The outer electron radiation belt is mostly produced by inward radial diffusion and local acceleration due to transfer of energy from whistler-mode plasma waves to radiation belt electrons. If the total number of charge carriers is known, the carrier density can be found by simply dividing by the volume. At position at time , the distribution of charge flowing is described by the current density: \mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) where * is the current density vector; * is the particles' average drift velocity (SI unit: m∙s−1); *\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) is the charge density (SI unit: coulombs per cubic metre), in which ** is the number of particles per unit volume (""number density"") (SI unit: m−3); ** is the charge of the individual particles with density (SI unit: coulombs). The outer Van Allen belt consists mainly of electrons. Spacecraft charging is what happens when charged particles from the surrounding energetic environment stop on either the exterior of a spacecraft or the interior, such as in conductors. ==References== Category:Spaceflight The charge density of the system at a point r is a sum of the charge densities for each charge qi at position ri, where : \rho_q(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R: Q = \int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by \rho_q(\mathbf{r}) = q n(\mathbf{r})\,. Material Number of valence electrons Carrier density (1/cm3) at 300K Copper 1 Silver 1 Gold 1 Beryllium 2 Magnesium 2 Calcium 2 Strontium 2 Barium 2 Niobium 1 Iron 2 Manganese 2 Zinc 2 Cadmium 2 Aluminum 3 Gallium 3 Indium 3 Thallium 3 Tin 4 Lead 4 Bismuth 5 Antimony 5 The values for n among metals inferred for example by the Hall effect are often on the same orders of magnitude, but this simple model cannot predict carrier density to very high accuracy. ==Measurement== The density of charge carriers can be determined in many cases using the Hall effect, the voltage of which depends inversely on the carrier density. ==References== Category:Density Category:Charge carriers thumb|320px|A cross section of Van Allen radiation belts A Van Allen radiation belt is a zone of energetic charged particles, most of which originate from the solar wind, that are captured by and held around a planet by that planet's magnetosphere. As of 2014, it remains uncertain if there are any negative unintended consequences to removing these radiation belts. ==See also== * Dipole model of the Earth's magnetic field * L-shell * List of artificial radiation belts * List of plasma (physics) articles * Space weather == Explanatory notes == == Citations == ==Additional sources== * * * Part I: Radial transport, pp. 1679–1693, ; Part II: Local acceleration and loss, pp. 1694–1713, . == External links == * An explanation of the belts by David P. Stern and Mauricio Peredo * Background: Trapped particle radiation models—Introduction to the trapped radiation belts by SPENVIS * SPENVIS—Space Environment, Effects, and Education System—Gateway to the SPENVIS orbital dose calculation software *The Van Allen Probes Web Site Johns Hopkins University Applied Physics Laboratory Category:1958 in science Category:Articles containing video clips Category:Geomagnetism Category:Space physics Category:Space plasmas The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. == Discrete charges == For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function: \rho_q(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_0) where r is the position to calculate the charge. Charge carrier density, also known as carrier concentration, denotes the number of charge carriers in per volume. Miniaturization and digitization of electronics and logic circuits have made satellites more vulnerable to radiation, as the total electric charge in these circuits is now small enough so as to be comparable with the charge of incoming ions. ",+11,3930,"""4.5""",5.4,2,C
+ A charge of $20 \mathrm{nC}$ is uniformly distributed along a straight rod of length $4.0 \mathrm{~m}$ that is bent into a circular arc with a radius of $2.0 \mathrm{~m}$. What is the magnitude of the electric field at the center of curvature of the arc?,"The radius of such a curve is 5729.57795. Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is , where is degree and is radius. The distance from the vertex to the center of curvature is the radius of curvature of the surface. Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose without kinking it, damaging it, or shortening its life. thumb|A concave mirror with light rays thumb|400px|Center of curvature In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. Degree of curve or degree of curvature is a measure of curvature of a circular arc used in civil engineering for its easy use in layout surveying. ==Definition== The degree of curvature is defined as the central angle to the ends of an agreed length of either an arc or a chord; various lengths are commonly used in different areas of practice. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. * If the vertex lies to the right of the center of curvature, the radius of curvature is negative. The is called a station, used to define length along a road or other alignment, annotated as stations plus feet 1+00, 2+00, etc. Metric work may use similar notation, such as kilometers plus meters 1+000. ==Formulas for radius of curvature== alt=Degree of Curvature Formula Explanation|thumb|Diagram showing different parts of the curve used in the formula Degree of curvature can be converted to radius of curvature by the following formulae: ===Formula from arc length=== r = \frac{180^\circ A}{\pi D_\text{C}} where A is arc length, r is radius of curvature, and D_\text{C} is degree of curvature, arc definition Substitute deflection angle for degree of curvature or make arc length equal to 100 feet. ===Formula from chord length=== r = \frac{C}{2 \sin \left( \frac{D_\text{C}}{2} \right) } where C is chord length, r is radius of curvature and D_\text{C} is degree of curvature, chord definition ===Formula from radius=== D_\text{C} = 5729.58/r === Example === As an example, a curve with an arc length of 600 units that has an overall sweep of 6 degrees is a 1-degree curve: For every 100 feet of arc, the bearing changes by 1 degree. Cauchy defined the center of curvature C as the intersection point of two infinitely close normal lines to the curve.* The osculating circle to the curve is centered at the centre of curvature. By using degrees of curvature, curve setting can be easily done with the help of a transit or theodolite and a chain, tape, or rope of a prescribed length. === Length selection === The usual distance used to compute degree of curvature in North American road work is of arc. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. For metal tubing, bend radius is to the centerline of tubing, not the exterior. ==References== Category:Cables Category:Fiber optics Category:Plumbing vi:Bán kính cong The sign convention for the optical radius of curvature is as follows: * If the vertex lies to the left of the center of curvature, the radius of curvature is positive. Suppose this arc includes point E within it. The minimum bending radius will vary with different cable designs. The locus of centers of curvature for each point on the curve comprise the evolute of the curve. In an n-degree curve, the forward bearing changes by n degrees over the standard length of arc or chord. ==Usage== Curvature is usually measured in radius of curvature. thumb|upright=1.3|Radius of curvature sign convention for optical design Radius of curvature (ROC) has specific meaning and sign convention in optical design. The smaller the bend radius, the greater the material flexibility (as the radius of curvature decreases, the curvature increases). If the chord definition is used, each 100-unit chord length will sweep 1 degree with a radius of 5729.651 units, and the chord of the whole curve will be slightly shorter than 600 units. == See also == * Geometric design of roads * Highway engineering * Lateral motion device * Minimum railway curve radius * Radius of curvature (applications) * Railway systems engineering * Track geometry * Track transition curve * Transition curve * Turning radius == References == == External links == * * http://www.tpub.com/content/engineering/14071/css/14071_242.htm * * * * * * * Category:Surveying Category:Transportation engineering Category:Track geometry ",38,-214,"""0.333333""",2.19,4.16,A
+Calculate the number of coulombs of positive charge in 250 $\mathrm{cm}^3$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.),"One coulomb is the charge of approximately , where the number is the reciprocal of This is also 160.2176634 zC of charge. In this case, the effective nuclear charge can be calculated by Coulomb's law. For example, a neutral chlorine atom has 17 protons and 17 electrons, whereas a Cl− anion has 17 protons and 18 electrons for a total charge of −1. The internationally accepted value of a proton's charge radius is . In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). It is positive (repulsive) to a radial distance of about 0.6 fm, negative (attractive) at greater distances, and very weak beyond about 2 fm. === Charge radius in solvated proton, hydronium === The radius of the hydrated proton appears in the Born equation for calculating the hydration enthalpy of hydronium. == Interaction of free protons with ordinary matter == Although protons have affinity for oppositely charged electrons, this is a relatively low-energy interaction and so free protons must lose sufficient velocity (and kinetic energy) in order to become closely associated and bound to electrons. It is impossible to realize exactly 1 C of charge, since the number of elementary charges is not an integer. In this case, one says that the ""elementary charge"" is three times as large as the ""quantum of charge"". Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). The 4s electrons in iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge. In January 2013, an updated value for the charge radius of a proton——was published. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. Protons are composed of two up quarks of charge +e and one down quark of charge −e. The effective nuclear charge on such an electron is given by the following equation: Z_\mathrm{eff} = Z - S where *Z is the number of protons in the nucleus (atomic number), and *S is the shielding constant. A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 e (elementary charge). By measuring the charges of many different oil drops, it can be seen that the charges are all integer multiples of a single small charge, namely e. It is possible to determine the strength of the nuclear charge by the oxidation number of the atom. In atomic physics, the effective nuclear charge is the actual amount of positive (nuclear) charge experienced by an electron in a multi-electron atom. The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). Their measurement of the root-mean-square charge radius of a proton is "", which differs by 5.0 standard deviations from the CODATA value of "". The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. The chemical properties of each atom are determined by the number of (negatively charged) electrons, which for neutral atoms is equal to the number of (positive) protons so that the total charge is zero. ",−1.642876,1.3,"""5040.0""",38,+7.3,B
+A charged cloud system produces an electric field in the air near Earth's surface. A particle of charge $-2.0 \times 10^{-9} \mathrm{C}$ is acted on by a downward electrostatic force of $3.0 \times 10^{-6} \mathrm{~N}$ when placed in this field. What is the magnitude of the electric field? ,"Near the surface of the Earth, the magnitude of the field is on average around 100 V/m. Because the atmospheric electric field is negatively directed in fair weather, the convention is to refer to the potential gradient, which has the opposite sign and is about 100 V/m at the surface, away from thunderstorms. From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge. This Carnegie curve variation has been described as ""the fundamental electrical heartbeat of the planet"".Liz Kalaugher, Atmospheric electricity affects cloud height 3 March 2013, physicsworld.com accessed 15 April 2021 Even away from thunderstorms, atmospheric electricity can be highly variable, but, generally, the electric field is enhanced in fogs and dust whereas the atmospheric electrical conductivity is diminished. === Links with biology === The atmospheric potential gradient leads to an ion flow from the positively charged atmosphere to the negatively charged earth surface. The electric field is a vector field... and has a magnitude and direction."" This aura is the electric field due to the charge. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Atmospheric electricity is an interdisciplinary topic with a long history, involving concepts from electrostatics, atmospheric physics, meteorology and Earth science. Atmospheric electricity is the study of electrical charges in the Earth's atmosphere (or that of another planet). Thunderstorms act as a giant battery in the atmosphere, charging up the electrosphere to about 400,000 volts with respect to the surface. The movement of charge between the Earth's surface, the atmosphere, and the ionosphere is known as the global atmospheric electrical circuit. thumb|upright=1.25|Convective cloud's thickness, between its base and top, shown on the background scale at different stages of its life The cloud height, more commonly known as cloud thickness or depth, is the distance between the cloud base and the cloud top. The Coulomb force on a charge of magnitude q at any point in space is equal to the product of the charge and the electric field at that point \mathbf{F} = q\mathbf{E} The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1. ===Superposition principle=== Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. Over a flat field on a day with clear skies, the atmospheric potential gradient is approximately 120 V/m. The derived SI unit for the electric field is the volt per meter (V/m), which is equal to the newton per coulomb (N/C)., p. 23 ==Description== The electric field is defined at each point in space as the force per unit charge that would be experienced by a vanishingly small positive test charge if held stationary at that point. This is the electric field at point \mathbf{x}_0 due to the point charge q_1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf{x}_0. There is a weak conduction current of atmospheric ions moving in the atmospheric electric field, about 2 picoamperes per square meter, and the air is weakly conductive due to the presence of these atmospheric ions. ===Variations=== Global daily cycles in the atmospheric electric field, with a minimum around 03 UT and peaking roughly 16 hours later, were researched by the Carnegie Institution of Washington in the 20th century. Discoveries about the electrification of the atmosphere via sensitive electrical instruments and ideas on how the Earth's negative charge is maintained were developed mainly in the 20th century, with CTR Wilson playing an important part.Encyclopedia of Geomagnetism and Paleomagnetism - Page 359 Current research on atmospheric electricity focuses mainly on lightning, particularly high-energy particles and transient luminous events, and the role of non-thunderstorm electrical processes in weather and climate. ==Description== Atmospheric electricity is always present, and during fine weather away from thunderstorms, the air above the surface of Earth is positively charged, while the Earth's surface charge is negative. As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. To make it easy to calculate the Coulomb force on any charge at position \mathbf{x}_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the source charge) \mathbf{E}_{1} (\mathbf{x}_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1 }{4\pi\varepsilon_0} \frac{\hat \mathbf{r}_{01}}{r_{01}^2} Where *\mathbf{E}_{1} (\mathbf{x}_0) is the component of the electric field at q_0 due to q_1 . An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them.Browne, p 225: ""... around every charge there is an aura that fills all space. ",6.2,2.74,"""2.8108""",1.1,1.5,E
 " An electric dipole with dipole moment
 $$
 \vec{p}=(3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}})\left(1.24 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}\right)
 $$
-is in an electric field $\vec{E}=(4000 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{i}}$. What is the potential energy of the electric dipole? (b) What is the torque acting on it?","An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{p} \cdot \mathbf{E},\qquad\ \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, The scalar dot """" product and the negative sign shows the potential energy minimises when the dipole is parallel with field and is maximum when antiparallel while zero when perpendicular. This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. The E-field vector and the dipole vector define a plane, and the torque is directed normal to that plane with the direction given by the right-hand rule. The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole: \left|\mathbf{\tau}\right| = \left|q\mathbf{r}\right| \left|\mathbf{E}\right|\sin\theta. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some angle with it. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. In the case of two classical point charges, +q and -q, with a displacement vector, \mathbf{r}, pointing from the negative charge to the positive charge, the electric dipole moment is given by \mathbf{p} = q\mathbf{r}. The torque tends to align the dipole with the field. The potential energy of a magnet or magnetic moment \mathbf{m} in a magnetic field \mathbf{B} is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to: E_\text{p,m} = -\mathbf{m} \cdot \mathbf{B} while the energy stored in an inductor (of inductance L) when a current I flows through it is given by: E_\text{p,m} = \frac{1}{2} LI^2. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). In the presence of an electric field, such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net torque on the dipole. The potential at a position r is: \phi (\mathbf{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\rho \left(\mathbf{r}_0\right)}{\left|\mathbf{r} - \mathbf{r}_0\right|} d^3 \mathbf{r}_0 \ + \frac {1}{4 \pi \varepsilon_0}\int \frac{\mathbf{p} \left(\mathbf{r}_0\right) \cdot \left(\mathbf{r} - \mathbf{r}_0\right)} {| \mathbf{r} - \mathbf{r}_0 |^3 } d^3 \mathbf{ r}_0 , where ρ(r) is the unpaired charge density, and p(r) is the dipole moment density.For example, for a system of ideal dipoles with dipole moment p confined within some closed surface, the dipole density p(r) is equal to p inside the surface, but is zero outside. It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. (See electron electric dipole moment). The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The electric dipole moment vector also points from the negative charge to the positive charge. ",-3.8,2500,0.0526315789,-1.49,-50,D
-What is the total charge in coulombs of $75.0 \mathrm{~kg}$ of electrons?,"One coulomb is the charge of approximately , where the number is the reciprocal of This is also 160.2176634 zC of charge. For an electron, it has a value of . (In other words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.) Electrons have an electric charge of coulombs,The original source for CODATA is :Individual physical constants from the CODATA are available at: which is used as a standard unit of charge for subatomic particles, and is also called the elementary charge. Ten years later, he switched to electron to describe these elementary charges, writing in 1894: ""... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron"". In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). The formal charge of any atom in a molecule can be calculated by the following equation: q^{*} = V - L - \frac{B}{2} where is the number of valence electrons of the neutral atom in isolation (in its ground state); is the number of non-bonding valence electrons assigned to this atom in the Lewis structure of the molecule; and is the total number of electrons shared in bonds with other atoms in the molecule. By carefully analyzing the noise of a current, the charge of an electron can be calculated. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. * Subtract the number of electrons in the circle from the number of valence electrons of the neutral atom in isolation (in its ground state) to determine the formal charge. :center|350px * The formal charges computed for the remaining atoms in this Lewis structure of carbon dioxide are shown below. :center|450px It is important to keep in mind that formal charges are just that – formal, in the sense that this system is a formalism. Somewhat confusingly, in atomic physics, e sometimes denotes the electron charge, i.e. the negative of the elementary charge. Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. In atomic physics, a partial charge (or net atomic charge) is a non-integer charge value when measured in elementary charge units. Later, the name electron was assigned to the particle and the unit of charge e lost its name. There are different ways to draw the Lewis structure **Carbon single bonded to both oxygen atoms (carbon = +2, oxygens = −1 each, total formal charge = 0) **Carbon single bonded to one oxygen and double bonded to another (carbon = +1, oxygendouble = 0, oxygensingle = −1, total formal charge = 0) **Carbon double bonded to both oxygen atoms (carbon = 0, oxygens = 0, total formal charge = 0) Even though all three structures gave us a total charge of zero, the final structure is the superior one because there are no charges in the molecule at all. === Pictorial method === The following is equivalent: *Draw a circle around the atom for which the formal charge is requested (as with carbon dioxide, below) :center|150px * Count up the number of electrons in the atom's ""circle."" It is also impossible to realize charge at the yoctocoulomb scale. ==SI prefixes== Like other SI units, the coulomb can be modified by adding a prefix that multiplies it by a power of 10. ==Conversions== *The magnitude of the electrical charge of one mole of elementary charges (approximately , the Avogadro number) is known as a faraday unit of charge (closely related to the Faraday constant). Therefore, the ""quantum of charge"" is e. However, the unit of energy electronvolt (eV) is a remnant of the fact that the elementary charge was once called electron. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). The electron (symbol e) is on the left. In simple terms, formal charge is the difference between the number of valence electrons of an atom in a neutral free state and the number assigned to that atom in a Lewis structure. ",-0.38,0.15,-1.32,-0.10,+80,C
-A uniformly charged conducting sphere of $1.2 \mathrm{~m}$ diameter has surface charge density $8.1 \mu \mathrm{C} / \mathrm{m}^2$. Find the net charge on the sphere.,"It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. A surface charge is an electric charge present on a two-dimensional surface. In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric: : \begin{matrix} \phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\\ \phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\\ \vdots \\\ \phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n \end{matrix}. where is the surface charge on conductor . thumb|Viviani's curve: intersection of a sphere with a tangent cylinder. thumb|upright=0.75|The light blue part of the half sphere can be squared. In this case, the surface charge density decreases upon approach. Within this double layer, the first layer corresponds to the charged surface. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor: :\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, or : \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. In this case, one must solve the PB equation together with an appropriate model of the surface charging process. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The ability of the surface to regulate its charge can be quantified by the regulation parameter : p = \frac{C_{\rm D}}{C_{\rm I}+C_{\rm D}} where CD = ε0 ε κ is the diffuse layer capacitance and CI the inner (or regulation) capacitance. Thereby is the electrical charge enclosed by the Gaussian surface. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. Instead, the entirety of the charge of the conductor resides on the surface, and can be expressed by the equation: \sigma = E\varepsilon_0 where E is the electric field caused by the charge on the conductor and \varepsilon_0 is the permittivity of the free space. In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. ",76,3.52,4.979,9,37,E
-The magnitude of the electrostatic force between two identical ions that are separated by a distance of $5.0 \times 10^{-10} \mathrm{~m}$ is $3.7 \times 10^{-9}$ N.  What is the charge of each ion? ,"The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. At large impact parameters, the charge of the ion is shielded by the tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. In passing through a field of ions with density n, an electron will have many such encounters simultaneously, with various impact parameters (distance to the ion) and directions. The CODATA recommended value for an electron is ==Origin== When charged particles move in electric and magnetic fields the following two laws apply: *Lorentz force law: \mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), *Newton's second law of motion:\mathbf{F}=m\mathbf{a} = m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} where F is the force applied to the ion, m is the mass of the particle, a is the acceleration, Q is the electric charge, E is the electric field, and v × B is the cross product of the ion's velocity and the magnetic flux density. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). Thus, the m/z of an ion alone neither infers mass nor the number of charges. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. thumb|540x540px|Oppositely charged particles interact as they are moved through a column. A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. List of orders of magnitude for electric charge Factor [Coulomb] SI prefix8th edition of the official brochure of the BIPM (SI units and prefixes). Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. ",0.33333333,35.2,3.2,0.123,0.88,C
-How many megacoulombs of positive charge are in $1.00 \mathrm{~mol}$ of neutral molecular-hydrogen gas $\left(\mathrm{H}_2\right)$ ?,"The ionization energy of the hydrogen molecule is 15.603 eV. The charge transfer gives 0.011 electron charge units to each helium atom. Because the two 1s electrons screen the protons to give an effective atomic number for the 2s electron close to 1, we can treat this 2s valence electron with a hydrogenic model. In this case, the effective nuclear charge can be calculated by Coulomb's law. Figure 2 - A molecular orbital diagram for open and closed hydrogen bridged cations with carbon is shown above. A hydrogen molecular ion cluster or hydrogen cluster ion is a positively charged cluster of hydrogen molecules. A free energy change of dissociation of −360 kJ/mol is equivalent to a pKa of −63 at 298 K. ===Other helium-hydrogen ions=== Additional helium atoms can attach to HeH+ to form larger clusters such as He2H+, He3H+, He4H+, He5H+ and He6H+. The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . The helium hydride ion or hydridohelium(1+) ion or helonium is a cation (positively charged ion) with chemical formula HeH+. He22+ is the smallest possible molecule with a double positive charge. The calculated dipole moment of HeH+ is 2.26 or 2.84 D. The length of the covalent bond in the ion is 0.772 Å. ===Isotopologues=== The helium hydride ion has six relatively stable isotopologues, that differ in the isotopes of the two elements, and hence in the total atomic mass number (A) and the total number of neutrons (N) in the two nuclei: * or (A = 4, N = 1) * or (A = 5, N = 2) * or (A = 6, N = 3; radioactive) * or (A = 5, N = 2) * or (A = 6, N = 3) * or (A = 7, N = 4; radioactive) They all have three protons and two electrons. The amount of the dimer formed in the gas beam is of the order of one percent. ==Molecular ions== He2+ is a related ion bonded by a half covalent bond. It consists of two hydrogen nuclei (protons) sharing a single electron. Negative hydrogen clusters have not been found to exist. is theoretically unstable, but in theory is bound at 0.003 eV. ==Decay== in the free gas state decays by giving off H atoms and molecules. However hydrogen also forms singly charged clusters () with n up to 120. ==Experiments== Hydrogen ion clusters can be formed in liquid helium or with lesser cluster size in pure hydrogen. is far more common than higher even numbered clusters. is stable in solid hydrogen. High speed electrons also cause ionization of hydrogen molecules with a peak cross section around 50 eV. The ion can be formed from the ionization of a neutral hydrogen molecule . The He2 molecule has a large separation distance between the atoms of about 5200 pm (= 52 ångström). Due to the increased number of bonds that the hydrogen atom is part of, the higher electron density around hydrogen shields the 1H nucleus, causing its chemical shift to appear at negative ppm. The molecular helium anion is also found in liquid helium that has been excited by electrons with an energy level higher than 22 eV. Hydrogen-bridged cations are a type of charged species in which a hydrogen atom is simultaneously bonded to two atoms through partial sigma bonds. ",0.19,0.68,0.69,0.132,15.1,A
-A charge (uniform linear density $=9.0 \mathrm{nC} / \mathrm{m}$ ) lies on a string that is stretched along an $x$ axis from $x=0$ to $x=3.0 \mathrm{~m}$. Determine the magnitude of the electric field at $x=4.0 \mathrm{~m}$ on the $x$ axis.,"We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The electric field is a vector field... and has a magnitude and direction."" Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Assuming infinite planes, the magnitude of the electric field E is: E = - \frac{\Delta V}{d} where ΔV is the potential difference between the plates and d is the distance separating the plates. To make it easy to calculate the Coulomb force on any charge at position \mathbf{x}_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the source charge) \mathbf{E}_{1} (\mathbf{x}_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1 }{4\pi\varepsilon_0} \frac{\hat \mathbf{r}_{01}}{r_{01}^2} Where *\mathbf{E}_{1} (\mathbf{x}_0) is the component of the electric field at q_0 due to q_1 . From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. Analysis of the physics of string bending suggests that the resultant pitch of a string bent at its midpoint is given by u = \frac{1}{2L} \sqrt{\frac{T + \cos\theta (T - EA)}{\mu_{o}}} where L is the length of the vibrating element, T is the tension of the string prior to bending, \theta is the bend angle, E is the Young's Modulus of the string material, A is the string cross sectional area and \mu_{o} is the linear density of the string material. This is the electric field at point \mathbf{x}_0 due to the point charge q_1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf{x}_0. The Coulomb force on a charge of magnitude q at any point in space is equal to the product of the charge and the electric field at that point \mathbf{F} = q\mathbf{E} The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1. ===Superposition principle=== Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. The electric field of such a uniformly moving point charge is hence given by: \mathbf{E} = \frac q {4 \pi \epsilon_0 r^3} \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \mathbf{r} where q is the charge of the point source, \mathbf{r} is the position vector from the point source to the point in space, \beta is the ratio of observed speed of the charge particle to the speed of light and \theta is the angle between \mathbf{r} and the observed velocity of the charged particle. 300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1). By considering the charge \rho(\mathbf{x}')dV in each small volume of space dV at point \mathbf{x}' as a point charge, the resulting electric field, d\mathbf{E}(\mathbf{x}), at point \mathbf{x} can be calculated as d\mathbf{E}(\mathbf{x}) = \frac{\rho(\mathbf{x}')}{4\pi\varepsilon_0}\frac{\hat \mathbf{r}'}{{r'}^2} dV where *\hat \mathbf{r}' is the unit vector pointing from \mathbf{x}' to \mathbf{x}. *r' is the distance from \mathbf{x}' to \mathbf{x}. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. The force required to bend a string at its midpoint to a given angle \theta is given by F_{B} = 2\left(T + EA\left(\frac{1 - \cos\theta}{\cos\theta} \right) \right)\sin\theta . thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. In that case, Coulomb's law fully describes the field.Purcell, pp. 5-7. ===Parallels between electrostatic and gravitational fields=== Coulomb's law, which describes the interaction of electric charges: \mathbf{F} = q \left(\frac{Q}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = q \mathbf{E} is similar to Newton's law of universal gravitation: \mathbf{F} = m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = m\mathbf{g} (where \mathbf{\hat{r}} = \mathbf{\frac{r}{|r|}}). Alternatively the electric field of uniformly moving point charges can be derived from the Lorentz transformation of four-force experienced by test charges in the source's rest frame given by Coulomb's law and assigning electric field and magnetic field by their definition given by the form of Lorentz force. The above equation, although consistent with that of uniformly moving point charges as well as its non-relativistic limit, are not corrected for quantum- mechanical effects. == Some common electric field values == * Infinite wire having uniform charge density \lambda has electric field at a distance x from it as \frac{\lambda}{2\pi\epsilon_0x} \hat{x} * Infinitely large surface having charge density \sigma has electric field at a distance x from it as \frac{\sigma}{2\epsilon_0} \hat{x} * Infinitely long cylinder having Uniform charge density \lambda that is charge contained along unit length of the cylinder has electric field at a distance x from it as \frac{\lambda}{2\pi\epsilon_0x} \hat{x} while it is 0 everywhere inside the cylinder * Uniformly charged non-conducting sphere of radius R, volume charge density \rho and total charge Q has electric field at a distance x from it as \frac{Q}{4\pi\epsilon_0x^2} \hat{x} while the electric field at a point \vec{r} inside sphere from its center is given by \frac{Q}{4\pi\epsilon_0R^3}\vec{r} * Uniformly charged conducting sphere of radius R, surface charge density \sigma and total charge Q has electric field at a distance x from it as \frac{Q}{4\pi\epsilon_0x^2} \hat{x} while the electric field inside is 0 * Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density \sigma at that point is \frac{\sigma}{\epsilon_0} \hat{x} * Uniformly charged ring having total charge Q has electric field at a distance x along its axis as \frac{Qx}{4\pi\epsilon_0(R^2+x^2)^{3/2}} \hat{x}' * Uniformly charged disc of radius R and charge density \sigma has electric field at a distance x along its axis from it as \frac{\sigma}{2\epsilon_0} \left[1-\left(\frac{R^2}{x^2}-1\right)^{-1/2}\right] \hat{x} * Electric field due to dipole of dipole moment \vec{p} at a distance x from their center along equatorial plane is given as -\frac{\vec{p}}{4\pi\epsilon_0x^3} and the same along the axial line is approximated to \frac{\vec{p}}{2\pi\epsilon_0x^3} for x much bigger than the distance between dipoles. 275px|thumb|Stick figure of 1.75 meters standing next to a violin string of .33 meters and a long string instrument string of 10 meters. ",13,5840,61.0,+10,131,C
+is in an electric field $\vec{E}=(4000 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{i}}$. What is the potential energy of the electric dipole? (b) What is the torque acting on it?","An object with an electric dipole moment p is subject to a torque τ when placed in an external electric field E. Here, the electric dipole moment p is, as above: \mathbf{p} = q\mathbf{d}\, . The electric field of the dipole is the negative gradient of the potential, leading to: \mathbf E\left(\mathbf R\right) = \frac{3\left(\mathbf{p} \cdot \hat{\mathbf{R}}\right) \hat{\mathbf{R}} - \mathbf{p}}{4 \pi \varepsilon_0 R^3}\, . For a spatially uniform electric field across the small region occupied by the dipole, the energy U and the torque \boldsymbol{\tau} are given by U = - \mathbf{p} \cdot \mathbf{E},\qquad\ \boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}, The scalar dot """" product and the negative sign shows the potential energy minimises when the dipole is parallel with field and is maximum when antiparallel while zero when perpendicular. This quantity is used in the definition of polarization density. ==Energy and torque== thumb|187x187px|Electric dipole p and its torque τ in a uniform E field. The E-field vector and the dipole vector define a plane, and the torque is directed normal to that plane with the direction given by the right-hand rule. The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_{\rm e} \cdot \mathbf E. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The magnitude of the torque is proportional to both the magnitude of the charges and the separation between them, and varies with the relative angles of the field and the dipole: \left|\mathbf{\tau}\right| = \left|q\mathbf{r}\right| \left|\mathbf{E}\right|\sin\theta. A dipole aligned parallel to an electric field has lower potential energy than a dipole making some angle with it. The dipole is represented by a vector from the negative charge towards the positive charge. ==Elementary definition== thumb|Quantities defining the electric dipole moment of two point charges. For this case, the electric dipole moment has a magnitude p = qd and is directed from the negative charge to the positive one. In the case of two classical point charges, +q and -q, with a displacement vector, \mathbf{r}, pointing from the negative charge to the positive charge, the electric dipole moment is given by \mathbf{p} = q\mathbf{r}. The torque tends to align the dipole with the field. The potential energy of a magnet or magnetic moment \mathbf{m} in a magnetic field \mathbf{B} is defined as the mechanical work of the magnetic force (actually magnetic torque) on the re-alignment of the vector of the magnetic dipole moment and is equal to: E_\text{p,m} = -\mathbf{m} \cdot \mathbf{B} while the energy stored in an inductor (of inductance L) when a current I flows through it is given by: E_\text{p,m} = \frac{1}{2} LI^2. To the accuracy of this dipole approximation, as shown in the previous section, the dipole moment density p(r) (which includes not only p but the location of p) serves as P(r). In the presence of an electric field, such as that due to an electromagnetic wave, the two charges will experience a force in opposite directions, leading to a net torque on the dipole. The potential at a position r is: \phi (\mathbf{r}) = \frac{1}{4 \pi \varepsilon_0} \int \frac{\rho \left(\mathbf{r}_0\right)}{\left|\mathbf{r} - \mathbf{r}_0\right|} d^3 \mathbf{r}_0 \ + \frac {1}{4 \pi \varepsilon_0}\int \frac{\mathbf{p} \left(\mathbf{r}_0\right) \cdot \left(\mathbf{r} - \mathbf{r}_0\right)} {| \mathbf{r} - \mathbf{r}_0 |^3 } d^3 \mathbf{ r}_0 , where ρ(r) is the unpaired charge density, and p(r) is the dipole moment density.For example, for a system of ideal dipoles with dipole moment p confined within some closed surface, the dipole density p(r) is equal to p inside the surface, but is zero outside. It can be shown that this net force is generally parallel to the dipole moment. ==Expression (general case)== More generally, for a continuous distribution of charge confined to a volume V, the corresponding expression for the dipole moment is: \mathbf{p}(\mathbf{r}) = \int_{V} \rho(\mathbf{r}') \left(\mathbf{r}' - \mathbf{r}\right) d^3 \mathbf{r}', where r locates the point of observation and d3r′ denotes an elementary volume in V. (See electron electric dipole moment). The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The electric dipole moment vector also points from the negative charge to the positive charge. ",-3.8,2500,"""0.0526315789""",-1.49,-50,D
+What is the total charge in coulombs of $75.0 \mathrm{~kg}$ of electrons?,"One coulomb is the charge of approximately , where the number is the reciprocal of This is also 160.2176634 zC of charge. For an electron, it has a value of . (In other words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.) Electrons have an electric charge of coulombs,The original source for CODATA is :Individual physical constants from the CODATA are available at: which is used as a standard unit of charge for subatomic particles, and is also called the elementary charge. Ten years later, he switched to electron to describe these elementary charges, writing in 1894: ""... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron"". In the SI system of units, the value of the elementary charge is exactly defined as e = coulombs, or 160.2176634 zeptocoulombs (zC). The formal charge of any atom in a molecule can be calculated by the following equation: q^{*} = V - L - \frac{B}{2} where is the number of valence electrons of the neutral atom in isolation (in its ground state); is the number of non-bonding valence electrons assigned to this atom in the Lewis structure of the molecule; and is the total number of electrons shared in bonds with other atoms in the molecule. By carefully analyzing the noise of a current, the charge of an electron can be calculated. The elementary charge, usually denoted by , is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 .The symbol e has many other meanings. * Subtract the number of electrons in the circle from the number of valence electrons of the neutral atom in isolation (in its ground state) to determine the formal charge. :center|350px * The formal charges computed for the remaining atoms in this Lewis structure of carbon dioxide are shown below. :center|450px It is important to keep in mind that formal charges are just that – formal, in the sense that this system is a formalism. Somewhat confusingly, in atomic physics, e sometimes denotes the electron charge, i.e. the negative of the elementary charge. Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. In atomic physics, a partial charge (or net atomic charge) is a non-integer charge value when measured in elementary charge units. Later, the name electron was assigned to the particle and the unit of charge e lost its name. There are different ways to draw the Lewis structure **Carbon single bonded to both oxygen atoms (carbon = +2, oxygens = −1 each, total formal charge = 0) **Carbon single bonded to one oxygen and double bonded to another (carbon = +1, oxygendouble = 0, oxygensingle = −1, total formal charge = 0) **Carbon double bonded to both oxygen atoms (carbon = 0, oxygens = 0, total formal charge = 0) Even though all three structures gave us a total charge of zero, the final structure is the superior one because there are no charges in the molecule at all. === Pictorial method === The following is equivalent: *Draw a circle around the atom for which the formal charge is requested (as with carbon dioxide, below) :center|150px * Count up the number of electrons in the atom's ""circle."" It is also impossible to realize charge at the yoctocoulomb scale. ==SI prefixes== Like other SI units, the coulomb can be modified by adding a prefix that multiplies it by a power of 10. ==Conversions== *The magnitude of the electrical charge of one mole of elementary charges (approximately , the Avogadro number) is known as a faraday unit of charge (closely related to the Faraday constant). Therefore, the ""quantum of charge"" is e. However, the unit of energy electronvolt (eV) is a remnant of the fact that the elementary charge was once called electron. Thus, an object's charge can be exactly 0 e, or exactly 1 e, −1 e, 2 e, etc., but not e, or −3.8 e, etc. The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). The electron (symbol e) is on the left. In simple terms, formal charge is the difference between the number of valence electrons of an atom in a neutral free state and the number assigned to that atom in a Lewis structure. ",-0.38,0.15,"""-1.32""",-0.10,+80,C
+A uniformly charged conducting sphere of $1.2 \mathrm{~m}$ diameter has surface charge density $8.1 \mu \mathrm{C} / \mathrm{m}^2$. Find the net charge on the sphere.,"It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. This non-trivial result shows that any spherical distribution of charge acts as a point charge when observed from the outside of the charge distribution; this is in fact a verification of Coulomb's law. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. A surface charge is an electric charge present on a two-dimensional surface. In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric: : \begin{matrix} \phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\\ \phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\\ \vdots \\\ \phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n \end{matrix}. where is the surface charge on conductor . thumb|Viviani's curve: intersection of a sphere with a tangent cylinder. thumb|upright=0.75|The light blue part of the half sphere can be squared. In this case, the surface charge density decreases upon approach. Within this double layer, the first layer corresponds to the charged surface. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor: :\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, or : \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. In this case, one must solve the PB equation together with an appropriate model of the surface charging process. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. The ability of the surface to regulate its charge can be quantified by the regulation parameter : p = \frac{C_{\rm D}}{C_{\rm I}+C_{\rm D}} where CD = ε0 ε κ is the diffuse layer capacitance and CI the inner (or regulation) capacitance. Thereby is the electrical charge enclosed by the Gaussian surface. Overall, two layers of charge and a potential drop from the electrode to the edge of the outer layer (outer Helmholtz Plane) are observed. Instead, the entirety of the charge of the conductor resides on the surface, and can be expressed by the equation: \sigma = E\varepsilon_0 where E is the electric field caused by the charge on the conductor and \varepsilon_0 is the permittivity of the free space. In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. ",76,3.52,"""4.979""",9,37,E
+The magnitude of the electrostatic force between two identical ions that are separated by a distance of $5.0 \times 10^{-10} \mathrm{~m}$ is $3.7 \times 10^{-9}$ N.  What is the charge of each ion? ,"The ratio of electrostatic to gravitational forces between two particles will be proportional to the product of their charge-to-mass ratios. For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. thumb|right|350px|Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m2 suspended in a monovalent electrolyte solutions of different molar concentrations as indicated. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . The range of these forces is typically below 1 nm. ==Like-charge attraction controversy== Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances. At large impact parameters, the charge of the ion is shielded by the tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). This approximation has been used to estimate the force between two charged colloidal particles as shown in the first figure of this article. This model treats the electrostatic and hard-core interactions between all individual ions explicitly. In passing through a field of ions with density n, an electron will have many such encounters simultaneously, with various impact parameters (distance to the ion) and directions. The CODATA recommended value for an electron is ==Origin== When charged particles move in electric and magnetic fields the following two laws apply: *Lorentz force law: \mathbf{F} = Q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), *Newton's second law of motion:\mathbf{F}=m\mathbf{a} = m \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} where F is the force applied to the ion, m is the mass of the particle, a is the acceleration, Q is the electric charge, E is the electric field, and v × B is the cross product of the ion's velocity and the magnetic flux density. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). Thus, the m/z of an ion alone neither infers mass nor the number of charges. thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. thumb|540x540px|Oppositely charged particles interact as they are moved through a column. A Coulomb collision is a binary elastic collision between two charged particles interacting through their own electric field. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. Such forces between atoms are much weaker than the attractive electrical forces that hold the atoms themselves together (i.e., that bind electrons to the nucleus), and their range between atoms is shorter, because they arise from small separation of charges inside the neutral atom. List of orders of magnitude for electric charge Factor [Coulomb] SI prefix8th edition of the official brochure of the BIPM (SI units and prefixes). Three-body forces: The interactions between weakly charged objects are pair- wise additive due to the linear nature of the DH approximation. ",0.33333333,35.2,"""3.2""",0.123,0.88,C
+How many megacoulombs of positive charge are in $1.00 \mathrm{~mol}$ of neutral molecular-hydrogen gas $\left(\mathrm{H}_2\right)$ ?,"The ionization energy of the hydrogen molecule is 15.603 eV. The charge transfer gives 0.011 electron charge units to each helium atom. Because the two 1s electrons screen the protons to give an effective atomic number for the 2s electron close to 1, we can treat this 2s valence electron with a hydrogenic model. In this case, the effective nuclear charge can be calculated by Coulomb's law. Figure 2 - A molecular orbital diagram for open and closed hydrogen bridged cations with carbon is shown above. A hydrogen molecular ion cluster or hydrogen cluster ion is a positively charged cluster of hydrogen molecules. A free energy change of dissociation of −360 kJ/mol is equivalent to a pKa of −63 at 298 K. ===Other helium-hydrogen ions=== Additional helium atoms can attach to HeH+ to form larger clusters such as He2H+, He3H+, He4H+, He5H+ and He6H+. The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . The helium hydride ion or hydridohelium(1+) ion or helonium is a cation (positively charged ion) with chemical formula HeH+. He22+ is the smallest possible molecule with a double positive charge. The calculated dipole moment of HeH+ is 2.26 or 2.84 D. The length of the covalent bond in the ion is 0.772 Å. ===Isotopologues=== The helium hydride ion has six relatively stable isotopologues, that differ in the isotopes of the two elements, and hence in the total atomic mass number (A) and the total number of neutrons (N) in the two nuclei: * or (A = 4, N = 1) * or (A = 5, N = 2) * or (A = 6, N = 3; radioactive) * or (A = 5, N = 2) * or (A = 6, N = 3) * or (A = 7, N = 4; radioactive) They all have three protons and two electrons. The amount of the dimer formed in the gas beam is of the order of one percent. ==Molecular ions== He2+ is a related ion bonded by a half covalent bond. It consists of two hydrogen nuclei (protons) sharing a single electron. Negative hydrogen clusters have not been found to exist. is theoretically unstable, but in theory is bound at 0.003 eV. ==Decay== in the free gas state decays by giving off H atoms and molecules. However hydrogen also forms singly charged clusters () with n up to 120. ==Experiments== Hydrogen ion clusters can be formed in liquid helium or with lesser cluster size in pure hydrogen. is far more common than higher even numbered clusters. is stable in solid hydrogen. High speed electrons also cause ionization of hydrogen molecules with a peak cross section around 50 eV. The ion can be formed from the ionization of a neutral hydrogen molecule . The He2 molecule has a large separation distance between the atoms of about 5200 pm (= 52 ångström). Due to the increased number of bonds that the hydrogen atom is part of, the higher electron density around hydrogen shields the 1H nucleus, causing its chemical shift to appear at negative ppm. The molecular helium anion is also found in liquid helium that has been excited by electrons with an energy level higher than 22 eV. Hydrogen-bridged cations are a type of charged species in which a hydrogen atom is simultaneously bonded to two atoms through partial sigma bonds. ",0.19,0.68,"""0.69""",0.132,15.1,A
+A charge (uniform linear density $=9.0 \mathrm{nC} / \mathrm{m}$ ) lies on a string that is stretched along an $x$ axis from $x=0$ to $x=3.0 \mathrm{~m}$. Determine the magnitude of the electric field at $x=4.0 \mathrm{~m}$ on the $x$ axis.,"We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The electric field is a vector field... and has a magnitude and direction."" Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. Assuming infinite planes, the magnitude of the electric field E is: E = - \frac{\Delta V}{d} where ΔV is the potential difference between the plates and d is the distance separating the plates. To make it easy to calculate the Coulomb force on any charge at position \mathbf{x}_0 this expression can be divided by q_0 leaving an expression that only depends on the other charge (the source charge) \mathbf{E}_{1} (\mathbf{x}_0) = \frac{ \mathbf{F}_{01} } {q_0} = \frac{q_1 }{4\pi\varepsilon_0} \frac{\hat \mathbf{r}_{01}}{r_{01}^2} Where *\mathbf{E}_{1} (\mathbf{x}_0) is the component of the electric field at q_0 due to q_1 . From the above formula it can be seen that the electric field due to a point charge is everywhere directed away from the charge if it is positive, and toward the charge if it is negative, and its magnitude decreases with the inverse square of the distance from the charge. The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. Analysis of the physics of string bending suggests that the resultant pitch of a string bent at its midpoint is given by u = \frac{1}{2L} \sqrt{\frac{T + \cos\theta (T - EA)}{\mu_{o}}} where L is the length of the vibrating element, T is the tension of the string prior to bending, \theta is the bend angle, E is the Young's Modulus of the string material, A is the string cross sectional area and \mu_{o} is the linear density of the string material. This is the electric field at point \mathbf{x}_0 due to the point charge q_1; it is a vector-valued function equal to the Coulomb force per unit charge that a positive point charge would experience at the position \mathbf{x}_0. The Coulomb force on a charge of magnitude q at any point in space is equal to the product of the charge and the electric field at that point \mathbf{F} = q\mathbf{E} The SI unit of the electric field is the newton per coulomb (N/C), or volt per meter (V/m); in terms of the SI base units it is kg⋅m⋅s−3⋅A−1. ===Superposition principle=== Due to the linearity of Maxwell's equations, electric fields satisfy the superposition principle, which states that the total electric field, at a point, due to a collection of charges is equal to the vector sum of the electric fields at that point due to the individual charges. The electric field of such a uniformly moving point charge is hence given by: \mathbf{E} = \frac q {4 \pi \epsilon_0 r^3} \frac {1- \beta^2} {(1- \beta^2 \sin^2 \theta)^{3/2}} \mathbf{r} where q is the charge of the point source, \mathbf{r} is the position vector from the point source to the point in space, \beta is the ratio of observed speed of the charge particle to the speed of light and \theta is the angle between \mathbf{r} and the observed velocity of the charged particle. 300px|thumb|right|A string half the length (1/2), four times the tension (4), or one-quarter the mass per length (1/4) is an octave higher (2/1). By considering the charge \rho(\mathbf{x}')dV in each small volume of space dV at point \mathbf{x}' as a point charge, the resulting electric field, d\mathbf{E}(\mathbf{x}), at point \mathbf{x} can be calculated as d\mathbf{E}(\mathbf{x}) = \frac{\rho(\mathbf{x}')}{4\pi\varepsilon_0}\frac{\hat \mathbf{r}'}{{r'}^2} dV where *\hat \mathbf{r}' is the unit vector pointing from \mathbf{x}' to \mathbf{x}. *r' is the distance from \mathbf{x}' to \mathbf{x}. 220px|thumb|right|Illustration of the electric field surrounding a positive (red) and a negative (blue) charge. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. The force required to bend a string at its midpoint to a given angle \theta is given by F_{B} = 2\left(T + EA\left(\frac{1 - \cos\theta}{\cos\theta} \right) \right)\sin\theta . thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. As the electric field is defined in terms of force, and force is a vector (i.e. having both magnitude and direction), it follows that an electric field is a vector field. In that case, Coulomb's law fully describes the field.Purcell, pp. 5-7. ===Parallels between electrostatic and gravitational fields=== Coulomb's law, which describes the interaction of electric charges: \mathbf{F} = q \left(\frac{Q}{4\pi\varepsilon_0} \frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = q \mathbf{E} is similar to Newton's law of universal gravitation: \mathbf{F} = m\left(-GM\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}\right) = m\mathbf{g} (where \mathbf{\hat{r}} = \mathbf{\frac{r}{|r|}}). Alternatively the electric field of uniformly moving point charges can be derived from the Lorentz transformation of four-force experienced by test charges in the source's rest frame given by Coulomb's law and assigning electric field and magnetic field by their definition given by the form of Lorentz force. The above equation, although consistent with that of uniformly moving point charges as well as its non-relativistic limit, are not corrected for quantum- mechanical effects. == Some common electric field values == * Infinite wire having uniform charge density \lambda has electric field at a distance x from it as \frac{\lambda}{2\pi\epsilon_0x} \hat{x} * Infinitely large surface having charge density \sigma has electric field at a distance x from it as \frac{\sigma}{2\epsilon_0} \hat{x} * Infinitely long cylinder having Uniform charge density \lambda that is charge contained along unit length of the cylinder has electric field at a distance x from it as \frac{\lambda}{2\pi\epsilon_0x} \hat{x} while it is 0 everywhere inside the cylinder * Uniformly charged non-conducting sphere of radius R, volume charge density \rho and total charge Q has electric field at a distance x from it as \frac{Q}{4\pi\epsilon_0x^2} \hat{x} while the electric field at a point \vec{r} inside sphere from its center is given by \frac{Q}{4\pi\epsilon_0R^3}\vec{r} * Uniformly charged conducting sphere of radius R, surface charge density \sigma and total charge Q has electric field at a distance x from it as \frac{Q}{4\pi\epsilon_0x^2} \hat{x} while the electric field inside is 0 * Electric field infinitely close to a conducting surface in electrostatic equilibrium having charge density \sigma at that point is \frac{\sigma}{\epsilon_0} \hat{x} * Uniformly charged ring having total charge Q has electric field at a distance x along its axis as \frac{Qx}{4\pi\epsilon_0(R^2+x^2)^{3/2}} \hat{x}' * Uniformly charged disc of radius R and charge density \sigma has electric field at a distance x along its axis from it as \frac{\sigma}{2\epsilon_0} \left[1-\left(\frac{R^2}{x^2}-1\right)^{-1/2}\right] \hat{x} * Electric field due to dipole of dipole moment \vec{p} at a distance x from their center along equatorial plane is given as -\frac{\vec{p}}{4\pi\epsilon_0x^3} and the same along the axial line is approximated to \frac{\vec{p}}{2\pi\epsilon_0x^3} for x much bigger than the distance between dipoles. 275px|thumb|Stick figure of 1.75 meters standing next to a violin string of .33 meters and a long string instrument string of 10 meters. ",13,5840,"""61.0""",+10,131,C
 "A long, straight wire has fixed negative charge with a linear charge density of magnitude $3.6 \mathrm{nC} / \mathrm{m}$. The wire is to be enclosed by a coaxial, thin-walled nonconducting cylindrical shell of radius $1.5 \mathrm{~cm}$. The shell is to have positive charge on its outside surface with a surface charge density $\sigma$ that makes the net external electric field zero. Calculate $\sigma$.
-","It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. With the same example, using a larger Gaussian surface outside the shell where , Gauss's law will produce a non-zero electric field. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example ""field near infinite line charge"" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor: :\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, or : \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Given the electrical potential on a conductor surface (the equipotential surface or the point chosen on surface ) contained in a system of conductors : :\phi_i = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{ji}} \mbox{ (i=1, 2..., n)}, where , i.e. the distance from the area- element to a particular point on conductor . is not, in general, uniformly distributed across the surface. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric: : \begin{matrix} \phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\\ \phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\\ \vdots \\\ \phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n \end{matrix}. where is the surface charge on conductor . Thereby is the electrical charge enclosed by the Gaussian surface. Surface Charging and Points of Zero Charge. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. If the electrode is polarizable, then its surface charge depends on the electrode potential. On a capacitor, the charge on the two conductors is equal and opposite: . Hence, with :p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}, we have :\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}. ==Example== In this example, we employ the method of coefficients of potential to determine the capacitance on a two-conductor system. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The electrostatic potential at point is \phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j}}. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. ",4,3.8,-1270.0,38,8.7,B
-"Beams of high-speed protons can be produced in ""guns"" using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun's electric field were $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$ ? (b) What speed would the proton attain if the field accelerated the proton through a distance of $1.00 \mathrm{~cm}$ ?","An electrostatic particle accelerator is a particle accelerator in which charged particles are accelerated to a high energy by a static high voltage potential. A special application of electrostatic particle accelerator are dust accelerators in which nanometer to micrometer sized electrically charged dust particles are accelerated to speeds up to 100 km/s. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. The advantages of electrostatic accelerators over oscillating field machines include lower cost, the ability to produce continuous beams, and higher beam currents that make them useful to industry. The accelerating voltage on electrostatic machines is in the range 0.1 to 25 MV and the charge on particles is a few elementary charges, so the particle energy is in the low MeV range. When performing a modeling task for any accelerator operation, the results of charged particle beam dynamics simulations must feed into the associated application. High energy oscillating field accelerators usually incorporate an electrostatic machine as their first stage, to accelerate particles to a high enough velocity to inject into the main accelerator. See particle- beam weapon for more information on this type of weapon. ==See also== * Ion source * Ion thruster * Ion wind ==References== ==External links== * Stopping parameters of ion beams in solids calculated by MELF-GOS model * ISOLDE – Facility dedicated to the production of a large variety of radioactive ion beams located at CERN Category:Plasma physics Category:Semiconductor device fabrication Category:Semiconductor analysis Category:Thin film deposition Category:Ions Category:Accelerator physics In particle accelerators, a common mechanism for accelerating a charged particle beam is via copper resonant cavities in which electric and magnetic fields form a standing wave, the mode of which is designed so that the E field points along the axis of the accelerator, producing forward acceleration of the particles when in the correct phase. A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Accelerator physics is a field of physics encompassing all the aspects required to design and operate the equipment and to understand the resulting dynamics of the charged particles. Oscillating accelerators do not have this limitation, so they can achieve higher particle energies than electrostatic machines. While all linacs accelerate particles in a straight line, electrostatic accelerators use a fixed accelerating field from a single high voltage source, while radiofrequency linacs use oscillating electric fields across a series of accelerating gaps. == Applications == Electrostatic accelerators have a wide array of applications in science and industry. This innovative propulsion technique named Ion Beam Shepherd has been shown to be effective in the area of active space debris removal as well as asteroid deflection. ===High-energy ion beams=== High-energy ion beams produced by particle accelerators are used in atomic physics, nuclear physics and particle physics. ===Weaponry=== The use of ion beams as a particle-beam weapon is theoretically possible, but has not been demonstrated. This contrasts with the other major category of particle accelerator, oscillating field particle accelerators, in which the particles are accelerated by oscillating electric fields. The maximum particle energy produced by electrostatic accelerators is limited by the maximum voltage which can be achieved the machine. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. Omitting practical problems, if the platform is positively charged, it will repel the ions of the same electric polarity, accelerating them. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . The ion current density j that can be accelerated using a gridded ion source is limited by the space charge effect, which is described by Child's law: j \approx \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{m}} \frac{(\Delta V)^{\frac{3}{2}}}{d^2}, where \Delta V is the voltage between the grids, d is the distance between the grids, and m is the ion mass. Electrostatic accelerators are a subset of linear accelerators (linacs). Many universities worldwide have electrostatic accelerators for research purposes. ",1.92,7200,0.42,4,0.0547,A
+","It is immediately apparent that for a spherical Gaussian surface of radius the enclosed charge is zero: hence the net flux is zero and the magnitude of the electric field on the Gaussian surface is also 0 (by letting in Gauss's law, where is the charge enclosed by the Gaussian surface). thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. With the same example, using a larger Gaussian surface outside the shell where , Gauss's law will produce a non-zero electric field. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example ""field near infinite line charge"" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. Let us introduce the factor that describes how the actual charge density differs from the average and itself on a position on the surface of the -th conductor: :\frac{\sigma_j}{\langle\sigma_j\rangle} = f_j, or : \sigma_j = \langle\sigma_j\rangle f_j = \frac{Q_j}{S_j}f_j. The sum of the electric flux through each component of the surface is proportional to the enclosed charge of the pillbox, as dictated by Gauss's Law. Given the electrical potential on a conductor surface (the equipotential surface or the point chosen on surface ) contained in a system of conductors : :\phi_i = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{ji}} \mbox{ (i=1, 2..., n)}, where , i.e. the distance from the area- element to a particular point on conductor . is not, in general, uniformly distributed across the surface. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. In electrostatics, the coefficients of potential determine the relationship between the charge and electrostatic potential (electrical potential), which is purely geometric: : \begin{matrix} \phi_1 = p_{11}Q_1 + \cdots + p_{1n}Q_n \\\ \phi_2 = p_{21}Q_1 + \cdots + p_{2n}Q_n \\\ \vdots \\\ \phi_n = p_{n1}Q_1 + \cdots + p_{nn}Q_n \end{matrix}. where is the surface charge on conductor . Thereby is the electrical charge enclosed by the Gaussian surface. Surface Charging and Points of Zero Charge. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. This is Gauss's law, combining both the divergence theorem and Coulomb's law. === Spherical surface === A spherical Gaussian surface is used when finding the electric field or the flux produced by any of the following:Physics for Scientists and Engineers - with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, * a point charge * a uniformly distributed spherical shell of charge * any other charge distribution with spherical symmetry The spherical Gaussian surface is chosen so that it is concentric with the charge distribution. The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. If the electrode is polarizable, then its surface charge depends on the electrode potential. On a capacitor, the charge on the two conductors is equal and opposite: . Hence, with :p_{ij} = \frac{1}{4\pi\epsilon_0 S_j}\int_{S_j}\frac{f_j da_j}{R_{ji}}, we have :\phi_i=\sum_{j = 1}^n p_{ij}Q_j \mbox{ (i = 1, 2, ..., n)}. ==Example== In this example, we employ the method of coefficients of potential to determine the capacitance on a two-conductor system. Using this and Coulomb's law tells us that the electric field due to a single charged particle is : \mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}. The electrostatic potential at point is \phi_P = \sum_{j = 1}^{n}\frac{1}{4\pi\epsilon_0}\int_{S_j}\frac{\sigma_j da_j}{R_{j}}. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. ",4,3.8,"""-1270.0""",38,8.7,B
+"Beams of high-speed protons can be produced in ""guns"" using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun's electric field were $2.00 \times 10^4 \mathrm{~N} / \mathrm{C}$ ? (b) What speed would the proton attain if the field accelerated the proton through a distance of $1.00 \mathrm{~cm}$ ?","An electrostatic particle accelerator is a particle accelerator in which charged particles are accelerated to a high energy by a static high voltage potential. A special application of electrostatic particle accelerator are dust accelerators in which nanometer to micrometer sized electrically charged dust particles are accelerated to speeds up to 100 km/s. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. The advantages of electrostatic accelerators over oscillating field machines include lower cost, the ability to produce continuous beams, and higher beam currents that make them useful to industry. The accelerating voltage on electrostatic machines is in the range 0.1 to 25 MV and the charge on particles is a few elementary charges, so the particle energy is in the low MeV range. When performing a modeling task for any accelerator operation, the results of charged particle beam dynamics simulations must feed into the associated application. High energy oscillating field accelerators usually incorporate an electrostatic machine as their first stage, to accelerate particles to a high enough velocity to inject into the main accelerator. See particle- beam weapon for more information on this type of weapon. ==See also== * Ion source * Ion thruster * Ion wind ==References== ==External links== * Stopping parameters of ion beams in solids calculated by MELF-GOS model * ISOLDE – Facility dedicated to the production of a large variety of radioactive ion beams located at CERN Category:Plasma physics Category:Semiconductor device fabrication Category:Semiconductor analysis Category:Thin film deposition Category:Ions Category:Accelerator physics In particle accelerators, a common mechanism for accelerating a charged particle beam is via copper resonant cavities in which electric and magnetic fields form a standing wave, the mode of which is designed so that the E field points along the axis of the accelerator, producing forward acceleration of the particles when in the correct phase. A charged particle accelerator is a complex machine that takes elementary charged particles and accelerates them to very high energies. Accelerator physics is a field of physics encompassing all the aspects required to design and operate the equipment and to understand the resulting dynamics of the charged particles. Oscillating accelerators do not have this limitation, so they can achieve higher particle energies than electrostatic machines. While all linacs accelerate particles in a straight line, electrostatic accelerators use a fixed accelerating field from a single high voltage source, while radiofrequency linacs use oscillating electric fields across a series of accelerating gaps. == Applications == Electrostatic accelerators have a wide array of applications in science and industry. This innovative propulsion technique named Ion Beam Shepherd has been shown to be effective in the area of active space debris removal as well as asteroid deflection. ===High-energy ion beams=== High-energy ion beams produced by particle accelerators are used in atomic physics, nuclear physics and particle physics. ===Weaponry=== The use of ion beams as a particle-beam weapon is theoretically possible, but has not been demonstrated. This contrasts with the other major category of particle accelerator, oscillating field particle accelerators, in which the particles are accelerated by oscillating electric fields. The maximum particle energy produced by electrostatic accelerators is limited by the maximum voltage which can be achieved the machine. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. Omitting practical problems, if the platform is positively charged, it will repel the ions of the same electric polarity, accelerating them. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . The ion current density j that can be accelerated using a gridded ion source is limited by the space charge effect, which is described by Child's law: j \approx \frac{4\epsilon_0}{9} \sqrt{\frac{2e}{m}} \frac{(\Delta V)^{\frac{3}{2}}}{d^2}, where \Delta V is the voltage between the grids, d is the distance between the grids, and m is the ion mass. Electrostatic accelerators are a subset of linear accelerators (linacs). Many universities worldwide have electrostatic accelerators for research purposes. ",1.92,7200,"""0.42""",4,0.0547,A
 "An infinite line of charge produces a field of magnitude $4.5 \times$ $10^4 \mathrm{~N} / \mathrm{C}$ at distance $2.0 \mathrm{~m}$. Find the linear charge density.
-","Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Like mass density, charge density can vary with position. The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element, \lambda_q = \frac{d Q}{d \ell}\,, similarly the surface charge density uses a surface area element dS \sigma_q = \frac{d Q}{d S}\,, and the volume charge density uses a volume element dV \rho_q =\frac{d Q}{d V} \, , Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C, Q = \int_L \lambda_q(\mathbf{r}) \, d\ell similarly a surface integral of the surface charge density σq(r) over a surface S, Q = \int_S \sigma_q(\mathbf{r}) \, dS and a volume integral of the volume charge density ρq(r) over a volume V, Q = \int_V \rho_q(\mathbf{r}) \, dV where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity. As always, the integral of the charge density over a region of space is the charge contained in that region. Similar equations are used for the linear and surface charge densities. == Charge density in special relativity == In special relativity, the length of a segment of wire depends on velocity of observer because of length contraction, so charge density will also depend on velocity. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. == Discrete charges == For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function: \rho_q(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_0) where r is the position to calculate the charge. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Derivation: qvB = mv\frac{v}{r} or Since F_\text{electric} = F_\text{magnetic}, E q = B q v or Equations () and () yield \frac{q}{m}=\frac{E}{B^2r} === Significance === In some experiments, the charge-to-mass ratio is the only quantity that can be measured directly. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. Charge density can be either positive or negative, since electric charge can be either positive or negative. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example ""field near infinite line charge"" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The value of nuclear magneton System of units Value Unit SI J·T CGS Erg·G eV eV·T MHz/T (per h) MHz/T The nuclear magneton (symbol μ) is a physical constant of magnetic moment, defined in SI units by: :\mu_\text{N} = {{e \hbar} \over {2 m_\text{p}}} and in Gaussian CGS units by: :\mu_\text{N} = {{e \hbar} \over{2 m_\text{p}c}} where: :e is the elementary charge, :ħ is the reduced Planck constant, :m is the proton rest mass, and :c is the speed of light In SI units, its value is approximately: :μ = In Gaussian CGS units, its value can be given in convenient units as :μ = The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . ",nan,5.0,0.000226,-0.16,449,B
+","Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. By Gauss's law \Phi_E = \frac{q}{\varepsilon_0} equating for yields E 2 \pi rh = \frac{\lambda h}{\varepsilon_0} \quad \Rightarrow \quad E = \frac{\lambda}{2 \pi\varepsilon_0 r} === Gaussian pillbox === This surface is most often used to determine the electric field due to an infinite sheet of charge with uniform charge density, or a slab of charge with some finite thickness. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The charge density at any point is equal to the charge carrier density multiplied by the elementary charge on the particles. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Like mass density, charge density can vary with position. The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element, \lambda_q = \frac{d Q}{d \ell}\,, similarly the surface charge density uses a surface area element dS \sigma_q = \frac{d Q}{d S}\,, and the volume charge density uses a volume element dV \rho_q =\frac{d Q}{d V} \, , Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C, Q = \int_L \lambda_q(\mathbf{r}) \, d\ell similarly a surface integral of the surface charge density σq(r) over a surface S, Q = \int_S \sigma_q(\mathbf{r}) \, dS and a volume integral of the volume charge density ρq(r) over a volume V, Q = \int_V \rho_q(\mathbf{r}) \, dV where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity. As always, the integral of the charge density over a region of space is the charge contained in that region. Similar equations are used for the linear and surface charge densities. == Charge density in special relativity == In special relativity, the length of a segment of wire depends on velocity of observer because of length contraction, so charge density will also depend on velocity. }} The mass-to-charge ratio (m/Q) is a physical quantity relating the mass (quantity of matter) and the electric charge of a given particle, expressed in units of kilograms per coulomb (kg/C). The equivalent proofs for linear charge density and surface charge density follow the same arguments as above. == Discrete charges == For a single point charge q at position r0 inside a region of 3d space R, like an electron, the volume charge density can be expressed by the Dirac delta function: \rho_q(\mathbf{r}) = q \delta(\mathbf{r} - \mathbf{r}_0) where r is the position to calculate the charge. thumb|A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, 'ideal' wire. The cyclotron equation, combined with other information such as the kinetic energy of the particle, will give the charge-to-mass ratio. Derivation: qvB = mv\frac{v}{r} or Since F_\text{electric} = F_\text{magnetic}, E q = B q v or Equations () and () yield \frac{q}{m}=\frac{E}{B^2r} === Significance === In some experiments, the charge-to-mass ratio is the only quantity that can be measured directly. The charge-to-mass ratio (Q/m) of an object is, as its name implies, the charge of an object divided by the mass of the same object. Since all charge is carried by subatomic particles, which can be idealized as points, the concept of a continuous charge distribution is an approximation, which becomes inaccurate at small length scales. It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. Charge density can be either positive or negative, since electric charge can be either positive or negative. And, as mentioned, any exterior charges do not count. === Cylindrical surface === A cylindrical Gaussian surface is used when finding the electric field or the flux produced by any of the following: * an infinitely long line of uniform charge * an infinite plane of uniform charge * an infinitely long cylinder of uniform charge As example ""field near infinite line charge"" is given below; Consider a point P at a distance from an infinite line charge having charge density (charge per unit length) λ. We can use Gauss's law to find the magnitude of the resultant electric field at a distance from the center of the charged shell. The value of nuclear magneton System of units Value Unit SI J·T CGS Erg·G eV eV·T MHz/T (per h) MHz/T The nuclear magneton (symbol μ) is a physical constant of magnetic moment, defined in SI units by: :\mu_\text{N} = {{e \hbar} \over {2 m_\text{p}}} and in Gaussian CGS units by: :\mu_\text{N} = {{e \hbar} \over{2 m_\text{p}c}} where: :e is the elementary charge, :ħ is the reduced Planck constant, :m is the proton rest mass, and :c is the speed of light In SI units, its value is approximately: :μ = In Gaussian CGS units, its value can be given in convenient units as :μ = The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei. An ion with a mass of 100 Da (daltons) () carrying two charges () will be observed at . ",,5.0,"""0.000226""",-0.16,449,B
 "A charged nonconducting rod, with a length of $2.00 \mathrm{~m}$ and a cross-sectional area of $4.00 \mathrm{~cm}^2$, lies along the positive side of an $x$ axis with one end at the origin. The volume charge density $\rho$ is charge per unit volume in coulombs per cubic meter. How many excess electrons are on the rod if $\rho$ is uniform, with a value of $-4.00 \mu \mathrm{C} / \mathrm{m}^3$?
-","Since the field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 732–733 J = \frac{I}{a}. In these situations, the density simplifies to :\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2. == General properties == From its definition, the electron density is a non-negative function integrating to the total number of electrons. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The charge density of the system at a point r is a sum of the charge densities for each charge qi at position ri, where : \rho_q(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R: Q = \int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by \rho_q(\mathbf{r}) = q n(\mathbf{r})\,. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into ""free"" and ""bound"" charges. At position at time , the distribution of charge flowing is described by the current density: \mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) where * is the current density vector; * is the particles' average drift velocity (SI unit: m∙s−1); *\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) is the charge density (SI unit: coulombs per cubic metre), in which ** is the number of particles per unit volume (""number density"") (SI unit: m−3); ** is the charge of the individual particles with density (SI unit: coulombs). This charge is equal to d q = \rho \, v\, dt \, dA, where is the charge density at . It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Charge density can be either positive or negative, since electric charge can be either positive or negative. In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. As always, the integral of the charge density over a region of space is the charge contained in that region. In quantum chemical calculations, the electron density, ρ(r), is a function of the coordinates r, defined so ρ(r)dr is the number of electrons in a small volume dr. The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element, \lambda_q = \frac{d Q}{d \ell}\,, similarly the surface charge density uses a surface area element dS \sigma_q = \frac{d Q}{d S}\,, and the volume charge density uses a volume element dV \rho_q =\frac{d Q}{d V} \, , Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C, Q = \int_L \lambda_q(\mathbf{r}) \, d\ell similarly a surface integral of the surface charge density σq(r) over a surface S, Q = \int_S \sigma_q(\mathbf{r}) \, dS and a volume integral of the volume charge density ρq(r) over a volume V, Q = \int_V \rho_q(\mathbf{r}) \, dV where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The electric current is dI = dq/dt = \rho v dA, it follows that the current density vector is the vector normal dA (i.e. parallel to ) and of magnitude dI/dA = \rho v \mathbf{j} = \rho \mathbf{v}. The electrical resistance of a uniform conductor is given in terms of resistivity by: {R} = \rho \frac{\ell}{a} where ℓ is the length of the conductor in SI units of meters, is the cross-sectional area (for a round wire if is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters. If (SI unit: A) is the electric current flowing through , then electric current density at is given by the limit: j = \lim_{A \to 0} \frac{I_A}{A} = \left.\frac{\partial I}{\partial A} \right|_{A=0}, with surface remaining centered at and orthogonal to the motion of the charges during the limit process. The delta function has the sifting property for any function f: \int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0) so the delta function ensures that when the charge density is integrated over R, the total charge in R is q: Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q This can be extended to N discrete point-like charge carriers. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. ",2.00,122,10.4,0, -31.95,A
-"A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured.  (b) Find the probability that the measured value is between $x=0$ and $x=2 \mathrm{~nm}$.","Hence, at a given time , is the probability density function of the particle's position. If the oscillator spends an infinitesimal amount of time in the vicinity of a given -value, then the probability of being in that vicinity will be :P(x)\, dx \propto dt. If it corresponds to a non-degenerate eigenvalue of , then |\psi (x)|^2 gives the probability of the corresponding value of for the initial state . Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Thus the probability that the particle is in the volume at is :\mathbf{P}(V)=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}} = \int_V |\psi(\mathbf {x}, t_0)|^2\, \mathrm{d\mathbf {x}}. If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of ) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. The probability of measuring |u\rangle is given by :P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} Which agrees with experiment. ==Normalization== In the example above, the measurement must give either or , so the total probability of measuring or must be 1. In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. Plugging this into the expression for yields :P(x) = \frac{1}{\pi}\frac{1}{\sqrt{A^2-x^2}}. Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position , for time ). Note that if any solution to the wave equation is normalisable at some time , then the defined above is always normalised, so that :\rho_t(\mathbf x)=\left |\psi(\mathbf x, t)\right |^2 = \left|\frac{\psi_0(\mathbf x, t)}{a}\right|^2 is always a probability density function for all . Plugging this into our expression for yields :P(x) = \frac{1}{T} \sqrt{\frac{2m}{E-U(x)}}. If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of , then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In this system, all momenta are equally probable. ==See also== *Probability density function *Correspondence principle *Classical limit *Wave function ==References== Category:Concepts in physics Category:Classical mechanics Category:Theoretical physics Then \psi (x) is the ""probability amplitude"" for the eigenstate . Once this is done, is readily obtained for any allowed energy . ==Examples== ===Simple harmonic oscillator=== thumb|300px|right|The probability density function of the state of the quantum harmonic oscillator. Therefore, if the system is known to be in some eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. Following a similar argument as above, the result is :P(p) = \frac{2}{T}\frac{1}{|F(x)|}, where is the force acting on the particle as a function of position. ", 7.42,0.4908,252.8,0,0.23333333333,B
-Calculate the ground-state energy of the hydrogen atom using SI units and convert the result to electronvolts.,"To convert from ""value of ionization energy"" to the corresponding ""value of molar ionization energy"", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. ""use"" and ""WEL"" give ionization energy in the unit kJ/mol; ""CRC"" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. The Prout is an obsolete unit of energy, whose value is: 1 Prout = 2.9638 \times 10^{-14} J This is equal to one twelfth of the binding energy of the deuteron. This page shows the electron configurations of the neutral gaseous atoms in their ground states. This website is also cited in the CRC Handbook as source of Section 1, subsection Electron Configuration of Neutral Atoms in the Ground State. *91 Pa : [Rn] 5f2(3H4) 6d 7s2 *92 U : [Rn] 5f3(4Io9/2) 6d 7s2 *93 Np : [Rn] 5f4(5I4) 6d 7s2 *103 Lr : [Rn] 5f14 7s2 7p1 question-marked *104 Rf : [Rn] 5f14 6d2 7s2 question-marked ===CRC=== *David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition, online version. After ionization, the electron and proton recombine to form a new hydrogen atom. Boca Raton, Florida, 2003; Section 1, Basic Constants, Units, and Conversion Factors; Electron Configuration of Neutral Atoms in the Ground State. (elements 1-104) *Also subsection Periodic Table of the Elements, (elements 1-103) based on: **G. J. Leigh, Editor, Nomenclature of Inorganic Chemistry, Blackwell Scientific Publications, Oxford, 1990. **Atomic Weights of the Elements, 1999, Pure Appl. Chem., 73, 667, 2001. ===WebElements=== *http://www.webelements.com/ ; retrieved July 2005, electron configurations based on: **Atomic, Molecular, & Optical Physics Handbook, Ed. alt=|thumb|upright=1.2|Hemispherical electron energy analyzer. Therefore, the H-alpha line occurs where hydrogen is being ionized. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. These tables list values of molar ionization energies, measured in kJ⋅mol−1. (Elements 1-106) *58 Ce : [Xe] 4f2 6s2 *103 Lr : [Rn] 5f14 6d1 7s2 *104 Rf : [Rn] 5f14 6d2 7s2 (agrees with guess above) *105 Db : [Rn] 5f14 6d3 7s2 *106 Sg : [Rn] 5f14 6d4 7s2 ===Hoffman, Lee, and Pershina=== This book contains predicted electron configurations for the elements up to 172, as well as 184, based on relativistic Dirac–Fock calculations by B. Fricke in * Category:Chemical element data pages * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. == External links == * NIST Atomic Spectra Database Ionization Energies == See also == *Molar ionization energies of the elements Category:Properties of chemical elements Category:Chemical element data pages The first molar ionization energy applies to the neutral atoms. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. Note that these electron configurations are given for neutral atoms in the gas phase, which are not the same as the electron configurations for the same atoms in chemical environments. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. ",2500,0.18162,1.51,15.757,-13.598 ,E
-Find the probability that the electron in the ground-state $\mathrm{H}$ atom is less than a distance $a$ from the nucleus.,"The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. * Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913). The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. The expected value of the radial distance of the electron, by contrast, . == Related constants == The Bohr radius is one of a trio of related units of length, the other two being the reduced Compton wavelength of the electron ( \lambda_{\mathrm{e}} / 2\pi ) and the classical electron radius ( r_{\mathrm{e}} ). H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. This is because there are more energy levels and therefore a greater distance between protons and electrons. The value of the radius may depend on the atom's state and context. The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy. The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrödinger equation as published in 1926. Therefore, the radius of an atom is more than 10,000 times the radius of its nucleus (1–10 fm), and less than 1/1000 of the wavelength of visible light (400–700 nm). ",5300,0.23333333333,0.19,0.323,9,D
-"A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured. (a) Find the probability that the measured value lies between $x=1.5000 \mathrm{~nm}$ and $x=1.5001 \mathrm{~nm}$.","Hence, at a given time , is the probability density function of the particle's position. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. If it corresponds to a non-degenerate eigenvalue of , then |\psi (x)|^2 gives the probability of the corresponding value of for the initial state . Thus the probability that the particle is in the volume at is :\mathbf{P}(V)=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}} = \int_V |\psi(\mathbf {x}, t_0)|^2\, \mathrm{d\mathbf {x}}. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of ) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). The probability of measuring |u\rangle is given by :P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} Which agrees with experiment. ==Normalization== In the example above, the measurement must give either or , so the total probability of measuring or must be 1. In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. Note that if any solution to the wave equation is normalisable at some time , then the defined above is always normalised, so that :\rho_t(\mathbf x)=\left |\psi(\mathbf x, t)\right |^2 = \left|\frac{\psi_0(\mathbf x, t)}{a}\right|^2 is always a probability density function for all . Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position , for time ). If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of , then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. The distribution is expressed in the form: :p_i \propto \exp\left(- \frac{\varepsilon_i}{kT} \right) where is the probability of the system being in state , is the exponential function, is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . Then \psi (x) is the ""probability amplitude"" for the eigenstate . Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In this case, if the vector has the norm 1, then is just the probability that the quantum system resides in the state . The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in to obtain probability values – as was stated above, the system can't be in some state with a positive probability. Therefore, if the system is known to be in some eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. upright=1.75|right|thumb|Boltzmann's distribution is an exponential distribution. upright=1.75|right|thumb|Boltzmann factor (vertical axis) as a function of temperature for several energy differences . If we have a system consisting of many particles, the probability of a particle being in state is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state . ",1.88,234.4,-4.37,22,4.979,E
-"In this example, $2.50 \mathrm{~mol}$ of an ideal gas with $C_{V, m}=12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ is expanded adiabatically against a constant external pressure of 1.00 bar. The initial temperature and pressure of the gas are $325 \mathrm{~K}$ and $2.50 \mathrm{bar}$, respectively. The final pressure is 1.25 bar. Calculate the final temperature, $q, w, \Delta U$.","The adiabatic constant remains the same, but with the resulting pressure unknown : P_2 V_2^\gamma = \mathrm{constant}_1 = 6.31~\text{Pa}\,\text{m}^{21/5} = P \times (0.0001~\text{m}^3)^\frac75, We can now solve for the final pressure : P_2 = P_1\left (\frac{V_1}{V_2}\right)^\gamma = 100\,000~\text{Pa} \times \text{10}^{7/5} = 2.51 \times 10^6~\text{Pa} or 25.1 bar. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm3 = 0.001 m3); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so ); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure). : \begin{align} & P_1 V_1^\gamma = \mathrm{constant}_1 = 100\,000~\text{Pa} \times (0.001~\text{m}^3)^\frac75 \\\ & = 10^5 \times 6.31 \times 10^{-5}~\text{Pa}\,\text{m}^{21/5} = 6.31~\text{Pa}\,\text{m}^{21/5}, \end{align} so the adiabatic constant for this example is about 6.31 Pa m4.2. Adiabatic heating occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (b4) gives : P = P_1 \left(\frac{V_1}{V} \right)^\gamma. For such an adiabatic process, the modulus of elasticity (Young's modulus) can be expressed as , where is the ratio of specific heats at constant pressure and at constant volume () and is the pressure of the gas. === Various applications of the adiabatic assumption === For a closed system, one may write the first law of thermodynamics as , where denotes the change of the system's internal energy, the quantity of energy added to it as heat, and the work done by the system on its surroundings. The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. In addition, through the use of the Euler chain relation it can be shown that \left ( \frac{\partial U}{\partial V} \right )_T = - \left ( \frac{\partial U}{\partial T} \right )_V \left ( \frac{\partial T}{\partial V} \right )_U Defining \mu_J = \left ( \frac{\partial T}{\partial V} \right )_U as the ""Joule coefficient"" J. Westin, A Course in Thermodynamics, Volume 1, Taylor and Francis, New York (1979). and recognizing \left ( \frac{\partial U}{\partial T} \right )_V as the heat capacity at constant volume = C_V , we have \pi_T = - C_V \mu_J The coefficient \mu_J can be obtained by measuring the temperature change for a constant-U experiment, i.e., an adiabatic free expansion (see below). The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient ""adiabatic approximation"".Bailyn, M. (1994), pp. 52–53. We know the compressed gas has = 0.1 L and = , so we can solve for temperature: : T = \frac{P V}{\mathrm{constant}_2} = \frac{2.51 \times 10^6~\text{Pa} \times 10^{-4}~\text{m}^3}{0.333~\text{Pa}\,\text{m}^3\text{K}^{-1}} = 753~\text{K}. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible. ===Derivation of P–V relation for adiabatic heating and cooling=== The definition of an adiabatic process is that heat transfer to the system is zero, . For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to: : dh = v \, dp. *If the system walls are adiabatic () but not rigid (), and, in a fictive idealized process, energy is added to the system in the form of frictionless, non-viscous pressure–volume work (), and there is no phase change, then the temperature of the system will rise. Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. In the constant volume adiabatic flame temperature case, the volume of the system is held constant and hence there is no work occurring: : {}_RW_P = \int\limits_R^P {pdV} = 0 There is also no heat transfer because the process is defined to be adiabatic: {}_RQ_P = 0 . For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame temperature by assuming combustion loses no heat to its surroundings. Simplifying, : T_2 - T_1 = T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right), : \frac{T_2}{T_1} - 1 = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1, : T_2 = T_1 \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}}. ===Derivation of discrete formula and work expression=== The change in internal energy of a system, measured from state 1 to state 2, is equal to : At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (c4) gives : P = P_1 \left(\frac{V_1}{V} \right)^\gamma. Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant (nR) is: : \frac{PV}{T} = \mathrm{constant}_2 = \frac{10^5~\text{Pa} \times 10^{-3}~\text{m}^3}{300~\text{K}} = 0.333~\text{Pa}\,\text{m}^3\text{K}^{-1}. ",+3.03,+2.9,46.7,92,-1.78,E
-"Find $Y_l^m(\theta, \phi)$  for  $l=0$.","The solutions are usually written in terms of complex exponentials: Y_{\ell, m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ Y_{\ell, m}^*(\theta, \phi) = (-1)^m Y_{\ell, -m}(\theta, \phi). The functions Y_{\ell, m}(\theta, \phi) are the spherical harmonics, and the quantity in the square root is a normalizing factor. P_\ell^{m}(\cos \theta)\ e^{im\phi}\qquad -\ell \le m \le \ell. When the partial differential equation \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2} + \lambda \psi = 0 is solved by the method of separation of variables, one gets a φ-dependent part \sin(m\phi) or \cos(m\phi) for integer m≥0, and an equation for the θ-dependent part \frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\, for which the solutions are P_\ell^{m}(\cos \theta) with \ell{\ge}m and \lambda = \ell(\ell+1). Therefore, the equation abla^2\psi + \lambda\psi = 0 has nonsingular separated solutions only when \lambda = \ell(\ell+1), and those solutions are proportional to P_\ell^{m}(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell and P_\ell^{m}(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 < m \le \ell. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for :. \left(1-x^2\right) \frac{d^2}{dx^2}P_\ell(x) -2x\frac{d}{dx}P_\ell(x)+ \ell(\ell+1)P_\ell(x) = 0. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is abla^2\psi = \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2}. The longitude angle, \phi, appears in a multiplying factor. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. P_\ell ^{m} (This followed from the Rodrigues' formula definition. L.M.L. is the second English album and fifth overall studio album by Nu Virgos. == Content == The title of the album comes from the song in the album titled ""L.M.L."". == Release == The album was released in Russia on September 13, 2007, and in Asia on September 19, 2007. In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. It is also commonly denoted as zn(u,k) :\Theta(u)=\Theta_{4}\left(\frac{\pi u}{2K}\right) :Z(u)=\frac{\partial}{\partial u}\ln\Theta(u) =\frac{\Theta'(u)}{\Theta(u)} :Z(\phi|m)=E(\phi|m)-\frac{E(m)}{K(m)}F(\phi|m) :Where E, K, and F are generic Incomplete Elliptical Integrals of the first and second kind. Moreover, since by Rodrigues' formula, P_\ell(x) = \frac{1}{2^\ell\,\ell!} \ \frac{d^\ell}{dx^\ell}\left[(x^2-1)^\ell\right], the P can be expressed in the form P_\ell^{m}(x) = \frac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell. This definition also makes the various recurrence formulas work for positive or negative .) \text{If}\quad |m| > \ell\,\quad\text{then}\quad P_\ell^{m} = 0.\, The differential equation is also invariant under a change from to , and the functions for negative are defined by P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, \dots). ==Parity== From their definition, one can verify that the Associated Legendre functions are either even or odd according to P_\ell ^{m} (-x) = (-1)^{\ell + m} P_\ell ^{m}(x) ==The first few associated Legendre functions== thumb|300px|Associated Legendre functions for m = 0 thumb|300px|Associated Legendre functions for m = 1 thumb|300px|Associated Legendre functions for m = 2 The first few associated Legendre functions, including those for negative values of m, are: P_{0}^{0}(x)=1 \begin{align} P_{1}^{-1}(x)&=-\tfrac{1}{2}P_{1}^{1}(x) \\\ P_{1}^{0}(x)&=x \\\ P_{1}^{1}(x)&=-(1-x^2)^{1/2} \end{align} \begin{align} P_{2}^{-2}(x)&=\tfrac{1}{24}P_{2}^{2}(x) \\\ P_{2}^{-1}(x)&=-\tfrac{1}{6}P_{2}^{1}(x) \\\ P_{2}^{0}(x)&=\tfrac{1}{2}(3x^{2}-1) \\\ P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} \\\ P_{2}^{2}(x)&=3(1-x^2) \end{align} \begin{align} P_{3}^{-3}(x)&=-\tfrac{1}{720}P_{3}^{3}(x) \\\ P_{3}^{-2}(x)&=\tfrac{1}{120}P_{3}^{2}(x) \\\ P_{3}^{-1}(x)&=-\tfrac{1}{12}P_{3}^{1}(x) \\\ P_{3}^{0}(x)&=\tfrac{1}{2}(5x^3-3x) \\\ P_{3}^{1}(x)&=\tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} \\\ P_{3}^{2}(x)&=15x(1-x^2) \\\ P_{3}^{3}(x)&=-15(1-x^2)^{3/2} \end{align} \begin{align} P_{4}^{-4}(x)&=\tfrac{1}{40320}P_{4}^{4}(x) \\\ P_{4}^{-3}(x)&=-\tfrac{1}{5040}P_{4}^{3}(x) \\\ P_{4}^{-2}(x)&=\tfrac{1}{360}P_{4}^{2}(x) \\\ P_{4}^{-1}(x)&=-\tfrac{1}{20}P_{4}^{1}(x) \\\ P_{4}^{0}(x)&=\tfrac{1}{8}(35x^{4}-30x^{2}+3) \\\ P_{4}^{1}(x)&=-\tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} \\\ P_{4}^{2}(x)&=\tfrac{15}{2}(7x^2-1)(1-x^2) \\\ P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} \\\ P_{4}^{4}(x)&=105(1-x^2)^{2} \end{align} ==Recurrence formula== These functions have a number of recurrence properties: (\ell-m-1)(\ell-m)P_{\ell}^{m}(x) = -P_{\ell}^{m+2}(x) + P_{\ell-2}^{m+2}(x) + (\ell+m)(\ell+m-1)P_{\ell-2}^{m}(x) (\ell-m+1)P_{\ell+1}^{m}(x) = (2\ell+1)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) 2mxP_{\ell}^{m}(x)=-\sqrt{1-x^2}\left[P_{\ell}^{m+1}(x)+(\ell+m)(\ell-m+1)P_{\ell}^{m-1}(x)\right] \frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell-1}^{m+1}(x) + (\ell+m-1)(\ell+m)P_{\ell-1}^{m-1}(x) \right] \frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell+1}^{m+1}(x) + (\ell-m+1)(\ell-m+2)P_{\ell+1}^{m-1}(x) \right] \sqrt{1-x^2}P_\ell^m(x) = \frac1{2\ell+1} \left[ (\ell-m+1)(\ell-m+2) P_{\ell+1}^{m-1}(x) - (\ell+m-1)(\ell+m) P_{\ell-1}^{m-1}(x) \right] \sqrt{1-x^2}P_\ell^m(x) = \frac{-1}{2\ell+1} \left[ P_{\ell+1}^{m+1}(x) - P_{\ell-1}^{m+1}(x) \right] \sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) \sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m+1)P_{\ell+1}^m(x) - (\ell+m+1)xP_\ell^m(x) \sqrt{1-x^2}\frac{d}{dx}{P_\ell^m}(x) = \frac12 \left[ (\ell+m)(\ell-m+1)P_\ell^{m-1}(x) - P_\ell^{m+1}(x) \right] (1-x^2)\frac{d}{dx}{P_\ell^m}(x) = \frac1{2\ell+1} \left[ (\ell+1)(\ell+m)P_{\ell-1}^m(x) - \ell(\ell-m+1)P_{\ell+1}^m(x) \right] (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = {\ell}xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = -(\ell+1)xP_{\ell}^{m}(x) + (\ell-m+1)P_{\ell+1}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = \sqrt{1-x^2}P_{\ell}^{m+1}(x) + mxP_{\ell}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = -(\ell+m)(\ell-m+1)\sqrt{1-x^2}P_{\ell}^{m-1}(x) - mxP_{\ell}^{m}(x) Helpful identities (initial values for the first recursion): P_{\ell +1}^{\ell +1}(x) = - (2\ell+1) \sqrt{1-x^2} P_{\ell}^{\ell}(x) P_{\ell}^{\ell}(x) = (-1)^\ell (2\ell-1)!! (1- x^2)^{(\ell/2)} P_{\ell +1}^{\ell}(x) = x (2\ell+1) P_{\ell}^{\ell}(x) with the double factorial. ==Gaunt's formula== The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. For each choice of ℓ, there are functions for the various values of m and choices of sine and cosine. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m: \int_{-1}^{1} P_k ^{m} P_\ell ^{m} dx = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell} Where is the Kronecker delta. P^{m}_\ell(x). ===Alternative notations=== The following alternative notations are also used in literature: P_{\ell m}(x) = (-1)^m P_\ell^{m}(x) ===Closed Form=== The Associated Legendre Polynomial can also be written as: P_l^m(x)=(-1)^{m} \cdot 2^{l} \cdot (1-x^2)^{m/2} \cdot \sum_{k=m}^l \frac{k!}{(k-m)!}\cdot x^{k-m} \cdot \binom{l}{k} \binom{\frac{l+k-1}{2}}{l} with simple monomials and the generalized form of the binomial coefficient. ==Orthogonality== The associated Legendre polynomials are not mutually orthogonal in general. Indeed, equate the coefficients of equal powers on the left and right hand side of \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell}, then it follows that the proportionality constant is c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} , so that P^{-m}_\ell(x) = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ",0.28209479,0.85,28.0,1.2,2.2,A
-The lowest-frequency pure-rotational absorption line of ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$ occurs at $48991.0 \mathrm{MHz}$. Find the bond distance in ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$.,"The C-S-C angles are 102° and C-S bond distance is 177 picometers. Since one atomic unit of length(i.e., a Bohr radius) is 52.9177 pm, the C–C bond length is 2.91 atomic units, or approximately three Bohr radii long. The bond lengths of these so- called ""pancake bonds"" are up to 305 pm. Shorter than average C–C bond distances are also possible: alkenes and alkynes have bond lengths of respectively 133 and 120 pm due to increased s-character of the sigma bond. The smallest theoretical C–C single bond obtained in this study is 131 pm for a hypothetical tetrahedrane derivative. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. Stretching or squeezing the same bond by 15 pm required an estimated 21.9 or 37.7 kJ/mol. Bond lengths in organic compounds C–H Length (pm) C–C Length (pm) Multiple-bonds Length (pm) sp3–H 110 sp3–sp3 154 Benzene 140 sp2–H 109 sp3–sp2 150 Alkene 134 sp–H 108 sp2–sp2 147 Alkyne 120 sp3–sp 146 Allene 130 sp2–sp 143 sp–sp 137 ==References== == External links == * Bond length tutorial Length Category:Molecular geometry Category:Length In benzene all bonds have the same length: 139 pm. Carbon–carbon single bonds increased s-character is also notable in the central bond of diacetylene (137 pm) and that of a certain tetrahedrane dimer (144 pm). Bond distance of carbon to other elements Bonded element Bond length (pm) Group H 106–112 group 1 Be 193 group 2 Mg 207 group 2 B 156 group 13 Al 224 group 13 In 216 group 13 C 120–154 group 14 Si 186 group 14 Sn 214 group 14 Pb 229 group 14 N 147–210 group 15 P 187 group 15 As 198 group 15 Sb 220 group 15 Bi 230 group 15 O 143–215 group 16 S 181–255 group 16 Cr 192 group 6 Se 198–271 group 16 Te 205 group 16 Mo 208 group 6 W 206 group 6 F 134 group 17 Cl 176 group 17 Br 193 group 17 I 213 group 17 ==Bond lengths in organic compounds== The bond length between two atoms in a molecule depends not only on the atoms but also on such factors as the orbital hybridization and the electronic and steric nature of the substituents. Bond is located between carbons C1 and C2 as depicted in a picture below. thumb|center|150px| Hexaphenylethane skeleton based derivative containing longest known C-C bond between atoms C1 and C2 with a length of 186.2 pm Another notable compound with an extraordinary C-C bond length is tricyclobutabenzene, in which a bond length of 160 pm is reported. The carbon–carbon (C–C) bond length in diamond is 154 pm. By approximation the bond distance between two different atoms is the sum of the individual covalent radii (these are given in the chemical element articles for each element). In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. In an in silico experiment a bond distance of 136 pm was estimated for neopentane locked up in fullerene. Bond lengths are given in picometers. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. By examining a large number of structures of molecular crystals, it is possible to find a minimum radius for each type of atom such that other non-bonded atoms do not encroach any closer. In this type of compound the cyclobutane ring would force 90° angles on the carbon atoms connected to the benzene ring where they ordinarily have angles of 120°. thumb|center|200px|Cyclobutabenzene with a bond length in red of 174 pm The existence of a very long C–C bond length of up to 290 pm is claimed in a dimer of two tetracyanoethylene dianions, although this concerns a 2-electron-4-center bond. Current record holder for the longest C-C bond with a length of 186.2 pm is 1,8-Bis(5-hydroxydibenzo[a,d]cycloheptatrien-5-yl)naphthalene, one of many molecules within a category of hexaaryl ethanes, which are derivatives based on hexaphenylethane skeleton. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. The 12.17×44mm RF is also known as ""12×44RF Norwegian Remington Model 1871"" and ""12.7×44RF Norwegian"". ==12.17×42mm RF and its subvariety the 12.17×44mm RF== thumb|250px|Model 1867 Remington rolling block chambered for the 12.17×42mm RF. The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2). ",+17.7,0.925,1.6,+5.41,1.5377,E
-The strongest infrared band of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$ occurs at $\widetilde{\nu}=2143 \mathrm{~cm}^{-1}$. Find the force constant of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$. ,"The C-band is located between the short wavelengths (S) band (1460–1530 nm) and the long wavelengths (L) band (1565–1625 nm). The nuclear force is powerfully attractive between nucleons at distances of about 0.8 femtometre (fm, or 0.8×10−15 metre), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. O16 or O-16 may refer to: * Curtiss O-16 Falcon, an observation aircraft of the United States Army Air Corps * Garberville Airport, in Humboldt County, California, United States * , a submarine of the Royal Netherlands Navy * Oxygen-16, an isotope of oxygen * , a submarine of the United States Navy The molecular formula for C10H12FNO (molar mass: 181.21 g/mol, exact mass: 181.0903 u) may refer to: * Flephedrone, also known as 4-fluoromethcathinone (4-FMC) * 3-Fluoromethcathinone In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase-shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. ==The nuclear force as a residual of the strong force== The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. thumb|Absorption in fiber in the range 900–1700 nm with a minimum at the C-band thumb|Transmittance of the atmosphere around the C-band In infrared optical communications, C-band (C for ""conventional"") refers to the wavelength range 1530–1565 nm, which corresponds to the amplification range of erbium doped fiber amplifiers (EDFAs). By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). The molecular formula C8H16 (molar mass: 112.21 g/mol, exact mass: 112.1252 u) may refer to: * Cyclooctane * Methylcycloheptane * Dimethylcyclohexanes * * Octenes ** 1-Octene Category:Molecular formulas thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. These nuclear forces are very weak compared to direct gluon forces (""color forces"" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. At distances less than 0.7 fm, the nuclear force becomes repulsive. The force depends on whether the spins of the nucleons are parallel or antiparallel, as it has a non-central or tensor component. After the verification of the quark model, strong interaction has come to mean QCD. ==Nucleon–nucleon potentials== Two- nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. The model also gave good predictions for the binding energy of nuclei. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). At small separations between nucleons (less than ~ 0.7 fm between their centers, depending upon spin alignment) the force becomes repulsive, which keeps the nucleons at a certain average separation. Nucleons have a radius of about 0.8 fm. A more recent approach is to develop effective field theories for a consistent description of nucleon–nucleon and three-nucleon forces. Sometimes, the nuclear force is called the residual strong force, in contrast to the strong interactions which arise from QCD. The constants are determined empirically. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Note: 1 fm = 1E-15 m. ",9,92,0.0526315789,1855,257,D
-Calculate the de Broglie wavelength for (a) an electron with a kinetic energy of $100 \mathrm{eV}$,"In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides incorrect results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above. But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. One of the methods is to use the concept of dressed particle. == See also == * Energy level * Mode (electromagnetism) == References == Category:Electron The energy released is equal to the difference in energy levels between the electron energy states. The correction to the Rydberg formula for these atoms is known as the quantum defect. ==See also== * Balmer series * Hydrogen line * Rydberg–Ritz combination principle ==References== * * Category:Atomic physics Category:Foundational quantum physics Category:Hydrogen physics Category:History of physics Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. Later models found that the values for n1 and n2 corresponded to the principal quantum numbers of the two orbitals. ==For hydrogen== \frac{1}{\lambda_{\mathrm{vac}}} = R_\text{H}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) , where *\lambda_{\mathrm{vac}} is the wavelength of electromagnetic radiation emitted in vacuum, *R_\text{H} is the Rydberg constant for hydrogen, approximately , *n_1 is the principal quantum number of an energy level, and *n_2 is the principal quantum number of an energy level for the atomic electron transition. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. When these deviations from the central trajectory are expressed in terms of the small parameters \varepsilon, \sigma defined as E_k=(1+\varepsilon)E_\textrm{P}, r_0=(1+\sigma)R_\textrm{P}, and having in mind that \alpha itself is small (of the order of 1°), the final radius of the electron's trajectory, r(\pi), can be expressed as :r_\pi\approx R_\textrm{P}(1+2\varepsilon-\sigma-2\alpha^2+2\varepsilon^2-6\alpha^2\varepsilon). Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. ", 9.73,1.01,0.123,310,228,C
+","Since the field is uniform in the direction of wire length, for a conductor having uniformly consistent resistivity ρ, the current density will also be uniform in any cross-sectional area and oriented in the direction of wire length, so we may write:Lerner L, Physics for scientists and engineers, Jones & Bartlett, 1997, pp. 732–733 J = \frac{I}{a}. In these situations, the density simplifies to :\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2. == General properties == From its definition, the electron density is a non-negative function integrating to the total number of electrons. Using the divergence theorem, the bound volume charge density within the material is q_b = \int \rho_b \, dV = -\oint_S \mathbf{P} \cdot \hat\mathbf{n} \, dS = -\int abla \cdot \mathbf{P} \, dV hence: \rho_b = - abla\cdot\mathbf{P}\,. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. The charge density of the system at a point r is a sum of the charge densities for each charge qi at position ri, where : \rho_q(\mathbf{r})=\sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) The delta function for each charge qi in the sum, δ(r − ri), ensures the integral of charge density over R returns the total charge in R: Q = \int_R d^3 \mathbf{r} \sum_{i=1}^N\ q_i\delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i \int_R d^3 \mathbf{r} \delta(\mathbf{r} - \mathbf{r}_i) = \sum_{i=1}^N\ q_i If all charge carriers have the same charge q (for electrons q = −e, the electron charge) the charge density can be expressed through the number of charge carriers per unit volume, n(r), by \rho_q(\mathbf{r}) = q n(\mathbf{r})\,. The total charge divided by the length, surface area, or volume will be the average charge densities: \langle\lambda_q \rangle = \frac{Q}{\ell}\,,\quad \langle\sigma_q\rangle = \frac{Q}{S}\,,\quad\langle\rho_q\rangle = \frac{Q}{V}\,. == Free, bound and total charge == In dielectric materials, the total charge of an object can be separated into ""free"" and ""bound"" charges. At position at time , the distribution of charge flowing is described by the current density: \mathbf{j}(\mathbf{r}, t) = \rho(\mathbf{r},t) \; \mathbf{v}_\text{d} (\mathbf{r},t) where * is the current density vector; * is the particles' average drift velocity (SI unit: m∙s−1); *\rho(\mathbf{r}, t) = q \, n(\mathbf{r},t) is the charge density (SI unit: coulombs per cubic metre), in which ** is the number of particles per unit volume (""number density"") (SI unit: m−3); ** is the charge of the individual particles with density (SI unit: coulombs). This charge is equal to d q = \rho \, v\, dt \, dA, where is the charge density at . It turns out the charge density ρ and current density J transform together as a four-current vector under Lorentz transformations. == Charge density in quantum mechanics== In quantum mechanics, charge density ρq is related to wavefunction ψ(r) by the equation \rho_q(\mathbf{r}) = q |\psi(\mathbf r)|^2 where q is the charge of the particle and is the probability density function i.e. probability per unit volume of a particle located at r. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Charge density can be either positive or negative, since electric charge can be either positive or negative. In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. As always, the integral of the charge density over a region of space is the charge contained in that region. In quantum chemical calculations, the electron density, ρ(r), is a function of the coordinates r, defined so ρ(r)dr is the number of electrons in a small volume dr. The linear charge density is the ratio of an infinitesimal electric charge dQ (SI unit: C) to an infinitesimal line element, \lambda_q = \frac{d Q}{d \ell}\,, similarly the surface charge density uses a surface area element dS \sigma_q = \frac{d Q}{d S}\,, and the volume charge density uses a volume element dV \rho_q =\frac{d Q}{d V} \, , Integrating the definitions gives the total charge Q of a region according to line integral of the linear charge density λq(r) over a line or 1d curve C, Q = \int_L \lambda_q(\mathbf{r}) \, d\ell similarly a surface integral of the surface charge density σq(r) over a surface S, Q = \int_S \sigma_q(\mathbf{r}) \, dS and a volume integral of the volume charge density ρq(r) over a volume V, Q = \int_V \rho_q(\mathbf{r}) \, dV where the subscript q is to clarify that the density is for electric charge, not other densities like mass density, number density, probability density, and prevent conflict with the many other uses of λ, σ, ρ in electromagnetism for wavelength, electrical resistivity and conductivity. The charge carrier density in a conductor is equal to the number of mobile charge carriers (electrons, ions, etc.) per unit volume. The electric current is dI = dq/dt = \rho v dA, it follows that the current density vector is the vector normal dA (i.e. parallel to ) and of magnitude dI/dA = \rho v \mathbf{j} = \rho \mathbf{v}. The electrical resistance of a uniform conductor is given in terms of resistivity by: {R} = \rho \frac{\ell}{a} where ℓ is the length of the conductor in SI units of meters, is the cross-sectional area (for a round wire if is radius) in units of meters squared, and ρ is the resistivity in units of ohm·meters. If (SI unit: A) is the electric current flowing through , then electric current density at is given by the limit: j = \lim_{A \to 0} \frac{I_A}{A} = \left.\frac{\partial I}{\partial A} \right|_{A=0}, with surface remaining centered at and orthogonal to the motion of the charges during the limit process. The delta function has the sifting property for any function f: \int_R d^3 \mathbf{r} f(\mathbf{r})\delta(\mathbf{r} - \mathbf{r}_0) = f(\mathbf{r}_0) so the delta function ensures that when the charge density is integrated over R, the total charge in R is q: Q =\int_R d^3 \mathbf{r} \, \rho_q =\int_R d^3 \mathbf{r} \, q \delta(\mathbf{r} - \mathbf{r}_0) = q \int_R d^3 \mathbf{r} \, \delta(\mathbf{r} - \mathbf{r}_0) = q This can be extended to N discrete point-like charge carriers. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. ",2.00,122,"""10.4""",0, -31.95,A
+"A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured.  (b) Find the probability that the measured value is between $x=0$ and $x=2 \mathrm{~nm}$.","Hence, at a given time , is the probability density function of the particle's position. If the oscillator spends an infinitesimal amount of time in the vicinity of a given -value, then the probability of being in that vicinity will be :P(x)\, dx \propto dt. If it corresponds to a non-degenerate eigenvalue of , then |\psi (x)|^2 gives the probability of the corresponding value of for the initial state . Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. Thus the probability that the particle is in the volume at is :\mathbf{P}(V)=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}} = \int_V |\psi(\mathbf {x}, t_0)|^2\, \mathrm{d\mathbf {x}}. If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of ) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. The probability of measuring |u\rangle is given by :P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} Which agrees with experiment. ==Normalization== In the example above, the measurement must give either or , so the total probability of measuring or must be 1. In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. Plugging this into the expression for yields :P(x) = \frac{1}{\pi}\frac{1}{\sqrt{A^2-x^2}}. Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position , for time ). Note that if any solution to the wave equation is normalisable at some time , then the defined above is always normalised, so that :\rho_t(\mathbf x)=\left |\psi(\mathbf x, t)\right |^2 = \left|\frac{\psi_0(\mathbf x, t)}{a}\right|^2 is always a probability density function for all . Plugging this into our expression for yields :P(x) = \frac{1}{T} \sqrt{\frac{2m}{E-U(x)}}. If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of , then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In this system, all momenta are equally probable. ==See also== *Probability density function *Correspondence principle *Classical limit *Wave function ==References== Category:Concepts in physics Category:Classical mechanics Category:Theoretical physics Then \psi (x) is the ""probability amplitude"" for the eigenstate . Once this is done, is readily obtained for any allowed energy . ==Examples== ===Simple harmonic oscillator=== thumb|300px|right|The probability density function of the state of the quantum harmonic oscillator. Therefore, if the system is known to be in some eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. Following a similar argument as above, the result is :P(p) = \frac{2}{T}\frac{1}{|F(x)|}, where is the force acting on the particle as a function of position. ", 7.42,0.4908,"""252.8""",0,0.23333333333,B
+Calculate the ground-state energy of the hydrogen atom using SI units and convert the result to electronvolts.,"To convert from ""value of ionization energy"" to the corresponding ""value of molar ionization energy"", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. ""use"" and ""WEL"" give ionization energy in the unit kJ/mol; ""CRC"" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. The Prout is an obsolete unit of energy, whose value is: 1 Prout = 2.9638 \times 10^{-14} J This is equal to one twelfth of the binding energy of the deuteron. This page shows the electron configurations of the neutral gaseous atoms in their ground states. This website is also cited in the CRC Handbook as source of Section 1, subsection Electron Configuration of Neutral Atoms in the Ground State. *91 Pa : [Rn] 5f2(3H4) 6d 7s2 *92 U : [Rn] 5f3(4Io9/2) 6d 7s2 *93 Np : [Rn] 5f4(5I4) 6d 7s2 *103 Lr : [Rn] 5f14 7s2 7p1 question-marked *104 Rf : [Rn] 5f14 6d2 7s2 question-marked ===CRC=== *David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition, online version. After ionization, the electron and proton recombine to form a new hydrogen atom. Boca Raton, Florida, 2003; Section 1, Basic Constants, Units, and Conversion Factors; Electron Configuration of Neutral Atoms in the Ground State. (elements 1-104) *Also subsection Periodic Table of the Elements, (elements 1-103) based on: **G. J. Leigh, Editor, Nomenclature of Inorganic Chemistry, Blackwell Scientific Publications, Oxford, 1990. **Atomic Weights of the Elements, 1999, Pure Appl. Chem., 73, 667, 2001. ===WebElements=== *http://www.webelements.com/ ; retrieved July 2005, electron configurations based on: **Atomic, Molecular, & Optical Physics Handbook, Ed. alt=|thumb|upright=1.2|Hemispherical electron energy analyzer. Therefore, the H-alpha line occurs where hydrogen is being ionized. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. These tables list values of molar ionization energies, measured in kJ⋅mol−1. (Elements 1-106) *58 Ce : [Xe] 4f2 6s2 *103 Lr : [Rn] 5f14 6d1 7s2 *104 Rf : [Rn] 5f14 6d2 7s2 (agrees with guess above) *105 Db : [Rn] 5f14 6d3 7s2 *106 Sg : [Rn] 5f14 6d4 7s2 ===Hoffman, Lee, and Pershina=== This book contains predicted electron configurations for the elements up to 172, as well as 184, based on relativistic Dirac–Fock calculations by B. Fricke in * Category:Chemical element data pages * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. == External links == * NIST Atomic Spectra Database Ionization Energies == See also == *Molar ionization energies of the elements Category:Properties of chemical elements Category:Chemical element data pages The first molar ionization energy applies to the neutral atoms. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. Note that these electron configurations are given for neutral atoms in the gas phase, which are not the same as the electron configurations for the same atoms in chemical environments. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. ",2500,0.18162,"""1.51""",15.757,-13.598 ,E
+Find the probability that the electron in the ground-state $\mathrm{H}$ atom is less than a distance $a$ from the nucleus.,"The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. In the simplest atom, hydrogen, a single electron orbits the nucleus, and its smallest possible orbit, with the lowest energy, has an orbital radius almost equal to the Bohr radius. In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. * Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913). The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. The expected value of the radial distance of the electron, by contrast, . == Related constants == The Bohr radius is one of a trio of related units of length, the other two being the reduced Compton wavelength of the electron ( \lambda_{\mathrm{e}} / 2\pi ) and the classical electron radius ( r_{\mathrm{e}} ). H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. An atom is composed of a positively charged nucleus, with a cloud of negatively charged electrons surrounding it, bound together by electrostatic force. The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. This is because there are more energy levels and therefore a greater distance between protons and electrons. The value of the radius may depend on the atom's state and context. The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The electric repulsion between each pair of protons in a nucleus contributes toward decreasing its binding energy. The Bohr model of the atom was superseded by an electron probability cloud obeying the Schrödinger equation as published in 1926. Therefore, the radius of an atom is more than 10,000 times the radius of its nucleus (1–10 fm), and less than 1/1000 of the wavelength of visible light (400–700 nm). ",5300,0.23333333333,"""0.19""",0.323,9,D
+"A one-particle, one-dimensional system has $\Psi=a^{-1 / 2} e^{-|x| / a}$ at $t=0$, where $a=1.0000 \mathrm{~nm}$. At $t=0$, the particle's position is measured. (a) Find the probability that the measured value lies between $x=1.5000 \mathrm{~nm}$ and $x=1.5001 \mathrm{~nm}$.","Hence, at a given time , is the probability density function of the particle's position. We may use the Boltzmann distribution to find this probability that is, as we have seen, equal to the fraction of particles that are in state i. If it corresponds to a non-degenerate eigenvalue of , then |\psi (x)|^2 gives the probability of the corresponding value of for the initial state . Thus the probability that the particle is in the volume at is :\mathbf{P}(V)=\int_V \rho(\mathbf {x})\, \mathrm{d\mathbf {x}} = \int_V |\psi(\mathbf {x}, t_0)|^2\, \mathrm{d\mathbf {x}}. Under the standard Copenhagen interpretation, the normalized wavefunction gives probability amplitudes for the position of the particle. If the system is known to be in some eigenstate of (e.g. after an observation of the corresponding eigenvalue of ) the probability of observing that eigenvalue becomes equal to 1 (certain) for all subsequent measurements of (so long as no other important forces act between the measurements). The probability of measuring |u\rangle is given by :P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} Which agrees with experiment. ==Normalization== In the example above, the measurement must give either or , so the total probability of measuring or must be 1. In other words the probability amplitudes are zero for all the other eigenstates, and remain zero for the future measurements. Note that if any solution to the wave equation is normalisable at some time , then the defined above is always normalised, so that :\rho_t(\mathbf x)=\left |\psi(\mathbf x, t)\right |^2 = \left|\frac{\psi_0(\mathbf x, t)}{a}\right|^2 is always a probability density function for all . Suppose a wavefunction is a solution of the wave equation, giving a description of the particle (position , for time ). If the set of eigenstates to which the system can jump upon measurement of is the same as the set of eigenstates for measurement of , then subsequent measurements of either or always produce the same values with probability of 1, no matter the order in which they are applied. The distribution is expressed in the form: :p_i \propto \exp\left(- \frac{\varepsilon_i}{kT} \right) where is the probability of the system being in state , is the exponential function, is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . Then \psi (x) is the ""probability amplitude"" for the eigenstate . Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Generally, it is the case when the motion of a particle is described in the position space, where the corresponding probability amplitude function is the wave function. Indeed, which of the above eigenstates the system jumps to is given by a probabilistic law: the probability of the system jumping to the state is proportional to the absolute value of the corresponding numerical weight squared. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In this case, if the vector has the norm 1, then is just the probability that the quantum system resides in the state . The difference of a density function from simply a numerical probability means that one should integrate this modulus-squared function over some (small) domains in to obtain probability values – as was stated above, the system can't be in some state with a positive probability. Therefore, if the system is known to be in some eigenstate of (all probability amplitudes zero except for one eigenstate), then when is observed the probability amplitudes are changed. upright=1.75|right|thumb|Boltzmann's distribution is an exponential distribution. upright=1.75|right|thumb|Boltzmann factor (vertical axis) as a function of temperature for several energy differences . If we have a system consisting of many particles, the probability of a particle being in state is practically the probability that, if we pick a random particle from that system and check what state it is in, we will find it is in state . ",1.88,234.4,"""-4.37""",22,4.979,E
+"In this example, $2.50 \mathrm{~mol}$ of an ideal gas with $C_{V, m}=12.47 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ is expanded adiabatically against a constant external pressure of 1.00 bar. The initial temperature and pressure of the gas are $325 \mathrm{~K}$ and $2.50 \mathrm{bar}$, respectively. The final pressure is 1.25 bar. Calculate the final temperature, $q, w, \Delta U$.","The adiabatic constant remains the same, but with the resulting pressure unknown : P_2 V_2^\gamma = \mathrm{constant}_1 = 6.31~\text{Pa}\,\text{m}^{21/5} = P \times (0.0001~\text{m}^3)^\frac75, We can now solve for the final pressure : P_2 = P_1\left (\frac{V_1}{V_2}\right)^\gamma = 100\,000~\text{Pa} \times \text{10}^{7/5} = 2.51 \times 10^6~\text{Pa} or 25.1 bar. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cm3 = 0.001 m3); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so ); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure). : \begin{align} & P_1 V_1^\gamma = \mathrm{constant}_1 = 100\,000~\text{Pa} \times (0.001~\text{m}^3)^\frac75 \\\ & = 10^5 \times 6.31 \times 10^{-5}~\text{Pa}\,\text{m}^{21/5} = 6.31~\text{Pa}\,\text{m}^{21/5}, \end{align} so the adiabatic constant for this example is about 6.31 Pa m4.2. Adiabatic heating occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (b4) gives : P = P_1 \left(\frac{V_1}{V} \right)^\gamma. For such an adiabatic process, the modulus of elasticity (Young's modulus) can be expressed as , where is the ratio of specific heats at constant pressure and at constant volume () and is the pressure of the gas. === Various applications of the adiabatic assumption === For a closed system, one may write the first law of thermodynamics as , where denotes the change of the system's internal energy, the quantity of energy added to it as heat, and the work done by the system on its surroundings. The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. In addition, through the use of the Euler chain relation it can be shown that \left ( \frac{\partial U}{\partial V} \right )_T = - \left ( \frac{\partial U}{\partial T} \right )_V \left ( \frac{\partial T}{\partial V} \right )_U Defining \mu_J = \left ( \frac{\partial T}{\partial V} \right )_U as the ""Joule coefficient"" J. Westin, A Course in Thermodynamics, Volume 1, Taylor and Francis, New York (1979). and recognizing \left ( \frac{\partial U}{\partial T} \right )_V as the heat capacity at constant volume = C_V , we have \pi_T = - C_V \mu_J The coefficient \mu_J can be obtained by measuring the temperature change for a constant-U experiment, i.e., an adiabatic free expansion (see below). The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient ""adiabatic approximation"".Bailyn, M. (1994), pp. 52–53. We know the compressed gas has = 0.1 L and = , so we can solve for temperature: : T = \frac{P V}{\mathrm{constant}_2} = \frac{2.51 \times 10^6~\text{Pa} \times 10^{-4}~\text{m}^3}{0.333~\text{Pa}\,\text{m}^3\text{K}^{-1}} = 753~\text{K}. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible. ===Derivation of P–V relation for adiabatic heating and cooling=== The definition of an adiabatic process is that heat transfer to the system is zero, . For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to: : dh = v \, dp. *If the system walls are adiabatic () but not rigid (), and, in a fictive idealized process, energy is added to the system in the form of frictionless, non-viscous pressure–volume work (), and there is no phase change, then the temperature of the system will rise. Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. In the constant volume adiabatic flame temperature case, the volume of the system is held constant and hence there is no work occurring: : {}_RW_P = \int\limits_R^P {pdV} = 0 There is also no heat transfer because the process is defined to be adiabatic: {}_RQ_P = 0 . For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame temperature by assuming combustion loses no heat to its surroundings. Simplifying, : T_2 - T_1 = T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right), : \frac{T_2}{T_1} - 1 = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1, : T_2 = T_1 \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}}. ===Derivation of discrete formula and work expression=== The change in internal energy of a system, measured from state 1 to state 2, is equal to : At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (c4) gives : P = P_1 \left(\frac{V_1}{V} \right)^\gamma. Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant (nR) is: : \frac{PV}{T} = \mathrm{constant}_2 = \frac{10^5~\text{Pa} \times 10^{-3}~\text{m}^3}{300~\text{K}} = 0.333~\text{Pa}\,\text{m}^3\text{K}^{-1}. ",+3.03,+2.9,"""46.7""",92,-1.78,E
+"Find $Y_l^m(\theta, \phi)$  for  $l=0$.","The solutions are usually written in terms of complex exponentials: Y_{\ell, m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}}\ Y_{\ell, m}^*(\theta, \phi) = (-1)^m Y_{\ell, -m}(\theta, \phi). The functions Y_{\ell, m}(\theta, \phi) are the spherical harmonics, and the quantity in the square root is a normalizing factor. P_\ell^{m}(\cos \theta)\ e^{im\phi}\qquad -\ell \le m \le \ell. When the partial differential equation \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2} + \lambda \psi = 0 is solved by the method of separation of variables, one gets a φ-dependent part \sin(m\phi) or \cos(m\phi) for integer m≥0, and an equation for the θ-dependent part \frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\, for which the solutions are P_\ell^{m}(\cos \theta) with \ell{\ge}m and \lambda = \ell(\ell+1). Therefore, the equation abla^2\psi + \lambda\psi = 0 has nonsingular separated solutions only when \lambda = \ell(\ell+1), and those solutions are proportional to P_\ell^{m}(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell and P_\ell^{m}(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 < m \le \ell. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for :. \left(1-x^2\right) \frac{d^2}{dx^2}P_\ell(x) -2x\frac{d}{dx}P_\ell(x)+ \ell(\ell+1)P_\ell(x) = 0. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is abla^2\psi = \frac{\partial^2\psi}{\partial\theta^2} + \cot \theta \frac{\partial \psi}{\partial \theta} + \csc^2 \theta\frac{\partial^2\psi}{\partial\phi^2}. The longitude angle, \phi, appears in a multiplying factor. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. P_\ell ^{m} (This followed from the Rodrigues' formula definition. L.M.L. is the second English album and fifth overall studio album by Nu Virgos. == Content == The title of the album comes from the song in the album titled ""L.M.L."". == Release == The album was released in Russia on September 13, 2007, and in Asia on September 19, 2007. In mathematics and computing, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method, is used to solve non-linear least squares problems. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. It is also commonly denoted as zn(u,k) :\Theta(u)=\Theta_{4}\left(\frac{\pi u}{2K}\right) :Z(u)=\frac{\partial}{\partial u}\ln\Theta(u) =\frac{\Theta'(u)}{\Theta(u)} :Z(\phi|m)=E(\phi|m)-\frac{E(m)}{K(m)}F(\phi|m) :Where E, K, and F are generic Incomplete Elliptical Integrals of the first and second kind. Moreover, since by Rodrigues' formula, P_\ell(x) = \frac{1}{2^\ell\,\ell!} \ \frac{d^\ell}{dx^\ell}\left[(x^2-1)^\ell\right], the P can be expressed in the form P_\ell^{m}(x) = \frac{(-1)^m}{2^\ell \ell!} (1-x^2)^{m/2}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell. This definition also makes the various recurrence formulas work for positive or negative .) \text{If}\quad |m| > \ell\,\quad\text{then}\quad P_\ell^{m} = 0.\, The differential equation is also invariant under a change from to , and the functions for negative are defined by P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, \dots). ==Parity== From their definition, one can verify that the Associated Legendre functions are either even or odd according to P_\ell ^{m} (-x) = (-1)^{\ell + m} P_\ell ^{m}(x) ==The first few associated Legendre functions== thumb|300px|Associated Legendre functions for m = 0 thumb|300px|Associated Legendre functions for m = 1 thumb|300px|Associated Legendre functions for m = 2 The first few associated Legendre functions, including those for negative values of m, are: P_{0}^{0}(x)=1 \begin{align} P_{1}^{-1}(x)&=-\tfrac{1}{2}P_{1}^{1}(x) \\\ P_{1}^{0}(x)&=x \\\ P_{1}^{1}(x)&=-(1-x^2)^{1/2} \end{align} \begin{align} P_{2}^{-2}(x)&=\tfrac{1}{24}P_{2}^{2}(x) \\\ P_{2}^{-1}(x)&=-\tfrac{1}{6}P_{2}^{1}(x) \\\ P_{2}^{0}(x)&=\tfrac{1}{2}(3x^{2}-1) \\\ P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} \\\ P_{2}^{2}(x)&=3(1-x^2) \end{align} \begin{align} P_{3}^{-3}(x)&=-\tfrac{1}{720}P_{3}^{3}(x) \\\ P_{3}^{-2}(x)&=\tfrac{1}{120}P_{3}^{2}(x) \\\ P_{3}^{-1}(x)&=-\tfrac{1}{12}P_{3}^{1}(x) \\\ P_{3}^{0}(x)&=\tfrac{1}{2}(5x^3-3x) \\\ P_{3}^{1}(x)&=\tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} \\\ P_{3}^{2}(x)&=15x(1-x^2) \\\ P_{3}^{3}(x)&=-15(1-x^2)^{3/2} \end{align} \begin{align} P_{4}^{-4}(x)&=\tfrac{1}{40320}P_{4}^{4}(x) \\\ P_{4}^{-3}(x)&=-\tfrac{1}{5040}P_{4}^{3}(x) \\\ P_{4}^{-2}(x)&=\tfrac{1}{360}P_{4}^{2}(x) \\\ P_{4}^{-1}(x)&=-\tfrac{1}{20}P_{4}^{1}(x) \\\ P_{4}^{0}(x)&=\tfrac{1}{8}(35x^{4}-30x^{2}+3) \\\ P_{4}^{1}(x)&=-\tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} \\\ P_{4}^{2}(x)&=\tfrac{15}{2}(7x^2-1)(1-x^2) \\\ P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} \\\ P_{4}^{4}(x)&=105(1-x^2)^{2} \end{align} ==Recurrence formula== These functions have a number of recurrence properties: (\ell-m-1)(\ell-m)P_{\ell}^{m}(x) = -P_{\ell}^{m+2}(x) + P_{\ell-2}^{m+2}(x) + (\ell+m)(\ell+m-1)P_{\ell-2}^{m}(x) (\ell-m+1)P_{\ell+1}^{m}(x) = (2\ell+1)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) 2mxP_{\ell}^{m}(x)=-\sqrt{1-x^2}\left[P_{\ell}^{m+1}(x)+(\ell+m)(\ell-m+1)P_{\ell}^{m-1}(x)\right] \frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell-1}^{m+1}(x) + (\ell+m-1)(\ell+m)P_{\ell-1}^{m-1}(x) \right] \frac{1}{\sqrt{1-x^2}}P_\ell^m(x) = \frac{-1}{2m} \left[ P_{\ell+1}^{m+1}(x) + (\ell-m+1)(\ell-m+2)P_{\ell+1}^{m-1}(x) \right] \sqrt{1-x^2}P_\ell^m(x) = \frac1{2\ell+1} \left[ (\ell-m+1)(\ell-m+2) P_{\ell+1}^{m-1}(x) - (\ell+m-1)(\ell+m) P_{\ell-1}^{m-1}(x) \right] \sqrt{1-x^2}P_\ell^m(x) = \frac{-1}{2\ell+1} \left[ P_{\ell+1}^{m+1}(x) - P_{\ell-1}^{m+1}(x) \right] \sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m)xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) \sqrt{1-x^2}P_\ell^{m+1}(x) = (\ell-m+1)P_{\ell+1}^m(x) - (\ell+m+1)xP_\ell^m(x) \sqrt{1-x^2}\frac{d}{dx}{P_\ell^m}(x) = \frac12 \left[ (\ell+m)(\ell-m+1)P_\ell^{m-1}(x) - P_\ell^{m+1}(x) \right] (1-x^2)\frac{d}{dx}{P_\ell^m}(x) = \frac1{2\ell+1} \left[ (\ell+1)(\ell+m)P_{\ell-1}^m(x) - \ell(\ell-m+1)P_{\ell+1}^m(x) \right] (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = {\ell}xP_{\ell}^{m}(x) - (\ell+m)P_{\ell-1}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = -(\ell+1)xP_{\ell}^{m}(x) + (\ell-m+1)P_{\ell+1}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = \sqrt{1-x^2}P_{\ell}^{m+1}(x) + mxP_{\ell}^{m}(x) (x^2-1)\frac{d}{dx}{P_{\ell}^{m}}(x) = -(\ell+m)(\ell-m+1)\sqrt{1-x^2}P_{\ell}^{m-1}(x) - mxP_{\ell}^{m}(x) Helpful identities (initial values for the first recursion): P_{\ell +1}^{\ell +1}(x) = - (2\ell+1) \sqrt{1-x^2} P_{\ell}^{\ell}(x) P_{\ell}^{\ell}(x) = (-1)^\ell (2\ell-1)!! (1- x^2)^{(\ell/2)} P_{\ell +1}^{\ell}(x) = x (2\ell+1) P_{\ell}^{\ell}(x) with the double factorial. ==Gaunt's formula== The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. For each choice of ℓ, there are functions for the various values of m and choices of sine and cosine. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m: \int_{-1}^{1} P_k ^{m} P_\ell ^{m} dx = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell} Where is the Kronecker delta. P^{m}_\ell(x). ===Alternative notations=== The following alternative notations are also used in literature: P_{\ell m}(x) = (-1)^m P_\ell^{m}(x) ===Closed Form=== The Associated Legendre Polynomial can also be written as: P_l^m(x)=(-1)^{m} \cdot 2^{l} \cdot (1-x^2)^{m/2} \cdot \sum_{k=m}^l \frac{k!}{(k-m)!}\cdot x^{k-m} \cdot \binom{l}{k} \binom{\frac{l+k-1}{2}}{l} with simple monomials and the generalized form of the binomial coefficient. ==Orthogonality== The associated Legendre polynomials are not mutually orthogonal in general. Indeed, equate the coefficients of equal powers on the left and right hand side of \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell}, then it follows that the proportionality constant is c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} , so that P^{-m}_\ell(x) = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ",0.28209479,0.85,"""28.0""",1.2,2.2,A
+The lowest-frequency pure-rotational absorption line of ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$ occurs at $48991.0 \mathrm{MHz}$. Find the bond distance in ${ }^{12} \mathrm{C}^{32} \mathrm{~S}$.,"The C-S-C angles are 102° and C-S bond distance is 177 picometers. Since one atomic unit of length(i.e., a Bohr radius) is 52.9177 pm, the C–C bond length is 2.91 atomic units, or approximately three Bohr radii long. The bond lengths of these so- called ""pancake bonds"" are up to 305 pm. Shorter than average C–C bond distances are also possible: alkenes and alkynes have bond lengths of respectively 133 and 120 pm due to increased s-character of the sigma bond. The smallest theoretical C–C single bond obtained in this study is 131 pm for a hypothetical tetrahedrane derivative. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. Stretching or squeezing the same bond by 15 pm required an estimated 21.9 or 37.7 kJ/mol. Bond lengths in organic compounds C–H Length (pm) C–C Length (pm) Multiple-bonds Length (pm) sp3–H 110 sp3–sp3 154 Benzene 140 sp2–H 109 sp3–sp2 150 Alkene 134 sp–H 108 sp2–sp2 147 Alkyne 120 sp3–sp 146 Allene 130 sp2–sp 143 sp–sp 137 ==References== == External links == * Bond length tutorial Length Category:Molecular geometry Category:Length In benzene all bonds have the same length: 139 pm. Carbon–carbon single bonds increased s-character is also notable in the central bond of diacetylene (137 pm) and that of a certain tetrahedrane dimer (144 pm). Bond distance of carbon to other elements Bonded element Bond length (pm) Group H 106–112 group 1 Be 193 group 2 Mg 207 group 2 B 156 group 13 Al 224 group 13 In 216 group 13 C 120–154 group 14 Si 186 group 14 Sn 214 group 14 Pb 229 group 14 N 147–210 group 15 P 187 group 15 As 198 group 15 Sb 220 group 15 Bi 230 group 15 O 143–215 group 16 S 181–255 group 16 Cr 192 group 6 Se 198–271 group 16 Te 205 group 16 Mo 208 group 6 W 206 group 6 F 134 group 17 Cl 176 group 17 Br 193 group 17 I 213 group 17 ==Bond lengths in organic compounds== The bond length between two atoms in a molecule depends not only on the atoms but also on such factors as the orbital hybridization and the electronic and steric nature of the substituents. Bond is located between carbons C1 and C2 as depicted in a picture below. thumb|center|150px| Hexaphenylethane skeleton based derivative containing longest known C-C bond between atoms C1 and C2 with a length of 186.2 pm Another notable compound with an extraordinary C-C bond length is tricyclobutabenzene, in which a bond length of 160 pm is reported. The carbon–carbon (C–C) bond length in diamond is 154 pm. By approximation the bond distance between two different atoms is the sum of the individual covalent radii (these are given in the chemical element articles for each element). In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. In an in silico experiment a bond distance of 136 pm was estimated for neopentane locked up in fullerene. Bond lengths are given in picometers. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. By examining a large number of structures of molecular crystals, it is possible to find a minimum radius for each type of atom such that other non-bonded atoms do not encroach any closer. In this type of compound the cyclobutane ring would force 90° angles on the carbon atoms connected to the benzene ring where they ordinarily have angles of 120°. thumb|center|200px|Cyclobutabenzene with a bond length in red of 174 pm The existence of a very long C–C bond length of up to 290 pm is claimed in a dimer of two tetracyanoethylene dianions, although this concerns a 2-electron-4-center bond. Current record holder for the longest C-C bond with a length of 186.2 pm is 1,8-Bis(5-hydroxydibenzo[a,d]cycloheptatrien-5-yl)naphthalene, one of many molecules within a category of hexaaryl ethanes, which are derivatives based on hexaphenylethane skeleton. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. The 12.17×44mm RF is also known as ""12×44RF Norwegian Remington Model 1871"" and ""12.7×44RF Norwegian"". ==12.17×42mm RF and its subvariety the 12.17×44mm RF== thumb|250px|Model 1867 Remington rolling block chambered for the 12.17×42mm RF. The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2). ",+17.7,0.925,"""1.6""",+5.41,1.5377,E
+The strongest infrared band of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$ occurs at $\widetilde{\nu}=2143 \mathrm{~cm}^{-1}$. Find the force constant of ${ }^{12} \mathrm{C}^{16} \mathrm{O}$. ,"The C-band is located between the short wavelengths (S) band (1460–1530 nm) and the long wavelengths (L) band (1565–1625 nm). The nuclear force is powerfully attractive between nucleons at distances of about 0.8 femtometre (fm, or 0.8×10−15 metre), but it rapidly decreases to insignificance at distances beyond about 2.5 fm. O16 or O-16 may refer to: * Curtiss O-16 Falcon, an observation aircraft of the United States Army Air Corps * Garberville Airport, in Humboldt County, California, United States * , a submarine of the Royal Netherlands Navy * Oxygen-16, an isotope of oxygen * , a submarine of the United States Navy The molecular formula for C10H12FNO (molar mass: 181.21 g/mol, exact mass: 181.0903 u) may refer to: * Flephedrone, also known as 4-fluoromethcathinone (4-FMC) * 3-Fluoromethcathinone In recent years, experimenters have concentrated on the subtleties of the nuclear force, such as its charge dependence, the precise value of the πNN coupling constant, improved phase-shift analysis, high-precision NN data, high-precision NN potentials, NN scattering at intermediate and high energies, and attempts to derive the nuclear force from QCD. ==The nuclear force as a residual of the strong force== The nuclear force is a residual effect of the more fundamental strong force, or strong interaction. thumb|Absorption in fiber in the range 900–1700 nm with a minimum at the C-band thumb|Transmittance of the atmosphere around the C-band In infrared optical communications, C-band (C for ""conventional"") refers to the wavelength range 1530–1565 nm, which corresponds to the amplification range of erbium doped fiber amplifiers (EDFAs). By this new model, the nuclear force, resulting from the exchange of mesons between neighboring nucleons, is a residual effect of the strong force. ==Description== thumb|Comparison between the Nuclear Force and the Coulomb Force. a - residual strong force (nuclear force), rapidly decreases to insignificance at distances beyond about 2.5 fm, b - at distances less than ~ 0.7 fm between nucleons centers the nuclear force becomes repulsive, c - coulomb repulsion force between two protons (over 3 fm force becomes the main), d - equilibrium position for proton - proton, r - radius of a nucleon (a cloud composed of three quarks). The molecular formula C8H16 (molar mass: 112.21 g/mol, exact mass: 112.1252 u) may refer to: * Cyclooctane * Methylcycloheptane * Dimethylcyclohexanes * * Octenes ** 1-Octene Category:Molecular formulas thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. These nuclear forces are very weak compared to direct gluon forces (""color forces"" or strong forces) inside nucleons, and the nuclear forces extend only over a few nuclear diameters, falling exponentially with distance. At distances less than 0.7 fm, the nuclear force becomes repulsive. The force depends on whether the spins of the nucleons are parallel or antiparallel, as it has a non-central or tensor component. After the verification of the quark model, strong interaction has come to mean QCD. ==Nucleon–nucleon potentials== Two- nucleon systems such as the deuteron, the nucleus of a deuterium atom, as well as proton–proton or neutron–proton scattering are ideal for studying the NN force. The model also gave good predictions for the binding energy of nuclei. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). At small separations between nucleons (less than ~ 0.7 fm between their centers, depending upon spin alignment) the force becomes repulsive, which keeps the nucleons at a certain average separation. Nucleons have a radius of about 0.8 fm. A more recent approach is to develop effective field theories for a consistent description of nucleon–nucleon and three-nucleon forces. Sometimes, the nuclear force is called the residual strong force, in contrast to the strong interactions which arise from QCD. The constants are determined empirically. The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between the protons and neutrons of atoms. Note: 1 fm = 1E-15 m. ",9,92,"""0.0526315789""",1855,257,D
+Calculate the de Broglie wavelength for (a) an electron with a kinetic energy of $100 \mathrm{eV}$,"In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides incorrect results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above. But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. One of the methods is to use the concept of dressed particle. == See also == * Energy level * Mode (electromagnetism) == References == Category:Electron The energy released is equal to the difference in energy levels between the electron energy states. The correction to the Rydberg formula for these atoms is known as the quantum defect. ==See also== * Balmer series * Hydrogen line * Rydberg–Ritz combination principle ==References== * * Category:Atomic physics Category:Foundational quantum physics Category:Hydrogen physics Category:History of physics Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. Later models found that the values for n1 and n2 corresponded to the principal quantum numbers of the two orbitals. ==For hydrogen== \frac{1}{\lambda_{\mathrm{vac}}} = R_\text{H}\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) , where *\lambda_{\mathrm{vac}} is the wavelength of electromagnetic radiation emitted in vacuum, *R_\text{H} is the Rydberg constant for hydrogen, approximately , *n_1 is the principal quantum number of an energy level, and *n_2 is the principal quantum number of an energy level for the atomic electron transition. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. When these deviations from the central trajectory are expressed in terms of the small parameters \varepsilon, \sigma defined as E_k=(1+\varepsilon)E_\textrm{P}, r_0=(1+\sigma)R_\textrm{P}, and having in mind that \alpha itself is small (of the order of 1°), the final radius of the electron's trajectory, r(\pi), can be expressed as :r_\pi\approx R_\textrm{P}(1+2\varepsilon-\sigma-2\alpha^2+2\varepsilon^2-6\alpha^2\varepsilon). Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. ", 9.73,1.01,"""0.123""",310,228,C
 "The threshold wavelength for potassium metal is $564 \mathrm{~nm}$. What is its work function? 
-","alt=Potassium ferricyanide milled|thumb|Potassium ferricyanide when milled has lighter color Potassium ferricyanide is the chemical compound with the formula K3[Fe(CN)6]. Potassium is the chemical element with the symbol K (from Neo-Latin kalium) and atomic number19. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. Element K may refer to: * The chemical element Potassium given symbol K (Latin kalium) * An educational software package owned by Skillsoft Using the equations given above one can then translate the electron energy E into the threshold energy T. In infrared astronomy, the K band is an atmospheric transmission window centered on 2.2 μm (in the near-infrared 136 THz range). Metallic potassium is used in several types of magnetometers. ==Precautions== Potassium metal can react violently with water producing KOH and hydrogen gas. : thumb|left|alt=A piece of potassium metal is dropped into a clear container of water and skates around, burning with a bright pinkish or lilac flame for a short time until finishing with a pop and splash.|A reaction of potassium metal with water. Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. Potassium is silvery in appearance, but it begins to tarnish toward gray immediately on exposure to air.Greenwood, p. 76 In a flame test, potassium and its compounds emit a lilac color with a peak emission wavelength of 766.5 nanometers.Greenwood, p. 75 Neutral potassium atoms have 19 electrons, one more than the configuration of the noble gas argon. Potassium ferricyanide separates from the solution: :2 K4[Fe(CN)6] + Cl2 → 2 K3[Fe(CN)6] + 2 KCl ==Structure== Like other metal cyanides, solid potassium ferricyanide has a complicated polymeric structure. This filtering involves about 600g of sodium and 33g of potassium. Potassium peroxochromate, potassium tetraperoxochromate(V), or simply potassium perchromate, is an inorganic chemical having the chemical formula K3[Cr(O2)4]. Potassium ions are present in a wide variety of proteins and enzymes. ===Biochemical function=== Potassium levels influence multiple physiological processes, including *resting cellular-membrane potential and the propagation of action potentials in neuronal, muscular, and cardiac tissue. The stable isotopes of potassium can be laser cooled and used to probe fundamental and technological problems in quantum physics. This is the fundamental (""primary damage"") threshold displacement energy, and also the one usually simulated by molecular dynamics computer simulations. It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Pure potassium metal can be isolated by electrolysis of its hydroxide in a process that has changed little since it was first used by Humphry Davy in 1807. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. In contrast, the second ionization energy is very high (3052kJ/mol). ===Chemical=== Potassium reacts with oxygen, water, and carbon dioxide components in air. The K+\---NCFe linkages break when the solid is dissolved in water. ==Applications== The compound is also used to harden iron and steel, in electroplating, dyeing wool, as a laboratory reagent, and as a mild oxidizing agent in organic chemistry. ===Photography=== ==== Blueprint, cyanotype, toner ==== The compound has widespread use in blueprint drawing and in photography (Cyanotype process). Potassium ferricyanide is used to determine the ferric reducing power potential of a sample (extract, chemical compound, etc.).Nakajima, Y., Sato, Y., & Konishi, T. (2007). It is also used to bleach textiles and straw, and in the tanning of leathers. ===Industrial=== Major potassium chemicals are potassium hydroxide, potassium carbonate, potassium sulfate, and potassium chloride. ",8,37,3.52,1.5,475,C
+","alt=Potassium ferricyanide milled|thumb|Potassium ferricyanide when milled has lighter color Potassium ferricyanide is the chemical compound with the formula K3[Fe(CN)6]. Potassium is the chemical element with the symbol K (from Neo-Latin kalium) and atomic number19. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. Element K may refer to: * The chemical element Potassium given symbol K (Latin kalium) * An educational software package owned by Skillsoft Using the equations given above one can then translate the electron energy E into the threshold energy T. In infrared astronomy, the K band is an atmospheric transmission window centered on 2.2 μm (in the near-infrared 136 THz range). Metallic potassium is used in several types of magnetometers. ==Precautions== Potassium metal can react violently with water producing KOH and hydrogen gas. : thumb|left|alt=A piece of potassium metal is dropped into a clear container of water and skates around, burning with a bright pinkish or lilac flame for a short time until finishing with a pop and splash.|A reaction of potassium metal with water. Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. Potassium is silvery in appearance, but it begins to tarnish toward gray immediately on exposure to air.Greenwood, p. 76 In a flame test, potassium and its compounds emit a lilac color with a peak emission wavelength of 766.5 nanometers.Greenwood, p. 75 Neutral potassium atoms have 19 electrons, one more than the configuration of the noble gas argon. Potassium ferricyanide separates from the solution: :2 K4[Fe(CN)6] + Cl2 → 2 K3[Fe(CN)6] + 2 KCl ==Structure== Like other metal cyanides, solid potassium ferricyanide has a complicated polymeric structure. This filtering involves about 600g of sodium and 33g of potassium. Potassium peroxochromate, potassium tetraperoxochromate(V), or simply potassium perchromate, is an inorganic chemical having the chemical formula K3[Cr(O2)4]. Potassium ions are present in a wide variety of proteins and enzymes. ===Biochemical function=== Potassium levels influence multiple physiological processes, including *resting cellular-membrane potential and the propagation of action potentials in neuronal, muscular, and cardiac tissue. The stable isotopes of potassium can be laser cooled and used to probe fundamental and technological problems in quantum physics. This is the fundamental (""primary damage"") threshold displacement energy, and also the one usually simulated by molecular dynamics computer simulations. It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Pure potassium metal can be isolated by electrolysis of its hydroxide in a process that has changed little since it was first used by Humphry Davy in 1807. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. In contrast, the second ionization energy is very high (3052kJ/mol). ===Chemical=== Potassium reacts with oxygen, water, and carbon dioxide components in air. The K+\---NCFe linkages break when the solid is dissolved in water. ==Applications== The compound is also used to harden iron and steel, in electroplating, dyeing wool, as a laboratory reagent, and as a mild oxidizing agent in organic chemistry. ===Photography=== ==== Blueprint, cyanotype, toner ==== The compound has widespread use in blueprint drawing and in photography (Cyanotype process). Potassium ferricyanide is used to determine the ferric reducing power potential of a sample (extract, chemical compound, etc.).Nakajima, Y., Sato, Y., & Konishi, T. (2007). It is also used to bleach textiles and straw, and in the tanning of leathers. ===Industrial=== Major potassium chemicals are potassium hydroxide, potassium carbonate, potassium sulfate, and potassium chloride. ",8,37,"""3.52""",1.5,475,C
 "Evaluate the series
 $$
 S=\sum_{n=0}^{\infty} \frac{1}{3^n}
-$$","It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is :1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}. The series \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots is known as the alternating harmonic series. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Explicitly, the asymptotic expansion of the series is \frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{2n} + O(n^{-2}) Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. ===Riemann zeta function=== The Riemann zeta function is defined for real x>1 by the convergent series \zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots, which for x=1 would be the harmonic series. That is, it is the sum : {\sideset{}{'}\sum_{n=1}^\infty} \frac{1}{n} where the prime indicates that n takes only values whose decimal expansion has no nines. The closed form geometric series 1 / (1 - r) is the black dashed line. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. One can derive that closed-form formula for the partial sum, sn, by subtracting out the many self-similar terms as follows: \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1},\\\ rs_n &= ar^1 + ar^2 + \cdots + ar^{n},\\\ s_n - rs_n &= ar^0 - ar^{n},\\\ s_n\left(1-r\right) &= a\left(1-r^{n}\right),\\\ s_n &= a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r eq 1. \end{align} As approaches infinity, the absolute value of must be less than one for the series to converge. The sum is :\frac{1}{1 -r}\;=\;\frac{1}{1 -\frac{1}{4}}\;=\;\frac{4}{3}. Its value can then be computed from the finite sum formula \sum_{k=0}^\infty ar^k = \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k} = \lim_{n\to\infty}\frac{a(1-r^{n+1})}{1-r}= \frac{a}{1-r} - \lim_{n\to\infty}{\frac{ar^{n+1}}{1-r}} thumb|350px|Animation, showing convergence of partial sums of geometric progression \sum\limits_{k=0}^{n}q^k (red line) to its sum {1\over 1-q} (blue line) for |q|<1. thumb|350px|Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ⋯ which converges to 2. In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as :\sum^{\infty}_{k=0} a r^k and a closed form of the geometric series written as :\frac{a}{1-r} \text{ for } |r|<1\. When r = 1, all of the terms of the series are the same and the series is infinite. Given that the last term is arn and the previous series remainder is s - sn-1 = arn / (1 - r)), this measure of the convergence rate of the geometric series is arn / (arn / (1 - r)) = 1 - r, if 0 ≤ r < 1\. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots. The following table shows several geometric series: a r Example series 4 10 4 + 40 + 400 + 4000 + 40,000 + ··· 3 1 3 + 3 + 3 + 3 + 3 + ··· 1 2/3 1 + 2/3 + 4/9 + 8/27 + 16/81 + ··· 1/2 1/2 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ··· 9 1/3 9 + 3 + 1 + 1/3 + 1/9 + ··· 7 1/10 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ··· 1 −1/2 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ··· 3 −1 3 − 3 + 3 − 3 + 3 − ··· The convergence of the geometric series depends on the value of the common ratio r: :* If |r| < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude), and the series converges to the sum a / (1 - r). The sum of the first n terms of a geometric series, up to and including the r n-1 term, is given by the closed-form formula: \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1}\\\ &= \sum_{k=0}^{n-1} ar^k = \sum_{k=1}^{n} ar^{k-1}\\\ &= \begin{cases} a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r eq 1\\\ an, \text{ for } r = 1 \end{cases} \end{align} where is the common ratio. The sum then becomes \begin{align} s &= a+ar+ar^2+ar^3+ar^4+\cdots\\\ &= \sum_{k=0}^\infty ar^{k} = \sum_{k=1}^\infty ar^{k-1}\\\ &= \frac{a}{1-r}, \text{ for } |r|<1\. \end{align} The formula also holds for complex , with the corresponding restriction that the modulus of is strictly less than one. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. ",1.5,-3.8,362880.0,0.16,6,A
+$$","It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. The first term of the geometric series is a = 3(1/9) = 1/3, so the sum is :1\,+\,\frac{a}{1-r}\;=\;1\,+\,\frac{\frac{1}{3}}{1-\frac{4}{9}}\;=\;\frac{8}{5}. The series \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots is known as the alternating harmonic series. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Explicitly, the asymptotic expansion of the series is \frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{2n} + O(n^{-2}) Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. ===Riemann zeta function=== The Riemann zeta function is defined for real x>1 by the convergent series \zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots, which for x=1 would be the harmonic series. That is, it is the sum : {\sideset{}{'}\sum_{n=1}^\infty} \frac{1}{n} where the prime indicates that n takes only values whose decimal expansion has no nines. The closed form geometric series 1 / (1 - r) is the black dashed line. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. Excluding the initial 1, this series is geometric with constant ratio r = 4/9. One can derive that closed-form formula for the partial sum, sn, by subtracting out the many self-similar terms as follows: \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1},\\\ rs_n &= ar^1 + ar^2 + \cdots + ar^{n},\\\ s_n - rs_n &= ar^0 - ar^{n},\\\ s_n\left(1-r\right) &= a\left(1-r^{n}\right),\\\ s_n &= a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r eq 1. \end{align} As approaches infinity, the absolute value of must be less than one for the series to converge. The sum is :\frac{1}{1 -r}\;=\;\frac{1}{1 -\frac{1}{4}}\;=\;\frac{4}{3}. Its value can then be computed from the finite sum formula \sum_{k=0}^\infty ar^k = \lim_{n\to\infty}{\sum_{k=0}^{n} ar^k} = \lim_{n\to\infty}\frac{a(1-r^{n+1})}{1-r}= \frac{a}{1-r} - \lim_{n\to\infty}{\frac{ar^{n+1}}{1-r}} thumb|350px|Animation, showing convergence of partial sums of geometric progression \sum\limits_{k=0}^{n}q^k (red line) to its sum {1\over 1-q} (blue line) for |q|<1. thumb|350px|Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + ⋯ which converges to 2. In addition to the expanded form of the geometric series, there is a generator form of the geometric series written as :\sum^{\infty}_{k=0} a r^k and a closed form of the geometric series written as :\frac{a}{1-r} \text{ for } |r|<1\. When r = 1, all of the terms of the series are the same and the series is infinite. Given that the last term is arn and the previous series remainder is s - sn-1 = arn / (1 - r)), this measure of the convergence rate of the geometric series is arn / (arn / (1 - r)) = 1 - r, if 0 ≤ r < 1\. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots. The following table shows several geometric series: a r Example series 4 10 4 + 40 + 400 + 4000 + 40,000 + ··· 3 1 3 + 3 + 3 + 3 + 3 + ··· 1 2/3 1 + 2/3 + 4/9 + 8/27 + 16/81 + ··· 1/2 1/2 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ··· 9 1/3 9 + 3 + 1 + 1/3 + 1/9 + ··· 7 1/10 7 + 0.7 + 0.07 + 0.007 + 0.0007 + ··· 1 −1/2 1 − 1/2 + 1/4 − 1/8 + 1/16 − 1/32 + ··· 3 −1 3 − 3 + 3 − 3 + 3 − ··· The convergence of the geometric series depends on the value of the common ratio r: :* If |r| < 1, the terms of the series approach zero in the limit (becoming smaller and smaller in magnitude), and the series converges to the sum a / (1 - r). The sum of the first n terms of a geometric series, up to and including the r n-1 term, is given by the closed-form formula: \begin{align} s_n &= ar^0 + ar^1 + \cdots + ar^{n-1}\\\ &= \sum_{k=0}^{n-1} ar^k = \sum_{k=1}^{n} ar^{k-1}\\\ &= \begin{cases} a\left(\frac{1-r^{n}}{1-r}\right), \text{ for } r eq 1\\\ an, \text{ for } r = 1 \end{cases} \end{align} where is the common ratio. The sum then becomes \begin{align} s &= a+ar+ar^2+ar^3+ar^4+\cdots\\\ &= \sum_{k=0}^\infty ar^{k} = \sum_{k=1}^\infty ar^{k-1}\\\ &= \frac{a}{1-r}, \text{ for } |r|<1\. \end{align} The formula also holds for complex , with the corresponding restriction that the modulus of is strictly less than one. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. ",1.5,-3.8,"""362880.0""",0.16,6,A
 "Evaluate the series
 $$
 S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2^n}
-$$","The Mercator series provides an analytic expression of the natural logarithm: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n \;=\; \ln (1+x). It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. As another example, by Mercator series \ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. In mathematics, an alternating series is an infinite series of the form \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n with for all . This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. For any \varepsilon \in (0,2), one thus finds ::\sum_{n=0}^\infty (-1+\varepsilon)^n=\frac{1}{1-(-1+\varepsilon)}=\frac{1}{2-\varepsilon}, and so the limit \varepsilon\to 0 of series evaluations is ::\lim_{\varepsilon\to 0}\lim_{N\to\infty}\sum_{n=0}^N (-1+\varepsilon)^n=\frac{1}{2}. In mathematics, the infinite series , also written : \sum_{n=0}^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. The series converges to the natural logarithm (shifted by 1) whenever -1 . ==History== The series was discovered independently by Johannes Hudde and Isaac Newton. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an ""endless"" and ""violent"" dispute between mathematicians.Kline 1983 p.307Knopp p.457 == Relation to the geometric series == For any number r in the interval (-1,1), the sum to infinity of a geometric series can be evaluated via ::\lim_{N\to\infty}\sum_{n=0}^N r^n = \sum_{n=0}^\infty r^n=\frac{1}{1-r}. The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1.Devlin p.77 The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. :\sum_{k=0}^{n} (-2)^k As a series of real numbers it diverges, so in the usual sense it has no sum. In the terms of complex analysis, \tfrac{1}{2} is thus seen to be the value at z=-1 of the analytic continuation of the series \sum_{n=0}^N z^n, which is only defined on the complex unit disk, |z|<1. ==Early ideas== ==Divergence== In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value: :S = 1 − 1 + 1 − 1 + ..., so :1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S :1 − S = S :1 = 2S, resulting in S = . In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for \tfrac 1 2 \ln(2): \begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\\\[8pt] & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\\\[8pt] & = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2). \end{align} == Series acceleration == In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7). In a much broader sense, the series is associated with another value besides ∞, namely , which is the limit of the series using the 2-adic metric. ==Historical arguments== Gottfried Leibniz considered the divergent alternating series as early as 1673. ",1,13.2,4.86,2.3613,0.65625,A
-"The relationship introduced in Problem $1-48$ has been interpreted to mean that a particle of mass $m\left(E=m c^2\right)$ can materialize from nothing provided that it returns to nothing within a time $\Delta t \leq h / m c^2$. Particles that last for time $\Delta t$ or more are called real particles; particles that last less than time $\Delta t$ are called virtual particles. The mass of the charged pion, a subatomic particle, is $2.5 \times 10^{-28} \mathrm{~kg}$. What is the minimum lifetime if the pion is to be considered a real particle?","It decays via the electromagnetic force, which explains why its mean lifetime is much smaller than that of the charged pion (which can only decay via the weak force). December 18, 2013 ===Neutral pion decays=== The meson has a mass of and a mean lifetime of . In particle physics, the pion decay constant is the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action. Each pion has isospin (I = 1) and third-component isospin equal to its charge (Iz = +1, 0 or −1). ===Charged pion decays=== The mesons have a mass of and a mean lifetime of . They are unstable, with the charged pions and decaying after a mean lifetime of 26.033 nanoseconds ( seconds), and the neutral pion decaying after a much shorter lifetime of 85 attoseconds ( seconds). Pions Particle name Particle symbol Antiparticle symbol Quark content Rest mass (MeV/c2) IG JPC S C B' Mean lifetime (s) Commonly decays to (>5% of decays) Pion 1− 0− 0 0 0 Pion Self \tfrac{\mathrm{u\bar{u}} - \mathrm{d\bar{d}}}{\sqrt 2} 1− 0−+ 0 0 0 [a] Make-up inexact due to non-zero quark masses. ==See also== *Pionium *Quark model *Static forces and virtual- particle exchange *Sanford-Wang parameterisation ==References== == Further reading == * Gerald Edward Brown and A. D. Jackson, The Nucleon-Nucleon Interaction (1976), North-Holland Publishing, Amsterdam ==External links== * * Mesons at the Particle Data Group Category:Mesons Also observed, for charged pions only, is the very rare ""pion beta decay"" (with branching fraction of about 10−8) into a neutral pion, an electron and an electron antineutrino (or for positive pions, a neutral pion, a positron, and electron neutrino). : → + + → + + The rate at which pions decay is a prominent quantity in many sub-fields of particle physics, such as chiral perturbation theory. In a series of articles published in Nature, they identified a cosmic particle having an average mass close to 200 times the mass of electron, today known as pions. This rate is parametrized by the pion decay constant (ƒπ), related to the wave function overlap of the quark and antiquark, which is about .Leptonic decays of charged pseudo- scalar mesons J. L. Rosner and S. Stone. Pions are not produced in radioactive decay, but commonly are in high-energy collisions between hadrons. The pion is one of the particles that mediate the residual strong interaction between a pair of nucleons. The existence of the neutral pion was inferred from observing its decay products from cosmic rays, a so-called ""soft component"" of slow electrons with photons. In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . The neutral pion has also been observed to decay into positronium with a branching fraction on the order of . According to Brown–Rho scaling, the masses of nucleons and most light mesons decrease at finite density as the ratio of the in-medium pion decay rate to the free-space pion decay constant. Giuseppe Paolo Stanislao ""Beppo"" Occhialini ForMemRS (; 5 December 1907 – 30 December 1993) was an Italian physicist who contributed to the discovery of the pion or pi-meson decay in 1947 with César Lattes and Cecil Frank Powell, the latter winning the Nobel Prize in Physics for this work. The pion mass is an exception to Brown-Rho scaling because the pion's mass is protected by its Goldstone boson nature. == References == # # # Particle Data Group: Decay constants of charged pseudoscalar mesons == External links == # Particle Data Group & WWW edition of Review of Particle Physics Category:Quantum chromodynamics Charged pions most often decay into muons and muon neutrinos, while neutral pions generally decay into gamma rays. The pion also plays a crucial role in cosmology, by imposing an upper limit on the energies of cosmic rays surviving collisions with the cosmic microwave background, through the Greisen–Zatsepin–Kuzmin limit. ==History== Theoretical work by Hideki Yukawa in 1935 had predicted the existence of mesons as the carrier particles of the strong nuclear force. Pions also result from some matter–antimatter annihilation events. If it does decay via a positron, the proton's half-life is constrained to be at least years. The exchange of virtual pions, along with vector, rho and omega mesons, provides an explanation for the residual strong force between nucleons. ",15,2.00,2.9,-2.99, 13.45,C
-A household lightbulb is a blackbody radiator. Many lightbulbs use tungsten filaments that are heated by an electric current. What temperature is needed so that $\lambda_{\max }=550 \mathrm{~nm}$ ?,"An incandescent lamp's light is thermal radiation, and the bulb approximates an ideal black-body radiator, so its color temperature is essentially the temperature of the filament. According to the Stefan–Boltzmann law, a black body at the Draper point emits 23 kW of radiation per square metre, almost exclusively infrared. ==See also== *Incandescence == References == Category:Heat transfer Category:Thermodynamics Category:Electromagnetic radiation Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. The value of the Draper point can be calculated using Wien's displacement law: the peak frequency u_\text{peak} (in hertz) emitted by a blackbody relates to temperature as follows: u_\text{peak} = 2.821 \frac{kT}{h}, where * is Boltzmann's constant, * is Planck's constant, * is temperature (in kelvins). The specific heat capacity has a sharp peak as the temperature approaches the lambda point. The temperature of the ideal emitter that matches the color most closely is defined as the color temperature of the original visible light source. That makes the reciprocal of the brightness temperature: ::T_b^{-1} = \frac{k}{h u}\, \text{ln}\left[1 + \frac{e^{\frac{h u}{kT}}-1}{\epsilon}\right] At low frequency and high temperatures, when h u \ll kT, we can use the Rayleigh–Jeans law: ::I_{ u} = \frac{2 u^2k T}{c^2} so that the brightness temperature can be simply written as: ::T_b=\epsilon T\, In general, the brightness temperature is a function of u, and only in the case of blackbody radiation it is the same at all frequencies. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). To the extent that a hot surface emits thermal radiation but is not an ideal black- body radiator, the color temperature of the light is not the actual temperature of the surface. The actual temperature will be higher than the brightness temperature if the emissivity of the object is greater than 1. Because such an approximation is not required for incandescent light, the CCT for an incandescent light is simply its unadjusted temperature, derived from comparison to a black-body radiator. ===The Sun=== The Sun closely approximates a black-body radiator. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation. == Calculating by frequency == The brightness temperature of a source with known spectral radiance can be expressed as: : T_b=\frac{h u}{k} \ln^{-1}\left( 1 + \frac{2h u^3}{I_{ u}c^2} \right) When h u \ll kT we can use the Rayleigh–Jeans law: : T_b=\frac{I_{ u}c^2}{2k u^2} For narrowband radiation with very low relative spectral linewidth \Delta u \ll u and known radiance I we can calculate the brightness temperature as: : T_b=\frac{I c^2}{2k u^2\Delta u} == Calculating by wavelength == Spectral radiance of black-body radiation is expressed by wavelength as: : I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1} So, the brightness temperature can be calculated as: : T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right) For long-wave radiation hc/\lambda \ll kT the brightness temperature is: : T_b=\frac{I_{\lambda}\lambda^4}{2kc} For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c: : T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} } ==In oceanography== In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature of the water. ==References== Category:Temperature Category:Radio astronomy Category:Planetary science Thus a relatively low temperature emits a dull red and a high temperature emits the almost white of the traditional incandescent light bulb. The effective temperature, defined by the total radiative power per square unit, is 5772 K. The Draper point is the approximate temperature above which almost all solid materials visibly glow as a result of blackbody radiation. Wet-bulb potential temperature, sometimes referred to as pseudo wet-bulb potential temperature, is the temperature that a parcel of air at any level would have if, starting at the wet-bulb temperature, it were brought at the saturated adiabatic lapse rate to the standard pressure of 1,000 mbar. When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. For radiation emitted by a non-thermal source such as a pulsar, synchrotron, maser, or a laser, the brightness temperature may be far higher than the actual temperature of the source. The fact that ""warm"" lighting in this sense actually has a ""cooler"" color temperature often leads to confusion.See the comments section of this LightNowBlog.com article on the recommendations of the American Medical Association to prefer LED- lighting with cooler color temperatures (i.e. warmer color). ==Categorizing different lighting== The color temperature of the electromagnetic radiation emitted from an ideal black body is defined as its surface temperature in kelvins, or alternatively in micro reciprocal degrees (mired). When the electromagnetic radiation observed is thermal radiation emitted by an object simply by virtue of its temperature, then the actual temperature of the object will always be equal to or higher than the brightness temperature. Color temperature is usually measured in kelvins. Color temperature is a parameter describing the color of a visible light source by comparing it to the color of light emitted by an idealized opaque, non-reflective body. ",24,0.22222222,7.82,-1,5300,E
+$$","The Mercator series provides an analytic expression of the natural logarithm: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n \;=\; \ln (1+x). It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. As another example, by Mercator series \ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. In mathematics, an alternating series is an infinite series of the form \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n with for all . This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. For any \varepsilon \in (0,2), one thus finds ::\sum_{n=0}^\infty (-1+\varepsilon)^n=\frac{1}{1-(-1+\varepsilon)}=\frac{1}{2-\varepsilon}, and so the limit \varepsilon\to 0 of series evaluations is ::\lim_{\varepsilon\to 0}\lim_{N\to\infty}\sum_{n=0}^N (-1+\varepsilon)^n=\frac{1}{2}. In mathematics, the infinite series , also written : \sum_{n=0}^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. The series converges to the natural logarithm (shifted by 1) whenever -1 . ==History== The series was discovered independently by Johannes Hudde and Isaac Newton. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an ""endless"" and ""violent"" dispute between mathematicians.Kline 1983 p.307Knopp p.457 == Relation to the geometric series == For any number r in the interval (-1,1), the sum to infinity of a geometric series can be evaluated via ::\lim_{N\to\infty}\sum_{n=0}^N r^n = \sum_{n=0}^\infty r^n=\frac{1}{1-r}. The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1.Devlin p.77 The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. :\sum_{k=0}^{n} (-2)^k As a series of real numbers it diverges, so in the usual sense it has no sum. In the terms of complex analysis, \tfrac{1}{2} is thus seen to be the value at z=-1 of the analytic continuation of the series \sum_{n=0}^N z^n, which is only defined on the complex unit disk, |z|<1. ==Early ideas== ==Divergence== In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value: :S = 1 − 1 + 1 − 1 + ..., so :1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S :1 − S = S :1 = 2S, resulting in S = . In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for \tfrac 1 2 \ln(2): \begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\\\[8pt] & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\\\[8pt] & = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2). \end{align} == Series acceleration == In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7). In a much broader sense, the series is associated with another value besides ∞, namely , which is the limit of the series using the 2-adic metric. ==Historical arguments== Gottfried Leibniz considered the divergent alternating series as early as 1673. ",1,13.2,"""4.86""",2.3613,0.65625,A
+"The relationship introduced in Problem $1-48$ has been interpreted to mean that a particle of mass $m\left(E=m c^2\right)$ can materialize from nothing provided that it returns to nothing within a time $\Delta t \leq h / m c^2$. Particles that last for time $\Delta t$ or more are called real particles; particles that last less than time $\Delta t$ are called virtual particles. The mass of the charged pion, a subatomic particle, is $2.5 \times 10^{-28} \mathrm{~kg}$. What is the minimum lifetime if the pion is to be considered a real particle?","It decays via the electromagnetic force, which explains why its mean lifetime is much smaller than that of the charged pion (which can only decay via the weak force). December 18, 2013 ===Neutral pion decays=== The meson has a mass of and a mean lifetime of . In particle physics, the pion decay constant is the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action. Each pion has isospin (I = 1) and third-component isospin equal to its charge (Iz = +1, 0 or −1). ===Charged pion decays=== The mesons have a mass of and a mean lifetime of . They are unstable, with the charged pions and decaying after a mean lifetime of 26.033 nanoseconds ( seconds), and the neutral pion decaying after a much shorter lifetime of 85 attoseconds ( seconds). Pions Particle name Particle symbol Antiparticle symbol Quark content Rest mass (MeV/c2) IG JPC S C B' Mean lifetime (s) Commonly decays to (>5% of decays) Pion 1− 0− 0 0 0 Pion Self \tfrac{\mathrm{u\bar{u}} - \mathrm{d\bar{d}}}{\sqrt 2} 1− 0−+ 0 0 0 [a] Make-up inexact due to non-zero quark masses. ==See also== *Pionium *Quark model *Static forces and virtual- particle exchange *Sanford-Wang parameterisation ==References== == Further reading == * Gerald Edward Brown and A. D. Jackson, The Nucleon-Nucleon Interaction (1976), North-Holland Publishing, Amsterdam ==External links== * * Mesons at the Particle Data Group Category:Mesons Also observed, for charged pions only, is the very rare ""pion beta decay"" (with branching fraction of about 10−8) into a neutral pion, an electron and an electron antineutrino (or for positive pions, a neutral pion, a positron, and electron neutrino). : → + + → + + The rate at which pions decay is a prominent quantity in many sub-fields of particle physics, such as chiral perturbation theory. In a series of articles published in Nature, they identified a cosmic particle having an average mass close to 200 times the mass of electron, today known as pions. This rate is parametrized by the pion decay constant (ƒπ), related to the wave function overlap of the quark and antiquark, which is about .Leptonic decays of charged pseudo- scalar mesons J. L. Rosner and S. Stone. Pions are not produced in radioactive decay, but commonly are in high-energy collisions between hadrons. The pion is one of the particles that mediate the residual strong interaction between a pair of nucleons. The existence of the neutral pion was inferred from observing its decay products from cosmic rays, a so-called ""soft component"" of slow electrons with photons. In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . The neutral pion has also been observed to decay into positronium with a branching fraction on the order of . According to Brown–Rho scaling, the masses of nucleons and most light mesons decrease at finite density as the ratio of the in-medium pion decay rate to the free-space pion decay constant. Giuseppe Paolo Stanislao ""Beppo"" Occhialini ForMemRS (; 5 December 1907 – 30 December 1993) was an Italian physicist who contributed to the discovery of the pion or pi-meson decay in 1947 with César Lattes and Cecil Frank Powell, the latter winning the Nobel Prize in Physics for this work. The pion mass is an exception to Brown-Rho scaling because the pion's mass is protected by its Goldstone boson nature. == References == # # # Particle Data Group: Decay constants of charged pseudoscalar mesons == External links == # Particle Data Group & WWW edition of Review of Particle Physics Category:Quantum chromodynamics Charged pions most often decay into muons and muon neutrinos, while neutral pions generally decay into gamma rays. The pion also plays a crucial role in cosmology, by imposing an upper limit on the energies of cosmic rays surviving collisions with the cosmic microwave background, through the Greisen–Zatsepin–Kuzmin limit. ==History== Theoretical work by Hideki Yukawa in 1935 had predicted the existence of mesons as the carrier particles of the strong nuclear force. Pions also result from some matter–antimatter annihilation events. If it does decay via a positron, the proton's half-life is constrained to be at least years. The exchange of virtual pions, along with vector, rho and omega mesons, provides an explanation for the residual strong force between nucleons. ",15,2.00,"""2.9""",-2.99, 13.45,C
+A household lightbulb is a blackbody radiator. Many lightbulbs use tungsten filaments that are heated by an electric current. What temperature is needed so that $\lambda_{\max }=550 \mathrm{~nm}$ ?,"An incandescent lamp's light is thermal radiation, and the bulb approximates an ideal black-body radiator, so its color temperature is essentially the temperature of the filament. According to the Stefan–Boltzmann law, a black body at the Draper point emits 23 kW of radiation per square metre, almost exclusively infrared. ==See also== *Incandescence == References == Category:Heat transfer Category:Thermodynamics Category:Electromagnetic radiation Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. The value of the Draper point can be calculated using Wien's displacement law: the peak frequency u_\text{peak} (in hertz) emitted by a blackbody relates to temperature as follows: u_\text{peak} = 2.821 \frac{kT}{h}, where * is Boltzmann's constant, * is Planck's constant, * is temperature (in kelvins). The specific heat capacity has a sharp peak as the temperature approaches the lambda point. The temperature of the ideal emitter that matches the color most closely is defined as the color temperature of the original visible light source. That makes the reciprocal of the brightness temperature: ::T_b^{-1} = \frac{k}{h u}\, \text{ln}\left[1 + \frac{e^{\frac{h u}{kT}}-1}{\epsilon}\right] At low frequency and high temperatures, when h u \ll kT, we can use the Rayleigh–Jeans law: ::I_{ u} = \frac{2 u^2k T}{c^2} so that the brightness temperature can be simply written as: ::T_b=\epsilon T\, In general, the brightness temperature is a function of u, and only in the case of blackbody radiation it is the same at all frequencies. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). To the extent that a hot surface emits thermal radiation but is not an ideal black- body radiator, the color temperature of the light is not the actual temperature of the surface. The actual temperature will be higher than the brightness temperature if the emissivity of the object is greater than 1. Because such an approximation is not required for incandescent light, the CCT for an incandescent light is simply its unadjusted temperature, derived from comparison to a black-body radiator. ===The Sun=== The Sun closely approximates a black-body radiator. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation. == Calculating by frequency == The brightness temperature of a source with known spectral radiance can be expressed as: : T_b=\frac{h u}{k} \ln^{-1}\left( 1 + \frac{2h u^3}{I_{ u}c^2} \right) When h u \ll kT we can use the Rayleigh–Jeans law: : T_b=\frac{I_{ u}c^2}{2k u^2} For narrowband radiation with very low relative spectral linewidth \Delta u \ll u and known radiance I we can calculate the brightness temperature as: : T_b=\frac{I c^2}{2k u^2\Delta u} == Calculating by wavelength == Spectral radiance of black-body radiation is expressed by wavelength as: : I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1} So, the brightness temperature can be calculated as: : T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right) For long-wave radiation hc/\lambda \ll kT the brightness temperature is: : T_b=\frac{I_{\lambda}\lambda^4}{2kc} For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c: : T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} } ==In oceanography== In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature of the water. ==References== Category:Temperature Category:Radio astronomy Category:Planetary science Thus a relatively low temperature emits a dull red and a high temperature emits the almost white of the traditional incandescent light bulb. The effective temperature, defined by the total radiative power per square unit, is 5772 K. The Draper point is the approximate temperature above which almost all solid materials visibly glow as a result of blackbody radiation. Wet-bulb potential temperature, sometimes referred to as pseudo wet-bulb potential temperature, is the temperature that a parcel of air at any level would have if, starting at the wet-bulb temperature, it were brought at the saturated adiabatic lapse rate to the standard pressure of 1,000 mbar. When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. For radiation emitted by a non-thermal source such as a pulsar, synchrotron, maser, or a laser, the brightness temperature may be far higher than the actual temperature of the source. The fact that ""warm"" lighting in this sense actually has a ""cooler"" color temperature often leads to confusion.See the comments section of this LightNowBlog.com article on the recommendations of the American Medical Association to prefer LED- lighting with cooler color temperatures (i.e. warmer color). ==Categorizing different lighting== The color temperature of the electromagnetic radiation emitted from an ideal black body is defined as its surface temperature in kelvins, or alternatively in micro reciprocal degrees (mired). When the electromagnetic radiation observed is thermal radiation emitted by an object simply by virtue of its temperature, then the actual temperature of the object will always be equal to or higher than the brightness temperature. Color temperature is usually measured in kelvins. Color temperature is a parameter describing the color of a visible light source by comparing it to the color of light emitted by an idealized opaque, non-reflective body. ",24,0.22222222,"""7.82""",-1,5300,E
 "Evaluate the series
 $$
 S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots
 $$
-","It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. Simplifying the fractions gives :1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots. It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. As it is a geometric series with first term and common ratio , its sum is :\sum_{n=1}^\infty \frac{1}{4^n}=\frac {\frac 1 4} {1 - \frac 1 4}=\frac 1 3. ==Visual demonstrations== left|thumb|3s = 1\. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. Since the sum of an infinite series is defined as the limit of its partial sums, :1+\frac14+\frac{1}{4^2}+\frac{1}{4^3}+\cdots = \frac43. ==Notes== ==References== * * Page images at HTML with figures and commentary at * * * * * Category:Geometric series Category:Proof without words Since these three areas cover the unit square, the figure demonstrates that :3\left(\frac14+\frac{1}{4^2}+\frac{1}{4^3}+\frac{1}{4^4}+\cdots\right) = 1. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots. Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series are: :1+\frac{1}{4}+\frac{1}{4^2}+\cdots+\frac{1}{4^n}=\frac{1-\left(\frac14\right)^{n+1}}{1-\frac14}. It is a geometric series whose first term is and whose common ratio is −, so its sum is :\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2^n}=\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{\frac12}{1-(-\frac12)} = \frac13. ==Hackenbush and the surreals== frame|right|Demonstration of via a zero-value game A slight rearrangement of the series reads :1-\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac13. Archimedes' own illustration, adapted at top,Heath p. 250 was slightly different, being closer to the equation right|thumb|3s = 1 again :\sum_{n=1}^\infty \frac{3}{4^n}=\frac34+\frac{3}{4^2}+\frac{3}{4^3}+\frac{3}{4^4}+\cdots = 1. The series \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots is known as the alternating harmonic series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of , which is expressed by a famous formula. : 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}, where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. Explicitly, the asymptotic expansion of the series is \frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{2n} + O(n^{-2}) Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. ===Riemann zeta function=== The Riemann zeta function is defined for real x>1 by the convergent series \zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots, which for x=1 would be the harmonic series. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. thumb|right|The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Two of the best-known are listed below. ===Comparison test=== One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two: \begin{alignat}{8} 1 & \+ \frac{1}{2} && \+ \frac{1}{3} && \+ \frac{1}{4} && \+ \frac{1}{5} && \+ \frac{1}{6} && \+ \frac{1}{7} && \+ \frac{1}{8} && \+ \frac{1}{9} && \+ \cdots \\\\[5pt] {} \geq 1 & \+ \frac{1}{2} && \+ \frac{1}{\color{red}{\mathbf{4}}} && \+ \frac{1}{4} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{8} && \+ \frac{1}{\color{red}{\mathbf{16}}} && \+ \cdots \\\\[5pt] \end{alignat} Grouping equal terms shows that the second series diverges (because every grouping of convergent series is only convergent): \begin{align} & 1 + \left(\frac{1}{2}\right) + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \left(\frac{1}{16} + \cdots + \frac{1}{16}\right) + \cdots \\\\[5pt] {} = {} & 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots. \end{align} Because each term of the harmonic series is greater than or equal to the corresponding term of the second series (and the terms are all positive), and since the second series diverges, it follows (by the comparison test) that the harmonic series diverges as well. Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. This list of mathematical series contains formulae for finite and infinite sums. ",0.132,7.654,1.0,0.38,2.3,C
-Through what potential must a proton initially at rest fall so that its de Broglie wavelength is $1.0 \times 10^{-10} \mathrm{~m}$ ?,"As a muon is 200 times heavier than an electron, its de Broglie wavelength is correspondingly shorter. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. When applied to the Planck relation, this gives: :\lambda = \frac {1}{ u} \cdot c = \frac {h}{E} \cdot c \approx \frac{\; 4.1357 \cdot 10^{-15} \ \mathrm{eV}\cdot\text{s} \;}{5.874\,33 \cdot 10^{-6}\ \mathrm{eV}}\, \cdot\, 2.9979 \cdot 10^8 \ \mathrm{m} \cdot \mathrm{s}^{-1} \approx 0.211\,06\ \mathrm{m} = 21.106\ \mathrm{cm}\; where is the wavelength of an emitted photon, is its frequency, is the photon energy, is the Planck constant, and is the speed of light. In kaonic hydrogen this strong contribution was found to be repulsive, shifting the ground state energy by 283 ± 36 (statistical) ± 6 (systematic) eV, thus making the system unstable with a resonance width of 541 ± 89 (stat) ± 22 (syst) eV (decay into Λπ and ΣπYiguang Yan, Kaonic hydrogen atom and kaon-proton scattering length, ). This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 e (elementary charge). Kaonic hydrogen is an exotic atom consisting of a negatively charged kaon orbiting a proton. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. The result is again ~5% smaller than the previously-accepted proton radius. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. In the inertial frame, the accelerating proton should decay according to the formula above. In 2019, two different studies, using different techniques, found this radius to be 0.833 fm, with an uncertainty of ±0.010 fm. Free protons occur occasionally on Earth: thunderstorms can produce protons with energies of up to several tens of MeV. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. The internationally accepted value of a proton's charge radius is . Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). ",0.0547,0.082,1068.0,24,1.5377,B
+","It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. Simplifying the fractions gives :1 \,+\, \frac{1}{4} \,+\, \frac{1}{16} \,+\, \frac{1}{64} \,+\, \cdots. It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. As it is a geometric series with first term and common ratio , its sum is :\sum_{n=1}^\infty \frac{1}{4^n}=\frac {\frac 1 4} {1 - \frac 1 4}=\frac 1 3. ==Visual demonstrations== left|thumb|3s = 1\. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. Since the sum of an infinite series is defined as the limit of its partial sums, :1+\frac14+\frac{1}{4^2}+\frac{1}{4^3}+\cdots = \frac43. ==Notes== ==References== * * Page images at HTML with figures and commentary at * * * * * Category:Geometric series Category:Proof without words Since these three areas cover the unit square, the figure demonstrates that :3\left(\frac14+\frac{1}{4^2}+\frac{1}{4^3}+\frac{1}{4^4}+\cdots\right) = 1. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots. Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series are: :1+\frac{1}{4}+\frac{1}{4^2}+\cdots+\frac{1}{4^n}=\frac{1-\left(\frac14\right)^{n+1}}{1-\frac14}. It is a geometric series whose first term is and whose common ratio is −, so its sum is :\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2^n}=\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{\frac12}{1-(-\frac12)} = \frac13. ==Hackenbush and the surreals== frame|right|Demonstration of via a zero-value game A slight rearrangement of the series reads :1-\frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac13. Archimedes' own illustration, adapted at top,Heath p. 250 was slightly different, being closer to the equation right|thumb|3s = 1 again :\sum_{n=1}^\infty \frac{3}{4^n}=\frac34+\frac{3}{4^2}+\frac{3}{4^3}+\frac{3}{4^4}+\cdots = 1. The series \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots is known as the alternating harmonic series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of , which is expressed by a famous formula. : 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}, where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. Explicitly, the asymptotic expansion of the series is \frac{1}{1} - \frac{1}{2} +\cdots + \frac{1}{2n-1} - \frac{1}{2n} = H_{2n} - H_n = \ln 2 - \frac{1}{2n} + O(n^{-2}) Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}. ===Riemann zeta function=== The Riemann zeta function is defined for real x>1 by the convergent series \zeta(x)=\sum_{n=1}^{\infty}\frac{1}{n^x}=\frac1{1^x}+\frac1{2^x}+\frac1{3^x}+\cdots, which for x=1 would be the harmonic series. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. The first dimension is horizontal, in the bottom row showing the geometric series S = 1/2 + 1/4 + 1/8 + 1/16 + ... , which is the geometric series with coefficient a = 1/2 and common ratio r = 1/2 that converges to S = a / (1-r) = (1/2) / (1-1/2) = 1. thumb|right|The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Two of the best-known are listed below. ===Comparison test=== One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two: \begin{alignat}{8} 1 & \+ \frac{1}{2} && \+ \frac{1}{3} && \+ \frac{1}{4} && \+ \frac{1}{5} && \+ \frac{1}{6} && \+ \frac{1}{7} && \+ \frac{1}{8} && \+ \frac{1}{9} && \+ \cdots \\\\[5pt] {} \geq 1 & \+ \frac{1}{2} && \+ \frac{1}{\color{red}{\mathbf{4}}} && \+ \frac{1}{4} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{\color{red}{\mathbf{8}}} && \+ \frac{1}{8} && \+ \frac{1}{\color{red}{\mathbf{16}}} && \+ \cdots \\\\[5pt] \end{alignat} Grouping equal terms shows that the second series diverges (because every grouping of convergent series is only convergent): \begin{align} & 1 + \left(\frac{1}{2}\right) + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) + \left(\frac{1}{16} + \cdots + \frac{1}{16}\right) + \cdots \\\\[5pt] {} = {} & 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \cdots. \end{align} Because each term of the harmonic series is greater than or equal to the corresponding term of the second series (and the terms are all positive), and since the second series diverges, it follows (by the comparison test) that the harmonic series diverges as well. Among his insights into infinite series, in addition to his elegantly simple proof of the divergence of the harmonic series, Nicole Oresme proved that the series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. This list of mathematical series contains formulae for finite and infinite sums. ",0.132,7.654,"""1.0""",0.38,2.3,C
+Through what potential must a proton initially at rest fall so that its de Broglie wavelength is $1.0 \times 10^{-10} \mathrm{~m}$ ?,"As a muon is 200 times heavier than an electron, its de Broglie wavelength is correspondingly shorter. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. When applied to the Planck relation, this gives: :\lambda = \frac {1}{ u} \cdot c = \frac {h}{E} \cdot c \approx \frac{\; 4.1357 \cdot 10^{-15} \ \mathrm{eV}\cdot\text{s} \;}{5.874\,33 \cdot 10^{-6}\ \mathrm{eV}}\, \cdot\, 2.9979 \cdot 10^8 \ \mathrm{m} \cdot \mathrm{s}^{-1} \approx 0.211\,06\ \mathrm{m} = 21.106\ \mathrm{cm}\; where is the wavelength of an emitted photon, is its frequency, is the photon energy, is the Planck constant, and is the speed of light. In kaonic hydrogen this strong contribution was found to be repulsive, shifting the ground state energy by 283 ± 36 (statistical) ± 6 (systematic) eV, thus making the system unstable with a resonance width of 541 ± 89 (stat) ± 22 (syst) eV (decay into Λπ and ΣπYiguang Yan, Kaonic hydrogen atom and kaon-proton scattering length, ). This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 e (elementary charge). Kaonic hydrogen is an exotic atom consisting of a negatively charged kaon orbiting a proton. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. The result is again ~5% smaller than the previously-accepted proton radius. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. In the inertial frame, the accelerating proton should decay according to the formula above. In 2019, two different studies, using different techniques, found this radius to be 0.833 fm, with an uncertainty of ±0.010 fm. Free protons occur occasionally on Earth: thunderstorms can produce protons with energies of up to several tens of MeV. However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. The internationally accepted value of a proton's charge radius is . Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). thumb|350px|Corresponding potential energy (in units of MeV) of two nucleons as a function of distance as computed from the Reid potential. (The size of an atom, measured in angstroms (Å, or 10−10 m), is five orders of magnitude larger). ",0.0547,0.082,"""1068.0""",24,1.5377,B
 "Example 5-3 shows that a Maclaurin expansion of a Morse potential leads to
 $$
 V(x)=D \beta^2 x^2+\cdots
 $$
-Given that $D=7.31 \times 10^{-19} \mathrm{~J} \cdot$ molecule ${ }^{-1}$ and $\beta=1.81 \times 10^{10} \mathrm{~m}^{-1}$ for $\mathrm{HCl}$, calculate the force constant of $\mathrm{HCl}$.","In fact, the real molecular spectra are generally fit to the form1 : E_n / hc = \omega_e (n+1/2) - \omega_e\chi_e (n+1/2)^2\, in which the constants \omega_e and \omega_e\chi_e can be directly related to the parameters for the Morse potential. More sophisticated versions are used for polyatomic molecules. ==See also== *Lennard-Jones potential *Molecular mechanics ==References== *1 CRC Handbook of chemistry and physics, Ed David R. Lide, 87th ed, Section 9, SPECTROSCOPIC CONSTANTS OF DIATOMIC MOLECULES pp. 9–82 * * * * * * * * I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry, Wiley, 2003, p207. The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. The Hamaker constant provides the means to determine the interaction parameter from the vdW-pair potential, :w(r) = \frac{-C}{r^6}. The molecular formula C12H12O4 (molar mass: 220.22 g/mol, exact mass: 220.0736 u) may refer to: * Eugenitin * Hispolon * Siderin The molecular formula C16H19BrN2 (molar mass: 319.24 g/mol, exact mass: 318.0732 u) may refer to: * Brompheniramine * Dexbrompheniramine Category:Molecular formulas Mathematically, the spacing of Morse levels is :E_{n+1} - E_n = h u_0 - (n+1) (h u_0)^2/2D_e.\, This trend matches the anharmonicity found in real molecules. The Morse potential energy function is of the form :V(r) = D_e ( 1-e^{-a(r-r_e)} )^2 Here r is the distance between the atoms, r_e is the equilibrium bond distance, D_e is the well depth (defined relative to the dissociated atoms), and a controls the 'width' of the potential (the smaller a is, the larger the well). Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ħω, the Morse potential level spacing decreases as the energy approaches the dissociation energy. A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs2, and this potential has since been used on HF, HCl, HBr and HI. == Function == The Morse/Long-range potential energy function is of the form V(r) = \mathfrak{D}_e \left( 1- \frac{u(r)}{u(r_e)} e^{-\beta(r) y_p^{r_{\rm{eq}}}(r)} \right)^2 where for large r, V(r) \simeq \mathfrak{D}_e - u(r) + \frac{u(r)^2}{4\mathfrak{D}_e}, so u(r) is defined according to the theoretically correct long-range behavior expected for the interatomic interaction. In molecular physics, the Hamaker constant (denoted ; named for H. C. Hamaker) is a physical constant that can be defined for a van der Waals (vdW) body–body interaction: :A=\pi^2C\rho_1\rho_2, where are the number densities of the two interacting kinds of particles, and is the London coefficient in the particle–particle pair interaction. The magnitude of this constant reflects the strength of the vdW-force between two particles, or between a particle and a substrate. The force constant (stiffness) of the bond can be found by Taylor expansion of V'(r) around r=r_e to the second derivative of the potential energy function, from which it can be shown that the parameter, a, is :a=\sqrt{k_e/2D_e}, where k_e is the force constant at the minimum of the well. right|thumb|illustrative example of C-C length molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C angle molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C-C torsion molecular energy dependence, numerical accuracy is not guaranteed A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration. The Morse/Long-range potential (MLR potential) is an interatomic interaction model for the potential energy of a diatomic molecule. Hamaker's method and the associated Hamaker constant ignores the influence of an intervening medium between the two particles of interaction. Since the zero of potential energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. When it is used to model the atom-surface interaction, the energy zero can be redefined so that the Morse potential becomes :V(r)= V'(r)-D_e = D_e ( 1-e^{-a(r-r_e)} )^2 -D_e which is usually written as :V(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} ) where r is now the coordinate perpendicular to the surface. It clearly shows that the Morse potential is the combination of a short-range repulsion term (the former) and a long-range attractive term (the latter), analogous to the Lennard-Jones potential. ==Vibrational states and energies== 500px|thumb|Harmonic oscillator (grey) and Morse (black) potentials curves are shown along with their eigenfunctions (respectively green and blue for harmonic oscillator and morse) for the same vibrational levels for nitrogen. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. The latter is mathematically related to the particle mass, m, and the Morse constants via : u_0 = \frac{a}{2\pi} \sqrt{2D_e/m}. As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation in which \omega_e represents a wavenumber obeying E=hc\omega, and not an angular frequency given by E=\hbar\omega. == Morse/Long-range potential == An extension of the Morse potential that made the Morse form useful for modern (high-resolution) spectroscopy is the MLR (Morse/Long-range) potential. ",9.8,1.45,-194.0,3.2,479,E
-A line in the Lyman series of hydrogen has a wavelength of $1.03 \times 10^{-7} \mathrm{~m}$. Find the original energy level of the electron.,"In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain : \frac{1}{\lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) which is Rydberg's formula for the Lyman series. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. The Lyman limit is the short-wavelength end of the hydrogen Lyman series, at . The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. ==History== thumb|upright=1.3|The Lyman series The first line in the spectrum of the Lyman series was discovered in 1906 by physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. This energy is equivalent to the Rydberg constant. == See also == * Balmer Limit * Lyman-alpha emitter * Lyman-alpha forest * Lyman-break galaxy * Lyman series * Rydberg formula ==References== Category:Atomic physics Bohr found that the electron bound to the hydrogen atom must have quantized energy levels described by the following formula, : E_n = - \frac{m_e e^4}{2(4\pi\varepsilon_0\hbar)^2}\,\frac{1}{n^2} = - \frac{13.6\,\text{eV}}{n^2}. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. For example, the line is called ""Lyman-alpha"" (Ly-α), while the line is called ""Paschen-delta"" (Pa-δ). thumb|Energy level diagram of electrons in hydrogen atom There are emission lines from hydrogen that fall outside of these series, such as the 21 cm line. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. DOI: https://doi.org/10.18434/T4W30F 3 102.57220 4 97.253650 5 94.974287 6 93.780331 7 93.0748142 8 92.6225605 9 92.3150275 10 92.0963006 11 91.9351334 ∞, the Lyman limit 91.1753 ==Explanation and derivation== In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. ==The Lyman series== The version of the Rydberg formula that generated the Lyman series was: {1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} \approx 1.0968{\times}10^7\,\text{m}^{-1} \approx \frac{13.6\,\text{eV}}{hc} \right) where n is a natural number greater than or equal to 2 (i.e., ). All the wavelengths in the Lyman series are in the ultraviolet band.. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). For the Lyman series the naming convention is: *n = 2 to n = 1 is called Lyman- alpha, *n = 3 to n = 1 is called Lyman-beta, etc. H-alpha has a wavelength of 656.281 nm, is visible in the red part of the electromagnetic spectrum, and is the easiest way for astronomers to trace the ionized hydrogen content of gas clouds. ",+116.0,26.9,0.166666666,4.16,3,E
-A helium-neon laser (used in supermarket scanners) emits light at $632.8 \mathrm{~nm}$. Calculate the frequency of this light.,"thumb|Helium–neon laser at the University of Chemnitz, Germany A helium–neon laser or He-Ne laser, is a type of gas laser whose high energetic medium gain medium consists of a mixture of ratio(between 5:1 and 20:1) of helium and neon at a total pressure of about 1 torr inside of a small electrical discharge. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. ==History of He-Ne laser development== The first He-Ne lasers emitted infrared at 1150 nm, and were the first gas lasers and the first lasers with continuous wave output. The excited helium atoms collide with neon atoms, exciting some of them to the state that radiates 632.8 nm. The precise wavelength of red He-Ne lasers is 632.991 nm in a vacuum, which is refracted to about 632.816 nm in air. Absolute stabilization of the laser's frequency (or wavelength) as fine as 2.5 parts in 1011 can be obtained through use of an iodine absorption cell. thumb|Energy levels in a He-Ne Laser|512x330px thumb|Ring He-Ne Laser The mechanism producing population inversion and light amplification in a He-Ne laser plasma originates with inelastic collision of energetic electrons with ground-state helium atoms in the gas mixture. However, a laser that operated at visible wavelengths was much more in demand, and a number of other neon transitions were investigated to identify ones in which a population inversion can be achieved. Without helium, the neon atoms would be excited mostly to lower excited states, responsible for non-laser lines. thumb|right|300 px|A 337nm wavelength and 170 μJ pulse energy 20 Hz cartridge nitrogen laser A nitrogen laser is a gas laser operating in the ultraviolet rangeC. The optical cavity of the laser usually consists of two concave mirrors or one plane and one concave mirror: one having very high (typically 99.9%) reflectance, and the output coupler mirror allowing approximately 1% transmission. frame|Schematic diagram of a helium–neon laser Commercial He-Ne lasers are relatively small devices compared to other gas lasers, having cavity lengths usually ranging from 15 to 50 cm (but sometimes up to about 1 meter to achieve the highest powers), and optical output power levels ranging from 0.5 to 50 mW. It was developed at Bell Telephone Laboratories in 1962, 18 months after the pioneering demonstration at the same laboratory of the first continuous infrared He-Ne gas laser in December 1960. ==Construction and operation== The gain medium of the laser, as suggested by its name, is a mixture of helium and neon gases, in approximately a 10:1 ratio, contained at low pressure in a glass envelope. The 633 nm line was found to have the highest gain in the visible spectrum, making this the wavelength of choice for most He-Ne lasers. A blue laser emits electromagnetic radiation with a wavelength between 400 and 500 nanometers, which the human eye sees in the visible spectrum as blue or violet. The Nike laser at the United States Naval Research Laboratory in Washington, DC is a 56-beam, 4–5 kJ per pulse electron beam pumped krypton fluoride excimer laser which operates in the ultraviolet at 248 nm with pulsewidths of a few nanoseconds. A neon laser with no helium can be constructed, but it is much more difficult without this means of energy coupling. Violet light's 405nm short wavelength, on the visible spectrum, causes fluorescence in some chemicals, like radiation in the ultraviolet (""black light"") spectrum (wavelengths less than 400 nm). == History == thumb|445nm - 450nm Blue Laser (middle) Prior to the 1960s and until the late 1990s, gas and argon-ion lasers were common; suffering from poor efficiencies(0.01%) and large sizes. Laser gyroscopes have employed He-Ne lasers operating at 633 nm in a ring laser configuration. * Frequency-resolved electro-absorption gating (FREAG) ==References== * *R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, ""Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating,"" Review of Scientific Instruments 68, 3277-3295 (1997). ==External links== *FROG Page by Rick Trebino (co-inventor of FROG) Category:Nonlinear optics Category:Lasers Category:Optical metrology They most commonly emit light at 473 nm, which is produced by frequency doubling of 946 nm laser radiation from a diode-pumped Nd:YAG or Nd:YVO4 crystal. Conversion efficiency for producing 473 nm laser radiation is inefficient with some of the best lab produced results coming in at 10-15% efficient at converting 946 nm laser radiation to 473 nm laser radiation. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. Stimulated emissions are known from over 100 μm in the far infrared to 540 nm in the visible. ",0.444444444444444 ,5654.86677646,14.0,4.738,4500,D
-What is the uncertainty of the momentum of an electron if we know its position is somewhere in a $10 \mathrm{pm}$ interval?,"However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. One way to quantify the precision of the position and momentum is the standard deviation σ. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. * Problem 1 – If the photon has a short wavelength, and therefore, a large momentum, the position can be measured accurately. Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. On the other hand, the standard deviation of the position is \sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2} such that the uncertainty product can only increase with time as \sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2} ==Additional uncertainty relations== ===Systematic and statistical errors=== The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation \sigma. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. In doing so, we find the momentum of the particle to > arbitrary accuracy by conservation of momentum. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it. This precision may be quantified by the standard deviations, \sigma_x=\sqrt{\langle \hat{x}^2 \rangle-\langle \hat{x}\rangle^2} \sigma_p=\sqrt{\langle \hat{p}^2 \rangle-\langle \hat{p}\rangle^2}. Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely. H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. A more robust representation of measurement uncertainty in such cases can be fashioned from intervals.Manski, C.F. (2003); Partial Identification of Probability Distributions, Springer Series in Statistics, Springer, New YorkFerson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007); Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, Sandia National Laboratories SAND 2007-0939 An interval [a, b] is different from a rectangular or uniform probability distribution over the same range in that the latter suggests that the true value lies inside the right half of the range [(a + b)/2, b] with probability one half, and within any subinterval of [a, b] with probability equal to the width of the subinterval divided by b − a. ",0.33333333,37.9,0.3359, 6.6,1.92,D
-"Using the Bohr theory, calculate the ionization energy (in electron volts and in $\mathrm{kJ} \cdot \mathrm{mol}^{-1}$ ) of singly ionized helium.","To convert from ""value of ionization energy"" to the corresponding ""value of molar ionization energy"", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. These tables list values of molar ionization energies, measured in kJ⋅mol−1. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or ""HOMO"" and the lowest unoccupied molecular orbital or ""LUMO"", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. ""use"" and ""WEL"" give ionization energy in the unit kJ/mol; ""CRC"" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. This in turn makes its ionization energies increase by 18 kJ/mol−1. There are two main ways in which ionization energy is calculated. * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. == External links == * NIST Atomic Spectra Database Ionization Energies == See also == *Molar ionization energies of the elements Category:Properties of chemical elements Category:Chemical element data pages That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable. ==Analogs of ionization energy to other systems== While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems. ===Electron binding energy=== thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph). ===Solid surfaces: work function=== Work function is the minimum amount of energy required to remove an electron from a solid surface, where the work function for a given surface is defined by the difference :W = -e\phi - E_{\rm F}, where is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level (electrochemical potential of electrons) inside the material. ==Note== ==See also== * Rydberg equation, a calculation that could determine the ionization energies of hydrogen and hydrogen-like elements. In general, the computation for the Nth ionization energy requires calculating the energies of Z-N+1 and Z-N electron systems. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. Calculating these energies exactly is not possible except for the simplest systems (i.e. hydrogen and hydrogen-like elements), primarily because of difficulties in integrating the electron correlation terms. * Electron pairing energies: Half-filled subshells usually result in higher ionization energies. The adiabatic ionization is the diagonal transition to the vibrational ground state of the ion. In chemistry, it is expressed as the energy to ionize a mole of atoms or molecules, usually as kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol). Some values for elements of the third period are given in the following table: Successive ionization energy values / kJ mol−1 (96.485 kJ mol−1 ≡ 1 eV) Element First Second Third Fourth Fifth Sixth Seventh Na 496 4,560 Mg 738 1,450 7,730 Al 577 1,816 2,881 11,600 Si 786 1,577 3,228 4,354 16,100 P 1,060 1,890 2,905 4,950 6,270 21,200 S 1,000 2,295 3,375 4,565 6,950 8,490 27,107 Cl 1,256 2,260 3,850 5,160 6,560 9,360 11,000 Ar 1,520 2,665 3,945 5,770 7,230 8,780 12,000 Large jumps in the successive molar ionization energies occur when passing noble gas configurations. ",0.54,292,0.2553,"89,034.79",54.394,E
-"When an excited nucleus decays, it emits a $\gamma$ ray. The lifetime of an excited state of a nucleus is of the order of $10^{-12} \mathrm{~s}$. What is the uncertainty in the energy of the $\gamma$ ray produced?","The emission of a gamma ray from an excited nucleus typically requires only 10−12 seconds. Gamma rays from radioactive decay are in the energy range from a few kiloelectronvolts (keV) to approximately 8 megaelectronvolts (MeV), corresponding to the typical energy levels in nuclei with reasonably long lifetimes. In this type of decay, an excited nucleus emits a gamma ray almost immediately upon formation.It is now understood that a nuclear isomeric transition, however, can produce inhibited gamma decay with a measurable and much longer half-life. As in optical spectroscopy (see Franck–Condon effect) the absorption of gamma rays by a nucleus is especially likely (i.e., peaks in a ""resonance"") when the energy of the gamma ray is the same as that of an energy transition in the nucleus. Those excited states that lie below the separation energy for protons (Sp) decay by γ emission towards the ground state of daughter B. Then the excited decays to the ground state (see nuclear shell model) by emitting gamma rays in succession of 1.17 MeV followed by . Gamma decay is also a mode of relaxation of many excited states of atomic nuclei following other types of radioactive decay, such as beta decay, so long as these states possess the necessary component of nuclear spin. Because subatomic particles mostly have far shorter wavelengths than atomic nuclei, particle physics gamma rays are generally several orders of magnitude more energetic than nuclear decay gamma rays. Such nuclei have half-lifes that are more easily measurable, and rare nuclear isomers are able to stay in their excited state for minutes, hours, days, or occasionally far longer, before emitting a gamma ray. The decay energy is the energy change of a nucleus having undergone a radioactive decay. In some cases, the gamma emission spectrum of the daughter nucleus is quite simple, (e.g. /) while in other cases, such as with (/ and /), the gamma emission spectrum is complex, revealing that a series of nuclear energy levels exist. ===Particle physics=== Gamma rays are produced in many processes of particle physics. The rate of gamma decay is also slowed when the energy of excitation of the nucleus is small. thumb|350px|The decay of a proton rich nucleus A populates excited states of a daughter nucleus B by β+ emission or electron capture (EC). It is thought to be the principal absorption mechanism for gamma rays in the intermediate energy range 100 keV to 10 MeV. Gamma rays are produced by a number of astronomical processes in which very high-energy electrons are produced. Any gamma energy in excess of the equivalent rest mass of the two particles (totaling at least 1.02 MeV) appears as the kinetic energy of the pair and in the recoil of the emitting nucleus. The energy spectrum of gamma rays can be used to identify the decaying radionuclides using gamma spectroscopy. This is part and parcel of the general realization that many gamma rays produced in astronomical processes result not from radioactive decay or particle annihilation, but rather in non-radioactive processes similar to X-rays. However, when emitted gamma rays carry essentially all of the energy of the atomic nuclear de-excitation that produces them, this energy is also sufficient to excite the same energy state in a second immobilized nucleus of the same type. ==Applications== Gamma rays provide information about some of the most energetic phenomena in the universe; however, they are largely absorbed by the Earth's atmosphere. If the annihilating electron and positron are at rest, each of the resulting gamma rays has an energy of ~ 511 keV and frequency of ~ . Gamma rays are approximately 50% of the total energy output. There is no lower limit to the energy of photons produced by nuclear reactions, and thus ultraviolet or lower energy photons produced by these processes would also be defined as ""gamma rays"". ",71,2.567,1.154700538,460.5,7,E
-Calculate the wavelength and the energy of a photon associated with the series limit of the Lyman series.,"The Lyman limit is the short-wavelength end of the hydrogen Lyman series, at . This energy is equivalent to the Rydberg constant. == See also == * Balmer Limit * Lyman-alpha emitter * Lyman-alpha forest * Lyman-break galaxy * Lyman series * Rydberg formula ==References== Category:Atomic physics All the wavelengths in the Lyman series are in the ultraviolet band.. The wavelengths in the Lyman series are all ultraviolet: n Wavelength (nm) 2 121.56701Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2019). In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain : \frac{1}{\lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) which is Rydberg's formula for the Lyman series. This also means that the inverse of the Rydberg constant is equal to the Lyman limit. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. ==The Lyman series== The version of the Rydberg formula that generated the Lyman series was: {1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} \approx 1.0968{\times}10^7\,\text{m}^{-1} \approx \frac{13.6\,\text{eV}}{hc} \right) where n is a natural number greater than or equal to 2 (i.e., ). In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. ==History== thumb|upright=1.3|The Lyman series The first line in the spectrum of the Lyman series was discovered in 1906 by physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. DOI: https://doi.org/10.18434/T4W30F 3 102.57220 4 97.253650 5 94.974287 6 93.780331 7 93.0748142 8 92.6225605 9 92.3150275 10 92.0963006 11 91.9351334 ∞, the Lyman limit 91.1753 ==Explanation and derivation== In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. There is also a more comfortable notation when dealing with energy in units of electronvolts and wavelengths in units of angstroms, : \lambda = \frac{12398.4\,\text{eV}}{E_\text{i} - E_\text{f}} Å. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. Because of the spin–orbit interaction, the Lyman-alpha line splits into a fine-structure doublet with the wavelengths of 1215.668 and 1215.674 angstroms. In physical cosmology, the photon epoch was the period in the evolution of the early universe in which photons dominated the energy of the universe. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. The energy of an emitted photon corresponds to the energy difference between the two states. ",22,228,91.17,200,5.1,C
+Given that $D=7.31 \times 10^{-19} \mathrm{~J} \cdot$ molecule ${ }^{-1}$ and $\beta=1.81 \times 10^{10} \mathrm{~m}^{-1}$ for $\mathrm{HCl}$, calculate the force constant of $\mathrm{HCl}$.","In fact, the real molecular spectra are generally fit to the form1 : E_n / hc = \omega_e (n+1/2) - \omega_e\chi_e (n+1/2)^2\, in which the constants \omega_e and \omega_e\chi_e can be directly related to the parameters for the Morse potential. More sophisticated versions are used for polyatomic molecules. ==See also== *Lennard-Jones potential *Molecular mechanics ==References== *1 CRC Handbook of chemistry and physics, Ed David R. Lide, 87th ed, Section 9, SPECTROSCOPIC CONSTANTS OF DIATOMIC MOLECULES pp. 9–82 * * * * * * * * I.G. Kaplan, in Handbook of Molecular Physics and Quantum Chemistry, Wiley, 2003, p207. The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. The Hamaker constant provides the means to determine the interaction parameter from the vdW-pair potential, :w(r) = \frac{-C}{r^6}. The molecular formula C12H12O4 (molar mass: 220.22 g/mol, exact mass: 220.0736 u) may refer to: * Eugenitin * Hispolon * Siderin The molecular formula C16H19BrN2 (molar mass: 319.24 g/mol, exact mass: 318.0732 u) may refer to: * Brompheniramine * Dexbrompheniramine Category:Molecular formulas Mathematically, the spacing of Morse levels is :E_{n+1} - E_n = h u_0 - (n+1) (h u_0)^2/2D_e.\, This trend matches the anharmonicity found in real molecules. The Morse potential energy function is of the form :V(r) = D_e ( 1-e^{-a(r-r_e)} )^2 Here r is the distance between the atoms, r_e is the equilibrium bond distance, D_e is the well depth (defined relative to the dissociated atoms), and a controls the 'width' of the potential (the smaller a is, the larger the well). Unlike the energy levels of the harmonic oscillator potential, which are evenly spaced by ħω, the Morse potential level spacing decreases as the energy approaches the dissociation energy. A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs2, and this potential has since been used on HF, HCl, HBr and HI. == Function == The Morse/Long-range potential energy function is of the form V(r) = \mathfrak{D}_e \left( 1- \frac{u(r)}{u(r_e)} e^{-\beta(r) y_p^{r_{\rm{eq}}}(r)} \right)^2 where for large r, V(r) \simeq \mathfrak{D}_e - u(r) + \frac{u(r)^2}{4\mathfrak{D}_e}, so u(r) is defined according to the theoretically correct long-range behavior expected for the interatomic interaction. In molecular physics, the Hamaker constant (denoted ; named for H. C. Hamaker) is a physical constant that can be defined for a van der Waals (vdW) body–body interaction: :A=\pi^2C\rho_1\rho_2, where are the number densities of the two interacting kinds of particles, and is the London coefficient in the particle–particle pair interaction. The magnitude of this constant reflects the strength of the vdW-force between two particles, or between a particle and a substrate. The force constant (stiffness) of the bond can be found by Taylor expansion of V'(r) around r=r_e to the second derivative of the potential energy function, from which it can be shown that the parameter, a, is :a=\sqrt{k_e/2D_e}, where k_e is the force constant at the minimum of the well. right|thumb|illustrative example of C-C length molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C angle molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C-C torsion molecular energy dependence, numerical accuracy is not guaranteed A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration. The Morse/Long-range potential (MLR potential) is an interatomic interaction model for the potential energy of a diatomic molecule. Hamaker's method and the associated Hamaker constant ignores the influence of an intervening medium between the two particles of interaction. Since the zero of potential energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. When it is used to model the atom-surface interaction, the energy zero can be redefined so that the Morse potential becomes :V(r)= V'(r)-D_e = D_e ( 1-e^{-a(r-r_e)} )^2 -D_e which is usually written as :V(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} ) where r is now the coordinate perpendicular to the surface. It clearly shows that the Morse potential is the combination of a short-range repulsion term (the former) and a long-range attractive term (the latter), analogous to the Lennard-Jones potential. ==Vibrational states and energies== 500px|thumb|Harmonic oscillator (grey) and Morse (black) potentials curves are shown along with their eigenfunctions (respectively green and blue for harmonic oscillator and morse) for the same vibrational levels for nitrogen. The Morse potential can also be used to model other interactions such as the interaction between an atom and a surface. The latter is mathematically related to the particle mass, m, and the Morse constants via : u_0 = \frac{a}{2\pi} \sqrt{2D_e/m}. As is clear from dimensional analysis, for historical reasons the last equation uses spectroscopic notation in which \omega_e represents a wavenumber obeying E=hc\omega, and not an angular frequency given by E=\hbar\omega. == Morse/Long-range potential == An extension of the Morse potential that made the Morse form useful for modern (high-resolution) spectroscopy is the MLR (Morse/Long-range) potential. ",9.8,1.45,"""-194.0""",3.2,479,E
+A line in the Lyman series of hydrogen has a wavelength of $1.03 \times 10^{-7} \mathrm{~m}$. Find the original energy level of the electron.,"In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain : \frac{1}{\lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) which is Rydberg's formula for the Lyman series. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. The Lyman limit is the short-wavelength end of the hydrogen Lyman series, at . The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. ==History== thumb|upright=1.3|The Lyman series The first line in the spectrum of the Lyman series was discovered in 1906 by physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. This energy is equivalent to the Rydberg constant. == See also == * Balmer Limit * Lyman-alpha emitter * Lyman-alpha forest * Lyman-break galaxy * Lyman series * Rydberg formula ==References== Category:Atomic physics Bohr found that the electron bound to the hydrogen atom must have quantized energy levels described by the following formula, : E_n = - \frac{m_e e^4}{2(4\pi\varepsilon_0\hbar)^2}\,\frac{1}{n^2} = - \frac{13.6\,\text{eV}}{n^2}. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. For example, the line is called ""Lyman-alpha"" (Ly-α), while the line is called ""Paschen-delta"" (Pa-δ). thumb|Energy level diagram of electrons in hydrogen atom There are emission lines from hydrogen that fall outside of these series, such as the 21 cm line. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. DOI: https://doi.org/10.18434/T4W30F 3 102.57220 4 97.253650 5 94.974287 6 93.780331 7 93.0748142 8 92.6225605 9 92.3150275 10 92.0963006 11 91.9351334 ∞, the Lyman limit 91.1753 ==Explanation and derivation== In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. ==The Lyman series== The version of the Rydberg formula that generated the Lyman series was: {1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} \approx 1.0968{\times}10^7\,\text{m}^{-1} \approx \frac{13.6\,\text{eV}}{hc} \right) where n is a natural number greater than or equal to 2 (i.e., ). All the wavelengths in the Lyman series are in the ultraviolet band.. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). For the Lyman series the naming convention is: *n = 2 to n = 1 is called Lyman- alpha, *n = 3 to n = 1 is called Lyman-beta, etc. H-alpha has a wavelength of 656.281 nm, is visible in the red part of the electromagnetic spectrum, and is the easiest way for astronomers to trace the ionized hydrogen content of gas clouds. ",+116.0,26.9,"""0.166666666""",4.16,3,E
+A helium-neon laser (used in supermarket scanners) emits light at $632.8 \mathrm{~nm}$. Calculate the frequency of this light.,"thumb|Helium–neon laser at the University of Chemnitz, Germany A helium–neon laser or He-Ne laser, is a type of gas laser whose high energetic medium gain medium consists of a mixture of ratio(between 5:1 and 20:1) of helium and neon at a total pressure of about 1 torr inside of a small electrical discharge. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. ==History of He-Ne laser development== The first He-Ne lasers emitted infrared at 1150 nm, and were the first gas lasers and the first lasers with continuous wave output. The excited helium atoms collide with neon atoms, exciting some of them to the state that radiates 632.8 nm. The precise wavelength of red He-Ne lasers is 632.991 nm in a vacuum, which is refracted to about 632.816 nm in air. Absolute stabilization of the laser's frequency (or wavelength) as fine as 2.5 parts in 1011 can be obtained through use of an iodine absorption cell. thumb|Energy levels in a He-Ne Laser|512x330px thumb|Ring He-Ne Laser The mechanism producing population inversion and light amplification in a He-Ne laser plasma originates with inelastic collision of energetic electrons with ground-state helium atoms in the gas mixture. However, a laser that operated at visible wavelengths was much more in demand, and a number of other neon transitions were investigated to identify ones in which a population inversion can be achieved. Without helium, the neon atoms would be excited mostly to lower excited states, responsible for non-laser lines. thumb|right|300 px|A 337nm wavelength and 170 μJ pulse energy 20 Hz cartridge nitrogen laser A nitrogen laser is a gas laser operating in the ultraviolet rangeC. The optical cavity of the laser usually consists of two concave mirrors or one plane and one concave mirror: one having very high (typically 99.9%) reflectance, and the output coupler mirror allowing approximately 1% transmission. frame|Schematic diagram of a helium–neon laser Commercial He-Ne lasers are relatively small devices compared to other gas lasers, having cavity lengths usually ranging from 15 to 50 cm (but sometimes up to about 1 meter to achieve the highest powers), and optical output power levels ranging from 0.5 to 50 mW. It was developed at Bell Telephone Laboratories in 1962, 18 months after the pioneering demonstration at the same laboratory of the first continuous infrared He-Ne gas laser in December 1960. ==Construction and operation== The gain medium of the laser, as suggested by its name, is a mixture of helium and neon gases, in approximately a 10:1 ratio, contained at low pressure in a glass envelope. The 633 nm line was found to have the highest gain in the visible spectrum, making this the wavelength of choice for most He-Ne lasers. A blue laser emits electromagnetic radiation with a wavelength between 400 and 500 nanometers, which the human eye sees in the visible spectrum as blue or violet. The Nike laser at the United States Naval Research Laboratory in Washington, DC is a 56-beam, 4–5 kJ per pulse electron beam pumped krypton fluoride excimer laser which operates in the ultraviolet at 248 nm with pulsewidths of a few nanoseconds. A neon laser with no helium can be constructed, but it is much more difficult without this means of energy coupling. Violet light's 405nm short wavelength, on the visible spectrum, causes fluorescence in some chemicals, like radiation in the ultraviolet (""black light"") spectrum (wavelengths less than 400 nm). == History == thumb|445nm - 450nm Blue Laser (middle) Prior to the 1960s and until the late 1990s, gas and argon-ion lasers were common; suffering from poor efficiencies(0.01%) and large sizes. Laser gyroscopes have employed He-Ne lasers operating at 633 nm in a ring laser configuration. * Frequency-resolved electro-absorption gating (FREAG) ==References== * *R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, and D. J. Kane, ""Measuring Ultrashort Laser Pulses in the Time-Frequency Domain Using Frequency-Resolved Optical Gating,"" Review of Scientific Instruments 68, 3277-3295 (1997). ==External links== *FROG Page by Rick Trebino (co-inventor of FROG) Category:Nonlinear optics Category:Lasers Category:Optical metrology They most commonly emit light at 473 nm, which is produced by frequency doubling of 946 nm laser radiation from a diode-pumped Nd:YAG or Nd:YVO4 crystal. Conversion efficiency for producing 473 nm laser radiation is inefficient with some of the best lab produced results coming in at 10-15% efficient at converting 946 nm laser radiation to 473 nm laser radiation. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. Stimulated emissions are known from over 100 μm in the far infrared to 540 nm in the visible. ",0.444444444444444 ,5654.86677646,"""14.0""",4.738,4500,D
+What is the uncertainty of the momentum of an electron if we know its position is somewhere in a $10 \mathrm{pm}$ interval?,"However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. One way to quantify the precision of the position and momentum is the standard deviation σ. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. * Problem 1 – If the photon has a short wavelength, and therefore, a large momentum, the position can be measured accurately. Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. On the other hand, the standard deviation of the position is \sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2} such that the uncertainty product can only increase with time as \sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2} ==Additional uncertainty relations== ===Systematic and statistical errors=== The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation \sigma. In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. In doing so, we find the momentum of the particle to > arbitrary accuracy by conservation of momentum. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. He imagines an experimenter trying to measure the position and momentum of an electron by shooting a photon at it. This precision may be quantified by the standard deviations, \sigma_x=\sqrt{\langle \hat{x}^2 \rangle-\langle \hat{x}\rangle^2} \sigma_p=\sqrt{\langle \hat{p}^2 \rangle-\langle \hat{p}\rangle^2}. Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely. H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. A more robust representation of measurement uncertainty in such cases can be fashioned from intervals.Manski, C.F. (2003); Partial Identification of Probability Distributions, Springer Series in Statistics, Springer, New YorkFerson, S., V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg (2007); Experimental Uncertainty Estimation and Statistics for Data Having Interval Uncertainty, Sandia National Laboratories SAND 2007-0939 An interval [a, b] is different from a rectangular or uniform probability distribution over the same range in that the latter suggests that the true value lies inside the right half of the range [(a + b)/2, b] with probability one half, and within any subinterval of [a, b] with probability equal to the width of the subinterval divided by b − a. ",0.33333333,37.9,"""0.3359""", 6.6,1.92,D
+"Using the Bohr theory, calculate the ionization energy (in electron volts and in $\mathrm{kJ} \cdot \mathrm{mol}^{-1}$ ) of singly ionized helium.","To convert from ""value of ionization energy"" to the corresponding ""value of molar ionization energy"", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. These tables list values of molar ionization energies, measured in kJ⋅mol−1. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or ""HOMO"" and the lowest unoccupied molecular orbital or ""LUMO"", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. ""use"" and ""WEL"" give ionization energy in the unit kJ/mol; ""CRC"" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. This in turn makes its ionization energies increase by 18 kJ/mol−1. There are two main ways in which ionization energy is calculated. * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. == External links == * NIST Atomic Spectra Database Ionization Energies == See also == *Molar ionization energies of the elements Category:Properties of chemical elements Category:Chemical element data pages That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable. ==Analogs of ionization energy to other systems== While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems. ===Electron binding energy=== thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph). ===Solid surfaces: work function=== Work function is the minimum amount of energy required to remove an electron from a solid surface, where the work function for a given surface is defined by the difference :W = -e\phi - E_{\rm F}, where is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level (electrochemical potential of electrons) inside the material. ==Note== ==See also== * Rydberg equation, a calculation that could determine the ionization energies of hydrogen and hydrogen-like elements. In general, the computation for the Nth ionization energy requires calculating the energies of Z-N+1 and Z-N electron systems. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. Calculating these energies exactly is not possible except for the simplest systems (i.e. hydrogen and hydrogen-like elements), primarily because of difficulties in integrating the electron correlation terms. * Electron pairing energies: Half-filled subshells usually result in higher ionization energies. The adiabatic ionization is the diagonal transition to the vibrational ground state of the ion. In chemistry, it is expressed as the energy to ionize a mole of atoms or molecules, usually as kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol). Some values for elements of the third period are given in the following table: Successive ionization energy values / kJ mol−1 (96.485 kJ mol−1 ≡ 1 eV) Element First Second Third Fourth Fifth Sixth Seventh Na 496 4,560 Mg 738 1,450 7,730 Al 577 1,816 2,881 11,600 Si 786 1,577 3,228 4,354 16,100 P 1,060 1,890 2,905 4,950 6,270 21,200 S 1,000 2,295 3,375 4,565 6,950 8,490 27,107 Cl 1,256 2,260 3,850 5,160 6,560 9,360 11,000 Ar 1,520 2,665 3,945 5,770 7,230 8,780 12,000 Large jumps in the successive molar ionization energies occur when passing noble gas configurations. ",0.54,292,"""0.2553""","89,034.79",54.394,E
+"When an excited nucleus decays, it emits a $\gamma$ ray. The lifetime of an excited state of a nucleus is of the order of $10^{-12} \mathrm{~s}$. What is the uncertainty in the energy of the $\gamma$ ray produced?","The emission of a gamma ray from an excited nucleus typically requires only 10−12 seconds. Gamma rays from radioactive decay are in the energy range from a few kiloelectronvolts (keV) to approximately 8 megaelectronvolts (MeV), corresponding to the typical energy levels in nuclei with reasonably long lifetimes. In this type of decay, an excited nucleus emits a gamma ray almost immediately upon formation.It is now understood that a nuclear isomeric transition, however, can produce inhibited gamma decay with a measurable and much longer half-life. As in optical spectroscopy (see Franck–Condon effect) the absorption of gamma rays by a nucleus is especially likely (i.e., peaks in a ""resonance"") when the energy of the gamma ray is the same as that of an energy transition in the nucleus. Those excited states that lie below the separation energy for protons (Sp) decay by γ emission towards the ground state of daughter B. Then the excited decays to the ground state (see nuclear shell model) by emitting gamma rays in succession of 1.17 MeV followed by . Gamma decay is also a mode of relaxation of many excited states of atomic nuclei following other types of radioactive decay, such as beta decay, so long as these states possess the necessary component of nuclear spin. Because subatomic particles mostly have far shorter wavelengths than atomic nuclei, particle physics gamma rays are generally several orders of magnitude more energetic than nuclear decay gamma rays. Such nuclei have half-lifes that are more easily measurable, and rare nuclear isomers are able to stay in their excited state for minutes, hours, days, or occasionally far longer, before emitting a gamma ray. The decay energy is the energy change of a nucleus having undergone a radioactive decay. In some cases, the gamma emission spectrum of the daughter nucleus is quite simple, (e.g. /) while in other cases, such as with (/ and /), the gamma emission spectrum is complex, revealing that a series of nuclear energy levels exist. ===Particle physics=== Gamma rays are produced in many processes of particle physics. The rate of gamma decay is also slowed when the energy of excitation of the nucleus is small. thumb|350px|The decay of a proton rich nucleus A populates excited states of a daughter nucleus B by β+ emission or electron capture (EC). It is thought to be the principal absorption mechanism for gamma rays in the intermediate energy range 100 keV to 10 MeV. Gamma rays are produced by a number of astronomical processes in which very high-energy electrons are produced. Any gamma energy in excess of the equivalent rest mass of the two particles (totaling at least 1.02 MeV) appears as the kinetic energy of the pair and in the recoil of the emitting nucleus. The energy spectrum of gamma rays can be used to identify the decaying radionuclides using gamma spectroscopy. This is part and parcel of the general realization that many gamma rays produced in astronomical processes result not from radioactive decay or particle annihilation, but rather in non-radioactive processes similar to X-rays. However, when emitted gamma rays carry essentially all of the energy of the atomic nuclear de-excitation that produces them, this energy is also sufficient to excite the same energy state in a second immobilized nucleus of the same type. ==Applications== Gamma rays provide information about some of the most energetic phenomena in the universe; however, they are largely absorbed by the Earth's atmosphere. If the annihilating electron and positron are at rest, each of the resulting gamma rays has an energy of ~ 511 keV and frequency of ~ . Gamma rays are approximately 50% of the total energy output. There is no lower limit to the energy of photons produced by nuclear reactions, and thus ultraviolet or lower energy photons produced by these processes would also be defined as ""gamma rays"". ",71,2.567,"""1.154700538""",460.5,7,E
+Calculate the wavelength and the energy of a photon associated with the series limit of the Lyman series.,"The Lyman limit is the short-wavelength end of the hydrogen Lyman series, at . This energy is equivalent to the Rydberg constant. == See also == * Balmer Limit * Lyman-alpha emitter * Lyman-alpha forest * Lyman-break galaxy * Lyman series * Rydberg formula ==References== Category:Atomic physics All the wavelengths in the Lyman series are in the ultraviolet band.. The wavelengths in the Lyman series are all ultraviolet: n Wavelength (nm) 2 121.56701Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2019). In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For the connection between Bohr, Rydberg, and Lyman, one must replace m with 1 to obtain : \frac{1}{\lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) which is Rydberg's formula for the Lyman series. This also means that the inverse of the Rydberg constant is equal to the Lyman limit. Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable. ==The Lyman series== The version of the Rydberg formula that generated the Lyman series was: {1 \over \lambda} = R_\text{H} \left( 1 - \frac{1}{n^2} \right) \qquad \left( R_\text{H} \approx 1.0968{\times}10^7\,\text{m}^{-1} \approx \frac{13.6\,\text{eV}}{hc} \right) where n is a natural number greater than or equal to 2 (i.e., ). In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. The greater the difference in the principal quantum numbers, the higher the energy of the electromagnetic emission. ==History== thumb|upright=1.3|The Lyman series The first line in the spectrum of the Lyman series was discovered in 1906 by physicist Theodore Lyman, who was studying the ultraviolet spectrum of electrically excited hydrogen gas. DOI: https://doi.org/10.18434/T4W30F 3 102.57220 4 97.253650 5 94.974287 6 93.780331 7 93.0748142 8 92.6225605 9 92.3150275 10 92.0963006 11 91.9351334 ∞, the Lyman limit 91.1753 ==Explanation and derivation== In 1914, when Niels Bohr produced his Bohr model theory, the reason why hydrogen spectral lines fit Rydberg's formula was explained. Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. There is also a more comfortable notation when dealing with energy in units of electronvolts and wavelengths in units of angstroms, : \lambda = \frac{12398.4\,\text{eV}}{E_\text{i} - E_\text{f}} Å. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. Because of the spin–orbit interaction, the Lyman-alpha line splits into a fine-structure doublet with the wavelengths of 1215.668 and 1215.674 angstroms. In physical cosmology, the photon epoch was the period in the evolution of the early universe in which photons dominated the energy of the universe. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. The energy of an emitted photon corresponds to the energy difference between the two states. ",22,228,"""91.17""",200,5.1,C
 "Another application of the relationship given in Problem $1-48$ has to do with the excitedstate energies and lifetimes of atoms and molecules. If we know that the lifetime of an excited state is $10^{-9} \mathrm{~s}$, then what is the uncertainty in the energy of this state?
-","After about 85 years of existence of the uncertainty relation this problem was solved recently by Lorenzo Maccone and Arun K. Pati. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. The Heisenberg–Robertson–Schrödinger uncertainty relation was proved at the dawn of quantum formalism and is ever-present in the teaching and research on quantum mechanics. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. A system of highly excited atoms can form a long-lived condensed excited state e.g. a condensed phase made completely of excited atoms: Rydberg matter. == Perturbed gas excitation == A collection of molecules forming a gas can be considered in an excited state if one or more molecules are elevated to kinetic energy levels such that the resulting velocity distribution departs from the equilibrium Boltzmann distribution. This phenomenon has been studied in the case of a two-dimensional gas in some detail, analyzing the time taken to relax to equilibrium. == Calculation of excited states == Excited states are often calculated using coupled cluster, Møller–Plesset perturbation theory, multi-configurational self-consistent field, configuration interaction, and time-dependent density functional theory. ==Excited-state absorption== The excitation of a system (an atom or molecule) from one excited state to a higher-energy excited state with the absorption of a photon is called excited-state absorption (ESA). In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. Everett's Dissertation proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. ""One-parameter class of uncertainty relations based on entropy power"". By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). An atom in a high excited state is termed a Rydberg atom. One can prove an improved version of the Heisenberg–Robertson uncertainty relation which reads as : \Delta A \Delta B \ge \frac{ \pm \frac{i}{2} \langle \Psi|[A, B]|\Psi \rangle }{1- \frac{1}{2} | \langle \Psi|( \frac{A}{\Delta A} \pm i \frac{B}{\Delta B} )| {\bar \Psi} \rangle|^2 }. The other non-trivial stronger uncertainty relation is given by : \Delta A^2 + \Delta B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} |(A + B)| \Psi \rangle|^2, where | {\bar \Psi}_{A+B} \rangle is a unit vector orthogonal to |\Psi \rangle . The Heisenberg–Robertson uncertainty relation follows from the above uncertainty relation. ==Remarks== In quantum theory, one should distinguish between the uncertainty relation and the uncertainty principle. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. And since the whole oscillatory process (from r_{min} to infinity and back) is periodic, it is logical that this quantum mechanical problem has a stationary solution. === Second case === thumb|(a) Potential and wave function (on an arbitrary scale along the vertical axis) corresponding to zero energy, for the second case of the Wigner von Neumann SSC, A=1.5 (b) A=15. The potential would then be equal (with the corrected arithmetical error in the original article):Stillinger, F. H. & Herrick, D. R. Bound states in the continuum. ",7,0.366,1.4907,35,3.9,A
-"One of the most powerful modern techniques for studying structure is neutron diffraction. This technique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by $v_{\mathrm{n}}=\left(3 k_{\mathrm{B}} T / m\right)^{1 / 2}$, where $m$ is the mass of a neutron, then what temperature is needed so that the neutrons have a de Broglie wavelength of $50 \mathrm{pm}$ ?","The following is a detailed classification: === Thermal === A thermal neutron is a free neutron with a kinetic energy of about 0.025 eV (about 4.0×10−21 J or 2.4 MJ/kg, hence a speed of 2.19 km/s), which is the energy corresponding to the most probable speed at a temperature of 290 K (17 °C or 62 °F), the mode of the Maxwell–Boltzmann distribution for this temperature, Epeak = 1/2 k T. The momentum and wavelength of the neutron are related through the de Broglie relation. The large wavelength of slow neutrons allows for the large cross section. == Neutron energy distribution ranges == Neutron energy range names Neutron energy Energy range 0.0 – 0.025 eV Cold (slow) neutrons 0.025 eV Thermal neutrons (at 20°C) 0.025–0.4 eV Epithermal neutrons 0.4–0.5 eV Cadmium neutrons 0.5–10 eV Epicadmium neutrons 10–300 eV Resonance neutrons 300 eV–1 MeV Intermediate neutrons 1–20 MeV Fast neutrons > 20 MeV Ultrafast neutrons But different ranges with different names are observed in other sources. Efficient neutron optical components are being developed and optimized to remedy this lack. ===Resonance=== :*Refers to neutrons which are strongly susceptible to non-fission capture by U-238. :*1 eV to 300 eV ===Intermediate=== :*Neutrons that are between slow and fast :*Few hundred eV to 0.5 MeV. ===Fast=== : A fast neutron is a free neutron with a kinetic energy level close to 1 MeV (100 TJ/kg), hence a speed of 14,000 km/s or higher. Engineering diffraction refers to a sub-field of neutron scattering which investigates microstructural features that influence the mechanical properties of materials. Hydrogen Deuterium Beryllium Carbon Oxygen Uranium Mass of kernels u 1 2 9 12 16 238 Energy decrement \xi 1 0.7261 0.2078 0.1589 0.1209 0.0084 Number of Collisions 18 25 86 114 150 2172 ===Distribution of neutron velocities once moderated=== After sufficient impacts, the speed of the neutron will be comparable to the speed of the nuclei given by thermal motion; this neutron is then called a thermal neutron, and the process may also be termed thermalization. The wavelength of the neutrons used for reflectivity are typically on the order of 0.2 to 1 nm (2 to 10 Å). The characteristic neutron temperature of several-MeV neutrons is several tens of billions kelvin. Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. After a number of collisions with nuclei (scattering) in a medium (neutron moderator) at this temperature, those neutrons which are not absorbed reach about this energy level. According to the equipartition theorem, the average kinetic energy, \bar{E}, can be related to temperature, T, via: :\bar{E}=\frac{1}{2}m_n \langle v^2 \rangle=\frac{3}{2}k_B T, where m_n is the neutron mass, \langle v^2 \rangle is the average squared neutron speed, and k_B is the Boltzmann constant. However the range of neutrons from fission follows a Maxwell–Boltzmann distribution from 0 to about 14 MeV in the center of momentum frame of the disintegration, and the mode of the energy is only 0.75 MeV, meaning that fewer than half of fission neutrons qualify as ""fast"" even by the 1 MeV criterion.Byrne, J. Neutrons, Nuclei, and Matter, Dover Publications, Mineola, New York, 2011, (pbk.) The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. The main limitations of the use of slow neutrons is the low flux and the lack of efficient optical devices (in the case of CN and VCN). Neutron is a medium-lift two-stage launch vehicle under development by Rocket Lab. Announced on 1 March 2021, the vehicle is being designed to be capable of delivering a payload of to low Earth orbit in a partially reusable configuration, and will focus on the growing megaconstellation satellite delivery market. This is only slightly modified in a real moderator due to the speed (energy) dependence of the absorption cross-section of most materials, so that low-speed neutrons are preferentially absorbed,Neutron scattering lengths and cross sections V.F. Sears, Neutron News 3, No. 3, 26-37 (1992) so that the true neutron velocity distribution in the core would be slightly hotter than predicted. ==Reactor moderators== In a thermal-neutron reactor, the nucleus of a heavy fuel element such as uranium absorbs a slow-moving free neutron, becomes unstable, and then splits (""fissions"") into two smaller atoms (""fission products""). The neutron energy distribution is then adapted to the Maxwell distribution known for thermal motion. :Cold (slow) neutrons are subclassified into cold (CN), very cold (VCN), and ultra-cold (UCN) neutrons, each having particular characteristics in terms of their optical interactions with matter. This is done through numerous collisions with (in general) slower-moving and thus lower- temperature particles like atomic nuclei and other neutrons. A neutron research facility is most commonly a big laboratory operating a large-scale neutron source that provides thermal neutrons to a suite of research instruments. ",0.1800,2500,56.0,817.90,432.07,B
-The temperature of the fireball in a thermonuclear explosion can reach temperatures of approximately $10^7 \mathrm{~K}$. What value of $\lambda_{\max }$ does this correspond to? ,"The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The specific heat capacity has a sharp peak as the temperature approaches the lambda point. The entropy of vaporization of at its boiling point has the extraordinarily high value of 136.9 J/(K·mol). Hence, \lambda_{\rm th} = \frac{h}{\sqrt{2\pi m k_{\mathrm B} T}} , where h is the Planck constant, is the mass of a gas particle, k_{\mathrm B} is the Boltzmann constant, and is the temperature of the gas. The lambda point is the temperature at which normal fluid helium (helium I) makes the transition to superfluid helium II (approximately 2.17 K at 1 atmosphere). The thermometer indicates the current temperature, and the highest and lowest temperatures since the last reset. ==Description== Six's Maximum and Minimum thermometer consists of a U-shaped glass tube with two separate temperature scales set along each arm of the U. The current temperature is 23 degrees Celsius, the maximum recorded is 25, and the minimum is 15; both read from the base of the small markers in each arm of the U tube. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. The plasma parameter is a dimensionless number, denoted by capital Lambda, Λ. In thermodynamics, Trouton's rule states that the entropy of vaporization is almost the same value, about 85–88 J/(K·mol), for various kinds of liquids at their boiling points.Compare 85 J/(K·mol) in and 88 J/(K·mol) in The entropy of vaporization is defined as the ratio between the enthalpy of vaporization and the boiling temperature. Accessed April 2011 thumb|200px|right|Detail of the thermometer bulbs of the maximum-minimum thermometer shown above. The kelvin, symbol K, is a unit of measurement for temperature. The point's name derives from the graph (pictured) that results from plotting the specific heat capacity as a function of temperature (for a given pressure in the above range, in the example shown, at 1 atmosphere), which resembles the Greek letter lambda \lambda. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. In this case, the plasma parameter is given by:Chen, F.F., Introduction to Plasma Physics and Controlled Fusion, (Springer, New York, 2006) \Lambda = 4\pi n_\text{e}\lambda_\text{D}^3 where * ne is the number density of electrons, * λD is the Debye length. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. Electronic thermometers often include a maximum-minimum registering feature. ==See also== * Maximum minimum temperature system ==References== * Two hundred years of the Six's Self Registering Thermometer Austin and McConnell, Notes and Records of the Royal Society of London * Volume. 35, No. 1, July., 1980 at JSTOR * A History of the Thermometer and Its Uses in Meteorology by Amit Batra, Johns Hopkins University Press, 1966; * The Construction of a Thermometer by James Six, Nimbus Publishing Ltd,1980; ==External links== * Article on Six's thermometer at the Museum of the History of Science at Florence, Italy * Explanation of the working of Six's thermometer * Category:Thermometers If the temperature rises, the maximum scale marker will be pushed. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. A similar way of stating this (Trouton's ratio) is that the latent heat is connected to boiling point roughly as : \frac{L_\text{vap}}{T_\text{boiling}} \approx 85{-}88\ \frac{\text{J}}{\text{K} \cdot \text{mol}}. Six's maximum and minimum thermometer is a registering thermometer that can record the maximum and minimum temperatures reached over a period of time, for example 24 hours. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. ",4943,4.86,3.0,0.65625,0.8561,C
+","After about 85 years of existence of the uncertainty relation this problem was solved recently by Lorenzo Maccone and Arun K. Pati. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. The Heisenberg–Robertson–Schrödinger uncertainty relation was proved at the dawn of quantum formalism and is ever-present in the teaching and research on quantum mechanics. It turns out that Heisenberg's uncertainty principle can be expressed as a lower bound on the sum of these entropies. A system of highly excited atoms can form a long-lived condensed excited state e.g. a condensed phase made completely of excited atoms: Rydberg matter. == Perturbed gas excitation == A collection of molecules forming a gas can be considered in an excited state if one or more molecules are elevated to kinetic energy levels such that the resulting velocity distribution departs from the equilibrium Boltzmann distribution. This phenomenon has been studied in the case of a two-dimensional gas in some detail, analyzing the time taken to relax to equilibrium. == Calculation of excited states == Excited states are often calculated using coupled cluster, Møller–Plesset perturbation theory, multi-configurational self-consistent field, configuration interaction, and time-dependent density functional theory. ==Excited-state absorption== The excitation of a system (an atom or molecule) from one excited state to a higher-energy excited state with the absorption of a photon is called excited-state absorption (ESA). In quantum mechanics, information theory, and Fourier analysis, the entropic uncertainty or Hirschman uncertainty is defined as the sum of the temporal and spectral Shannon entropies. Everett's Dissertation proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. ""One-parameter class of uncertainty relations based on entropy power"". By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). An atom in a high excited state is termed a Rydberg atom. One can prove an improved version of the Heisenberg–Robertson uncertainty relation which reads as : \Delta A \Delta B \ge \frac{ \pm \frac{i}{2} \langle \Psi|[A, B]|\Psi \rangle }{1- \frac{1}{2} | \langle \Psi|( \frac{A}{\Delta A} \pm i \frac{B}{\Delta B} )| {\bar \Psi} \rangle|^2 }. The other non-trivial stronger uncertainty relation is given by : \Delta A^2 + \Delta B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} |(A + B)| \Psi \rangle|^2, where | {\bar \Psi}_{A+B} \rangle is a unit vector orthogonal to |\Psi \rangle . The Heisenberg–Robertson uncertainty relation follows from the above uncertainty relation. ==Remarks== In quantum theory, one should distinguish between the uncertainty relation and the uncertainty principle. This is stronger than the usual statement of the uncertainty principle in terms of the product of standard deviations. And since the whole oscillatory process (from r_{min} to infinity and back) is periodic, it is logical that this quantum mechanical problem has a stationary solution. === Second case === thumb|(a) Potential and wave function (on an arbitrary scale along the vertical axis) corresponding to zero energy, for the second case of the Wigner von Neumann SSC, A=1.5 (b) A=15. The potential would then be equal (with the corrected arithmetical error in the original article):Stillinger, F. H. & Herrick, D. R. Bound states in the continuum. ",7,0.366,"""1.4907""",35,3.9,A
+"One of the most powerful modern techniques for studying structure is neutron diffraction. This technique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by $v_{\mathrm{n}}=\left(3 k_{\mathrm{B}} T / m\right)^{1 / 2}$, where $m$ is the mass of a neutron, then what temperature is needed so that the neutrons have a de Broglie wavelength of $50 \mathrm{pm}$ ?","The following is a detailed classification: === Thermal === A thermal neutron is a free neutron with a kinetic energy of about 0.025 eV (about 4.0×10−21 J or 2.4 MJ/kg, hence a speed of 2.19 km/s), which is the energy corresponding to the most probable speed at a temperature of 290 K (17 °C or 62 °F), the mode of the Maxwell–Boltzmann distribution for this temperature, Epeak = 1/2 k T. The momentum and wavelength of the neutron are related through the de Broglie relation. The large wavelength of slow neutrons allows for the large cross section. == Neutron energy distribution ranges == Neutron energy range names Neutron energy Energy range 0.0 – 0.025 eV Cold (slow) neutrons 0.025 eV Thermal neutrons (at 20°C) 0.025–0.4 eV Epithermal neutrons 0.4–0.5 eV Cadmium neutrons 0.5–10 eV Epicadmium neutrons 10–300 eV Resonance neutrons 300 eV–1 MeV Intermediate neutrons 1–20 MeV Fast neutrons > 20 MeV Ultrafast neutrons But different ranges with different names are observed in other sources. Efficient neutron optical components are being developed and optimized to remedy this lack. ===Resonance=== :*Refers to neutrons which are strongly susceptible to non-fission capture by U-238. :*1 eV to 300 eV ===Intermediate=== :*Neutrons that are between slow and fast :*Few hundred eV to 0.5 MeV. ===Fast=== : A fast neutron is a free neutron with a kinetic energy level close to 1 MeV (100 TJ/kg), hence a speed of 14,000 km/s or higher. Engineering diffraction refers to a sub-field of neutron scattering which investigates microstructural features that influence the mechanical properties of materials. Hydrogen Deuterium Beryllium Carbon Oxygen Uranium Mass of kernels u 1 2 9 12 16 238 Energy decrement \xi 1 0.7261 0.2078 0.1589 0.1209 0.0084 Number of Collisions 18 25 86 114 150 2172 ===Distribution of neutron velocities once moderated=== After sufficient impacts, the speed of the neutron will be comparable to the speed of the nuclei given by thermal motion; this neutron is then called a thermal neutron, and the process may also be termed thermalization. The wavelength of the neutrons used for reflectivity are typically on the order of 0.2 to 1 nm (2 to 10 Å). The characteristic neutron temperature of several-MeV neutrons is several tens of billions kelvin. Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. After a number of collisions with nuclei (scattering) in a medium (neutron moderator) at this temperature, those neutrons which are not absorbed reach about this energy level. According to the equipartition theorem, the average kinetic energy, \bar{E}, can be related to temperature, T, via: :\bar{E}=\frac{1}{2}m_n \langle v^2 \rangle=\frac{3}{2}k_B T, where m_n is the neutron mass, \langle v^2 \rangle is the average squared neutron speed, and k_B is the Boltzmann constant. However the range of neutrons from fission follows a Maxwell–Boltzmann distribution from 0 to about 14 MeV in the center of momentum frame of the disintegration, and the mode of the energy is only 0.75 MeV, meaning that fewer than half of fission neutrons qualify as ""fast"" even by the 1 MeV criterion.Byrne, J. Neutrons, Nuclei, and Matter, Dover Publications, Mineola, New York, 2011, (pbk.) The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. The main limitations of the use of slow neutrons is the low flux and the lack of efficient optical devices (in the case of CN and VCN). Neutron is a medium-lift two-stage launch vehicle under development by Rocket Lab. Announced on 1 March 2021, the vehicle is being designed to be capable of delivering a payload of to low Earth orbit in a partially reusable configuration, and will focus on the growing megaconstellation satellite delivery market. This is only slightly modified in a real moderator due to the speed (energy) dependence of the absorption cross-section of most materials, so that low-speed neutrons are preferentially absorbed,Neutron scattering lengths and cross sections V.F. Sears, Neutron News 3, No. 3, 26-37 (1992) so that the true neutron velocity distribution in the core would be slightly hotter than predicted. ==Reactor moderators== In a thermal-neutron reactor, the nucleus of a heavy fuel element such as uranium absorbs a slow-moving free neutron, becomes unstable, and then splits (""fissions"") into two smaller atoms (""fission products""). The neutron energy distribution is then adapted to the Maxwell distribution known for thermal motion. :Cold (slow) neutrons are subclassified into cold (CN), very cold (VCN), and ultra-cold (UCN) neutrons, each having particular characteristics in terms of their optical interactions with matter. This is done through numerous collisions with (in general) slower-moving and thus lower- temperature particles like atomic nuclei and other neutrons. A neutron research facility is most commonly a big laboratory operating a large-scale neutron source that provides thermal neutrons to a suite of research instruments. ",0.1800,2500,"""56.0""",817.90,432.07,B
+The temperature of the fireball in a thermonuclear explosion can reach temperatures of approximately $10^7 \mathrm{~K}$. What value of $\lambda_{\max }$ does this correspond to? ,"The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The specific heat capacity has a sharp peak as the temperature approaches the lambda point. The entropy of vaporization of at its boiling point has the extraordinarily high value of 136.9 J/(K·mol). Hence, \lambda_{\rm th} = \frac{h}{\sqrt{2\pi m k_{\mathrm B} T}} , where h is the Planck constant, is the mass of a gas particle, k_{\mathrm B} is the Boltzmann constant, and is the temperature of the gas. The lambda point is the temperature at which normal fluid helium (helium I) makes the transition to superfluid helium II (approximately 2.17 K at 1 atmosphere). The thermometer indicates the current temperature, and the highest and lowest temperatures since the last reset. ==Description== Six's Maximum and Minimum thermometer consists of a U-shaped glass tube with two separate temperature scales set along each arm of the U. The current temperature is 23 degrees Celsius, the maximum recorded is 25, and the minimum is 15; both read from the base of the small markers in each arm of the U tube. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. The plasma parameter is a dimensionless number, denoted by capital Lambda, Λ. In thermodynamics, Trouton's rule states that the entropy of vaporization is almost the same value, about 85–88 J/(K·mol), for various kinds of liquids at their boiling points.Compare 85 J/(K·mol) in and 88 J/(K·mol) in The entropy of vaporization is defined as the ratio between the enthalpy of vaporization and the boiling temperature. Accessed April 2011 thumb|200px|right|Detail of the thermometer bulbs of the maximum-minimum thermometer shown above. The kelvin, symbol K, is a unit of measurement for temperature. The point's name derives from the graph (pictured) that results from plotting the specific heat capacity as a function of temperature (for a given pressure in the above range, in the example shown, at 1 atmosphere), which resembles the Greek letter lambda \lambda. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. In this case, the plasma parameter is given by:Chen, F.F., Introduction to Plasma Physics and Controlled Fusion, (Springer, New York, 2006) \Lambda = 4\pi n_\text{e}\lambda_\text{D}^3 where * ne is the number density of electrons, * λD is the Debye length. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. Electronic thermometers often include a maximum-minimum registering feature. ==See also== * Maximum minimum temperature system ==References== * Two hundred years of the Six's Self Registering Thermometer Austin and McConnell, Notes and Records of the Royal Society of London * Volume. 35, No. 1, July., 1980 at JSTOR * A History of the Thermometer and Its Uses in Meteorology by Amit Batra, Johns Hopkins University Press, 1966; * The Construction of a Thermometer by James Six, Nimbus Publishing Ltd,1980; ==External links== * Article on Six's thermometer at the Museum of the History of Science at Florence, Italy * Explanation of the working of Six's thermometer * Category:Thermometers If the temperature rises, the maximum scale marker will be pushed. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. A similar way of stating this (Trouton's ratio) is that the latent heat is connected to boiling point roughly as : \frac{L_\text{vap}}{T_\text{boiling}} \approx 85{-}88\ \frac{\text{J}}{\text{K} \cdot \text{mol}}. Six's maximum and minimum thermometer is a registering thermometer that can record the maximum and minimum temperatures reached over a period of time, for example 24 hours. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. ",4943,4.86,"""3.0""",0.65625,0.8561,C
 "Show that l'Hôpital's rule amounts to forming a Taylor expansion of both the numerator and the denominator. Evaluate the limit
 $$
 \lim _{x \rightarrow 0} \frac{\ln (1+x)-x}{x^2}
 $$
-both ways.","Applying L'Hôpital's rule and finding the derivatives with respect to of the numerator and the denominator yields as expected. Applying L'Hôpital's rule a single time still results in an indeterminate form. Limitations of the Taylor rule include. The \ln(1 + x) = x approximation is used here. Applying L'Hopital's rule shows that f'(a) := \lim_{x\to a}\frac{f(x)-f(a)}{x-a} = \lim_{x\to a}\frac{h(x)}{g(x)} = \lim_{x\to a}f'(x). == See also == * L'Hôpital controversy == Notes == == References == === Sources === * * * * * Category:Articles containing proofs Category:Theorems in calculus Category:Theorems in real analysis Category:Limits (mathematics) For example, to evaluate a limit involving , convert the difference of two functions to a quotient: : \begin{align} \lim_{x\to 1}\left(\frac{x}{x-1}-\frac1{\ln x}\right) & = \lim_{x\to 1}\frac{x\cdot\ln x -x+1}{(x-1)\cdot\ln x} & \quad (1) \\\\[6pt] & = \lim_{x\to 1}\frac{\ln x}{\frac{x-1}{x}+\ln x} & \quad (2) \\\\[6pt] & = \lim_{x\to 1}\frac{x\cdot\ln x}{x-1+x\cdot\ln x} & \quad (3) \\\\[6pt] & = \lim_{x\to 1}\frac{1+\ln x}{1+1+\ln x} & \quad (4) \\\\[6pt] & = \lim_{x\to 1}\frac{1+\ln x}{2+\ln x} \\\\[6pt] & = \frac{1}{2}, \end{align} where L'Hôpital's rule is applied when going from (1) to (2) and again when going from (3) to (4). L'Hôpital's rule can be used on indeterminate forms involving exponents by using logarithms to ""move the exponent down"". The limit \lim_{x\to 0^+}x\cdot\ln x is of the indeterminate form , but as shown in an example above, l'Hôpital's rule may be used to determine that :\lim_{x\to 0^+}x\cdot\ln x = 0. * Here is an example involving the indeterminate form (see below), which is rewritten as the form : \lim_{x\to 0^+}x \ln x =\lim_{x\to 0^+} \frac{\ln x}{\frac{1}{x}} = \lim_{x\to 0^+} \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim_{x\to 0^+} -x = 0. If is twice-differentiable in a neighborhood of and that its second derivative is continuous on this neighbourhood, then \begin{align} \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2} &= \lim_{h\to 0}\frac{f'(x+h)-f'(x-h)}{2h} \\\\[4pt] &= \lim_{h\to 0}\frac{f(x+h) + f(x-h)}{2} \\\\[4pt] &= f(x). \end{align} * Sometimes L'Hôpital's rule is invoked in a tricky way: suppose converges as and that e^x\cdot f(x) converges to positive or negative infinity. L'Hôpital's rule states that for functions and which are differentiable on an open interval except possibly at a point contained in , if \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0 \text{ or } \pm\infty, and g'(x) e 0 for all in with , and \lim_{x\to c}\frac{f'(x)}{g'(x)} exists, then :\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}. Again, an alternative approach is to multiply numerator and denominator by x^{1/2} before applying L'Hôpital's rule: \lim_{x\to\infty} \frac{x^\frac{1}{2}+x^{-\frac{1}{2}}}{x^\frac{1}{2}-x^{-\frac{1}{2}}} = \lim_{x\to\infty} \frac{x+1}{x-1} = \lim_{x\to\infty} \frac{1}{1} = 1. L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Extract of page 321 == General form == The general form of L'Hôpital's rule covers many cases. L'Hôpital's rule then states that the slope of the curve when is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined. == Proof of L'Hôpital's rule == ===Special case=== The proof of L'Hôpital's rule is simple in the case where and are continuously differentiable at the point and where a finite limit is found after the first round of differentiation. A common pitfall is using L'Hôpital's rule with some circular reasoning to compute a derivative via a difference quotient. It is not a proof of the general L'Hôpital's rule because it is stricter in its definition, requiring both differentiability and that c be a real number. Thus, since \lim_{x\to c} \frac{f(x)}{g(x)} = \frac{0}{0} and \lim_{x\to c} \frac{f'(x)}{g'(x)} exists, L'Hôpital's rule still holds. === Derivative of denominator is zero === The necessity of the condition that g'(x) e 0 near c can be seen by the following counterexample due to Otto Stolz. The limit in the conclusion is not indeterminate because g'(c) e 0. It follows that :\int_0^x \frac{dt}{1+t}=\int_0^x \left(1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}\right)\ dt and by termwise integration, :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{n-1}\frac{x^n}{n}+(-1)^n \int_0^x \frac{t^n}{1+t}\ dt. Repeatedly apply L'Hôpital's rule until the exponent is zero (if is an integer) or negative (if is fractional) to conclude that the limit is zero. In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. ",-0.5,0.178,4.85,7200,7,A
+both ways.","Applying L'Hôpital's rule and finding the derivatives with respect to of the numerator and the denominator yields as expected. Applying L'Hôpital's rule a single time still results in an indeterminate form. Limitations of the Taylor rule include. The \ln(1 + x) = x approximation is used here. Applying L'Hopital's rule shows that f'(a) := \lim_{x\to a}\frac{f(x)-f(a)}{x-a} = \lim_{x\to a}\frac{h(x)}{g(x)} = \lim_{x\to a}f'(x). == See also == * L'Hôpital controversy == Notes == == References == === Sources === * * * * * Category:Articles containing proofs Category:Theorems in calculus Category:Theorems in real analysis Category:Limits (mathematics) For example, to evaluate a limit involving , convert the difference of two functions to a quotient: : \begin{align} \lim_{x\to 1}\left(\frac{x}{x-1}-\frac1{\ln x}\right) & = \lim_{x\to 1}\frac{x\cdot\ln x -x+1}{(x-1)\cdot\ln x} & \quad (1) \\\\[6pt] & = \lim_{x\to 1}\frac{\ln x}{\frac{x-1}{x}+\ln x} & \quad (2) \\\\[6pt] & = \lim_{x\to 1}\frac{x\cdot\ln x}{x-1+x\cdot\ln x} & \quad (3) \\\\[6pt] & = \lim_{x\to 1}\frac{1+\ln x}{1+1+\ln x} & \quad (4) \\\\[6pt] & = \lim_{x\to 1}\frac{1+\ln x}{2+\ln x} \\\\[6pt] & = \frac{1}{2}, \end{align} where L'Hôpital's rule is applied when going from (1) to (2) and again when going from (3) to (4). L'Hôpital's rule can be used on indeterminate forms involving exponents by using logarithms to ""move the exponent down"". The limit \lim_{x\to 0^+}x\cdot\ln x is of the indeterminate form , but as shown in an example above, l'Hôpital's rule may be used to determine that :\lim_{x\to 0^+}x\cdot\ln x = 0. * Here is an example involving the indeterminate form (see below), which is rewritten as the form : \lim_{x\to 0^+}x \ln x =\lim_{x\to 0^+} \frac{\ln x}{\frac{1}{x}} = \lim_{x\to 0^+} \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim_{x\to 0^+} -x = 0. If is twice-differentiable in a neighborhood of and that its second derivative is continuous on this neighbourhood, then \begin{align} \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2} &= \lim_{h\to 0}\frac{f'(x+h)-f'(x-h)}{2h} \\\\[4pt] &= \lim_{h\to 0}\frac{f(x+h) + f(x-h)}{2} \\\\[4pt] &= f(x). \end{align} * Sometimes L'Hôpital's rule is invoked in a tricky way: suppose converges as and that e^x\cdot f(x) converges to positive or negative infinity. L'Hôpital's rule states that for functions and which are differentiable on an open interval except possibly at a point contained in , if \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0 \text{ or } \pm\infty, and g'(x) e 0 for all in with , and \lim_{x\to c}\frac{f'(x)}{g'(x)} exists, then :\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}. Again, an alternative approach is to multiply numerator and denominator by x^{1/2} before applying L'Hôpital's rule: \lim_{x\to\infty} \frac{x^\frac{1}{2}+x^{-\frac{1}{2}}}{x^\frac{1}{2}-x^{-\frac{1}{2}}} = \lim_{x\to\infty} \frac{x+1}{x-1} = \lim_{x\to\infty} \frac{1}{1} = 1. L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Extract of page 321 == General form == The general form of L'Hôpital's rule covers many cases. L'Hôpital's rule then states that the slope of the curve when is the limit of the slope of the tangent to the curve as the curve approaches the origin, provided that this is defined. == Proof of L'Hôpital's rule == ===Special case=== The proof of L'Hôpital's rule is simple in the case where and are continuously differentiable at the point and where a finite limit is found after the first round of differentiation. A common pitfall is using L'Hôpital's rule with some circular reasoning to compute a derivative via a difference quotient. It is not a proof of the general L'Hôpital's rule because it is stricter in its definition, requiring both differentiability and that c be a real number. Thus, since \lim_{x\to c} \frac{f(x)}{g(x)} = \frac{0}{0} and \lim_{x\to c} \frac{f'(x)}{g'(x)} exists, L'Hôpital's rule still holds. === Derivative of denominator is zero === The necessity of the condition that g'(x) e 0 near c can be seen by the following counterexample due to Otto Stolz. The limit in the conclusion is not indeterminate because g'(c) e 0. It follows that :\int_0^x \frac{dt}{1+t}=\int_0^x \left(1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}\right)\ dt and by termwise integration, :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots+(-1)^{n-1}\frac{x^n}{n}+(-1)^n \int_0^x \frac{t^n}{1+t}\ dt. Repeatedly apply L'Hôpital's rule until the exponent is zero (if is an integer) or negative (if is fractional) to conclude that the limit is zero. In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. ",-0.5,0.178,"""4.85""",7200,7,A
 "Evaluate the series
 $$
 S=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2^n}
-$$","The Mercator series provides an analytic expression of the natural logarithm: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n \;=\; \ln (1+x). It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. The constant can also be explicitly defined by the following infinite sums: : 0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right). As another example, by Mercator series \ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. The constant relates to the divergent series: :\sum_{k=1}^{\infty} (-1)^k k^{1/k}. In mathematics, an alternating series is an infinite series of the form \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n with for all . In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. The series converges to the natural logarithm (shifted by 1) whenever -1 . ==History== The series was discovered independently by Johannes Hudde and Isaac Newton. But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for \tfrac 1 2 \ln(2): \begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\\\[8pt] & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\\\[8pt] & = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2). \end{align} == Series acceleration == In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. :\sum_{k=0}^{n} (-2)^k As a series of real numbers it diverges, so in the usual sense it has no sum. Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7). In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. The closed form geometric series 1 / (1 - r) is the black dashed line. See for example Grandi's series: 1 − 1 + 1 − 1 + ···. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Alternatively, one can start with the finite geometric series (t e -1) :1-t+t^2-\cdots+(-t)^{n-1}=\frac{1-(-t)^n}{1+t} which gives :\frac1{1+t}=1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}. That flipping behavior near r = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with a = 1 and |r| < 1\. ",1.07,1.22,-36.5,-1.00,0.3333333,E
-Calculate the percentage difference between $\ln (1+x)$ and $x$ for $x=0.0050$,"thumb|400px|The logarithmic decrement can be obtained e.g. as ln(x1/x3). A percentage point or percent point is the unit for the arithmetic difference between two percentages. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). == Related units == * Percentage (%) 1 part in 100 * Per mille (‰) 1 part in 1,000 *Basis point (bp) difference of 1 part in 10,000 *Permyriad (‱) 1 part in 10,000 * Per cent mille (pcm) 1 part in 100,000 * Baker percentage == See also == * Parts-per notation * Per-unit system * Percent point function * Relative change and difference == References == Category:Mathematical terminology Category:Probability assessment Category:Units of measurement ru:Процент#Процентный пункт In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, :(\ln f)' = \frac{f'}{f} \quad \implies \quad f' = f \cdot (\ln f)'. thumbnail|upright=1.3|Plot of logit(x) in the domain of 0 to 1, where the base of the logarithm is e. In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : -\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions. Mathematically, the logit is the inverse of the standard logistic function \sigma(x) = 1/(1+e^{-x}), so the logit is defined as :\operatorname{logit} p = \sigma^{-1}(p) = \ln \frac{p}{1-p} \quad \text{for} \quad p \in (0,1). Percentage-point differences are one way to express a risk or probability. After the first occurrence, some writers abbreviate by using just ""point"" or ""points"". ==Differences between percentages and percentage points== Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. Differentiating by applying the chain and the sum rules yields :\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}, and, after rearranging, yields :f'(x) = f(x)\times \Bigg\\{\frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}\Bigg\\}= g(x)h(x)\times \Bigg\\{\frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}\Bigg\\}=g'(x)h(x)+g(x)h'(x), which is the product rule for derivatives. ===Quotients=== A natural logarithm is applied to a quotient of two functions :f(x)=\frac{g(x)}{h(x)}\,\\! to transform the division into a subtraction :\ln(f(x))=\ln\Bigg(\frac{g(x)}{h(x)}\Bigg)=\ln(g(x))-\ln(h(x))\,\\! For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. Percentage solution may refer to: * Mass fraction (or ""% w/w"" or ""wt.%""), for percent mass * Volume fraction (or ""% v/v"" or ""vol.%""), volume concentration, for percent volume * ""Mass/volume percentage"" (or ""% m/v"") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). Fisher described the logarithmic distribution in a paper that used it to model relative species abundance. ==See also== * Poisson distribution (also derived from a Maclaurin series) ==References== ==Further reading== * * Category:Discrete distributions Category:Logarithms Thus, the logit is a type of function that maps probability values from (0, 1) to real numbers in (-\infty, +\infty), akin to the probit function. ==Definition== If is a probability, then is the corresponding odds; the of the probability is the logarithm of the odds, i.e.: :\operatorname{logit}(p)=\ln\left( \frac{p}{1-p} \right) =\ln(p)-\ln(1-p)=-\ln\left( \frac{1}{p}-1\right)=2\operatorname{atanh}(2p-1) The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base is the one most often used. The cumulative distribution function is : F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)} where B is the incomplete beta function. The application of natural logarithms results in (with capital sigma notation) :\ln (f(x))=\sum_i\alpha_i(x)\cdot \ln(f_i(x)), and after differentiation, :\frac{f'(x)}{f(x)}=\sum_i\left[\alpha_i'(x)\cdot \ln(f_i(x))+\alpha_i(x)\cdot \frac{f_i'(x)}{f_i(x)}\right]. Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds \frac{p}{1-p} where is a probability. In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. In fact, the is the quantile function of the logistic distribution, while the is the quantile function of the normal distribution. ",0.3359,0.249,540.0,1.5,3.8,B
-Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00.,"The molecular formula C14H14 (molar mass: 182.26 g/mol, exact mass: 182.1096 u) may refer to: * Bibenzyl * Cyclotetradecaheptaene, or [14]annulene Category:Molecular formulas The molecular formula C11H14N2 (molar mass : 174.24 g/mol, exact mass : 174.115698) may refer to : * 6-(2-Aminopropyl)indole * Gramine * 5-IT * α-Methyltryptamine * N-Methyltryptamine The molecular formula C24H14 (molar mass: 302.37 g/mol, exact mass: 302.1096 u) may refer to: * Dibenzopyrenes * Zethrene, or dibenzo[de,mn]naphthacene The molecular formula CH4N2S (molar mass: 76.12 g/mol, exact mass: 76.0095 u) may refer to: * Ammonium thiocyanate * Thiourea The reduction of nitro compounds are chemical reactions of wide interest in organic chemistry. The reaction can also be effected through radical reaction with tributyltin hydride and a radical initiator, AIBN as an example.T. V. (Babu) RajanBabu, Philip C. Bulman Page, Benjamin R. Buckley, ""Tri-n-butylstannane"" Encyclopedia of Reagents for Organic Synthesis 2004, John Wiley & Sons. doi:10.1002/047084289X.rt181.pub2 ===Reduction to amines=== 180px|Generalization of the reduction of a nitroalkane to an amine Aliphatic nitro compounds can be reduced to aliphatic amines by several reagents: * Catalytic hydrogenation using platinum(IV) oxide (PtO2) or Raney nickel * Iron metal in refluxing acetic acid * Samarium diiodide *Raney nickel, platinum on carbon, or zinc dust and formic acid or ammonium formate α,β-Unsaturated nitro compounds can be reduced to saturated amines by: * Catalytic hydrogenation over palladium-on-carbon * Iron metal * Lithium aluminium hydride (Note: Hydroxylamines and oximes are typical impurities.) *Lithium borohydride or sodium borohydride and trimethylsilyl chloride *Red-Al ===Reduction to hydroxylamines=== Aliphatic nitro compounds can be reduced to aliphatic hydroxylamines using diborane. :180px|Generalization of the reduction of a nitroalkane to a hydroxylamine The reaction can also be carried out with zinc dust and ammonium chloride: : R-NO2 \+ 4 NH4Cl + 2 Zn → R-NH-OH + 2 ZnCl2 \+ 4 NH3 \+ H2O ===Reduction to oximes=== 180px|Generalization of the reduction of a nitroalkane to an oxime Nitro compounds are typically reduced to oximes using metal salts, such as tin(II) chloride or chromium(II) chloride. The nitro group was one of the first functional groups to be reduced. Illustrated by the selective reduction of dinitrophenol to the nitroaminophenol. (Excess zinc will reduce the azo group to a hydrazino compound.) ==Aliphatic nitro compounds== ===Reduction to hydrocarbons=== 180px|Generalization of the reduction of a nitroalkane to an alkane Hydrodenitration (replacement of a nitro group with hydrogen) is difficult to achieve but can be effected by catalytic hydrogenation over platinum on silica gel at high temperatures. Most useful is the reduction of aryl nitro compounds. ==Aromatic nitro compounds== ===Reduction to anilines=== :200px|Generalization of the reduction of a nitroarene to aniline The reduction of nitroaromatics is conducted on an industrial scale. Alkyl and aryl nitro compounds behave differently. (See below) ===Reduction to hydroxylamines=== Several methods have been described for the production of aryl hydroxylamines from aryl nitro compounds: * Raney nickel and hydrazine at 0-10 °C * Electrolytic reduction * Zinc metal in aqueous ammonium chloride * Catalytic Rhodium on carbon with excess hydrazine monohydrate at room temperature ===Reduction to hydrazine compounds=== Treatment of nitroarenes with excess zinc metal results in the formation of N,N'-diarylhydrazine. ===Reduction to azo compounds=== 240px|Generalization of the reduction of a nitroarene to an azo compound Treatment of aromatic nitro compounds with metal hydrides gives good yields of azo compounds. * Tin(II) chloride * Titanium(III) chloride * Samarium *Hydroiodic acid Metal hydrides are typically not used to reduce aryl nitro compounds to anilines because they tend to produce azo compounds. Many methods exist, such as: * Catalytic hydrogenation using: Raney nickel or palladium-on- carbon, platinum(IV) oxide, or Urushibara nickel. The conversion can be effected by many reagents. For example, one could use: * Lithium aluminium hydride * Zinc metal with sodium hydroxide. * Sodium hydrosulfite * Sodium sulfide (or hydrogen sulfide and base). Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions * Iron in acidic media. Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions ",0.75,0.88,7.654,-0.347,7.00,E
+$$","The Mercator series provides an analytic expression of the natural logarithm: \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n \;=\; \ln (1+x). It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is \frac12-\frac14+\frac18-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)} = \frac13. ====Complex series==== The summation formula for geometric series remains valid even when the common ratio is a complex number. It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is \frac12+\frac14+\frac18+\frac{1}{16}+\cdots=\frac{1/2}{1-(+1/2)} = 1. The constant can also be explicitly defined by the following infinite sums: : 0.187859\ldots = \sum_{k=1}^{\infty} (-1)^k (k^{1/k} - 1) = \sum_{k=1}^{\infty} \left((2k)^{1/(2k)} - (2k-1)^{1/(2k-1)}\right). As another example, by Mercator series \ln(2) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots. This is a geometric series with common ratio and the fractional part is equal to :\sum_{n=0}^\infty 4^{-n} = 1 + 4^{-1} + 4^{-2} + 4^{-3} + \cdots = {4\over 3}. In contrast, as r approaches −1 the sum of the first several terms of the geometric series starts to converge to 1/2 but slightly flips up or down depending on whether the most recently added term has a power of r that is even or odd. The constant relates to the divergent series: :\sum_{k=1}^{\infty} (-1)^k k^{1/k}. In mathematics, an alternating series is an infinite series of the form \sum_{n=0}^\infty (-1)^n a_n or \sum_{n=0}^\infty (-1)^{n+1} a_n with for all . In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: :\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots In summation notation, :\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} x^n. The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + ⋯ is a simple example of an alternating series that converges absolutely. The series converges to the natural logarithm (shifted by 1) whenever -1 . ==History== The series was discovered independently by Johannes Hudde and Isaac Newton. But, since the series does not converge absolutely, we can rearrange the terms to obtain a series for \tfrac 1 2 \ln(2): \begin{align} & {} \quad \left(1-\frac{1}{2}\right)-\frac{1}{4} +\left(\frac{1}{3}-\frac{1}{6}\right) -\frac{1}{8}+\left(\frac{1}{5} -\frac{1}{10}\right)-\frac{1}{12}+\cdots \\\\[8pt] & = \frac{1}{2}-\frac{1}{4}+\frac{1}{6} -\frac{1}{8}+\frac{1}{10}-\frac{1}{12} +\cdots \\\\[8pt] & = \frac{1}{2}\left(1-\frac{1}{2} + \frac{1}{3} -\frac{1}{4}+\frac{1}{5}- \frac{1}{6}+ \cdots\right)= \frac{1}{2} \ln(2). \end{align} == Series acceleration == In practice, the numerical summation of an alternating series may be sped up using any one of a variety of series acceleration techniques. Using calculus, the same area could be found by a definite integral. ===Nicole Oresme (c.1323 – 1382)=== thumb|A two dimensional geometric series diagram Nicole Oresme used to determine that the infinite series 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + 7/128 + ... converges to 2. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. :\sum_{k=0}^{n} (-2)^k As a series of real numbers it diverges, so in the usual sense it has no sum. Therefore, an alternating series is also a unit series when -1 < r < 0 and a + r = 1 (for example, coefficient a = 1.7 and common ratio r = -0.7). In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. The closed form geometric series 1 / (1 - r) is the black dashed line. See for example Grandi's series: 1 − 1 + 1 − 1 + ···. Using the geometric series closed form as before :0.7777\ldots \;=\; 0b0.110001110001110001\ldots \;=\; \frac{a}{1-r} \;=\; \frac{49/64}{1-1/64} \;=\; \frac{49/64}{63/64} \;=\; \frac{49}{63} \;=\; \frac{7}{9}. Alternatively, one can start with the finite geometric series (t e -1) :1-t+t^2-\cdots+(-t)^{n-1}=\frac{1-(-t)^n}{1+t} which gives :\frac1{1+t}=1-t+t^2-\cdots+(-t)^{n-1}+\frac{(-t)^n}{1+t}. That flipping behavior near r = −1 is illustrated in the adjacent image showing the first 11 terms of the geometric series with a = 1 and |r| < 1\. ",1.07,1.22,"""-36.5""",-1.00,0.3333333,E
+Calculate the percentage difference between $\ln (1+x)$ and $x$ for $x=0.0050$,"thumb|400px|The logarithmic decrement can be obtained e.g. as ln(x1/x3). A percentage point or percent point is the unit for the arithmetic difference between two percentages. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). == Related units == * Percentage (%) 1 part in 100 * Per mille (‰) 1 part in 1,000 *Basis point (bp) difference of 1 part in 10,000 *Permyriad (‱) 1 part in 10,000 * Per cent mille (pcm) 1 part in 100,000 * Baker percentage == See also == * Parts-per notation * Per-unit system * Percent point function * Relative change and difference == References == Category:Mathematical terminology Category:Probability assessment Category:Units of measurement ru:Процент#Процентный пункт In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, :(\ln f)' = \frac{f'}{f} \quad \implies \quad f' = f \cdot (\ln f)'. thumbnail|upright=1.3|Plot of logit(x) in the domain of 0 to 1, where the base of the logarithm is e. In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion : -\ln(1-p) = p + \frac{p^2}{2} + \frac{p^3}{3} + \cdots. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions. Mathematically, the logit is the inverse of the standard logistic function \sigma(x) = 1/(1+e^{-x}), so the logit is defined as :\operatorname{logit} p = \sigma^{-1}(p) = \ln \frac{p}{1-p} \quad \text{for} \quad p \in (0,1). Percentage-point differences are one way to express a risk or probability. After the first occurrence, some writers abbreviate by using just ""point"" or ""points"". ==Differences between percentages and percentage points== Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. Differentiating by applying the chain and the sum rules yields :\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}, and, after rearranging, yields :f'(x) = f(x)\times \Bigg\\{\frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}\Bigg\\}= g(x)h(x)\times \Bigg\\{\frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}\Bigg\\}=g'(x)h(x)+g(x)h'(x), which is the product rule for derivatives. ===Quotients=== A natural logarithm is applied to a quotient of two functions :f(x)=\frac{g(x)}{h(x)}\,\\! to transform the division into a subtraction :\ln(f(x))=\ln\Bigg(\frac{g(x)}{h(x)}\Bigg)=\ln(g(x))-\ln(h(x))\,\\! For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. Percentage solution may refer to: * Mass fraction (or ""% w/w"" or ""wt.%""), for percent mass * Volume fraction (or ""% v/v"" or ""vol.%""), volume concentration, for percent volume * ""Mass/volume percentage"" (or ""% m/v"") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). Fisher described the logarithmic distribution in a paper that used it to model relative species abundance. ==See also== * Poisson distribution (also derived from a Maclaurin series) ==References== ==Further reading== * * Category:Discrete distributions Category:Logarithms Thus, the logit is a type of function that maps probability values from (0, 1) to real numbers in (-\infty, +\infty), akin to the probit function. ==Definition== If is a probability, then is the corresponding odds; the of the probability is the logarithm of the odds, i.e.: :\operatorname{logit}(p)=\ln\left( \frac{p}{1-p} \right) =\ln(p)-\ln(1-p)=-\ln\left( \frac{1}{p}-1\right)=2\operatorname{atanh}(2p-1) The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base is the one most often used. The cumulative distribution function is : F(k) = 1 + \frac{\Beta(p; k+1,0)}{\ln(1-p)} where B is the incomplete beta function. The application of natural logarithms results in (with capital sigma notation) :\ln (f(x))=\sum_i\alpha_i(x)\cdot \ln(f_i(x)), and after differentiation, :\frac{f'(x)}{f(x)}=\sum_i\left[\alpha_i'(x)\cdot \ln(f_i(x))+\alpha_i(x)\cdot \frac{f_i'(x)}{f_i(x)}\right]. Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds \frac{p}{1-p} where is a probability. In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. In fact, the is the quantile function of the logistic distribution, while the is the quantile function of the normal distribution. ",0.3359,0.249,"""540.0""",1.5,3.8,B
+Calculate the reduced mass of a nitrogen molecule in which both nitrogen atoms have an atomic mass of 14.00.,"The molecular formula C14H14 (molar mass: 182.26 g/mol, exact mass: 182.1096 u) may refer to: * Bibenzyl * Cyclotetradecaheptaene, or [14]annulene Category:Molecular formulas The molecular formula C11H14N2 (molar mass : 174.24 g/mol, exact mass : 174.115698) may refer to : * 6-(2-Aminopropyl)indole * Gramine * 5-IT * α-Methyltryptamine * N-Methyltryptamine The molecular formula C24H14 (molar mass: 302.37 g/mol, exact mass: 302.1096 u) may refer to: * Dibenzopyrenes * Zethrene, or dibenzo[de,mn]naphthacene The molecular formula CH4N2S (molar mass: 76.12 g/mol, exact mass: 76.0095 u) may refer to: * Ammonium thiocyanate * Thiourea The reduction of nitro compounds are chemical reactions of wide interest in organic chemistry. The reaction can also be effected through radical reaction with tributyltin hydride and a radical initiator, AIBN as an example.T. V. (Babu) RajanBabu, Philip C. Bulman Page, Benjamin R. Buckley, ""Tri-n-butylstannane"" Encyclopedia of Reagents for Organic Synthesis 2004, John Wiley & Sons. doi:10.1002/047084289X.rt181.pub2 ===Reduction to amines=== 180px|Generalization of the reduction of a nitroalkane to an amine Aliphatic nitro compounds can be reduced to aliphatic amines by several reagents: * Catalytic hydrogenation using platinum(IV) oxide (PtO2) or Raney nickel * Iron metal in refluxing acetic acid * Samarium diiodide *Raney nickel, platinum on carbon, or zinc dust and formic acid or ammonium formate α,β-Unsaturated nitro compounds can be reduced to saturated amines by: * Catalytic hydrogenation over palladium-on-carbon * Iron metal * Lithium aluminium hydride (Note: Hydroxylamines and oximes are typical impurities.) *Lithium borohydride or sodium borohydride and trimethylsilyl chloride *Red-Al ===Reduction to hydroxylamines=== Aliphatic nitro compounds can be reduced to aliphatic hydroxylamines using diborane. :180px|Generalization of the reduction of a nitroalkane to a hydroxylamine The reaction can also be carried out with zinc dust and ammonium chloride: : R-NO2 \+ 4 NH4Cl + 2 Zn → R-NH-OH + 2 ZnCl2 \+ 4 NH3 \+ H2O ===Reduction to oximes=== 180px|Generalization of the reduction of a nitroalkane to an oxime Nitro compounds are typically reduced to oximes using metal salts, such as tin(II) chloride or chromium(II) chloride. The nitro group was one of the first functional groups to be reduced. Illustrated by the selective reduction of dinitrophenol to the nitroaminophenol. (Excess zinc will reduce the azo group to a hydrazino compound.) ==Aliphatic nitro compounds== ===Reduction to hydrocarbons=== 180px|Generalization of the reduction of a nitroalkane to an alkane Hydrodenitration (replacement of a nitro group with hydrogen) is difficult to achieve but can be effected by catalytic hydrogenation over platinum on silica gel at high temperatures. Most useful is the reduction of aryl nitro compounds. ==Aromatic nitro compounds== ===Reduction to anilines=== :200px|Generalization of the reduction of a nitroarene to aniline The reduction of nitroaromatics is conducted on an industrial scale. Alkyl and aryl nitro compounds behave differently. (See below) ===Reduction to hydroxylamines=== Several methods have been described for the production of aryl hydroxylamines from aryl nitro compounds: * Raney nickel and hydrazine at 0-10 °C * Electrolytic reduction * Zinc metal in aqueous ammonium chloride * Catalytic Rhodium on carbon with excess hydrazine monohydrate at room temperature ===Reduction to hydrazine compounds=== Treatment of nitroarenes with excess zinc metal results in the formation of N,N'-diarylhydrazine. ===Reduction to azo compounds=== 240px|Generalization of the reduction of a nitroarene to an azo compound Treatment of aromatic nitro compounds with metal hydrides gives good yields of azo compounds. * Tin(II) chloride * Titanium(III) chloride * Samarium *Hydroiodic acid Metal hydrides are typically not used to reduce aryl nitro compounds to anilines because they tend to produce azo compounds. Many methods exist, such as: * Catalytic hydrogenation using: Raney nickel or palladium-on- carbon, platinum(IV) oxide, or Urushibara nickel. The conversion can be effected by many reagents. For example, one could use: * Lithium aluminium hydride * Zinc metal with sodium hydroxide. * Sodium hydrosulfite * Sodium sulfide (or hydrogen sulfide and base). Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions * Iron in acidic media. Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions Additionally, catalytic hydrogenation using a controlled amount of hydrogen can generate oximes. ==References== Category:Organic redox reactions ",0.75,0.88,"""7.654""",-0.347,7.00,E
 "Two narrow slits are illuminated with red light of wavelength $694.3 \mathrm{~nm}$ from a laser, producing a set of evenly placed bright bands on a screen located $3.00 \mathrm{~m}$ beyond the slits. If the distance between the bands is $1.50 \mathrm{~cm}$, then what is the distance between the slits?
-","In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be \Delta S={a} \sin \theta Maxima in the intensity occur if this path length difference is an integer number of wavelengths. a \sin \theta = n \lambda where * n is an integer that labels the order of each maximum, * \lambda is the wavelength, * a is the distance between the slits, and * \theta is the angle at which constructive interference occurs. For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper, as can be seen in the image. right|frame|2-slit and 5-slit diffraction of red laser light ===Mathematical description=== To calculate this intensity pattern, one needs to introduce some more sophisticated methods. As the distance between the measured point of diffraction and the obstruction point increases, the diffraction patterns or results predicted converge towards those of Fraunhofer diffraction, which is more often observed in nature due to the extremely small wavelength of visible light. ==Multiple narrow slits== ===A simple quantitative description=== right|thumb|Diagram of a two slit diffraction problem, showing the angle to the first minimum, where a path length difference of a half wavelength causes destructive interference. The wave at a screen some distance away from the plane of the slits is given by the sum of the waves emanating from each of the slits. thumb|right|Interference pattern of double slits, where the slit width is one third the wavelength. The simplest case is that of two narrow slits, spaced a distance \ a apart. The result is the Fraunhofer approximation, which is only valid very far away from the object S \approx L + \frac{x^2}{2L}+\frac{x a}{2L} Depending on the size of the diffraction object, the distance to the object and the wavelength of the wave, the Fresnel approximation, the Fraunhofer approximation or neither approximation may be valid. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. If is the location at which the intensity of the diffraction pattern is being computed, the slit extends from x' = -a/2 to +a/2\,, and from y'=-\infty to \infty. Now, substituting in \frac{2\pi}{\lambda} = k, the intensity (squared amplitude) I of the diffracted waves at an angle θ is given by: I(\theta) = I_0 {\left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right] }^2 ==Multiple slits== right|frame|Double-slit diffraction of red laser light right|frame|2-slit and 5-slit diffraction Let us again start with the mathematical representation of Huygens' principle. To determine the maxima and minima in the amplitude we must determine the path difference to the first slit and to the second one. In other words, the distance to the target is much larger than the diffraction width on the target. The distance r from the slot is: r = \sqrt{\left(x - x^\prime\right)^2 + y^{\prime2} + z^2} r = z \left(1 + \frac{\left(x - x^\prime\right)^2 + y^{\prime2}}{z^2}\right)^\frac{1}{2} Assuming Fraunhofer diffraction will result in the conclusion z \gg \big|\left(x - x^\prime\right)\big|. The C-band is located between the short wavelengths (S) band (1460–1530 nm) and the long wavelengths (L) band (1565–1625 nm). The W band of the microwave part of the electromagnetic spectrum ranges from 75 to 110 GHz, wavelength ≈2.7–4 mm. Consider a monochromatic complex plane wave \Psi^\prime of wavelength λ incident on a slit of width a. Optical Fiber Communications article in rp- photonics' Encyclopedia of Laser Physics and Technology (accessed Nov. 11 2010) The C-band is located around the absorption minimum in optical fiber, where the loss reaches values as good as 0.2 dB/km, as well as an atmospheric transmission window (see figures). Since the pulse duration from these oscillators is about 3 ns, the laser linewidth performance is near the limit allowed by the Heisenberg uncertainty principle. ==See also== *Laser *Fabry–Perot interferometer *Beam divergence *Multiple-prism dispersion theory *Multiple-prism grating laser oscillator *N-slit interferometric equation *Oscillator linewidth *Solid state dye lasers ==References== Linewidth thumb|Absorption in fiber in the range 900–1700 nm with a minimum at the C-band thumb|Transmittance of the atmosphere around the C-band In infrared optical communications, C-band (C for ""conventional"") refers to the wavelength range 1530–1565 nm, which corresponds to the amplification range of erbium doped fiber amplifiers (EDFAs). For definiteness let us say we are diffracting light and we are interested in what the intensity looks like on a screen a distance L away from the object. Laser linewidth is the spectral linewidth of a laser beam. ",1.51,8.3147,0.5,167,0.139,E
-Calculate the energy and wavelength associated with an $\alpha$ particle that has fallen through a potential difference of $4.0 \mathrm{~V}$. Take the mass of an $\alpha$ particle to be $6.64 \times 10^{-27} \mathrm{~kg}$.,"Alpha particles have a typical kinetic energy of 5 MeV (or ≈ 0.13% of their total energy, 110 TJ/kg) and have a speed of about 15,000,000 m/s, or 5% of the speed of light. When an atom emits an alpha particle in alpha decay, the atom's mass number decreases by four due to the loss of the four nucleons in the alpha particle. Due to the mechanism of their production in standard alpha radioactive decay, alpha particles generally have a kinetic energy of about 5 MeV, and a velocity in the vicinity of 4% of the speed of light. With a typical kinetic energy of 5 MeV; the speed of emitted alpha particles is 15,000 km/s, which is 5% of the speed of light. However, decay alpha particles only have energies of around 4 to 9 MeV above the potential at infinity, far less than the energy needed to overcome the barrier and escape. The energy of alpha particles emitted varies, with higher energy alpha particles being emitted from larger nuclei, but most alpha particles have energies of between 3 and 7 MeV (mega- electron-volts), corresponding to extremely long and extremely short half- lives of alpha-emitting nuclides, respectively. For example, performing the calculation for uranium-232 shows that alpha particle emission releases 5.4 MeV of energy, while a single proton emission would require 6.1 MeV. This energy is roughly the weight of the alpha (4 u) divided by the weight of the parent (typically about 200 u) times the total energy of the alpha. Computing the total disintegration energy given by the equation E_{di} = (m_\text{i} - m_\text{f} - m_\text{p})c^2 where is the initial mass of the nucleus, is the mass of the nucleus after particle emission, and is the mass of the emitted (alpha-)particle, one finds that in certain cases it is positive and so alpha particle emission is possible, whereas other decay modes would require energy to be added. This energy is a substantial amount of energy for a single particle, but their high mass means alpha particles have a lower speed than any other common type of radiation, e.g. β particles, neutrons.N.B. The energy needed to bring an alpha particle from infinity to a point near the nucleus just outside the range of the nuclear force's influence is generally in the range of about 25 MeV. Most of the disintegration energy becomes the kinetic energy of the alpha particle, although to fulfill conservation of momentum, part of the energy goes to the recoil of the nucleus itself (see atomic recoil). However, since the mass numbers of most alpha-emitting radioisotopes exceed 210, far greater than the mass number of the alpha particle (4), the fraction of the energy going to the recoil of the nucleus is generally quite small, less than 2%. An alpha particle is identical to the nucleus of a helium-4 atom, which consists of two protons and two neutrons. The radiated energy is approximately 2.8MeV. Such alpha particles are termed ""long range alphas"" since at their typical energy of 16 MeV, they are at far higher energy than is ever produced by alpha decay. An alpha particle with a speed of 1.5×107 m/s within a nuclear diameter of approximately 10−14 m will collide with the barrier more than 1021 times per second. These disintegration energies, however, are substantially smaller than the repulsive potential barrier created by the interplay between the strong nuclear and the electromagnetic force, which prevents the alpha particle from escaping. For example, one of the heaviest naturally occurring isotopes, ^238U -> ^234Th + ^4He (ignoring charges): : Qα = -931.5 (234.043 601 + 4.002 603 254 13 - 238.050 788 2) = 4.2699 MeV Note that the decay energy will be divided between the alpha-particle and the heavy recoiling daughter so that the kinetic energy of the alpha particle (Tα) will be slightly less: Tα = (234.043 601 / 238.050 788 2) 4.2699 = 4.198 MeV, (note this is for the 238gU to 238gTh reaction, which in this case has the branching ratio of 79%). To the adjacent pictures: According to the energy-loss curve by Bragg, it is recognizable that the alpha particle indeed loses more energy on the end of the trace.Magazine ""nuclear energy"" (III/18 (203) special edition, Volume 10, Issue 2 /1967. ==Anti-alpha particle== In 2011, members of the international STAR collaboration using the Relativistic Heavy Ion Collider at the U.S. Department of Energy's Brookhaven National Laboratory detected the antimatter partner of the helium nucleus, also known as the anti-alpha. . The symbol for the alpha particle is α or α2+. Alpha radiation has a high linear energy transfer (LET) coefficient, which is about one ionization of a molecule/atom for every angstrom of travel by the alpha particle. ",635013559600,2.9,-32.0,1.3,226,D
+","In the Fraunhofer approximation, with the observer far away from the slits, the difference in path length to the two slits can be seen from the image to be \Delta S={a} \sin \theta Maxima in the intensity occur if this path length difference is an integer number of wavelengths. a \sin \theta = n \lambda where * n is an integer that labels the order of each maximum, * \lambda is the wavelength, * a is the distance between the slits, and * \theta is the angle at which constructive interference occurs. For an array of slits, positions of the minima and maxima are not changed, the fringes visible on a screen however do become sharper, as can be seen in the image. right|frame|2-slit and 5-slit diffraction of red laser light ===Mathematical description=== To calculate this intensity pattern, one needs to introduce some more sophisticated methods. As the distance between the measured point of diffraction and the obstruction point increases, the diffraction patterns or results predicted converge towards those of Fraunhofer diffraction, which is more often observed in nature due to the extremely small wavelength of visible light. ==Multiple narrow slits== ===A simple quantitative description=== right|thumb|Diagram of a two slit diffraction problem, showing the angle to the first minimum, where a path length difference of a half wavelength causes destructive interference. The wave at a screen some distance away from the plane of the slits is given by the sum of the waves emanating from each of the slits. thumb|right|Interference pattern of double slits, where the slit width is one third the wavelength. The simplest case is that of two narrow slits, spaced a distance \ a apart. The result is the Fraunhofer approximation, which is only valid very far away from the object S \approx L + \frac{x^2}{2L}+\frac{x a}{2L} Depending on the size of the diffraction object, the distance to the object and the wavelength of the wave, the Fresnel approximation, the Fraunhofer approximation or neither approximation may be valid. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. If the distance to each source is an integer plus one half of a wavelength, there will be complete destructive interference. If is the location at which the intensity of the diffraction pattern is being computed, the slit extends from x' = -a/2 to +a/2\,, and from y'=-\infty to \infty. Now, substituting in \frac{2\pi}{\lambda} = k, the intensity (squared amplitude) I of the diffracted waves at an angle θ is given by: I(\theta) = I_0 {\left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right] }^2 ==Multiple slits== right|frame|Double-slit diffraction of red laser light right|frame|2-slit and 5-slit diffraction Let us again start with the mathematical representation of Huygens' principle. To determine the maxima and minima in the amplitude we must determine the path difference to the first slit and to the second one. In other words, the distance to the target is much larger than the diffraction width on the target. The distance r from the slot is: r = \sqrt{\left(x - x^\prime\right)^2 + y^{\prime2} + z^2} r = z \left(1 + \frac{\left(x - x^\prime\right)^2 + y^{\prime2}}{z^2}\right)^\frac{1}{2} Assuming Fraunhofer diffraction will result in the conclusion z \gg \big|\left(x - x^\prime\right)\big|. The C-band is located between the short wavelengths (S) band (1460–1530 nm) and the long wavelengths (L) band (1565–1625 nm). The W band of the microwave part of the electromagnetic spectrum ranges from 75 to 110 GHz, wavelength ≈2.7–4 mm. Consider a monochromatic complex plane wave \Psi^\prime of wavelength λ incident on a slit of width a. Optical Fiber Communications article in rp- photonics' Encyclopedia of Laser Physics and Technology (accessed Nov. 11 2010) The C-band is located around the absorption minimum in optical fiber, where the loss reaches values as good as 0.2 dB/km, as well as an atmospheric transmission window (see figures). Since the pulse duration from these oscillators is about 3 ns, the laser linewidth performance is near the limit allowed by the Heisenberg uncertainty principle. ==See also== *Laser *Fabry–Perot interferometer *Beam divergence *Multiple-prism dispersion theory *Multiple-prism grating laser oscillator *N-slit interferometric equation *Oscillator linewidth *Solid state dye lasers ==References== Linewidth thumb|Absorption in fiber in the range 900–1700 nm with a minimum at the C-band thumb|Transmittance of the atmosphere around the C-band In infrared optical communications, C-band (C for ""conventional"") refers to the wavelength range 1530–1565 nm, which corresponds to the amplification range of erbium doped fiber amplifiers (EDFAs). For definiteness let us say we are diffracting light and we are interested in what the intensity looks like on a screen a distance L away from the object. Laser linewidth is the spectral linewidth of a laser beam. ",1.51,8.3147,"""0.5""",167,0.139,E
+Calculate the energy and wavelength associated with an $\alpha$ particle that has fallen through a potential difference of $4.0 \mathrm{~V}$. Take the mass of an $\alpha$ particle to be $6.64 \times 10^{-27} \mathrm{~kg}$.,"Alpha particles have a typical kinetic energy of 5 MeV (or ≈ 0.13% of their total energy, 110 TJ/kg) and have a speed of about 15,000,000 m/s, or 5% of the speed of light. When an atom emits an alpha particle in alpha decay, the atom's mass number decreases by four due to the loss of the four nucleons in the alpha particle. Due to the mechanism of their production in standard alpha radioactive decay, alpha particles generally have a kinetic energy of about 5 MeV, and a velocity in the vicinity of 4% of the speed of light. With a typical kinetic energy of 5 MeV; the speed of emitted alpha particles is 15,000 km/s, which is 5% of the speed of light. However, decay alpha particles only have energies of around 4 to 9 MeV above the potential at infinity, far less than the energy needed to overcome the barrier and escape. The energy of alpha particles emitted varies, with higher energy alpha particles being emitted from larger nuclei, but most alpha particles have energies of between 3 and 7 MeV (mega- electron-volts), corresponding to extremely long and extremely short half- lives of alpha-emitting nuclides, respectively. For example, performing the calculation for uranium-232 shows that alpha particle emission releases 5.4 MeV of energy, while a single proton emission would require 6.1 MeV. This energy is roughly the weight of the alpha (4 u) divided by the weight of the parent (typically about 200 u) times the total energy of the alpha. Computing the total disintegration energy given by the equation E_{di} = (m_\text{i} - m_\text{f} - m_\text{p})c^2 where is the initial mass of the nucleus, is the mass of the nucleus after particle emission, and is the mass of the emitted (alpha-)particle, one finds that in certain cases it is positive and so alpha particle emission is possible, whereas other decay modes would require energy to be added. This energy is a substantial amount of energy for a single particle, but their high mass means alpha particles have a lower speed than any other common type of radiation, e.g. β particles, neutrons.N.B. The energy needed to bring an alpha particle from infinity to a point near the nucleus just outside the range of the nuclear force's influence is generally in the range of about 25 MeV. Most of the disintegration energy becomes the kinetic energy of the alpha particle, although to fulfill conservation of momentum, part of the energy goes to the recoil of the nucleus itself (see atomic recoil). However, since the mass numbers of most alpha-emitting radioisotopes exceed 210, far greater than the mass number of the alpha particle (4), the fraction of the energy going to the recoil of the nucleus is generally quite small, less than 2%. An alpha particle is identical to the nucleus of a helium-4 atom, which consists of two protons and two neutrons. The radiated energy is approximately 2.8MeV. Such alpha particles are termed ""long range alphas"" since at their typical energy of 16 MeV, they are at far higher energy than is ever produced by alpha decay. An alpha particle with a speed of 1.5×107 m/s within a nuclear diameter of approximately 10−14 m will collide with the barrier more than 1021 times per second. These disintegration energies, however, are substantially smaller than the repulsive potential barrier created by the interplay between the strong nuclear and the electromagnetic force, which prevents the alpha particle from escaping. For example, one of the heaviest naturally occurring isotopes, ^238U -> ^234Th + ^4He (ignoring charges): : Qα = -931.5 (234.043 601 + 4.002 603 254 13 - 238.050 788 2) = 4.2699 MeV Note that the decay energy will be divided between the alpha-particle and the heavy recoiling daughter so that the kinetic energy of the alpha particle (Tα) will be slightly less: Tα = (234.043 601 / 238.050 788 2) 4.2699 = 4.198 MeV, (note this is for the 238gU to 238gTh reaction, which in this case has the branching ratio of 79%). To the adjacent pictures: According to the energy-loss curve by Bragg, it is recognizable that the alpha particle indeed loses more energy on the end of the trace.Magazine ""nuclear energy"" (III/18 (203) special edition, Volume 10, Issue 2 /1967. ==Anti-alpha particle== In 2011, members of the international STAR collaboration using the Relativistic Heavy Ion Collider at the U.S. Department of Energy's Brookhaven National Laboratory detected the antimatter partner of the helium nucleus, also known as the anti-alpha. . The symbol for the alpha particle is α or α2+. Alpha radiation has a high linear energy transfer (LET) coefficient, which is about one ionization of a molecule/atom for every angstrom of travel by the alpha particle. ",635013559600,2.9,"""-32.0""",1.3,226,D
 "Calculate the number of photons in a $2.00 \mathrm{~mJ}$ light pulse at (a) $1.06 \mu \mathrm{m}$
-","Retrieved 9 December 2015 Using the intensity distribution together with Mandel's formula which describes the probability of the number of photon counts registered by a photodetector, the statistical distribution of photons in thermal light can be obtained. The formula describes the probability of observing n photon counts and is given by : P(n) = \int_{0}^{\infty} \frac{{\left ( \epsilon I \right )}^{n}}{n!} e^{-\epsilon I} P(I) dI The factor \epsilon = \frac{\eta}{h u} where \eta is the quantum efficiency describes the efficiency of the photon counter. In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts. SI units for these quantities are summarized in the table below. ==See also== *Single-photon source *Visible Light Photon Counter *Transition edge sensor *Superconducting nanowire single-photon detector *Time-correlated single photon counting *Oversampled binary image sensor ==References== Category:Optical metrology Category:Photonics Category:Particle detectors Photon counting is a technique in which individual photons are counted using a single-photon detector (SPD). A single-photon detector emits a pulse of signal for each detected photon. thumb|470px|Diagramm of operation of a photonic radar. Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source. Evidence of the sub-Poissonian nature of light is shown by obtaining a negative intensity correlation as was shown in. ==References== Category:Optical metrology Category:Photonics thumb|right|220 px|Schematic of an ion-to-photon detector with a conversion dynode. A study that compared photon-counting mammography to the state-wide average of the North Rhine-Westphalian mammography screening program in Germany reported a slightly improved diagnostic performance at a dose that was 40% of conventional technologies. However, if it is known that a single photon was detected, the center of the impulse response can be evaluated to precisely determine its arrival time. These photons are then detected by the photomultiplier tube. Photon-counting mammography was introduced commercially in 2003 and was the first widely available application of photon-counting detector technology in medical x-ray imaging. Photon- counting mammography was introduced commercially in 2003. In photon-counting mammography, contrast-enhanced imaging has been focused on iodine imaging. === Tomosynthesis === Photon-counting breast tomosynthesis has been developed to a prototype state. thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. Thus, the excess noise factor of a photon-counting detector is unity, and the achievable signal-to-noise ratio for a fixed number of photons is generally higher than the same detector without photon counting. Photon-counting detectors, on the other hand, are fast enough to register single photon events. The flux per unit solid angle is the photon intensity. The advent of ultrafast photodetectors has made it possible to measure the sub-Poissonian nature of light. Photons that arrive during this interval may not be detected. ",-191.2,8,1.07,58.2,7,C
-The force constant of ${ }^{35} \mathrm{Cl}^{35} \mathrm{Cl}$ is $319 \mathrm{~N} \cdot \mathrm{m}^{-1}$. Calculate the fundamental vibrational frequency,"The fundamental is the frequency at which the entire wave vibrates. The Rayleigh's quotient represents a quick method to estimate the natural frequency of a multi-degree-of-freedom vibration system, in which the mass and the stiffness matrices are known. The fundamental frequency is defined as its reciprocal: When the units of time are seconds, the frequency is in s^{-1}, also known as Hertz. ===Fundamental frequency of a pipe=== For a pipe of length L with one end closed and the other end open the wavelength of the fundamental harmonic is 4L, as indicated by the first two animations. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The natural frequency, or fundamental frequency, 0, can be found using the following equation: \, }} where: * = stiffness of the spring * = mass * 0 = natural frequency in radians per second. Each step represents a frequency ratio of , or 70.6 cents. 17-ET is the tuning of the regular diatonic tuning in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label ""17-TET""). ==History and use== Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale.Ellis, Alexander J. (1863). Hence, Therefore, using the relation where v is the speed of the wave, the fundamental frequency can be found in terms of the speed of the wave and the length of the pipe: If the ends of the same pipe are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2L . Conversely 34-ET is a subdivision of 17-ET. ==References== Sources * == External links == * ""The 17-tone Puzzle — And the Neo-medieval Key that Unlocks It"" by George Secor * Libro y Programa Tonalismo, heptadecatonic system applications (in Spanish) * Georg Hajdu's 1992 ICMC paper on the 17-tone piano project * , by Wongi Hwang Category:Equal temperaments Category:Microtonality Chladni's law, named after Ernst Chladni, relates the frequency of modes of vibration for flat circular surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes. The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In other words, 2^{24/41} \approx 1.50042 is a better approximation to the ratio 3/2 = 1.5 than either 2^{17/29} \approx 1.50129 or 2^{7/12} \approx 1.49831. ==History and use== Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET , pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague. It is stated as the equation : f = C (m + 2n)^p where C and p are coefficients which depend on the properties of the plate.. thumb|Chladni figures, used for studying vibrationsFor flat circular plates, p is roughly 2, but Chladni's law can also be used to describe the vibrations of cymbals, handbells, and church bells in which case p can vary from 1.4 to 2.4.. The latter can be elucidated by the following 3-DOF example. == Example – 3DOF == As an example, we can consider a 3-degree-of-freedom system in which the mass and the stiffness matrices of them are known as follows: M = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 3 \end{bmatrix} \; , \quad K = \begin{bmatrix} 3 & -1 & 0 \\\ -1 & 3 & -2 \\\ 0 & -2 & 2 \end{bmatrix} To get an estimation of the lowest natural frequency we choose a trial vector of static displacement obtained by loading the system with a force proportional to the masses: \textbf{F} = k\begin{bmatrix} m_1 \\\ m_2 \\\ m_3 \end{bmatrix} = 1 \begin{bmatrix} 1 \\\ 1 \\\ 3 \end{bmatrix} Thus, the trial vector will become \textbf{u} = K^{-1}\textbf{F} = \begin{bmatrix} 2.5 \\\ 6.5 \\\ 8 \end{bmatrix} that allow us to calculate the Rayleigh's quotient: R = \frac{\textbf{u}^T\,K\,\textbf{u}}{\textbf{u}^T\,M\,\textbf{u}} = \cdots = 0.137214 Thus, the lowest natural frequency, calculated by means of Rayleigh's quotient is: w_\text{Ray} = 0.370424 Using a calculation tool is pretty fast to verify how much it differs from the ""real"" one. This is also expressed as: \,}} where: * 0 = natural frequency (SI unit: hertz) * = length of the string (SI unit: metre) * = mass per unit length of the string (SI unit: kg/m) * = tension on the string (SI unit: newton) ==See also== *Greatest common divisor *Hertz *Missing fundamental *Natural frequency *Oscillation *Harmonic series (music)#Terminology *Pitch detection algorithm *Scale of harmonics ==References== Category:Musical tuning Category:Acoustics Category:Fourier analysis While doing a modal analysis, the frequency of the 1st mode is the fundamental frequency. Moreover, the equation allow us to calculate the natural frequency only if the eigenvector (as well as any other displacement vector) \textbf{u}_{m} is known. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. The fundamental may be created by vibration over the full length of a string or air column, or a higher harmonic chosen by the player. To determine the natural frequency in Hz, the omega value is divided by 2. By the same method as above, the fundamental frequency is found to be ==In music== In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. The fundamental frequency is considered the first harmonic and the first partial. In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). ",2.14,556,4.16,52,2688,B
+","Retrieved 9 December 2015 Using the intensity distribution together with Mandel's formula which describes the probability of the number of photon counts registered by a photodetector, the statistical distribution of photons in thermal light can be obtained. The formula describes the probability of observing n photon counts and is given by : P(n) = \int_{0}^{\infty} \frac{{\left ( \epsilon I \right )}^{n}}{n!} e^{-\epsilon I} P(I) dI The factor \epsilon = \frac{\eta}{h u} where \eta is the quantum efficiency describes the efficiency of the photon counter. In these experiments, light incident on the photodetector generates photoelectrons and a counter registers electrical pulses generating a statistical distribution of photon counts. SI units for these quantities are summarized in the table below. ==See also== *Single-photon source *Visible Light Photon Counter *Transition edge sensor *Superconducting nanowire single-photon detector *Time-correlated single photon counting *Oversampled binary image sensor ==References== Category:Optical metrology Category:Photonics Category:Particle detectors Photon counting is a technique in which individual photons are counted using a single-photon detector (SPD). A single-photon detector emits a pulse of signal for each detected photon. thumb|470px|Diagramm of operation of a photonic radar. Photon statistics is the theoretical and experimental study of the statistical distributions produced in photon counting experiments, which use photodetectors to analyze the intrinsic statistical nature of photons in a light source. Evidence of the sub-Poissonian nature of light is shown by obtaining a negative intensity correlation as was shown in. ==References== Category:Optical metrology Category:Photonics thumb|right|220 px|Schematic of an ion-to-photon detector with a conversion dynode. A study that compared photon-counting mammography to the state-wide average of the North Rhine-Westphalian mammography screening program in Germany reported a slightly improved diagnostic performance at a dose that was 40% of conventional technologies. However, if it is known that a single photon was detected, the center of the impulse response can be evaluated to precisely determine its arrival time. These photons are then detected by the photomultiplier tube. Photon-counting mammography was introduced commercially in 2003 and was the first widely available application of photon-counting detector technology in medical x-ray imaging. Photon- counting mammography was introduced commercially in 2003. In photon-counting mammography, contrast-enhanced imaging has been focused on iodine imaging. === Tomosynthesis === Photon-counting breast tomosynthesis has been developed to a prototype state. thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. Thus, the excess noise factor of a photon-counting detector is unity, and the achievable signal-to-noise ratio for a fixed number of photons is generally higher than the same detector without photon counting. Photon-counting detectors, on the other hand, are fast enough to register single photon events. The flux per unit solid angle is the photon intensity. The advent of ultrafast photodetectors has made it possible to measure the sub-Poissonian nature of light. Photons that arrive during this interval may not be detected. ",-191.2,8,"""1.07""",58.2,7,C
+The force constant of ${ }^{35} \mathrm{Cl}^{35} \mathrm{Cl}$ is $319 \mathrm{~N} \cdot \mathrm{m}^{-1}$. Calculate the fundamental vibrational frequency,"The fundamental is the frequency at which the entire wave vibrates. The Rayleigh's quotient represents a quick method to estimate the natural frequency of a multi-degree-of-freedom vibration system, in which the mass and the stiffness matrices are known. The fundamental frequency is defined as its reciprocal: When the units of time are seconds, the frequency is in s^{-1}, also known as Hertz. ===Fundamental frequency of a pipe=== For a pipe of length L with one end closed and the other end open the wavelength of the fundamental harmonic is 4L, as indicated by the first two animations. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The natural frequency, or fundamental frequency, 0, can be found using the following equation: \, }} where: * = stiffness of the spring * = mass * 0 = natural frequency in radians per second. Each step represents a frequency ratio of , or 70.6 cents. 17-ET is the tuning of the regular diatonic tuning in which the tempered perfect fifth is equal to 705.88 cents, as shown in Figure 1 (look for the label ""17-TET""). ==History and use== Alexander J. Ellis refers to a tuning of seventeen tones based on perfect fourths and fifths as the Arabic scale.Ellis, Alexander J. (1863). Hence, Therefore, using the relation where v is the speed of the wave, the fundamental frequency can be found in terms of the speed of the wave and the length of the pipe: If the ends of the same pipe are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2L . Conversely 34-ET is a subdivision of 17-ET. ==References== Sources * == External links == * ""The 17-tone Puzzle — And the Neo-medieval Key that Unlocks It"" by George Secor * Libro y Programa Tonalismo, heptadecatonic system applications (in Spanish) * Georg Hajdu's 1992 ICMC paper on the 17-tone piano project * , by Wongi Hwang Category:Equal temperaments Category:Microtonality Chladni's law, named after Ernst Chladni, relates the frequency of modes of vibration for flat circular surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes. The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In other words, 2^{24/41} \approx 1.50042 is a better approximation to the ratio 3/2 = 1.5 than either 2^{17/29} \approx 1.50129 or 2^{7/12} \approx 1.49831. ==History and use== Although 41-ET has not seen as wide use as other temperaments such as 19-ET or 31-ET , pianist and engineer Paul von Janko built a piano using this tuning, which is on display at the Gemeentemuseum in The Hague. It is stated as the equation : f = C (m + 2n)^p where C and p are coefficients which depend on the properties of the plate.. thumb|Chladni figures, used for studying vibrationsFor flat circular plates, p is roughly 2, but Chladni's law can also be used to describe the vibrations of cymbals, handbells, and church bells in which case p can vary from 1.4 to 2.4.. The latter can be elucidated by the following 3-DOF example. == Example – 3DOF == As an example, we can consider a 3-degree-of-freedom system in which the mass and the stiffness matrices of them are known as follows: M = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 3 \end{bmatrix} \; , \quad K = \begin{bmatrix} 3 & -1 & 0 \\\ -1 & 3 & -2 \\\ 0 & -2 & 2 \end{bmatrix} To get an estimation of the lowest natural frequency we choose a trial vector of static displacement obtained by loading the system with a force proportional to the masses: \textbf{F} = k\begin{bmatrix} m_1 \\\ m_2 \\\ m_3 \end{bmatrix} = 1 \begin{bmatrix} 1 \\\ 1 \\\ 3 \end{bmatrix} Thus, the trial vector will become \textbf{u} = K^{-1}\textbf{F} = \begin{bmatrix} 2.5 \\\ 6.5 \\\ 8 \end{bmatrix} that allow us to calculate the Rayleigh's quotient: R = \frac{\textbf{u}^T\,K\,\textbf{u}}{\textbf{u}^T\,M\,\textbf{u}} = \cdots = 0.137214 Thus, the lowest natural frequency, calculated by means of Rayleigh's quotient is: w_\text{Ray} = 0.370424 Using a calculation tool is pretty fast to verify how much it differs from the ""real"" one. This is also expressed as: \,}} where: * 0 = natural frequency (SI unit: hertz) * = length of the string (SI unit: metre) * = mass per unit length of the string (SI unit: kg/m) * = tension on the string (SI unit: newton) ==See also== *Greatest common divisor *Hertz *Missing fundamental *Natural frequency *Oscillation *Harmonic series (music)#Terminology *Pitch detection algorithm *Scale of harmonics ==References== Category:Musical tuning Category:Acoustics Category:Fourier analysis While doing a modal analysis, the frequency of the 1st mode is the fundamental frequency. Moreover, the equation allow us to calculate the natural frequency only if the eigenvector (as well as any other displacement vector) \textbf{u}_{m} is known. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. The fundamental may be created by vibration over the full length of a string or air column, or a higher harmonic chosen by the player. To determine the natural frequency in Hz, the omega value is divided by 2. By the same method as above, the fundamental frequency is found to be ==In music== In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. The fundamental frequency is considered the first harmonic and the first partial. In music, 41 equal temperament, abbreviated 41-TET, 41-EDO, or 41-ET, is the tempered scale derived by dividing the octave into 41 equally sized steps (equal frequency ratios). ",2.14,556,"""4.16""",52,2688,B
 "$$
 \text {Calculate the energy of a photon for a wavelength of } 100 \mathrm{pm} \text { (about one atomic diameter). }
 $$
-","To find the photon energy in electronvolts using the wavelength in micrometres, the equation is approximately :E\text{ (eV)} = \frac{1.2398}{\lambda\text{ (μm)}} since hc/e=1.239 \; 841 \; 984... \times 10^{-6} eVm where h is Planck's constant, c is the speed of light in m/sec, and e is the electron charge. The photon energy of near infrared radiation at 1 μm wavelength is approximately 1.2398 eV. ==Examples== An FM radio station transmitting at 100 MHz emits photons with an energy of about 4.1357 × 10−7 eV. Additionally, E = \frac{hc}{\lambda} where *E is photon energy *λ is the photon's wavelength *c is the speed of light in vacuum *h is the Planck constant The photon energy at 1 Hz is equal to 6.62607015 × 10−34 J That is equal to 4.135667697 × 10−15 eV === Electronvolt === Energy is often measured in electronvolts. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. Photon energy can be expressed using any unit of energy. Very-high-energy gamma rays have photon energies of 100 GeV to over 1 PeV (1011 to 1015 electronvolts) or 16 nanojoules to 160 microjoules. Photon energy is the energy carried by a single photon. These wavelengths correspond to photon energies of down to . Equivalently, the longer the photon's wavelength, the lower its energy. The higher the photon's frequency, the higher its energy. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. This minuscule amount of energy is approximately 8 × 10−13 times the electron's mass (via mass-energy equivalence). Light's wavenumber is proportional to frequency \textstyle \frac{1}{\lambda}=\frac{f}{c}, and therefore also proportional to light's quantum energy E. As one joule equals 6.24 × 1018 eV, the larger units may be more useful in denoting the energy of photons with higher frequency and higher energy, such as gamma rays, as opposed to lower energy photons as in the optical and radio frequency regions of the electromagnetic spectrum. ==Formulas== === Physics === Photon energy is directly proportional to frequency. The centre wavelength is the power-weighted mean wavelength: : \lambda_c = \frac {1}{P_{total}} \int p(\lambda) \lambda\, d\lambda And the total power is: :P_{total}=\int p(\lambda) d\lambda where p(\lambda) is the power spectral density, for example in W/nm. The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. During photosynthesis, specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm in the photosystem I, corresponding to an energy of each photon of ≈ 2 eV ≈ 3 × 10−19 J ≈ 75 kBT, where kBT denotes the thermal energy. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. Among the units commonly used to denote photon energy are the electronvolt (eV) and the joule (as well as its multiples, such as the microjoule). Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. Spectral irradiance of wavelengths in the solar spectrum. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5 × 10−18 J) with a maximal energy conversion efficiency of 35%. ==See also== *Photon *Electromagnetic radiation *Electromagnetic spectrum *Planck constant *Planck–Einstein relation *Soft photon ==References== Category:Foundational quantum physics Category:Electromagnetic spectrum Category:Photons ",140,0.318,2.0,76,16,C
-"A proton and a negatively charged $\mu$ meson (called a muon) can form a short-lived species called a mesonic atom. The charge of a muon is the same as that on an electron and the mass of a muon is $207 m_{\mathrm{e}}$. Assume that the Bohr theory can be applied to such a mesonic atom and calculate the ground-state energy, the radius of the first Bohr orbit, and the energy and frequency associated with the $n=1$ to $n=2$ transition in a mesonic atom.","Unlike baryonic molecules, which form the nuclei of all elements in nature save hydrogen-1, a mesonic molecule has yet to be definitively observed. The Bohr model also has difficulty with, or else fails to explain: * Much of the spectra of larger atoms. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). In kaonic hydrogen this strong contribution was found to be repulsive, shifting the ground state energy by 283 ± 36 (statistical) ± 6 (systematic) eV, thus making the system unstable with a resonance width of 541 ± 89 (stat) ± 22 (syst) eV (decay into Λπ and ΣπYiguang Yan, Kaonic hydrogen atom and kaon-proton scattering length, ). System Radius Hydrogen 1.00054\, a_0 Positronium 2 a_0 Muonium 1.0048\, a_0 He+ a_0/2 Li2+ a_0/3 ==See also== * Bohr magneton * Rydberg energy ==References== == External links == * Length Scales in Physics: the Bohr Radius Category:Atomic physics Category:Physical constants Category:Units of length Category:Niels Bohr Category:Atomic radius In atomic physics, the Bohr model or Rutherford–Bohr model of the atom, presented by Niels Bohr and Ernest Rutherford in 1913, consists of a small, dense nucleus surrounded by orbiting electrons. Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. A heavy Rydberg system consists of a weakly bound positive and negative ion orbiting their common centre of mass. Kaonic hydrogen is an exotic atom consisting of a negatively charged kaon orbiting a proton. The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model. This outer electron should be at nearly one Bohr radius from the nucleus. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. A mesonic molecule is a set of two or more mesons bound together by the strong force. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a hydrogen-like atom, by being trapped by the potential of the positive nuclear charge. == Electron energy levels == thumb|Models depicting electron energy levels in hydrogen, helium, lithium, and neon The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron, :m_\text{red} = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = m_\mathrm{e} \frac{1}{1 + m_\mathrm{e}/m_\mathrm{p}}. :The smallest possible value of r in the hydrogen atom () is called the Bohr radius and is equal to: :: r_1 = \frac{\hbar^2}{k_\mathrm{e} e^2 m_\mathrm{e}} \approx 5.29 \times 10^{-11}~\mathrm{m}. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. In 1921, following the work of chemists and others involved in work on the periodic table, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This result can be generalized to other systems, such as positronium (an electron orbiting a positron) and muonium (an electron orbiting an anti-muon) by using the reduced mass of the system and considering the possible change in charge. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions. ",2.25,0.4772,0.405,1.69,-59.24,D
+","To find the photon energy in electronvolts using the wavelength in micrometres, the equation is approximately :E\text{ (eV)} = \frac{1.2398}{\lambda\text{ (μm)}} since hc/e=1.239 \; 841 \; 984... \times 10^{-6} eVm where h is Planck's constant, c is the speed of light in m/sec, and e is the electron charge. The photon energy of near infrared radiation at 1 μm wavelength is approximately 1.2398 eV. ==Examples== An FM radio station transmitting at 100 MHz emits photons with an energy of about 4.1357 × 10−7 eV. Additionally, E = \frac{hc}{\lambda} where *E is photon energy *λ is the photon's wavelength *c is the speed of light in vacuum *h is the Planck constant The photon energy at 1 Hz is equal to 6.62607015 × 10−34 J That is equal to 4.135667697 × 10−15 eV === Electronvolt === Energy is often measured in electronvolts. The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. Photon energy can be expressed using any unit of energy. Very-high-energy gamma rays have photon energies of 100 GeV to over 1 PeV (1011 to 1015 electronvolts) or 16 nanojoules to 160 microjoules. Photon energy is the energy carried by a single photon. These wavelengths correspond to photon energies of down to . Equivalently, the longer the photon's wavelength, the lower its energy. The higher the photon's frequency, the higher its energy. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. This minuscule amount of energy is approximately 8 × 10−13 times the electron's mass (via mass-energy equivalence). Light's wavenumber is proportional to frequency \textstyle \frac{1}{\lambda}=\frac{f}{c}, and therefore also proportional to light's quantum energy E. As one joule equals 6.24 × 1018 eV, the larger units may be more useful in denoting the energy of photons with higher frequency and higher energy, such as gamma rays, as opposed to lower energy photons as in the optical and radio frequency regions of the electromagnetic spectrum. ==Formulas== === Physics === Photon energy is directly proportional to frequency. The centre wavelength is the power-weighted mean wavelength: : \lambda_c = \frac {1}{P_{total}} \int p(\lambda) \lambda\, d\lambda And the total power is: :P_{total}=\int p(\lambda) d\lambda where p(\lambda) is the power spectral density, for example in W/nm. The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. During photosynthesis, specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm in the photosystem I, corresponding to an energy of each photon of ≈ 2 eV ≈ 3 × 10−19 J ≈ 75 kBT, where kBT denotes the thermal energy. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. Among the units commonly used to denote photon energy are the electronvolt (eV) and the joule (as well as its multiples, such as the microjoule). Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. Spectral irradiance of wavelengths in the solar spectrum. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5 × 10−18 J) with a maximal energy conversion efficiency of 35%. ==See also== *Photon *Electromagnetic radiation *Electromagnetic spectrum *Planck constant *Planck–Einstein relation *Soft photon ==References== Category:Foundational quantum physics Category:Electromagnetic spectrum Category:Photons ",140,0.318,"""2.0""",76,16,C
+"A proton and a negatively charged $\mu$ meson (called a muon) can form a short-lived species called a mesonic atom. The charge of a muon is the same as that on an electron and the mass of a muon is $207 m_{\mathrm{e}}$. Assume that the Bohr theory can be applied to such a mesonic atom and calculate the ground-state energy, the radius of the first Bohr orbit, and the energy and frequency associated with the $n=1$ to $n=2$ transition in a mesonic atom.","Unlike baryonic molecules, which form the nuclei of all elements in nature save hydrogen-1, a mesonic molecule has yet to be definitively observed. The Bohr model also has difficulty with, or else fails to explain: * Much of the spectra of larger atoms. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). In kaonic hydrogen this strong contribution was found to be repulsive, shifting the ground state energy by 283 ± 36 (statistical) ± 6 (systematic) eV, thus making the system unstable with a resonance width of 541 ± 89 (stat) ± 22 (syst) eV (decay into Λπ and ΣπYiguang Yan, Kaonic hydrogen atom and kaon-proton scattering length, ). System Radius Hydrogen 1.00054\, a_0 Positronium 2 a_0 Muonium 1.0048\, a_0 He+ a_0/2 Li2+ a_0/3 ==See also== * Bohr magneton * Rydberg energy ==References== == External links == * Length Scales in Physics: the Bohr Radius Category:Atomic physics Category:Physical constants Category:Units of length Category:Niels Bohr Category:Atomic radius In atomic physics, the Bohr model or Rutherford–Bohr model of the atom, presented by Niels Bohr and Ernest Rutherford in 1913, consists of a small, dense nucleus surrounded by orbiting electrons. Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). The CODATA value of the Bohr radius (in SI units) is ==History== In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. A heavy Rydberg system consists of a weakly bound positive and negative ion orbiting their common centre of mass. Kaonic hydrogen is an exotic atom consisting of a negatively charged kaon orbiting a proton. The Bohr model is a relatively primitive model of the hydrogen atom, compared to the valence shell model. This outer electron should be at nearly one Bohr radius from the nucleus. A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. A mesonic molecule is a set of two or more mesons bound together by the strong force. Schrödinger employed de Broglie's matter waves, but sought wave solutions of a three-dimensional wave equation describing electrons that were constrained to move about the nucleus of a hydrogen-like atom, by being trapped by the potential of the positive nuclear charge. == Electron energy levels == thumb|Models depicting electron energy levels in hydrogen, helium, lithium, and neon The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. The Bohr formula properly uses the reduced mass of electron and proton in all situations, instead of the mass of the electron, :m_\text{red} = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = m_\mathrm{e} \frac{1}{1 + m_\mathrm{e}/m_\mathrm{p}}. :The smallest possible value of r in the hydrogen atom () is called the Bohr radius and is equal to: :: r_1 = \frac{\hbar^2}{k_\mathrm{e} e^2 m_\mathrm{e}} \approx 5.29 \times 10^{-11}~\mathrm{m}. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. In 1921, following the work of chemists and others involved in work on the periodic table, Bohr extended the model of hydrogen to give an approximate model for heavier atoms. This result can be generalized to other systems, such as positronium (an electron orbiting a positron) and muonium (an electron orbiting an anti-muon) by using the reduced mass of the system and considering the possible change in charge. Starting from the angular momentum quantum rule as Bohr admits is previously given by Nicholson in his 1912 paper, Bohr was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions. ",2.25,0.4772,"""0.405""",1.69,-59.24,D
 "$$
 \beta=2 \pi c \tilde{\omega}_{\mathrm{obs}}\left(\frac{\mu}{2 D}\right)^{1 / 2}
 $$
-Given that $\tilde{\omega}_{\mathrm{obs}}=2886 \mathrm{~cm}^{-1}$ and $D=440.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$ for $\mathrm{H}^{35} \mathrm{Cl}$, calculate $\beta$.","But these variables can be calculated with GSW. thumb|440x440px|This is the 2020 average for the haline contraction coefficient β. The GSW beta(SA,CT,p) function can calculate β when the absolute salinity (SA), conserved temperature (CT) and the pressure are known. \\\ & = \left ( \frac{(1 - 0.0436) (3.827 \times 10^{26}\ \mbox{W})} {0.9 (5.670 \times 10^{-8}\ \mbox{W/m}^2\mbox{K}^4) 16 \cdot 3.142 (3.959 \times 10^{11}\ \mbox{m})^2} \right )^{\frac{1}{4}} \\\ & = 173.7\ \mbox{K} \end{align} See: K | mean_motion= / day | observation_arc=130.38 yr (47622 d) | uncertainty=0 | moid= | jupiter_moid= | tisserand=3.331 }} Mathilde (minor planet designation: 253 Mathilde) is an asteroid in the intermediate asteroid belt, approximately 50 kilometers in diameter, that was discovered by Austrian astronomer Johann Palisa at Vienna Observatory on 12 November 1885. The molecular formula C25H27N (molar mass: 341.49 g/mol, exact mass: 341.2143 u) may refer to: * JWH-184 * JWH-196 The haline contraction coefficient is constant when a water parcel moves adiabatically along the isobars. == Application == The amount that density is influenced by a change in salinity or temperature can be computed from the density formula that is derived from the thermal wind balance. \rho = \rho_0 \left( \alpha \Theta + \beta S_A \right) The Brunt–Väisälä frequency can also be defined when β is known, in combination with α, Θ and S_A. A high β means that the increase in density is more than when β is low.thumb|435x435px|This graph shows the 2020 average salinity in an intersection in the Atlantic ocean at 30W. HD 142415 b is an exoplanet with the semi-amplitude of 51.3 ± 2.3 m/s. With these two coefficients, the density ratio can be calculated. The molecular formula C25H38O2 (molar mass: 370.57 g/mol, exact mass: 370.2872 u) may refer to: * CBD-DMH, or DMH-CBD * Dimethylheptylpyran * JWH-051 * Penmesterol, or penmestrol * Variecolol The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The Haline contraction coefficient, abbreviated as β, is a coefficient that describes the change in ocean density due to a salinity change, while the potential temperature and the pressure are kept constant. It is a parameter in the Equation Of State (EOS) of the ocean. β is also described as the saline contraction coefficient and is measured in [kg]/[g] in the EOS that describes the ocean. The subscripts Θ and p indicate that β is defined at constant potential temperature Θ and constant pressure p. This determines the contribution of the temperature and salinity to the density of a water parcel. β is called a contraction coefficient, because when salinity increases, water becomes denser, and if the temperature increases, water becomes less dense. == Definition == Τhe haline contraction coefficient is defined as: \beta = \frac{1}{\rho} \frac{\partial \rho}{\partial S_A}\Bigg |_{\Theta,p} where ρ is the density of a water parcel in the ocean and S_A is the absolute salinity. This is the thermodynamic equation of state. β is the salinity variant of the thermal expansion coefficient α, where the density changes due to a change in temperature instead of salinity. The direction of the mixing and whether the mixing is temperature- or salinity-driven can be determined from the density difference and the Brunt-Väisälä frequency. == Computation == β can be computed when the conserved temperature, the absolute salinity and the pressure are known from a water parcel. This equation relates the thermodynamic properties of the ocean (density, temperature, salinity and pressure). This means that changing the salinity will have a large effect on the density when the haline contraction coefficient is high. === Physical examples === β is not a constant, it mostly changes with latitude and depth. This frequency is a measure of the stratification of a fluid column and is defined over depth as: N^2 = g \left( \alpha \frac{\partial \Theta}{\partial z} - \beta \frac{\partial S_A}{\partial z} \right). These equations are based on empirical thermodynamic properties. This water is dense, because it is cold. β around Antarctica is relatively high. At locations where salinity is high, as in the tropics, β is low and where salinity is low, β is high. ",-233,1.81,35.64,7,0.14,B
-"Two narrow slits separated by $0.10 \mathrm{~mm}$ are illuminated by light of wavelength $600 \mathrm{~nm}$. What is the angular position of the first maximum in the interference pattern? If a detector is located $2.00 \mathrm{~m}$ beyond the slits, what is the distance between the central maximum and the first maximum?","thumb|right|Interference pattern of double slits, where the slit width is one third the wavelength. Angular distance or angular separation, also known as apparent distance or apparent separation, denoted \theta, is the angle between the two sightlines, or between two point objects as viewed from an observer. Given that \delta_A-\delta_B\ll 1 and \alpha_A-\alpha_B\ll 1, at a second-order development it turns that \cos\delta_A\cos\delta_B \frac{(\alpha_A - \alpha_B)^2}{2} \approx \cos^2\delta_A \frac{(\alpha_A - \alpha_B)^2}{2}, so that :\theta \approx \sqrt{\left[(\alpha_A - \alpha_B)\cos\delta_A\right]^2 + (\delta_A-\delta_B)^2} === Small angular distance: planar approximation === thumb|Planar approximation of angular distance on sky If we consider a detector imaging a small sky field (dimension much less than one radian) with the y-axis pointing up, parallel to the meridian of right ascension \alpha, and the x-axis along the parallel of declination \delta, the angular separation can be written as: : \theta \approx \sqrt{\delta x^2 + \delta y^2} where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B. Note that the y-axis is equal to the declination, whereas the x-axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure). ==See also== * Milliradian * Gradian * Hour angle * Central angle * Angle of rotation * Angular diameter * Angular displacement * Great-circle distance * ==References== * CASTOR, author(s) unknown. thumb|250px|An illustration of how position angle is estimated through a telescope eyepiece; the primary star is at center. 300px|right|thumb|Jack Huntington .500 Maximum The .500 Maximum, also known as .500 Linebaugh Maximum and .500 Linebaugh Long, is a revolver cartridge developed by John Linebaugh. thumb|Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1 In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. The minimum distance is the distance along a great circle that runs through and . thumb|upright=1.35|The five red points are the maxima of the set of all the red and yellow points. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative 2x. If it is, output the point as one of the maximal points, and remember its -coordinate as the greatest seen so far. If it is, output the point as one of the maximal points, and remember its -coordinate as the greatest seen so far. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. It is calculated in a plane that contains the sphere center and the great circle, :: d_{s,t}=R\theta_{s,t} where is the angular distance of two points viewed from the center of the sphere, measured in radians. A position along the great circle is ::\mathbf{s}(\theta) = \cos\theta \mathbf{s}+\sin\theta \mathbf{s}_\perp,\quad 0\le\theta\le 2\pi. The maxima of a point set are all the maximal points of . Therefore, \mathbf{n_A} \cdot \mathbf{n_B} = \cos\delta_A \cos\alpha_A \cos\delta_B \cos\alpha_B + \cos\delta_A \sin\alpha_A \cos\delta_B \sin\alpha_B + \sin\delta_A \sin\delta_B \equiv \cos\theta then: :\theta = \cos^{-1}\left[\sin\delta_A \sin\delta_B + \cos\delta_A \cos\delta_B \cos(\alpha_A - \alpha_B)\right] === Small angular distance approximation === The above expression is valid for any position of A and B on the sphere. Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is ==Functions of more than one variable== thumb|right|The global maximum is the point at the top thumb|right|Counterexample: The red dot shows a local minimum that is not a global minimum For functions of more than one variable, similar conditions apply. In general, if an ordered set S has a greatest element m, then m is a maximal element of the set, also denoted as \max(S). The problem of finding all maximal points, sometimes called the problem of the maxima or maxima set problem, has been studied as a variant of the convex hull and orthogonal convex hull problems. Let O indicate the observer on Earth, assumed to be located at the center of the celestial sphere. The value of the function at a maximum point is called the of the function, denoted \max(f(x)), and the value of the function at a minimum point is called the of the function. ",12,0.02828,-0.1,25.6773,435,A
+Given that $\tilde{\omega}_{\mathrm{obs}}=2886 \mathrm{~cm}^{-1}$ and $D=440.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$ for $\mathrm{H}^{35} \mathrm{Cl}$, calculate $\beta$.","But these variables can be calculated with GSW. thumb|440x440px|This is the 2020 average for the haline contraction coefficient β. The GSW beta(SA,CT,p) function can calculate β when the absolute salinity (SA), conserved temperature (CT) and the pressure are known. \\\ & = \left ( \frac{(1 - 0.0436) (3.827 \times 10^{26}\ \mbox{W})} {0.9 (5.670 \times 10^{-8}\ \mbox{W/m}^2\mbox{K}^4) 16 \cdot 3.142 (3.959 \times 10^{11}\ \mbox{m})^2} \right )^{\frac{1}{4}} \\\ & = 173.7\ \mbox{K} \end{align} See: K | mean_motion= / day | observation_arc=130.38 yr (47622 d) | uncertainty=0 | moid= | jupiter_moid= | tisserand=3.331 }} Mathilde (minor planet designation: 253 Mathilde) is an asteroid in the intermediate asteroid belt, approximately 50 kilometers in diameter, that was discovered by Austrian astronomer Johann Palisa at Vienna Observatory on 12 November 1885. The molecular formula C25H27N (molar mass: 341.49 g/mol, exact mass: 341.2143 u) may refer to: * JWH-184 * JWH-196 The haline contraction coefficient is constant when a water parcel moves adiabatically along the isobars. == Application == The amount that density is influenced by a change in salinity or temperature can be computed from the density formula that is derived from the thermal wind balance. \rho = \rho_0 \left( \alpha \Theta + \beta S_A \right) The Brunt–Väisälä frequency can also be defined when β is known, in combination with α, Θ and S_A. A high β means that the increase in density is more than when β is low.thumb|435x435px|This graph shows the 2020 average salinity in an intersection in the Atlantic ocean at 30W. HD 142415 b is an exoplanet with the semi-amplitude of 51.3 ± 2.3 m/s. With these two coefficients, the density ratio can be calculated. The molecular formula C25H38O2 (molar mass: 370.57 g/mol, exact mass: 370.2872 u) may refer to: * CBD-DMH, or DMH-CBD * Dimethylheptylpyran * JWH-051 * Penmesterol, or penmestrol * Variecolol The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The Haline contraction coefficient, abbreviated as β, is a coefficient that describes the change in ocean density due to a salinity change, while the potential temperature and the pressure are kept constant. It is a parameter in the Equation Of State (EOS) of the ocean. β is also described as the saline contraction coefficient and is measured in [kg]/[g] in the EOS that describes the ocean. The subscripts Θ and p indicate that β is defined at constant potential temperature Θ and constant pressure p. This determines the contribution of the temperature and salinity to the density of a water parcel. β is called a contraction coefficient, because when salinity increases, water becomes denser, and if the temperature increases, water becomes less dense. == Definition == Τhe haline contraction coefficient is defined as: \beta = \frac{1}{\rho} \frac{\partial \rho}{\partial S_A}\Bigg |_{\Theta,p} where ρ is the density of a water parcel in the ocean and S_A is the absolute salinity. This is the thermodynamic equation of state. β is the salinity variant of the thermal expansion coefficient α, where the density changes due to a change in temperature instead of salinity. The direction of the mixing and whether the mixing is temperature- or salinity-driven can be determined from the density difference and the Brunt-Väisälä frequency. == Computation == β can be computed when the conserved temperature, the absolute salinity and the pressure are known from a water parcel. This equation relates the thermodynamic properties of the ocean (density, temperature, salinity and pressure). This means that changing the salinity will have a large effect on the density when the haline contraction coefficient is high. === Physical examples === β is not a constant, it mostly changes with latitude and depth. This frequency is a measure of the stratification of a fluid column and is defined over depth as: N^2 = g \left( \alpha \frac{\partial \Theta}{\partial z} - \beta \frac{\partial S_A}{\partial z} \right). These equations are based on empirical thermodynamic properties. This water is dense, because it is cold. β around Antarctica is relatively high. At locations where salinity is high, as in the tropics, β is low and where salinity is low, β is high. ",-233,1.81,"""35.64""",7,0.14,B
+"Two narrow slits separated by $0.10 \mathrm{~mm}$ are illuminated by light of wavelength $600 \mathrm{~nm}$. What is the angular position of the first maximum in the interference pattern? If a detector is located $2.00 \mathrm{~m}$ beyond the slits, what is the distance between the central maximum and the first maximum?","thumb|right|Interference pattern of double slits, where the slit width is one third the wavelength. Angular distance or angular separation, also known as apparent distance or apparent separation, denoted \theta, is the angle between the two sightlines, or between two point objects as viewed from an observer. Given that \delta_A-\delta_B\ll 1 and \alpha_A-\alpha_B\ll 1, at a second-order development it turns that \cos\delta_A\cos\delta_B \frac{(\alpha_A - \alpha_B)^2}{2} \approx \cos^2\delta_A \frac{(\alpha_A - \alpha_B)^2}{2}, so that :\theta \approx \sqrt{\left[(\alpha_A - \alpha_B)\cos\delta_A\right]^2 + (\delta_A-\delta_B)^2} === Small angular distance: planar approximation === thumb|Planar approximation of angular distance on sky If we consider a detector imaging a small sky field (dimension much less than one radian) with the y-axis pointing up, parallel to the meridian of right ascension \alpha, and the x-axis along the parallel of declination \delta, the angular separation can be written as: : \theta \approx \sqrt{\delta x^2 + \delta y^2} where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B. Note that the y-axis is equal to the declination, whereas the x-axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure). ==See also== * Milliradian * Gradian * Hour angle * Central angle * Angle of rotation * Angular diameter * Angular displacement * Great-circle distance * ==References== * CASTOR, author(s) unknown. thumb|250px|An illustration of how position angle is estimated through a telescope eyepiece; the primary star is at center. 300px|right|thumb|Jack Huntington .500 Maximum The .500 Maximum, also known as .500 Linebaugh Maximum and .500 Linebaugh Long, is a revolver cartridge developed by John Linebaugh. thumb|Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1 In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. The minimum distance is the distance along a great circle that runs through and . thumb|upright=1.35|The five red points are the maxima of the set of all the red and yellow points. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative 2x. If it is, output the point as one of the maximal points, and remember its -coordinate as the greatest seen so far. If it is, output the point as one of the maximal points, and remember its -coordinate as the greatest seen so far. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the largest (or smallest) one. Although the first derivative (3x2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) \sqrt[x]{x} Unique global maximum at x = e. It is calculated in a plane that contains the sphere center and the great circle, :: d_{s,t}=R\theta_{s,t} where is the angular distance of two points viewed from the center of the sphere, measured in radians. A position along the great circle is ::\mathbf{s}(\theta) = \cos\theta \mathbf{s}+\sin\theta \mathbf{s}_\perp,\quad 0\le\theta\le 2\pi. The maxima of a point set are all the maximal points of . Therefore, \mathbf{n_A} \cdot \mathbf{n_B} = \cos\delta_A \cos\alpha_A \cos\delta_B \cos\alpha_B + \cos\delta_A \sin\alpha_A \cos\delta_B \sin\alpha_B + \sin\delta_A \sin\delta_B \equiv \cos\theta then: :\theta = \cos^{-1}\left[\sin\delta_A \sin\delta_B + \cos\delta_A \cos\delta_B \cos(\alpha_A - \alpha_B)\right] === Small angular distance approximation === The above expression is valid for any position of A and B on the sphere. Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is ==Functions of more than one variable== thumb|right|The global maximum is the point at the top thumb|right|Counterexample: The red dot shows a local minimum that is not a global minimum For functions of more than one variable, similar conditions apply. In general, if an ordered set S has a greatest element m, then m is a maximal element of the set, also denoted as \max(S). The problem of finding all maximal points, sometimes called the problem of the maxima or maxima set problem, has been studied as a variant of the convex hull and orthogonal convex hull problems. Let O indicate the observer on Earth, assumed to be located at the center of the celestial sphere. The value of the function at a maximum point is called the of the function, denoted \max(f(x)), and the value of the function at a minimum point is called the of the function. ",12,0.02828,"""-0.1""",25.6773,435,A
 "$$
 \text { If we locate an electron to within } 20 \mathrm{pm} \text {, then what is the uncertainty in its speed? }
-$$","* Grabe, M ., Measurement Uncertainties in Science and Technology, Springer 2005. Uncertainty of measurement results. Evaluating the Uncertainty of Measurement. The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The measurement uncertainty strongly depends on the size of the measured value itself, e.g. amplitude-proportional. == See also == * measurement uncertainty * uncertainty * accuracy and precision * confidence Interval Category:Measurement In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. Uncertainty principle of Heisenberg, 1927. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. One way to quantify the precision of the position and momentum is the standard deviation σ. The measurement uncertainty is often taken as the standard deviation of a state-of-knowledge probability distribution over the possible values that could be attributed to a measured quantity. * Possolo A and Iyer H K 2017 Concepts and tools for the evaluation of measurement uncertainty Rev. Sci. Instrum.,88 011301 (2017). * Introduction to evaluating uncertainty of measurement * JCGM 200:2008. ""Quantifying uncertainty in analytical measurement"". Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The book draws its name from the uncertainty principle, which states that, in quantum mechanics, there exists pairs of quantities, such as position and velocity, in which you cannot know the precise value of both at the same time. In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. The uncertainty has two components, namely, bias (related to accuracy) and the unavoidable random variation that occurs when making repeated measurements (related to precision). In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle."" In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. ",0.042,3.7,25.6773,9.14,840,B
-The mean temperature of the earth's surface is $288 \mathrm{~K}$. What is the maximum wavelength of the earth's blackbody radiation?,"This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrumsPrinciples of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. Notice that a gray (flat spectrum) ball where ({1-\alpha}) ={\overline{\varepsilon}} comes to the same temperature as a black body no matter how dark or light gray. ====Effective temperature of Earth==== Substituting the measured values for the Sun and Earth yields: :T_{\rm S} = 5778 \ \mathrm{K},NASA Sun Fact Sheet :R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m}, :D = 1.496 \times 10^{11} \ \mathrm{m}, :\alpha = 0.306 \ With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: :T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. Solving for the wavelength \lambda in millimetres, and using kelvins for the temperature yields: : ===Parameterization by frequency=== Another common parameterization is by frequency. Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law. thumb|upright=1.45|Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K. Notice that for a given temperature, different parameterizations imply different maximal wavelengths. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. The formula is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and is the Stefan–Boltzmann constant. ==Equations== ===Planck's law of black-body radiation=== Planck's law states that :B_ u(T) = \frac{2h u^3}{c^2}\frac{1}{e^{h u/kT} - 1}, where :B_{ u}(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency u radiation per unit frequency at thermal equilibrium at temperature T. Units: power / [area × solid angle × frequency]. :h is the Planck constant; :c is the speed of light in vacuum; :k is the Boltzmann constant; : u is the frequency of the electromagnetic radiation; :T is the absolute temperature of the body. An Introductory Survey, second edition, Elsevier, Amsterdam, , exercise 4.6, pages 119–120. ===Cosmology=== The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K. Application of Wien's law to human-body emission results in a peak wavelength of :\lambda_\text{peak} = \frac{2.898 \times 10^{-3}~\text{K} \cdot \text{m}}{305~\text{K}} = 9.50~\mu\text{m}. The relevant math is detailed in the next section. ==Derivation from Planck's law== ===Parameterization by wavelength=== Planck's law for the spectrum of black body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also Wien's law). thumb|303px|As the temperature increases, the peak of the emitted black-body radiation curve moves to higher intensities and shorter wavelengths. An equilibrium temperature of 255 K on Earth yields a skin temperature of 214 K, which compares with a tropopause temperature of 209 K. == References == Category:Temperature Category:Atmospheric radiation At a typical room temperature of 293 K (20 °C), the maximum intensity is for . ===Stefan–Boltzmann law=== By integrating B_ u(T)\cos(\theta) over the frequency the radiance L (units: power / [area * solid angle] ) is : L=\frac{2\pi^5}{15} \frac{k^4 T^4}{c^2h^3} \frac{1}{\pi}= \sigma T^4 \frac{\cos(\theta)}{\pi} by using \int_0^\infty dx\, \frac{x^3}{e^x - 1}=\frac{\pi^4}{15} with x \equiv \frac{h u}{k T} and with \sigma \equiv \frac{2\pi^5}{15} \frac{k^4}{c^2h^3}=5.670373 \times 10^{-8} \frac{W}{m^2 K^4} being the Stefan–Boltzmann constant. The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. This is an inverse relationship between wavelength and temperature. *The effective temperature of the Sun is 5778 Kelvin. The intensity maximum for this is given by : u_\text{peak} = T \times 5.879 \times 10^{10} \ \mathrm{Hz}/\mathrm{K}. For the same temperature, but parameterizing by frequency, the frequency for maximal spectral radiance is with corresponding wavelength . For wavelength λ, it is: B_{\lambda} (T) = \frac{2 ck_{\mathrm{B}} T}{\lambda^4}, where B_{\lambda} is the spectral radiance, the power emitted per unit emitting area, per steradian, per unit wavelength; c is the speed of light; k_{\mathrm{B}} is the Boltzmann constant; and T is the temperature in kelvin. At a typical room temperature of 293 K (20 °C), the maximum intensity is at . The two factors combined give the characteristic maximum wavelength. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is: :u_{\lambda}(\lambda,T) = {2 h c^2\over \lambda^5}{1\over e^{h c/\lambda kT}-1}. ",1.01,6.0,6.283185307,4.85,11,A
-"The power output of a laser is measured in units of watts (W), where one watt is equal to one joule per second. $\left(1 \mathrm{~W}=1 \mathrm{~J} \cdot \mathrm{s}^{-1}\right.$.) What is the number of photons emitted per second by a $1.00 \mathrm{~mW}$ nitrogen laser? The wavelength emitted by a nitrogen laser is $337 \mathrm{~nm}$.","thumb|right|300 px|A 337nm wavelength and 170 μJ pulse energy 20 Hz cartridge nitrogen laser A nitrogen laser is a gas laser operating in the ultraviolet rangeC. The wall-plug efficiency of the nitrogen laser is low, typically 0.1% or less, though nitrogen lasers with efficiency of up to 3% have been reported in the literature. Duarte and L. W. Hillman, Dye Laser Principles (Academic, New York, 1990) Chapter 6. * measurement of air pollution (Lidar) * Matrix-assisted laser desorption/ionization * List of laser articles ==External links== *Professor Mark Csele's Homebuilt Lasers Page *Example of TEA Laser prototype *Sam's lasers FAQ/Home Built nitrogen (N2) laser * an update of the Amateur Scientist column, on page 122 of the June, 1974 issue of Scientific American *Compact High-Power N2 Laser: Circuit Theory and Design Adolph Schwab & Fritz Hollinger IEEE Journal of Quantum Electronics, QE-12, No. 3, March 1966, p.183 ==References== Category:Gas lasers Air, which is 78% nitrogen, can be used, but more than 0.5% oxygen poisons the laser. == Optics == Nitrogen lasers can operate within a resonator cavity, but due to the typical gain of 2 every 20 mm they more often operate on superluminescence alone; though it is common to put a mirror at one end such that the output is emitted from the opposite end. Pulse durations vary from a few hundred picoseconds (at 1 atmosphere partial pressure of nitrogen) to about 30 nanoseconds at reduced pressure (typically some dozens of Torr), though FWHM pulsewidths of 6 to 8 ns are typical. === Amateur construction === The transverse discharge nitrogen laser has long been a popular choice for amateur home construction, owing to its simple construction and simple gas handling. With about 11 ns the UV generation, ionisation, and electron capture are in a similar speed regime as the nitrogen laser pulse duration and thus as fast electric must be applied. === Excitation by electron impact === The upper laser level is excited efficiently by electrons with more than 11 eV, best energy is 15 eV. The wall-plug efficiency is the product of the following three efficiencies: * electrical: TEA laser * gain medium: This is the same for all nitrogen lasers and thus has to be at least 3% ** inversion by electron impact is 10 to 1 due to Franck–Condon principle ** energy lost in the lower laser level: 40% * optical: More stimulated emission than spontaneous emission ==Gain medium== The gain medium is nitrogen molecules in the gas phase. This nicely matches the typical rise times of 1×10−8 s and typical currents of 1×103 A occurring in nitrogen lasers. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. ==History of He-Ne laser development== The first He-Ne lasers emitted infrared at 1150 nm, and were the first gas lasers and the first lasers with continuous wave output. Measured nitrogen laser pulses are so long that the second step is unimportant. The precise wavelength of red He-Ne lasers is 632.991 nm in a vacuum, which is refracted to about 632.816 nm in air. The nitrogen laser is a three-level laser. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. Absolute stabilization of the laser's frequency (or wavelength) as fine as 2.5 parts in 1011 can be obtained through use of an iodine absorption cell. thumb|Energy levels in a He-Ne Laser|512x330px thumb|Ring He-Ne Laser The mechanism producing population inversion and light amplification in a He-Ne laser plasma originates with inelastic collision of energetic electrons with ground-state helium atoms in the gas mixture. In contrast to more typical four-level lasers, the upper laser level of nitrogen is directly pumped, imposing no speed limits on the pump. Higher voltages mean shorter pulses. ==Typical devices== The gas pressure in a nitrogen laser ranges from a few mbar to as much as several bar. The laser diode rate equations model the electrical and optical performance of a laser diode. thumb|250px|Laser modules (bottom to top: 405, 445, 520, 532, 635, and 660 nm) Laser science or laser physics is a branch of optics that describes the theory and practice of lasers. thumb|Helium–neon laser at the University of Chemnitz, Germany A helium–neon laser or He-Ne laser, is a type of gas laser whose high energetic medium gain medium consists of a mixture of ratio(between 5:1 and 20:1) of helium and neon at a total pressure of about 1 torr inside of a small electrical discharge. However, in high-power He-Ne lasers having a particularly long cavity, superluminescence at 3.39 μm can become a nuisance, robbing power from the stimulated emission medium, often requiring additional suppression. Willett, Introduction to Gas Lasers: Population Inversion Mechanisms (Pergamon, New York,1974). (typically 337.1 nm) using molecular nitrogen as its gain medium, pumped by an electrical discharge. ",1.70,3.2,0.54,-13.598 ,4.16,A
+$$","* Grabe, M ., Measurement Uncertainties in Science and Technology, Springer 2005. Uncertainty of measurement results. Evaluating the Uncertainty of Measurement. The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The measurement uncertainty strongly depends on the size of the measured value itself, e.g. amplitude-proportional. == See also == * measurement uncertainty * uncertainty * accuracy and precision * confidence Interval Category:Measurement In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. Uncertainty principle of Heisenberg, 1927. Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology. One way to quantify the precision of the position and momentum is the standard deviation σ. The measurement uncertainty is often taken as the standard deviation of a state-of-knowledge probability distribution over the possible values that could be attributed to a measured quantity. * Possolo A and Iyer H K 2017 Concepts and tools for the evaluation of measurement uncertainty Rev. Sci. Instrum.,88 011301 (2017). * Introduction to evaluating uncertainty of measurement * JCGM 200:2008. ""Quantifying uncertainty in analytical measurement"". Introduced first in 1927 by the German physicist Werner Heisenberg, the uncertainty principle states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted from initial conditions, and vice versa. The book draws its name from the uncertainty principle, which states that, in quantum mechanics, there exists pairs of quantities, such as position and velocity, in which you cannot know the precise value of both at the same time. In metrology, measurement uncertainty is the expression of the statistical dispersion of the values attributed to a measured quantity. Since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. The uncertainty has two components, namely, bias (related to accuracy) and the unavoidable random variation that occurs when making repeated measurements (related to precision). In this manner, said Einstein, one could measure the energy emitted and the time it was released with any desired precision, in contradiction to the uncertainty principle."" In other words, the particle position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. ",0.042,3.7,"""25.6773""",9.14,840,B
+The mean temperature of the earth's surface is $288 \mathrm{~K}$. What is the maximum wavelength of the earth's blackbody radiation?,"This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrumsPrinciples of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. Notice that a gray (flat spectrum) ball where ({1-\alpha}) ={\overline{\varepsilon}} comes to the same temperature as a black body no matter how dark or light gray. ====Effective temperature of Earth==== Substituting the measured values for the Sun and Earth yields: :T_{\rm S} = 5778 \ \mathrm{K},NASA Sun Fact Sheet :R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m}, :D = 1.496 \times 10^{11} \ \mathrm{m}, :\alpha = 0.306 \ With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: :T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. Solving for the wavelength \lambda in millimetres, and using kelvins for the temperature yields: : ===Parameterization by frequency=== Another common parameterization is by frequency. Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law. thumb|upright=1.45|Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K. Notice that for a given temperature, different parameterizations imply different maximal wavelengths. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. The formula is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and is the Stefan–Boltzmann constant. ==Equations== ===Planck's law of black-body radiation=== Planck's law states that :B_ u(T) = \frac{2h u^3}{c^2}\frac{1}{e^{h u/kT} - 1}, where :B_{ u}(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency u radiation per unit frequency at thermal equilibrium at temperature T. Units: power / [area × solid angle × frequency]. :h is the Planck constant; :c is the speed of light in vacuum; :k is the Boltzmann constant; : u is the frequency of the electromagnetic radiation; :T is the absolute temperature of the body. An Introductory Survey, second edition, Elsevier, Amsterdam, , exercise 4.6, pages 119–120. ===Cosmology=== The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K. Application of Wien's law to human-body emission results in a peak wavelength of :\lambda_\text{peak} = \frac{2.898 \times 10^{-3}~\text{K} \cdot \text{m}}{305~\text{K}} = 9.50~\mu\text{m}. The relevant math is detailed in the next section. ==Derivation from Planck's law== ===Parameterization by wavelength=== Planck's law for the spectrum of black body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also Wien's law). thumb|303px|As the temperature increases, the peak of the emitted black-body radiation curve moves to higher intensities and shorter wavelengths. An equilibrium temperature of 255 K on Earth yields a skin temperature of 214 K, which compares with a tropopause temperature of 209 K. == References == Category:Temperature Category:Atmospheric radiation At a typical room temperature of 293 K (20 °C), the maximum intensity is for . ===Stefan–Boltzmann law=== By integrating B_ u(T)\cos(\theta) over the frequency the radiance L (units: power / [area * solid angle] ) is : L=\frac{2\pi^5}{15} \frac{k^4 T^4}{c^2h^3} \frac{1}{\pi}= \sigma T^4 \frac{\cos(\theta)}{\pi} by using \int_0^\infty dx\, \frac{x^3}{e^x - 1}=\frac{\pi^4}{15} with x \equiv \frac{h u}{k T} and with \sigma \equiv \frac{2\pi^5}{15} \frac{k^4}{c^2h^3}=5.670373 \times 10^{-8} \frac{W}{m^2 K^4} being the Stefan–Boltzmann constant. The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the Stefan–Boltzmann law. This is an inverse relationship between wavelength and temperature. *The effective temperature of the Sun is 5778 Kelvin. The intensity maximum for this is given by : u_\text{peak} = T \times 5.879 \times 10^{10} \ \mathrm{Hz}/\mathrm{K}. For the same temperature, but parameterizing by frequency, the frequency for maximal spectral radiance is with corresponding wavelength . For wavelength λ, it is: B_{\lambda} (T) = \frac{2 ck_{\mathrm{B}} T}{\lambda^4}, where B_{\lambda} is the spectral radiance, the power emitted per unit emitting area, per steradian, per unit wavelength; c is the speed of light; k_{\mathrm{B}} is the Boltzmann constant; and T is the temperature in kelvin. At a typical room temperature of 293 K (20 °C), the maximum intensity is at . The two factors combined give the characteristic maximum wavelength. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is: :u_{\lambda}(\lambda,T) = {2 h c^2\over \lambda^5}{1\over e^{h c/\lambda kT}-1}. ",1.01,6.0,"""6.283185307""",4.85,11,A
+"The power output of a laser is measured in units of watts (W), where one watt is equal to one joule per second. $\left(1 \mathrm{~W}=1 \mathrm{~J} \cdot \mathrm{s}^{-1}\right.$.) What is the number of photons emitted per second by a $1.00 \mathrm{~mW}$ nitrogen laser? The wavelength emitted by a nitrogen laser is $337 \mathrm{~nm}$.","thumb|right|300 px|A 337nm wavelength and 170 μJ pulse energy 20 Hz cartridge nitrogen laser A nitrogen laser is a gas laser operating in the ultraviolet rangeC. The wall-plug efficiency of the nitrogen laser is low, typically 0.1% or less, though nitrogen lasers with efficiency of up to 3% have been reported in the literature. Duarte and L. W. Hillman, Dye Laser Principles (Academic, New York, 1990) Chapter 6. * measurement of air pollution (Lidar) * Matrix-assisted laser desorption/ionization * List of laser articles ==External links== *Professor Mark Csele's Homebuilt Lasers Page *Example of TEA Laser prototype *Sam's lasers FAQ/Home Built nitrogen (N2) laser * an update of the Amateur Scientist column, on page 122 of the June, 1974 issue of Scientific American *Compact High-Power N2 Laser: Circuit Theory and Design Adolph Schwab & Fritz Hollinger IEEE Journal of Quantum Electronics, QE-12, No. 3, March 1966, p.183 ==References== Category:Gas lasers Air, which is 78% nitrogen, can be used, but more than 0.5% oxygen poisons the laser. == Optics == Nitrogen lasers can operate within a resonator cavity, but due to the typical gain of 2 every 20 mm they more often operate on superluminescence alone; though it is common to put a mirror at one end such that the output is emitted from the opposite end. Pulse durations vary from a few hundred picoseconds (at 1 atmosphere partial pressure of nitrogen) to about 30 nanoseconds at reduced pressure (typically some dozens of Torr), though FWHM pulsewidths of 6 to 8 ns are typical. === Amateur construction === The transverse discharge nitrogen laser has long been a popular choice for amateur home construction, owing to its simple construction and simple gas handling. With about 11 ns the UV generation, ionisation, and electron capture are in a similar speed regime as the nitrogen laser pulse duration and thus as fast electric must be applied. === Excitation by electron impact === The upper laser level is excited efficiently by electrons with more than 11 eV, best energy is 15 eV. The wall-plug efficiency is the product of the following three efficiencies: * electrical: TEA laser * gain medium: This is the same for all nitrogen lasers and thus has to be at least 3% ** inversion by electron impact is 10 to 1 due to Franck–Condon principle ** energy lost in the lower laser level: 40% * optical: More stimulated emission than spontaneous emission ==Gain medium== The gain medium is nitrogen molecules in the gas phase. This nicely matches the typical rise times of 1×10−8 s and typical currents of 1×103 A occurring in nitrogen lasers. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. ==History of He-Ne laser development== The first He-Ne lasers emitted infrared at 1150 nm, and were the first gas lasers and the first lasers with continuous wave output. Measured nitrogen laser pulses are so long that the second step is unimportant. The precise wavelength of red He-Ne lasers is 632.991 nm in a vacuum, which is refracted to about 632.816 nm in air. The nitrogen laser is a three-level laser. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. The best-known and most widely used He-Ne laser operates at a wavelength of 632.8 nm, in the red part of the visible spectrum. Absolute stabilization of the laser's frequency (or wavelength) as fine as 2.5 parts in 1011 can be obtained through use of an iodine absorption cell. thumb|Energy levels in a He-Ne Laser|512x330px thumb|Ring He-Ne Laser The mechanism producing population inversion and light amplification in a He-Ne laser plasma originates with inelastic collision of energetic electrons with ground-state helium atoms in the gas mixture. In contrast to more typical four-level lasers, the upper laser level of nitrogen is directly pumped, imposing no speed limits on the pump. Higher voltages mean shorter pulses. ==Typical devices== The gas pressure in a nitrogen laser ranges from a few mbar to as much as several bar. The laser diode rate equations model the electrical and optical performance of a laser diode. thumb|250px|Laser modules (bottom to top: 405, 445, 520, 532, 635, and 660 nm) Laser science or laser physics is a branch of optics that describes the theory and practice of lasers. thumb|Helium–neon laser at the University of Chemnitz, Germany A helium–neon laser or He-Ne laser, is a type of gas laser whose high energetic medium gain medium consists of a mixture of ratio(between 5:1 and 20:1) of helium and neon at a total pressure of about 1 torr inside of a small electrical discharge. However, in high-power He-Ne lasers having a particularly long cavity, superluminescence at 3.39 μm can become a nuisance, robbing power from the stimulated emission medium, often requiring additional suppression. Willett, Introduction to Gas Lasers: Population Inversion Mechanisms (Pergamon, New York,1974). (typically 337.1 nm) using molecular nitrogen as its gain medium, pumped by an electrical discharge. ",1.70,3.2,"""0.54""",-13.598 ,4.16,A
 " Sirius, one of the hottest known stars, has approximately a blackbody spectrum with $\lambda_{\max }=260 \mathrm{~nm}$. Estimate the surface temperature of Sirius.
-","The brighter component, termed Sirius A, is a main-sequence star of spectral type early A, with an estimated surface temperature of 9,940 K. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. Because there is no internal heat source, Sirius B will steadily cool as the remaining heat is radiated into space over the next two billion years or so. Sirius is the brightest star in the night sky. To the naked eye, it often appears to be flashing with red, white, and blue hues when near the horizon. == Observation == With an apparent magnitude of −1.46, Sirius is the brightest star in the night sky, almost twice as bright as the second-brightest star, Canopus. This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrumsPrinciples of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. An Introductory Survey, second edition, Elsevier, Amsterdam, , exercise 4.6, pages 119–120. ===Cosmology=== The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K. This is a list of hottest stars so far discovered (excluding degenerate stars), arranged by decreasing temperature. Because of its declination of roughly −17°, Sirius is a circumpolar star from latitudes south of 73° S. The effective temperature or effective radiative emission temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation.Stull, R. (2000). Sirius is gradually moving closer to the Solar System; it is expected to increase in brightness slightly over the next 60,000 years to reach a peak magnitude of −1.68. Notice that a gray (flat spectrum) ball where ({1-\alpha}) ={\overline{\varepsilon}} comes to the same temperature as a black body no matter how dark or light gray. ====Effective temperature of Earth==== Substituting the measured values for the Sun and Earth yields: :T_{\rm S} = 5778 \ \mathrm{K},NASA Sun Fact Sheet :R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m}, :D = 1.496 \times 10^{11} \ \mathrm{m}, :\alpha = 0.306 \ With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: :T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. In that year, Sirius will come to within 1.6 degrees of the south celestial pole. The outer atmosphere of Sirius B is now almost pure hydrogen—the element with the lowest mass—and no other elements are seen in its spectrum. === Apparent third star === Since 1894, irregularities have been tentatively observed in the orbits of Sirius A and B with an apparent periodicity of 6–6.4 years. With a visual apparent magnitude of −1.46, Sirius is almost twice as bright as Canopus, the next brightest star. Sirius appears bright because of its intrinsic luminosity and its proximity to the Solar System. A star near the middle of the spectrum, such as the modest Sun or the giant Capella radiates more energy per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as Vega or Rigel. == Planet == ===Blackbody temperature=== To find the effective (blackbody) temperature of a planet, it can be calculated by equating the power received by the planet to the known power emitted by a blackbody of temperature . Sirius A is about twice as massive as the Sun () and has an absolute visual magnitude of +1.43. The visible star is now sometimes known as Sirius A. When compared to the Sun, the proportion of iron in the atmosphere of Sirius A relative to hydrogen is given by \textstyle\ \left[\frac{\ce{Fe}}{\ce{H}}\right] = 0.5\ , meaning iron is 316% as abundant as in the Sun's atmosphere. The stars with temperatures higher than 60,000 K are included. ==List== Star name Effective Temperature (K) Mass () Luminosity () Spectral type Distance Ref. WR 102 210,000 16.1 380,000 WO2 8,610 WR 142 200,000 28.6 912,000 WO2 5,400 LMC195-1 200,000 WO2 160,000 BAT99-123 170,000 158,000 WO3 ~160,000 WR 93b 160,000 8.1 110,000 WO3 7,470 [HC2007] 31 160,000? ", 135.36,234.4,11000.0,130.400766848,0.264,C
-A ground-state hydrogen atom absorbs a photon of light that has a wavelength of $97.2 \mathrm{~nm}$. It then gives off a photon that has a wavelength of $486 \mathrm{~nm}$. What is the final state of the hydrogen atom?,"In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state. == Hydrogenic potential == thumb|right|Figure 3. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). The record wavelength for hydrogen is λ = 73 cm for H253α, implying atomic diameters of a few microns, and for carbon, λ = 18 metres, from C732α, from atoms with a diameter of 57 micron. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. The transition from the state 1^2S_{1/2} with mj=-1/2 to the state 2^2P_{3/2} with mj=-1/2 is only allowed for light with polarization along the z axis (quantization axis) of the atom. This is a quantum state change between the two hyperfine levels of the hydrogen 1 s ground state. A hydrogen atom consists of an electron orbiting its nucleus. In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. This two-step photodissociation process, known as the Solomon process, is one of the main mechanisms by which molecular hydrogen is destroyed in the interstellar medium. thumb|Electronic and vibrational levels of the hydrogen molecule In reference to the figure shown, Lyman-Werner photons are emitted as described below: *A hydrogen molecule can absorb a far- ultraviolet photon (11.2 eV < energy of the photon < 13.6 eV) and make a transition from the ground electronic state X to excited state B (Lyman) or C (Werner). A state that cannot absorb an incident photon is called a dark state. The new state with energy E_3 of the atom no longer interacts with the laser simply because no photons of the right frequency are present to induce a transition to a different level. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. An atom in a Rydberg state has a valence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almost hydrogenic, Coulomb potential, UC from a compact ion core consisting of a nucleus with Z protons and the lower electron shells filled with Z-1 electrons. In hydrogen the binding energy is given by: : E_\text{B} = -\frac{\rm Ry}{n^2}, where Ry = 13.6 eV is the Rydberg constant. Electromagnetically induced transparency was used in combination with strong interactions between two atoms excited in Rydberg state to provide medium that exhibits strongly nonlinear behaviour at the level of individual optical photons. However, the atom can still decay spontaneously into a third state by emitting a photon of a different frequency. ",0.69,0.132,2.0,1.61,0.11,C
+","The brighter component, termed Sirius A, is a main-sequence star of spectral type early A, with an estimated surface temperature of 9,940 K. The Sun, with an effective temperature of approximately 5800 K, is an approximate black body with an emission spectrum peaked in the central, yellow-green part of the visible spectrum, but with significant power in the ultraviolet as well. Because there is no internal heat source, Sirius B will steadily cool as the remaining heat is radiated into space over the next two billion years or so. Sirius is the brightest star in the night sky. To the naked eye, it often appears to be flashing with red, white, and blue hues when near the horizon. == Observation == With an apparent magnitude of −1.46, Sirius is the brightest star in the night sky, almost twice as bright as the second-brightest star, Canopus. This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrumsPrinciples of Planetary Climate by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. When the star's or planet's net emissivity in the relevant wavelength band is less than unity (less than that of a black body), the actual temperature of the body will be higher than the effective temperature. An Introductory Survey, second edition, Elsevier, Amsterdam, , exercise 4.6, pages 119–120. ===Cosmology=== The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K. This is a list of hottest stars so far discovered (excluding degenerate stars), arranged by decreasing temperature. Because of its declination of roughly −17°, Sirius is a circumpolar star from latitudes south of 73° S. The effective temperature or effective radiative emission temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation.Stull, R. (2000). Sirius is gradually moving closer to the Solar System; it is expected to increase in brightness slightly over the next 60,000 years to reach a peak magnitude of −1.68. Notice that a gray (flat spectrum) ball where ({1-\alpha}) ={\overline{\varepsilon}} comes to the same temperature as a black body no matter how dark or light gray. ====Effective temperature of Earth==== Substituting the measured values for the Sun and Earth yields: :T_{\rm S} = 5778 \ \mathrm{K},NASA Sun Fact Sheet :R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m}, :D = 1.496 \times 10^{11} \ \mathrm{m}, :\alpha = 0.306 \ With the average emissivity \overline{\varepsilon} set to unity, the effective temperature of the Earth is: :T_{\rm E} = 254.356\ \mathrm{K} or −18.8 °C. In that year, Sirius will come to within 1.6 degrees of the south celestial pole. The outer atmosphere of Sirius B is now almost pure hydrogen—the element with the lowest mass—and no other elements are seen in its spectrum. === Apparent third star === Since 1894, irregularities have been tentatively observed in the orbits of Sirius A and B with an apparent periodicity of 6–6.4 years. With a visual apparent magnitude of −1.46, Sirius is almost twice as bright as Canopus, the next brightest star. Sirius appears bright because of its intrinsic luminosity and its proximity to the Solar System. A star near the middle of the spectrum, such as the modest Sun or the giant Capella radiates more energy per unit of surface area than the feeble red dwarf stars or the bloated supergiants, but much less than such a white or blue star as Vega or Rigel. == Planet == ===Blackbody temperature=== To find the effective (blackbody) temperature of a planet, it can be calculated by equating the power received by the planet to the known power emitted by a blackbody of temperature . Sirius A is about twice as massive as the Sun () and has an absolute visual magnitude of +1.43. The visible star is now sometimes known as Sirius A. When compared to the Sun, the proportion of iron in the atmosphere of Sirius A relative to hydrogen is given by \textstyle\ \left[\frac{\ce{Fe}}{\ce{H}}\right] = 0.5\ , meaning iron is 316% as abundant as in the Sun's atmosphere. The stars with temperatures higher than 60,000 K are included. ==List== Star name Effective Temperature (K) Mass () Luminosity () Spectral type Distance Ref. WR 102 210,000 16.1 380,000 WO2 8,610 WR 142 200,000 28.6 912,000 WO2 5,400 LMC195-1 200,000 WO2 160,000 BAT99-123 170,000 158,000 WO3 ~160,000 WR 93b 160,000 8.1 110,000 WO3 7,470 [HC2007] 31 160,000? ", 135.36,234.4,"""11000.0""",130.400766848,0.264,C
+A ground-state hydrogen atom absorbs a photon of light that has a wavelength of $97.2 \mathrm{~nm}$. It then gives off a photon that has a wavelength of $486 \mathrm{~nm}$. What is the final state of the hydrogen atom?,"In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state. == Hydrogenic potential == thumb|right|Figure 3. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). The record wavelength for hydrogen is λ = 73 cm for H253α, implying atomic diameters of a few microns, and for carbon, λ = 18 metres, from C732α, from atoms with a diameter of 57 micron. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. The transition from the state 1^2S_{1/2} with mj=-1/2 to the state 2^2P_{3/2} with mj=-1/2 is only allowed for light with polarization along the z axis (quantization axis) of the atom. This is a quantum state change between the two hyperfine levels of the hydrogen 1 s ground state. A hydrogen atom consists of an electron orbiting its nucleus. In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. This two-step photodissociation process, known as the Solomon process, is one of the main mechanisms by which molecular hydrogen is destroyed in the interstellar medium. thumb|Electronic and vibrational levels of the hydrogen molecule In reference to the figure shown, Lyman-Werner photons are emitted as described below: *A hydrogen molecule can absorb a far- ultraviolet photon (11.2 eV < energy of the photon < 13.6 eV) and make a transition from the ground electronic state X to excited state B (Lyman) or C (Werner). A state that cannot absorb an incident photon is called a dark state. The new state with energy E_3 of the atom no longer interacts with the laser simply because no photons of the right frequency are present to induce a transition to a different level. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. An atom in a Rydberg state has a valence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almost hydrogenic, Coulomb potential, UC from a compact ion core consisting of a nucleus with Z protons and the lower electron shells filled with Z-1 electrons. In hydrogen the binding energy is given by: : E_\text{B} = -\frac{\rm Ry}{n^2}, where Ry = 13.6 eV is the Rydberg constant. Electromagnetically induced transparency was used in combination with strong interactions between two atoms excited in Rydberg state to provide medium that exhibits strongly nonlinear behaviour at the level of individual optical photons. However, the atom can still decay spontaneously into a third state by emitting a photon of a different frequency. ",0.69,0.132,"""2.0""",1.61,0.11,C
 "It turns out that the solution of the Schrödinger equation for the Morse potential can be expressed as
 $$
 G(v)=\tilde{\omega}_{\mathrm{e}}\left(v+\frac{1}{2}\right)-\tilde{\omega}_{\mathrm{e}} \tilde{x}_{\mathrm{e}}\left(v+\frac{1}{2}\right)^2
@@ -332,446 +332,446 @@ where
 $$
 \tilde{x}_{\mathrm{e}}=\frac{h c \tilde{\omega}_{\mathrm{e}}}{4 D}
 $$
-Given that $\tilde{\omega}_{\mathrm{e}}=2886 \mathrm{~cm}^{-1}$ and $D=440.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$ for $\mathrm{H}^{35} \mathrm{Cl}$, calculate $\tilde{x}_{\mathrm{e}}$ and $\tilde{\omega}_{\mathrm{e}} \tilde{x}_{\mathrm{e}}$.","With that, the one-dimensional Schrödinger equation that describes on S^3 the quantum motion of an electric charge dipole perturbed by the trigonometric Rosen–Morse potential, produced by another electric charge dipole, takes the form of {{NumBlk|:| \left(-\frac{\hbar^2 c^2}{R^2}\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+V^{(\ell+1,\alpha Z/2,1)}_{\mbox{tRM}}(\chi)\right)U_{K\ell}^{(\alpha Z)}(\chi)=\frac{\hbar^2c^2}{R^2}\left({\epsilon}_{\ell n}^{(\alpha Z)}\right)^2 U_{K\ell }^{(\alpha Z)}(\chi), |}} (\chi)=\frac{\hbar^2 c^2}{R^2}\frac{\ell(\ell+1)}{\sin^2\chi}-2\frac{\hbar^2 c^2}{R^2}\alpha Z\cot\chi, |}} Because of the relationship, K-\ell=n, with n being the node number of the wave function, one could change labeling of the wave functions, U^{(b)}_{K\ell}(\chi), to the more familiar in the literature, U^{(b)}_{\ell n}(\chi). For this reason, the wave equation which transforms upon the variable change, \Psi_{K\ell m}(\chi,\theta,\varphi)=\frac{U^{(b)}_{K\ell}(\chi)}{\sin\chi}Y_{\ell}^m(\theta,\varphi), into the familiar one-dimensional Schrödinger equation with the V_{tRM}^{(\ell+1,b,1)}(\chi) trigonometric Rosen–Morse potential, {{NumBlk|:| -\frac{\hbar^2c^2}{R^2}\left[\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+\frac{\ell(\ell+1)}{\sin^2\chi}\right]U^{(b)}_{K\ell}(\chi)-2b\cot\chi U^{(b)}_{K\ell}(\chi)=\frac{\hbar^2c^2}{R^2}\left[(K+1)^2-\frac{b^2}{(K+1)^2}\right]U^{(b)}_{K\ell}(\chi), |}} in reality describes quantum motion of a charge dipole perturbed by the field due to another charge dipole, and not the motion of a single charge within the field produced by another charge. Changing in () variables as one observes that the \psi_{K\ell}(\chi) function satisfies the one-dimensional Schrödinger equation with the \csc^2\chi potential according to {{NumBlk|:| -\frac{\hbar^2c^2}{R^2}\left[\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+\frac{\ell(\ell+1)}{\sin^2\chi}\right]\psi_{K\ell}(\chi)=\frac{\hbar^2c^2}{R^2}(K+1)^2\psi_{K\ell}(\chi). |}} The one-dimensional potential in the latter equation, in coinciding with the Rosen–Morse potential in () for a=\ell+1 and b=0, clearly reveals that for integer a values, the first term of this potential takes its origin from the centrifugal barrier on S^3. This potential is approximately a Morse potential with 16\pi^{2} e^{8|x|} The asymptotic of the energies depend on the quantum number as E_n = \frac{4\pi^2 n^2}{W^2(ne^{-1})} , where is the Lambert W function. == References == * * G. Sierra, A physics pathway to the Riemann hypothesis, arXiv:math-ph/1012.4264, 2010. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. For the case of the Riemann zeros Wu and Sprung and others have shown that the potential can be written implicitly in terms of the Gamma function and zeroth- order Bessel function. f^{-1} (x)=\frac{2}{\sqrt{4x+1} } +\frac{1}{4\pi } \int_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right) -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{2\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right) and that the density of states of this Hamiltonian is just the Delsarte's formula for the Riemann zeta function and defined semiclassically as \frac{1}{\sqrt \pi} \frac{d^{1/2}}{dx^{1/2}}f^{-1}(x)= \sum_{n=0}^{\infty}\delta (x-E_{n}) \begin{align} \sum_{n=0}^{\infty }\delta \left( x-\gamma _{n} \right) + \sum_{n=0}^{\infty }\delta \left( x+\gamma _{n} \right) ={}& \frac{1}{2\pi } \frac{\zeta }{\zeta } \left( \frac{1}{2} +ix\right) +\frac{1}{2\pi } \frac{\zeta '}{\zeta } \left( \frac{1}{2} -ix\right) -\frac{\ln \pi }{2\pi } \\\\[10pt] &{} +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} -i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{1}{\pi } \delta \left( x-\frac{i}{2} \right) + \frac{1}{\pi } \delta \left( x + \frac{i}{2} \right) \end{align} here they have taken the derivative of the Euler product on the critical line \frac{1}{2}+is ; also they use the Dirichlet generating function \frac{\zeta ' (s)}{\zeta(s)}= -\sum_{n=1}^{\infty} \Lambda (n) e^{-slnn} . \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. Wu and Sprung also showed that the zeta-regularized functional determinant is the Riemann Xi-function \frac{ \xi(s)}{\xi(0)} = \frac{\det(H-s(1-s)+\frac{1}{4})}{\det(H+\frac{1}{4})} The main idea inside this problem is to recover the potential from spectral data as in some inverse spectral problems in this case the spectral data is the Eigenvalue staircase, which is a quantum property of the system, the inverse of the potential then, satisfies an Abel integral equation (fractional calculus) which can be immediately solved to obtain the potential. == Asymptotics== For large x if we take only the smooth part of the eigenvalue staircase N(E) \sim \frac{\sqrt{E} }{2\pi } \log \left( \frac{\sqrt{E} }{2\pi e} \right) , then the potential as |x| \to \infty is positive and it is given by the asymptotic expression f(-x) = f(x) \sim 4\pi^2 e^2 \left( \frac{2\epsilon \sqrt{\pi } x+B}{A(\epsilon )} \right) ^{2 / \epsilon } with A(\epsilon ) = \frac{\Gamma{\left( \frac{3+\epsilon }{2} \right)}}{\Gamma{\left( 1 + \frac{\epsilon }{2} \right)}} and B = A(0) in the limit \epsilon \to 0 . Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)} \frac{1}{\sqrt{2 \pi}} e^{-\frac{t^2}{2}}\ dt| cdf =\Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right) T(h,a) is Owen's T function| mean =\xi + \omega\delta\sqrt{\frac{2}{\pi}} where \delta = \frac{\alpha}{\sqrt{1+\alpha^2}}| median =| mode =\xi + \omega m_o(\alpha) | variance =\omega^2\left(1 - \frac{2\delta^2}{\pi}\right)| skewness =\gamma_1 = \frac{4-\pi}{2} \frac{\left(\delta\sqrt{2/\pi}\right)^3}{ \left(1-2\delta^2/\pi\right)^{3/2}}| kurtosis =2(\pi - 3)\frac{\left(\delta\sqrt{2/\pi}\right)^4}{\left(1-2\delta^2/\pi\right)^2}| entropy =| mgf =M_X\left(t\right)=2\exp\left(\xi t+\frac{\omega^2t^2}{2}\right)\Phi\left(\omega\delta t\right)| cf =M_X\left(i\delta\omega t\right)| char =e^{i t \xi -t^2\omega^2/2}\left(1+i\, \textrm{Erfi}\left(\frac{\delta\omega t}{\sqrt{2}}\right)\right)| }} In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness. ==Definition== Let \phi(x) denote the standard normal probability density function :\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} with the cumulative distribution function given by :\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right], where ""erf"" is the error function. {\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}\\!| cdf =\Gamma\\!\left(\frac{ u}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\\!\left(\frac{ u}{2}\right)\\!| mean =\frac{1}{ u-2}\\! for u >2\\!| median = \approx \dfrac{1}{ u\bigg(1-\dfrac{2}{9 u}\bigg)^3}| mode =\frac{1}{ u+2}\\!| variance =\frac{2}{( u-2)^2 ( u-4)}\\! for u >4\\!| skewness =\frac{4}{ u-6}\sqrt{2( u-4)}\\! for u >6\\!| kurtosis =\frac{12(5 u-22)}{( u-6)( u-8)}\\! for u >8\\!| entropy =\frac{ u}{2} \\!+\\!\ln\\!\left(\frac{ u}{2}\Gamma\\!\left(\frac{ u}{2}\right)\right) \\!-\\!\left(1\\!+\\!\frac{ u}{2}\right)\psi\\!\left(\frac{ u}{2}\right)| mgf =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-t}{2i}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2t}\right); does not exist as real valued function| char =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-it}{2}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2it}\right)| }} In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-valued random variable. In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian H = p^2 + f(x) with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies E_n and the potential f(x) . \oint p \, dq =2 \pi n(E)= 4\int_0^a dx \sqrt{E_n - f(x)} here is a classical turning point so E = f(a) = f(-a) , the quantum energies of the model are the roots of the Riemann Xi function \xi{\left( \frac{1}{2} + i \sqrt{E_n}\right)} = 0 and f(x)=f(-x) . The trigonometric Rosen–Morse potential, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials. ==Definition== In dimensionless units and modulo additive constants, it is defined as where r is a relative distance, \lambda is an angle rescaling parameter, and R is so far a matching length parameter. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. In eqs. ()-() one recognizes the one-dimensional wave equation with the trigonometric Rosen–Morse potential in () for a=\ell+1 and 2b=\alpha Z. > In this way, the cotangent term of the trigonometric Rosen–Morse potential > could be derived from the Gauss law on S^3 in combination with the > superposition principle, and could be interpreted as a dipole potential > generated by a system consisting of two opposite fundamental charges. In this manner, the complete trigonometric > Rosen–Morse potential could be derived from first principles. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. Introducing elliptic coordinates, : x = a \cosh \xi \cos \eta, : y = a \sinh \xi \sin \eta, the potential energy can be written as : V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)}, and the kinetic energy as : T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right). In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by f^{-1}(x)= \sqrt \pi \frac{d^{1/2}}{dx^{1/2}}N(x) ; with being the eigenvalue staircase N(x) = \sum_{n=0}^\infty H(x - E_{n}) and is the Heaviside step function. Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics When transcribed to the current notations and units, the partition function in presents itself as, , \quad p=\beta \frac{\hbar c}{R}\alpha^2Z^2. \end{align} |}} The infinite integral has first been treated by means of partial integration giving, & = \int_0^\infty x^2e^{-ax^2}e^{\frac{p}{x^2}}{\mathrm d}x \\\ & = -\frac{1}{2a}\int_0^\infty x e^{\frac{p}{x^2}}{\mathrm d}e^{-ax^2}\\\ & = -\frac{1}{2a}xe^{\frac{p}{x^2}}e^{-ax^2}|_0^\infty + \frac{1}{2a}\int_0^\infty e^{-ax^2}{\mathrm d}\, \left( x e^{\frac{p}{x^2}}\right)\\\ & = \frac{1}{2a} \int_0^\infty e^{-ax^2 +\frac{p}{x^2}}{\mathrm d}x + \frac{2p}{2a}\int_0^\infty e^{-ax^2 +\frac{p}{x^2}}{\mathrm d}\left(\frac{1}{x}\right). \end{align} |}} Then the argument of the exponential under the sign of the integral has been cast as, {x}, \end{align} |}} thus reaching the following intermediate result, \int_0^\infty e^{-z^2}{\mathrm d} \left( x + \frac{2p}{x}\right) . \end{align} |}} As a next step the differential has been represented as -2i\sqrt{p} \right) z + \frac{1}{2} \left(\frac{1}{\sqrt{a}}+2i\sqrt{p} \right) z^\ast, \end{align} |}} an algebraic manipulation which allows to express the partition function in () in terms of the \mbox{erf}(u) function of complex argument according to, -2i\sqrt{p}\right) e^{2i\sqrt{ap}}\int_{\Gamma} e^{-z^2}{\mathrm d}z + \left( \frac{1}{\sqrt{a}} +2i\sqrt{p} \right) e^{-2i\sqrt{ap}}\int_{\Gamma} e^{- \left( z^\ast\right)^2} {\mathrm d}z^\ast \right], \\\ \end{align} |}} where \Gamma is an arbitrary path on the complex plane starting in zero and ending in u\to \infty. ",-24, 0.01961,1.11,0.020,11000,B
-" In the infrared spectrum of $\mathrm{H}^{127} \mathrm{I}$, there is an intense line at $2309 \mathrm{~cm}^{-1}$. Calculate the force constant of $\mathrm{H}^{127} \mathrm{I}$.","In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. Such a signal would not be overwhelmed by the H I line itself, or by any of its harmonics. ==See also== * Balmer series * Chronology of the universe * Dark Ages Radio Explorer * Hydrogen spectral series * H-alpha, the visible red spectral line with wavelength of 656.28 nanometers * Rydberg formula * Timeline of the Big Bang ==Footnotes== ==References== ==Further reading== ===Cosmology=== * * * * * * * * ==External links== * * — PAST experiment description * * * Category:Hydrogen physics Category:Emission spectroscopy Category:Radio astronomy Category:Physical cosmology Category:Astrochemistry Category:Hydrogen Available: http://physics.nist.gov/constants. The value of this constant is given here as 1/137.035999206 (note the difference in the last three digits). The constant is expressed for either hydrogen as R_\text{H}, or at the limit of infinite nuclear mass as R_\infty. Approximate value of Value of In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter alpha), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between elementary charged particles. Since the measurement of an absolute intensity in an experiment can be challenging, the ratio of different spectral line intensities can be used to achieve information about the plasma, as well. == Theory == The emission intensity density of an atomic transition from the upper state to the lower state is: P_{u \rightarrow l} = N_u \ \hbar \omega_{u \rightarrow l} \ A_{u \rightarrow l} , where: * N_u is the density of ions in the upper state, * \hbar \omega_{u \rightarrow l} is the energy of the emitted photon, which is the product of the Planck constant and the transition frequency, * A_{u \rightarrow l} is the Einstein coefficient for the specific transition. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants according to his model of the atom. In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. At higher energies, such as the scale of the Z boson, about 90 GeV, one instead measures an effective ≈ 1/127. In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen atom from its ground state. Its numerical value is approximately , with a relative uncertainty of The constant was named by Arnold Sommerfeld, who introduced it in 1916 Equation 12a, ""rund 7·"" (about ...) when extending the Bohr model of the atom. quantified the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Michelson and Morley in 1887. In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text{H} for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. When applied to the Planck relation, this gives: :\lambda = \frac {1}{ u} \cdot c = \frac {h}{E} \cdot c \approx \frac{\; 4.1357 \cdot 10^{-15} \ \mathrm{eV}\cdot\text{s} \;}{5.874\,33 \cdot 10^{-6}\ \mathrm{eV}}\, \cdot\, 2.9979 \cdot 10^8 \ \mathrm{m} \cdot \mathrm{s}^{-1} \approx 0.211\,06\ \mathrm{m} = 21.106\ \mathrm{cm}\; where is the wavelength of an emitted photon, is its frequency, is the photon energy, is the Planck constant, and is the speed of light. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. The electromagnetic radiation producing this line has a frequency of (1.42 GHz), which is equivalent to a wavelength of in a vacuum. Why the constant should have this value is not understood, but there are a number of ways to measure its value. ==Definition== In terms of other fundamental physical constants, may be defined as: \alpha = \frac{e^2}{2 \varepsilon_0 h c} = \frac{e^2}{4 \pi \varepsilon_0 \hbar c} , where * is the elementary charge (); * is the Planck constant (); * is the reduced Planck constant, (6.62607015×10−34 J⋅Hz−1/2π) * is the speed of light (); * is the electric constant (). Specifically, they found that :\frac{\ \Delta \alpha\ }{\alpha} ~~ \overset{\underset{\mathsf{~def~}}{}}{=} ~~ \frac{\ \alpha _\mathrm{prev}-\alpha _\mathrm{now}\ }{\alpha_\mathrm{now}} ~~=~~ \left(-5.7\pm 1.0 \right) \times 10^{-6} ~. The first physical interpretation of the fine-structure constant was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen R_\text{H} and the Rydberg formula. The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron: : R_\text{H} = R_\infty \frac{ m_\text{e} m_\text{p} }{ m_\text{e}+m_\text{p} } \approx 1.09678 \times 10^7 \text{ m}^{-1} , where * m_\text{e} is the mass of the electron, * m_\text{p} is the mass of the nucleus (a proton). === Rydberg unit of energy === The Rydberg unit of energy is equivalent to joules and electronvolts in the following manner: :1 \ \text{Ry} \equiv h c R_\infty = \frac{m_\text{e} e^4}{8 \varepsilon_{0}^{2} h^2} = \frac{e^2}{8 \pi \varepsilon_{0} a_0} = 2.179\;872\;361\;1035(42) \times 10^{-18}\ \text{J} \ = 13.605\;693\;122\;994(26)\ \text{eV}. === Rydberg frequency === :c R_\infty = 3.289\;841\;960\;2508(64) \times 10^{15}\ \text{Hz} . === Rydberg wavelength === :\frac 1 {R_\infty} = 9.112\;670\;505\;824(17) \times 10^{-8}\ \text{m}. The strength of the electromagnetic interaction varies with the strength of the energy field. | In the fields of electrical engineering and solid-state physics, the fine- structure constant is one fourth the product of the characteristic impedance of free space, ~ Z_0 = \mu_0 c = \sqrt{\frac{\mu_0}{\varepsilon_0}} , and the conductance quantum, G_0 = \frac{2 e^2}{ h }: \alpha = \tfrac{1}{4} Z_0 G_0\ . ",-2.99,3.0,313.0,2.57,0.11,C
-Calculate the percentage difference between $e^x$ and $1+x$ for $x=0.0050$,"It follows that is transcendental over . ==Computation== When computing (an approximation of) the exponential function near the argument , the result will be close to 1, and computing the value of the difference e^x-1 with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result. That is, \frac{d}{dx}e^x = e^x \quad\text{and}\quad e^0=1. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). == Related units == * Percentage (%) 1 part in 100 * Per mille (‰) 1 part in 1,000 *Basis point (bp) difference of 1 part in 10,000 *Permyriad (‱) 1 part in 10,000 * Per cent mille (pcm) 1 part in 100,000 * Baker percentage == See also == * Parts-per notation * Per-unit system * Percent point function * Relative change and difference == References == Category:Mathematical terminology Category:Probability assessment Category:Units of measurement ru:Процент#Процентный пункт thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. For example, if the exponential is computed by using its Taylor series e^x = 1 + x + \frac {x^2}2 + \frac{x^3}6 + \cdots + \frac{x^n}{n!} + \cdots, one may use the Taylor series of e^x-1: e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots. The equation \tfrac{d}{dx}e^x = e^x means that the slope of the tangent to the graph at each point is equal to its -coordinate at that point. ==Relation to more general exponential functions== The exponential function f(x) = e^x is sometimes called the natural exponential function for distinguishing it from the other exponential functions. A percentage point or percent point is the unit for the arithmetic difference between two percentages. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. If v e 0, the relative error is : \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, and the percent error (an expression of the relative error) is :\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. 420px|thumb| As x approaches zero from the right, the magnitude of the rate of change of 1/x increases. The derivative (rate of change) of the exponential function is the exponential function itself. Percentage-point differences are one way to express a risk or probability. For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. thumb|200px|right|Exponential functions with bases 2 and 1/2 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). The base of the exponential function, its value at 1, e = \exp(1), is a ubiquitous mathematical constant called Euler's number. After the first occurrence, some writers abbreviate by using just ""point"" or ""points"". ==Differences between percentages and percentage points== Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. Euler's number is the unique base for which the constant of proportionality is 1, since \ln(e) = 1, so that the function is its own derivative: \frac{d}{dx} e^x = e^x \ln (e) = e^x. Exponential constant may refer to: * e (mathematical constant) * The growth or decay constant in exponential growth or exponential decay, respectively. For real numbers and , a function of the form f(x) = a b^{cx + d} is also an exponential function, since it can be rewritten as a b^{c x + d} = \left(a b^d\right) \left(b^c\right)^x. ==Formal definition== right|thumb|The exponential function (in blue), and the sum of the first terms of its power series (in red). The natural exponential is hence denoted by x\mapsto e^x or x\mapsto \exp x. Percentage solution may refer to: * Mass fraction (or ""% w/w"" or ""wt.%""), for percent mass * Volume fraction (or ""% v/v"" or ""vol.%""), volume concentration, for percent volume * ""Mass/volume percentage"" (or ""% m/v"") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). ",0.3085,1855,-1.46,1.51,1.25,E
- Calculate (a) the wavelength and kinetic energy of an electron in a beam of electrons accelerated by a voltage increment of $100 \mathrm{~V}$ ,"# Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I–V curves. Following Kunio Fujiwara and Archibald Howie, the relationship between the total energy of the electrons and the wavevector is written as: :E=\frac{h^2k^2}{2m^*} with m^*=m_0 + \frac{E}{2c^2} where h is Planck's constant, m^* is a relativistic effective mass used to cancel out the relativistic terms for electrons of energy E with c the speed of light and m_0 the rest mass of the electron. Typically the energy of the electrons is written in electronvolts (eV), the voltage used to accelerate the electrons; the actual energy of each electron is this voltage times the electron charge. framed|Geometry of electron beam in precession electron diffraction. Relativistic electron beams are streams of electrons moving at relativistic speeds. The Schrödinger equation combines the kinetic energy of waves and the potential energy due to, for electrons, the Coulomb potential. Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. First, as previously noted, the electron must absorb an amount of energy equivalent to the energy difference between the electron's current energy level and an unoccupied, higher energy level in order to be promoted to that energy level. The energy released is equal to the difference in energy levels between the electron energy states. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. Combining this with the aforementioned Lorentz correction yields: I_\mathbf{g}^{kinematical} \propto I_\mathbf{g}^{experimental} \cdot g\sqrt{1-\frac{g}{2R_o}} \cdot \int\limits_0^{A_\mathbf{g}}J_0(2x)\, dx where A_\mathbf{g} = \frac{2 \pi t F_\mathbf{g}}{k} , t is the sample thickness, and k is the wave-vector of the electron beam. Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . These include: # The simplest approximation using the de Broglie wavelength for electrons, where only the geometry is considered and often Bragg's law is invoked, a far- field or Fraunhofer approach. One of the methods is to use the concept of dressed particle. == See also == * Energy level * Mode (electromagnetism) == References == Category:Electron *Developments in the convergent-beam electron diffraction approach. ",144,226,0.3085,1.602,355.1,D
-Suppose that $10.0 \mathrm{~mol} \mathrm{C}_2 \mathrm{H}_6(\mathrm{~g})$ is confined to $4.860 \mathrm{dm}^3$ at $27^{\circ} \mathrm{C}$. Predict the pressure exerted by the ethane from the perfect gas.,"Natural gas from different gas fields varies in ethane content from less than 1% to more than 6% by volume. The bond parameters of ethane have been measured to high precision by microwave spectroscopy and electron diffraction: rC−C = 1.528(3) Å, rC−H = 1.088(5) Å, and ∠CCH = 111.6(5)° by microwave and rC−C = 1.524(3) Å, rC−H = 1.089(5) Å, and ∠CCH = 111.9(5)° by electron diffraction (the numbers in parentheses represents the uncertainties in the final digits). ===Atmospheric and extraterrestrial=== Ethane occurs as a trace gas in the Earth's atmosphere, currently having a concentration at sea level of 0.5 ppb,Trace gases (archived). At standard temperature and pressure, ethane is a colorless, odorless gas. Today, ethane is an important petrochemical feedstock and is separated from the other components of natural gas in most well-developed gas fields. Global ethane quantities have varied over time, likely due to flaring at natural gas fields. Atmosphere.mpg.de. Retrieved on 2011-12-08. though its preindustrial concentration is likely to have been only around 0.25 part per billion since a significant proportion of the ethane in today's atmosphere may have originated as fossil fuels. Ethane was discovered dissolved in Pennsylvanian light crude oil by Edmund Ronalds in 1864. ==Properties== At standard temperature and pressure, ethane is a colorless, odorless gas. As far back as 1890–1891, chemists suggested that ethane molecules preferred the staggered conformation with the two ends of the molecule askew from each other. ==Production== After methane, ethane is the second-largest component of natural gas. : C2H5O• → CH3• + CH2O Some minor products in the incomplete combustion of ethane include acetaldehyde, methane, methanol, and ethanol. Ethane ( , ) is an organic chemical compound with chemical formula . Computer simulations of the chemical kinetics of ethane combustion have included hundreds of reactions. In 2006, Dale Cruikshank of NASA/Ames Research Center (a New Horizons co-investigator) and his colleagues announced the spectroscopic discovery of ethane on Pluto's surface. ==Chemistry== Ethane can be viewed as two methyl groups joined, that is, a dimer of methyl groups. This error was corrected in 1864 by Carl Schorlemmer, who showed that the product of all these reactions was in fact ethane. Solid ethane exists in several modifications. Ethane is most efficiently separated from methane by liquefying it at cryogenic temperatures. This page provides supplementary chemical data on ethane. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. : C2H5• + O2 → C2H5OO• : C2H5OO• + HR → C2H5OOH + •R : C2H5OOH → C2H5O• + •OH The principal carbon-containing products of incomplete ethane combustion are single-carbon compounds such as carbon monoxide and formaldehyde. Ethane can also be separated from petroleum gas, a mixture of gaseous hydrocarbons produced as a byproduct of petroleum refining. Like many hydrocarbons, ethane is isolated on an industrial scale from natural gas and as a petrochemical by-product of petroleum refining. The chemistry of ethane involves chiefly free radical reactions. They mistook the product of these reactions for the methyl radical (), of which ethane () is a dimer. In fact, ethane's global warming potential largely results from its conversion in the atmosphere to methane.Hodnebrog, Øivind; Dalsøren, Stig B. and Myrhe, Gunnar; ‘Lifetimes, direct and indirect radiative forcing, and globalwarming potentials of ethane (C2H6), propane (C3H8),and butane (C4H10)’; Atmospheric Science Letters; 2018;19:e804 It has been detected as a trace component in the atmospheres of all four giant planets, and in the atmosphere of Saturn's moon Titan. ",16.3923,-20,24.0,4.16,50.7,E
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the standard enthalpy of solution of $\mathrm{AgCl}(\mathrm{s})$ in water from the enthalpies of formation of the solid and the aqueous ions.","The solubility product, Ksp, for AgCl in water is at room temperature, which indicates that only 1.9 mg (that is, \sqrt{1.77\times 10^{-10}} \ \mathrm{mol}) of AgCl will dissolve per liter of water. The chloride content of an aqueous solution can be determined quantitatively by weighing the precipitated AgCl, which conveniently is non-hygroscopic since AgCl is one of the few transition metal chlorides that are unreactive toward water. The standard enthalpy of formation is then determined using Hess's law. The Saha equation describes the degree of ionization for any gas in thermal equilibrium as a function of the temperature, density, and ionization energies of the atoms. Silver chloride is a chemical compound with the chemical formula AgCl. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). :AgNO3 + NaCl -> AgCl(v) + NaNO3 :2 AgNO3 + CoCl2 -> 2 AgCl(v) + Co(NO3)2 It can also be produced by the reaction of silver metal and aqua regia, however, the insolubility of silver chloride decelerates the reaction. In physics, the Saha ionization equation is an expression that relates the ionization state of a gas in thermal equilibrium to the temperature and pressure. Examples are given in the following sections. == Ionic compounds: Born–Haber cycle == For ionic compounds, the standard enthalpy of formation is equivalent to the sum of several terms included in the Born–Haber cycle. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. Above 7.5 GPa, silver chloride transitions into a monoclinic KOH phase. K (? °C), ? K (? °C), ? This is true for all enthalpies of formation. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. For a gas composed of a single atomic species, the Saha equation is written: :\frac{n_{i+1}n_e}{n_i} = \frac{2}{\lambda^{3}}\frac{g_{i+1}}{g_i}\exp\left[-\frac{(\epsilon_{i+1}-\epsilon_i)}{k_B T}\right] where: * n_i is the density of atoms in the i-th state of ionization, that is with i electrons removed. * g_i is the degeneracy of states for the i-ions * \epsilon_i is the energy required to remove i electrons from a neutral atom, creating an i-level ion. * n_e is the electron density * \lambda is the thermal de Broglie wavelength of an electron ::\lambda \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{h^2}{2\pi m_e k_B T}} * m_e is the mass of an electron * T is the temperature of the gas * h is Planck's constant The expression (\epsilon_{i+1}-\epsilon_i) is the energy required to remove the (i+1)^{th} electron. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Silver chloride is also produced as a byproduct of the Miller process where silver metal is reacted with chlorine gas at elevated temperatures. ==History== Silver chloride was known since ancient times. ""Standard potential of the silver-silver chloride electrode"". ",+65.49,36,130.41,76,0.166666666,A
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in chemical potential of a perfect gas when its pressure is increased isothermally from $1.8 \mathrm{~atm}$ to $29.5 \mathrm{~atm}$ at $40^{\circ} \mathrm{C}$.,"For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. * A partial pressure of 101.325 kPa (absolute) (1 atm, 1.01325 bar) for each gaseous reagent. In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. (i) Indicates values calculated from ideal gas thermodynamic functions. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. When pressure and temperature are variable, only I-1 of I components have independent values for chemical potential and Gibbs' phase rule follows. The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . ",0.086,7166.67,7.3,-131.1,14.5115,C
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. 3.1(a) Calculate the change in entropy when $25 \mathrm{~kJ}$ of energy is transferred reversibly and isothermally as heat to a large block of iron at $100^{\circ} \mathrm{C}$.,"The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. * The heat capacity of the gas from the boiling point to room temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). The amount of energy added equals , with representing the change in temperature. Therefore, the heat capacity ratio in this example is 1.4. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Equilibrium occurs when the temperature is equal to the melting point T = T_f so that :\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T_f \times \Delta S_{\text{fus}} = 0, and the entropy of fusion is the heat of fusion divided by the melting point: : \Delta S_{\text{fus}} = \frac {\Delta H_{\text{fus}}} {T_f} ==Helium== Helium-3 has a negative entropy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative entropy of fusion below 0.8 K. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. ",240,14.34457,36.0,67,-2,D
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $q$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. In addition, other factors come into play and dominate during a compression cycle if they have a greater impact on the final calculated result than whether or not C_P was held constant. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. ",27,59.4,0.0029,+7.3,0,E
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. For the reaction $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{l})+3 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})+3 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$, $\Delta_{\mathrm{r}} U^\ominus=-1373 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}$, calculate $\Delta_{\mathrm{r}} H^{\ominus}$.","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics To convert from L^2bar/mol^2 to m^6 Pa/mol^2, divide by 10. a (L2bar/mol2) b (L/mol) Acetic acid 17.7098 0.1065 Acetic anhydride 20.158 0.1263 Acetone 16.02 0.1124 Acetonitrile 17.81 0.1168 Acetylene 4.516 0.0522 Ammonia 4.225 0.0371 Aniline 29.14 0.1486 Argon 1.355 0.03201 Benzene 18.24 0.1193 Bromobenzene 28.94 0.1539 Butane 14.66 0.1226 1-Butanol 20.94 0.1326 2-Butanone 19.97 0.1326 Carbon dioxide 3.640 0.04267 Carbon disulfide 11.77 0.07685 Carbon monoxide 1.505 0.0398500 Carbon tetrachloride 19.7483 0.1281 Chlorine 6.579 0.05622 Chlorobenzene 25.77 0.1453 Chloroethane 11.05 0.08651 Chloromethane 7.570 0.06483 Cyanogen 7.769 0.06901 Cyclohexane 23.11 0.1424 Cyclopropane 8.34 0.0747 Decane 52.74 0.3043 1-Decanol 59.51 0.3086 Diethyl ether 17.61 0.1344 Diethyl sulfide 19.00 0.1214 Dimethyl ether 8.180 0.07246 Dimethyl sulfide 13.04 0.09213 Dodecane 69.38 0.3758 1-Dodecanol 75.70 0.3750 Ethane 5.562 0.0638 Ethanethiol 11.39 0.08098 Ethanol 12.18 0.08407 Ethyl acetate 20.72 0.1412 Ethylamine 10.74 0.08409 Ethylene 4.612 0.0582 Fluorine 1.171 0.0290 Fluorobenzene 20.19 0.1286 Fluoromethane 4.692 0.05264 Freon 10.78 0.0998 Furan 12.74 0.0926 Germanium tetrachloride 22.90 0.1485 Helium 0.0346 0.0238 Heptane 31.06 0.2049 1-Heptanol 38.17 0.2150 Hexane 24.71 0.1735 1-Hexanol 31.79 0.1856 Hydrazine 8.46 0.0462 Hydrogen 0.2476 0.02661 Hydrogen bromide 4.510 0.04431 Hydrogen chloride 3.716 0.04081 Hydrogen cyanide 11.29 0.0881 Hydrogen fluoride 9.565 0.0739 Hydrogen iodide 6.309 0.0530 Hydrogen selenide 5.338 0.04637 Hydrogen sulfide 4.490 0.04287 Isobutane 13.32 0.1164 Iodobenzene 33.52 0.1656 Krypton 2.349 0.03978 Mercury 8.200 0.01696 Methane 2.283 0.04278 Methanol 9.649 0.06702 Methylamine 7.106 0.0588 Neon 0.2135 0.01709 Neopentane 17.17 0.1411 Nitric oxide 1.358 0.02789 Nitrogen 1.370 0.0387 Nitrogen dioxide 5.354 0.04424 Nitrogen trifluoride 3.58 0.0545 Nitrous oxide 3.832 0.04415 Octane 37.88 0.2374 1-Octanol 44.71 0.2442 Oxygen 1.382 0.03186 Ozone 3.570 0.0487 Pentane 19.26 0.146 1-Pentanol 25.88 0.1568 Phenol 22.93 0.1177 Phosphine 4.692 0.05156 Propane 8.779 0.08445 1-Propanol 16.26 0.1079 2-Propanol 15.82 0.1109 Propene 8.442 0.0824 Pyridine 19.77 0.1137 Pyrrole 18.82 0.1049 Radon 6.601 0.06239 Silane 4.377 0.05786 Silicon tetrafluoride 4.251 0.05571 Sulfur dioxide 6.803 0.05636 Sulfur hexafluoride 7.857 0.0879 Tetrachloromethane 20.01 0.1281 Tetrachlorosilane 20.96 0.1470 Tetrafluoroethylene 6.954 0.0809 Tetrafluoromethane 4.040 0.0633 Tetrafluorosilane 5.259 0.0724 Tetrahydrofuran 16.39 0.1082 Tin tetrachloride 27.27 0.1642 Thiophene 17.21 0.1058 Toluene 24.38 0.1463 1-1-1-Trichloroethane 20.15 0.1317 Trichloromethane 15.34 0.1019 Trifluoromethane 5.378 0.0640 Trimethylamine 13.37 0.1101 Water 5.536 0.03049 Xenon 4.250 0.05105 ==Units== 1 J·m3/mol2 = 1 m6·Pa/mol2 = 10 L2·bar/mol2 1 L2atm/mol2 = 0.101325 J·m3/mol2 = 0.101325 Pa·m6/mol2 1 dm3/mol = 1 L/mol = 1 m3/kmol (where kmol is kilomoles = 1000 moles) ==References== Category:Gas laws Constants (Data Page) :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. This disparity is not a significant departure from accuracy, and USSA1976 uses this value of R∗ for all the calculations of the standard atmosphere. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. ",0.11,209.1,-1368.0,1.51,1.88,C
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $w$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Therefore, the heat capacity ratio in this example is 1.4. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi- quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). ",0,11000,15.0,8.87,0.1792,A
-The density of a gaseous compound was found to be $1.23 \mathrm{kg} \mathrm{m}^{-3}$ at $330 \mathrm{K}$ and $20 \mathrm{kPa}$. What is the molar mass of the compound?,"The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule: :\begin{array}{lll} M(\ce{H2}) &= 2\times 1.00797(7) \times M_\mathrm{u} &= 2.01588(14) \text{ g/mol} \\\ M(\ce{S8}) &= 8\times 32.065(5) \times M_\mathrm{u} &= 256.52(4) \text{ g/mol} \\\ M(\ce{Cl2}) &= 2\times 35.453(2) \times M_\mathrm{u} &= 70.906(4) \text{ g/mol} \end{array} == Molar masses of compounds == The molar mass of a compound is given by the sum of the relative atomic mass of the atoms which form the compound multiplied by the molar mass constant : :M = M_{\rm u} M_{\rm r} = M_{\rm u} \sum_i {A_{\rm r}}_i. Examples are: \begin{array}{ll} M(\ce{NaCl}) &= \bigl[22.98976928(2) + 35.453(2)\bigr] \times 1 \text{ g/mol} \\\ &= 58.443(2) \text{ g/mol} \\\\[4pt] M(\ce{C12H22O11}) &= \bigl[12 \times 12.0107(8) + 22 \times 1.00794(7) + 11 \times 15.9994(3)\bigr] \times 1 \text{ g/mol} \\\ &= 342.297(14) \text{ g/mol} \end{array} An average molar mass may be defined for mixtures of compounds. In chemistry, the molar mass () of a chemical compound is defined as the ratio between the mass and the amount of substance (measured in moles) of any sample of said compound. The molar mass of a compound in g/mol thus is equal to the mass of this number of molecules of the compound in grams. == Molar masses of elements == The molar mass of atoms of an element is given by the relative atomic mass of the element multiplied by the molar mass constant, For normal samples from earth with typical isotope composition, the atomic weight can be approximated by the standard atomic weight or the conventional atomic weight. :\begin{array}{lll} M(\ce{H}) &= 1.00797(7) \times M_\mathrm{u} &= 1.00797(7) \text{ g/mol} \\\ M(\ce{S}) &= 32.065(5) \times M_\mathrm{u} &= 32.065(5) \text{ g/mol} \\\ M(\ce{Cl}) &= 35.453(2) \times M_\mathrm{u} &= 35.453(2) \text{ g/mol} \\\ M(\ce{Fe}) &= 55.845(2) \times M_\mathrm{u} &= 55.845(2) \text{ g/mol} \end{array} Multiplying by the molar mass constant ensures that the calculation is dimensionally correct: standard relative atomic masses are dimensionless quantities (i.e., pure numbers) whereas molar masses have units (in this case, grams per mole). The molar mass is an average of many instances of the compound, which often vary in mass due to the presence of isotopes. As an example, the average molar mass of dry air is 28.97 g/mol.The Engineering ToolBox Molecular Mass of Air == Related quantities == Molar mass is closely related to the relative molar mass () of a compound, to the older term formula weight (F.W.), and to the standard atomic masses of its constituent elements. Thus, for example, the average mass of a molecule of water is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol. Molar masses typically vary between: :1–238 g/mol for atoms of naturally occurring elements; : for simple chemical compounds; : for polymers, proteins, DNA fragments, etc. * Molar mass: chemistry second-level course. If represents the mass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by :M = {{wK_\text{b}}\over{\Delta T}}.\ == See also == * Mole map (chemistry) == References == ==External links== * HTML5 Molar Mass Calculator web and mobile application. The mass average molar mass is calculated by \bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} where is the number of molecules of molecular mass . The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities. * Online Molar Mass Calculator with the uncertainty of M and all the calculations shown * Molar Mass Calculator Online Molar Mass and Elemental Composition Calculator * Stoichiometry Add-In for Microsoft Excel for calculation of molecular weights, reaction coefficients and stoichiometry. The molecular formula C23H21NO (molar mass: 327.42 g/mol, exact mass: 327.1623 u) may refer to: * JWH-015 * JWH-073 * JWH-120 Category:Molecular formulas In the International System of Units (SI), the coherent unit of molar mass is kg/mol. The molecular formula C13H28O (molar mass: 200.36 g/mol, exact mass: 200.2140 u) may refer to: * 2,2,4,4-Tetramethyl-3-t-butyl-pentane-3-ol, or tri-tert- butylcarbinol * 1-Tridecanol In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species () and the molar mass () of that species.I. Katime ""Química Física Macromolecular"". The molar mass of any element or compound is its relative atomic mass (atomic weight) multiplied by the molar mass constant. Here, is the relative molar mass, also called formula weight. Gram atomic mass is another term for the mass, in grams, of one mole of atoms of that element. The molar mass constant was thus given by :M_{\text{u}} = {\text{molar mass }[M( ^{12}\mathrm{C} )]\over \text{relative atomic weight }[A_{\text{r}}( ^{12}\mathrm{C} )]} = {{12\ {\rm g/mol}}\over 12}=1\ \rm g/mol The molar mass constant is related to the mass of a carbon-12 atom in grams: :m({}^{12}{\text{C}}) = \frac{12 \times M_{\text{u}}}{N_{\text{A}}} The Avogadro constant being a fixed value, the mass of a carbon-12 atom depends on the accuracy and precision of the molar mass constant. The molecular formula C12H15N5O3 (molar mass: 277.28 g/mol, exact mass: 277.1175 u) may refer to: * Entecavir (ETV) * Queuine (Q) ",169, 4.56,258.14,1.8,57.2,A
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of $4.50 \mathrm{~g}$ of methane occupies $12.7 \mathrm{dm}^3$ at $310 \mathrm{~K}$. Calculate the work that would be done if the same expansion occurred reversibly.","In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. We assume the expansion occurs without exchange of heat (adiabatic expansion). The reaction depends on a delicate balance between methane pressure and catalyst concentration, and consequently more work is being done to further improve yields. ==References== Category:Organometallic chemistry Category:Organic chemistry Category:Chemistry Category:Methane The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. A methane reformer is a device based on steam reforming, autothermal reforming or partial oxidation and is a type of chemical synthesis which can produce pure hydrogen gas from methane using a catalyst. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. An experimental value should be used rather than one based on this approximation, where possible. Methane functionalization is the process of converting methane in its gaseous state to another molecule with a functional group, typically methanol or acetic acid, through the use of transition metal catalysts. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. For example, a comparison of calculations for one compression stage of an axial compressor (one with variable C_P and one with constant C_P) may produce a deviation small enough to support this approach. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . ",-167,1.8,6.64, 0.01961,0.396,A
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A strip of magnesium of mass $15 \mathrm{~g}$ is placed in a beaker of dilute hydrochloric acid. Calculate the work done by the system as a result of the reaction. The atmospheric pressure is 1.0 atm and the temperature $25^{\circ} \mathrm{C}$.","The quantity of thermodynamic work is defined as work done by the system on its surroundings. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. Such work done by compression is thermodynamic work as here defined. An electric discharge through hydrogen gas at low pressure (20 pascals) containing pieces of magnesium can produce MgH. The reaction that produces it is either or Mg + H → MgH. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. The reaction of Mg atoms with (dihydrogen gas) is actually endothermic and proceeds when magnesium atoms are excited electronically. Otherwise in these stars, below any magnesium silicate clouds where the temperature is hotter, the concentration of MgH is proportional to the square root of the pressure, and concentration of magnesium, and 10−4236/T. MgH is the second most abundant magnesium containing gas (after atomic magnesium) in the deeper hotter parts of planets and brown dwarfs. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. The work is due to change of system volume by expansion or contraction of the system. First, one assumes that the given reaction at constant temperature and pressure is the only one that is occurring. Bulk properties of the MgH gas include enthalpy of formation of 229.79 kJ mol−1, entropy 193.20 J K−1 mol−1 and heat capacity of 29.59 J K−1 mol−1. Magnesium monohydride is a molecular gas with formula MgH that exists at high temperatures, such as the atmospheres of the Sun and stars. A complete reaction takes 20 to 24 hours at 1,200 °C."" Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. Several kinds of thermodynamic work are especially important. Atmospheric Thermodynamics. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Both the temperature change ∆T of the water and the height of the fall ∆h of the weight mg were recorded. As a result, the work done by the system also depends on the initial and final states. ",17.4,-1.5,1.5,14.34457,48,B
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. For a van der Waals gas, $\pi_T=a / V_{\mathrm{m}}^2$. Calculate $\Delta U_{\mathrm{m}}$ for the isothermal expansion of nitrogen gas from an initial volume of $1.00 \mathrm{dm}^3$ to $24.8 \mathrm{dm}^3$ at $298 \mathrm{~K}$.","The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density. ===Statistical thermodynamics derivation=== The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is: : Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3} where \Lambda is the thermal de Broglie wavelength, : \Lambda = \sqrt{\frac{h^2}{2\pi m k T}} with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the Van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the p–V diagram as the ratio of vapour to liquid changes. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi- quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ",313,122,0.6,3.333333333,131,E
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Take nitrogen to be a van der Waals gas with $a=1.352 \mathrm{dm}^6 \mathrm{~atm} \mathrm{\textrm {mol } ^ { - 2 }}$ and $b=0.0387 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and calculate $\Delta H_{\mathrm{m}}$ when the pressure on the gas is decreased from $500 \mathrm{~atm}$ to $1.00 \mathrm{~atm}$ at $300 \mathrm{~K}$. For a van der Waals gas, $\mu=\{(2 a / R T)-b\} / C_{p, \mathrm{~m}}$. Assume $C_{p, \mathrm{~m}}=\frac{7}{2} R$.","Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",30,+3.60,0.3333333,140,3.07,B
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the molar entropy of a constant-volume sample of neon at $500 \mathrm{~K}$ given that it is $146.22 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}$.,"In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Substituting this into the above equation along with V=[\mathrm{g}]/\rho\, and \gamma = 5/3\, for an ideal monatomic gas one finds : K = \frac{k_{B}T}{(\rho/\mu m_{H})^{2/3}}, where \mu\, is the mean molecular weight of the gas or plasma; and m_{H}\, is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, m_{p}\,, the quantity more often used in astrophysical theory of galaxy clusters. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, i.e. the pressure–volume product, rather than energy per temperature increment per particle. As a consequence, the SI value of the molar gas constant is exactly . This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Instead of a mole the constant can be expressed by considering the normal cubic meter. * The heat capacity of the gas from the boiling point to room temperature. The law can also be written as a function of the total number of atoms N in the sample: :C/N = 3k_{\rm B}, where kB is Boltzmann constant. ==Application limits== thumb|upright=2.2|The molar heat capacity plotted of most elements at 25°C plotted as a function of atomic number. Otherwise, we can also say that: :\mathrm{force} = \frac{ \mathrm{mass} \times \mathrm{length} } { (\mathrm{time})^2 } Therefore, we can write R as: :R = \frac{ \mathrm{mass} \times \mathrm{length}^2 } { \mathrm{amount} \times \mathrm{temperature} \times (\mathrm{time})^2 } And so, in terms of SI base units: :R = . ==Relationship with the Boltzmann constant== The Boltzmann constant kB (alternatively k) may be used in place of the molar gas constant by working in pure particle count, N, rather than amount of substance, n, since :R = N_{\rm A} k_{\rm B},\, where NA is the Avogadro constant. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). ",38,2.3,152.67,-59.24,-2,C
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final temperature of a sample of argon of mass $12.0 \mathrm{~g}$ that is expanded reversibly and adiabatically from $1.0 \mathrm{dm}^3$ at $273.15 \mathrm{~K}$ to $3.0 \mathrm{dm}^3$.","The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). We assume the expansion occurs without exchange of heat (adiabatic expansion). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The following figure illustrates that the effects of dissociation tend to lower the adiabatic flame temperature. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. thumb|upright|Ethanol burning with its spectrum depicted In the study of combustion, the adiabatic flame temperature is the temperature reached by a flame under ideal conditions. Temperatures in the atmosphere decrease with height at an average rate of 6.5°C (11.7°F) per kilometer. Adiabatic flame temperature (constant pressure) of common fuels Fuel Oxidizer 1 bar 20 °C T_\text{ad} T_\text{ad} Fuel Oxidizer 1 bar 20 °C (°C) (°F) Acetylene () Air 2500 4532 Acetylene () Oxygen 3480 6296 Butane () Air 2231 4074 Cyanogen () Oxygen 4525 8177 Dicyanoacetylene () Oxygen 4990 9010 Ethane () Air 1955 3551 Ethanol () Air 2082 3779Flame Temperature Analysis and NOx Emissions for Different Fuels Gasoline Air 2138 3880 Hydrogen () Air 2254 4089 Magnesium (Mg) Air 1982 3600 Methane () Air 1963 3565CRC Handbook of Chemistry and Physics, 96th Edition, p. 15-51 Methanol () Air 1949 3540 Naphtha Air 2533 4591 Natural gas Air 1960 3562 Pentane () Air 1977 3591 Propane () Air 1980 3596 Methylacetylene () Air 2010 3650 Methylacetylene () Oxygen 2927 5301 Toluene () Air 2071 3760 Wood Air 1980 3596 Kerosene Air 2093Power Point Presentation: Flame Temperature, Hsin Chu, Department of Environmental Engineering, National Cheng Kung University, Taiwan 3801 Light fuel oil Air 2104 3820 Medium fuel oil Air 2101 3815 Heavy fuel oil Air 2102 3817 Bituminous Coal Air 2172 3943 Anthracite Air 2180 3957 Anthracite Oxygen ≈3500Analysis of oxy-fuel combustion power cycle utilizing a pressurized coal combustor by Jongsup Hong et al., MIT, which cites . ",131,2,3.51,432,14,A
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in entropy when $25 \mathrm{~kJ}$ of energy is transferred reversibly and isothermally as heat to a large block of iron at $0^{\circ} \mathrm{C}$.,"However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. * The heat capacity of the gas from the boiling point to room temperature. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Equilibrium occurs when the temperature is equal to the melting point T = T_f so that :\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T_f \times \Delta S_{\text{fus}} = 0, and the entropy of fusion is the heat of fusion divided by the melting point: : \Delta S_{\text{fus}} = \frac {\Delta H_{\text{fus}}} {T_f} ==Helium== Helium-3 has a negative entropy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative entropy of fusion below 0.8 K. In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. In thermodynamics, the entropy of fusion is the increase in entropy when melting a solid substance. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T1, through the ""working body"" of fluid, which was typically a body of steam, to the temperature T2 as shown below: If we make the assignment: : S= \frac {Q}{T} Then, the entropy change or ""equivalence-value"" for this transformation is: : \Delta S = S_{\rm final} - S_{\rm initial} \, which equals: : \Delta S = \left(\frac {Q}{T_2} - \frac {Q}{T_1}\right) and by factoring out Q, we have the following form, as was derived by Clausius: : \Delta S = Q\left(\frac {1}{T_2} - \frac {1}{T_1}\right) ==1856 definition== In 1856, Clausius stated what he called the ""second fundamental theorem in the mechanical theory of heat"" in the following form: :\int \frac{\delta Q}{T} = -N where N is the ""equivalence-value"" of all uncompensated transformations involved in a cyclical process. Changes in entropy are associated with phase transitions and chemical reactions. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics Jaynes (1957) Information theory and statistical mechanics II, Physical Review 108:171 the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate. ==Popular use== The term entropy is often used in popular language to denote a variety of unrelated phenomena. 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-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\Delta H$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Therefore, the heat capacity ratio in this example is 1.4. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. ",30,5,205.0,0.66666666666,+3.03,E
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Concerns over the harmful effects of chlorofluorocarbons on stratospheric ozone have motivated a search for new refrigerants. One such alternative is 2,2-dichloro-1,1,1-trifluoroethane (refrigerant 123). Younglove and McLinden published a compendium of thermophysical properties of this substance (J. Phys. Chem. Ref. Data 23, 7 (1994)), from which properties such as the Joule-Thomson coefficient $\mu$ can be computed. Compute $\mu$ at 1.00 bar and $50^{\circ} \mathrm{C}$ given that $(\partial H / \partial p)_T=-3.29 \times 10^3 \mathrm{~J} \mathrm{MPa}^{-1} \mathrm{~mol}^{-1}$ and $C_{p, \mathrm{~m}}=110.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$.","Property Value Ozone depletion potential (ODP) 0.44 (CCl3F = 1) Global warming potential (GWP: 100-year) 5,860 \- 7,670 (CO2 = 1) Atmospheric lifetime 1,020 \- 1,700 years == See also == * IPCC list of greenhouse gases * List of refrigerants == References == Category:Ozone depletion Category:Greenhouse gases Category:Refrigerants Category:Chlorofluorocarbons ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases 1,1,1,2-Tetrafluoroethane (also known as norflurane (INN), R-134a, Klea®134a, Freon 134a, Forane 134a, Genetron 134a, Green Gas, Florasol 134a, Suva 134a, or HFC-134a) is a hydrofluorocarbon (HFC) and haloalkane refrigerant with thermodynamic properties similar to R-12 (dichlorodifluoromethane) but with insignificant ozone depletion potential and a lower 100-year global warming potential (1,430, compared to R-12's GWP of 10,900). Even though 1,1,1,2-Tetrafluoroethane has insignificant ozone depletion potential (ozone layer) and negligible acidification potential (acid rain), it has a 100-year global warming potential (GWP) of 1430 and an approximate atmospheric lifetime of 14 years. Retrieved 21 August 2011. == History and environmental impacts == 1,1,1,2-Tetrafluoroethane was introduced in the early 1990s as a replacement for dichlorodifluoromethane (R-12), which has massive ozone depleting properties. Some of the properties of cyclic ozone have been predicted theoretically. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. This colorless gas is of interest as a more environmentally friendly (lower GWP; global warming potential) refrigerant in air conditioners. It has the formula CFCHF and a boiling point of −26.3 °C (−15.34 °F) at atmospheric pressure. The Society of Automotive Engineers (SAE) has proposed that it be best replaced by a new fluorochemical refrigerant HFO-1234yf (CFCF=CH) in automobile air-conditioning systems.HFO-1234yf A Low GWP Refrigerant For MAC . Chloropentafluoroethane is a chlorofluorocarbon (CFC) once used as a refrigerant and also known as R-115 and CFC-115. A phaseout and transition to HFO-1234yf and other refrigerants, with GWPs similar to CO2, began in 2012 within the automotive market. == Uses == 1,1,1,2-Tetrafluoroethane is a non-flammable gas used primarily as a ""high-temperature"" refrigerant for domestic refrigeration and automobile air conditioners. It has also been studied as a potential inhalational anesthetic, but it is nonanaesthetic at doses used in inhalers. == See also == * List of refrigerants * Tetrabromoethane * Tetrachloroethane == References == == External links == * * European Fluorocarbons Technical Committee (EFCTC) * MSDS at Oxford University * Concise International Chemical Assessment Document 11, at inchem.org * Pressure temperature calculator * * R134a 2 phase computer cooling Category:Fluoroalkanes Category:Refrigerants Category:Automotive chemicals Category:Propellants Category:Airsoft Category:Excipients Category:Greenhouse gases Category:GABAA receptor positive allosteric modulators Category:General anesthetics It should have more energy than ordinary ozone. It has been speculated that, if cyclic ozone could be made in bulk, and if it proved to have good stability properties, it could be added to liquid oxygen to improve the specific impulse of rocket fuel. 1-Chloro-3,3,3-trifluoropropene (HFO-1233zd) is the unsaturated chlorofluorocarbon with the formula HClC=C(H)CF3. Currently, the possibility of cyclic ozone is confirmed within diverse theoretical approaches. ==References== ==External links== * Category:Allotropes of oxygen Category:Hypothetical chemical compounds Category:Three-membered rings Category:Homonuclear triatomic molecules It would differ from ordinary ozone in how those three oxygen atoms are arranged. Its production and consumption has been banned since 1 January 1996 under the Montreal Protocol because of its high ozone depletion potential and very long lifetime when released into the environment.Ozone Depleting Substances List (Montreal Protocol) CFC-115 is also a potent greenhouse gas. ==Atmospheric properties== The atmospheric abundance of CFC-115 rose from 8.4 parts per trillion (ppt) in year 2010 to 8.7 ppt in 2020 based on analysis of air samples gathered from sites around the world. In ordinary ozone, the atoms are arranged in a bent line; in cyclic ozone, they would form an equilateral triangle. Thus it was included in the IPCC list of greenhouse gases. thumb|left|200px|HFC-134a atmospheric concentration since year 1995. Cyclic ozone has not been made in bulk, although at least one researcher has attempted to do so using lasers. ",200,29.9,252.8,6.6,0.84,B
-A gas at $250 \mathrm{~K}$ and $15 \mathrm{~atm}$ has a molar volume 12 per cent smaller than that calculated from the perfect gas law. Calculate the molar volume of the gas.,"The molar volume of an ideal gas at 100 kPa (1 bar) is : at 0 °C, : at 25 °C. The molar volume of an ideal gas at 1 atmosphere of pressure is : at 0 °C, : at 25 °C. == Crystalline solids == For crystalline solids, the molar volume can be measured by X-ray crystallography. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. This follows from above where the specific volume is the reciprocal of the density of a substance: V_{\rm m,i} = {M_i \over \rho_i^0} = M_i v_i == Ideal gases == For ideal gases, the molar volume is given by the ideal gas equation; this is a good approximation for many common gases at standard temperature and pressure. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas: :PV = k_5 T This can also be written as: : \frac {P_1V_1}{T_1}= \frac {P_2V_2}{T_2} With the addition of Avogadro's law, the combined gas law develops into the ideal gas law: :PV = nRT :where :*P is pressure :*V is volume :*n is the number of moles :*R is the universal gas constant :*T is temperature (K) :The proportionality constant, now named R, is the universal gas constant with a value of 8.3144598 (kPa∙L)/(mol∙K). The ideal gas equation can be rearranged to give an expression for the molar volume of an ideal gas: V_{\rm m} = \frac{V}{n} = \frac{RT}{P} Hence, for a given temperature and pressure, the molar volume is the same for all ideal gases and is based on the gas constant: R = , or about . The relation is given by :V \propto n\,, or :\frac{V_1}{n_1}=\frac{V_2}{n_2} \, :where n is equal to the number of molecules of gas (or the number of moles of gas). ==Combined and ideal gas laws== The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. An equivalent formulation of this law is: :PV = Nk_\text{B}T :where :*P is the pressure :*V is the volume :*N is the number of gas molecules :*kB is the Boltzmann constant (1.381×10−23J·K−1 in SI units) :*T is the temperature (K) These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). It is equal to the molar mass (M) divided by the mass density (ρ): V_{\text{m}} = \frac{M}{\rho} The molar volume has the SI unit of cubic metres per mole (m3/mol), although it is more typical to use the units cubic decimetres per mole (dm3/mol) for gases, and cubic centimetres per mole (cm3/mol) for liquids and solids. ==Definition== thumb|Change in volume with increasing ethanol fraction. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature. # If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. The molar volume of a substance i is defined as its molar mass divided by its density ρi0: V_{\rm m,i} = {M_i\over\rho_i^0} For an ideal mixture containing N components, the molar volume of the mixture is the weighted sum of the molar volumes of its individual components. The basic gas laws had been discovered by the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. In chemistry and related fields, the molar volume, symbol Vm, or \tilde V of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. # If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present. He observed that volume of a given mass of a gas is inversely proportional to its pressure at a constant temperature. This law has the following important consequences: # If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas. # If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature. ==Other gas laws== ;Graham's law: states that the rate at which gas molecules diffuse is inversely proportional to the square root of the gas density at constant temperature. The statement of Charles's law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T). In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. ",1.2,2,0.686,0,22.2036033112,A
-Calculate the mass of water vapour present in a room of volume $400 \mathrm{m}^3$ that contains air at $27^{\circ} \mathrm{C}$ on a day when the relative humidity is 60 per cent.,"The density of humid air is found by:Shelquist, R (2009) Equations - Air Density and Density Altitude \rho_\text{humid air} = \frac{p_\text{d}}{R_\text{d} T} + \frac{p_\text{v}}{R_\text{v} T} = \frac{p_\text{d}M_\text{d} + p_\text{v}M_\text{v}}{R T} where: *\rho_\text{humid air}, density of the humid air (kg/m3) *p_\text{d}, partial pressure of dry air (Pa) *R_\text{d}, specific gas constant for dry air, 287.058J/(kg·K) *T, temperature (K) *p_\text{v}, pressure of water vapor (Pa) *R_\text{v}, specific gas constant for water vapor, 461.495J/(kg·K) *M_\text{d}, molar mass of dry air, 0.0289652kg/mol *M_\text{v}, molar mass of water vapor, 0.018016kg/mol *R, universal gas constant, 8.31446J/(K·mol) The vapor pressure of water may be calculated from the saturation vapor pressure and relative humidity. The theoretical value for water vapor is 19.6, but due to vapor condensation the water vapor density dependence is highly variable and is not well approximated by this formula. This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\\ \end{align} where: *\rho, air density (kg/m3)In the SI unit system. The density of humid air may be calculated by treating it as a mixture of ideal gases. This occurs because the molar mass of water vapor (18g/mol) is less than the molar mass of dry airas dry air is a mixture of gases, its molar mass is the weighted average of the molar masses of its components (around 29g/mol). Air is given a vapour density of one. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. * At 70°F and 14.696psi, dry air has a density of 0.074887lb/ft3. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. It is found by: p_\text{v} = \phi p_\text{sat} where: *p_\text{v}, vapor pressure of water *\phi, relative humidity (0.0–1.0) *p_\text{sat}, saturation vapor pressure The saturation vapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%. The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa: ==Humid air== thumb|right|400px|Effect of temperature and relative humidity on air density The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. However, other units can be used. *p, absolute pressure (Pa) *T, absolute temperature (K) *R is the gas constant, in J⋅K−1⋅mol−1 *M is the molar mass of dry air, approximately in kg⋅mol−1. *k_{\rm B} is the Boltzmann constant, in J⋅K−1 *m is the molecular mass of dry air, approximately in kg. One formula is Tetens' equation fromShelquist, R (2009) Algorithms - Schlatter and Baker used to find the saturation vapor pressure is: p_\text{sat} = 6.1078 \times 10^{\frac{7.5 T}{T + 237.3}} where: *p_\text{sat}, saturation vapor pressure (hPa) *T, temperature (°C) See vapor pressure of water for other equations. Therefore: * At IUPAC standard temperature and pressure (0°C and 100kPa), dry air has a density of approximately 1.2754kg/m3. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). So when water molecules (water vapor) are added to a given volume of air, the dry air molecules must decrease by the same number, to keep the pressure or temperature from increasing. At 101.325kPa (abs) and , air has a density of approximately , which is about that of water, according to the International Standard Atmosphere (ISA). ",22,+116.0,3930.0,6.2,2,D
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The constant-volume heat capacity of a gas can be measured by observing the decrease in temperature when it expands adiabatically and reversibly. If the decrease in pressure is also measured, we can use it to infer the value of $\gamma=C_p / C_V$ and hence, by combining the two values, deduce the constant-pressure heat capacity. A fluorocarbon gas was allowed to expand reversibly and adiabatically to twice its volume; as a result, the temperature fell from $298.15 \mathrm{~K}$ to $248.44 \mathrm{~K}$ and its pressure fell from $202.94 \mathrm{kPa}$ to $81.840 \mathrm{kPa}$. Evaluate $C_p$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio () and the gas constant (): : C_P = \frac{\gamma n R}{\gamma - 1} \quad \text{and} \quad C_V = \frac{n R}{\gamma - 1}, === Relation with degrees of freedom === The classical equipartition theorem predicts that the heat capacity ratio () for an ideal gas can be related to the thermally accessible degrees of freedom () of a molecule by : \gamma = 1 + \frac{2}{f},\quad \text{or} \quad f = \frac{2}{\gamma - 1}. Therefore, the heat capacity ratio in this example is 1.4. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). This result can be explained through Le Chatelier's principle. ==See also== *Flame speed ==References== == External links == === General information === * * Computation of adiabatic flame temperature * Adiabatic flame temperature === Tables === * adiabatic flame temperature of hydrogen, methane, propane and octane with oxygen or air as oxidizers * * Temperature of a blue flame and common materials === Calculators === * Online adiabatic flame temperature calculator using Cantera * Adiabatic flame temperature program * Gaseq, program for performing chemical equilibrium calculations. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. ",41.40,0.36,0.66666666666,0.011,1.6,A
-Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. Estimate the standard reaction Gibbs energy of $\mathrm{N}_2(\mathrm{g})+3 \mathrm{H}_2(\mathrm{g}) \rightarrow$ $2 \mathrm{NH}_3$ (g) at $500 \mathrm{~K}$.,"Substance (state) ΔfG° ΔfG° Substance (state) (kJ/mol) (kcal/mol) NO(g) 87.6 20.9 NO2(g) 51.3 12.3 N2O(g) 103.7 24.78 H2O(g) −228.6 −54.64 H2O(l) −237.1 −56.67 CO2(g) −394.4 −94.26 CO(g) −137.2 −32.79 CH4(g) −50.5 −12.1 C2H6(g) −32.0 −7.65 C3H8(g) −23.4 −5.59 C6H6(g) 129.7 29.76 C6H6(l) 124.5 31.00 The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). This implies that at equilibrium Q_\text{r} = K_\text{eq} and \Delta_\text{r} G = 0. ==Standard Gibbs energy change of formation== Table of selected substancesCRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide. Moreover, we also have \begin{align} K_\text{eq} &= e^{-\frac{\Delta_\text{r} G^\circ}{RT}}, \\\ \Delta_\text{r} G^\circ &= -RT\left(\ln K_\text{eq}\right) = -2.303\,RT\left(\log_{10} K_\text{eq}\right), \end{align} which relates the equilibrium constant with Gibbs free energy. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The Gibbs energy of any system is and an infinitesimal change in G, at constant temperature and pressure, yields :dG=dU+pdV-TdS. The Gibbs free energy is expressed as : G(p,T) = U + pV - TS = H - TS where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure-volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. Value of R Unit SI units J��K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. ",7,205,0.0245, 9.73,58.2,A
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The enthalpy of vaporization of chloroform $\left(\mathrm{CHCl}_3\right)$ is $29.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at its normal boiling point of $334.88 \mathrm{~K}$. Calculate the entropy of vaporization of chloroform at this temperature.,"J/(mol K) Enthalpy of combustion –473.2 kJ/mol ΔcH ~~o~~ Heat capacity, cp 114.25 J/(mol K) Gas properties Std enthalpy change of formation, ΔfH ~~o~~ gas –103.18 kJ/mol Standard molar entropy, S ~~o~~ gas 295.6 J/(mol K) at 25 °C Heat capacity, cp 65.33 J/(mol K) at 25 °C van der Waals' constantsLange's Handbook of Chemistry 10th ed, pp. 1522–1524 a = 1537 L2 kPa/mol2 b = 0.1022 liter per mole ==Vapor pressure of liquid== P in mm Hg 1 10 40 100 400 760 1520 3800 7600 15200 30400 45600 T in °C –58.0 –29.7 –7.1 10.4 42.7 61.3 83.9 120.0 152.3 191.8 237.5 — Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. thumb|812px|left|log10 of Chloroform vapor pressure. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Uses formula: \scriptstyle \log_e P_{mmHg} =\scriptstyle \log_e (\frac {760} {101.325}) - 10.07089 \log_e(T+273.15) - \frac {6351.140} {T+273.15} + 81.14393 + 9.127608 \times 10^{-6} (T+273.15)^2 obtained from CHERIC ==Distillation data== {| border=""1"" cellspacing=""0"" cellpadding=""6"" style=""margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;"" Vapor-liquid Equilibrium for Chloroform/Ethanol P = 101.325 kPa BP Temp. °C % by mole chloroform liquid vapor 78.15 0.0 0.0 78.07 0.52 1.59 77.83 1.02 3.01 76.81 2.21 7.45 74.90 5.80 17.10 74.39 6.72 19.40 73.55 8.38 23.31 72.85 10.57 28.05 72.24 11.80 30.52 71.58 13.18 33.13 69.72 17.65 41.00 68.95 19.62 43.66 68.58 20.71 45.43 67.35 23.86 49.77 65.89 28.54 55.09 64.87 32.35 58.48 63.88 36.07 61.27 63.23 39.34 64.50 62.61 41.38 65.49 62.17 44.41 67.57 61.48 49.97 71.11 61.00 53.92 72.91 60.49 54.76 73.57 60.35 59.65 74.68 60.30 61.60 75.53 60.20 63.04 76.12 60.09 64.48 76.69 59.97 66.90 77.74 59.54 72.01 79.33 59.32 79.07 82.62 59.26 82.99 83.59 59.28 84.97 84.69 59.31 85.96 85.24 59.46 89.92 87.93 59.72 91.10 88.73 59.70 92.44 89.79 59.84 93.90 91.02 59.91 95.26 92.56 60.18 96.13 93.58 60.88 98.89 97.93 61.13 100.00 100.00 == Spectral data == UV-Vis λmax ? nm Extinction coefficient, ε ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Pa Critical point 537 K (264 °C), 5328.68 kPa Std enthalpy change of fusion, ΔfusH ~~o~~ 8.8 kJ/mol Std entropy change of fusion, ΔfusS ~~o~~ 42 J/(mol·K) Std enthalpy change of vaporization, ΔvapH ~~o~~ 31.4 kJ/mol Std entropy change of vaporization, ΔvapS ~~o~~ 105.3 J/(mol·K) Solid properties Std enthalpy change of formation, ΔfH ~~o~~ solid ? kJ/mol Standard molar entropy, S ~~o~~ solid ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Liquid properties Std enthalpy change of formation, ΔfH ~~o~~ liquid –134.3 kJ/mol Standard molar entropy, S ~~o~~ liquid ? In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. K (? °C), ? K (? °C), ? K (? °C), ? K (? °C), ? This page provides supplementary chemical data on chloroform. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). ", 258.14,+87.8,0.33333333,-0.029,24,B
+Given that $\tilde{\omega}_{\mathrm{e}}=2886 \mathrm{~cm}^{-1}$ and $D=440.2 \mathrm{~kJ} \cdot \mathrm{mol}^{-1}$ for $\mathrm{H}^{35} \mathrm{Cl}$, calculate $\tilde{x}_{\mathrm{e}}$ and $\tilde{\omega}_{\mathrm{e}} \tilde{x}_{\mathrm{e}}$.","With that, the one-dimensional Schrödinger equation that describes on S^3 the quantum motion of an electric charge dipole perturbed by the trigonometric Rosen–Morse potential, produced by another electric charge dipole, takes the form of {{NumBlk|:| \left(-\frac{\hbar^2 c^2}{R^2}\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+V^{(\ell+1,\alpha Z/2,1)}_{\mbox{tRM}}(\chi)\right)U_{K\ell}^{(\alpha Z)}(\chi)=\frac{\hbar^2c^2}{R^2}\left({\epsilon}_{\ell n}^{(\alpha Z)}\right)^2 U_{K\ell }^{(\alpha Z)}(\chi), |}} (\chi)=\frac{\hbar^2 c^2}{R^2}\frac{\ell(\ell+1)}{\sin^2\chi}-2\frac{\hbar^2 c^2}{R^2}\alpha Z\cot\chi, |}} Because of the relationship, K-\ell=n, with n being the node number of the wave function, one could change labeling of the wave functions, U^{(b)}_{K\ell}(\chi), to the more familiar in the literature, U^{(b)}_{\ell n}(\chi). For this reason, the wave equation which transforms upon the variable change, \Psi_{K\ell m}(\chi,\theta,\varphi)=\frac{U^{(b)}_{K\ell}(\chi)}{\sin\chi}Y_{\ell}^m(\theta,\varphi), into the familiar one-dimensional Schrödinger equation with the V_{tRM}^{(\ell+1,b,1)}(\chi) trigonometric Rosen–Morse potential, {{NumBlk|:| -\frac{\hbar^2c^2}{R^2}\left[\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+\frac{\ell(\ell+1)}{\sin^2\chi}\right]U^{(b)}_{K\ell}(\chi)-2b\cot\chi U^{(b)}_{K\ell}(\chi)=\frac{\hbar^2c^2}{R^2}\left[(K+1)^2-\frac{b^2}{(K+1)^2}\right]U^{(b)}_{K\ell}(\chi), |}} in reality describes quantum motion of a charge dipole perturbed by the field due to another charge dipole, and not the motion of a single charge within the field produced by another charge. Changing in () variables as one observes that the \psi_{K\ell}(\chi) function satisfies the one-dimensional Schrödinger equation with the \csc^2\chi potential according to {{NumBlk|:| -\frac{\hbar^2c^2}{R^2}\left[\frac{{\mathrm d}^2}{{\mathrm d}\chi^2}+\frac{\ell(\ell+1)}{\sin^2\chi}\right]\psi_{K\ell}(\chi)=\frac{\hbar^2c^2}{R^2}(K+1)^2\psi_{K\ell}(\chi). |}} The one-dimensional potential in the latter equation, in coinciding with the Rosen–Morse potential in () for a=\ell+1 and b=0, clearly reveals that for integer a values, the first term of this potential takes its origin from the centrifugal barrier on S^3. This potential is approximately a Morse potential with 16\pi^{2} e^{8|x|} The asymptotic of the energies depend on the quantum number as E_n = \frac{4\pi^2 n^2}{W^2(ne^{-1})} , where is the Lambert W function. == References == * * G. Sierra, A physics pathway to the Riemann hypothesis, arXiv:math-ph/1012.4264, 2010. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. For the case of the Riemann zeros Wu and Sprung and others have shown that the potential can be written implicitly in terms of the Gamma function and zeroth- order Bessel function. f^{-1} (x)=\frac{2}{\sqrt{4x+1} } +\frac{1}{4\pi } \int_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right) -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{2\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right) and that the density of states of this Hamiltonian is just the Delsarte's formula for the Riemann zeta function and defined semiclassically as \frac{1}{\sqrt \pi} \frac{d^{1/2}}{dx^{1/2}}f^{-1}(x)= \sum_{n=0}^{\infty}\delta (x-E_{n}) \begin{align} \sum_{n=0}^{\infty }\delta \left( x-\gamma _{n} \right) + \sum_{n=0}^{\infty }\delta \left( x+\gamma _{n} \right) ={}& \frac{1}{2\pi } \frac{\zeta }{\zeta } \left( \frac{1}{2} +ix\right) +\frac{1}{2\pi } \frac{\zeta '}{\zeta } \left( \frac{1}{2} -ix\right) -\frac{\ln \pi }{2\pi } \\\\[10pt] &{} +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{\Gamma '}{\Gamma } \left( \frac{1}{4} -i\frac{x}{2} \right) \frac{1}{4\pi } +\frac{1}{\pi } \delta \left( x-\frac{i}{2} \right) + \frac{1}{\pi } \delta \left( x + \frac{i}{2} \right) \end{align} here they have taken the derivative of the Euler product on the critical line \frac{1}{2}+is ; also they use the Dirichlet generating function \frac{\zeta ' (s)}{\zeta(s)}= -\sum_{n=1}^{\infty} \Lambda (n) e^{-slnn} . \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. Wu and Sprung also showed that the zeta-regularized functional determinant is the Riemann Xi-function \frac{ \xi(s)}{\xi(0)} = \frac{\det(H-s(1-s)+\frac{1}{4})}{\det(H+\frac{1}{4})} The main idea inside this problem is to recover the potential from spectral data as in some inverse spectral problems in this case the spectral data is the Eigenvalue staircase, which is a quantum property of the system, the inverse of the potential then, satisfies an Abel integral equation (fractional calculus) which can be immediately solved to obtain the potential. == Asymptotics== For large x if we take only the smooth part of the eigenvalue staircase N(E) \sim \frac{\sqrt{E} }{2\pi } \log \left( \frac{\sqrt{E} }{2\pi e} \right) , then the potential as |x| \to \infty is positive and it is given by the asymptotic expression f(-x) = f(x) \sim 4\pi^2 e^2 \left( \frac{2\epsilon \sqrt{\pi } x+B}{A(\epsilon )} \right) ^{2 / \epsilon } with A(\epsilon ) = \frac{\Gamma{\left( \frac{3+\epsilon }{2} \right)}}{\Gamma{\left( 1 + \frac{\epsilon }{2} \right)}} and B = A(0) in the limit \epsilon \to 0 . Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. e^{-\frac{(x-\xi)^2}{2\omega^2}} \int_{-\infty}^{\alpha\left(\frac{x-\xi}{\omega}\right)} \frac{1}{\sqrt{2 \pi}} e^{-\frac{t^2}{2}}\ dt| cdf =\Phi\left(\frac{x-\xi}{\omega}\right)-2T\left(\frac{x-\xi}{\omega},\alpha\right) T(h,a) is Owen's T function| mean =\xi + \omega\delta\sqrt{\frac{2}{\pi}} where \delta = \frac{\alpha}{\sqrt{1+\alpha^2}}| median =| mode =\xi + \omega m_o(\alpha) | variance =\omega^2\left(1 - \frac{2\delta^2}{\pi}\right)| skewness =\gamma_1 = \frac{4-\pi}{2} \frac{\left(\delta\sqrt{2/\pi}\right)^3}{ \left(1-2\delta^2/\pi\right)^{3/2}}| kurtosis =2(\pi - 3)\frac{\left(\delta\sqrt{2/\pi}\right)^4}{\left(1-2\delta^2/\pi\right)^2}| entropy =| mgf =M_X\left(t\right)=2\exp\left(\xi t+\frac{\omega^2t^2}{2}\right)\Phi\left(\omega\delta t\right)| cf =M_X\left(i\delta\omega t\right)| char =e^{i t \xi -t^2\omega^2/2}\left(1+i\, \textrm{Erfi}\left(\frac{\delta\omega t}{\sqrt{2}}\right)\right)| }} In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness. ==Definition== Let \phi(x) denote the standard normal probability density function :\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} with the cumulative distribution function given by :\Phi(x) = \int_{-\infty}^{x} \phi(t)\ dt = \frac{1}{2} \left[ 1 + \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right)\right], where ""erf"" is the error function. {\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}\\!| cdf =\Gamma\\!\left(\frac{ u}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\\!\left(\frac{ u}{2}\right)\\!| mean =\frac{1}{ u-2}\\! for u >2\\!| median = \approx \dfrac{1}{ u\bigg(1-\dfrac{2}{9 u}\bigg)^3}| mode =\frac{1}{ u+2}\\!| variance =\frac{2}{( u-2)^2 ( u-4)}\\! for u >4\\!| skewness =\frac{4}{ u-6}\sqrt{2( u-4)}\\! for u >6\\!| kurtosis =\frac{12(5 u-22)}{( u-6)( u-8)}\\! for u >8\\!| entropy =\frac{ u}{2} \\!+\\!\ln\\!\left(\frac{ u}{2}\Gamma\\!\left(\frac{ u}{2}\right)\right) \\!-\\!\left(1\\!+\\!\frac{ u}{2}\right)\psi\\!\left(\frac{ u}{2}\right)| mgf =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-t}{2i}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2t}\right); does not exist as real valued function| char =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-it}{2}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2it}\right)| }} In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-valued random variable. In mathematical physics, the Wu–Sprung potential, named after Hua Wu and Donald Sprung, is a potential function in one dimension inside a Hamiltonian H = p^2 + f(x) with the potential defined by solving a non-linear integral equation defined by the Bohr–Sommerfeld quantization conditions involving the spectral staircase, the energies E_n and the potential f(x) . \oint p \, dq =2 \pi n(E)= 4\int_0^a dx \sqrt{E_n - f(x)} here is a classical turning point so E = f(a) = f(-a) , the quantum energies of the model are the roots of the Riemann Xi function \xi{\left( \frac{1}{2} + i \sqrt{E_n}\right)} = 0 and f(x)=f(-x) . The trigonometric Rosen–Morse potential, named after the physicists Nathan Rosen and Philip M. Morse, is among the exactly solvable quantum mechanical potentials. ==Definition== In dimensionless units and modulo additive constants, it is defined as where r is a relative distance, \lambda is an angle rescaling parameter, and R is so far a matching length parameter. Similarly, denoting the sum of the ω functions by W : W = \omega_{1}(\varphi_{1}) + \omega_{2}(\varphi_{2}) + \cdots + \omega_{s}(\varphi_{s}), the potential energy V can be written as : V = \frac{W}{Y}. ===Lagrange equation=== The Lagrange equation for the rth variable \varphi_{r} is : \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{\varphi}_{r}} \right) = \frac{d}{dt} \left( Y \dot{\varphi}_{r} \right) = \frac{1}{2} F \frac{\partial Y}{\partial \varphi_{r}} -\frac{\partial V}{\partial \varphi_{r}}. In eqs. ()-() one recognizes the one-dimensional wave equation with the trigonometric Rosen–Morse potential in () for a=\ell+1 and 2b=\alpha Z. > In this way, the cotangent term of the trigonometric Rosen–Morse potential > could be derived from the Gauss law on S^3 in combination with the > superposition principle, and could be interpreted as a dipole potential > generated by a system consisting of two opposite fundamental charges. In this manner, the complete trigonometric > Rosen–Morse potential could be derived from first principles. This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. Introducing elliptic coordinates, : x = a \cosh \xi \cos \eta, : y = a \sinh \xi \sin \eta, the potential energy can be written as : V(\xi, \eta) = \frac{-\mu_{1}}{a\left( \cosh \xi - \cos \eta \right)} - \frac{\mu_{2}}{a\left( \cosh \xi + \cos \eta \right)} = \frac{-\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right)}{a\left( \cosh^{2} \xi - \cos^{2} \eta \right)}, and the kinetic energy as : T = \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right) \left( \dot{\xi}^{2} + \dot{\eta}^{2} \right). In general, although Wu and Sprung considered only the smooth part, the potential is defined implicitly by f^{-1}(x)= \sqrt \pi \frac{d^{1/2}}{dx^{1/2}}N(x) ; with being the eigenvalue staircase N(x) = \sum_{n=0}^\infty H(x - E_{n}) and is the Heaviside step function. Inverting, taking the square root and separating the variables yields a set of separably integrable equations: : \frac{\sqrt{2}}{Y} dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}}. ==References== ==Further reading== * Category:Classical mechanics When transcribed to the current notations and units, the partition function in presents itself as, , \quad p=\beta \frac{\hbar c}{R}\alpha^2Z^2. \end{align} |}} The infinite integral has first been treated by means of partial integration giving, & = \int_0^\infty x^2e^{-ax^2}e^{\frac{p}{x^2}}{\mathrm d}x \\\ & = -\frac{1}{2a}\int_0^\infty x e^{\frac{p}{x^2}}{\mathrm d}e^{-ax^2}\\\ & = -\frac{1}{2a}xe^{\frac{p}{x^2}}e^{-ax^2}|_0^\infty + \frac{1}{2a}\int_0^\infty e^{-ax^2}{\mathrm d}\, \left( x e^{\frac{p}{x^2}}\right)\\\ & = \frac{1}{2a} \int_0^\infty e^{-ax^2 +\frac{p}{x^2}}{\mathrm d}x + \frac{2p}{2a}\int_0^\infty e^{-ax^2 +\frac{p}{x^2}}{\mathrm d}\left(\frac{1}{x}\right). \end{align} |}} Then the argument of the exponential under the sign of the integral has been cast as, {x}, \end{align} |}} thus reaching the following intermediate result, \int_0^\infty e^{-z^2}{\mathrm d} \left( x + \frac{2p}{x}\right) . \end{align} |}} As a next step the differential has been represented as -2i\sqrt{p} \right) z + \frac{1}{2} \left(\frac{1}{\sqrt{a}}+2i\sqrt{p} \right) z^\ast, \end{align} |}} an algebraic manipulation which allows to express the partition function in () in terms of the \mbox{erf}(u) function of complex argument according to, -2i\sqrt{p}\right) e^{2i\sqrt{ap}}\int_{\Gamma} e^{-z^2}{\mathrm d}z + \left( \frac{1}{\sqrt{a}} +2i\sqrt{p} \right) e^{-2i\sqrt{ap}}\int_{\Gamma} e^{- \left( z^\ast\right)^2} {\mathrm d}z^\ast \right], \\\ \end{align} |}} where \Gamma is an arbitrary path on the complex plane starting in zero and ending in u\to \infty. ",-24, 0.01961,"""1.11""",0.020,11000,B
+" In the infrared spectrum of $\mathrm{H}^{127} \mathrm{I}$, there is an intense line at $2309 \mathrm{~cm}^{-1}$. Calculate the force constant of $\mathrm{H}^{127} \mathrm{I}$.","In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. Such a signal would not be overwhelmed by the H I line itself, or by any of its harmonics. ==See also== * Balmer series * Chronology of the universe * Dark Ages Radio Explorer * Hydrogen spectral series * H-alpha, the visible red spectral line with wavelength of 656.28 nanometers * Rydberg formula * Timeline of the Big Bang ==Footnotes== ==References== ==Further reading== ===Cosmology=== * * * * * * * * ==External links== * * — PAST experiment description * * * Category:Hydrogen physics Category:Emission spectroscopy Category:Radio astronomy Category:Physical cosmology Category:Astrochemistry Category:Hydrogen Available: http://physics.nist.gov/constants. The value of this constant is given here as 1/137.035999206 (note the difference in the last three digits). The constant is expressed for either hydrogen as R_\text{H}, or at the limit of infinite nuclear mass as R_\infty. Approximate value of Value of In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter alpha), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between elementary charged particles. Since the measurement of an absolute intensity in an experiment can be challenging, the ratio of different spectral line intensities can be used to achieve information about the plasma, as well. == Theory == The emission intensity density of an atomic transition from the upper state to the lower state is: P_{u \rightarrow l} = N_u \ \hbar \omega_{u \rightarrow l} \ A_{u \rightarrow l} , where: * N_u is the density of ions in the upper state, * \hbar \omega_{u \rightarrow l} is the energy of the emitted photon, which is the product of the Planck constant and the transition frequency, * A_{u \rightarrow l} is the Einstein coefficient for the specific transition. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants according to his model of the atom. In hydrogen, its wavelength of 1215.67 angstroms ( or ), corresponding to a frequency of about , places Lyman-alpha in the ultraviolet (UV) part of the electromagnetic spectrum. At higher energies, such as the scale of the Z boson, about 90 GeV, one instead measures an effective ≈ 1/127. In either case, the constant is used to express the limiting value of the highest wavenumber (inverse wavelength) of any photon that can be emitted from a hydrogen atom, or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing a hydrogen atom from its ground state. Its numerical value is approximately , with a relative uncertainty of The constant was named by Arnold Sommerfeld, who introduced it in 1916 Equation 12a, ""rund 7·"" (about ...) when extending the Bohr model of the atom. quantified the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Michelson and Morley in 1887. In spectroscopy, the Rydberg constant, symbol R_\infty for heavy atoms or R_\text{H} for hydrogen, named after the Swedish physicist Johannes Rydberg, is a physical constant relating to the electromagnetic spectra of an atom. When applied to the Planck relation, this gives: :\lambda = \frac {1}{ u} \cdot c = \frac {h}{E} \cdot c \approx \frac{\; 4.1357 \cdot 10^{-15} \ \mathrm{eV}\cdot\text{s} \;}{5.874\,33 \cdot 10^{-6}\ \mathrm{eV}}\, \cdot\, 2.9979 \cdot 10^8 \ \mathrm{m} \cdot \mathrm{s}^{-1} \approx 0.211\,06\ \mathrm{m} = 21.106\ \mathrm{cm}\; where is the wavelength of an emitted photon, is its frequency, is the photon energy, is the Planck constant, and is the speed of light. The Lyman-alpha line, typically denoted by Ly-α, is a spectral line of hydrogen (or, more generally, of any one-electron atom) in the Lyman series. The electromagnetic radiation producing this line has a frequency of (1.42 GHz), which is equivalent to a wavelength of in a vacuum. Why the constant should have this value is not understood, but there are a number of ways to measure its value. ==Definition== In terms of other fundamental physical constants, may be defined as: \alpha = \frac{e^2}{2 \varepsilon_0 h c} = \frac{e^2}{4 \pi \varepsilon_0 \hbar c} , where * is the elementary charge (); * is the Planck constant (); * is the reduced Planck constant, (6.62607015×10−34 J⋅Hz−1/2π) * is the speed of light (); * is the electric constant (). Specifically, they found that :\frac{\ \Delta \alpha\ }{\alpha} ~~ \overset{\underset{\mathsf{~def~}}{}}{=} ~~ \frac{\ \alpha _\mathrm{prev}-\alpha _\mathrm{now}\ }{\alpha_\mathrm{now}} ~~=~~ \left(-5.7\pm 1.0 \right) \times 10^{-6} ~. The first physical interpretation of the fine-structure constant was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum. The hydrogen spectral series can be expressed simply in terms of the Rydberg constant for hydrogen R_\text{H} and the Rydberg formula. The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron: : R_\text{H} = R_\infty \frac{ m_\text{e} m_\text{p} }{ m_\text{e}+m_\text{p} } \approx 1.09678 \times 10^7 \text{ m}^{-1} , where * m_\text{e} is the mass of the electron, * m_\text{p} is the mass of the nucleus (a proton). === Rydberg unit of energy === The Rydberg unit of energy is equivalent to joules and electronvolts in the following manner: :1 \ \text{Ry} \equiv h c R_\infty = \frac{m_\text{e} e^4}{8 \varepsilon_{0}^{2} h^2} = \frac{e^2}{8 \pi \varepsilon_{0} a_0} = 2.179\;872\;361\;1035(42) \times 10^{-18}\ \text{J} \ = 13.605\;693\;122\;994(26)\ \text{eV}. === Rydberg frequency === :c R_\infty = 3.289\;841\;960\;2508(64) \times 10^{15}\ \text{Hz} . === Rydberg wavelength === :\frac 1 {R_\infty} = 9.112\;670\;505\;824(17) \times 10^{-8}\ \text{m}. The strength of the electromagnetic interaction varies with the strength of the energy field. | In the fields of electrical engineering and solid-state physics, the fine- structure constant is one fourth the product of the characteristic impedance of free space, ~ Z_0 = \mu_0 c = \sqrt{\frac{\mu_0}{\varepsilon_0}} , and the conductance quantum, G_0 = \frac{2 e^2}{ h }: \alpha = \tfrac{1}{4} Z_0 G_0\ . ",-2.99,3.0,"""313.0""",2.57,0.11,C
+Calculate the percentage difference between $e^x$ and $1+x$ for $x=0.0050$,"It follows that is transcendental over . ==Computation== When computing (an approximation of) the exponential function near the argument , the result will be close to 1, and computing the value of the difference e^x-1 with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result. That is, \frac{d}{dx}e^x = e^x \quad\text{and}\quad e^0=1. Mistakenly using percentage as the unit for the standard deviation is confusing, since percentage is also used as a unit for the relative standard deviation, i.e. standard deviation divided by average value (coefficient of variation). == Related units == * Percentage (%) 1 part in 100 * Per mille (‰) 1 part in 1,000 *Basis point (bp) difference of 1 part in 10,000 *Permyriad (‱) 1 part in 10,000 * Per cent mille (pcm) 1 part in 100,000 * Baker percentage == See also == * Parts-per notation * Per-unit system * Percent point function * Relative change and difference == References == Category:Mathematical terminology Category:Probability assessment Category:Units of measurement ru:Процент#Процентный пункт thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. In this case, the reciprocal transform of the percentage-point difference would be 1/(20pp) = 1/0.20 = 5. For example, if the exponential is computed by using its Taylor series e^x = 1 + x + \frac {x^2}2 + \frac{x^3}6 + \cdots + \frac{x^n}{n!} + \cdots, one may use the Taylor series of e^x-1: e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots. The equation \tfrac{d}{dx}e^x = e^x means that the slope of the tangent to the graph at each point is equal to its -coordinate at that point. ==Relation to more general exponential functions== The exponential function f(x) = e^x is sometimes called the natural exponential function for distinguishing it from the other exponential functions. A percentage point or percent point is the unit for the arithmetic difference between two percentages. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (that is, percentage increase or decrease) in the dependent variable. If v e 0, the relative error is : \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, and the percent error (an expression of the relative error) is :\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. 420px|thumb| As x approaches zero from the right, the magnitude of the rate of change of 1/x increases. The derivative (rate of change) of the exponential function is the exponential function itself. Percentage-point differences are one way to express a risk or probability. For measurements involving percentages as a unit, such as, growth, yield, or ejection fraction, statistical deviations and related descriptive statistics, including the standard deviation and root-mean-square error, the result should be expressed in units of percentage points instead of percentage. thumb|200px|right|Exponential functions with bases 2 and 1/2 The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). The base of the exponential function, its value at 1, e = \exp(1), is a ubiquitous mathematical constant called Euler's number. After the first occurrence, some writers abbreviate by using just ""point"" or ""points"". ==Differences between percentages and percentage points== Consider the following hypothetical example: In 1980, 50 percent of the population smoked, and in 1990 only 40 percent of the population smoked. Euler's number is the unique base for which the constant of proportionality is 1, since \ln(e) = 1, so that the function is its own derivative: \frac{d}{dx} e^x = e^x \ln (e) = e^x. Exponential constant may refer to: * e (mathematical constant) * The growth or decay constant in exponential growth or exponential decay, respectively. For real numbers and , a function of the form f(x) = a b^{cx + d} is also an exponential function, since it can be rewritten as a b^{c x + d} = \left(a b^d\right) \left(b^c\right)^x. ==Formal definition== right|thumb|The exponential function (in blue), and the sum of the first terms of its power series (in red). The natural exponential is hence denoted by x\mapsto e^x or x\mapsto \exp x. Percentage solution may refer to: * Mass fraction (or ""% w/w"" or ""wt.%""), for percent mass * Volume fraction (or ""% v/v"" or ""vol.%""), volume concentration, for percent volume * ""Mass/volume percentage"" (or ""% m/v"") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). ",0.3085,1855,"""-1.46""",1.51,1.25,E
+ Calculate (a) the wavelength and kinetic energy of an electron in a beam of electrons accelerated by a voltage increment of $100 \mathrm{~V}$ ,"# Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I–V curves. Following Kunio Fujiwara and Archibald Howie, the relationship between the total energy of the electrons and the wavevector is written as: :E=\frac{h^2k^2}{2m^*} with m^*=m_0 + \frac{E}{2c^2} where h is Planck's constant, m^* is a relativistic effective mass used to cancel out the relativistic terms for electrons of energy E with c the speed of light and m_0 the rest mass of the electron. Typically the energy of the electrons is written in electronvolts (eV), the voltage used to accelerate the electrons; the actual energy of each electron is this voltage times the electron charge. framed|Geometry of electron beam in precession electron diffraction. Relativistic electron beams are streams of electrons moving at relativistic speeds. The Schrödinger equation combines the kinetic energy of waves and the potential energy due to, for electrons, the Coulomb potential. Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. First, as previously noted, the electron must absorb an amount of energy equivalent to the energy difference between the electron's current energy level and an unoccupied, higher energy level in order to be promoted to that energy level. The energy released is equal to the difference in energy levels between the electron energy states. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. Combining this with the aforementioned Lorentz correction yields: I_\mathbf{g}^{kinematical} \propto I_\mathbf{g}^{experimental} \cdot g\sqrt{1-\frac{g}{2R_o}} \cdot \int\limits_0^{A_\mathbf{g}}J_0(2x)\, dx where A_\mathbf{g} = \frac{2 \pi t F_\mathbf{g}}{k} , t is the sample thickness, and k is the wave-vector of the electron beam. Johannes (Janne) Robert Rydberg (; 8 November 1854 - 28 December 1919) was a Swedish physicist mainly known for devising the Rydberg formula, in 1888, which is used to describe the wavelengths of photons (of visible light and other electromagnetic radiation) emitted by changes in the energy level of an electron in a hydrogen atom. == Biography == Rydberg was born 8 November 1854 in Halmstad in southern Sweden, the only child of Sven Rydberg and Maria Anderson Rydberg. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. The acceleration voltage is an important quantity for the design of microwave cavities for particle accelerators. The following considerations are generalized for time-dependent fields. == Longitudinal voltage == The longitudinal effective acceleration voltage is given by the kinetic energy gain experienced by a particle with velocity \beta c along a defined straight path (path integral of the longitudinal Lorentz forces) divided by its charge, V_\parallel(\beta) = \frac 1 q \vec e_s \cdot \int \vec F_L(s,t) \,\mathrm{d}s = \frac 1 q \vec e_s \cdot \int \vec F_L(s, t = \frac{s}{\beta c}) \,\mathrm d s . These include: # The simplest approximation using the de Broglie wavelength for electrons, where only the geometry is considered and often Bragg's law is invoked, a far- field or Fraunhofer approach. One of the methods is to use the concept of dressed particle. == See also == * Energy level * Mode (electromagnetism) == References == Category:Electron *Developments in the convergent-beam electron diffraction approach. ",144,226,"""0.3085""",1.602,355.1,D
+Suppose that $10.0 \mathrm{~mol} \mathrm{C}_2 \mathrm{H}_6(\mathrm{~g})$ is confined to $4.860 \mathrm{dm}^3$ at $27^{\circ} \mathrm{C}$. Predict the pressure exerted by the ethane from the perfect gas.,"Natural gas from different gas fields varies in ethane content from less than 1% to more than 6% by volume. The bond parameters of ethane have been measured to high precision by microwave spectroscopy and electron diffraction: rC−C = 1.528(3) Å, rC−H = 1.088(5) Å, and ∠CCH = 111.6(5)° by microwave and rC−C = 1.524(3) Å, rC−H = 1.089(5) Å, and ∠CCH = 111.9(5)° by electron diffraction (the numbers in parentheses represents the uncertainties in the final digits). ===Atmospheric and extraterrestrial=== Ethane occurs as a trace gas in the Earth's atmosphere, currently having a concentration at sea level of 0.5 ppb,Trace gases (archived). At standard temperature and pressure, ethane is a colorless, odorless gas. Today, ethane is an important petrochemical feedstock and is separated from the other components of natural gas in most well-developed gas fields. Global ethane quantities have varied over time, likely due to flaring at natural gas fields. Atmosphere.mpg.de. Retrieved on 2011-12-08. though its preindustrial concentration is likely to have been only around 0.25 part per billion since a significant proportion of the ethane in today's atmosphere may have originated as fossil fuels. Ethane was discovered dissolved in Pennsylvanian light crude oil by Edmund Ronalds in 1864. ==Properties== At standard temperature and pressure, ethane is a colorless, odorless gas. As far back as 1890–1891, chemists suggested that ethane molecules preferred the staggered conformation with the two ends of the molecule askew from each other. ==Production== After methane, ethane is the second-largest component of natural gas. : C2H5O• → CH3• + CH2O Some minor products in the incomplete combustion of ethane include acetaldehyde, methane, methanol, and ethanol. Ethane ( , ) is an organic chemical compound with chemical formula . Computer simulations of the chemical kinetics of ethane combustion have included hundreds of reactions. In 2006, Dale Cruikshank of NASA/Ames Research Center (a New Horizons co-investigator) and his colleagues announced the spectroscopic discovery of ethane on Pluto's surface. ==Chemistry== Ethane can be viewed as two methyl groups joined, that is, a dimer of methyl groups. This error was corrected in 1864 by Carl Schorlemmer, who showed that the product of all these reactions was in fact ethane. Solid ethane exists in several modifications. Ethane is most efficiently separated from methane by liquefying it at cryogenic temperatures. This page provides supplementary chemical data on ethane. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. : C2H5• + O2 → C2H5OO• : C2H5OO• + HR → C2H5OOH + •R : C2H5OOH → C2H5O• + •OH The principal carbon-containing products of incomplete ethane combustion are single-carbon compounds such as carbon monoxide and formaldehyde. Ethane can also be separated from petroleum gas, a mixture of gaseous hydrocarbons produced as a byproduct of petroleum refining. Like many hydrocarbons, ethane is isolated on an industrial scale from natural gas and as a petrochemical by-product of petroleum refining. The chemistry of ethane involves chiefly free radical reactions. They mistook the product of these reactions for the methyl radical (), of which ethane () is a dimer. In fact, ethane's global warming potential largely results from its conversion in the atmosphere to methane.Hodnebrog, Øivind; Dalsøren, Stig B. and Myrhe, Gunnar; ‘Lifetimes, direct and indirect radiative forcing, and globalwarming potentials of ethane (C2H6), propane (C3H8),and butane (C4H10)’; Atmospheric Science Letters; 2018;19:e804 It has been detected as a trace component in the atmospheres of all four giant planets, and in the atmosphere of Saturn's moon Titan. ",16.3923,-20,"""24.0""",4.16,50.7,E
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the standard enthalpy of solution of $\mathrm{AgCl}(\mathrm{s})$ in water from the enthalpies of formation of the solid and the aqueous ions.","The solubility product, Ksp, for AgCl in water is at room temperature, which indicates that only 1.9 mg (that is, \sqrt{1.77\times 10^{-10}} \ \mathrm{mol}) of AgCl will dissolve per liter of water. The chloride content of an aqueous solution can be determined quantitatively by weighing the precipitated AgCl, which conveniently is non-hygroscopic since AgCl is one of the few transition metal chlorides that are unreactive toward water. The standard enthalpy of formation is then determined using Hess's law. The Saha equation describes the degree of ionization for any gas in thermal equilibrium as a function of the temperature, density, and ionization energies of the atoms. Silver chloride is a chemical compound with the chemical formula AgCl. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). :AgNO3 + NaCl -> AgCl(v) + NaNO3 :2 AgNO3 + CoCl2 -> 2 AgCl(v) + Co(NO3)2 It can also be produced by the reaction of silver metal and aqua regia, however, the insolubility of silver chloride decelerates the reaction. In physics, the Saha ionization equation is an expression that relates the ionization state of a gas in thermal equilibrium to the temperature and pressure. Examples are given in the following sections. == Ionic compounds: Born–Haber cycle == For ionic compounds, the standard enthalpy of formation is equivalent to the sum of several terms included in the Born–Haber cycle. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. Above 7.5 GPa, silver chloride transitions into a monoclinic KOH phase. K (? °C), ? K (? °C), ? This is true for all enthalpies of formation. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. For a gas composed of a single atomic species, the Saha equation is written: :\frac{n_{i+1}n_e}{n_i} = \frac{2}{\lambda^{3}}\frac{g_{i+1}}{g_i}\exp\left[-\frac{(\epsilon_{i+1}-\epsilon_i)}{k_B T}\right] where: * n_i is the density of atoms in the i-th state of ionization, that is with i electrons removed. * g_i is the degeneracy of states for the i-ions * \epsilon_i is the energy required to remove i electrons from a neutral atom, creating an i-level ion. * n_e is the electron density * \lambda is the thermal de Broglie wavelength of an electron ::\lambda \ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{h^2}{2\pi m_e k_B T}} * m_e is the mass of an electron * T is the temperature of the gas * h is Planck's constant The expression (\epsilon_{i+1}-\epsilon_i) is the energy required to remove the (i+1)^{th} electron. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Silver chloride is also produced as a byproduct of the Miller process where silver metal is reacted with chlorine gas at elevated temperatures. ==History== Silver chloride was known since ancient times. ""Standard potential of the silver-silver chloride electrode"". ",+65.49,36,"""130.41""",76,0.166666666,A
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in chemical potential of a perfect gas when its pressure is increased isothermally from $1.8 \mathrm{~atm}$ to $29.5 \mathrm{~atm}$ at $40^{\circ} \mathrm{C}$.,"For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. * A partial pressure of 101.325 kPa (absolute) (1 atm, 1.01325 bar) for each gaseous reagent. In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. (i) Indicates values calculated from ideal gas thermodynamic functions. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. When pressure and temperature are variable, only I-1 of I components have independent values for chemical potential and Gibbs' phase rule follows. The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . ",0.086,7166.67,"""7.3""",-131.1,14.5115,C
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. 3.1(a) Calculate the change in entropy when $25 \mathrm{~kJ}$ of energy is transferred reversibly and isothermally as heat to a large block of iron at $100^{\circ} \mathrm{C}$.,"The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. * The heat capacity of the gas from the boiling point to room temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). The amount of energy added equals , with representing the change in temperature. Therefore, the heat capacity ratio in this example is 1.4. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Equilibrium occurs when the temperature is equal to the melting point T = T_f so that :\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T_f \times \Delta S_{\text{fus}} = 0, and the entropy of fusion is the heat of fusion divided by the melting point: : \Delta S_{\text{fus}} = \frac {\Delta H_{\text{fus}}} {T_f} ==Helium== Helium-3 has a negative entropy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative entropy of fusion below 0.8 K. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. ",240,14.34457,"""36.0""",67,-2,D
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $q$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. In addition, other factors come into play and dominate during a compression cycle if they have a greater impact on the final calculated result than whether or not C_P was held constant. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. ",27,59.4,"""0.0029""",+7.3,0,E
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. For the reaction $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(\mathrm{l})+3 \mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})+3 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$, $\Delta_{\mathrm{r}} U^\ominus=-1373 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}$, calculate $\Delta_{\mathrm{r}} H^{\ominus}$.","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics To convert from L^2bar/mol^2 to m^6 Pa/mol^2, divide by 10. a (L2bar/mol2) b (L/mol) Acetic acid 17.7098 0.1065 Acetic anhydride 20.158 0.1263 Acetone 16.02 0.1124 Acetonitrile 17.81 0.1168 Acetylene 4.516 0.0522 Ammonia 4.225 0.0371 Aniline 29.14 0.1486 Argon 1.355 0.03201 Benzene 18.24 0.1193 Bromobenzene 28.94 0.1539 Butane 14.66 0.1226 1-Butanol 20.94 0.1326 2-Butanone 19.97 0.1326 Carbon dioxide 3.640 0.04267 Carbon disulfide 11.77 0.07685 Carbon monoxide 1.505 0.0398500 Carbon tetrachloride 19.7483 0.1281 Chlorine 6.579 0.05622 Chlorobenzene 25.77 0.1453 Chloroethane 11.05 0.08651 Chloromethane 7.570 0.06483 Cyanogen 7.769 0.06901 Cyclohexane 23.11 0.1424 Cyclopropane 8.34 0.0747 Decane 52.74 0.3043 1-Decanol 59.51 0.3086 Diethyl ether 17.61 0.1344 Diethyl sulfide 19.00 0.1214 Dimethyl ether 8.180 0.07246 Dimethyl sulfide 13.04 0.09213 Dodecane 69.38 0.3758 1-Dodecanol 75.70 0.3750 Ethane 5.562 0.0638 Ethanethiol 11.39 0.08098 Ethanol 12.18 0.08407 Ethyl acetate 20.72 0.1412 Ethylamine 10.74 0.08409 Ethylene 4.612 0.0582 Fluorine 1.171 0.0290 Fluorobenzene 20.19 0.1286 Fluoromethane 4.692 0.05264 Freon 10.78 0.0998 Furan 12.74 0.0926 Germanium tetrachloride 22.90 0.1485 Helium 0.0346 0.0238 Heptane 31.06 0.2049 1-Heptanol 38.17 0.2150 Hexane 24.71 0.1735 1-Hexanol 31.79 0.1856 Hydrazine 8.46 0.0462 Hydrogen 0.2476 0.02661 Hydrogen bromide 4.510 0.04431 Hydrogen chloride 3.716 0.04081 Hydrogen cyanide 11.29 0.0881 Hydrogen fluoride 9.565 0.0739 Hydrogen iodide 6.309 0.0530 Hydrogen selenide 5.338 0.04637 Hydrogen sulfide 4.490 0.04287 Isobutane 13.32 0.1164 Iodobenzene 33.52 0.1656 Krypton 2.349 0.03978 Mercury 8.200 0.01696 Methane 2.283 0.04278 Methanol 9.649 0.06702 Methylamine 7.106 0.0588 Neon 0.2135 0.01709 Neopentane 17.17 0.1411 Nitric oxide 1.358 0.02789 Nitrogen 1.370 0.0387 Nitrogen dioxide 5.354 0.04424 Nitrogen trifluoride 3.58 0.0545 Nitrous oxide 3.832 0.04415 Octane 37.88 0.2374 1-Octanol 44.71 0.2442 Oxygen 1.382 0.03186 Ozone 3.570 0.0487 Pentane 19.26 0.146 1-Pentanol 25.88 0.1568 Phenol 22.93 0.1177 Phosphine 4.692 0.05156 Propane 8.779 0.08445 1-Propanol 16.26 0.1079 2-Propanol 15.82 0.1109 Propene 8.442 0.0824 Pyridine 19.77 0.1137 Pyrrole 18.82 0.1049 Radon 6.601 0.06239 Silane 4.377 0.05786 Silicon tetrafluoride 4.251 0.05571 Sulfur dioxide 6.803 0.05636 Sulfur hexafluoride 7.857 0.0879 Tetrachloromethane 20.01 0.1281 Tetrachlorosilane 20.96 0.1470 Tetrafluoroethylene 6.954 0.0809 Tetrafluoromethane 4.040 0.0633 Tetrafluorosilane 5.259 0.0724 Tetrahydrofuran 16.39 0.1082 Tin tetrachloride 27.27 0.1642 Thiophene 17.21 0.1058 Toluene 24.38 0.1463 1-1-1-Trichloroethane 20.15 0.1317 Trichloromethane 15.34 0.1019 Trifluoromethane 5.378 0.0640 Trimethylamine 13.37 0.1101 Water 5.536 0.03049 Xenon 4.250 0.05105 ==Units== 1 J·m3/mol2 = 1 m6·Pa/mol2 = 10 L2·bar/mol2 1 L2atm/mol2 = 0.101325 J·m3/mol2 = 0.101325 Pa·m6/mol2 1 dm3/mol = 1 L/mol = 1 m3/kmol (where kmol is kilomoles = 1000 moles) ==References== Category:Gas laws Constants (Data Page) :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. This disparity is not a significant departure from accuracy, and USSA1976 uses this value of R∗ for all the calculations of the standard atmosphere. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. ",0.11,209.1,"""-1368.0""",1.51,1.88,C
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $w$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Therefore, the heat capacity ratio in this example is 1.4. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi- quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). ",0,11000,"""15.0""",8.87,0.1792,A
+The density of a gaseous compound was found to be $1.23 \mathrm{kg} \mathrm{m}^{-3}$ at $330 \mathrm{K}$ and $20 \mathrm{kPa}$. What is the molar mass of the compound?,"The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule: :\begin{array}{lll} M(\ce{H2}) &= 2\times 1.00797(7) \times M_\mathrm{u} &= 2.01588(14) \text{ g/mol} \\\ M(\ce{S8}) &= 8\times 32.065(5) \times M_\mathrm{u} &= 256.52(4) \text{ g/mol} \\\ M(\ce{Cl2}) &= 2\times 35.453(2) \times M_\mathrm{u} &= 70.906(4) \text{ g/mol} \end{array} == Molar masses of compounds == The molar mass of a compound is given by the sum of the relative atomic mass of the atoms which form the compound multiplied by the molar mass constant : :M = M_{\rm u} M_{\rm r} = M_{\rm u} \sum_i {A_{\rm r}}_i. Examples are: \begin{array}{ll} M(\ce{NaCl}) &= \bigl[22.98976928(2) + 35.453(2)\bigr] \times 1 \text{ g/mol} \\\ &= 58.443(2) \text{ g/mol} \\\\[4pt] M(\ce{C12H22O11}) &= \bigl[12 \times 12.0107(8) + 22 \times 1.00794(7) + 11 \times 15.9994(3)\bigr] \times 1 \text{ g/mol} \\\ &= 342.297(14) \text{ g/mol} \end{array} An average molar mass may be defined for mixtures of compounds. In chemistry, the molar mass () of a chemical compound is defined as the ratio between the mass and the amount of substance (measured in moles) of any sample of said compound. The molar mass of a compound in g/mol thus is equal to the mass of this number of molecules of the compound in grams. == Molar masses of elements == The molar mass of atoms of an element is given by the relative atomic mass of the element multiplied by the molar mass constant, For normal samples from earth with typical isotope composition, the atomic weight can be approximated by the standard atomic weight or the conventional atomic weight. :\begin{array}{lll} M(\ce{H}) &= 1.00797(7) \times M_\mathrm{u} &= 1.00797(7) \text{ g/mol} \\\ M(\ce{S}) &= 32.065(5) \times M_\mathrm{u} &= 32.065(5) \text{ g/mol} \\\ M(\ce{Cl}) &= 35.453(2) \times M_\mathrm{u} &= 35.453(2) \text{ g/mol} \\\ M(\ce{Fe}) &= 55.845(2) \times M_\mathrm{u} &= 55.845(2) \text{ g/mol} \end{array} Multiplying by the molar mass constant ensures that the calculation is dimensionally correct: standard relative atomic masses are dimensionless quantities (i.e., pure numbers) whereas molar masses have units (in this case, grams per mole). The molar mass is an average of many instances of the compound, which often vary in mass due to the presence of isotopes. As an example, the average molar mass of dry air is 28.97 g/mol.The Engineering ToolBox Molecular Mass of Air == Related quantities == Molar mass is closely related to the relative molar mass () of a compound, to the older term formula weight (F.W.), and to the standard atomic masses of its constituent elements. Thus, for example, the average mass of a molecule of water is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol. Molar masses typically vary between: :1–238 g/mol for atoms of naturally occurring elements; : for simple chemical compounds; : for polymers, proteins, DNA fragments, etc. * Molar mass: chemistry second-level course. If represents the mass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by :M = {{wK_\text{b}}\over{\Delta T}}.\ == See also == * Mole map (chemistry) == References == ==External links== * HTML5 Molar Mass Calculator web and mobile application. The mass average molar mass is calculated by \bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} where is the number of molecules of molecular mass . The molar mass is appropriate for converting between the mass of a substance and the amount of a substance for bulk quantities. * Online Molar Mass Calculator with the uncertainty of M and all the calculations shown * Molar Mass Calculator Online Molar Mass and Elemental Composition Calculator * Stoichiometry Add-In for Microsoft Excel for calculation of molecular weights, reaction coefficients and stoichiometry. The molecular formula C23H21NO (molar mass: 327.42 g/mol, exact mass: 327.1623 u) may refer to: * JWH-015 * JWH-073 * JWH-120 Category:Molecular formulas In the International System of Units (SI), the coherent unit of molar mass is kg/mol. The molecular formula C13H28O (molar mass: 200.36 g/mol, exact mass: 200.2140 u) may refer to: * 2,2,4,4-Tetramethyl-3-t-butyl-pentane-3-ol, or tri-tert- butylcarbinol * 1-Tridecanol In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species () and the molar mass () of that species.I. Katime ""Química Física Macromolecular"". The molar mass of any element or compound is its relative atomic mass (atomic weight) multiplied by the molar mass constant. Here, is the relative molar mass, also called formula weight. Gram atomic mass is another term for the mass, in grams, of one mole of atoms of that element. The molar mass constant was thus given by :M_{\text{u}} = {\text{molar mass }[M( ^{12}\mathrm{C} )]\over \text{relative atomic weight }[A_{\text{r}}( ^{12}\mathrm{C} )]} = {{12\ {\rm g/mol}}\over 12}=1\ \rm g/mol The molar mass constant is related to the mass of a carbon-12 atom in grams: :m({}^{12}{\text{C}}) = \frac{12 \times M_{\text{u}}}{N_{\text{A}}} The Avogadro constant being a fixed value, the mass of a carbon-12 atom depends on the accuracy and precision of the molar mass constant. The molecular formula C12H15N5O3 (molar mass: 277.28 g/mol, exact mass: 277.1175 u) may refer to: * Entecavir (ETV) * Queuine (Q) ",169, 4.56,"""258.14""",1.8,57.2,A
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of $4.50 \mathrm{~g}$ of methane occupies $12.7 \mathrm{dm}^3$ at $310 \mathrm{~K}$. Calculate the work that would be done if the same expansion occurred reversibly.","In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. We assume the expansion occurs without exchange of heat (adiabatic expansion). The reaction depends on a delicate balance between methane pressure and catalyst concentration, and consequently more work is being done to further improve yields. ==References== Category:Organometallic chemistry Category:Organic chemistry Category:Chemistry Category:Methane The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. A methane reformer is a device based on steam reforming, autothermal reforming or partial oxidation and is a type of chemical synthesis which can produce pure hydrogen gas from methane using a catalyst. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. An experimental value should be used rather than one based on this approximation, where possible. Methane functionalization is the process of converting methane in its gaseous state to another molecule with a functional group, typically methanol or acetic acid, through the use of transition metal catalysts. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. For example, a comparison of calculations for one compression stage of an axial compressor (one with variable C_P and one with constant C_P) may produce a deviation small enough to support this approach. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . ",-167,1.8,"""6.64""", 0.01961,0.396,A
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A strip of magnesium of mass $15 \mathrm{~g}$ is placed in a beaker of dilute hydrochloric acid. Calculate the work done by the system as a result of the reaction. The atmospheric pressure is 1.0 atm and the temperature $25^{\circ} \mathrm{C}$.","The quantity of thermodynamic work is defined as work done by the system on its surroundings. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. Such work done by compression is thermodynamic work as here defined. An electric discharge through hydrogen gas at low pressure (20 pascals) containing pieces of magnesium can produce MgH. The reaction that produces it is either or Mg + H → MgH. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. The reaction of Mg atoms with (dihydrogen gas) is actually endothermic and proceeds when magnesium atoms are excited electronically. Otherwise in these stars, below any magnesium silicate clouds where the temperature is hotter, the concentration of MgH is proportional to the square root of the pressure, and concentration of magnesium, and 10−4236/T. MgH is the second most abundant magnesium containing gas (after atomic magnesium) in the deeper hotter parts of planets and brown dwarfs. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. The work is due to change of system volume by expansion or contraction of the system. First, one assumes that the given reaction at constant temperature and pressure is the only one that is occurring. Bulk properties of the MgH gas include enthalpy of formation of 229.79 kJ mol−1, entropy 193.20 J K−1 mol−1 and heat capacity of 29.59 J K−1 mol−1. Magnesium monohydride is a molecular gas with formula MgH that exists at high temperatures, such as the atmospheres of the Sun and stars. A complete reaction takes 20 to 24 hours at 1,200 °C."" Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. Several kinds of thermodynamic work are especially important. Atmospheric Thermodynamics. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Both the temperature change ∆T of the water and the height of the fall ∆h of the weight mg were recorded. As a result, the work done by the system also depends on the initial and final states. ",17.4,-1.5,"""1.5""",14.34457,48,B
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. For a van der Waals gas, $\pi_T=a / V_{\mathrm{m}}^2$. Calculate $\Delta U_{\mathrm{m}}$ for the isothermal expansion of nitrogen gas from an initial volume of $1.00 \mathrm{dm}^3$ to $24.8 \mathrm{dm}^3$ at $298 \mathrm{~K}$.","The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density. ===Statistical thermodynamics derivation=== The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is: : Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3} where \Lambda is the thermal de Broglie wavelength, : \Lambda = \sqrt{\frac{h^2}{2\pi m k T}} with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the Van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the p–V diagram as the ratio of vapour to liquid changes. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Van der Waals' doctoral research culminated in an 1873 dissertation that provided a semi- quantitative theory describing the gas-liquid change of state and the origin of a critical temperature, Over de Continuïteit van den Gas- en Vloeistoftoestand (Dutch; in English, On the Continuity of the Gas and Liquid State). The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ",313,122,"""0.6""",3.333333333,131,E
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Take nitrogen to be a van der Waals gas with $a=1.352 \mathrm{dm}^6 \mathrm{~atm} \mathrm{\textrm {mol } ^ { - 2 }}$ and $b=0.0387 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and calculate $\Delta H_{\mathrm{m}}$ when the pressure on the gas is decreased from $500 \mathrm{~atm}$ to $1.00 \mathrm{~atm}$ at $300 \mathrm{~K}$. For a van der Waals gas, $\mu=\{(2 a / R T)-b\} / C_{p, \mathrm{~m}}$. Assume $C_{p, \mathrm{~m}}=\frac{7}{2} R$.","Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",30,+3.60,"""0.3333333""",140,3.07,B
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the molar entropy of a constant-volume sample of neon at $500 \mathrm{~K}$ given that it is $146.22 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}$.,"In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Substituting this into the above equation along with V=[\mathrm{g}]/\rho\, and \gamma = 5/3\, for an ideal monatomic gas one finds : K = \frac{k_{B}T}{(\rho/\mu m_{H})^{2/3}}, where \mu\, is the mean molecular weight of the gas or plasma; and m_{H}\, is the mass of the Hydrogen atom, which is extremely close to the mass of the proton, m_{p}\,, the quantity more often used in astrophysical theory of galaxy clusters. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, i.e. the pressure–volume product, rather than energy per temperature increment per particle. As a consequence, the SI value of the molar gas constant is exactly . This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Instead of a mole the constant can be expressed by considering the normal cubic meter. * The heat capacity of the gas from the boiling point to room temperature. The law can also be written as a function of the total number of atoms N in the sample: :C/N = 3k_{\rm B}, where kB is Boltzmann constant. ==Application limits== thumb|upright=2.2|The molar heat capacity plotted of most elements at 25°C plotted as a function of atomic number. Otherwise, we can also say that: :\mathrm{force} = \frac{ \mathrm{mass} \times \mathrm{length} } { (\mathrm{time})^2 } Therefore, we can write R as: :R = \frac{ \mathrm{mass} \times \mathrm{length}^2 } { \mathrm{amount} \times \mathrm{temperature} \times (\mathrm{time})^2 } And so, in terms of SI base units: :R = . ==Relationship with the Boltzmann constant== The Boltzmann constant kB (alternatively k) may be used in place of the molar gas constant by working in pure particle count, N, rather than amount of substance, n, since :R = N_{\rm A} k_{\rm B},\, where NA is the Avogadro constant. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). ",38,2.3,"""152.67""",-59.24,-2,C
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final temperature of a sample of argon of mass $12.0 \mathrm{~g}$ that is expanded reversibly and adiabatically from $1.0 \mathrm{dm}^3$ at $273.15 \mathrm{~K}$ to $3.0 \mathrm{dm}^3$.","The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). We assume the expansion occurs without exchange of heat (adiabatic expansion). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The following figure illustrates that the effects of dissociation tend to lower the adiabatic flame temperature. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. thumb|upright|Ethanol burning with its spectrum depicted In the study of combustion, the adiabatic flame temperature is the temperature reached by a flame under ideal conditions. Temperatures in the atmosphere decrease with height at an average rate of 6.5°C (11.7°F) per kilometer. Adiabatic flame temperature (constant pressure) of common fuels Fuel Oxidizer 1 bar 20 °C T_\text{ad} T_\text{ad} Fuel Oxidizer 1 bar 20 °C (°C) (°F) Acetylene () Air 2500 4532 Acetylene () Oxygen 3480 6296 Butane () Air 2231 4074 Cyanogen () Oxygen 4525 8177 Dicyanoacetylene () Oxygen 4990 9010 Ethane () Air 1955 3551 Ethanol () Air 2082 3779Flame Temperature Analysis and NOx Emissions for Different Fuels Gasoline Air 2138 3880 Hydrogen () Air 2254 4089 Magnesium (Mg) Air 1982 3600 Methane () Air 1963 3565CRC Handbook of Chemistry and Physics, 96th Edition, p. 15-51 Methanol () Air 1949 3540 Naphtha Air 2533 4591 Natural gas Air 1960 3562 Pentane () Air 1977 3591 Propane () Air 1980 3596 Methylacetylene () Air 2010 3650 Methylacetylene () Oxygen 2927 5301 Toluene () Air 2071 3760 Wood Air 1980 3596 Kerosene Air 2093Power Point Presentation: Flame Temperature, Hsin Chu, Department of Environmental Engineering, National Cheng Kung University, Taiwan 3801 Light fuel oil Air 2104 3820 Medium fuel oil Air 2101 3815 Heavy fuel oil Air 2102 3817 Bituminous Coal Air 2172 3943 Anthracite Air 2180 3957 Anthracite Oxygen ≈3500Analysis of oxy-fuel combustion power cycle utilizing a pressurized coal combustor by Jongsup Hong et al., MIT, which cites . ",131,2,"""3.51""",432,14,A
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in entropy when $25 \mathrm{~kJ}$ of energy is transferred reversibly and isothermally as heat to a large block of iron at $0^{\circ} \mathrm{C}$.,"However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. * The heat capacity of the gas from the boiling point to room temperature. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Equilibrium occurs when the temperature is equal to the melting point T = T_f so that :\Delta G_{\text{fus}} = \Delta H_{\text{fus}} - T_f \times \Delta S_{\text{fus}} = 0, and the entropy of fusion is the heat of fusion divided by the melting point: : \Delta S_{\text{fus}} = \frac {\Delta H_{\text{fus}}} {T_f} ==Helium== Helium-3 has a negative entropy of fusion at temperatures below 0.3 K. Helium-4 also has a very slightly negative entropy of fusion below 0.8 K. In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The concept of entropy developed in response to the observation that a certain amount of functional energy released from combustion reactions is always lost to dissipation or friction and is thus not transformed into useful work. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. In thermodynamics, the entropy of fusion is the increase in entropy when melting a solid substance. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). Thus, using the above description, we can calculate the entropy change ΔS for the passage of the quantity of heat Q from the temperature T1, through the ""working body"" of fluid, which was typically a body of steam, to the temperature T2 as shown below: If we make the assignment: : S= \frac {Q}{T} Then, the entropy change or ""equivalence-value"" for this transformation is: : \Delta S = S_{\rm final} - S_{\rm initial} \, which equals: : \Delta S = \left(\frac {Q}{T_2} - \frac {Q}{T_1}\right) and by factoring out Q, we have the following form, as was derived by Clausius: : \Delta S = Q\left(\frac {1}{T_2} - \frac {1}{T_1}\right) ==1856 definition== In 1856, Clausius stated what he called the ""second fundamental theorem in the mechanical theory of heat"" in the following form: :\int \frac{\delta Q}{T} = -N where N is the ""equivalence-value"" of all uncompensated transformations involved in a cyclical process. Changes in entropy are associated with phase transitions and chemical reactions. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics Jaynes (1957) Information theory and statistical mechanics II, Physical Review 108:171 the statistical thermodynamic entropy can be seen as just a particular application of Shannon's information entropy to the probabilities of particular microstates of a system occurring in order to produce a particular macrostate. ==Popular use== The term entropy is often used in popular language to denote a variety of unrelated phenomena. ",0,-3.0,"""5.85""",92,10,D
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\Delta H$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Therefore, the heat capacity ratio in this example is 1.4. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. ",30,5,"""205.0""",0.66666666666,+3.03,E
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Concerns over the harmful effects of chlorofluorocarbons on stratospheric ozone have motivated a search for new refrigerants. One such alternative is 2,2-dichloro-1,1,1-trifluoroethane (refrigerant 123). Younglove and McLinden published a compendium of thermophysical properties of this substance (J. Phys. Chem. Ref. Data 23, 7 (1994)), from which properties such as the Joule-Thomson coefficient $\mu$ can be computed. Compute $\mu$ at 1.00 bar and $50^{\circ} \mathrm{C}$ given that $(\partial H / \partial p)_T=-3.29 \times 10^3 \mathrm{~J} \mathrm{MPa}^{-1} \mathrm{~mol}^{-1}$ and $C_{p, \mathrm{~m}}=110.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$.","Property Value Ozone depletion potential (ODP) 0.44 (CCl3F = 1) Global warming potential (GWP: 100-year) 5,860 \- 7,670 (CO2 = 1) Atmospheric lifetime 1,020 \- 1,700 years == See also == * IPCC list of greenhouse gases * List of refrigerants == References == Category:Ozone depletion Category:Greenhouse gases Category:Refrigerants Category:Chlorofluorocarbons ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases 1,1,1,2-Tetrafluoroethane (also known as norflurane (INN), R-134a, Klea®134a, Freon 134a, Forane 134a, Genetron 134a, Green Gas, Florasol 134a, Suva 134a, or HFC-134a) is a hydrofluorocarbon (HFC) and haloalkane refrigerant with thermodynamic properties similar to R-12 (dichlorodifluoromethane) but with insignificant ozone depletion potential and a lower 100-year global warming potential (1,430, compared to R-12's GWP of 10,900). Even though 1,1,1,2-Tetrafluoroethane has insignificant ozone depletion potential (ozone layer) and negligible acidification potential (acid rain), it has a 100-year global warming potential (GWP) of 1430 and an approximate atmospheric lifetime of 14 years. Retrieved 21 August 2011. == History and environmental impacts == 1,1,1,2-Tetrafluoroethane was introduced in the early 1990s as a replacement for dichlorodifluoromethane (R-12), which has massive ozone depleting properties. Some of the properties of cyclic ozone have been predicted theoretically. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. This colorless gas is of interest as a more environmentally friendly (lower GWP; global warming potential) refrigerant in air conditioners. It has the formula CFCHF and a boiling point of −26.3 °C (−15.34 °F) at atmospheric pressure. The Society of Automotive Engineers (SAE) has proposed that it be best replaced by a new fluorochemical refrigerant HFO-1234yf (CFCF=CH) in automobile air-conditioning systems.HFO-1234yf A Low GWP Refrigerant For MAC . Chloropentafluoroethane is a chlorofluorocarbon (CFC) once used as a refrigerant and also known as R-115 and CFC-115. A phaseout and transition to HFO-1234yf and other refrigerants, with GWPs similar to CO2, began in 2012 within the automotive market. == Uses == 1,1,1,2-Tetrafluoroethane is a non-flammable gas used primarily as a ""high-temperature"" refrigerant for domestic refrigeration and automobile air conditioners. It has also been studied as a potential inhalational anesthetic, but it is nonanaesthetic at doses used in inhalers. == See also == * List of refrigerants * Tetrabromoethane * Tetrachloroethane == References == == External links == * * European Fluorocarbons Technical Committee (EFCTC) * MSDS at Oxford University * Concise International Chemical Assessment Document 11, at inchem.org * Pressure temperature calculator * * R134a 2 phase computer cooling Category:Fluoroalkanes Category:Refrigerants Category:Automotive chemicals Category:Propellants Category:Airsoft Category:Excipients Category:Greenhouse gases Category:GABAA receptor positive allosteric modulators Category:General anesthetics It should have more energy than ordinary ozone. It has been speculated that, if cyclic ozone could be made in bulk, and if it proved to have good stability properties, it could be added to liquid oxygen to improve the specific impulse of rocket fuel. 1-Chloro-3,3,3-trifluoropropene (HFO-1233zd) is the unsaturated chlorofluorocarbon with the formula HClC=C(H)CF3. Currently, the possibility of cyclic ozone is confirmed within diverse theoretical approaches. ==References== ==External links== * Category:Allotropes of oxygen Category:Hypothetical chemical compounds Category:Three-membered rings Category:Homonuclear triatomic molecules It would differ from ordinary ozone in how those three oxygen atoms are arranged. Its production and consumption has been banned since 1 January 1996 under the Montreal Protocol because of its high ozone depletion potential and very long lifetime when released into the environment.Ozone Depleting Substances List (Montreal Protocol) CFC-115 is also a potent greenhouse gas. ==Atmospheric properties== The atmospheric abundance of CFC-115 rose from 8.4 parts per trillion (ppt) in year 2010 to 8.7 ppt in 2020 based on analysis of air samples gathered from sites around the world. In ordinary ozone, the atoms are arranged in a bent line; in cyclic ozone, they would form an equilateral triangle. Thus it was included in the IPCC list of greenhouse gases. thumb|left|200px|HFC-134a atmospheric concentration since year 1995. Cyclic ozone has not been made in bulk, although at least one researcher has attempted to do so using lasers. ",200,29.9,"""252.8""",6.6,0.84,B
+A gas at $250 \mathrm{~K}$ and $15 \mathrm{~atm}$ has a molar volume 12 per cent smaller than that calculated from the perfect gas law. Calculate the molar volume of the gas.,"The molar volume of an ideal gas at 100 kPa (1 bar) is : at 0 °C, : at 25 °C. The molar volume of an ideal gas at 1 atmosphere of pressure is : at 0 °C, : at 25 °C. == Crystalline solids == For crystalline solids, the molar volume can be measured by X-ray crystallography. This gives rise to the molar volume of a gas, which at STP (273.15 K, 1 atm) is about 22.4 L. This follows from above where the specific volume is the reciprocal of the density of a substance: V_{\rm m,i} = {M_i \over \rho_i^0} = M_i v_i == Ideal gases == For ideal gases, the molar volume is given by the ideal gas equation; this is a good approximation for many common gases at standard temperature and pressure. It shows the relationship between the pressure, volume, and temperature for a fixed mass (quantity) of gas: :PV = k_5 T This can also be written as: : \frac {P_1V_1}{T_1}= \frac {P_2V_2}{T_2} With the addition of Avogadro's law, the combined gas law develops into the ideal gas law: :PV = nRT :where :*P is pressure :*V is volume :*n is the number of moles :*R is the universal gas constant :*T is temperature (K) :The proportionality constant, now named R, is the universal gas constant with a value of 8.3144598 (kPa∙L)/(mol∙K). The ideal gas equation can be rearranged to give an expression for the molar volume of an ideal gas: V_{\rm m} = \frac{V}{n} = \frac{RT}{P} Hence, for a given temperature and pressure, the molar volume is the same for all ideal gases and is based on the gas constant: R = , or about . The relation is given by :V \propto n\,, or :\frac{V_1}{n_1}=\frac{V_2}{n_2} \, :where n is equal to the number of molecules of gas (or the number of moles of gas). ==Combined and ideal gas laws== The Combined gas law or General Gas Equation is obtained by combining Boyle's Law, Charles's law, and Gay-Lussac's Law. An equivalent formulation of this law is: :PV = Nk_\text{B}T :where :*P is the pressure :*V is the volume :*N is the number of gas molecules :*kB is the Boltzmann constant (1.381×10−23J·K−1 in SI units) :*T is the temperature (K) These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). It is equal to the molar mass (M) divided by the mass density (ρ): V_{\text{m}} = \frac{M}{\rho} The molar volume has the SI unit of cubic metres per mole (m3/mol), although it is more typical to use the units cubic decimetres per mole (dm3/mol) for gases, and cubic centimetres per mole (cm3/mol) for liquids and solids. ==Definition== thumb|Change in volume with increasing ethanol fraction. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature. # If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. The molar volume of a substance i is defined as its molar mass divided by its density ρi0: V_{\rm m,i} = {M_i\over\rho_i^0} For an ideal mixture containing N components, the molar volume of the mixture is the weighted sum of the molar volumes of its individual components. The basic gas laws had been discovered by the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. In chemistry and related fields, the molar volume, symbol Vm, or \tilde V of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. # If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present. He observed that volume of a given mass of a gas is inversely proportional to its pressure at a constant temperature. This law has the following important consequences: # If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas. # If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature. ==Other gas laws== ;Graham's law: states that the rate at which gas molecules diffuse is inversely proportional to the square root of the gas density at constant temperature. The statement of Charles's law is as follows: the volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T). In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. ",1.2,2,"""0.686""",0,22.2036033112,A
+Calculate the mass of water vapour present in a room of volume $400 \mathrm{m}^3$ that contains air at $27^{\circ} \mathrm{C}$ on a day when the relative humidity is 60 per cent.,"The density of humid air is found by:Shelquist, R (2009) Equations - Air Density and Density Altitude \rho_\text{humid air} = \frac{p_\text{d}}{R_\text{d} T} + \frac{p_\text{v}}{R_\text{v} T} = \frac{p_\text{d}M_\text{d} + p_\text{v}M_\text{v}}{R T} where: *\rho_\text{humid air}, density of the humid air (kg/m3) *p_\text{d}, partial pressure of dry air (Pa) *R_\text{d}, specific gas constant for dry air, 287.058J/(kg·K) *T, temperature (K) *p_\text{v}, pressure of water vapor (Pa) *R_\text{v}, specific gas constant for water vapor, 461.495J/(kg·K) *M_\text{d}, molar mass of dry air, 0.0289652kg/mol *M_\text{v}, molar mass of water vapor, 0.018016kg/mol *R, universal gas constant, 8.31446J/(K·mol) The vapor pressure of water may be calculated from the saturation vapor pressure and relative humidity. The theoretical value for water vapor is 19.6, but due to vapor condensation the water vapor density dependence is highly variable and is not well approximated by this formula. This can be seen by using the ideal gas law as an approximation. ==Dry air== The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure: \begin{align} \rho &= \frac{p}{R_\text{specific} T}\\\ R_\text{specific} &= \frac{R}{M} = \frac{k_{\rm B}}{m}\\\ \rho &= \frac{pM}{RT} = \frac{pm}{k_{\rm B}T}\\\ \end{align} where: *\rho, air density (kg/m3)In the SI unit system. The density of humid air may be calculated by treating it as a mixture of ideal gases. This occurs because the molar mass of water vapor (18g/mol) is less than the molar mass of dry airas dry air is a mixture of gases, its molar mass is the weighted average of the molar masses of its components (around 29g/mol). Air is given a vapour density of one. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . * At 20°C and 101.325kPa, dry air has a density of 1.2041 kg/m3. * At 70°F and 14.696psi, dry air has a density of 0.074887lb/ft3. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. It is found by: p_\text{v} = \phi p_\text{sat} where: *p_\text{v}, vapor pressure of water *\phi, relative humidity (0.0–1.0) *p_\text{sat}, saturation vapor pressure The saturation vapor pressure of water at any given temperature is the vapor pressure when relative humidity is 100%. The following table illustrates the air density–temperature relationship at 1 atm or 101.325 kPa: ==Humid air== thumb|right|400px|Effect of temperature and relative humidity on air density The addition of water vapor to air (making the air humid) reduces the density of the air, which may at first appear counter-intuitive. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. Depending on the measuring instruments used, different sets of equations for the calculation of the density of air can be applied. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. However, other units can be used. *p, absolute pressure (Pa) *T, absolute temperature (K) *R is the gas constant, in J⋅K−1⋅mol−1 *M is the molar mass of dry air, approximately in kg⋅mol−1. *k_{\rm B} is the Boltzmann constant, in J⋅K−1 *m is the molecular mass of dry air, approximately in kg. One formula is Tetens' equation fromShelquist, R (2009) Algorithms - Schlatter and Baker used to find the saturation vapor pressure is: p_\text{sat} = 6.1078 \times 10^{\frac{7.5 T}{T + 237.3}} where: *p_\text{sat}, saturation vapor pressure (hPa) *T, temperature (°C) See vapor pressure of water for other equations. Therefore: * At IUPAC standard temperature and pressure (0°C and 100kPa), dry air has a density of approximately 1.2754kg/m3. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). So when water molecules (water vapor) are added to a given volume of air, the dry air molecules must decrease by the same number, to keep the pressure or temperature from increasing. At 101.325kPa (abs) and , air has a density of approximately , which is about that of water, according to the International Standard Atmosphere (ISA). ",22,+116.0,"""3930.0""",6.2,2,D
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The constant-volume heat capacity of a gas can be measured by observing the decrease in temperature when it expands adiabatically and reversibly. If the decrease in pressure is also measured, we can use it to infer the value of $\gamma=C_p / C_V$ and hence, by combining the two values, deduce the constant-pressure heat capacity. A fluorocarbon gas was allowed to expand reversibly and adiabatically to twice its volume; as a result, the temperature fell from $298.15 \mathrm{~K}$ to $248.44 \mathrm{~K}$ and its pressure fell from $202.94 \mathrm{kPa}$ to $81.840 \mathrm{kPa}$. Evaluate $C_p$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). This relation may be used to show the heat capacities may be expressed in terms of the heat capacity ratio () and the gas constant (): : C_P = \frac{\gamma n R}{\gamma - 1} \quad \text{and} \quad C_V = \frac{n R}{\gamma - 1}, === Relation with degrees of freedom === The classical equipartition theorem predicts that the heat capacity ratio () for an ideal gas can be related to the thermally accessible degrees of freedom () of a molecule by : \gamma = 1 + \frac{2}{f},\quad \text{or} \quad f = \frac{2}{\gamma - 1}. Therefore, the heat capacity ratio in this example is 1.4. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). This result can be explained through Le Chatelier's principle. ==See also== *Flame speed ==References== == External links == === General information === * * Computation of adiabatic flame temperature * Adiabatic flame temperature === Tables === * adiabatic flame temperature of hydrogen, methane, propane and octane with oxygen or air as oxidizers * * Temperature of a blue flame and common materials === Calculators === * Online adiabatic flame temperature calculator using Cantera * Adiabatic flame temperature program * Gaseq, program for performing chemical equilibrium calculations. The constant-pressure adiabatic flame temperature of such substances in air is in a relatively narrow range around 1950 °C. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. ",41.40,0.36,"""0.66666666666""",0.011,1.6,A
+Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. Estimate the standard reaction Gibbs energy of $\mathrm{N}_2(\mathrm{g})+3 \mathrm{H}_2(\mathrm{g}) \rightarrow$ $2 \mathrm{NH}_3$ (g) at $500 \mathrm{~K}$.,"Substance (state) ΔfG° ΔfG° Substance (state) (kJ/mol) (kcal/mol) NO(g) 87.6 20.9 NO2(g) 51.3 12.3 N2O(g) 103.7 24.78 H2O(g) −228.6 −54.64 H2O(l) −237.1 −56.67 CO2(g) −394.4 −94.26 CO(g) −137.2 −32.79 CH4(g) −50.5 −12.1 C2H6(g) −32.0 −7.65 C3H8(g) −23.4 −5.59 C6H6(g) 129.7 29.76 C6H6(l) 124.5 31.00 The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). This implies that at equilibrium Q_\text{r} = K_\text{eq} and \Delta_\text{r} G = 0. ==Standard Gibbs energy change of formation== Table of selected substancesCRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide. Moreover, we also have \begin{align} K_\text{eq} &= e^{-\frac{\Delta_\text{r} G^\circ}{RT}}, \\\ \Delta_\text{r} G^\circ &= -RT\left(\ln K_\text{eq}\right) = -2.303\,RT\left(\log_{10} K_\text{eq}\right), \end{align} which relates the equilibrium constant with Gibbs free energy. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The Gibbs energy of any system is and an infinitesimal change in G, at constant temperature and pressure, yields :dG=dU+pdV-TdS. The Gibbs free energy is expressed as : G(p,T) = U + pV - TS = H - TS where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure-volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. ",7,205,"""0.0245""", 9.73,58.2,A
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The enthalpy of vaporization of chloroform $\left(\mathrm{CHCl}_3\right)$ is $29.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at its normal boiling point of $334.88 \mathrm{~K}$. Calculate the entropy of vaporization of chloroform at this temperature.,"J/(mol K) Enthalpy of combustion –473.2 kJ/mol ΔcH ~~o~~ Heat capacity, cp 114.25 J/(mol K) Gas properties Std enthalpy change of formation, ΔfH ~~o~~ gas –103.18 kJ/mol Standard molar entropy, S ~~o~~ gas 295.6 J/(mol K) at 25 °C Heat capacity, cp 65.33 J/(mol K) at 25 °C van der Waals' constantsLange's Handbook of Chemistry 10th ed, pp. 1522–1524 a = 1537 L2 kPa/mol2 b = 0.1022 liter per mole ==Vapor pressure of liquid== P in mm Hg 1 10 40 100 400 760 1520 3800 7600 15200 30400 45600 T in °C –58.0 –29.7 –7.1 10.4 42.7 61.3 83.9 120.0 152.3 191.8 237.5 — Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. thumb|812px|left|log10 of Chloroform vapor pressure. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Uses formula: \scriptstyle \log_e P_{mmHg} =\scriptstyle \log_e (\frac {760} {101.325}) - 10.07089 \log_e(T+273.15) - \frac {6351.140} {T+273.15} + 81.14393 + 9.127608 \times 10^{-6} (T+273.15)^2 obtained from CHERIC ==Distillation data== {| border=""1"" cellspacing=""0"" cellpadding=""6"" style=""margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;"" Vapor-liquid Equilibrium for Chloroform/Ethanol P = 101.325 kPa BP Temp. °C % by mole chloroform liquid vapor 78.15 0.0 0.0 78.07 0.52 1.59 77.83 1.02 3.01 76.81 2.21 7.45 74.90 5.80 17.10 74.39 6.72 19.40 73.55 8.38 23.31 72.85 10.57 28.05 72.24 11.80 30.52 71.58 13.18 33.13 69.72 17.65 41.00 68.95 19.62 43.66 68.58 20.71 45.43 67.35 23.86 49.77 65.89 28.54 55.09 64.87 32.35 58.48 63.88 36.07 61.27 63.23 39.34 64.50 62.61 41.38 65.49 62.17 44.41 67.57 61.48 49.97 71.11 61.00 53.92 72.91 60.49 54.76 73.57 60.35 59.65 74.68 60.30 61.60 75.53 60.20 63.04 76.12 60.09 64.48 76.69 59.97 66.90 77.74 59.54 72.01 79.33 59.32 79.07 82.62 59.26 82.99 83.59 59.28 84.97 84.69 59.31 85.96 85.24 59.46 89.92 87.93 59.72 91.10 88.73 59.70 92.44 89.79 59.84 93.90 91.02 59.91 95.26 92.56 60.18 96.13 93.58 60.88 98.89 97.93 61.13 100.00 100.00 == Spectral data == UV-Vis λmax ? nm Extinction coefficient, ε ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Pa Critical point 537 K (264 °C), 5328.68 kPa Std enthalpy change of fusion, ΔfusH ~~o~~ 8.8 kJ/mol Std entropy change of fusion, ΔfusS ~~o~~ 42 J/(mol·K) Std enthalpy change of vaporization, ΔvapH ~~o~~ 31.4 kJ/mol Std entropy change of vaporization, ΔvapS ~~o~~ 105.3 J/(mol·K) Solid properties Std enthalpy change of formation, ΔfH ~~o~~ solid ? kJ/mol Standard molar entropy, S ~~o~~ solid ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Liquid properties Std enthalpy change of formation, ΔfH ~~o~~ liquid –134.3 kJ/mol Standard molar entropy, S ~~o~~ liquid ? In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. K (? °C), ? K (? °C), ? K (? °C), ? K (? °C), ? This page provides supplementary chemical data on chloroform. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). ", 258.14,+87.8,"""0.33333333""",-0.029,24,B
 "Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K.  Given the reactions (1) and (2) below, determine $\Delta_{\mathrm{r}} H^{\ominus}$ for reaction (3).
 
 (1) $\mathrm{H}_2(\mathrm{g})+\mathrm{Cl}_2(\mathrm{g}) \rightarrow 2 \mathrm{HCl}(\mathrm{g})$, $\Delta_{\mathrm{r}} H^{\ominus}=-184.62 \mathrm{~kJ} \mathrm{~mol}^{-1}$
 
 (2) $2 \mathrm{H}_2(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$, $\Delta_{\mathrm{r}} H^\ominus=-483.64 \mathrm{~kJ} \mathrm{~mol}^{-1}$
 
-(3) $4 \mathrm{HCl}(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{Cl}_2(\mathrm{g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup == Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). K (? °C), ? K (? °C), ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Ellis (Ed.) in Nuffield Advanced Science Book of Data, Longman, London, UK, 1972. == See also == Category:Thermodynamic properties Category:Chemical element data pages Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? ",2.57,0.0761,0.8185,-114.40,4.4,D
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. An average human produces about $10 \mathrm{MJ}$ of heat each day through metabolic activity. If a human body were an isolated system of mass $65 \mathrm{~kg}$ with the heat capacity of water, what temperature rise would the body experience?","The normal human body temperature is often stated as . Human body temperature varies. Normal human body temperature varies slightly from person to person and by the time of day. In humans, the average internal temperature is widely accepted to be 37 °C (98.6 °F), a ""normal"" temperature established in the 1800s. The normal human body temperature range is typically stated as . An individual's body temperature typically changes by about between its highest and lowest points each day. Normal human body-temperature (normothermia, euthermia) is the typical temperature range found in humans. In adults a review of the literature has found a wider range of for normal temperatures, depending on the gender and location measured. While some people think of these averages as representing normal or ideal measurements, a wide range of temperatures has been found in healthy people. The range for normal human body temperatures, taken orally, is . The typical daytime temperatures among healthy adults are as follows: * Temperature in the anus (rectum/rectal), vagina, or in the ear (tympanic) is about * Temperature in the mouth (oral) is about * Temperature under the arm (axillary) is about Generally, oral, rectal, gut, and core body temperatures, although slightly different, are well-correlated. It has been found that physically active individuals have larger changes in body temperature throughout the day. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. The commonly given value 98.6 °F is simply the exact conversion of the nineteenth-century German standard of 37 °C. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. Rectal temperature is expected to be approximately 1 Fahrenheit (or 0.55 Celsius) degree higher than an oral temperature taken on the same person at the same time. The body temperature of a healthy person varies during the day by about with lower temperatures in the morning and higher temperatures in the late afternoon and evening, as the body's needs and activities change. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. The heat uptake results from a persistent warming imbalance in Earth's energy budget that is most fundamentally caused by the anthropogenic increase in atmospheric greenhouse gases. Springer, Cham, 2017. 441-493. == Historical understanding == In the 19th century, most books quoted ""blood heat"" as 98 °F, until a study published the mean (but not the variance) of a large sample as .Inwit Publishing, Inc. and Inwit, LLC – Writings, Links and Software Demonstrations – A Fahrenheit–Celsius Activity, inwit.com Subsequently, that mean was widely quoted as ""37 °C or 98.4 °F""Oxford Dictionary of English, 2010 edition, entry on ""blood heat""Collins English Dictionary, 1979 edition, entry on ""blood heat"" until editors realized 37 °C is equal to 98.6 °F, not 98.4 °F. thumb |upright=1.35 |Upper ocean heat content (OHC) has been increasing because oceans have absorbed a large portion of the excess heat trapped in the atmosphere by human-caused global warming. ",0.11,0.068,37.0,269,0.332,C
-"1.19(a) The critical constants of methane are $p_{\mathrm{c}}=45.6 \mathrm{~atm}, V_{\mathrm{c}}=98.7 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$, and $T_{\mathrm{c}}=190.6 \mathrm{~K}$. Estimate the radius of the molecules.","Methane has a boiling point of −161.5 °C at a pressure of one atmosphere. Methane's heat of combustion is 55.5 MJ/kg.Energy Content of some Combustibles (in MJ/kg) . Cooling methane at normal pressure results in the formation of methane I. However the authors stress ""our findings are preliminary with regard to the methane emission strength"". Note that these are all negative temperature values. thumb|432px|left|Methane vapor pressure vs. temperature. Uses formula \log_{10} P_\text{mm Hg} = 6.61184 - \frac{389.93}{266.00 + T_{^\circ\text{C}}} given in Lange's Handbook of Chemistry, 10th ed. Note that formula loses accuracy near Tcrit = −82.6 °C == Spectral data == thumb|right|450px|Methane infrared spectrum UV-Vis λmax ? nm Extinction coefficient, ε ? {{Chembox | ImageFile1 = File:Tris(dimethylamino)methan Struktur.svg | ImageSize1 = 150px | ImageFile2 = Tris(dimethylamino)methane-3D-balls-by- AHRLS-2012.png | ImageSize2 = 150px | ImageFile3 = Tris(dimethylamino)methane-3D-sticks-by-AHRLS-2012.png | ImageSize3 = 150px | ImageAlt = | PIN = N,N,N,N,N,N-Hexamethylmethanetriamine | OtherNames = N,N,N,N,N,N-hexamethylmethanetriamine [bis(dimethylamino)methyl]dimethylamine | Section1 = | Section2 = | Section3 = }} Tris(dimethylamino)methane (TDAM) is the simplest representative of the tris(dialkylamino)methanes of the general formula (R2N)3CH in which three of the four of methane's hydrogen atoms are replaced by dimethylamino groups (−N(CH3)2). Methane has also been detected on other planets, including Mars, which has implications for astrobiology research. ==Properties and bonding== Methane is a tetrahedral molecule with four equivalent C–H bonds. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Solid methane exists in several modifications. Temperatures in excess of 1200 °C are required to break the bonds of methane to produce Hydrogen gas and solid carbon. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The reaction is moderately endothermic as shown in the reaction equation below. : :( 74.8 kJ/mol) ==Generation== === Geological routes === left|thumb|upright=1.35|Abiotic sources of methane have been found in more than 20 countries and in several deep ocean regions so far.The two main routes for geological methane generation are (i) organic (thermally generated, or thermogenic) and (ii) inorganic (abiotic). Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09 Å for the van der Waals radius of hydrogen as opposed to Bondi's 1.20 Å. Methane at scales of the atmosphere is commonly measured in teragrams (Tg ) or millions of metric tons (MMT ), which mean the same thing. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Methane is an important greenhouse gas with a global warming potential of 40 compared to (potential of 1) over a 100-year period, and above 80 over a 20-year period. At about 891 kJ/mol, methane's heat of combustion is lower than that of any other hydrocarbon, but the ratio of the heat of combustion (891 kJ/mol) to the molecular mass (16.0 g/mol, of which 12.0 g/mol is carbon) shows that methane, being the simplest hydrocarbon, produces more heat per mass unit (55.7 kJ/g) than other complex hydrocarbons. The largest reservoir of methane is under the seafloor in the form of methane clathrates. Methane is a strong GHG with a global warming potential 84 times greater than CO2 in a 20-year time frame. The positions of the hydrogen atoms are not fixed in methane I, i.e. methane molecules may rotate freely. At high pressures, such as are found on the bottom of the ocean, methane forms a solid clathrate with water, known as methane hydrate. ",0.0408,0.05882352941,0.118,1260,22,C
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The enthalpy of vaporization of chloroform $\left(\mathrm{CHCl}_3\right)$ is $29.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at its normal boiling point of $334.88 \mathrm{~K}$. Calculate the entropy change of the surroundings.,"The entropy of vaporization of at its boiling point has the extraordinarily high value of 136.9 J/(K·mol). It is valid for many liquids; for instance, the entropy of vaporization of toluene is 87.30 J/(K·mol), that of benzene is 89.45 J/(K·mol), and that of chloroform is 87.92 J/(K·mol). The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. thumb|The Mollier enthalpy–entropy diagram for water and steam. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? In thermodynamics, Trouton's rule states that the entropy of vaporization is almost the same value, about 85–88 J/(K·mol), for various kinds of liquids at their boiling points.Compare 85 J/(K·mol) in and 88 J/(K·mol) in The entropy of vaporization is defined as the ratio between the enthalpy of vaporization and the boiling temperature. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. For example, the entropies of vaporization of water, ethanol, formic acid and hydrogen fluoride are far from the predicted values. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? The Mollier diagram coordinates are enthalpy h and humidity ratio x. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? K (? °C), ? K (? °C), ? K (? °C), ? ",-87.8,0.139,1.7,1.775,6.0,A
-"Recent communication with the inhabitants of Neptune has revealed that they have a Celsius-type temperature scale, but based on the melting point $(0^{\circ} \mathrm{N})$ and boiling point $(100^{\circ} \mathrm{N})$ of their most common substance, hydrogen. Further communications have revealed that the Neptunians know about perfect gas behaviour and they find that, in the limit of zero pressure, the value of $p V$ is $28 \mathrm{dm}^3$ atm at $0^{\circ} \mathrm{N}$ and $40 \mathrm{dm}^3$ atm at $100^{\circ} \mathrm{N}$. What is the value of the absolute zero of temperature on their temperature scale?","Below 0.9 kelvin at their saturated vapor pressure, a mixture of the two isotopes undergoes a phase separation into a normal fluid (mostly helium-3) that floats on a denser superfluid consisting mostly of helium-4. At these low temperatures, the melting pressure of helium-3 varies from about 2.9 MPa to nearly 4.0 MPa. thumb|0 °C = 32 °F A degree of frost is a non-standard unit of measure for air temperature meaning degrees below melting point (also known as ""freezing point"") of water (0 degrees Celsius or 32 degrees Fahrenheit). Its boiling point and critical point depend on which isotope of helium is present: the common isotope helium-4 or the rare isotope helium-3. Although this gives a disadvantage of non-monotonicity, in that two different temperatures can give the same pressure, the scale is otherwise robust since the melting pressure of helium-3 is insensitive to many experimental factors. == See also == * International Temperature Scale of 1990 (ITS-90) — the calibration standard used for all temperatures above 0.6 K * Leiden scale == References == Category:Temperature Category:Scales of temperature At standard pressure, the chemical element helium exists in a liquid form only at the extremely low temperature of . The zero point energy of liquid helium is less if its atoms are less confined by their neighbors. It is based on the melting pressure of solidified helium-3. When based on Celsius, 0 degrees of frost is the same as 0 °C, and any other value is simply the negative of the Celsius temperature. At the temperature of approximately 315 mK, a minimum of pressure (2.9 MPa) occurs. Because of the very weak interatomic forces in helium, the element remains a liquid at atmospheric pressure all the way from its liquefaction point down to absolute zero. Smaller gas planets and planets closer to their star will lose atmospheric mass more quickly via hydrodynamic escape than larger planets and planets farther out.Mass-radius relationships for exoplanets, Damian C. Swift, Jon Eggert, Damien G. Hicks, Sebastien Hamel, Kyle Caspersen, Eric Schwegler, and Gilbert W. Collins A low-mass gas planet can still have a radius resembling that of a gas giant if it has the right temperature.Mass-Radius Relationships for Very Low Mass Gaseous Planets, Konstantin Batygin, David J. Stevenson, 18 Apr 2013 Neptune-like planets are considerably rarer than sub-Neptunes, despite being only slightly bigger.Superabundance of Exoplanet Sub-Neptunes Explained by Fugacity Crisis, Edwin S. Kite, Bruce Fegley Jr., Laura Schaefer, Eric B. Ford, 5 Dec 2019 This ""radius cliff"" separates sub-Neptunes (radius < 3 Earth radii) from Neptunes (radius > 3 Earth radii). Liquid helium is a physical state of helium at very low temperatures at standard atmospheric pressures. When based on Fahrenheit, 0 degrees of frost is equal to 32 °F. The business was named after the Kelvin temperature scale. The density of liquid helium-4 at its boiling point and a pressure of one atmosphere (101.3 kilopascals) is about , or about one-eighth the density of liquid water. ==Liquefaction== Helium was first liquefied on July 10, 1908, by the Dutch physicist Heike Kamerlingh Onnes at the University of Leiden in the Netherlands. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. thumb|upright=1.1|Artist's conception of a mini-Neptune or ""gas dwarf"" A Mini- Neptune (sometimes known as a gas dwarf or transitional planet) is a planet less massive than Neptune but resembling Neptune in that it has a thick hydrogen–helium atmosphere, probably with deep layers of ice, rock or liquid oceans (made of water, ammonia, a mixture of both, or heavier volatiles). Important early work on the characteristics of liquid helium was done by the Soviet physicist Lev Landau, later extended by the American physicist Richard Feynman. ==Data== Properties of liquid helium Helium-4 Helium-3 Critical temperature Boiling point at one atmosphere Minimum melting pressure at Superfluid transition temperature at saturated vapor pressure 1 mK in the absence of a magnetic field ==Gallery== File:Liquid Helium.jpg|Liquid helium (in a vacuum bottle) at and boiling slowly. See the table below for the values of these physical quantities. The Provisional Low Temperature Scale of 2000 (PLTS-2000) is an equipment calibration standard for making measurements of very low temperatures, in the range of 0.9 mK (millikelvin) to 1 K, adopted by the International Committee for Weights and Measures in October 2000. Liquid helium can be solidified only under very low temperatures and high pressures. ", 135.36,4943,2283.63,-233,47,D
-A gas at $250 \mathrm{~K}$ and $15 \mathrm{~atm}$ has a molar volume 12 per cent smaller than that calculated from the perfect gas law. Calculate the compression factor under these conditions.,"For an ideal gas the compressibility factor is Z=1 per definition. In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. The compression ratio is the ratio between the volume of the cylinder and combustion chamber in an internal combustion engine at their maximum and minimum values. Experimental values for the compressibility factor confirm this. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . Compression ratio is a ratio of volumes. Figure 2 is an example of a generalized compressibility factor graph derived from hundreds of experimental PVT data points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, i-pentane, n-hexane, nitrogen, carbon dioxide and steam. The compressibility factor should not be confused with the compressibility (also known as coefficient of compressibility or isothermal compressibility) of a material, which is the measure of the relative volume change of a fluid or solid in response to a pressure change. ==Definition and physical significance== thumb|A graphical representation of the behavior of gases and how that behavior relates to compressibility factor. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. The closer the gas is to its critical point or its boiling point, the more Z deviates from the ideal case. === Fugacity === The compressibility factor is linked to the fugacity by the relation: : f = P \exp\left(\int \frac{Z-1}{P} dP\right) ==Generalized compressibility factor graphs for pure gases== thumb|400px|Generalized compressibility factor diagram. There are more detailed generalized compressibility factor graphs based on as many as 25 or more different pure gases, such as the Nelson-Obert graphs. * Real Gases includes a discussion of compressibility factors. For a gas that is a mixture of two or more pure gases (air or natural gas, for example), the gas composition must be known before compressibility can be calculated. Above the Boyle temperature, the compressibility factor is always greater than unity and increases slowly but steadily as pressure increases. ==Experimental values== It is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomes important. The reduced specific volume is defined by, : u_R = \frac{ u_\text{actual}}{RT_\text{cr}/P_\text{cr}} where u_\text{actual} is the specific volume. page 140 Once two of the three reduced properties are found, the compressibility chart can be used. The dynamic compression ratio accounts for these factors. In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature, P_r and T_r, are used to normalize the compressibility factor data. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document) In general, deviation from ideal behaviour becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure. Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound- specific empirical constants as input. ",0.5117,10,0.88, -6.04697,7.25,C
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Suppose that $3.0 \mathrm{mmol} \mathrm{N}_2$ (g) occupies $36 \mathrm{~cm}^3$ at $300 \mathrm{~K}$ and expands to $60 \mathrm{~cm}^3$. Calculate $\Delta G$ for the process.,"The value of ΔL must however be taken from the relevant cartridge data information in the C.I.P. Tables. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced ""delta fifteen n"") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. The delta L problem (ΔL problem) refers to certain firearm chambers and the incompatibility of some ammunition made for that chamber. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. Instead of a mole the constant can be expressed by considering the normal cubic meter. Firearms users that have to rely on their weapon under adverse conditions, such as big five and other dangerous game hunters, obviously have to check the correct functioning of the firearm and ammunition they intend to use before exposing themselves to potentially dangerous situations. ==Delta L (ΔL) problem== The length specification ""S"" is a basic dimension (or a datum reference) for the computation of the dimensions of firearms cartridges and chambers. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). In general the first law requires that : Q = \Delta U + W (work) If W is only pressure–volume work, then at constant pressure : Q_P = \Delta U + P \Delta V Assuming that the change in state variables is due solely to a chemical reaction, we have : Q_P = \sum U_{products} - \sum U_{reactants} + P \left(\sum V_{products} - \sum V_{reactants}\right) : Q_P = \sum \left(U_{products} + P V_{products} \right) - \sum \left(U_{reactants} + P V_{reactants} \right) As enthalpy or heat content is defined by H = U + PV , we have : Q_P = \sum H_{products} - \sum H_{reactants} = \Delta H By convention, the enthalpy of each element in its standard state is assigned a value of zero. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . When only expansion work is possible for a process we have \Delta U = Q_V; this implies that the heat of reaction at constant volume is equal to the change in the internal energy \Delta U of the reacting system. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . Since the atmosphere is 99.6337% 14N and 0.3663% 15N, a is 0.003663 in the former case and 0.003663/0.996337 = 0.003676 in the latter. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. For this reason it is important to note which standard concentration value is being used when consulting tables of enthalpies of formation. ==Introduction== Two initial thermodynamic systems, each isolated in their separate states of internal thermodynamic equilibrium, can, by a thermodynamic operation, be coalesced into a single new final isolated thermodynamic system. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. ",257,0.0761,0.082,13,-3.8,E
-1.18(a) A vessel of volume $22.4 \mathrm{dm}^3$ contains $2.0 \mathrm{~mol} \mathrm{H}_2$ and $1.0 \mathrm{~mol} \mathrm{~N}_2$ at $273.15 \mathrm{~K}$. Calculate their total pressure.,"With the ""area"" in the numerator and the ""area"" in the denominator canceling each other out, we are left with :\text{pressure} = \text{weight density} \times \text{depth}. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. * IS 2825–1969 (RE1977)_code_unfired_Pressure_vessels. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. The total pressure of a liquid, then, is ρgh plus the pressure of the atmosphere. Flows with a higher Mach number M cannot approximate the total pressure using the incompressible formula given above. * P_0 is the pressure at the surface. * \rho(z) is the density of the material above the depth z. * g is the gravity acceleration in m/s^2 . upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. Atmospheric pressure pressing on the surface of a liquid must be taken into account when trying to discover the total pressure acting on a liquid. Then we have :\text{pressure} = \frac{\text{force}}{\text{area}} = \frac{\text{weight}}{\text{area}} = \frac{\text{weight density} \times \text{volume}}{\text{area}}, :\text{pressure} = \frac{\text{weight density} \times \text{(area} \times \text{depth)}}{\text{area}}. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation , where g is the gravitational acceleration. Pressure is related to energy density and may be expressed in units such as joules per cubic metre (J/m3, which is equal to Pa). Derivation of this equation This is derived from the definitions of pressure and weight density. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. For a cylinder with hemispherical ends, :M = 2 \pi R^2 (R + W) P {\rho \over \sigma}, where *R is the Radius (m) *W is the middle cylinder width only, and the overall width is W + 2R (m) ====Cylindrical vessel with semi-elliptical ends==== In a vessel with an aspect ratio of middle cylinder width to radius of 2:1, :M = 6 \pi R^3 P {\rho \over \sigma}. ====Gas storage==== In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. The pressure coefficient is used in aerodynamics and hydrodynamics. The CGS unit of pressure is the barye (Ba), equal to 1 dyn·cm−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm2 or kg/cm2) and the like without properly identifying the force units. * EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC. Further the volume of the gas is (4πr3)/3. Almost all pressure vessel design standards contain variations of these two formulas with additional empirical terms to account for variation of stresses across thickness, quality control of welds and in-service corrosion allowances. Pressure is a scalar quantity. ",3930,0.21,524.0,3.0,-131.1,D
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the total change in entropy, when a sample of nitrogen gas of mass $14 \mathrm{~g}$ at $298 \mathrm{~K}$ and $1.00 \mathrm{bar}$ doubles its volume in an adiabatic reversible expansion.","After the removal of the partition, the n_i = nx_i moles of component i may explore the combined volume V\,, which causes an entropy increase equal to nx_i R \ln(V/V_i) = - nR x_i \ln x_i for each component gas. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. In this case, the increase in entropy is entirely due to the irreversible processes of expansion of the two gases, and involves no heat or work flow between the system and its surroundings. ===Gibbs free energy of mixing=== The Gibbs free energy change \Delta G_\text{mix} = \Delta H_\text{mix} - T\Delta S_\text{mix} determines whether mixing at constant (absolute) temperature T and pressure p is a spontaneous process. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). It states that the entropy of such ""mixing"" of perfect gases is zero. ====Mixing at constant total volume and changing partial volumes, with mechanically controlled varying pressure, and constant temperature==== An experimental demonstration may be considered. Upon removal of the dividing partition, they expand into a final common volume (the sum of the two initial volumes), and the entropy of mixing \Delta S_{mix} is given by :\Delta S_{mix} = -nR(x_1\ln x_1 + x_2\ln x_2)\,. where R is the gas constant, n the total number of moles and x_i the mole fraction of component i\,, which initially occupies volume V_i = x_iV\,. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . We can use Boltzmann's equation for the entropy change as applied to the mixing process :\Delta S_\text{mix}= k_\text{B} \ln\Omega where k_\text{B} is the Boltzmann constant. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The lowest value is when the mole fraction is 0.5 for a mixture of two components, or 1/n for a mixture of n components. thumb|The entropy of mixing for an ideal solution of two species is maximized when the mole fraction of each species is 0.5. ==Solutions and temperature dependence of miscibility== ===Ideal and regular solutions=== The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or solid) solutions—those formed by completely random mixing so that the components move independently in the total volume. This quantity combines two physical effects—the enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing considered here. For ideal gases, the entropy of mixing at prescribed common temperature and pressure has nothing to do with mixing in the sense of intermingling and interactions of molecular species, but is only to do with expansion into the common volume.Bailyn, M. (1994). A version of Statistical Mechanics for Students of Physics and Chemistry, Cambridge University Press, Cambridge UK, pages 163-164 the conflating of the just-mentioned two mechanisms for the entropy of mixing is well established in customary terminology, but can be confusing unless it is borne in mind that the independent variables are the common initial and final temperature and total pressure; if the respective partial pressures or the total volume are chosen as independent variables instead of the total pressure, the description is different. ====Mixing with each gas kept at constant partial volume, with changing total volume==== In contrast to the established customary usage, ""mixing"" might be conducted reversibly at constant volume for each of two fixed masses of gases of equal volume, being mixed by gradually merging their initially separate volumes by use of two ideal semipermeable membranes, each permeable only to one of the respective gases, so that the respective volumes available to each gas remain constant during the merge. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. We again obtain the entropy of mixing on multiplying by the Boltzmann constant k_\text{B}. ",+4.1, 9.73,0.0,7.136,-1.46,C
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Concerns over the harmful effects of chlorofluorocarbons on stratospheric ozone have motivated a search for new refrigerants. One such alternative is 2,2-dichloro-1,1,1-trifluoroethane (refrigerant 123). Younglove and McLinden published a compendium of thermophysical properties of this substance (J. Phys. Chem. Ref. Data 23, 7 (1994)), from which properties such as the Joule-Thomson coefficient $\mu$ can be computed. Compute the temperature change that would accompany adiabatic expansion of $2.0 \mathrm{~mol}$ of this refrigerant from $1.5 \mathrm{bar}$ to 0.5 bar at $50^{\circ} \mathrm{C}$.","Note the trend of the CClF2-X series in the table below. ==Ozone depleting potential of common compounds== Compound R No. ODP Trichlorofluoromethane (CCl3F) R-11 1.00 1,1,1,2-Tetrafluoroethane (CF3-CH2F) R-134a 0.000015 Chlorodifluoromethane (CClF2-H) R-22 0.05 Chlorotrifluoromethane (CClF2-F) R-13 1.00 Dichlorodifluoromethane (CClF2-Cl) R-12 1.00 Bromochlorodifluoromethane (CClF2-Br) R-12B1 7.9 Carbon tetrachloride (CCl4) R-10 0.82 Nitrous oxide (N2O) R-744A 0.017 Alkanes (Propane, Isobutane, etc.) 0 Ammonia (NH3) R-717 0 Carbon dioxide (CO2) R-744 0 Nitrogen (N2) R-728 0 ==References== ==External links== * List of ozone depleting substances with their ODPs * Scientific Assessment of Ozone Depletion: 1991 Category:Ozone depletion Property Value Ozone depletion potential (ODP) 0.44 (CCl3F = 1) Global warming potential (GWP: 100-year) 5,860 \- 7,670 (CO2 = 1) Atmospheric lifetime 1,020 \- 1,700 years == See also == * IPCC list of greenhouse gases * List of refrigerants == References == Category:Ozone depletion Category:Greenhouse gases Category:Refrigerants Category:Chlorofluorocarbons 1,1,1,2-Tetrafluoroethane (also known as norflurane (INN), R-134a, Klea®134a, Freon 134a, Forane 134a, Genetron 134a, Green Gas, Florasol 134a, Suva 134a, or HFC-134a) is a hydrofluorocarbon (HFC) and haloalkane refrigerant with thermodynamic properties similar to R-12 (dichlorodifluoromethane) but with insignificant ozone depletion potential and a lower 100-year global warming potential (1,430, compared to R-12's GWP of 10,900). Even though 1,1,1,2-Tetrafluoroethane has insignificant ozone depletion potential (ozone layer) and negligible acidification potential (acid rain), it has a 100-year global warming potential (GWP) of 1430 and an approximate atmospheric lifetime of 14 years. ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases Retrieved 21 August 2011. == History and environmental impacts == 1,1,1,2-Tetrafluoroethane was introduced in the early 1990s as a replacement for dichlorodifluoromethane (R-12), which has massive ozone depleting properties. Hydrochlorofluorocarbons have ODPs mostly in range 0.005 - 0.2 due to the presence of the hydrogen which causes them to react readily in the troposphere, therefore reducing their chance to reach the stratosphere where the ozone layer is present. This colorless gas is of interest as a more environmentally friendly (lower GWP; global warming potential) refrigerant in air conditioners. Some of the properties of cyclic ozone have been predicted theoretically. A phaseout and transition to HFO-1234yf and other refrigerants, with GWPs similar to CO2, began in 2012 within the automotive market. == Uses == 1,1,1,2-Tetrafluoroethane is a non-flammable gas used primarily as a ""high-temperature"" refrigerant for domestic refrigeration and automobile air conditioners. It should have more energy than ordinary ozone. Chloropentafluoroethane is a chlorofluorocarbon (CFC) once used as a refrigerant and also known as R-115 and CFC-115. It has been speculated that, if cyclic ozone could be made in bulk, and if it proved to have good stability properties, it could be added to liquid oxygen to improve the specific impulse of rocket fuel. Chlorofluorocarbons have ODPs roughly equal to 1. The Society of Automotive Engineers (SAE) has proposed that it be best replaced by a new fluorochemical refrigerant HFO-1234yf (CFCF=CH) in automobile air-conditioning systems.HFO-1234yf A Low GWP Refrigerant For MAC . It was defined as a measure of destructive effects of a substance compared to a reference substance.Ozone-Depletion and Chlorine- Loading Potential of Chlorofluorocarbon Alternatives Precisely, ODP of a given substance is defined as the ratio of global loss of ozone due to the given substance to the global loss of ozone due to CFC-11 of the same mass. It has the formula CFCHF and a boiling point of −26.3 °C (−15.34 °F) at atmospheric pressure. Currently, the possibility of cyclic ozone is confirmed within diverse theoretical approaches. ==References== ==External links== * Category:Allotropes of oxygen Category:Hypothetical chemical compounds Category:Three-membered rings Category:Homonuclear triatomic molecules The ozone depletion potential (ODP) of a chemical compound is the relative amount of degradation to the ozone layer it can cause, with trichlorofluoromethane (R-11 or CFC-11) being fixed at an ODP of 1.0. It would differ from ordinary ozone in how those three oxygen atoms are arranged. It has also been studied as a potential inhalational anesthetic, but it is nonanaesthetic at doses used in inhalers. == See also == * List of refrigerants * Tetrabromoethane * Tetrachloroethane == References == == External links == * * European Fluorocarbons Technical Committee (EFCTC) * MSDS at Oxford University * Concise International Chemical Assessment Document 11, at inchem.org * Pressure temperature calculator * * R134a 2 phase computer cooling Category:Fluoroalkanes Category:Refrigerants Category:Automotive chemicals Category:Propellants Category:Airsoft Category:Excipients Category:Greenhouse gases Category:GABAA receptor positive allosteric modulators Category:General anesthetics The ozone depletion potential increases with the heavier halogens since the C-X bond strength is lower. ",0.23,-2.99,10.7598,1.1,0.2115,B
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the Carnot efficiency of a modern steam turbine that operates with steam at $300^{\circ} \mathrm{C}$ and discharges at $80^{\circ} \mathrm{C}$.,"Latest generation gas turbine engines have achieved an efficiency of 46% in simple cycle and 61% when used in combined cycle. ==External combustion engines== ===Steam engine=== ::See also: Steam engine#Efficiency ::See also: Timeline of steam power ====Piston engine==== Steam engines and turbines operate on the Rankine cycle which has a maximum Carnot efficiency of 63% for practical engines, with steam turbine power plants able to achieve efficiency in the mid 40% range. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. *Thermal engineering by Rathore and Mahesh; Tata McGraw Hill publications. ==Further reading== * Basic concepts in Turbo machinery by Ingarm. * http://www.turbinesinfo.com/steam-turbine-efficiency. * http://www.physicsforums.com › Physics › General Physics/ * https://web.archive.org/web/20150219211612/http://www.techloud.net/2012/04/losses- in-steam-turbines.html * http://www.learnthermo.com/examples/ch05/p-5c-2.php Category:Steam turbines Its theoretical efficiency depends on the compression ratio r of the engine and the specific heat ratio γ of the gas in the combustion chamber. \eta_{\rm th} = 1 - \frac{1}{r^{\gamma-1}} Thus, the efficiency increases with the compression ratio. The efficiency of steam engines is primarily related to the steam temperature and pressure and the number of stages or expansions. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. See graph:Steam Engine Efficiency In earliest steam engines the boiler was considered part of the engine. As is the case with the gas turbine, the steam turbine works most efficiently at full power, and poorly at slower speeds. The efficiency depends largely on the ratio of the pressure inside the combustion chamber p2 to the pressure outside p1 \eta_{\rm th} = 1 - \left(\frac{p_2}{p_1}\right)^\frac{1-\gamma}{\gamma} ===Other inefficiencies=== One should not confuse thermal efficiency with other efficiencies that are used when discussing engines. In a combined cycle plant, thermal efficiencies approach 60%.GE Power’s H Series Turbine Such a real-world value may be used as a figure of merit for the device. One other factor negatively affecting the gas turbine efficiency is the ambient air temperature. Steam engine efficiency improved as the operating principles were discovered, which led to the development of the science of thermodynamics. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. A very well-designed and built steam locomotive used to get around 7-8% efficiency in its heyday. Today they are considered separate, so it is necessary to know whether stated efficiency is overall, which includes the boiler, or just of the engine. Engines in large diesel trucks, buses, and newer diesel cars can achieve peak efficiencies around 45%. ===Gas turbine=== The gas turbine is most efficient at maximum power output in the same way reciprocating engines are most efficient at maximum load. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature T_{\rm H}, and removed at the minimum temperature T_{\rm C}. Comparisons of efficiency and power of the early steam engines is difficult for several reasons: 1) there was no standard weight for a bushel of coal, which could be anywhere from 82 to 96 pounds (37 to 44 kg). Despite of being at very low pressure the exhaust coming out of the turbine and entering the condenser carries some of kinetic energy and useful enthalpy, which is direct energy loss. ==Radiation and convection losses== The steam turbine operates at a relatively high temperature; therefore some of the heat energy of steam is radiated and convected from the body of the turbine to its surrounding. In thermodynamics, the thermal efficiency (\eta_{\rm th}) is a dimensionless performance measure of a device that uses thermal energy, such as an internal combustion engine, steam turbine, steam engine, boiler, furnace, refrigerator, ACs etc. The most efficient reciprocating steam engine design (per stage) was the uniflow engine, but by the time it appeared steam was being displaced by diesel engines, which were even more efficient and had the advantages of requiring less labor (for coal handling and oiling), being a more dense fuel, and displaced less cargo. ====Steam turbine==== The steam turbine is the most efficient steam engine and for this reason is universally used for electrical generation. ",0.9731,22,0.38,35.64, 6.6,C
-"The discovery of the element argon by Lord Rayleigh and Sir William Ramsay had its origins in Rayleigh's measurements of the density of nitrogen with an eye toward accurate determination of its molar mass. Rayleigh prepared some samples of nitrogen by chemical reaction of nitrogencontaining compounds; under his standard conditions, a glass globe filled with this 'chemical nitrogen' had a mass of $2.2990 \mathrm{~g}$. He prepared other samples by removing oxygen, carbon dioxide, and water vapour from atmospheric air; under the same conditions, this 'atmospheric nitrogen' had a mass of $2.3102 \mathrm{~g}$ (Lord Rayleigh, Royal Institution Proceedings 14, 524 (1895)). With the hindsight of knowing accurate values for the molar masses of nitrogen and argon, compute the mole fraction of argon in the latter sample on the assumption that the former was pure nitrogen and the latter a mixture of nitrogen and argon.","Argon was first isolated from air in 1894 by Lord Rayleigh and Sir William Ramsay at University College London by removing oxygen, carbon dioxide, water, and nitrogen from a sample of clean air. After the two men identified argon, Ramsay investigated other atmospheric gases. Until 1957, the symbol for argon was ""A"", but now it is ""Ar"". ==Occurrence== Argon constitutes 0.934% by volume and 1.288% by mass of Earth's atmosphere. Most of the argon in Earth's atmosphere was produced by electron capture of long-lived ( + e− → + ν) present in natural potassium within Earth. Before isolating the gas, they had determined that nitrogen produced from chemical compounds was 0.5% lighter than nitrogen from the atmosphere. It forms at pressures between 4.3 and 220 GPa, though Raman measurements suggest that the H2 molecules in Ar(H2)2 dissociate above 175 GPa. ==Production== Argon is extracted industrially by the fractional distillation of liquid air in a cryogenic air separation unit; a process that separates liquid nitrogen, which boils at 77.3 K, from argon, which boils at 87.3 K, and liquid oxygen, which boils at 90.2 K. The content of 39Ar in natural argon is measured to be of (8.0±0.6)×10−16 g/g, or (1.01±0.08) Bq/kg of 36, 38, 40Ar. Almost all of the argon in the Earth's atmosphere is the product of 40K decay, since 99.6% of Earth atmospheric argon is 40Ar, whereas in the Sun and presumably in primordial star-forming clouds, argon consists of < 15% 38Ar and mostly (85%) 36Ar. Argon is the most abundant noble gas in Earth's crust, comprising 0.00015% of the crust. Sir William Ramsay (; 2 October 1852 – 23 July 1916) was a Scottish chemist who discovered the noble gases and received the Nobel Prize in Chemistry in 1904 ""in recognition of his services in the discovery of the inert gaseous elements in air"" along with his collaborator, John William Strutt, 3rd Baron Rayleigh, who received the Nobel Prize in Physics that same year for their discovery of argon. The predominance of radiogenic is the reason the standard atomic weight of terrestrial argon is greater than that of the next element, potassium, a fact that was puzzling when argon was discovered. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). Earth's crust and seawater contain 1.2 ppm and 0.45 ppm of argon, respectively. ==Isotopes== The main isotopes of argon found on Earth are (99.6%), (0.34%), and (0.06%). Argon is a chemical element with the symbol Ar and atomic number 18. This discovery caused the recognition that argon could form weakly bound compounds, even though it was not the first. Argon (18Ar) has 26 known isotopes, from 29Ar to 54Ar and 1 isomer (32mAr), of which three are stable (36Ar, 38Ar, and 40Ar). Before 1962, argon and the other noble gases were considered to be chemically inert and unable to form compounds; however, compounds of the heavier noble gases have since been synthesized. Nearly all of the argon in Earth's atmosphere is radiogenic argon-40, derived from the decay of potassium-40 in Earth's crust. Correspondingly, solar argon contains 84.6% (according to solar wind measurements), and the ratio of the three isotopes 36Ar : 38Ar : 40Ar in the atmospheres of the outer planets is 8400 : 1600 : 1. * On triple point pressure at 83.8058 K. ==External links== * Argon at The Periodic Table of Videos (University of Nottingham) * USGS Periodic Table – Argon * Diving applications: Why Argon? Rayleigh had noticed a discrepancy between the density of nitrogen made by chemical synthesis and nitrogen isolated from the air by removal of the other known components. That proposed element was named gnomium. ",5.1,200,0.011,8,4.5,C
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $w$.","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. A carbon dioxide generator or CO2 generator is a machine used to enhance carbon dioxide levels in order to promote plant growth in greenhouses or other enclosed areas. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. Accumulation of carbon dioxide is predominantly a result of hypoperfusion and not hypoxia. Gastric tonometry describes the measurement of the carbon dioxide level inside the stomach in order to assess the degree of blood flow to the stomach and bowel. Normocapnia or normocarbia is a state of normal arterial carbon dioxide pressure, usually about 40 mmHg. == See also == * * * == References == *The Free Dictionary - Normocapnia Category:Diving medicine Category:Pulmonology A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. For gases that remain gaseous under ambient storage conditions, the upstream pressure gauge can be used to estimate how much gas is left in the cylinder according to pressure. The manifold may be arranged to allow simultaneous flow from all the cylinders, or, for a cascade filling system, where gas is tapped off cylinders according to the lowest positive pressure difference between storage and destination cylinder, being a more efficient use of pressurised gas. === Gas storage quads === thumb|Helium quad for surface-supplied diving gas A gas quad is a group of high pressure cylinders mounted on a transport and storage frame. Each cylinder shall be painted externally in the colours corresponding to its gaseous contents. == Common sizes == The below are example cylinder sizes and do not constitute an industry standard. size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) Internal volume, 70 °F (21 °C), 1atm Internal volume, 70 °F (21 °C), 1atm U.S. DOT specs size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) (liters) (cu.ft) U.S. DOT specs 2HP 43.3 1.53 3AA3500 K 110 49.9 1.76 3AA2400 A 96 43.8 1.55 3AA2015 B 37.9 17.2 0.61 3AA2015 C 15.2 6.88 0.24 3AA2015 D 4.9 2.24 0.08 3AA2015 AL 64.8 29.5 1.04 3AL2015 BL 34.6 15.7 0.55 3AL2216 CL 13 5.9 0.21 3AL2216 XL 238 108 3.83 4BA240 SSB 41.6 18.9 0.67 3A1800 10S 8.3 3.8 0.13 3A1800 LB 1 0.44 0.016 3E1800 XF 60.9 2.15 8AL XG 278 126.3 4.46 4AA480 XM 120 54.3 1.92 3A480 XP 124 55.7 1.98 4BA300 QT (includes 4.5 inches for valve) (includes 1.5 lb for valve) 2.0 0.900 0.0318 4B-240ET LP5 47.7 21.68 0.76 4BW240 Medical E (excludes valve and cap) (excludes valve and cap) 9.9 4.5 0.16 3AA2015 (US DOT specs define material, making, and maximum pressure in psi. Carbon tetroxide or Oxygen carbonate (in its C2v isomer) is a highly unstable oxide of carbon with formula . Because the introduction of a nasogastric tube is almost routine in critically ill patients, the measurement of gastric carbon dioxide can be an easy method to monitor tissue perfusion. It was proposed as an intermediate in the O-atom exchange between carbon dioxide () and oxygen () at high temperatures. Tonometry is based on the principle that at equilibrium the partial pressure of a diffusible gas such as CO2 is the same in both the wall and lumen of a viscus. Using compressed CO2 is an alternative to generators. == See also == ==References== Category:Horticulture Category:Industrial gases A typical gas cylinder design is elongated, standing upright on a flattened bottom end, with the valve and fitting at the top for connecting to the receiving apparatus. * ISO 11439: Gas cylinders — High-pressure cylinders for the on-board storage of natural gas as a fuel for automotive vehicles * ISO 15500-5: Road vehicles — Compressed natural gas (CNG) fuel system components — Part 5: Manual cylinder valve * US DOT CFR Title 49, part 178, Subpart C — Specification for CylindersUS DOT e-CFR (Electronic Code of Federal Regulations) Title 49, part 178, Subpart C — Specification for Cylinders — eg DOT 3AL = seamless aluminum * US DOT Aluminum Tank Alloy 6351-T6 amendment for SCUBA, SCBA, Oxygen Service — Visual Eddy inspectionFederal Register / Vol. 71, No. 167 / Tuesday, August 29, 2006 / Rules and Regulations Title 49 CFR Parts 173 and 180 Visual Edddy * AS 2896-2011:Medical gas systems—Installation and testing of non-flammable medical gas pipeline systems pipeline systems (Australian Standards). === Color coding === Gas cylinders are often color- coded, but the codes are not standard across different jurisdictions, and sometimes are not regulated. High-pressure gas cylinders are also called bottles. The regulator is adjusted to control the downstream pressure, which will limit the maximum flow of gas out of the cylinder at the pressure shown by the downstream gauge. Pressure vessels for gas storage may also be classified by volume. ",4.16,-20,0.264, -2.5,15.425,B
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When a certain freon used in refrigeration was expanded adiabatically from an initial pressure of $32 \mathrm{~atm}$ and $0^{\circ} \mathrm{C}$ to a final pressure of $1.00 \mathrm{~atm}$, the temperature fell by $22 \mathrm{~K}$. Calculate the Joule-Thomson coefficient, $\mu$, at $0^{\circ} \mathrm{C}$, assuming it remains constant over this temperature range.","The temperature change produced during a Joule–Thomson expansion is quantified by the Joule–Thomson coefficient, \mu_{\mathrm{JT}}. This equation can be used to obtain Joule–Thomson coefficients from the more easily measured isothermal Joule–Thomson coefficient. The temperature of this point, the Joule–Thomson inversion temperature, depends on the pressure of the gas before expansion. Since this is true at all temperatures for ideal gases (see expansion in gases), the Joule–Thomson coefficient of an ideal gas is zero at all temperatures. ==Joule's second law== It is easy to verify that for an ideal gas defined by suitable microscopic postulates that αT = 1, so the temperature change of such an ideal gas at a Joule–Thomson expansion is zero. Thus, for N2 gas below 621 K, a Joule–Thomson expansion can be used to cool the gas until liquid N2 forms. ==Physical mechanism== There are two factors that can change the temperature of a fluid during an adiabatic expansion: a change in internal energy or the conversion between potential and kinetic internal energy. This expression can now replace \mu_{\mathrm{T}} in the earlier equation for \mu_{\mathrm{JT}} to obtain: :\mu_{\mathrm{JT}} \equiv \left( \frac{\partial T}{\partial P} \right)_H = \frac V {C_{\mathrm{p}}} (\alpha T - 1).\, This provides an expression for the Joule–Thomson coefficient in terms of the commonly available properties heat capacity, molar volume, and thermal expansion coefficient. The first step in obtaining these results is to note that the Joule–Thomson coefficient involves the three variables T, P, and H. In a Joule–Thomson expansion the enthalpy remains constant. The physical mechanism associated with the Joule–Thomson effect is closely related to that of a shock wave, although a shock wave differs in that the change in bulk kinetic energy of the gas flow is not negligible. ==The Joule–Thomson (Kelvin) coefficient== thumb|400px|Fig. 1 – Joule–Thomson coefficients for various gases at atmospheric pressure The rate of change of temperature T with respect to pressure P in a Joule–Thomson process (that is, at constant enthalpy H) is the Joule–Thomson (Kelvin) coefficient \mu_{\mathrm{JT}}. At room temperature, all gases except hydrogen, helium, and neon cool upon expansion by the Joule–Thomson process when being throttled through an orifice; these three gases experience the same effect but only at lower temperatures. This produces a decrease in temperature and results in a positive Joule–Thomson coefficient. The cooling produced in the Joule–Thomson expansion makes it a valuable tool in refrigeration.Keenan, J.H. (1970). This means that the mass fraction of the liquid in the liquid–gas mixture leaving the throttling valve is 40%. ==Derivation of the Joule–Thomson coefficient== It is difficult to think physically about what the Joule–Thomson coefficient, \mu_{\mathrm{JT}}, represents. For an ideal gas, \mu_\text{JT} is always equal to zero: ideal gases neither warm nor cool upon being expanded at constant enthalpy. ==Applications== In practice, the Joule–Thomson effect is achieved by allowing the gas to expand through a throttling device (usually a valve) which must be very well insulated to prevent any heat transfer to or from the gas. At high temperature, Z and PV decrease as the gas expands; if the decrease is large enough, the Joule–Thomson coefficient will be negative. Thus at low temperature, Z and PV will increase as the gas expands, resulting in a positive Joule–Thomson coefficient. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). This coefficient can be expressed in terms of the gas's volume V, its heat capacity at constant pressure C_{\mathrm{p}}, and its coefficient of thermal expansion \alpha as: :\mu_{\mathrm{JT}} = \left( {\partial T \over \partial P} \right)_H = \frac V {C_{\mathrm{p}}}(\alpha T - 1)\, See the below for the proof of this relation. With that in mind, the following table explains when the Joule–Thomson effect cools or warms a real gas: If the gas temperature is then \mu_\text{JT} is since \partial P is thus \partial T must be so the gas below the inversion temperature positive always negative negative cools above the inversion temperature negative always negative positive warms Helium and hydrogen are two gases whose Joule–Thomson inversion temperatures at a pressure of one atmosphere are very low (e.g., about 45 K, −228 °C for helium). Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). In a Joule–Thomson process the specific enthalpy h remains constant.See e.g. M.J. Moran and H.N. Shapiro ""Fundamentals of Engineering Thermodynamics"" 5th Edition (2006) John Wiley & Sons, Inc. page 147 To prove this, the first step is to compute the net work done when a mass m of the gas moves through the plug. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. ",0.444444444444444 ,0.9992093669,3.2,-1.32,0.71,E
+(3) $4 \mathrm{HCl}(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{Cl}_2(\mathrm{g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup == Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). K (? °C), ? K (? °C), ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Ellis (Ed.) in Nuffield Advanced Science Book of Data, Longman, London, UK, 1972. == See also == Category:Thermodynamic properties Category:Chemical element data pages Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? ",2.57,0.0761,"""0.8185""",-114.40,4.4,D
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. An average human produces about $10 \mathrm{MJ}$ of heat each day through metabolic activity. If a human body were an isolated system of mass $65 \mathrm{~kg}$ with the heat capacity of water, what temperature rise would the body experience?","The normal human body temperature is often stated as . Human body temperature varies. Normal human body temperature varies slightly from person to person and by the time of day. In humans, the average internal temperature is widely accepted to be 37 °C (98.6 °F), a ""normal"" temperature established in the 1800s. The normal human body temperature range is typically stated as . An individual's body temperature typically changes by about between its highest and lowest points each day. Normal human body-temperature (normothermia, euthermia) is the typical temperature range found in humans. In adults a review of the literature has found a wider range of for normal temperatures, depending on the gender and location measured. While some people think of these averages as representing normal or ideal measurements, a wide range of temperatures has been found in healthy people. The range for normal human body temperatures, taken orally, is . The typical daytime temperatures among healthy adults are as follows: * Temperature in the anus (rectum/rectal), vagina, or in the ear (tympanic) is about * Temperature in the mouth (oral) is about * Temperature under the arm (axillary) is about Generally, oral, rectal, gut, and core body temperatures, although slightly different, are well-correlated. It has been found that physically active individuals have larger changes in body temperature throughout the day. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. The commonly given value 98.6 °F is simply the exact conversion of the nineteenth-century German standard of 37 °C. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. Rectal temperature is expected to be approximately 1 Fahrenheit (or 0.55 Celsius) degree higher than an oral temperature taken on the same person at the same time. The body temperature of a healthy person varies during the day by about with lower temperatures in the morning and higher temperatures in the late afternoon and evening, as the body's needs and activities change. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. The heat uptake results from a persistent warming imbalance in Earth's energy budget that is most fundamentally caused by the anthropogenic increase in atmospheric greenhouse gases. Springer, Cham, 2017. 441-493. == Historical understanding == In the 19th century, most books quoted ""blood heat"" as 98 °F, until a study published the mean (but not the variance) of a large sample as .Inwit Publishing, Inc. and Inwit, LLC – Writings, Links and Software Demonstrations – A Fahrenheit–Celsius Activity, inwit.com Subsequently, that mean was widely quoted as ""37 °C or 98.4 °F""Oxford Dictionary of English, 2010 edition, entry on ""blood heat""Collins English Dictionary, 1979 edition, entry on ""blood heat"" until editors realized 37 °C is equal to 98.6 °F, not 98.4 °F. thumb |upright=1.35 |Upper ocean heat content (OHC) has been increasing because oceans have absorbed a large portion of the excess heat trapped in the atmosphere by human-caused global warming. ",0.11,0.068,"""37.0""",269,0.332,C
+"1.19(a) The critical constants of methane are $p_{\mathrm{c}}=45.6 \mathrm{~atm}, V_{\mathrm{c}}=98.7 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$, and $T_{\mathrm{c}}=190.6 \mathrm{~K}$. Estimate the radius of the molecules.","Methane has a boiling point of −161.5 °C at a pressure of one atmosphere. Methane's heat of combustion is 55.5 MJ/kg.Energy Content of some Combustibles (in MJ/kg) . Cooling methane at normal pressure results in the formation of methane I. However the authors stress ""our findings are preliminary with regard to the methane emission strength"". Note that these are all negative temperature values. thumb|432px|left|Methane vapor pressure vs. temperature. Uses formula \log_{10} P_\text{mm Hg} = 6.61184 - \frac{389.93}{266.00 + T_{^\circ\text{C}}} given in Lange's Handbook of Chemistry, 10th ed. Note that formula loses accuracy near Tcrit = −82.6 °C == Spectral data == thumb|right|450px|Methane infrared spectrum UV-Vis λmax ? nm Extinction coefficient, ε ? {{Chembox | ImageFile1 = File:Tris(dimethylamino)methan Struktur.svg | ImageSize1 = 150px | ImageFile2 = Tris(dimethylamino)methane-3D-balls-by- AHRLS-2012.png | ImageSize2 = 150px | ImageFile3 = Tris(dimethylamino)methane-3D-sticks-by-AHRLS-2012.png | ImageSize3 = 150px | ImageAlt = | PIN = N,N,N,N,N,N-Hexamethylmethanetriamine | OtherNames = N,N,N,N,N,N-hexamethylmethanetriamine [bis(dimethylamino)methyl]dimethylamine | Section1 = | Section2 = | Section3 = }} Tris(dimethylamino)methane (TDAM) is the simplest representative of the tris(dialkylamino)methanes of the general formula (R2N)3CH in which three of the four of methane's hydrogen atoms are replaced by dimethylamino groups (−N(CH3)2). Methane has also been detected on other planets, including Mars, which has implications for astrobiology research. ==Properties and bonding== Methane is a tetrahedral molecule with four equivalent C–H bonds. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Solid methane exists in several modifications. Temperatures in excess of 1200 °C are required to break the bonds of methane to produce Hydrogen gas and solid carbon. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The reaction is moderately endothermic as shown in the reaction equation below. : :( 74.8 kJ/mol) ==Generation== === Geological routes === left|thumb|upright=1.35|Abiotic sources of methane have been found in more than 20 countries and in several deep ocean regions so far.The two main routes for geological methane generation are (i) organic (thermally generated, or thermogenic) and (ii) inorganic (abiotic). Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09 Å for the van der Waals radius of hydrogen as opposed to Bondi's 1.20 Å. Methane at scales of the atmosphere is commonly measured in teragrams (Tg ) or millions of metric tons (MMT ), which mean the same thing. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Methane is an important greenhouse gas with a global warming potential of 40 compared to (potential of 1) over a 100-year period, and above 80 over a 20-year period. At about 891 kJ/mol, methane's heat of combustion is lower than that of any other hydrocarbon, but the ratio of the heat of combustion (891 kJ/mol) to the molecular mass (16.0 g/mol, of which 12.0 g/mol is carbon) shows that methane, being the simplest hydrocarbon, produces more heat per mass unit (55.7 kJ/g) than other complex hydrocarbons. The largest reservoir of methane is under the seafloor in the form of methane clathrates. Methane is a strong GHG with a global warming potential 84 times greater than CO2 in a 20-year time frame. The positions of the hydrogen atoms are not fixed in methane I, i.e. methane molecules may rotate freely. At high pressures, such as are found on the bottom of the ocean, methane forms a solid clathrate with water, known as methane hydrate. ",0.0408,0.05882352941,"""0.118""",1260,22,C
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The enthalpy of vaporization of chloroform $\left(\mathrm{CHCl}_3\right)$ is $29.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at its normal boiling point of $334.88 \mathrm{~K}$. Calculate the entropy change of the surroundings.,"The entropy of vaporization of at its boiling point has the extraordinarily high value of 136.9 J/(K·mol). It is valid for many liquids; for instance, the entropy of vaporization of toluene is 87.30 J/(K·mol), that of benzene is 89.45 J/(K·mol), and that of chloroform is 87.92 J/(K·mol). The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. thumb|The Mollier enthalpy–entropy diagram for water and steam. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? In thermodynamics, Trouton's rule states that the entropy of vaporization is almost the same value, about 85–88 J/(K·mol), for various kinds of liquids at their boiling points.Compare 85 J/(K·mol) in and 88 J/(K·mol) in The entropy of vaporization is defined as the ratio between the enthalpy of vaporization and the boiling temperature. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. For example, the entropies of vaporization of water, ethanol, formic acid and hydrogen fluoride are far from the predicted values. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? The Mollier diagram coordinates are enthalpy h and humidity ratio x. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? K (? °C), ? K (? °C), ? K (? °C), ? ",-87.8,0.139,"""1.7""",1.775,6.0,A
+"Recent communication with the inhabitants of Neptune has revealed that they have a Celsius-type temperature scale, but based on the melting point $(0^{\circ} \mathrm{N})$ and boiling point $(100^{\circ} \mathrm{N})$ of their most common substance, hydrogen. Further communications have revealed that the Neptunians know about perfect gas behaviour and they find that, in the limit of zero pressure, the value of $p V$ is $28 \mathrm{dm}^3$ atm at $0^{\circ} \mathrm{N}$ and $40 \mathrm{dm}^3$ atm at $100^{\circ} \mathrm{N}$. What is the value of the absolute zero of temperature on their temperature scale?","Below 0.9 kelvin at their saturated vapor pressure, a mixture of the two isotopes undergoes a phase separation into a normal fluid (mostly helium-3) that floats on a denser superfluid consisting mostly of helium-4. At these low temperatures, the melting pressure of helium-3 varies from about 2.9 MPa to nearly 4.0 MPa. thumb|0 °C = 32 °F A degree of frost is a non-standard unit of measure for air temperature meaning degrees below melting point (also known as ""freezing point"") of water (0 degrees Celsius or 32 degrees Fahrenheit). Its boiling point and critical point depend on which isotope of helium is present: the common isotope helium-4 or the rare isotope helium-3. Although this gives a disadvantage of non-monotonicity, in that two different temperatures can give the same pressure, the scale is otherwise robust since the melting pressure of helium-3 is insensitive to many experimental factors. == See also == * International Temperature Scale of 1990 (ITS-90) — the calibration standard used for all temperatures above 0.6 K * Leiden scale == References == Category:Temperature Category:Scales of temperature At standard pressure, the chemical element helium exists in a liquid form only at the extremely low temperature of . The zero point energy of liquid helium is less if its atoms are less confined by their neighbors. It is based on the melting pressure of solidified helium-3. When based on Celsius, 0 degrees of frost is the same as 0 °C, and any other value is simply the negative of the Celsius temperature. At the temperature of approximately 315 mK, a minimum of pressure (2.9 MPa) occurs. Because of the very weak interatomic forces in helium, the element remains a liquid at atmospheric pressure all the way from its liquefaction point down to absolute zero. Smaller gas planets and planets closer to their star will lose atmospheric mass more quickly via hydrodynamic escape than larger planets and planets farther out.Mass-radius relationships for exoplanets, Damian C. Swift, Jon Eggert, Damien G. Hicks, Sebastien Hamel, Kyle Caspersen, Eric Schwegler, and Gilbert W. Collins A low-mass gas planet can still have a radius resembling that of a gas giant if it has the right temperature.Mass-Radius Relationships for Very Low Mass Gaseous Planets, Konstantin Batygin, David J. Stevenson, 18 Apr 2013 Neptune-like planets are considerably rarer than sub-Neptunes, despite being only slightly bigger.Superabundance of Exoplanet Sub-Neptunes Explained by Fugacity Crisis, Edwin S. Kite, Bruce Fegley Jr., Laura Schaefer, Eric B. Ford, 5 Dec 2019 This ""radius cliff"" separates sub-Neptunes (radius < 3 Earth radii) from Neptunes (radius > 3 Earth radii). Liquid helium is a physical state of helium at very low temperatures at standard atmospheric pressures. When based on Fahrenheit, 0 degrees of frost is equal to 32 °F. The business was named after the Kelvin temperature scale. The density of liquid helium-4 at its boiling point and a pressure of one atmosphere (101.3 kilopascals) is about , or about one-eighth the density of liquid water. ==Liquefaction== Helium was first liquefied on July 10, 1908, by the Dutch physicist Heike Kamerlingh Onnes at the University of Leiden in the Netherlands. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. thumb|upright=1.1|Artist's conception of a mini-Neptune or ""gas dwarf"" A Mini- Neptune (sometimes known as a gas dwarf or transitional planet) is a planet less massive than Neptune but resembling Neptune in that it has a thick hydrogen–helium atmosphere, probably with deep layers of ice, rock or liquid oceans (made of water, ammonia, a mixture of both, or heavier volatiles). Important early work on the characteristics of liquid helium was done by the Soviet physicist Lev Landau, later extended by the American physicist Richard Feynman. ==Data== Properties of liquid helium Helium-4 Helium-3 Critical temperature Boiling point at one atmosphere Minimum melting pressure at Superfluid transition temperature at saturated vapor pressure 1 mK in the absence of a magnetic field ==Gallery== File:Liquid Helium.jpg|Liquid helium (in a vacuum bottle) at and boiling slowly. See the table below for the values of these physical quantities. The Provisional Low Temperature Scale of 2000 (PLTS-2000) is an equipment calibration standard for making measurements of very low temperatures, in the range of 0.9 mK (millikelvin) to 1 K, adopted by the International Committee for Weights and Measures in October 2000. Liquid helium can be solidified only under very low temperatures and high pressures. ", 135.36,4943,"""2283.63""",-233,47,D
+A gas at $250 \mathrm{~K}$ and $15 \mathrm{~atm}$ has a molar volume 12 per cent smaller than that calculated from the perfect gas law. Calculate the compression factor under these conditions.,"For an ideal gas the compressibility factor is Z=1 per definition. In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. The compression ratio is the ratio between the volume of the cylinder and combustion chamber in an internal combustion engine at their maximum and minimum values. Experimental values for the compressibility factor confirm this. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . Compression ratio is a ratio of volumes. Figure 2 is an example of a generalized compressibility factor graph derived from hundreds of experimental PVT data points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, i-pentane, n-hexane, nitrogen, carbon dioxide and steam. The compressibility factor should not be confused with the compressibility (also known as coefficient of compressibility or isothermal compressibility) of a material, which is the measure of the relative volume change of a fluid or solid in response to a pressure change. ==Definition and physical significance== thumb|A graphical representation of the behavior of gases and how that behavior relates to compressibility factor. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. The closer the gas is to its critical point or its boiling point, the more Z deviates from the ideal case. === Fugacity === The compressibility factor is linked to the fugacity by the relation: : f = P \exp\left(\int \frac{Z-1}{P} dP\right) ==Generalized compressibility factor graphs for pure gases== thumb|400px|Generalized compressibility factor diagram. There are more detailed generalized compressibility factor graphs based on as many as 25 or more different pure gases, such as the Nelson-Obert graphs. * Real Gases includes a discussion of compressibility factors. For a gas that is a mixture of two or more pure gases (air or natural gas, for example), the gas composition must be known before compressibility can be calculated. Above the Boyle temperature, the compressibility factor is always greater than unity and increases slowly but steadily as pressure increases. ==Experimental values== It is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomes important. The reduced specific volume is defined by, : u_R = \frac{ u_\text{actual}}{RT_\text{cr}/P_\text{cr}} where u_\text{actual} is the specific volume. page 140 Once two of the three reduced properties are found, the compressibility chart can be used. The dynamic compression ratio accounts for these factors. In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature, P_r and T_r, are used to normalize the compressibility factor data. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document) In general, deviation from ideal behaviour becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure. Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound- specific empirical constants as input. ",0.5117,10,"""0.88""", -6.04697,7.25,C
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Suppose that $3.0 \mathrm{mmol} \mathrm{N}_2$ (g) occupies $36 \mathrm{~cm}^3$ at $300 \mathrm{~K}$ and expands to $60 \mathrm{~cm}^3$. Calculate $\Delta G$ for the process.,"The value of ΔL must however be taken from the relevant cartridge data information in the C.I.P. Tables. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced ""delta fifteen n"") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. The delta L problem (ΔL problem) refers to certain firearm chambers and the incompatibility of some ammunition made for that chamber. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. Instead of a mole the constant can be expressed by considering the normal cubic meter. Firearms users that have to rely on their weapon under adverse conditions, such as big five and other dangerous game hunters, obviously have to check the correct functioning of the firearm and ammunition they intend to use before exposing themselves to potentially dangerous situations. ==Delta L (ΔL) problem== The length specification ""S"" is a basic dimension (or a datum reference) for the computation of the dimensions of firearms cartridges and chambers. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). In general the first law requires that : Q = \Delta U + W (work) If W is only pressure–volume work, then at constant pressure : Q_P = \Delta U + P \Delta V Assuming that the change in state variables is due solely to a chemical reaction, we have : Q_P = \sum U_{products} - \sum U_{reactants} + P \left(\sum V_{products} - \sum V_{reactants}\right) : Q_P = \sum \left(U_{products} + P V_{products} \right) - \sum \left(U_{reactants} + P V_{reactants} \right) As enthalpy or heat content is defined by H = U + PV , we have : Q_P = \sum H_{products} - \sum H_{reactants} = \Delta H By convention, the enthalpy of each element in its standard state is assigned a value of zero. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . When only expansion work is possible for a process we have \Delta U = Q_V; this implies that the heat of reaction at constant volume is equal to the change in the internal energy \Delta U of the reacting system. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . Since the atmosphere is 99.6337% 14N and 0.3663% 15N, a is 0.003663 in the former case and 0.003663/0.996337 = 0.003676 in the latter. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. For this reason it is important to note which standard concentration value is being used when consulting tables of enthalpies of formation. ==Introduction== Two initial thermodynamic systems, each isolated in their separate states of internal thermodynamic equilibrium, can, by a thermodynamic operation, be coalesced into a single new final isolated thermodynamic system. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. ",257,0.0761,"""0.082""",13,-3.8,E
+1.18(a) A vessel of volume $22.4 \mathrm{dm}^3$ contains $2.0 \mathrm{~mol} \mathrm{H}_2$ and $1.0 \mathrm{~mol} \mathrm{~N}_2$ at $273.15 \mathrm{~K}$. Calculate their total pressure.,"With the ""area"" in the numerator and the ""area"" in the denominator canceling each other out, we are left with :\text{pressure} = \text{weight density} \times \text{depth}. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. * IS 2825–1969 (RE1977)_code_unfired_Pressure_vessels. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. The total pressure of a liquid, then, is ρgh plus the pressure of the atmosphere. Flows with a higher Mach number M cannot approximate the total pressure using the incompressible formula given above. * P_0 is the pressure at the surface. * \rho(z) is the density of the material above the depth z. * g is the gravity acceleration in m/s^2 . upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. Atmospheric pressure pressing on the surface of a liquid must be taken into account when trying to discover the total pressure acting on a liquid. Then we have :\text{pressure} = \frac{\text{force}}{\text{area}} = \frac{\text{weight}}{\text{area}} = \frac{\text{weight density} \times \text{volume}}{\text{area}}, :\text{pressure} = \frac{\text{weight density} \times \text{(area} \times \text{depth)}}{\text{area}}. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation , where g is the gravitational acceleration. Pressure is related to energy density and may be expressed in units such as joules per cubic metre (J/m3, which is equal to Pa). Derivation of this equation This is derived from the definitions of pressure and weight density. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. For a cylinder with hemispherical ends, :M = 2 \pi R^2 (R + W) P {\rho \over \sigma}, where *R is the Radius (m) *W is the middle cylinder width only, and the overall width is W + 2R (m) ====Cylindrical vessel with semi-elliptical ends==== In a vessel with an aspect ratio of middle cylinder width to radius of 2:1, :M = 6 \pi R^3 P {\rho \over \sigma}. ====Gas storage==== In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. The pressure coefficient is used in aerodynamics and hydrodynamics. The CGS unit of pressure is the barye (Ba), equal to 1 dyn·cm−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm2 or kg/cm2) and the like without properly identifying the force units. * EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC. Further the volume of the gas is (4πr3)/3. Almost all pressure vessel design standards contain variations of these two formulas with additional empirical terms to account for variation of stresses across thickness, quality control of welds and in-service corrosion allowances. Pressure is a scalar quantity. ",3930,0.21,"""524.0""",3.0,-131.1,D
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the total change in entropy, when a sample of nitrogen gas of mass $14 \mathrm{~g}$ at $298 \mathrm{~K}$ and $1.00 \mathrm{bar}$ doubles its volume in an adiabatic reversible expansion.","After the removal of the partition, the n_i = nx_i moles of component i may explore the combined volume V\,, which causes an entropy increase equal to nx_i R \ln(V/V_i) = - nR x_i \ln x_i for each component gas. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. In this case, the increase in entropy is entirely due to the irreversible processes of expansion of the two gases, and involves no heat or work flow between the system and its surroundings. ===Gibbs free energy of mixing=== The Gibbs free energy change \Delta G_\text{mix} = \Delta H_\text{mix} - T\Delta S_\text{mix} determines whether mixing at constant (absolute) temperature T and pressure p is a spontaneous process. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). It states that the entropy of such ""mixing"" of perfect gases is zero. ====Mixing at constant total volume and changing partial volumes, with mechanically controlled varying pressure, and constant temperature==== An experimental demonstration may be considered. Upon removal of the dividing partition, they expand into a final common volume (the sum of the two initial volumes), and the entropy of mixing \Delta S_{mix} is given by :\Delta S_{mix} = -nR(x_1\ln x_1 + x_2\ln x_2)\,. where R is the gas constant, n the total number of moles and x_i the mole fraction of component i\,, which initially occupies volume V_i = x_iV\,. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . We can use Boltzmann's equation for the entropy change as applied to the mixing process :\Delta S_\text{mix}= k_\text{B} \ln\Omega where k_\text{B} is the Boltzmann constant. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The lowest value is when the mole fraction is 0.5 for a mixture of two components, or 1/n for a mixture of n components. thumb|The entropy of mixing for an ideal solution of two species is maximized when the mole fraction of each species is 0.5. ==Solutions and temperature dependence of miscibility== ===Ideal and regular solutions=== The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or solid) solutions—those formed by completely random mixing so that the components move independently in the total volume. This quantity combines two physical effects—the enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing considered here. For ideal gases, the entropy of mixing at prescribed common temperature and pressure has nothing to do with mixing in the sense of intermingling and interactions of molecular species, but is only to do with expansion into the common volume.Bailyn, M. (1994). A version of Statistical Mechanics for Students of Physics and Chemistry, Cambridge University Press, Cambridge UK, pages 163-164 the conflating of the just-mentioned two mechanisms for the entropy of mixing is well established in customary terminology, but can be confusing unless it is borne in mind that the independent variables are the common initial and final temperature and total pressure; if the respective partial pressures or the total volume are chosen as independent variables instead of the total pressure, the description is different. ====Mixing with each gas kept at constant partial volume, with changing total volume==== In contrast to the established customary usage, ""mixing"" might be conducted reversibly at constant volume for each of two fixed masses of gases of equal volume, being mixed by gradually merging their initially separate volumes by use of two ideal semipermeable membranes, each permeable only to one of the respective gases, so that the respective volumes available to each gas remain constant during the merge. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. We again obtain the entropy of mixing on multiplying by the Boltzmann constant k_\text{B}. ",+4.1, 9.73,"""0.0""",7.136,-1.46,C
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. Concerns over the harmful effects of chlorofluorocarbons on stratospheric ozone have motivated a search for new refrigerants. One such alternative is 2,2-dichloro-1,1,1-trifluoroethane (refrigerant 123). Younglove and McLinden published a compendium of thermophysical properties of this substance (J. Phys. Chem. Ref. Data 23, 7 (1994)), from which properties such as the Joule-Thomson coefficient $\mu$ can be computed. Compute the temperature change that would accompany adiabatic expansion of $2.0 \mathrm{~mol}$ of this refrigerant from $1.5 \mathrm{bar}$ to 0.5 bar at $50^{\circ} \mathrm{C}$.","Note the trend of the CClF2-X series in the table below. ==Ozone depleting potential of common compounds== Compound R No. ODP Trichlorofluoromethane (CCl3F) R-11 1.00 1,1,1,2-Tetrafluoroethane (CF3-CH2F) R-134a 0.000015 Chlorodifluoromethane (CClF2-H) R-22 0.05 Chlorotrifluoromethane (CClF2-F) R-13 1.00 Dichlorodifluoromethane (CClF2-Cl) R-12 1.00 Bromochlorodifluoromethane (CClF2-Br) R-12B1 7.9 Carbon tetrachloride (CCl4) R-10 0.82 Nitrous oxide (N2O) R-744A 0.017 Alkanes (Propane, Isobutane, etc.) 0 Ammonia (NH3) R-717 0 Carbon dioxide (CO2) R-744 0 Nitrogen (N2) R-728 0 ==References== ==External links== * List of ozone depleting substances with their ODPs * Scientific Assessment of Ozone Depletion: 1991 Category:Ozone depletion Property Value Ozone depletion potential (ODP) 0.44 (CCl3F = 1) Global warming potential (GWP: 100-year) 5,860 \- 7,670 (CO2 = 1) Atmospheric lifetime 1,020 \- 1,700 years == See also == * IPCC list of greenhouse gases * List of refrigerants == References == Category:Ozone depletion Category:Greenhouse gases Category:Refrigerants Category:Chlorofluorocarbons 1,1,1,2-Tetrafluoroethane (also known as norflurane (INN), R-134a, Klea®134a, Freon 134a, Forane 134a, Genetron 134a, Green Gas, Florasol 134a, Suva 134a, or HFC-134a) is a hydrofluorocarbon (HFC) and haloalkane refrigerant with thermodynamic properties similar to R-12 (dichlorodifluoromethane) but with insignificant ozone depletion potential and a lower 100-year global warming potential (1,430, compared to R-12's GWP of 10,900). Even though 1,1,1,2-Tetrafluoroethane has insignificant ozone depletion potential (ozone layer) and negligible acidification potential (acid rain), it has a 100-year global warming potential (GWP) of 1430 and an approximate atmospheric lifetime of 14 years. ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases Retrieved 21 August 2011. == History and environmental impacts == 1,1,1,2-Tetrafluoroethane was introduced in the early 1990s as a replacement for dichlorodifluoromethane (R-12), which has massive ozone depleting properties. Hydrochlorofluorocarbons have ODPs mostly in range 0.005 - 0.2 due to the presence of the hydrogen which causes them to react readily in the troposphere, therefore reducing their chance to reach the stratosphere where the ozone layer is present. This colorless gas is of interest as a more environmentally friendly (lower GWP; global warming potential) refrigerant in air conditioners. Some of the properties of cyclic ozone have been predicted theoretically. A phaseout and transition to HFO-1234yf and other refrigerants, with GWPs similar to CO2, began in 2012 within the automotive market. == Uses == 1,1,1,2-Tetrafluoroethane is a non-flammable gas used primarily as a ""high-temperature"" refrigerant for domestic refrigeration and automobile air conditioners. It should have more energy than ordinary ozone. Chloropentafluoroethane is a chlorofluorocarbon (CFC) once used as a refrigerant and also known as R-115 and CFC-115. It has been speculated that, if cyclic ozone could be made in bulk, and if it proved to have good stability properties, it could be added to liquid oxygen to improve the specific impulse of rocket fuel. Chlorofluorocarbons have ODPs roughly equal to 1. The Society of Automotive Engineers (SAE) has proposed that it be best replaced by a new fluorochemical refrigerant HFO-1234yf (CFCF=CH) in automobile air-conditioning systems.HFO-1234yf A Low GWP Refrigerant For MAC . It was defined as a measure of destructive effects of a substance compared to a reference substance.Ozone-Depletion and Chlorine- Loading Potential of Chlorofluorocarbon Alternatives Precisely, ODP of a given substance is defined as the ratio of global loss of ozone due to the given substance to the global loss of ozone due to CFC-11 of the same mass. It has the formula CFCHF and a boiling point of −26.3 °C (−15.34 °F) at atmospheric pressure. Currently, the possibility of cyclic ozone is confirmed within diverse theoretical approaches. ==References== ==External links== * Category:Allotropes of oxygen Category:Hypothetical chemical compounds Category:Three-membered rings Category:Homonuclear triatomic molecules The ozone depletion potential (ODP) of a chemical compound is the relative amount of degradation to the ozone layer it can cause, with trichlorofluoromethane (R-11 or CFC-11) being fixed at an ODP of 1.0. It would differ from ordinary ozone in how those three oxygen atoms are arranged. It has also been studied as a potential inhalational anesthetic, but it is nonanaesthetic at doses used in inhalers. == See also == * List of refrigerants * Tetrabromoethane * Tetrachloroethane == References == == External links == * * European Fluorocarbons Technical Committee (EFCTC) * MSDS at Oxford University * Concise International Chemical Assessment Document 11, at inchem.org * Pressure temperature calculator * * R134a 2 phase computer cooling Category:Fluoroalkanes Category:Refrigerants Category:Automotive chemicals Category:Propellants Category:Airsoft Category:Excipients Category:Greenhouse gases Category:GABAA receptor positive allosteric modulators Category:General anesthetics The ozone depletion potential increases with the heavier halogens since the C-X bond strength is lower. ",0.23,-2.99,"""10.7598""",1.1,0.2115,B
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the Carnot efficiency of a modern steam turbine that operates with steam at $300^{\circ} \mathrm{C}$ and discharges at $80^{\circ} \mathrm{C}$.,"Latest generation gas turbine engines have achieved an efficiency of 46% in simple cycle and 61% when used in combined cycle. ==External combustion engines== ===Steam engine=== ::See also: Steam engine#Efficiency ::See also: Timeline of steam power ====Piston engine==== Steam engines and turbines operate on the Rankine cycle which has a maximum Carnot efficiency of 63% for practical engines, with steam turbine power plants able to achieve efficiency in the mid 40% range. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. *Thermal engineering by Rathore and Mahesh; Tata McGraw Hill publications. ==Further reading== * Basic concepts in Turbo machinery by Ingarm. * http://www.turbinesinfo.com/steam-turbine-efficiency. * http://www.physicsforums.com › Physics › General Physics/ * https://web.archive.org/web/20150219211612/http://www.techloud.net/2012/04/losses- in-steam-turbines.html * http://www.learnthermo.com/examples/ch05/p-5c-2.php Category:Steam turbines Its theoretical efficiency depends on the compression ratio r of the engine and the specific heat ratio γ of the gas in the combustion chamber. \eta_{\rm th} = 1 - \frac{1}{r^{\gamma-1}} Thus, the efficiency increases with the compression ratio. The efficiency of steam engines is primarily related to the steam temperature and pressure and the number of stages or expansions. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. See graph:Steam Engine Efficiency In earliest steam engines the boiler was considered part of the engine. As is the case with the gas turbine, the steam turbine works most efficiently at full power, and poorly at slower speeds. The efficiency depends largely on the ratio of the pressure inside the combustion chamber p2 to the pressure outside p1 \eta_{\rm th} = 1 - \left(\frac{p_2}{p_1}\right)^\frac{1-\gamma}{\gamma} ===Other inefficiencies=== One should not confuse thermal efficiency with other efficiencies that are used when discussing engines. In a combined cycle plant, thermal efficiencies approach 60%.GE Power’s H Series Turbine Such a real-world value may be used as a figure of merit for the device. One other factor negatively affecting the gas turbine efficiency is the ambient air temperature. Steam engine efficiency improved as the operating principles were discovered, which led to the development of the science of thermodynamics. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. A very well-designed and built steam locomotive used to get around 7-8% efficiency in its heyday. Today they are considered separate, so it is necessary to know whether stated efficiency is overall, which includes the boiler, or just of the engine. Engines in large diesel trucks, buses, and newer diesel cars can achieve peak efficiencies around 45%. ===Gas turbine=== The gas turbine is most efficient at maximum power output in the same way reciprocating engines are most efficient at maximum load. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature T_{\rm H}, and removed at the minimum temperature T_{\rm C}. Comparisons of efficiency and power of the early steam engines is difficult for several reasons: 1) there was no standard weight for a bushel of coal, which could be anywhere from 82 to 96 pounds (37 to 44 kg). Despite of being at very low pressure the exhaust coming out of the turbine and entering the condenser carries some of kinetic energy and useful enthalpy, which is direct energy loss. ==Radiation and convection losses== The steam turbine operates at a relatively high temperature; therefore some of the heat energy of steam is radiated and convected from the body of the turbine to its surrounding. In thermodynamics, the thermal efficiency (\eta_{\rm th}) is a dimensionless performance measure of a device that uses thermal energy, such as an internal combustion engine, steam turbine, steam engine, boiler, furnace, refrigerator, ACs etc. The most efficient reciprocating steam engine design (per stage) was the uniflow engine, but by the time it appeared steam was being displaced by diesel engines, which were even more efficient and had the advantages of requiring less labor (for coal handling and oiling), being a more dense fuel, and displaced less cargo. ====Steam turbine==== The steam turbine is the most efficient steam engine and for this reason is universally used for electrical generation. ",0.9731,22,"""0.38""",35.64, 6.6,C
+"The discovery of the element argon by Lord Rayleigh and Sir William Ramsay had its origins in Rayleigh's measurements of the density of nitrogen with an eye toward accurate determination of its molar mass. Rayleigh prepared some samples of nitrogen by chemical reaction of nitrogencontaining compounds; under his standard conditions, a glass globe filled with this 'chemical nitrogen' had a mass of $2.2990 \mathrm{~g}$. He prepared other samples by removing oxygen, carbon dioxide, and water vapour from atmospheric air; under the same conditions, this 'atmospheric nitrogen' had a mass of $2.3102 \mathrm{~g}$ (Lord Rayleigh, Royal Institution Proceedings 14, 524 (1895)). With the hindsight of knowing accurate values for the molar masses of nitrogen and argon, compute the mole fraction of argon in the latter sample on the assumption that the former was pure nitrogen and the latter a mixture of nitrogen and argon.","Argon was first isolated from air in 1894 by Lord Rayleigh and Sir William Ramsay at University College London by removing oxygen, carbon dioxide, water, and nitrogen from a sample of clean air. After the two men identified argon, Ramsay investigated other atmospheric gases. Until 1957, the symbol for argon was ""A"", but now it is ""Ar"". ==Occurrence== Argon constitutes 0.934% by volume and 1.288% by mass of Earth's atmosphere. Most of the argon in Earth's atmosphere was produced by electron capture of long-lived ( + e− → + ν) present in natural potassium within Earth. Before isolating the gas, they had determined that nitrogen produced from chemical compounds was 0.5% lighter than nitrogen from the atmosphere. It forms at pressures between 4.3 and 220 GPa, though Raman measurements suggest that the H2 molecules in Ar(H2)2 dissociate above 175 GPa. ==Production== Argon is extracted industrially by the fractional distillation of liquid air in a cryogenic air separation unit; a process that separates liquid nitrogen, which boils at 77.3 K, from argon, which boils at 87.3 K, and liquid oxygen, which boils at 90.2 K. The content of 39Ar in natural argon is measured to be of (8.0±0.6)×10−16 g/g, or (1.01±0.08) Bq/kg of 36, 38, 40Ar. Almost all of the argon in the Earth's atmosphere is the product of 40K decay, since 99.6% of Earth atmospheric argon is 40Ar, whereas in the Sun and presumably in primordial star-forming clouds, argon consists of < 15% 38Ar and mostly (85%) 36Ar. Argon is the most abundant noble gas in Earth's crust, comprising 0.00015% of the crust. Sir William Ramsay (; 2 October 1852 – 23 July 1916) was a Scottish chemist who discovered the noble gases and received the Nobel Prize in Chemistry in 1904 ""in recognition of his services in the discovery of the inert gaseous elements in air"" along with his collaborator, John William Strutt, 3rd Baron Rayleigh, who received the Nobel Prize in Physics that same year for their discovery of argon. The predominance of radiogenic is the reason the standard atomic weight of terrestrial argon is greater than that of the next element, potassium, a fact that was puzzling when argon was discovered. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). Earth's crust and seawater contain 1.2 ppm and 0.45 ppm of argon, respectively. ==Isotopes== The main isotopes of argon found on Earth are (99.6%), (0.34%), and (0.06%). Argon is a chemical element with the symbol Ar and atomic number 18. This discovery caused the recognition that argon could form weakly bound compounds, even though it was not the first. Argon (18Ar) has 26 known isotopes, from 29Ar to 54Ar and 1 isomer (32mAr), of which three are stable (36Ar, 38Ar, and 40Ar). Before 1962, argon and the other noble gases were considered to be chemically inert and unable to form compounds; however, compounds of the heavier noble gases have since been synthesized. Nearly all of the argon in Earth's atmosphere is radiogenic argon-40, derived from the decay of potassium-40 in Earth's crust. Correspondingly, solar argon contains 84.6% (according to solar wind measurements), and the ratio of the three isotopes 36Ar : 38Ar : 40Ar in the atmospheres of the outer planets is 8400 : 1600 : 1. * On triple point pressure at 83.8058 K. ==External links== * Argon at The Periodic Table of Videos (University of Nottingham) * USGS Periodic Table – Argon * Diving applications: Why Argon? Rayleigh had noticed a discrepancy between the density of nitrogen made by chemical synthesis and nitrogen isolated from the air by removal of the other known components. That proposed element was named gnomium. ",5.1,200,"""0.011""",8,4.5,C
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $w$.","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. A carbon dioxide generator or CO2 generator is a machine used to enhance carbon dioxide levels in order to promote plant growth in greenhouses or other enclosed areas. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. Accumulation of carbon dioxide is predominantly a result of hypoperfusion and not hypoxia. Gastric tonometry describes the measurement of the carbon dioxide level inside the stomach in order to assess the degree of blood flow to the stomach and bowel. Normocapnia or normocarbia is a state of normal arterial carbon dioxide pressure, usually about 40 mmHg. == See also == * * * == References == *The Free Dictionary - Normocapnia Category:Diving medicine Category:Pulmonology A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. For gases that remain gaseous under ambient storage conditions, the upstream pressure gauge can be used to estimate how much gas is left in the cylinder according to pressure. The manifold may be arranged to allow simultaneous flow from all the cylinders, or, for a cascade filling system, where gas is tapped off cylinders according to the lowest positive pressure difference between storage and destination cylinder, being a more efficient use of pressurised gas. === Gas storage quads === thumb|Helium quad for surface-supplied diving gas A gas quad is a group of high pressure cylinders mounted on a transport and storage frame. Each cylinder shall be painted externally in the colours corresponding to its gaseous contents. == Common sizes == The below are example cylinder sizes and do not constitute an industry standard. size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) Internal volume, 70 °F (21 °C), 1atm Internal volume, 70 °F (21 °C), 1atm U.S. DOT specs size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) (liters) (cu.ft) U.S. DOT specs 2HP 43.3 1.53 3AA3500 K 110 49.9 1.76 3AA2400 A 96 43.8 1.55 3AA2015 B 37.9 17.2 0.61 3AA2015 C 15.2 6.88 0.24 3AA2015 D 4.9 2.24 0.08 3AA2015 AL 64.8 29.5 1.04 3AL2015 BL 34.6 15.7 0.55 3AL2216 CL 13 5.9 0.21 3AL2216 XL 238 108 3.83 4BA240 SSB 41.6 18.9 0.67 3A1800 10S 8.3 3.8 0.13 3A1800 LB 1 0.44 0.016 3E1800 XF 60.9 2.15 8AL XG 278 126.3 4.46 4AA480 XM 120 54.3 1.92 3A480 XP 124 55.7 1.98 4BA300 QT (includes 4.5 inches for valve) (includes 1.5 lb for valve) 2.0 0.900 0.0318 4B-240ET LP5 47.7 21.68 0.76 4BW240 Medical E (excludes valve and cap) (excludes valve and cap) 9.9 4.5 0.16 3AA2015 (US DOT specs define material, making, and maximum pressure in psi. Carbon tetroxide or Oxygen carbonate (in its C2v isomer) is a highly unstable oxide of carbon with formula . Because the introduction of a nasogastric tube is almost routine in critically ill patients, the measurement of gastric carbon dioxide can be an easy method to monitor tissue perfusion. It was proposed as an intermediate in the O-atom exchange between carbon dioxide () and oxygen () at high temperatures. Tonometry is based on the principle that at equilibrium the partial pressure of a diffusible gas such as CO2 is the same in both the wall and lumen of a viscus. Using compressed CO2 is an alternative to generators. == See also == ==References== Category:Horticulture Category:Industrial gases A typical gas cylinder design is elongated, standing upright on a flattened bottom end, with the valve and fitting at the top for connecting to the receiving apparatus. * ISO 11439: Gas cylinders — High-pressure cylinders for the on-board storage of natural gas as a fuel for automotive vehicles * ISO 15500-5: Road vehicles — Compressed natural gas (CNG) fuel system components — Part 5: Manual cylinder valve * US DOT CFR Title 49, part 178, Subpart C — Specification for CylindersUS DOT e-CFR (Electronic Code of Federal Regulations) Title 49, part 178, Subpart C — Specification for Cylinders — eg DOT 3AL = seamless aluminum * US DOT Aluminum Tank Alloy 6351-T6 amendment for SCUBA, SCBA, Oxygen Service — Visual Eddy inspectionFederal Register / Vol. 71, No. 167 / Tuesday, August 29, 2006 / Rules and Regulations Title 49 CFR Parts 173 and 180 Visual Edddy * AS 2896-2011:Medical gas systems—Installation and testing of non-flammable medical gas pipeline systems pipeline systems (Australian Standards). === Color coding === Gas cylinders are often color- coded, but the codes are not standard across different jurisdictions, and sometimes are not regulated. High-pressure gas cylinders are also called bottles. The regulator is adjusted to control the downstream pressure, which will limit the maximum flow of gas out of the cylinder at the pressure shown by the downstream gauge. Pressure vessels for gas storage may also be classified by volume. ",4.16,-20,"""0.264""", -2.5,15.425,B
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When a certain freon used in refrigeration was expanded adiabatically from an initial pressure of $32 \mathrm{~atm}$ and $0^{\circ} \mathrm{C}$ to a final pressure of $1.00 \mathrm{~atm}$, the temperature fell by $22 \mathrm{~K}$. Calculate the Joule-Thomson coefficient, $\mu$, at $0^{\circ} \mathrm{C}$, assuming it remains constant over this temperature range.","The temperature change produced during a Joule–Thomson expansion is quantified by the Joule–Thomson coefficient, \mu_{\mathrm{JT}}. This equation can be used to obtain Joule–Thomson coefficients from the more easily measured isothermal Joule–Thomson coefficient. The temperature of this point, the Joule–Thomson inversion temperature, depends on the pressure of the gas before expansion. Since this is true at all temperatures for ideal gases (see expansion in gases), the Joule–Thomson coefficient of an ideal gas is zero at all temperatures. ==Joule's second law== It is easy to verify that for an ideal gas defined by suitable microscopic postulates that αT = 1, so the temperature change of such an ideal gas at a Joule–Thomson expansion is zero. Thus, for N2 gas below 621 K, a Joule–Thomson expansion can be used to cool the gas until liquid N2 forms. ==Physical mechanism== There are two factors that can change the temperature of a fluid during an adiabatic expansion: a change in internal energy or the conversion between potential and kinetic internal energy. This expression can now replace \mu_{\mathrm{T}} in the earlier equation for \mu_{\mathrm{JT}} to obtain: :\mu_{\mathrm{JT}} \equiv \left( \frac{\partial T}{\partial P} \right)_H = \frac V {C_{\mathrm{p}}} (\alpha T - 1).\, This provides an expression for the Joule–Thomson coefficient in terms of the commonly available properties heat capacity, molar volume, and thermal expansion coefficient. The first step in obtaining these results is to note that the Joule–Thomson coefficient involves the three variables T, P, and H. In a Joule–Thomson expansion the enthalpy remains constant. The physical mechanism associated with the Joule–Thomson effect is closely related to that of a shock wave, although a shock wave differs in that the change in bulk kinetic energy of the gas flow is not negligible. ==The Joule–Thomson (Kelvin) coefficient== thumb|400px|Fig. 1 – Joule–Thomson coefficients for various gases at atmospheric pressure The rate of change of temperature T with respect to pressure P in a Joule–Thomson process (that is, at constant enthalpy H) is the Joule–Thomson (Kelvin) coefficient \mu_{\mathrm{JT}}. At room temperature, all gases except hydrogen, helium, and neon cool upon expansion by the Joule–Thomson process when being throttled through an orifice; these three gases experience the same effect but only at lower temperatures. This produces a decrease in temperature and results in a positive Joule–Thomson coefficient. The cooling produced in the Joule–Thomson expansion makes it a valuable tool in refrigeration.Keenan, J.H. (1970). This means that the mass fraction of the liquid in the liquid–gas mixture leaving the throttling valve is 40%. ==Derivation of the Joule–Thomson coefficient== It is difficult to think physically about what the Joule–Thomson coefficient, \mu_{\mathrm{JT}}, represents. For an ideal gas, \mu_\text{JT} is always equal to zero: ideal gases neither warm nor cool upon being expanded at constant enthalpy. ==Applications== In practice, the Joule–Thomson effect is achieved by allowing the gas to expand through a throttling device (usually a valve) which must be very well insulated to prevent any heat transfer to or from the gas. At high temperature, Z and PV decrease as the gas expands; if the decrease is large enough, the Joule–Thomson coefficient will be negative. Thus at low temperature, Z and PV will increase as the gas expands, resulting in a positive Joule–Thomson coefficient. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). This coefficient can be expressed in terms of the gas's volume V, its heat capacity at constant pressure C_{\mathrm{p}}, and its coefficient of thermal expansion \alpha as: :\mu_{\mathrm{JT}} = \left( {\partial T \over \partial P} \right)_H = \frac V {C_{\mathrm{p}}}(\alpha T - 1)\, See the below for the proof of this relation. With that in mind, the following table explains when the Joule–Thomson effect cools or warms a real gas: If the gas temperature is then \mu_\text{JT} is since \partial P is thus \partial T must be so the gas below the inversion temperature positive always negative negative cools above the inversion temperature negative always negative positive warms Helium and hydrogen are two gases whose Joule–Thomson inversion temperatures at a pressure of one atmosphere are very low (e.g., about 45 K, −228 °C for helium). Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). In a Joule–Thomson process the specific enthalpy h remains constant.See e.g. M.J. Moran and H.N. Shapiro ""Fundamentals of Engineering Thermodynamics"" 5th Edition (2006) John Wiley & Sons, Inc. page 147 To prove this, the first step is to compute the net work done when a mass m of the gas moves through the plug. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. ",0.444444444444444 ,0.9992093669,"""3.2""",-1.32,0.71,E
 "Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The volume of a certain liquid varies with temperature as
 $$
 V=V^{\prime}\left\{0.75+3.9 \times 10^{-4}(T / \mathrm{K})+1.48 \times 10^{-6}(T / \mathrm{K})^2\right\}
 $$
-where $V^{\prime}$ is its volume at $300 \mathrm{~K}$. Calculate its expansion coefficient, $\alpha$, at $320 \mathrm{~K}$.","== Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The expansion ratio of a liquefied and cryogenic substance is the volume of a given amount of that substance in liquid form compared to the volume of the same amount of substance in gaseous form, at room temperature and normal atmospheric pressure. We assume the expansion occurs without exchange of heat (adiabatic expansion). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Therefore, the heat capacity ratio in this example is 1.4. An experimental value should be used rather than one based on this approximation, where possible. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. The analysis here shows the simple unattributed formula and the Antoine equation are reasonably accurate at 100 °C, but quite poor for lower temperatures above freezing. The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. Hence the use of pressure relief valves and vent valves are important.Safetygram-27 Cryogenic Liquid Containers The expansion ratio of liquefied and cryogenic from the boiling point to ambient is: *nitrogen – 1 to 696 *liquid helium – 1 to 745 *argon – 1 to 842 *liquid hydrogen – 1 to 850 *liquid oxygen – 1 to 860 *neon – Neon has the highest expansion ratio with 1 to 1445. ==See also== *Liquid-to-gas ratio *Boiling liquid expanding vapor explosion *Thermal expansion ==References== ==External links== *cryogenic-gas- hazards Category:Cryogenics Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. ",0.00131,5840,12.0,29.36,0.03,A
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The standard enthalpy of formation of the metallocene bis(benzene)chromium was measured in a calorimeter. It was found for the reaction $\mathrm{Cr}\left(\mathrm{C}_6 \mathrm{H}_6\right)_2(\mathrm{~s}) \rightarrow \mathrm{Cr}(\mathrm{s})+2 \mathrm{C}_6 \mathrm{H}_6(\mathrm{~g})$ that $\Delta_{\mathrm{r}} U^{\bullet}(583 \mathrm{~K})=+8.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Estimate the standard enthalpy of formation of the compound at $583 \mathrm{~K}$. The constant-pressure molar heat capacity of benzene is $136.1 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ in its liquid range and $81.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ as a gas.","Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. The standard enthalpy of formation is then determined using Hess's law. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. The heat of reaction is then minus the sum of the standard enthalpies of formation of the reactants (each being multiplied by its respective stoichiometric coefficient, ) plus the sum of the standard enthalpies of formation of the products (each also multiplied by its respective stoichiometric coefficient), as shown in the equation below: :\Delta_{\text{r}} H^{\ominus } = \sum u \Delta_{\text{f}} H^{\ominus }(\text{products}) - \sum u \Delta_{\text{f}} H^{\ominus}(\text{reactants}). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. From these sources: :*a - K.K. Kelley, Bur. Mines Bull. 383, (1935). :*b - :*c - Brewer, The thermodynamic and physical properties of the elements, Report for the Manhattan Project, (1946). :*d - :*e - :*f - :*g - ; :*h - :*i - . :*j - :*k - Int. National Critical Tables, vol. 3, p. 306, (1928). :*l - :*m - :*n - :*o - :*p - :*q - :*r - :*s - H.A. Jones, I. Langmuir, Gen. Electric Rev., vol. 30, p. 354, (1927). == See also == Category:Properties of chemical elements Category:Chemical element data pages This is true for all enthalpies of formation. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. ",0.33333333,7200,116.0,1068,4,C
-"A car tyre (i.e. an automobile tire) was inflated to a pressure of $24 \mathrm{lb} \mathrm{in}^{-2}$ $(1.00 \mathrm{atm}=14.7 \mathrm{lb} \mathrm{in}^{-2})$ on a winter's day when the temperature was $-5^{\circ} \mathrm{C}$. What pressure will be found, assuming no leaks have occurred and that the volume is constant, on a subsequent summer's day when the temperature is $35^{\circ} \mathrm{C}$?","So if the tire was filled at 80 °F to 32 psi (or 47 psi absolute when we add atmospheric pressure), the change would be 4.7 psi for this 30 Celsius degree change, or 0.16 psi per Celsius degree or 0.1 psi per Fahrenheit degree or 1 psi for every 10 Fahrenheit degrees. Hence, for a tire filled to 32 psi, the approximation usually made is that within the range of normal atmospheric temperatures and pressures: Tire pressure increases 1 psi for each 10 Fahrenheit degree increase in temperature, or conversely decreases 1 psi for each 10 Fahrenheit degree decrease in temperature and in SI units, tire pressure increases 1.1 kPa for each 1 Celsius degree increase in temperature, or conversely decreases 1.1 kPa for each 1 Celsius degree decrease in temperature. Ambient temperature affects the cold tire pressure. To understand this, assume the tire was filled when it was 300 kelvin (approximately 27 degrees Celsius or 80 degrees Fahrenheit). From the table below, one can see that these are only approximations: == Variation of tire pressure with temperature in Fahrenheit and Celsius == (Assuming atmospheric pressure is 14.696 psi, or 101.3 kPA.) Most passenger cars are recommended to have a tire pressure of 30 to 35 pounds per square inch when not warmed by driving. 40% of passenger cars have at least one tire under-inflated by 6 psi or more. Cold tire absolute pressure (gauge pressure plus atmospheric pressure) varies directly with the absolute temperature, measured in kelvin. For tires that need inflation greater than 32 psi it might be easier to use a Rule of Thumb of 2% pressure change for a change of 10 degrees Fahrenheit. Cold inflation pressure is the inflation pressure of tires before a car is driven and the tires (tyres) warmed up. From physics, the ideal gas law states that PV = nRT, where P is absolute pressure, T is absolute temperature, V is the volume (assumed to be relatively constant in the case of a tire), and nR is constant for a given number of molecules of gas. The European Union concludes that tire under-inflation today is responsible for over 20 million liters of unnecessarily-burned fuel, dumping over 2 million tonnes of CO2 into the atmosphere, and for 200 million tires being prematurely wasted worldwide. If the temperature varies 10% (i.e., by 30 kelvins [also 30 degrees Celsius or 54 degrees Fahrenheit]), the pressure varies 10%. Tires do not only leak air if punctured, they also leak air naturally, and over a year, even a typical new, properly mounted tire can lose from 20 to 60 kPa (3 to 9 psi), roughly 10% or even more of its initial pressure. * Environmental efficiency: Under-inflated tires, as estimated by the US Department of Transportation, release over 26 billion kilograms (57.5 billion pounds) of unnecessary carbon-monoxide pollutants into the atmosphere each year in the United States alone. Cold inflation may refer to: * Cold inflation pressure, the pressure in tires before they are warmed up by the car's motion; * One of the two dynamical realizations of cosmological inflation the other being warm inflation. Some units also measure and alert temperatures of the tire as well. These systems can identify under-inflation for each individual tire. Extreme under-inflation can even lead to thermal and mechanical overload caused by overheating and subsequent, sudden destruction of the tire itself. Further, a difference of in pressure on a set of duals literally drags the lower pressured tire 2.5 metres per kilometre (13 feet per mile). The European Union reports that an average under-inflation of 40 kPa produces an increase of fuel consumption of 2% and a decrease of tire life of 25%. The frost point for a given parcel of air is always higher than the dew point, as breaking the stronger bonding between water molecules on the surface of ice compared to the surface of (supercooled) liquid water requires a higher temperature. == See also == * Bubble point * Carburetor heat * Hydrocarbon dew point * Psychrometrics * Thermodynamic diagrams ==References== ==External links== * Often Needed Answers about Temp, Humidity & Dew Point from the sci.geo.meteorology Category:Atmospheric thermodynamics Category:Temperature Category:Psychrometrics Category:Threshold temperatures Category:Gases Category:Meteorological data and networks Category:Humidity and hygrometry sv:Luftfuktighet#Daggpunkt The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. ",30,4.49,0.00131,2,0.5768,A
-Suppose that $10.0 \mathrm{~mol} \mathrm{C}_2 \mathrm{H}_6(\mathrm{~g})$ is confined to $4.860 \mathrm{dm}^3$ at $27^{\circ} \mathrm{C}$. Predict the pressure exerted by the ethane from the van der Waals equations of state.,"The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. These van der Waals interactions are up to 40 times stronger than in H2, which has only one valence electron, and they are still not strong enough to achieve an aggregate state other than gas for Xe under standard conditions. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. Even with its acknowledged shortcomings, the pervasive use of the Van der Waals equation in standard university physical chemistry textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. * Van der Waals forces are independent of temperature except for dipole-dipole interactions. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . J/(mol K) Liquid properties Std enthalpy change of formation, ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy, S ~~o~~ liquid 126.7 J/(mol K) Heat capacity, cp 68.5 J/(mol K) at −179 °C Gas properties Std enthalpy change of formation, ΔfH ~~o~~ gas −83.8 kJ/mol Standard molar entropy, S ~~o~~ gas 229.6 J/(mol K) Enthalpy of combustion, ΔcH ~~o~~ −1560.7 kJ/mol Heat capacity, cp 52.49 J/(mol K) at 25 °C van der Waals' constantsLange's Handbook of Chemistry 10th ed, pp 1522-1524 a = 556.2 L2 kPa/mol2 b = 0.06380 L/mol ==Vapor pressure of liquid== P in mm Hg 1 10 40 100 400 760 1520 3800 7600 15200 30400 45600 T in °C −159.5 −142.9 −129.8 −119.3 −99.6 −88.6 −75.0 −52.8 −32.0 −6.4 23.6 — Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. == Melting point data == Mean value for acceptable data: −183.01 °C (90.14 K). The VRT spectroscopic study of Van der Waals molecules is one of the most direct routes to the understanding of intermolecular forces. == See also == * Van der Waals radius * Van der Waals strain * Van der Waals surface * –articles about specific chemicals * Researchers active in this field: ** Donald Levy ** Richard J. Saykally ** Richard Smalley ** William Klemperer == References == == Further reading == * So far three special issues of Chemical Reviews have been devoted to vdW molecules: I. Vol. 88(6) (1988). * Early reviews of vdW molecules: G. E. Ewing, Accounts of Chemical Research, Vol. 8, pp. 185-192, (1975): Structure and Properties of Van der Waals molecules. A Van der Waals molecule is a weakly bound complex of atoms or molecules held together by intermolecular attractions such as Van der Waals forces or by hydrogen bonds. ",-11.2,1410,1.4,35.2,4.5,D
-Use the van der Waals parameters for chlorine to calculate approximate values of the radius of a $\mathrm{Cl}_2$ molecule regarded as a sphere.,"Values from other sources may differ significantly (see text) The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It may be calculated for atoms if the Van der Waals radius is known, and for molecules if its atoms radii and the inter- atomic distances and angles are known. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. Petitjean, On the Analytical Calculation of Van der Waals Surfaces and Volumes: Some Numerical Aspects, Journal of Computational Chemistry, Volume 15, Number 5, 1994, pp. 507–523. == External links == * VSAs for various molecules by Anton Antonov, The Wolfram Demonstrations Project, 2007. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. * Analytical calculation of Van der Waals surfaces and volumes. Van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 Van der Waals radii taken from Bondi's compilation (1964). van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Useful tabulated values of Van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will be seen to present different values for the Van der Waals radius of the same atom. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. As well, it has been argued that the Van der Waals radius is not a fixed property of an atom in all circumstances, rather, that it will vary with the chemical environment of the atom. == Gallery == File:3D ammoniac.PNG|alt=Ammonia, van-der-Waals-based model|Ammonia, NH3, space- filling, Van der Waal's-based representation, nitrogen (N) in blue, hydrogen (H) in white. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. For a molecule, it is the volume enclosed by the van der Waals surface. Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",2,210,22.0,0.8561,0.139,E
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $\Delta U$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). * Values from CRC refer to ""100 kPa (1 bar or 0.987 standard atmospheres)"". Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. This error is corrected in the above equation. ==Measurement== The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. Lange indirectly defines the values to be at a standard state pressure of ""1 atm (101325 Pa)"", although citing the same NBS and JANAF sources among others. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: : \gamma = \frac{5}{3} = 1.6666\ldots, As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. ",524,+65.49,0.33333333,635.7,+4.1,E
-"1.7(a) In an attempt to determine an accurate value of the gas constant, $R$, a student heated a container of volume $20.000 \mathrm{dm}^3$ filled with $0.25132 \mathrm{g}$ of helium gas to $500^{\circ} \mathrm{C}$ and measured the pressure as $206.402 \mathrm{cm}$ of water in a manometer at $25^{\circ} \mathrm{C}$. Calculate the value of $R$ from these data. (The density of water at $25^{\circ} \mathrm{C}$ is $0.99707 \mathrm{g} \mathrm{cm}^{-3}$; a manometer consists of a U-shaped tube containing a liquid. One side is connected to the apparatus and the other is open to the atmosphere. The pressure inside the apparatus is then determined from the difference in heights of the liquid.)","Note that data could have been collected with three different amounts of the same gas, which would have rendered this experiment easy to do in the eighteenth century. ==History== == See also == * Thermodynamic instruments * Boyle's law * Combined gas law * Gay-Lussac's law * Avogadro's law * Ideal gas law ==References== Category:Thermometers Category:Gases fr:Thermomètre#Thermomètre à gaz thumb|400px|Diagram showing pressure difference induced by a temperature difference. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. :V \propto T\, or :\frac{V}{T}=k V is the volume, T is the thermodynamic temperature, k is the constant for the system. k is not a fixed constant across all systems and therefore needs to be found experimentally for a given system through testing with known temperature values. ==Pressure Thermometer and Absolute Zero== thumb|left|180px|Plots of pressure vs temperature for three different gas samples extrapolate to absolute zero. thumb|500px|Two variants of a gas thermometer A gas thermometer is a thermometer that measures temperature by the variation in volume or pressure of a gas. ==Volume Thermometer== This thermometer functions by Charles's Law. Gas volume corrector - device for calculating, summing and determining increments of gas volume, measured by gas meter if it were operating base conditions. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. Using Charles's Law, the temperature can be measured by knowing the volume of gas at a certain temperature by using the formula, written below. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The constant volume gas thermometer plays a crucial role in understanding how absolute zero could be discovered long before the advent of cryogenics. Consider a graph of pressure versus temperature made not far from standard conditions (well above absolute zero) for three different samples of any ideal gas (a, b, c). For this purpose, uses as input the gas volume, measured by the gas meter and other parameters such as: gas pressure and temperature. To the extent that the gas is ideal, the pressure depends linearly on temperature, and the extrapolation to zero pressure occurs at absolute zero. Charles's Law states that when the temperature of a gas increases, so does the volume. . The prover determines the meter factor, which is the volume of air passed divided by the volume of air measured.Fundamentals of Meter Provers and Proving Methods https://asgmt.com/wp-content/uploads/2018/05/070.pdf ==Types== ===Manual bell prover=== Since the early 1900s, bell provers have been the most common reference standard used in gas meter proving, and has provided standards for the gas industry that is unfortunately susceptible to a myriad of immeasurable uncertainties. Since atmospheric pressure, P, depends upon altitude, so does \gamma. There are two types of gas volume correctors: Type 1- gas volume corrector with specific types of transducers for pressure and temperature or temperature only. Although \left( c_p \right)_{H_2 O} is constant, varied air composition results in varied \left( c_p \right)_{air} . A gas meter prover is a device to verify the accuracy of a gas meter. Translating it to the correct levels of the device that is holding the gas. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus When proving a meter using a manually controlled bell, an operator must first fill the bell with a controlled air supply or raise it manually by opening a valve and pulling a chained mechanism, seal the bell and meter and check the sealed system for leaks, determine the flow rate needed for the meter, install a special flow-rate cap on the meter outlet, note the starting points of both the bell scale and meter index, release the bell valve to pass air through the meter, observe the meter index and calculate the time it takes to pass the predetermined amount of air, then manually calculate the meter's proof accounting for bell air and meter temperature and in some cases other environmental factors. ",-6.42,0.66666666666,15.0,8.3147,1260,D
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The standard enthalpy of combustion of solid phenol $\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}\right)$ is $-3054 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}_{\text {and }}$ its standard molar entropy is $144.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Calculate the standard Gibbs energy of formation of phenol at $298 \mathrm{~K}$.,"Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. The standard enthalpy of formation is then determined using Hess's law. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. ",-50,0.3085,537.0,200, 258.14,A
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $\Delta U$.","In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We assume the expansion occurs without exchange of heat (adiabatic expansion). The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. For real gases the equation of state will depart from the simpler one, and the result above derived for an ideal gas will only be a good approximation provided that (a) the typical size of the molecule is negligible compared to the average distance between the individual molecules, and (b) the short range behavior of the inter-molecular potential can be neglected, i.e., when the molecules can be considered to rebound elastically off each other during molecular collisions. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Since the piston cannot move, the volume is constant. CO-0.40-0.22 is a high velocity compact gas cloud near the centre of the Milky Way. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. The fugacities commonly obey a similar law called the Lewis and Randall rule: f_i = y_i f^*_i, where is the fugacity that component would have if the entire gas had that composition at the same temperature and pressure. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. thumb|Jets of liquid carbon dioxide Liquid carbon dioxide is the liquid state of carbon dioxide (), which cannot occur under atmospheric pressure. The solubility turned out to be very low: from 0.02 to 0.10 %. thumb|Carbon dioxide pressure-temperature phase diagram == Uses == thumb|219x219px|Fire extinguisher Uses of liquid carbon dioxide include the preservation of food, in fire extinguishers, and in commercial food processes. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. ",−1.642876,0.042,16.0,-59.24,-20,E
-"A manometer consists of a U-shaped tube containing a liquid. One side is connected to the apparatus and the other is open to the atmosphere. The pressure inside the apparatus is then determined from the difference in heights of the liquid. Suppose the liquid is water, the external pressure is 770 Torr, and the open side is $10.0 \mathrm{cm}$ lower than the side connected to the apparatus. What is the pressure in the apparatus? (The density of water at $25^{\circ} \mathrm{C}$ is $0.99707 \mathrm{g} \mathrm{cm}^{-3}$.)","The open end of the manometer is then connected to a pressure measuring device. Therefore, the pressure difference between the applied pressure Pa and the reference pressure P0 in a U-tube manometer can be found by solving . The pressure at the bottom of the barometer, Point B, is equal to the atmospheric pressure. It consists of a submerged manometer and container holding the substance whose vapor pressure is being measured. Manometric measurement is the subject of pressure head calculations. A single-limb liquid-column manometer has a larger reservoir instead of one side of the U-tube and has a scale beside the narrower column. Simple hydrostatic gauges can measure pressures ranging from a few torrs (a few 100 Pa) to a few atmospheres (approximately ). Bourdon tube pressure gauges. If the fluid being measured is significantly dense, hydrostatic corrections may have to be made for the height between the moving surface of the manometer working fluid and the location where the pressure measurement is desired, except when measuring differential pressure of a fluid (for example, across an orifice plate or venturi), in which case the density ρ should be corrected by subtracting the density of the fluid being measured. There are three different types of pressuremeters. The difference in liquid levels represents the applied pressure. The pressuremeter has two major components. Typically, atmospheric pressure is measured between and of Hg. A very simple version is a U-shaped tube half-full of liquid, one side of which is connected to the region of interest while the reference pressure (which might be the atmospheric pressure or a vacuum) is applied to the other. They have poor dynamic response. ====Piston==== Piston-type gauges counterbalance the pressure of a fluid with a spring (for example tire- pressure gauges of comparatively low accuracy) or a solid weight, in which case it is known as a deadweight tester and may be used for calibration of other gauges. ====Liquid column (manometer)==== thumb|upright|The difference in fluid height in a liquid-column manometer is proportional to the pressure difference: h = \frac{P_a - P_o}{g \rho} Liquid-column gauges consist of a column of liquid in a tube whose ends are exposed to different pressures. The pressure inside the probe is held constant for a specific period of time and the increase in volume required to maintain the pressure is recorded. thumb|Barometer A barometer is a scientific instrument that is used to measure air pressure in a certain environment. Gauges that rely on a change in capacitance are often referred to as capacitance manometers. ====Bourdon tube==== thumb|Membrane-type manometer The Bourdon pressure gauge uses the principle that a flattened tube tends to straighten or regain its circular form in cross-section when pressurized. A pressuremeter is a meter constructed to measure the “at-rest horizontal earth pressure”. The word ""gauge"" or ""vacuum"" may be added to such a measurement to distinguish between a pressure above or below the atmospheric pressure. Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure. It has a pressure resolution of approximately 1mm of water when measuring pressure at a depth of several kilometers. ",32,0.9984,102.0,-100,4.8,C
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The standard enthalpy of formation of the metallocene bis(benzene)chromium was measured in a calorimeter. It was found for the reaction $\mathrm{Cr}\left(\mathrm{C}_6 \mathrm{H}_6\right)_2(\mathrm{~s}) \rightarrow \mathrm{Cr}(\mathrm{s})+2 \mathrm{C}_6 \mathrm{H}_6(\mathrm{~g})$ that $\Delta_{\mathrm{r}} U^{\bullet}(583 \mathrm{~K})=+8.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Find the corresponding reaction enthalpy and estimate the standard enthalpy of formation of the compound at $583 \mathrm{~K}$. The constant-pressure molar heat capacity of benzene is $136.1 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ in its liquid range and $81.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ as a gas.","Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). The standard enthalpy of formation is then determined using Hess's law. * Standard enthalpy of formation is the enthalpy change when one mole of any compound is formed from its constituent elements in their standard states. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. The standard enthalpy of formation, which has been determined for a vast number of substances, is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements, with all substances in their standard states. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. ",+17.7,3.38,0.2553,0.11,1.95 ,A
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the Carnot efficiency of a primitive steam engine operating on steam at $100^{\circ} \mathrm{C}$ and discharging at $60^{\circ} \mathrm{C}$.,"Latest generation gas turbine engines have achieved an efficiency of 46% in simple cycle and 61% when used in combined cycle. ==External combustion engines== ===Steam engine=== ::See also: Steam engine#Efficiency ::See also: Timeline of steam power ====Piston engine==== Steam engines and turbines operate on the Rankine cycle which has a maximum Carnot efficiency of 63% for practical engines, with steam turbine power plants able to achieve efficiency in the mid 40% range. While gasoline-powered ICE cars have an operational thermal efficiency of 15% to 30%, early automotive steam units were capable of only about half this efficiency. Its theoretical efficiency depends on the compression ratio r of the engine and the specific heat ratio γ of the gas in the combustion chamber. \eta_{\rm th} = 1 - \frac{1}{r^{\gamma-1}} Thus, the efficiency increases with the compression ratio. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The efficiency depends largely on the ratio of the pressure inside the combustion chamber p2 to the pressure outside p1 \eta_{\rm th} = 1 - \left(\frac{p_2}{p_1}\right)^\frac{1-\gamma}{\gamma} ===Other inefficiencies=== One should not confuse thermal efficiency with other efficiencies that are used when discussing engines. Steam engine efficiency improved as the operating principles were discovered, which led to the development of the science of thermodynamics. The efficiency of steam engines is primarily related to the steam temperature and pressure and the number of stages or expansions. See graph:Steam Engine Efficiency In earliest steam engines the boiler was considered part of the engine. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The first piston steam engine, developed by Thomas Newcomen around 1710, was slightly over one half percent (0.5%) efficient. Some steam enthusiasts feel steam has not received its share of attention in the field of automobile efficiency. A very well-designed and built steam locomotive used to get around 7-8% efficiency in its heyday. The above efficiency formulas are based on simple idealized mathematical models of engines, with no friction and working fluids that obey simple thermodynamic rules called the ideal gas law. Today they are considered separate, so it is necessary to know whether stated efficiency is overall, which includes the boiler, or just of the engine. Comparisons of efficiency and power of the early steam engines is difficult for several reasons: 1) there was no standard weight for a bushel of coal, which could be anywhere from 82 to 96 pounds (37 to 44 kg). The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature T_{\rm H}, and removed at the minimum temperature T_{\rm C}. From Carnot's theorem, for any engine working between these two temperatures: :\eta_{\rm th} \le 1 - \frac{T_{\rm C}}{T_{\rm H}} This limiting value is called the Carnot cycle efficiency because it is the efficiency of an unattainable, ideal, reversible engine cycle called the Carnot cycle. Practical engine cycles are irreversible and thus have inherently lower efficiency than the Carnot efficiency when operated between the same temperatures T_{\rm H} and T_{\rm C}. For example, the average automobile engine is less than 35% efficient. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. ",52,5.51,4500.0,129,0.11,E
-"In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ at a constant volume of $1.000 \mathrm{~m}^3$. The gas enters the container at $300 \mathrm{~K}$ and $100 \mathrm{~atm}$. The mass of the gas is $92.4 \mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \mathrm{~K}$. For nitrogen, $a=1.352 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}, b=0.0387 \mathrm{dm}^3 \mathrm{~mol}^{-1}$.","Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. * Analytical calculation of Van der Waals surfaces and volumes. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . ",-11.875,140,131.0,655, 13.45,B
+where $V^{\prime}$ is its volume at $300 \mathrm{~K}$. Calculate its expansion coefficient, $\alpha$, at $320 \mathrm{~K}$.","== Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The expansion ratio of a liquefied and cryogenic substance is the volume of a given amount of that substance in liquid form compared to the volume of the same amount of substance in gaseous form, at room temperature and normal atmospheric pressure. We assume the expansion occurs without exchange of heat (adiabatic expansion). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Therefore, the heat capacity ratio in this example is 1.4. An experimental value should be used rather than one based on this approximation, where possible. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. The analysis here shows the simple unattributed formula and the Antoine equation are reasonably accurate at 100 °C, but quite poor for lower temperatures above freezing. The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. Hence the use of pressure relief valves and vent valves are important.Safetygram-27 Cryogenic Liquid Containers The expansion ratio of liquefied and cryogenic from the boiling point to ambient is: *nitrogen – 1 to 696 *liquid helium – 1 to 745 *argon – 1 to 842 *liquid hydrogen – 1 to 850 *liquid oxygen – 1 to 860 *neon – Neon has the highest expansion ratio with 1 to 1445. ==See also== *Liquid-to-gas ratio *Boiling liquid expanding vapor explosion *Thermal expansion ==References== ==External links== *cryogenic-gas- hazards Category:Cryogenics Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. ",0.00131,5840,"""12.0""",29.36,0.03,A
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The standard enthalpy of formation of the metallocene bis(benzene)chromium was measured in a calorimeter. It was found for the reaction $\mathrm{Cr}\left(\mathrm{C}_6 \mathrm{H}_6\right)_2(\mathrm{~s}) \rightarrow \mathrm{Cr}(\mathrm{s})+2 \mathrm{C}_6 \mathrm{H}_6(\mathrm{~g})$ that $\Delta_{\mathrm{r}} U^{\bullet}(583 \mathrm{~K})=+8.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Estimate the standard enthalpy of formation of the compound at $583 \mathrm{~K}$. The constant-pressure molar heat capacity of benzene is $136.1 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ in its liquid range and $81.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ as a gas.","Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. The standard enthalpy of formation is then determined using Hess's law. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. The heat of reaction is then minus the sum of the standard enthalpies of formation of the reactants (each being multiplied by its respective stoichiometric coefficient, ) plus the sum of the standard enthalpies of formation of the products (each also multiplied by its respective stoichiometric coefficient), as shown in the equation below: :\Delta_{\text{r}} H^{\ominus } = \sum u \Delta_{\text{f}} H^{\ominus }(\text{products}) - \sum u \Delta_{\text{f}} H^{\ominus}(\text{reactants}). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. From these sources: :*a - K.K. Kelley, Bur. Mines Bull. 383, (1935). :*b - :*c - Brewer, The thermodynamic and physical properties of the elements, Report for the Manhattan Project, (1946). :*d - :*e - :*f - :*g - ; :*h - :*i - . :*j - :*k - Int. National Critical Tables, vol. 3, p. 306, (1928). :*l - :*m - :*n - :*o - :*p - :*q - :*r - :*s - H.A. Jones, I. Langmuir, Gen. Electric Rev., vol. 30, p. 354, (1927). == See also == Category:Properties of chemical elements Category:Chemical element data pages This is true for all enthalpies of formation. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. ",0.33333333,7200,"""116.0""",1068,4,C
+"A car tyre (i.e. an automobile tire) was inflated to a pressure of $24 \mathrm{lb} \mathrm{in}^{-2}$ $(1.00 \mathrm{atm}=14.7 \mathrm{lb} \mathrm{in}^{-2})$ on a winter's day when the temperature was $-5^{\circ} \mathrm{C}$. What pressure will be found, assuming no leaks have occurred and that the volume is constant, on a subsequent summer's day when the temperature is $35^{\circ} \mathrm{C}$?","So if the tire was filled at 80 °F to 32 psi (or 47 psi absolute when we add atmospheric pressure), the change would be 4.7 psi for this 30 Celsius degree change, or 0.16 psi per Celsius degree or 0.1 psi per Fahrenheit degree or 1 psi for every 10 Fahrenheit degrees. Hence, for a tire filled to 32 psi, the approximation usually made is that within the range of normal atmospheric temperatures and pressures: Tire pressure increases 1 psi for each 10 Fahrenheit degree increase in temperature, or conversely decreases 1 psi for each 10 Fahrenheit degree decrease in temperature and in SI units, tire pressure increases 1.1 kPa for each 1 Celsius degree increase in temperature, or conversely decreases 1.1 kPa for each 1 Celsius degree decrease in temperature. Ambient temperature affects the cold tire pressure. To understand this, assume the tire was filled when it was 300 kelvin (approximately 27 degrees Celsius or 80 degrees Fahrenheit). From the table below, one can see that these are only approximations: == Variation of tire pressure with temperature in Fahrenheit and Celsius == (Assuming atmospheric pressure is 14.696 psi, or 101.3 kPA.) Most passenger cars are recommended to have a tire pressure of 30 to 35 pounds per square inch when not warmed by driving. 40% of passenger cars have at least one tire under-inflated by 6 psi or more. Cold tire absolute pressure (gauge pressure plus atmospheric pressure) varies directly with the absolute temperature, measured in kelvin. For tires that need inflation greater than 32 psi it might be easier to use a Rule of Thumb of 2% pressure change for a change of 10 degrees Fahrenheit. Cold inflation pressure is the inflation pressure of tires before a car is driven and the tires (tyres) warmed up. From physics, the ideal gas law states that PV = nRT, where P is absolute pressure, T is absolute temperature, V is the volume (assumed to be relatively constant in the case of a tire), and nR is constant for a given number of molecules of gas. The European Union concludes that tire under-inflation today is responsible for over 20 million liters of unnecessarily-burned fuel, dumping over 2 million tonnes of CO2 into the atmosphere, and for 200 million tires being prematurely wasted worldwide. If the temperature varies 10% (i.e., by 30 kelvins [also 30 degrees Celsius or 54 degrees Fahrenheit]), the pressure varies 10%. Tires do not only leak air if punctured, they also leak air naturally, and over a year, even a typical new, properly mounted tire can lose from 20 to 60 kPa (3 to 9 psi), roughly 10% or even more of its initial pressure. * Environmental efficiency: Under-inflated tires, as estimated by the US Department of Transportation, release over 26 billion kilograms (57.5 billion pounds) of unnecessary carbon-monoxide pollutants into the atmosphere each year in the United States alone. Cold inflation may refer to: * Cold inflation pressure, the pressure in tires before they are warmed up by the car's motion; * One of the two dynamical realizations of cosmological inflation the other being warm inflation. Some units also measure and alert temperatures of the tire as well. These systems can identify under-inflation for each individual tire. Extreme under-inflation can even lead to thermal and mechanical overload caused by overheating and subsequent, sudden destruction of the tire itself. Further, a difference of in pressure on a set of duals literally drags the lower pressured tire 2.5 metres per kilometre (13 feet per mile). The European Union reports that an average under-inflation of 40 kPa produces an increase of fuel consumption of 2% and a decrease of tire life of 25%. The frost point for a given parcel of air is always higher than the dew point, as breaking the stronger bonding between water molecules on the surface of ice compared to the surface of (supercooled) liquid water requires a higher temperature. == See also == * Bubble point * Carburetor heat * Hydrocarbon dew point * Psychrometrics * Thermodynamic diagrams ==References== ==External links== * Often Needed Answers about Temp, Humidity & Dew Point from the sci.geo.meteorology Category:Atmospheric thermodynamics Category:Temperature Category:Psychrometrics Category:Threshold temperatures Category:Gases Category:Meteorological data and networks Category:Humidity and hygrometry sv:Luftfuktighet#Daggpunkt The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. ",30,4.49,"""0.00131""",2,0.5768,A
+Suppose that $10.0 \mathrm{~mol} \mathrm{C}_2 \mathrm{H}_6(\mathrm{~g})$ is confined to $4.860 \mathrm{dm}^3$ at $27^{\circ} \mathrm{C}$. Predict the pressure exerted by the ethane from the van der Waals equations of state.,"The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. These van der Waals interactions are up to 40 times stronger than in H2, which has only one valence electron, and they are still not strong enough to achieve an aggregate state other than gas for Xe under standard conditions. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. Even with its acknowledged shortcomings, the pervasive use of the Van der Waals equation in standard university physical chemistry textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. * Van der Waals forces are independent of temperature except for dipole-dipole interactions. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . J/(mol K) Liquid properties Std enthalpy change of formation, ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy, S ~~o~~ liquid 126.7 J/(mol K) Heat capacity, cp 68.5 J/(mol K) at −179 °C Gas properties Std enthalpy change of formation, ΔfH ~~o~~ gas −83.8 kJ/mol Standard molar entropy, S ~~o~~ gas 229.6 J/(mol K) Enthalpy of combustion, ΔcH ~~o~~ −1560.7 kJ/mol Heat capacity, cp 52.49 J/(mol K) at 25 °C van der Waals' constantsLange's Handbook of Chemistry 10th ed, pp 1522-1524 a = 556.2 L2 kPa/mol2 b = 0.06380 L/mol ==Vapor pressure of liquid== P in mm Hg 1 10 40 100 400 760 1520 3800 7600 15200 30400 45600 T in °C −159.5 −142.9 −129.8 −119.3 −99.6 −88.6 −75.0 −52.8 −32.0 −6.4 23.6 — Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. == Melting point data == Mean value for acceptable data: −183.01 °C (90.14 K). The VRT spectroscopic study of Van der Waals molecules is one of the most direct routes to the understanding of intermolecular forces. == See also == * Van der Waals radius * Van der Waals strain * Van der Waals surface * –articles about specific chemicals * Researchers active in this field: ** Donald Levy ** Richard J. Saykally ** Richard Smalley ** William Klemperer == References == == Further reading == * So far three special issues of Chemical Reviews have been devoted to vdW molecules: I. Vol. 88(6) (1988). * Early reviews of vdW molecules: G. E. Ewing, Accounts of Chemical Research, Vol. 8, pp. 185-192, (1975): Structure and Properties of Van der Waals molecules. A Van der Waals molecule is a weakly bound complex of atoms or molecules held together by intermolecular attractions such as Van der Waals forces or by hydrogen bonds. ",-11.2,1410,"""1.4""",35.2,4.5,D
+Use the van der Waals parameters for chlorine to calculate approximate values of the radius of a $\mathrm{Cl}_2$ molecule regarded as a sphere.,"Values from other sources may differ significantly (see text) The van der Waals radius, r, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom. It may be calculated for atoms if the Van der Waals radius is known, and for molecules if its atoms radii and the inter- atomic distances and angles are known. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. Petitjean, On the Analytical Calculation of Van der Waals Surfaces and Volumes: Some Numerical Aspects, Journal of Computational Chemistry, Volume 15, Number 5, 1994, pp. 507–523. == External links == * VSAs for various molecules by Anton Antonov, The Wolfram Demonstrations Project, 2007. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. * Analytical calculation of Van der Waals surfaces and volumes. Van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 Van der Waals radii taken from Bondi's compilation (1964). van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Useful tabulated values of Van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will be seen to present different values for the Van der Waals radius of the same atom. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. As well, it has been argued that the Van der Waals radius is not a fixed property of an atom in all circumstances, rather, that it will vary with the chemical environment of the atom. == Gallery == File:3D ammoniac.PNG|alt=Ammonia, van-der-Waals-based model|Ammonia, NH3, space- filling, Van der Waal's-based representation, nitrogen (N) in blue, hydrogen (H) in white. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. For a molecule, it is the volume enclosed by the van der Waals surface. Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",2,210,"""22.0""",0.8561,0.139,E
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $\Delta U$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). * Values from CRC refer to ""100 kPa (1 bar or 0.987 standard atmospheres)"". Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. This error is corrected in the above equation. ==Measurement== The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. Lange indirectly defines the values to be at a standard state pressure of ""1 atm (101325 Pa)"", although citing the same NBS and JANAF sources among others. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: : \gamma = \frac{5}{3} = 1.6666\ldots, As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. ",524,+65.49,"""0.33333333""",635.7,+4.1,E
+"1.7(a) In an attempt to determine an accurate value of the gas constant, $R$, a student heated a container of volume $20.000 \mathrm{dm}^3$ filled with $0.25132 \mathrm{g}$ of helium gas to $500^{\circ} \mathrm{C}$ and measured the pressure as $206.402 \mathrm{cm}$ of water in a manometer at $25^{\circ} \mathrm{C}$. Calculate the value of $R$ from these data. (The density of water at $25^{\circ} \mathrm{C}$ is $0.99707 \mathrm{g} \mathrm{cm}^{-3}$; a manometer consists of a U-shaped tube containing a liquid. One side is connected to the apparatus and the other is open to the atmosphere. The pressure inside the apparatus is then determined from the difference in heights of the liquid.)","Note that data could have been collected with three different amounts of the same gas, which would have rendered this experiment easy to do in the eighteenth century. ==History== == See also == * Thermodynamic instruments * Boyle's law * Combined gas law * Gay-Lussac's law * Avogadro's law * Ideal gas law ==References== Category:Thermometers Category:Gases fr:Thermomètre#Thermomètre à gaz thumb|400px|Diagram showing pressure difference induced by a temperature difference. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. :V \propto T\, or :\frac{V}{T}=k V is the volume, T is the thermodynamic temperature, k is the constant for the system. k is not a fixed constant across all systems and therefore needs to be found experimentally for a given system through testing with known temperature values. ==Pressure Thermometer and Absolute Zero== thumb|left|180px|Plots of pressure vs temperature for three different gas samples extrapolate to absolute zero. thumb|500px|Two variants of a gas thermometer A gas thermometer is a thermometer that measures temperature by the variation in volume or pressure of a gas. ==Volume Thermometer== This thermometer functions by Charles's Law. Gas volume corrector - device for calculating, summing and determining increments of gas volume, measured by gas meter if it were operating base conditions. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. Using Charles's Law, the temperature can be measured by knowing the volume of gas at a certain temperature by using the formula, written below. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The constant volume gas thermometer plays a crucial role in understanding how absolute zero could be discovered long before the advent of cryogenics. Consider a graph of pressure versus temperature made not far from standard conditions (well above absolute zero) for three different samples of any ideal gas (a, b, c). For this purpose, uses as input the gas volume, measured by the gas meter and other parameters such as: gas pressure and temperature. To the extent that the gas is ideal, the pressure depends linearly on temperature, and the extrapolation to zero pressure occurs at absolute zero. Charles's Law states that when the temperature of a gas increases, so does the volume. . The prover determines the meter factor, which is the volume of air passed divided by the volume of air measured.Fundamentals of Meter Provers and Proving Methods https://asgmt.com/wp-content/uploads/2018/05/070.pdf ==Types== ===Manual bell prover=== Since the early 1900s, bell provers have been the most common reference standard used in gas meter proving, and has provided standards for the gas industry that is unfortunately susceptible to a myriad of immeasurable uncertainties. Since atmospheric pressure, P, depends upon altitude, so does \gamma. There are two types of gas volume correctors: Type 1- gas volume corrector with specific types of transducers for pressure and temperature or temperature only. Although \left( c_p \right)_{H_2 O} is constant, varied air composition results in varied \left( c_p \right)_{air} . A gas meter prover is a device to verify the accuracy of a gas meter. Translating it to the correct levels of the device that is holding the gas. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus When proving a meter using a manually controlled bell, an operator must first fill the bell with a controlled air supply or raise it manually by opening a valve and pulling a chained mechanism, seal the bell and meter and check the sealed system for leaks, determine the flow rate needed for the meter, install a special flow-rate cap on the meter outlet, note the starting points of both the bell scale and meter index, release the bell valve to pass air through the meter, observe the meter index and calculate the time it takes to pass the predetermined amount of air, then manually calculate the meter's proof accounting for bell air and meter temperature and in some cases other environmental factors. ",-6.42,0.66666666666,"""15.0""",8.3147,1260,D
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The standard enthalpy of combustion of solid phenol $\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}\right)$ is $-3054 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298 \mathrm{~K}_{\text {and }}$ its standard molar entropy is $144.0 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Calculate the standard Gibbs energy of formation of phenol at $298 \mathrm{~K}$.,"Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. The standard enthalpy of formation is then determined using Hess's law. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. ",-50,0.3085,"""537.0""",200, 258.14,A
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $\Delta U$.","In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We assume the expansion occurs without exchange of heat (adiabatic expansion). The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. For real gases the equation of state will depart from the simpler one, and the result above derived for an ideal gas will only be a good approximation provided that (a) the typical size of the molecule is negligible compared to the average distance between the individual molecules, and (b) the short range behavior of the inter-molecular potential can be neglected, i.e., when the molecules can be considered to rebound elastically off each other during molecular collisions. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Since the piston cannot move, the volume is constant. CO-0.40-0.22 is a high velocity compact gas cloud near the centre of the Milky Way. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. The fugacities commonly obey a similar law called the Lewis and Randall rule: f_i = y_i f^*_i, where is the fugacity that component would have if the entire gas had that composition at the same temperature and pressure. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. thumb|Jets of liquid carbon dioxide Liquid carbon dioxide is the liquid state of carbon dioxide (), which cannot occur under atmospheric pressure. The solubility turned out to be very low: from 0.02 to 0.10 %. thumb|Carbon dioxide pressure-temperature phase diagram == Uses == thumb|219x219px|Fire extinguisher Uses of liquid carbon dioxide include the preservation of food, in fire extinguishers, and in commercial food processes. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. ",−1.642876,0.042,"""16.0""",-59.24,-20,E
+"A manometer consists of a U-shaped tube containing a liquid. One side is connected to the apparatus and the other is open to the atmosphere. The pressure inside the apparatus is then determined from the difference in heights of the liquid. Suppose the liquid is water, the external pressure is 770 Torr, and the open side is $10.0 \mathrm{cm}$ lower than the side connected to the apparatus. What is the pressure in the apparatus? (The density of water at $25^{\circ} \mathrm{C}$ is $0.99707 \mathrm{g} \mathrm{cm}^{-3}$.)","The open end of the manometer is then connected to a pressure measuring device. Therefore, the pressure difference between the applied pressure Pa and the reference pressure P0 in a U-tube manometer can be found by solving . The pressure at the bottom of the barometer, Point B, is equal to the atmospheric pressure. It consists of a submerged manometer and container holding the substance whose vapor pressure is being measured. Manometric measurement is the subject of pressure head calculations. A single-limb liquid-column manometer has a larger reservoir instead of one side of the U-tube and has a scale beside the narrower column. Simple hydrostatic gauges can measure pressures ranging from a few torrs (a few 100 Pa) to a few atmospheres (approximately ). Bourdon tube pressure gauges. If the fluid being measured is significantly dense, hydrostatic corrections may have to be made for the height between the moving surface of the manometer working fluid and the location where the pressure measurement is desired, except when measuring differential pressure of a fluid (for example, across an orifice plate or venturi), in which case the density ρ should be corrected by subtracting the density of the fluid being measured. There are three different types of pressuremeters. The difference in liquid levels represents the applied pressure. The pressuremeter has two major components. Typically, atmospheric pressure is measured between and of Hg. A very simple version is a U-shaped tube half-full of liquid, one side of which is connected to the region of interest while the reference pressure (which might be the atmospheric pressure or a vacuum) is applied to the other. They have poor dynamic response. ====Piston==== Piston-type gauges counterbalance the pressure of a fluid with a spring (for example tire- pressure gauges of comparatively low accuracy) or a solid weight, in which case it is known as a deadweight tester and may be used for calibration of other gauges. ====Liquid column (manometer)==== thumb|upright|The difference in fluid height in a liquid-column manometer is proportional to the pressure difference: h = \frac{P_a - P_o}{g \rho} Liquid-column gauges consist of a column of liquid in a tube whose ends are exposed to different pressures. The pressure inside the probe is held constant for a specific period of time and the increase in volume required to maintain the pressure is recorded. thumb|Barometer A barometer is a scientific instrument that is used to measure air pressure in a certain environment. Gauges that rely on a change in capacitance are often referred to as capacitance manometers. ====Bourdon tube==== thumb|Membrane-type manometer The Bourdon pressure gauge uses the principle that a flattened tube tends to straighten or regain its circular form in cross-section when pressurized. A pressuremeter is a meter constructed to measure the “at-rest horizontal earth pressure”. The word ""gauge"" or ""vacuum"" may be added to such a measurement to distinguish between a pressure above or below the atmospheric pressure. Pressure instruments connected to the system will indicate pressures relative to the current atmospheric pressure. It has a pressure resolution of approximately 1mm of water when measuring pressure at a depth of several kilometers. ",32,0.9984,"""102.0""",-100,4.8,C
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. The standard enthalpy of formation of the metallocene bis(benzene)chromium was measured in a calorimeter. It was found for the reaction $\mathrm{Cr}\left(\mathrm{C}_6 \mathrm{H}_6\right)_2(\mathrm{~s}) \rightarrow \mathrm{Cr}(\mathrm{s})+2 \mathrm{C}_6 \mathrm{H}_6(\mathrm{~g})$ that $\Delta_{\mathrm{r}} U^{\bullet}(583 \mathrm{~K})=+8.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Find the corresponding reaction enthalpy and estimate the standard enthalpy of formation of the compound at $583 \mathrm{~K}$. The constant-pressure molar heat capacity of benzene is $136.1 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ in its liquid range and $81.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ as a gas.","Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). The standard enthalpy of formation is then determined using Hess's law. * Standard enthalpy of formation is the enthalpy change when one mole of any compound is formed from its constituent elements in their standard states. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. The standard enthalpy of formation, which has been determined for a vast number of substances, is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements, with all substances in their standard states. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. ",+17.7,3.38,"""0.2553""",0.11,1.95 ,A
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the Carnot efficiency of a primitive steam engine operating on steam at $100^{\circ} \mathrm{C}$ and discharging at $60^{\circ} \mathrm{C}$.,"Latest generation gas turbine engines have achieved an efficiency of 46% in simple cycle and 61% when used in combined cycle. ==External combustion engines== ===Steam engine=== ::See also: Steam engine#Efficiency ::See also: Timeline of steam power ====Piston engine==== Steam engines and turbines operate on the Rankine cycle which has a maximum Carnot efficiency of 63% for practical engines, with steam turbine power plants able to achieve efficiency in the mid 40% range. While gasoline-powered ICE cars have an operational thermal efficiency of 15% to 30%, early automotive steam units were capable of only about half this efficiency. Its theoretical efficiency depends on the compression ratio r of the engine and the specific heat ratio γ of the gas in the combustion chamber. \eta_{\rm th} = 1 - \frac{1}{r^{\gamma-1}} Thus, the efficiency increases with the compression ratio. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The efficiency depends largely on the ratio of the pressure inside the combustion chamber p2 to the pressure outside p1 \eta_{\rm th} = 1 - \left(\frac{p_2}{p_1}\right)^\frac{1-\gamma}{\gamma} ===Other inefficiencies=== One should not confuse thermal efficiency with other efficiencies that are used when discussing engines. Steam engine efficiency improved as the operating principles were discovered, which led to the development of the science of thermodynamics. The efficiency of steam engines is primarily related to the steam temperature and pressure and the number of stages or expansions. See graph:Steam Engine Efficiency In earliest steam engines the boiler was considered part of the engine. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The first piston steam engine, developed by Thomas Newcomen around 1710, was slightly over one half percent (0.5%) efficient. Some steam enthusiasts feel steam has not received its share of attention in the field of automobile efficiency. A very well-designed and built steam locomotive used to get around 7-8% efficiency in its heyday. The above efficiency formulas are based on simple idealized mathematical models of engines, with no friction and working fluids that obey simple thermodynamic rules called the ideal gas law. Today they are considered separate, so it is necessary to know whether stated efficiency is overall, which includes the boiler, or just of the engine. Comparisons of efficiency and power of the early steam engines is difficult for several reasons: 1) there was no standard weight for a bushel of coal, which could be anywhere from 82 to 96 pounds (37 to 44 kg). The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature T_{\rm H}, and removed at the minimum temperature T_{\rm C}. From Carnot's theorem, for any engine working between these two temperatures: :\eta_{\rm th} \le 1 - \frac{T_{\rm C}}{T_{\rm H}} This limiting value is called the Carnot cycle efficiency because it is the efficiency of an unattainable, ideal, reversible engine cycle called the Carnot cycle. Practical engine cycles are irreversible and thus have inherently lower efficiency than the Carnot efficiency when operated between the same temperatures T_{\rm H} and T_{\rm C}. For example, the average automobile engine is less than 35% efficient. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. ",52,5.51,"""4500.0""",129,0.11,E
+"In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ at a constant volume of $1.000 \mathrm{~m}^3$. The gas enters the container at $300 \mathrm{~K}$ and $100 \mathrm{~atm}$. The mass of the gas is $92.4 \mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \mathrm{~K}$. For nitrogen, $a=1.352 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}, b=0.0387 \mathrm{dm}^3 \mathrm{~mol}^{-1}$.","Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. * Analytical calculation of Van der Waals surfaces and volumes. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . ",-11.875,140,"""131.0""",655, 13.45,B
 "Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K.
 Silylene $\left(\mathrm{SiH}_2\right)$ is a key intermediate in the thermal decomposition of silicon hydrides such as silane $\left(\mathrm{SiH}_4\right)$ and disilane $\left(\mathrm{Si}_2 \mathrm{H}_6\right)$. Moffat et al. (J. Phys. Chem. 95, 145 (1991)) report $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{SiH}_2\right)=+274 \mathrm{~kJ} \mathrm{~mol}^{-1}$. If $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{SiH}_4\right)=+34.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{Si}_2 \mathrm{H}_6\right)=+80.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$(CRC Handbook (2008)), compute the standard enthalpies of the following reaction:
-$\mathrm{SiH}_4 (\mathrm{g}) \rightarrow \mathrm{SiH}_2(\mathrm{g})+\mathrm{H}_2(\mathrm{g})$","Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? \\!| cdf =| mean =\mu + \frac{\delta \beta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)}| median =| mode =| variance =\frac{\delta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left( \frac{K_{\lambda+2}(\delta\gamma)}{K_{\lambda}(\delta\gamma)} - \frac{K_{\lambda+1}^2(\delta\gamma)}{K_{\lambda}^2(\delta\gamma)} \right)| skewness =| kurtosis =| entropy =| mgf =\frac{e^{\mu z}\gamma^\lambda}{(\sqrt{\alpha^2 -(\beta +z)^2})^\lambda} \frac{K_\lambda(\delta \sqrt{\alpha^2 -(\beta +z)^2})}{K_\lambda (\delta \gamma)}| char =| }} The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution (GIG). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2} | cdf = \Phi \left(\gamma + \delta \sinh^{-1} \left( \frac{x-\xi}{\lambda} \right) \right) | mean = \xi - \lambda \exp \frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right) | median = \xi + \lambda \sinh \left( - \frac{\gamma}{\delta} \right) | mode = | variance = \frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right) | skewness = -\frac{\lambda^3\sqrt{e^{\delta^{-2}}}(e^{\delta^{-2}}-1)^{2}((e^{\delta^{-2}})(e^{\delta^{-2}}+2)\sinh(\frac{3\gamma}{\delta})+3\sinh(\frac{2\gamma}{\delta}))}{4(\operatorname{Variance}X)^{1.5}} | kurtosis = \frac{\lambda^4(e^{\delta^{-2}}-1)^2(K_{1}+K_2+K_3)}{8(\operatorname{Variance}X)^2} K_{1}=\left( e^{\delta^{-2}} \right)^{2}\left( \left( e^{\delta^{-2}} \right)^{4}+2\left( e^{\delta^{-2}} \right)^{3}+3\left( e^{\delta^{-2}} \right)^{2}-3 \right)\cosh\left( \frac{4\gamma}{\delta} \right) K_2=4\left( e^{\delta^{-2}} \right)^2 \left( \left( e^{\delta^{-2}} \right)+2 \right)\cosh\left( \frac{3\gamma}{\delta} \right) K_3=3\left( 2\left( e^{\delta^{-2}} \right)+1 \right) | entropy = | pgf = | mgf = | char = }} The Johnson's SU-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? K (? °C), ? K (? °C), ? * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. ==See also== * List of thermal conductivities Category:Properties of chemical elements Category:Chemical element data pages LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? * J.A. Dean (ed) in Lange's Handbook of Chemistry, McGraw-Hill, New York, USA, 14th edition, 1992. This page provides supplementary chemical data on Lutetium(III) oxide == Thermodynamic properties == Phase behavior Triple point ? * X \sim \mathrm{GH}(1, \alpha, \beta, \delta, \mu)\, has a hyperbolic distribution. ",5.1,240,-8.0,2.25,0.9731,B
+$\mathrm{SiH}_4 (\mathrm{g}) \rightarrow \mathrm{SiH}_2(\mathrm{g})+\mathrm{H}_2(\mathrm{g})$","Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? \\!| cdf =| mean =\mu + \frac{\delta \beta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)}| median =| mode =| variance =\frac{\delta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left( \frac{K_{\lambda+2}(\delta\gamma)}{K_{\lambda}(\delta\gamma)} - \frac{K_{\lambda+1}^2(\delta\gamma)}{K_{\lambda}^2(\delta\gamma)} \right)| skewness =| kurtosis =| entropy =| mgf =\frac{e^{\mu z}\gamma^\lambda}{(\sqrt{\alpha^2 -(\beta +z)^2})^\lambda} \frac{K_\lambda(\delta \sqrt{\alpha^2 -(\beta +z)^2})}{K_\lambda (\delta \gamma)}| char =| }} The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution (GIG). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2} | cdf = \Phi \left(\gamma + \delta \sinh^{-1} \left( \frac{x-\xi}{\lambda} \right) \right) | mean = \xi - \lambda \exp \frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right) | median = \xi + \lambda \sinh \left( - \frac{\gamma}{\delta} \right) | mode = | variance = \frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right) | skewness = -\frac{\lambda^3\sqrt{e^{\delta^{-2}}}(e^{\delta^{-2}}-1)^{2}((e^{\delta^{-2}})(e^{\delta^{-2}}+2)\sinh(\frac{3\gamma}{\delta})+3\sinh(\frac{2\gamma}{\delta}))}{4(\operatorname{Variance}X)^{1.5}} | kurtosis = \frac{\lambda^4(e^{\delta^{-2}}-1)^2(K_{1}+K_2+K_3)}{8(\operatorname{Variance}X)^2} K_{1}=\left( e^{\delta^{-2}} \right)^{2}\left( \left( e^{\delta^{-2}} \right)^{4}+2\left( e^{\delta^{-2}} \right)^{3}+3\left( e^{\delta^{-2}} \right)^{2}-3 \right)\cosh\left( \frac{4\gamma}{\delta} \right) K_2=4\left( e^{\delta^{-2}} \right)^2 \left( \left( e^{\delta^{-2}} \right)+2 \right)\cosh\left( \frac{3\gamma}{\delta} \right) K_3=3\left( 2\left( e^{\delta^{-2}} \right)+1 \right) | entropy = | pgf = | mgf = | char = }} The Johnson's SU-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m��K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? K (? °C), ? K (? °C), ? * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. ==See also== * List of thermal conductivities Category:Properties of chemical elements Category:Chemical element data pages LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? * J.A. Dean (ed) in Lange's Handbook of Chemistry, McGraw-Hill, New York, USA, 14th edition, 1992. This page provides supplementary chemical data on Lutetium(III) oxide == Thermodynamic properties == Phase behavior Triple point ? * X \sim \mathrm{GH}(1, \alpha, \beta, \delta, \mu)\, has a hyperbolic distribution. ",5.1,240,"""-8.0""",2.25,0.9731,B
 "Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of $4.50 \mathrm{~g}$ of methane occupies $12.7 \mathrm{dm}^3$ at $310 \mathrm{~K}$. Calculate the work done when the gas expands isothermally against a constant external pressure of 200 Torr until its volume has increased by
-$3.3 \mathrm{dm}^3$.","In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Therefore, the heat capacity ratio in this example is 1.4. A methane reformer is a device based on steam reforming, autothermal reforming or partial oxidation and is a type of chemical synthesis which can produce pure hydrogen gas from methane using a catalyst. A gas heater is a space heater used to heat a room or outdoor area by burning natural gas, liquefied petroleum gas, propane, or butane. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. thumb|Large propane torch used for construction A propane torch is a tool normally used for the application of flame or heat which uses propane, a hydrocarbon gas, for its fuel and ambient air as its combustion medium. We assume the expansion occurs without exchange of heat (adiabatic expansion). Conventional steam reforming plants operate at pressures between 200 and 600 psi with outlet temperatures in the range of 815 to 925 °C. In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; is used. To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The stoichiometric equation for complete combustion of propane with 100% oxygen is: :C3H8 \+ 5 (O2) → 4 (H2O) + 3 (CO2) In this case, the only products are CO2 and water. The balanced equation shows to use 1 mole of propane for every 5 moles of oxygen. An example of incomplete combustion that uses 1 mole of propane for every 4 moles of oxygen: :C3H8 \+ 4 (O2) → 4 (H2O) + 2 (CO2) + 1 C The extra carbon product will cause soot to form, and the less oxygen used, the more soot will form. Oxygen-fed torches can be much hotter at up to . ==See also== * Butane torch * Blowtorch * Thermal lance ==References== ==Bibliography== * * ==External links== * How to Silver Solder Steel with a Propane Torch * How To properly Heat Up Copper Pipe Using A Propane Torch Category:Burners Category:Metalworking tools Category:Welding Torch Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Another way of understanding the difference between and is that applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move). applies only if P\,\mathrm{d}V = 0, that is, no work is done. ",16,3.07,7200.0,15.1,-88,E
-The barometric formula relates the pressure of a gas of molar mass $M$ at an altitude $h$ to its pressure $p_0$ at sea level. Derive this relation by showing that the change in pressure $\mathrm{d} p$ for an infinitesimal change in altitude $\mathrm{d} h$ where the density is $\rho$ is $\mathrm{d} p=-\rho g \mathrm{~d} h$. Remember that $\rho$ depends on the pressure. Evaluate the pressure difference between the top and bottom of a laboratory vessel of height 15 cm.,"The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude. == Pressure equations == thumb|300px|Pressure as a function of the height above the sea level There are two equations for computing pressure as a function of height. Subscript b Height Above Sea Level (h) Mass Density (\rho) Standard Temperature (T') (K) Temperature Lapse Rate (L) (m) (ft) (kg/m3) (slug/ft3) (K/m) (K/ft) 0 0 0 1.2250 288.15 0.0065 0.0019812 1 11 000 36,089.24 0.36391 216.65 0.0 0.0 2 20 000 65,616.79 0.08803 216.65 -0.001 -0.0003048 3 32 000 104,986.87 0.01322 228.65 -0.0028 -0.00085344 4 47 000 154,199.48 0.00143 270.65 0.0 0.0 5 51 000 167,322.83 0.00086 270.65 0.0028 0.00085344 6 71 000 232,939.63 0.000064 214.65 0.002 0.0006096 ==Derivation== The barometric formula can be derived using the ideal gas law: P = \frac{\rho}{M} {R^*} T Assuming that all pressure is hydrostatic: dP = - \rho g\,dz and dividing the dP by the P expression we get: \frac{dP}{P} = - \frac{M g\,dz}{R^*T} Integrating this expression from the surface to the altitude z we get: P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}} Assuming linear temperature change T = T_0 - L z and constant molar mass and gravitational acceleration, we get the first barometric formula: P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}} Instead, assuming constant temperature, integrating gives the second barometric formula: P = P_0 e^{-M g z/R^*T} In this formulation, R* is the gas constant, and the term R*T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere). This then yields a more accurate formula, of the form P_h = P_0 e^{-\frac{mgh}{kT}}, where * is the pressure at height , * is the pressure at reference point 0 (typically referring to sea level), * is the mass per air molecule, * is the acceleration due to gravity, * is height from reference point 0, * is the Boltzmann constant, * is the temperature in kelvins. Subscript b Height above sea level Static pressure Standard temperature (K) Temperature lapse rate Exponent g0 M / R L (m) (ft) (Pa) (inHg) (K/m) (K/ft) 0 0 0 101 325.00 29.92126 288.15 0.0065 0.0019812 5.2558 1 11 000 36,089 22 632.10 6.683245 216.65 0.0 0.0 -- 2 20 000 65,617 5474.89 1.616734 216.65 -0.001 -0.0003048 -34.1626 3 32 000 104,987 868.02 0.2563258 228.65 -0.0028 -0.00085344 -12.2009 4 47 000 154,199 110.91 0.0327506 270.65 0.0 0.0 -- 5 51 000 167,323 66.94 0.01976704 270.65 0.0028 0.00085344 12.2009 6 71 000 232,940 3.96 0.00116833 214.65 0.002 0.0006096 17.0813 ==Density equations== The expressions for calculating density are nearly identical to calculating pressure. Equation 1: \rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^* L_b}-1\right)} which is equivalent to the ratio of the relative pressure and temperature changes \rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} Equation 2: \rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where *{\rho} = mass density (kg/m3) *T_b = standard temperature (K) *L = standard temperature lapse rate (see table below) (K/m) in ISA *h = height above sea level (geopotential meters) *R^* = universal gas constant 8.3144598 N·m/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): *{\rho} = mass density (slug/ft3) *{T_b} = standard temperature (K) *{L} = standard temperature lapse rate (K/ft) *{h} = height above sea level (geopotential feet) *{R^*} = universal gas constant: 8.9494596×104 ft2/(s·K) *{g_0} = gravitational acceleration: 32.17405 ft/s2 *{M} = molar mass of Earth's air: 0.0289644 kg/mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. The equation is as follows: \frac{dP}{dh} = - \rho g , where * is pressure, * is density, * is acceleration of gravity, and * is height. When density and gravity are approximately constant (that is, for relatively small changes in height), simply multiplying height difference, gravity, and density will yield a good approximation of pressure difference. Different formulas are presented for different kinds of approximations; for comparison with the previous formula, the first referenced from the article will be the one applying the same constant-temperature approximation; in which case: z = -\frac{RT}{g} \ln \frac{P}{P_0} where (with values used in the article) * is the elevation in meters, * is the specific gas constant = * is the absolute temperature in kelvins = at sea level, * is the acceleration due to gravity = at sea level, * is the pressure at a given point at elevation in Pascals, and * is pressure at the reference point = at sea level. This may seem counter-intuitive, as pressure results from height rather than vice versa, but such a formula can be useful in finding height based on pressure difference when one knows the latter and not the former. However, the vertical variation is especially significant, as it results from the pull of gravity on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point. ==Basic formula== A relatively simple version of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and density. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. Vertical pressure variation is the variation in pressure as a function of elevation. Thus water's density can be more reasonably approximated as constant than that of air, and given the same height difference, the pressure differences in water are approximately equal at any height. ==Hydrostatic paradox== thumb|upright|Diagram illustrating the hydrostatic paradox The barometric formula depends only on the height of the fluid chamber, and not on its width or length. (The total air mass below a certain altitude is calculated by integrating over the density function.) Image:Liquid ocean atmosphere model.png ===Isothermal-barotropic approximation and scale height=== This atmospheric model assumes both molecular weight and temperature are constant over a wide range of altitude. Assuming density is constant, then a graph of pressure vs altitude will have a retained slope, since the weight of the ocean over head is directly proportional to its depth. An alternative derivation, shown by the Portland State Aerospace Society, is used to give height as a function of pressure instead. This is the recommended formula to use. ==See also== * Barometer * Hypsometric equation * Pascal's barrel * Ruina montium * Pressure gradient * Siphon ==References== * ==External links== Category:Pressure Category:Vertical position For the isothermal-barotropic model, density and pressure turn out to be exponential functions of altitude. If the pressure at one point in a liquid with uniform density ρ is known to be P0, then the pressure at another point is P1: :P_1=P_0 - \rho g (h_1 - h_0) where h1 \- h0 is the vertical distance between the two points.Streeter, Victor L. (1966). The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of L_b: P = P_b \left[\frac{T_b - \left(h - h_b\right) L_b}{T_b}\right]^{\tfrac{g_0 M}{R^* L_b}} The second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude: P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/m) in ISA *h = height at which pressure is calculated (m) *h_b = height of reference level b (meters; e.g., hb = 11 000 m) *R^* = universal gas constant: 8.3144598 J/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol Or converted to imperial units:Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. Since is negative, an increase in height will correspond to a decrease in pressure, which fits with the previously mentioned reasoning about the weight of a column of fluid. ", 0.01961,0.00017,0.2553,362880,0.6296296296,B
-"The mass density of water vapour at $327.6 \mathrm{~atm}$ and $776.4 \mathrm{~K}$ is $133.2 \mathrm{~kg} \mathrm{~m}^{-3}$. Given that for water $T_{\mathrm{c}}=647.4 \mathrm{~K}, p_{\mathrm{c}}=218.3 \mathrm{~atm}, a=5.464 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}$, $b=0.03049 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and $M=18.02 \mathrm{~g} \mathrm{~mol}^{-1}$, calculate the molar volume.","The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. Liquid water path - in units of g/m2 is a measure of the total amount of liquid water present between two points in the atmosphere. thumb|The Mollier enthalpy–entropy diagram for water and steam. Air is given a vapour density of one. Further g is acceleration due to gravity and ρ is the fluid density. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . Further ρ is the (constant) fluid density and g is the gravitational acceleration. The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. It may be defined as mass of a certain volume of a substance divided by mass of same volume of hydrogen. :vapour density = mass of n molecules of gas / mass of n molecules of hydrogen gas . :vapour density = molar mass of gas / molar mass of H2 :vapour density = molar mass of gas / 2.016 :vapour density = × molar mass (and thus: molar mass = ~2 × vapour density) For example, vapour density of mixture of NO2 and N2O4 is 38.3. U.S. Geological Survey Water-Supply Paper 1541-B' An example of a double mass analysis is a ""double mass plot"", or ""double mass curve"".Wilson, E.M. (1983) Engineering Hydrology, 3rd edition. Double mass analysis is a simple graphical method to evaluate the consistency of hydrological data. The first equation is derived from mass conservation, the second two from momentum conservation. ===Non-conservative form=== Expanding the derivatives in the above using the product rule, the non-conservative form of the shallow-water equations is obtained. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. For nadir observations and whole atmospheric column we have :LWP=\int_{z=0}^\infty \rho_{air} r_L dz' where is the liquid water mixing ratio and is the density of air (including water loading). X gives the fraction (by mass) of gaseous substance in the wet region, the remainder being colloidal liquid droplets. The Mollier diagram coordinates are enthalpy h and humidity ratio x. thumb|491px|right|Output from a shallow-water equation model of water in a bathtub. The shallow- water equations are thus derived. ",-1.32,-9.54,-13.598,4.49,0.1353,E
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $q$.","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. A 2012 study of the effects for the original hypothesis, based on a coupled climate–carbon cycle model (GCM) assessed a 1000-fold (from <1 to 1000 ppmv) methane increase—within a single pulse, from methane hydrates (based on carbon amount estimates for the PETM, with ~2000 GtC), and concluded it would increase atmospheric temperatures by more than 6 °C within 80 years. * Being a gaseous fuel, CNG mixes easily and evenly in air. A carbon dioxide generator or CO2 generator is a machine used to enhance carbon dioxide levels in order to promote plant growth in greenhouses or other enclosed areas. thumb|upright=1.6|Methane clathrate is released as gas into the surrounding water column or soils when ambient temperature increases thumb|upright=1.6|The impact of CH4 atmospheric methane concentrations on global temperature increase may be far greater than previously estimated. [http://regmorrison.edublogs.org/files/2013/02/METHANE-2-1sca3tx.pdf] The clathrate gun hypothesis is a proposed explanation for the periods of rapid warming during the Quaternary. The estimated amount of methane hydrate in this slope is 2.5 gigatonnes (about 0.2% of the amount required to cause the PETM), and it is unclear if the methane could reach the atmosphere. This would have had an immediate impact on the global temperature, as methane is a much more powerful greenhouse gas than carbon dioxide. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. A release on this scale would increase the methane content of the planet's atmosphere by a factor of twelve, equivalent in greenhouse effect to a doubling in the 2008 level of . Especially, for the CNG Type 1 and Type 2 cylinders, many countries are able to make reliable and cost effective cylinders for conversion need. ==Energy density== CNG's energy density is the same as liquefied natural gas at 53.6 MJ/kg. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). ** CNG produced from landfill biogas was found by CARB to have the lowest greenhouse gas emissions of any fuel analyzed, with a value of 11.26 gCO2e/MJ (more than 88 percent lower than conventional petrol) in the low-carbon fuel standard that went into effect on January 12, 2010. Gastric tonometry describes the measurement of the carbon dioxide level inside the stomach in order to assess the degree of blood flow to the stomach and bowel. Normocapnia or normocarbia is a state of normal arterial carbon dioxide pressure, usually about 40 mmHg. == See also == * * * == References == *The Free Dictionary - Normocapnia Category:Diving medicine Category:Pulmonology Since it is a compressed gas, rather than a liquid like petrol, CNG takes up more space for each GGE (petrol gallon equivalent). Accumulation of carbon dioxide is predominantly a result of hypoperfusion and not hypoxia. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. In this process known as high pressure ANG, a high pressure CNG tank is filled by absorbers such as activated carbon (which is an adsorbent with high surface area) and stores natural gas by both CNG and ANG mechanisms. ",0,1000,0.18,-1.5,1855,A
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $\Delta S$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: : \gamma = \frac{5}{3} = 1.6666\ldots, As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. For an imperfect or non-ideal gas, Chandrasekhar defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure: : \begin{align} \Gamma_1 &= \left.\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}, \\\\[2pt] \frac{\Gamma_2 - 1}{\Gamma_2} &= \left.\frac{\partial \ln T}{\partial \ln P}\right|_{S}, \\\\[2pt] \Gamma_3 - 1 &= \left.\frac{\partial \ln T}{\partial \ln \rho}\right|_{S}. \end{align} All of these are equal to \gamma in the case of an ideal gas. == See also == * Relations between heat capacities * Heat capacity * Specific heat capacity * Speed of sound * Thermodynamic equations * Thermodynamics * Volumetric heat capacity == References == == Notes == Category:Thermodynamic properties Category:Physical quantities Category:Ratios Category:Thought experiments in physics Sometimes, other distinctions are made, such as between thermally perfect gas and calorically perfect gas, or between imperfect, semi-perfect, and perfect gases, and as well as the characteristics of ideal gases. In addition, other factors come into play and dominate during a compression cycle if they have a greater impact on the final calculated result than whether or not C_P was held constant. (i) Indicates values calculated from ideal gas thermodynamic functions. ",0,650000,4.68,0.925,-0.38,A
-Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. Find an expression for the fugacity coefficient of a gas that obeys the equation of state $p V_{\mathrm{m}}=R T\left(1+B / V_{\mathrm{m}}+C / V_{\mathrm{m}}^2\right)$. Use the resulting expression to estimate the fugacity of argon at 1.00 atm and $100 \mathrm{~K}$ using $B=-21.13 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$ and $C=1054 \mathrm{~cm}^6 \mathrm{~mol}^{-2}$.,"For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. The fugacity coefficient is . Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. Using , where is the fugacity coefficient, f = \varphi_\mathrm{sat}P_\mathrm{sat}\exp\left(\frac{V\left(P-P_\mathrm{sat}\right)}{R T}\right). This error is corrected in the above equation. ==Measurement== The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. When using a fugacity capacity approach to calculate the concentrations of a chemical in each of several medias/phases/compartments, it is often convenient to calculate the prevailing fugacity of the system using the following equation if the total mass of target chemical (MT) and the volume of each compartment (Vm) are known: :f = M_T / \Sigma_m (V_m Z_m) Alternatively, if the target chemical is present as a pure phase at equilibrium, its vapor pressure will be the prevailing fugacity of the system. ==See also== * Multimedia fugacity model ==References== Category:Chemical thermodynamics Category:Environmental chemistry Category:Equilibrium chemistry The real gas pressure and fugacity are related through the dimensionless fugacity coefficient . \varphi = \frac{f}{P} For an ideal gas, fugacity and pressure are equal and so . If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar reaction quotient form (or law of mass action) except that the pressures are replaced by fugacities. Numerical example: Nitrogen gas (N2) at 0 °C and a pressure of atmospheres (atm) has a fugacity of atm. In addition, there are natural reference states for fugacity (for example, an ideal gas makes a natural reference state for gas mixtures since the fugacity and pressure converge at low pressure). ===Gases=== In a mixture of gases, the fugacity of each component has a similar definition, with partial molar quantities instead of molar quantities (e.g., instead of and instead of ): dG_i = R T \,d \ln f_i and \lim_{P\to 0} \frac{f_i}{P_i} = 1, where is the partial pressure of component . :C_m = Z_m \cdot f where Z is a proportional constant, termed fugacity capacity. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The constant of proportionality (a measured Henry's constant) depends on whether the concentration is represented by the mole fraction, molality or molarity. ==Temperature and pressure dependence== The pressure dependence of fugacity (at constant temperature) is given by \left(\frac{\partial \ln f}{\partial P}\right)_T = \frac{V_\mathrm{m}}{R T} and is always positive. The fugacities commonly obey a similar law called the Lewis and Randall rule: f_i = y_i f^*_i, where is the fugacity that component would have if the entire gas had that composition at the same temperature and pressure. Hemond and Hechner-Levy (2000) describe how to utilize the fugacity capacity to calculate the concentration of a chemical in a system. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. ",6.6,0.03,0.9974,-4564.7,2.3613,C
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the maximum non-expansion work per mole that may be obtained from a fuel cell in which the chemical reaction is the combustion of methane at $298 \mathrm{~K}$.,"For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Even more restricted is the calorically perfect gas for which, in addition, the heat capacity is assumed to be constant. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). Instead of a mole the constant can be expressed by considering the normal cubic meter. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics The contribution of nonideality to the molar Gibbs energy of a real gas is equal to . Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . ",54.7,91.7,-233.0,817.90,0,D
+$3.3 \mathrm{dm}^3$.","In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Therefore, the heat capacity ratio in this example is 1.4. A methane reformer is a device based on steam reforming, autothermal reforming or partial oxidation and is a type of chemical synthesis which can produce pure hydrogen gas from methane using a catalyst. A gas heater is a space heater used to heat a room or outdoor area by burning natural gas, liquefied petroleum gas, propane, or butane. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. thumb|Large propane torch used for construction A propane torch is a tool normally used for the application of flame or heat which uses propane, a hydrocarbon gas, for its fuel and ambient air as its combustion medium. We assume the expansion occurs without exchange of heat (adiabatic expansion). Conventional steam reforming plants operate at pressures between 200 and 600 psi with outlet temperatures in the range of 815 to 925 °C. In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; is used. To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The stoichiometric equation for complete combustion of propane with 100% oxygen is: :C3H8 \+ 5 (O2) → 4 (H2O) + 3 (CO2) In this case, the only products are CO2 and water. The balanced equation shows to use 1 mole of propane for every 5 moles of oxygen. An example of incomplete combustion that uses 1 mole of propane for every 4 moles of oxygen: :C3H8 \+ 4 (O2) → 4 (H2O) + 2 (CO2) + 1 C The extra carbon product will cause soot to form, and the less oxygen used, the more soot will form. Oxygen-fed torches can be much hotter at up to . ==See also== * Butane torch * Blowtorch * Thermal lance ==References== ==Bibliography== * * ==External links== * How to Silver Solder Steel with a Propane Torch * How To properly Heat Up Copper Pipe Using A Propane Torch Category:Burners Category:Metalworking tools Category:Welding Torch Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Another way of understanding the difference between and is that applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move). applies only if P\,\mathrm{d}V = 0, that is, no work is done. ",16,3.07,"""7200.0""",15.1,-88,E
+The barometric formula relates the pressure of a gas of molar mass $M$ at an altitude $h$ to its pressure $p_0$ at sea level. Derive this relation by showing that the change in pressure $\mathrm{d} p$ for an infinitesimal change in altitude $\mathrm{d} h$ where the density is $\rho$ is $\mathrm{d} p=-\rho g \mathrm{~d} h$. Remember that $\rho$ depends on the pressure. Evaluate the pressure difference between the top and bottom of a laboratory vessel of height 15 cm.,"The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude. == Pressure equations == thumb|300px|Pressure as a function of the height above the sea level There are two equations for computing pressure as a function of height. Subscript b Height Above Sea Level (h) Mass Density (\rho) Standard Temperature (T') (K) Temperature Lapse Rate (L) (m) (ft) (kg/m3) (slug/ft3) (K/m) (K/ft) 0 0 0 1.2250 288.15 0.0065 0.0019812 1 11 000 36,089.24 0.36391 216.65 0.0 0.0 2 20 000 65,616.79 0.08803 216.65 -0.001 -0.0003048 3 32 000 104,986.87 0.01322 228.65 -0.0028 -0.00085344 4 47 000 154,199.48 0.00143 270.65 0.0 0.0 5 51 000 167,322.83 0.00086 270.65 0.0028 0.00085344 6 71 000 232,939.63 0.000064 214.65 0.002 0.0006096 ==Derivation== The barometric formula can be derived using the ideal gas law: P = \frac{\rho}{M} {R^*} T Assuming that all pressure is hydrostatic: dP = - \rho g\,dz and dividing the dP by the P expression we get: \frac{dP}{P} = - \frac{M g\,dz}{R^*T} Integrating this expression from the surface to the altitude z we get: P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}} Assuming linear temperature change T = T_0 - L z and constant molar mass and gravitational acceleration, we get the first barometric formula: P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}} Instead, assuming constant temperature, integrating gives the second barometric formula: P = P_0 e^{-M g z/R^*T} In this formulation, R* is the gas constant, and the term R*T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere). This then yields a more accurate formula, of the form P_h = P_0 e^{-\frac{mgh}{kT}}, where * is the pressure at height , * is the pressure at reference point 0 (typically referring to sea level), * is the mass per air molecule, * is the acceleration due to gravity, * is height from reference point 0, * is the Boltzmann constant, * is the temperature in kelvins. Subscript b Height above sea level Static pressure Standard temperature (K) Temperature lapse rate Exponent g0 M / R L (m) (ft) (Pa) (inHg) (K/m) (K/ft) 0 0 0 101 325.00 29.92126 288.15 0.0065 0.0019812 5.2558 1 11 000 36,089 22 632.10 6.683245 216.65 0.0 0.0 -- 2 20 000 65,617 5474.89 1.616734 216.65 -0.001 -0.0003048 -34.1626 3 32 000 104,987 868.02 0.2563258 228.65 -0.0028 -0.00085344 -12.2009 4 47 000 154,199 110.91 0.0327506 270.65 0.0 0.0 -- 5 51 000 167,323 66.94 0.01976704 270.65 0.0028 0.00085344 12.2009 6 71 000 232,940 3.96 0.00116833 214.65 0.002 0.0006096 17.0813 ==Density equations== The expressions for calculating density are nearly identical to calculating pressure. Equation 1: \rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^* L_b}-1\right)} which is equivalent to the ratio of the relative pressure and temperature changes \rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} Equation 2: \rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where *{\rho} = mass density (kg/m3) *T_b = standard temperature (K) *L = standard temperature lapse rate (see table below) (K/m) in ISA *h = height above sea level (geopotential meters) *R^* = universal gas constant 8.3144598 N·m/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): *{\rho} = mass density (slug/ft3) *{T_b} = standard temperature (K) *{L} = standard temperature lapse rate (K/ft) *{h} = height above sea level (geopotential feet) *{R^*} = universal gas constant: 8.9494596×104 ft2/(s·K) *{g_0} = gravitational acceleration: 32.17405 ft/s2 *{M} = molar mass of Earth's air: 0.0289644 kg/mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. The equation is as follows: \frac{dP}{dh} = - \rho g , where * is pressure, * is density, * is acceleration of gravity, and * is height. When density and gravity are approximately constant (that is, for relatively small changes in height), simply multiplying height difference, gravity, and density will yield a good approximation of pressure difference. Different formulas are presented for different kinds of approximations; for comparison with the previous formula, the first referenced from the article will be the one applying the same constant-temperature approximation; in which case: z = -\frac{RT}{g} \ln \frac{P}{P_0} where (with values used in the article) * is the elevation in meters, * is the specific gas constant = * is the absolute temperature in kelvins = at sea level, * is the acceleration due to gravity = at sea level, * is the pressure at a given point at elevation in Pascals, and * is pressure at the reference point = at sea level. This may seem counter-intuitive, as pressure results from height rather than vice versa, but such a formula can be useful in finding height based on pressure difference when one knows the latter and not the former. However, the vertical variation is especially significant, as it results from the pull of gravity on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point. ==Basic formula== A relatively simple version of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and density. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. Vertical pressure variation is the variation in pressure as a function of elevation. Thus water's density can be more reasonably approximated as constant than that of air, and given the same height difference, the pressure differences in water are approximately equal at any height. ==Hydrostatic paradox== thumb|upright|Diagram illustrating the hydrostatic paradox The barometric formula depends only on the height of the fluid chamber, and not on its width or length. (The total air mass below a certain altitude is calculated by integrating over the density function.) Image:Liquid ocean atmosphere model.png ===Isothermal-barotropic approximation and scale height=== This atmospheric model assumes both molecular weight and temperature are constant over a wide range of altitude. Assuming density is constant, then a graph of pressure vs altitude will have a retained slope, since the weight of the ocean over head is directly proportional to its depth. An alternative derivation, shown by the Portland State Aerospace Society, is used to give height as a function of pressure instead. This is the recommended formula to use. ==See also== * Barometer * Hypsometric equation * Pascal's barrel * Ruina montium * Pressure gradient * Siphon ==References== * ==External links== Category:Pressure Category:Vertical position For the isothermal-barotropic model, density and pressure turn out to be exponential functions of altitude. If the pressure at one point in a liquid with uniform density ρ is known to be P0, then the pressure at another point is P1: :P_1=P_0 - \rho g (h_1 - h_0) where h1 \- h0 is the vertical distance between the two points.Streeter, Victor L. (1966). The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of L_b: P = P_b \left[\frac{T_b - \left(h - h_b\right) L_b}{T_b}\right]^{\tfrac{g_0 M}{R^* L_b}} The second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude: P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/m) in ISA *h = height at which pressure is calculated (m) *h_b = height of reference level b (meters; e.g., hb = 11 000 m) *R^* = universal gas constant: 8.3144598 J/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol Or converted to imperial units:Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. Since is negative, an increase in height will correspond to a decrease in pressure, which fits with the previously mentioned reasoning about the weight of a column of fluid. ", 0.01961,0.00017,"""0.2553""",362880,0.6296296296,B
+"The mass density of water vapour at $327.6 \mathrm{~atm}$ and $776.4 \mathrm{~K}$ is $133.2 \mathrm{~kg} \mathrm{~m}^{-3}$. Given that for water $T_{\mathrm{c}}=647.4 \mathrm{~K}, p_{\mathrm{c}}=218.3 \mathrm{~atm}, a=5.464 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}$, $b=0.03049 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and $M=18.02 \mathrm{~g} \mathrm{~mol}^{-1}$, calculate the molar volume.","The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. Liquid water path - in units of g/m2 is a measure of the total amount of liquid water present between two points in the atmosphere. thumb|The Mollier enthalpy–entropy diagram for water and steam. Air is given a vapour density of one. Further g is acceleration due to gravity and ρ is the fluid density. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . Further ρ is the (constant) fluid density and g is the gravitational acceleration. The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. It may be defined as mass of a certain volume of a substance divided by mass of same volume of hydrogen. :vapour density = mass of n molecules of gas / mass of n molecules of hydrogen gas . :vapour density = molar mass of gas / molar mass of H2 :vapour density = molar mass of gas / 2.016 :vapour density = × molar mass (and thus: molar mass = ~2 × vapour density) For example, vapour density of mixture of NO2 and N2O4 is 38.3. U.S. Geological Survey Water-Supply Paper 1541-B' An example of a double mass analysis is a ""double mass plot"", or ""double mass curve"".Wilson, E.M. (1983) Engineering Hydrology, 3rd edition. Double mass analysis is a simple graphical method to evaluate the consistency of hydrological data. The first equation is derived from mass conservation, the second two from momentum conservation. ===Non-conservative form=== Expanding the derivatives in the above using the product rule, the non-conservative form of the shallow-water equations is obtained. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. For nadir observations and whole atmospheric column we have :LWP=\int_{z=0}^\infty \rho_{air} r_L dz' where is the liquid water mixing ratio and is the density of air (including water loading). X gives the fraction (by mass) of gaseous substance in the wet region, the remainder being colloidal liquid droplets. The Mollier diagram coordinates are enthalpy h and humidity ratio x. thumb|491px|right|Output from a shallow-water equation model of water in a bathtub. The shallow- water equations are thus derived. ",-1.32,-9.54,"""-13.598""",4.49,0.1353,E
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $q$.","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. A 2012 study of the effects for the original hypothesis, based on a coupled climate–carbon cycle model (GCM) assessed a 1000-fold (from <1 to 1000 ppmv) methane increase—within a single pulse, from methane hydrates (based on carbon amount estimates for the PETM, with ~2000 GtC), and concluded it would increase atmospheric temperatures by more than 6 °C within 80 years. * Being a gaseous fuel, CNG mixes easily and evenly in air. A carbon dioxide generator or CO2 generator is a machine used to enhance carbon dioxide levels in order to promote plant growth in greenhouses or other enclosed areas. thumb|upright=1.6|Methane clathrate is released as gas into the surrounding water column or soils when ambient temperature increases thumb|upright=1.6|The impact of CH4 atmospheric methane concentrations on global temperature increase may be far greater than previously estimated. [http://regmorrison.edublogs.org/files/2013/02/METHANE-2-1sca3tx.pdf] The clathrate gun hypothesis is a proposed explanation for the periods of rapid warming during the Quaternary. The estimated amount of methane hydrate in this slope is 2.5 gigatonnes (about 0.2% of the amount required to cause the PETM), and it is unclear if the methane could reach the atmosphere. This would have had an immediate impact on the global temperature, as methane is a much more powerful greenhouse gas than carbon dioxide. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. A release on this scale would increase the methane content of the planet's atmosphere by a factor of twelve, equivalent in greenhouse effect to a doubling in the 2008 level of . Especially, for the CNG Type 1 and Type 2 cylinders, many countries are able to make reliable and cost effective cylinders for conversion need. ==Energy density== CNG's energy density is the same as liquefied natural gas at 53.6 MJ/kg. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). ** CNG produced from landfill biogas was found by CARB to have the lowest greenhouse gas emissions of any fuel analyzed, with a value of 11.26 gCO2e/MJ (more than 88 percent lower than conventional petrol) in the low-carbon fuel standard that went into effect on January 12, 2010. Gastric tonometry describes the measurement of the carbon dioxide level inside the stomach in order to assess the degree of blood flow to the stomach and bowel. Normocapnia or normocarbia is a state of normal arterial carbon dioxide pressure, usually about 40 mmHg. == See also == * * * == References == *The Free Dictionary - Normocapnia Category:Diving medicine Category:Pulmonology Since it is a compressed gas, rather than a liquid like petrol, CNG takes up more space for each GGE (petrol gallon equivalent). Accumulation of carbon dioxide is predominantly a result of hypoperfusion and not hypoxia. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. In this process known as high pressure ANG, a high pressure CNG tank is filled by absorbers such as activated carbon (which is an adsorbent with high surface area) and stores natural gas by both CNG and ANG mechanisms. ",0,1000,"""0.18""",-1.5,1855,A
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $\Delta S$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. *A conversion factor is included into the original first coefficients of the equations to provide the pressure in pascals (CR2: 5.006, SMI: -0.875). *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: : \gamma = \frac{5}{3} = 1.6666\ldots, As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. For an imperfect or non-ideal gas, Chandrasekhar defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure: : \begin{align} \Gamma_1 &= \left.\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}, \\\\[2pt] \frac{\Gamma_2 - 1}{\Gamma_2} &= \left.\frac{\partial \ln T}{\partial \ln P}\right|_{S}, \\\\[2pt] \Gamma_3 - 1 &= \left.\frac{\partial \ln T}{\partial \ln \rho}\right|_{S}. \end{align} All of these are equal to \gamma in the case of an ideal gas. == See also == * Relations between heat capacities * Heat capacity * Specific heat capacity * Speed of sound * Thermodynamic equations * Thermodynamics * Volumetric heat capacity == References == == Notes == Category:Thermodynamic properties Category:Physical quantities Category:Ratios Category:Thought experiments in physics Sometimes, other distinctions are made, such as between thermally perfect gas and calorically perfect gas, or between imperfect, semi-perfect, and perfect gases, and as well as the characteristics of ideal gases. In addition, other factors come into play and dominate during a compression cycle if they have a greater impact on the final calculated result than whether or not C_P was held constant. (i) Indicates values calculated from ideal gas thermodynamic functions. ",0,650000,"""4.68""",0.925,-0.38,A
+Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. Find an expression for the fugacity coefficient of a gas that obeys the equation of state $p V_{\mathrm{m}}=R T\left(1+B / V_{\mathrm{m}}+C / V_{\mathrm{m}}^2\right)$. Use the resulting expression to estimate the fugacity of argon at 1.00 atm and $100 \mathrm{~K}$ using $B=-21.13 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$ and $C=1054 \mathrm{~cm}^6 \mathrm{~mol}^{-2}$.,"For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. The fugacity coefficient is . Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. Using , where is the fugacity coefficient, f = \varphi_\mathrm{sat}P_\mathrm{sat}\exp\left(\frac{V\left(P-P_\mathrm{sat}\right)}{R T}\right). This error is corrected in the above equation. ==Measurement== The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. When using a fugacity capacity approach to calculate the concentrations of a chemical in each of several medias/phases/compartments, it is often convenient to calculate the prevailing fugacity of the system using the following equation if the total mass of target chemical (MT) and the volume of each compartment (Vm) are known: :f = M_T / \Sigma_m (V_m Z_m) Alternatively, if the target chemical is present as a pure phase at equilibrium, its vapor pressure will be the prevailing fugacity of the system. ==See also== * Multimedia fugacity model ==References== Category:Chemical thermodynamics Category:Environmental chemistry Category:Equilibrium chemistry The real gas pressure and fugacity are related through the dimensionless fugacity coefficient . \varphi = \frac{f}{P} For an ideal gas, fugacity and pressure are equal and so . If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar reaction quotient form (or law of mass action) except that the pressures are replaced by fugacities. Numerical example: Nitrogen gas (N2) at 0 °C and a pressure of atmospheres (atm) has a fugacity of atm. In addition, there are natural reference states for fugacity (for example, an ideal gas makes a natural reference state for gas mixtures since the fugacity and pressure converge at low pressure). ===Gases=== In a mixture of gases, the fugacity of each component has a similar definition, with partial molar quantities instead of molar quantities (e.g., instead of and instead of ): dG_i = R T \,d \ln f_i and \lim_{P\to 0} \frac{f_i}{P_i} = 1, where is the partial pressure of component . :C_m = Z_m \cdot f where Z is a proportional constant, termed fugacity capacity. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The constant of proportionality (a measured Henry's constant) depends on whether the concentration is represented by the mole fraction, molality or molarity. ==Temperature and pressure dependence== The pressure dependence of fugacity (at constant temperature) is given by \left(\frac{\partial \ln f}{\partial P}\right)_T = \frac{V_\mathrm{m}}{R T} and is always positive. The fugacities commonly obey a similar law called the Lewis and Randall rule: f_i = y_i f^*_i, where is the fugacity that component would have if the entire gas had that composition at the same temperature and pressure. Hemond and Hechner-Levy (2000) describe how to utilize the fugacity capacity to calculate the concentration of a chemical in a system. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. ",6.6,0.03,"""0.9974""",-4564.7,2.3613,C
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the maximum non-expansion work per mole that may be obtained from a fuel cell in which the chemical reaction is the combustion of methane at $298 \mathrm{~K}$.,"For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Even more restricted is the calorically perfect gas for which, in addition, the heat capacity is assumed to be constant. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. But even if the heat capacity is strictly a function of temperature for a given gas, it might be assumed constant for purposes of calculation if the temperature and heat capacity variations are not too large, which would lead to the assumption of a calorically perfect gas (see below). Instead of a mole the constant can be expressed by considering the normal cubic meter. * Individual Gas Constants and the Universal Gas Constant – Engineering Toolbox Category:Ideal gas Category:Physical constants Category:Amount of substance Category:Statistical mechanics Category:Thermodynamics The contribution of nonideality to the molar Gibbs energy of a real gas is equal to . Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . ",54.7,91.7,"""-233.0""",817.90,0,D
 "Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K.
 Silylene $\left(\mathrm{SiH}_2\right)$ is a key intermediate in the thermal decomposition of silicon hydrides such as silane $\left(\mathrm{SiH}_4\right)$ and disilane $\left(\mathrm{Si}_2 \mathrm{H}_6\right)$. Moffat et al. (J. Phys. Chem. 95, 145 (1991)) report $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{SiH}_2\right)=+274 \mathrm{~kJ} \mathrm{~mol}^{-1}$. If $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{SiH}_4\right)=+34.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{Si}_2 \mathrm{H}_6\right)=+80.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$(CRC Handbook (2008)), compute the standard enthalpies of the following reaction:
-$\mathrm{Si}_2 \mathrm{H}_6(\mathrm{g}) \rightarrow \mathrm{SiH}_2(\mathrm{g})+\mathrm{SiH}_4(\mathrm{g})$","Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2} | cdf = \Phi \left(\gamma + \delta \sinh^{-1} \left( \frac{x-\xi}{\lambda} \right) \right) | mean = \xi - \lambda \exp \frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right) | median = \xi + \lambda \sinh \left( - \frac{\gamma}{\delta} \right) | mode = | variance = \frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right) | skewness = -\frac{\lambda^3\sqrt{e^{\delta^{-2}}}(e^{\delta^{-2}}-1)^{2}((e^{\delta^{-2}})(e^{\delta^{-2}}+2)\sinh(\frac{3\gamma}{\delta})+3\sinh(\frac{2\gamma}{\delta}))}{4(\operatorname{Variance}X)^{1.5}} | kurtosis = \frac{\lambda^4(e^{\delta^{-2}}-1)^2(K_{1}+K_2+K_3)}{8(\operatorname{Variance}X)^2} K_{1}=\left( e^{\delta^{-2}} \right)^{2}\left( \left( e^{\delta^{-2}} \right)^{4}+2\left( e^{\delta^{-2}} \right)^{3}+3\left( e^{\delta^{-2}} \right)^{2}-3 \right)\cosh\left( \frac{4\gamma}{\delta} \right) K_2=4\left( e^{\delta^{-2}} \right)^2 \left( \left( e^{\delta^{-2}} \right)+2 \right)\cosh\left( \frac{3\gamma}{\delta} \right) K_3=3\left( 2\left( e^{\delta^{-2}} \right)+1 \right) | entropy = | pgf = | mgf = | char = }} The Johnson's SU-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). K (? °C), ? K (? °C), ? K (? °C), ? K (? °C), ? ",+37,0.9522,3.2,228,22.2036033112,D
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $\Delta S$.","In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We assume the expansion occurs without exchange of heat (adiabatic expansion). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. Since the piston cannot move, the volume is constant. The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. * Being a gaseous fuel, CNG mixes easily and evenly in air. ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). Especially, for the CNG Type 1 and Type 2 cylinders, many countries are able to make reliable and cost effective cylinders for conversion need. ==Energy density== CNG's energy density is the same as liquefied natural gas at 53.6 MJ/kg. ",30,+0.60,0.2553,0.14,0.1792,B
-"The mass density of water vapour at $327.6 \mathrm{~atm}$ and $776.4 \mathrm{~K}$ is $133.2 \mathrm{~kg} \mathrm{~m}^{-3}$. Given that for water $T_{\mathrm{c}}=647.4 \mathrm{~K}, p_{\mathrm{c}}=218.3 \mathrm{~atm}, a=5.464 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}$, $b=0.03049 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and $M=18.02 \mathrm{~g} \mathrm{~mol}^{-1}$, calculate the compression factor from the virial expansion of the van der Waals equation.","Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated. ===Reduced form=== Although the material constant a and b in the usual form of the Van der Waals equation differs for every single fluid considered, the equation can be recast into an invariant form applicable to all fluids. Even with its acknowledged shortcomings, the pervasive use of the Van der Waals equation in standard university physical chemistry textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the Van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the p–V diagram as the ratio of vapour to liquid changes. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. The corrected equation becomes : p = \frac{RT}{V_\mathrm{m}-b}. ",-167,0.7158,0.18,9.8,0.123,B
-Express the van der Waals parameters $a=0.751 \mathrm{~atm} \mathrm{dm}^6 \mathrm{~mol}^{-2}$ in SI base units.,"* Analytical calculation of Van der Waals surfaces and volumes. Petitjean, On the Analytical Calculation of Van der Waals Surfaces and Volumes: Some Numerical Aspects, Journal of Computational Chemistry, Volume 15, Number 5, 1994, pp. 507–523. == External links == * VSAs for various molecules by Anton Antonov, The Wolfram Demonstrations Project, 2007. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. Useful tabulated values of Van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will be seen to present different values for the Van der Waals radius of the same atom. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Van der Waals volumes of a single atom or molecules are arrived at by dividing the macroscopically determined volumes by the Avogadro constant. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). ",0.0761,0.3359,2.3,0.0625,9.30,A
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Estimate the change in the Gibbs energy of $1.0 \mathrm{dm}^3$ of benzene when the pressure acting on it is increased from $1.0 \mathrm{~atm}$ to $100 \mathrm{~atm}$.,"Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . thumb|2D model of a benzene molecule. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant is the constant of proportionality that relates the energy scale in physics to the temperature scale and the scale used for amount of substance. Uses formula: \scriptstyle \log_e P_{mmHg} =\scriptstyle \log_e (\frac {760} {101.325}) - 8.433613\log_e(T+273.15) - \frac {6281.040} {T+273.15} + 71.10718 + 6.198413 \times 10^{-06} (T+273.15)^2 obtained from CHERIC Note: yellow area is the region where the formula disagrees with tabulated data above. ==Distillation data== {| border=""1"" cellspacing=""0"" cellpadding=""6"" style=""margin: 0 0 0 0.5em; background: white; border- collapse: collapse; border-color: #C0C090;"" Vapor-liquid Equilibrium for Benzene/Ethanol P = 760 mm Hg BP Temp. °C % by mole ethanol liquid vapor 70.8 8.6 26.5 69.8 11.2 28.2 69.6 12.0 30.8 69.1 15.8 33.5 68.5 20.0 36.8 67.7 30.8 41.0 67.7 44.2 44.6 68.1 60.4 50.5 69.6 77.0 59.0 70.4 81.5 62.8 70.9 84.1 66.5 72.7 89.8 74.4 73.8 92.4 78.2 == Spectral data == UV-Vis Ionization potential 9.24 eV (74525.6 cm−1) S1 4.75 eV (38311.3 cm−1) S2 6.05 eV (48796.5 cm−1) λmax 255 nm Extinction coefficient, ε ? This page provides supplementary chemical data on benzene. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. (i) Indicates values calculated from ideal gas thermodynamic functions. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. On the elasticity of gases. 1875 (in Russian) Mendeleev also calculated it with high precision, within 0.3% of its modern value. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, i.e. the pressure–volume product, rather than energy per temperature increment per particle. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of units of energy, temperature and amount of substance. ",0.1591549431,+10,1.33,35.64, 0.0024,B
-"The mass density of water vapour at $327.6 \mathrm{~atm}$ and $776.4 \mathrm{~K}$ is $133.2 \mathrm{~kg} \mathrm{~m}^{-3}$. Given that for water $T_{\mathrm{c}}=647.4 \mathrm{~K}, p_{\mathrm{c}}=218.3 \mathrm{~atm}, a=5.464 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}$, $b=0.03049 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and $M=18.02 \mathrm{~g} \mathrm{~mol}^{-1}$, calculate the compression factor from the data.","The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Regarding the measurement of upper-air humidity, this publication also reads (in Section 12.5.1): The saturation with respect to water cannot be measured much below –50 °C, so manufacturers should use one of the following expressions for calculating saturation vapour pressure relative to water at the lowest temperatures – Wexler (1976, 1977), reported by Flatau et al. (1992)., Hyland and Wexler (1983) or Sonntag (1994) – and not the Goff- Gratch equation recommended in earlier WMO publications. ==Experimental correlation== The original Goff–Gratch (1945) experimental correlation reads as follows: : \log\ e^*\ = -7.90298(T_\mathrm{st}/T-1)\ +\ 5.02808\ \log(T_\mathrm{st}/T) -\ 1.3816\times10^{-7}(10^{11.344(1-T/T_\mathrm{st})}-1) +\ 8.1328\times10^{-3}(10^{-3.49149(T_\mathrm{st}/T-1)}-1)\ +\ \log\ e^*_\mathrm{st} where: :log refers to the logarithm in base 10 :e* is the saturation water vapor pressure (hPa) :T is the absolute air temperature in kelvins :Tst is the steam-point (i.e. boiling point at 1 atm.) temperature (373.15 K) :e*st is e* at the steam-point pressure (1 atm = 1013.25 hPa) Similarly, the correlation for the saturation water vapor pressure over ice is: : \log\ e^*_i\ = -9.09718(T_0/T-1)\ -\ 3.56654\ \log(T_0/T) +\ 0.876793(1-T/T_0) +\ \log\ e^*_{i0} where: :log stands for the logarithm in base 10 :e*i is the saturation water vapor pressure over ice (hPa) :T is the air temperature (K) :T0 is the ice-point (triple point) temperature (273.16 K) :e*i0 is e* at the ice-point pressure (6.1173 hPa) ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Tetens equation *Lee–Kesler method ==References== * Goff, J. A., and Gratch, S. (1946) Low- pressure properties of water from −160 to 212 °F, in Transactions of the American Society of Heating and Ventilating Engineers, pp 95–122, presented at the 52nd annual meeting of the American Society of Heating and Ventilating Engineers, New York, 1946. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. * * ;Notes ==External links== * * Free Windows Program, Moisture Units Conversion Calculator w/Goff-Gratch equation — PhyMetrix Category:Atmospheric thermodynamics Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . Air is given a vapour density of one. Further g is acceleration due to gravity and ρ is the fluid density. The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. The Goff–Gratch equation is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the saturation water vapor pressure at a given temperature. thumb|The Mollier enthalpy–entropy diagram for water and steam. Further ρ is the (constant) fluid density and g is the gravitational acceleration. * Goff, J. A. (1957) Saturation pressure of water on the new Kelvin temperature scale, Transactions of the American Society of Heating and Ventilating Engineers, pp 347–354, presented at the semi-annual meeting of the American Society of Heating and Ventilating Engineers, Murray Bay, Que. Canada. * After correction, repeat this process until all data points have the same slope. ==See also== *Statistics ==Notes== ==Further reading== * Dubreuil P. (1974) Initiation à l'analyse hydrologique Masson& Cie et ORSTOM, Paris. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. Another similar equation based on more recent data is the Arden Buck equation. ==Historical note== This equation is named after the authors of the original scientific article who described how to calculate the saturation water vapor pressure above a flat free water surface as a function of temperature (Goff and Gratch, 1946). Double mass analysis is a simple graphical method to evaluate the consistency of hydrological data. The Mollier diagram coordinates are enthalpy h and humidity ratio x. The temperature-vapour pressure relation inversely describes the relation between the boiling point of water and the pressure. An understanding of vapour pressure is also relevant in explaining high altitude breathing and cavitation. ==Approximation formulas == There are many published approximations for calculating saturated vapour pressure over water and over ice. ",+93.4,2.3,1.0,0.6957,311875200,D
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in Gibbs energy of $35 \mathrm{~g}$ of ethanol (mass density $0.789 \mathrm{~g} \mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from $1 \mathrm{~atm}$ to $3000 \mathrm{~atm}$.,"Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The partial pressures obey Dalton's law: P_i = y_i P, where is the total pressure and is the mole fraction of the component (so the partial pressures add up to the total pressure). It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to . *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., ) is given by d\mu = V_\mathrm{m}dP = RT \, \frac{dP}{P} = R T\, d \ln P,where ln p is the natural logarithm of p. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. The contribution of nonideality to the molar Gibbs energy of a real gas is equal to . The Reid vapor pressure (RVP) can differ substantially from the true vapor pressure (TVP) of a liquid mixture, since (1) RVP is the vapor pressure measured at 37.8 °C (100 °F) and the TVP is a function of the temperature; (2) RVP is defined as being measured at a vapor-to-liquid ratio of 4:1, whereas the TVP of mixtures can depend on the actual vapor-to-liquid ratio; (3) RVP will include the pressure associated with the presence of dissolved water and air in the sample (which is excluded by some but not all definitions of TVP); and (4) the RVP method is applied to a sample which has had the opportunity to volatilize somewhat prior to measurement: i.e., the sample container is required to be only 70-80% full of liquid ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Section 8.3(so that whatever volatilizes into the container headspace is lost prior to analysis); the sample then again volatilizes into the headspace of the D323 test chamber before it is heated to 37.8 degrees Celsius.Conversion between the two measures can be found here, from p. 7.1-54 onwards. ==See also== * Crude oil assay * Gasoline volatility * Vapor pressure ==External links== * ASTM D323 - 06 Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method) * Reid Vapor Pressure Requirements for Ethanol Congressional Research Service * USA's Environmental Protection Agency (EPA) publication AP-42, Compilation of Air Pollutant Emissions. Reid vapor pressure (RVP) is a common measure of the volatility of gasoline and other petroleum products.ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Section 1.1 It is defined as the absolute vapor pressure exerted by the vapor of the liquid and any dissolved gases/moisture at 37.8 °C (100 °F) as determined by the test method ASTM-D-323, which was first developed in 1930 ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), footnote 1 and has been revised several times (the latest version is ASTM D323-15a).ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method) The test method measures the vapor pressure of gasoline, volatile crude oil, jet fuels, naphtha, and other volatile petroleum products but is not applicable for liquefied petroleum gases.ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Sections 1.1 and 1.6 ASTM D323-15a requires that the sample be chilled to 0 to 1 degrees Celsius and then poured into the apparatus;ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Sections 11.1 and 11.1.2 for any material that solidifies at this temperature, this step cannot be performed. ",5,0.68,152.67,-167,12,E
-"The densities of air at $-85^{\circ} \mathrm{C}, 0^{\circ} \mathrm{C}$, and $100^{\circ} \mathrm{C}$ are $1.877 \mathrm{~g} \mathrm{dm}^{-3}, 1.294 \mathrm{~g}$ $\mathrm{dm}^{-3}$, and $0.946 \mathrm{~g} \mathrm{dm}^{-3}$, respectively. From these data, and assuming that air obeys Charles's law, determine a value for the absolute zero of temperature in degrees Celsius.","Rounding up 1.98°C to 2°C, this approximation simplifies to become :\begin{align} \text{DA} & \approx \text{PA} + 118.8 ~ \frac{\text{ft}}{^\circ \text{C}} \left[ T_\text{OA} + \frac{\text{PA}}{500 ~ \text{ft}} {^\circ \text{C}} - 15 ~ {^\circ \text{C}} \right] \\\\[3pt] & = 1.2376 \, \text{PA} + 118.8 ~ \frac{\text{ft}}{{}^\circ \text{C}} \, T_\text{OA} - 1782 ~ \text{ft}. \end{align} ==See also== *Outside air temperature *Barometric formula *Density of air *Hot and high *List of longest runways == Notes == ==References== * * * Advisory Circular AC 61-23C, Pilot's Handbook of Aeronautical Knowledge, U.S. Federal Aviation Administration, Revised 1997 * http://www.tpub.com/content/aerographer/14269/css/14269_74.htm * ==External links== *Density Altitude Calculator *Density Altitude influence on aircraft performance *NewByte Atmospheric Calculator Category:Altitudes in aviation Category:Atmospheric thermodynamics Unaware of the inaccuracies of mercury thermometers at the time, which were divided into equal portions between the fixed points, Dalton, after concluding in Essay II that in the case of vapours, “any elastic fluid expands nearly in a uniform manner into 1370 or 1380 parts by 180 degrees (Fahrenheit) of heat”, was unable to confirm it for gases. ==Relation to absolute zero== Charles's law appears to imply that the volume of a gas will descend to zero at a certain temperature (−266.66 °C according to Gay-Lussac's figures) or −273.15 °C. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In this formula, : \text{DA} , density altitude in meters (m); : P , (static) atmospheric pressure; : P_\text{SL} , standard sea-level atmospheric pressure, International Standard Atmosphere (ISA): 1013.25 hectopascals (hPa), or U.S. Standard Atmosphere: 29.92 inches of mercury (inHg); : T , outside air temperature in kelvins (K); : T_\text{SL} = 288.15K, ISA sea-level air temperature; : \Gamma = 0.0065K/m, ISA temperature lapse rate (below 11km); : R ≈ 8.3144598J/mol·K, ideal gas constant; : g ≈ 9.80665m/s, gravitational acceleration; : M ≈ 0.028964kg/mol, molar mass of dry air. ===The National Weather Service (NWS) formula=== The National Weather Service uses the following dry-air approximation to the formula for the density altitude above in its standard: : \text{DA}_\text{NWS} = 145442.16 ~ \text{ft} \left( 1 - \left[ 17.326 ~ \frac{^\circ \text{F}}{\text{inHg}} \ \frac{P}{459.67 ~ {{}^\circ \text{F}} + T} \right]^{0.235} \right). Subscript b Height above sea level Static pressure Standard temperature (K) Temperature lapse rate Exponent g0 M / R L (m) (ft) (Pa) (inHg) (K/m) (K/ft) 0 0 0 101 325.00 29.92126 288.15 0.0065 0.0019812 5.2558 1 11 000 36,089 22 632.10 6.683245 216.65 0.0 0.0 -- 2 20 000 65,617 5474.89 1.616734 216.65 -0.001 -0.0003048 -34.1626 3 32 000 104,987 868.02 0.2563258 228.65 -0.0028 -0.00085344 -12.2009 4 47 000 154,199 110.91 0.0327506 270.65 0.0 0.0 -- 5 51 000 167,323 66.94 0.01976704 270.65 0.0028 0.00085344 12.2009 6 71 000 232,940 3.96 0.00116833 214.65 0.002 0.0006096 17.0813 ==Density equations== The expressions for calculating density are nearly identical to calculating pressure. (Before going further, I should inform [you] that although I had recognized many times that the gases oxygen, nitrogen, hydrogen, and carbonic acid [i.e., carbon dioxide], and atmospheric air also expand from 0° to 80°, citizen Charles had noticed 15 years ago the same property in these gases; but having never published his results, it is by the merest chance that I knew of them.) although he credited the discovery to unpublished work from the 1780s by Jacques Charles. This equation does not contain the temperature and so is not what became known as Charles's Law. Equation 1: \rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^* L_b}-1\right)} which is equivalent to the ratio of the relative pressure and temperature changes \rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} Equation 2: \rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where *{\rho} = mass density (kg/m3) *T_b = standard temperature (K) *L = standard temperature lapse rate (see table below) (K/m) in ISA *h = height above sea level (geopotential meters) *R^* = universal gas constant 8.3144598 N·m/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): *{\rho} = mass density (slug/ft3) *{T_b} = standard temperature (K) *{L} = standard temperature lapse rate (K/ft) *{h} = height above sea level (geopotential feet) *{R^*} = universal gas constant: 8.9494596×104 ft2/(s·K) *{g_0} = gravitational acceleration: 32.17405 ft/s2 *{M} = molar mass of Earth's air: 0.0289644 kg/mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In this formula, : \text{PA} , pressure altitude in feet (ft) \approx \text{station elevation in feet} + 27 ~ \frac{\text{ft}}{\text{mb}} (1013 ~ \text{mb} - \text{QNH}) ; : \text{QNH} , atmospheric pressure in millibars (mb) adjusted to mean sea level; : T_\text{OA}, outside air temperature in degrees Celsius (°C); : T_\text{ISA} \approx 15 ~ {{}^\circ \text{C}} - 1.98 ~ {{}^\circ \text{C}} \, \frac{\text{PA}}{1000 ~ \text{ft}} , assuming that the outside air temperature falls at the rate of 1.98°C per 1,000ft of altitude until the tropopause (at ) is reached. The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of L_b: P = P_b \left[\frac{T_b - \left(h - h_b\right) L_b}{T_b}\right]^{\tfrac{g_0 M}{R^* L_b}} The second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude: P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/m) in ISA *h = height at which pressure is calculated (m) *h_b = height of reference level b (meters; e.g., hb = 11 000 m) *R^* = universal gas constant: 8.3144598 J/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol Or converted to imperial units:Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. In the absence of a firm record, the gas law relating volume to temperature cannot be attributed to Charles. A modern statement of Charles' law is: > When the pressure on a sample of a dry gas is held constant, the Kelvin > temperature and the volume will be in direct proportion.. Thomson did not assume that this was equal to the ""zero-volume point"" of Charles's law, merely that Charles's law provided the minimum temperature which could be attained. The first mention of a temperature at which the volume of a gas might descend to zero was by William Thomson (later known as Lord Kelvin) in 1848:. > This is what we might anticipate when we reflect that infinite cold must > correspond to a finite number of degrees of the air-thermometer below zero; > since if we push the strict principle of graduation, stated above, > sufficiently far, we should arrive at a point corresponding to the volume of > air being reduced to nothing, which would be marked as −273° of the scale > (−100/.366, if .366 be the coefficient of expansion); and therefore −273° of > the air-thermometer is a point which cannot be reached at any finite > temperature, however low. The values used for M, g0 and R* are in accordance with the U.S. Standard Atmosphere, 1976, and that the value for R* in particular does not agree with standard values for this constant. Subscript b Height Above Sea Level (h) Mass Density (\rho) Standard Temperature (T') (K) Temperature Lapse Rate (L) (m) (ft) (kg/m3) (slug/ft3) (K/m) (K/ft) 0 0 0 1.2250 288.15 0.0065 0.0019812 1 11 000 36,089.24 0.36391 216.65 0.0 0.0 2 20 000 65,616.79 0.08803 216.65 -0.001 -0.0003048 3 32 000 104,986.87 0.01322 228.65 -0.0028 -0.00085344 4 47 000 154,199.48 0.00143 270.65 0.0 0.0 5 51 000 167,322.83 0.00086 270.65 0.0028 0.00085344 6 71 000 232,939.63 0.000064 214.65 0.002 0.0006096 ==Derivation== The barometric formula can be derived using the ideal gas law: P = \frac{\rho}{M} {R^*} T Assuming that all pressure is hydrostatic: dP = - \rho g\,dz and dividing the dP by the P expression we get: \frac{dP}{P} = - \frac{M g\,dz}{R^*T} Integrating this expression from the surface to the altitude z we get: P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}} Assuming linear temperature change T = T_0 - L z and constant molar mass and gravitational acceleration, we get the first barometric formula: P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}} Instead, assuming constant temperature, integrating gives the second barometric formula: P = P_0 e^{-M g z/R^*T} In this formulation, R* is the gas constant, and the term R*T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere). The values used for M, g0, and R* are in accordance with the U.S. Standard Atmosphere, 1976, and the value for R* in particular does not agree with standard values for this constant.U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976. Note that the NWS standard specifies that the density altitude should be rounded to the nearest 100ft. ===Approximation formula for calculating the density altitude from the pressure altitude=== This is an easier formula to calculate (with great approximation) the density altitude from the pressure altitude and the ISA temperature deviation: : \text{DA} \approx \text{PA} + 118.8 ~ \frac{\text{ft}}{{^\circ \text{C}}} \left(T_\text{OA} - T_\text{ISA}\right). To derive Charles's law from kinetic theory, it is necessary to have a microscopic definition of temperature: this can be conveniently taken as the temperature being proportional to the average kinetic energy of the gas molecules, k: :T \propto \bar{E_{\rm k}}.\, Under this definition, the demonstration of Charles's law is almost trivial. This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. In this formula, : \text{DA}_\text{NWS} , National Weather Service density altitude in feet ( \text{ft} ); : P , station pressure (static atmospheric pressure) in inches of mercury (inHg); : T , station temperature (outside air temperature) in degrees Fahrenheit (°F). * Review of Amontons' findings: ""Sur une nouvelle proprieté de l'air, et une nouvelle construction de Thermométre"" (On a new property of the air and a new construction of thermometer), Histoire de l'Académie Royale des Sciences, 1–8 (submitted: 1702; published: 1743). and Francis Hauksbee* Englishman Francis Hauksbee (1660–1713) independently also discovered Charles's law: Francis Hauksbee (1708) ""An account of an experiment touching the different densities of air, from the greatest natural heat to the greatest natural cold in this climate,"" Philosophical Transactions of the Royal Society of London 26(315): 93–96. a century earlier. ",-100,-273,4.946,-7.5,+65.49,B
-A certain gas obeys the van der Waals equation with $a=0.50 \mathrm{~m}^6 \mathrm{~Pa}$ $\mathrm{mol}^{-2}$. Its volume is found to be $5.00 \times 10^{-4} \mathrm{~m}^3 \mathrm{~mol}^{-1}$ at $273 \mathrm{~K}$ and $3.0 \mathrm{MPa}$. From this information calculate the van der Waals constant $b$. What is the compression factor for this gas at the prevailing temperature and pressure?,"The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density. ===Statistical thermodynamics derivation=== The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is: : Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3} where \Lambda is the thermal de Broglie wavelength, : \Lambda = \sqrt{\frac{h^2}{2\pi m k T}} with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. For an ideal gas the compressibility factor is Z=1 per definition. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. Experimental values for the compressibility factor confirm this. According to van der Waals, the theorem of corresponding states (or principle/law of corresponding states) indicates that all fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree. page 141 Material constants that vary for each type of material are eliminated, in a recast reduced form of a constitutive equation. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. ",0.0761,7.136,4.0,2,0.66,E
-Calculate the pressure exerted by $1.0 \mathrm{~mol} \mathrm{Xe}$ when it is confined to $1.0 \mathrm{dm}^3$ at $25^{\circ} \mathrm{C}$.,"The CGS unit of pressure is the barye (Ba), equal to 1 dyn·cm−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm2 or kg/cm2) and the like without properly identifying the force units. :P=\frac {2 \sigma_\theta s} {D}thumb|252x252px|Cylinder, where :P : internal pressure, :\sigma_\theta : allowable stress, :s : wall thickness, :D : outside diameter. Derivation of this equation This is derived from the definitions of pressure and weight density. Pressure is related to energy density and may be expressed in units such as joules per cubic metre (J/m3, which is equal to Pa). With the ""area"" in the numerator and the ""area"" in the denominator canceling each other out, we are left with :\text{pressure} = \text{weight density} \times \text{depth}. Presently or formerly popular pressure units include the following: *atmosphere (atm) *manometric units: **centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury, **height of equivalent column of water, including millimetre (mm ), centimetre (cm ), metre, inch, and foot of water; *imperial and customary units: **kip, short ton-force, long ton-force, pound- force, ounce-force, and poundal per square inch, **short ton-force and long ton-force per square inch, **fsw (feet sea water) used in underwater diving, particularly in connection with diving pressure exposure and decompression; *non-SI metric units: **bar, decibar, millibar, ***msw (metres sea water), used in underwater diving, particularly in connection with diving pressure exposure and decompression, **kilogram-force, or kilopond, per square centimetre (technical atmosphere), **gram-force and tonne-force (metric ton- force) per square centimetre, **barye (dyne per square centimetre), **kilogram-force and tonne-force per square metre, **sthene per square metre (pieze). ===Examples=== 120px|thumbnail|right|The effects of an external pressure of 700 bar on an aluminum cylinder with wall thickness As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation , where g is the gravitational acceleration. thumb|259x259px|English Parliament of General Election 1702 The 1702 English general election was the first to be held during the reign of Queen Anne, and was necessitated by the demise of William III. * P_0 is the pressure at the surface. * \rho(z) is the density of the material above the depth z. * g is the gravity acceleration in m/s^2 . Pressure is force magnitude applied over an area. Barlow's formula (called ""Kesselformel"" in German) relates the internal pressure that a pipeOr pressure vessel, or other cylindrical pressure containment structure. can withstand to its dimensions and the strength of its material. The molecular formula C23H21NO (molar mass: 327.42 g/mol, exact mass: 327.1623 u) may refer to: * JWH-015 * JWH-073 * JWH-120 Category:Molecular formulas The molecular formula C13H14O3 (molar mass: 218.248 g/mol, exact mass: 218.0943 u) may refer to: * NCS-382 * Toxol Category:Molecular formulas Pressure force acts in all directions at a point inside a gas. Then we have :\text{pressure} = \frac{\text{force}}{\text{area}} = \frac{\text{weight}}{\text{area}} = \frac{\text{weight density} \times \text{volume}}{\text{area}}, :\text{pressure} = \frac{\text{weight density} \times \text{(area} \times \text{depth)}}{\text{area}}. Pressure in open conditions usually can be approximated as the pressure in ""static"" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. The negative gradient of pressure is called the force density. The pressure is the scalar proportionality constant that relates the two normal vectors: :d\mathbf{F}_n = -p\,d\mathbf{A} = -p\,\mathbf{n}\,dA. It is a fundamental parameter in thermodynamics, and it is conjugate to volume. ===Units=== thumb|right|Mercury column The SI unit for pressure is the pascal (Pa), equal to one newton per square metre (N/m2, or kg·m−1·s−2). In a stratigraphic layer that is in hydrostatic equilibrium; the overburden pressure at a depth z, assuming the magnitude of the gravity acceleration is approximately constant, is given by: P(z) = P_0 + g \int_{0}^{z} \rho(z) \, dz Where: * z is the depth in meters. This tensor may be expressed as the sum of the viscous stress tensor minus the hydrostatic pressure. The total pressure of a liquid, then, is ρgh plus the pressure of the atmosphere. ",21,4152,-20.0,24,-0.347,A
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the total change in entropy, when a sample of nitrogen gas of mass $14 \mathrm{~g}$ at $298 \mathrm{~K}$ and $1.00 \mathrm{bar}$ doubles its volume in an isothermal reversible expansion.","The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. ISO 15971 is an ISO standard for calorific value measurement of natural gas and its substitutes. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. The limit of this sum as N \rightarrow \infty becomes an integral: :S^\circ = \sum_{k=1}^N \Delta S_k = \sum_{k=1}^N \frac{dQ_k}{T} \rightarrow \int _0 ^{T_2} \frac{dS}{dT} dT = \int _0 ^{T_2} \frac {C_{p_k}}{T} dT In this example, T_2 =298.15 K and C_{p_k} is the molar heat capacity at a constant pressure of the substance in the reversible process . However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). * The heat capacity of the gas from the boiling point to room temperature. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. ",0.25,0,9.2e-06,0.2307692308,2,B
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate $\Delta H$ when two copper blocks, each of mass $10.0 \mathrm{~kg}$, one at $100^{\circ} \mathrm{C}$ and the other at $0^{\circ} \mathrm{C}$, are placed in contact in an isolated container. The specific heat capacity of copper is $0.385 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$ and may be assumed constant over the temperature range involved.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The formula is: :\alpha = \frac{ k }{ \rho c_{p} } where * is thermal conductivity (W/(m·K)) * is specific heat capacity (J/(kg·K)) * is density (kg/m3) Together, can be considered the volumetric heat capacity (J/(m3·K)). Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). thumb|250px|The plot of the specific heat capacity versus temperature. Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities. The amount of energy added equals , with representing the change in temperature. Boca Raton, Florida, 2003; Section 4, Properties of the Elements and Inorganic Compounds; Vapor Pressure of the Metallic Elements :The equations reproduce the observed pressures to an accuracy of ±5% or better. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. ",2.3613,0,65.49,14,0.086,B
-"A perfect gas undergoes isothermal compression, which reduces its volume by $2.20 \mathrm{dm}^3$. The final pressure and volume of the gas are $5.04 \mathrm{bar}$ and $4.65 \mathrm{dm}^3$, respectively. Calculate the original pressure of the gas in atm.","The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture (Dalton's Law). The partial pressures obey Dalton's law: P_i = y_i P, where is the total pressure and is the mole fraction of the component (so the partial pressures add up to the total pressure). It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules. That is, the mole fraction x_{\mathrm{i}} of an individual gas component in an ideal gas mixture can be expressed in terms of the component's partial pressure or the moles of the component: x_{\mathrm{i}} = \frac{p_{\mathrm{i}}}{p} = \frac{n_{\mathrm{i}}}{n} and the partial pressure of an individual gas component in an ideal gas can be obtained using this expression: p_{\mathrm{i}} = x_{\mathrm{i}} \cdot p where: x_{\mathrm{i}} = mole fraction of any individual gas component in a gas mixture p_{\mathrm{i}} = partial pressure of any individual gas component in a gas mixture n_{\mathrm{i}} = moles of any individual gas component in a gas mixture n = total moles of the gas mixture p = total pressure of the gas mixture The mole fraction of a gas component in a gas mixture is equal to the volumetric fraction of that component in a gas mixture.Frostberg State University's ""General Chemistry Online"" The ratio of partial pressures relies on the following isotherm relation: \frac{V_{\rm X}}{V_{\rm tot}} = \frac{p_{\rm X}}{p_{\rm tot}} = \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of any individual gas component (X) * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas (X) * ntot is the total amount of substance in gas mixture ==Partial volume (Amagat's law of additive volume)== The partial volume of a particular gas in a mixture is the volume of one component of the gas mixture. In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The standard atmosphere (symbol: atm) is a unit of pressure defined as Pa. Using diving terms, partial pressure is calculated as: :partial pressure = (total absolute pressure) × (volume fraction of gas component) For the component gas ""i"": :pi = P × Fi For example, at underwater, the total absolute pressure is (i.e., 1 bar of atmospheric pressure + 5 bar of water pressure) and the partial pressures of the main components of air, oxygen 21% by volume and nitrogen approximately 79% by volume are: :pN2 = 6 bar × 0.79 = 4.7 bar absolute :pO2 = 6 bar × 0.21 = 1.3 bar absolute where: pi = partial pressure of gas component i = P_{\mathrm{i}} in the terms used in this article P = total pressure = P in the terms used in this article Fi = volume fraction of gas component i = mole fraction, x_{\mathrm{i}}, in the terms used in this article pN2 = partial pressure of nitrogen = P_\mathrm{N_2} in the terms used in this article pO2 = partial pressure of oxygen = P_\mathrm{O_2} in the terms used in this article The minimum safe lower limit for the partial pressures of oxygen in a breathing gas mixture for diving is absolute. V_{\rm X} = V_{\rm tot} \times \frac{p_{\rm X}}{p_{\rm tot}} = V_{\rm tot} \times \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of an individual gas component X in the mixture * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas X * ntot is the total amount of substance in the gas mixture ==Vapor pressure== thumb|right|A log-lin vapor pressure chart for various liquids Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases (i.e., liquid or solid). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Typically, the maximum total partial pressure of narcotic gases used when planning for technical diving may be around 4.5 bar absolute, based on an equivalent narcotic depth of . In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . In chemistry and in various industries, the reference pressure referred to in standard temperature and pressure was commonly but standards have since diverged; in 1982, the International Union of Pure and Applied Chemistry recommended that for the purposes of specifying the physical properties of substances, standard pressure should be precisely .IUPAC.org, Gold Book, Standard Pressure ==Pressure units and equivalencies == A pressure of 1 atm can also be stated as: :≡ pascals (Pa) :≡ bar :≈ kgf/cm2 :≈ technical atmosphere :≈ m H2O, 4 °CThis is the customarily accepted value for cm–H2O, 4 °C. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. It is approximately equal to Earth's average atmospheric pressure at sea level. ==History== The standard atmosphere was originally defined as the pressure exerted by 760 mm of mercury at and standard gravity (gn = ). The partial pressure of oxygen also determines the maximum operating depth of a gas mixture. That is, at low pressures is the same as the pressure, so it has the same units as pressure. When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. ",7.00,3.38,0.4,0.6296296296,+5.41,B
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the total change in entropy, when a sample of nitrogen gas of mass $14 \mathrm{~g}$ at $298 \mathrm{~K}$ and $1.00 \mathrm{bar}$ doubles its volume in an isothermal irreversible expansion against $p_{\mathrm{ex}}=0$.","However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., ) is given by d\mu = V_\mathrm{m}dP = RT \, \frac{dP}{P} = R T\, d \ln P,where ln p is the natural logarithm of p. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Numerical example: Nitrogen gas (N2) at 0 °C and a pressure of atmospheres (atm) has a fugacity of atm. ISO 15971 is an ISO standard for calorific value measurement of natural gas and its substitutes. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. * The heat capacity of the gas from the boiling point to room temperature. ",0,-6.9,3.07,34,+2.9,E
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in the molar Gibbs energy of hydrogen gas when its pressure is increased isothermally from $1.0 \mathrm{~atm}$ to 100.0 atm at $298 \mathrm{~K}$.,"The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. * The heat capacity of the gas from the boiling point to room temperature. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. ", 4.56,0.24995,-3.8,+11,0.7854,D
-"A perfect gas undergoes isothermal compression, which reduces its volume by $2.20 \mathrm{dm}^3$. The final pressure and volume of the gas are $5.04 \mathrm{bar}$ and $4.65 \mathrm{dm}^3$, respectively. Calculate the original pressure of the gas in bar.","The partial pressures obey Dalton's law: P_i = y_i P, where is the total pressure and is the mole fraction of the component (so the partial pressures add up to the total pressure). The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture (Dalton's Law). The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules. That is, the mole fraction x_{\mathrm{i}} of an individual gas component in an ideal gas mixture can be expressed in terms of the component's partial pressure or the moles of the component: x_{\mathrm{i}} = \frac{p_{\mathrm{i}}}{p} = \frac{n_{\mathrm{i}}}{n} and the partial pressure of an individual gas component in an ideal gas can be obtained using this expression: p_{\mathrm{i}} = x_{\mathrm{i}} \cdot p where: x_{\mathrm{i}} = mole fraction of any individual gas component in a gas mixture p_{\mathrm{i}} = partial pressure of any individual gas component in a gas mixture n_{\mathrm{i}} = moles of any individual gas component in a gas mixture n = total moles of the gas mixture p = total pressure of the gas mixture The mole fraction of a gas component in a gas mixture is equal to the volumetric fraction of that component in a gas mixture.Frostberg State University's ""General Chemistry Online"" The ratio of partial pressures relies on the following isotherm relation: \frac{V_{\rm X}}{V_{\rm tot}} = \frac{p_{\rm X}}{p_{\rm tot}} = \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of any individual gas component (X) * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas (X) * ntot is the total amount of substance in gas mixture ==Partial volume (Amagat's law of additive volume)== The partial volume of a particular gas in a mixture is the volume of one component of the gas mixture. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. V_{\rm X} = V_{\rm tot} \times \frac{p_{\rm X}}{p_{\rm tot}} = V_{\rm tot} \times \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of an individual gas component X in the mixture * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas X * ntot is the total amount of substance in the gas mixture ==Vapor pressure== thumb|right|A log-lin vapor pressure chart for various liquids Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases (i.e., liquid or solid). Typically, the maximum total partial pressure of narcotic gases used when planning for technical diving may be around 4.5 bar absolute, based on an equivalent narcotic depth of . Using diving terms, partial pressure is calculated as: :partial pressure = (total absolute pressure) × (volume fraction of gas component) For the component gas ""i"": :pi = P × Fi For example, at underwater, the total absolute pressure is (i.e., 1 bar of atmospheric pressure + 5 bar of water pressure) and the partial pressures of the main components of air, oxygen 21% by volume and nitrogen approximately 79% by volume are: :pN2 = 6 bar × 0.79 = 4.7 bar absolute :pO2 = 6 bar × 0.21 = 1.3 bar absolute where: pi = partial pressure of gas component i = P_{\mathrm{i}} in the terms used in this article P = total pressure = P in the terms used in this article Fi = volume fraction of gas component i = mole fraction, x_{\mathrm{i}}, in the terms used in this article pN2 = partial pressure of nitrogen = P_\mathrm{N_2} in the terms used in this article pO2 = partial pressure of oxygen = P_\mathrm{O_2} in the terms used in this article The minimum safe lower limit for the partial pressures of oxygen in a breathing gas mixture for diving is absolute. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. VPT may refer to: * Valor por Tamaulipas * Vermont Public Television * Volume, temperature and pressure, the three parameters in the combined gas law The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude. == Pressure equations == thumb|300px|Pressure as a function of the height above the sea level There are two equations for computing pressure as a function of height. The partial pressure of oxygen also determines the maximum operating depth of a gas mixture. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. See real gas or perfect gas or gas for further understanding.) ==See also== *Hypsometric equation *NRLMSISE-00 *Vertical pressure variation == References == Category:Atmosphere Category:Vertical position Category:Pressure That is, at low pressures is the same as the pressure, so it has the same units as pressure. In aeronautical engineering, overall pressure ratio, or overall compression ratio, is the ratio of the stagnation pressure as measured at the front and rear of the compressor of a gas turbine engine. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. ", 10.7598,5654.86677646,1.43,3.42,2,D
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate $\Delta S$ (for the system) when the state of $3.00 \mathrm{~mol}$ of perfect gas atoms, for which $C_{p, \mathrm{~m}}=\frac{5}{2} R$, is changed from $25^{\circ} \mathrm{C}$ and 1.00 atm to $125^{\circ} \mathrm{C}$ and $5.00 \mathrm{~atm}$. How do you rationalize the $\operatorname{sign}$ of $\Delta S$?","The δ34S (pronounced delta 34 S) value is a standardized method for reporting measurements of the ratio of two stable isotopes of sulfur, 34S:32S, in a sample against the equivalent ratio in a known reference standard. With VCDT as the reference standard, natural δ34S value variations have been recorded between -72‰ and +147‰. The δ34S value refers to a measure of the ratio of the two most common stable sulfur isotopes, 34S:32S, as measured in a sample against that same ratio as measured in a known reference standard. In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced ""delta fifteen n"") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave. :\begin{align} \frac{50\log_2{\left(\frac32\right)} + 28\log_2{\left(\frac54\right)} + 23\log_2{\left(\frac65\right)}}{50^2+28^2+23^2} = 0.011\,621\,2701 \\\ 0.011\,621\,2701 \times 1200 = 13.945\,524\,1627 \end{align} () The Bohlen–Pierce delta scale is based on the tritave and the 7:5:3 ""wide"" triad () and the 9:7:5 ""narrow"" triad () (rather than the conventional 4:5:6 triad). For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. That value can be calculated in per mil (‰, parts per thousand) as: :\delta \ce{^{34}S} = \left( \frac{\left( \frac\ce{^{34}S}\ce{^{32}S} \right)_\mathrm{sample}}{\left( \frac\ce{^{34}S}\ce{^{32}S} \right)_\mathrm{standard}} - 1 \right) \times 1000 ‰ Less commonly, if the appropriate isotope abundances are measured, similar formulae can be used to quantify ratio variations between 33S and 32S, and 36S and 32S, reported as δ33S and δ36S, respectively. ===Reference standard=== thumb|right|Troilite from the Canyon Diablo meteorite was the first reference standard for δ34S.|alt=A worn brown-red-gold space rock covered in smoothed pock-marks sits mounted in a museum. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Since the atmosphere is 99.6337% 14N and 0.3663% 15N, a is 0.003663 in the former case and 0.003663/0.996337 = 0.003676 in the latter. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. Both have the form 1000\frac{s-a}a ‰ (‰ = permil or parts per thousand) where s and a are the relative abundances of 15N in respectively the sample and the atmosphere. The lowercase delta character is used by convention, to be consistent with use in other areas of stable isotope chemistry. ISO 15971 is an ISO standard for calorific value measurement of natural gas and its substitutes. However s varies similarly; for example if in the sample 15N is 0.385% and 14N is 99.615%, s is 0.003850 in the former case and 0.00385/0.99615 = 0.003865 in the latter. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. The value of 1000\frac{s-a}a is then 51.05‰ in the former case and 51.38‰ in the latter, an insignificant difference in practice given the typical range of -20 to 80 for . ==Applications== One use of 15N is as a tracer to determine the path taken by fertilizers applied to anything from pots to landscapes. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. Sometimes, other distinctions are made, such as between thermally perfect gas and calorically perfect gas, or between imperfect, semi-perfect, and perfect gases, and as well as the characteristics of ideal gases. The δ (delta) scale is a non-octave repeating musical scale. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. ",1.154700538,4.3,-22.1,169,2.567,C
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $\Delta H$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. This error is corrected in the above equation. ==Measurement== The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. ",+5.41,0.68,0.59,3.8,58.2,A
-A sample of $255 \mathrm{mg}$ of neon occupies $3.00 \mathrm{dm}^3$ at $122 \mathrm{K}$. Use the perfect gas law to calculate the pressure of the gas.,"For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. Billion cubic meters of natural gas (non SI abbreviation: bcm) or cubic kilometer of natural gas is a measure of natural gas production and trade. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. That means that 1 billion cubic metres of natural gas by the International Energy Agency standard is equivalent to 1.017 billion cubic metres of natural gas by the Russian standard. ==Energy based definitions== Some other organizations use energy equivalent-based standards. Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. The pressure inside is equal to atmospheric pressure. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. 8) or of pressure P (p. 9). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: : \gamma = \frac{5}{3} = 1.6666\ldots, As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. According to the Russian standard, the gas volume is measured at . Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Dalton's law is not strictly followed by real gases, with the deviation increasing with pressure. For example, terrestrial air is primarily made up of diatomic gases (around 78% nitrogen, N2, and 21% oxygen, O2), and at standard conditions it can be considered to be an ideal gas. Cedigaz uses a standard which is equivalent to per billion cubic metres. ==References== Category:Natural gas Category:Units of volume Category:Units of energy Category:Non-SI metric units Category:International Energy Agency Category:BP However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. ",1.2,0.042,152.67,9.90,0.9984,B
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A chemical reaction takes place in a container of cross-sectional area $100 \mathrm{~cm}^2$. As a result of the reaction, a piston is pushed out through $10 \mathrm{~cm}$ against an external pressure of $1.0 \mathrm{~atm}$. Calculate the work done by the system.","In this case the work is given by (where is the pressure at the surface, is the increase of the volume of the system). When a system, for example, moles of a gas of volume at pressure and temperature , is created or brought to its present state from absolute zero, energy must be supplied equal to its internal energy plus , where is the work done in pushing against the ambient (atmospheric) pressure. The quantity of thermodynamic work is defined as work done by the system on its surroundings. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. In chemistry, experiments are often conducted at constant atmospheric pressure, and the pressure–volume work represents a small, well-defined energy exchange with the atmosphere, so that is the appropriate expression for the heat of reaction. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). For systems at constant pressure, with no external work done other than the work, the change in enthalpy is the heat received by the system. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Such work done by compression is thermodynamic work as here defined. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. Real gases at common temperatures and pressures often closely approximate this behavior, which simplifies practical thermodynamic design and analysis. ==Definition== The enthalpy of a thermodynamic system is defined as the sum of its internal energy and the product of its pressure and volume: : , where is the internal energy, is pressure, and is the volume of the system; is sometimes referred to as the pressure energy . Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. The work is due to change of system volume by expansion or contraction of the system. To get an actual and precise physical measurement of a quantity of thermodynamic work, it is necessary to take account of the irreversibility by restoring the system to its initial condition by running a cycle, for example a Carnot cycle, that includes the target work as a step. As a result, the work done by the system also depends on the initial and final states. An Introduction to Thermal Physics, 2000, Addison Wesley Longman, San Francisco, CA, , p. 18 According to the first law of thermodynamics for a closed system, any net change in the internal energy U must be fully accounted for, in terms of heat Q entering the system and work W done by the system: :\Delta U = Q - W.\; Freedman, Roger A., and Young, Hugh D. (2008). 12th Edition. Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. Consequently, thermodynamic work is defined in terms of quantities that describe the states of materials, which appear as the usual thermodynamic state variables, such as volume, pressure, temperature, chemical composition, and electric polarization. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of mass into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of mass out as if it were driving a piston of fluid. ",-11.2,-100,6.2,3.54,+93.4,B
-Use the van der Waals parameters for chlorine to calculate approximate values of the Boyle temperature of chlorine.,"J/(mol K) Enthalpy of combustion –473.2 kJ/mol ΔcH ~~o~~ Heat capacity, cp 114.25 J/(mol K) Gas properties Std enthalpy change of formation, ΔfH ~~o~~ gas –103.18 kJ/mol Standard molar entropy, S ~~o~~ gas 295.6 J/(mol K) at 25 °C Heat capacity, cp 65.33 J/(mol K) at 25 °C van der Waals' constantsLange's Handbook of Chemistry 10th ed, pp. 1522–1524 a = 1537 L2 kPa/mol2 b = 0.1022 liter per mole ==Vapor pressure of liquid== P in mm Hg 1 10 40 100 400 760 1520 3800 7600 15200 30400 45600 T in °C –58.0 –29.7 –7.1 10.4 42.7 61.3 83.9 120.0 152.3 191.8 237.5 — Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. thumb|812px|left|log10 of Chloroform vapor pressure. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The molecular formula C3H4Cl2O2 (molar mass: 142.97 g/mol, exact mass: 141.9588 u) may refer to: * Chloroethyl chloroformate * Dalapon The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. Uses formula: \scriptstyle \log_e P_{mmHg} =\scriptstyle \log_e (\frac {760} {101.325}) - 10.07089 \log_e(T+273.15) - \frac {6351.140} {T+273.15} + 81.14393 + 9.127608 \times 10^{-6} (T+273.15)^2 obtained from CHERIC ==Distillation data== {| border=""1"" cellspacing=""0"" cellpadding=""6"" style=""margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;"" Vapor-liquid Equilibrium for Chloroform/Ethanol P = 101.325 kPa BP Temp. °C % by mole chloroform liquid vapor 78.15 0.0 0.0 78.07 0.52 1.59 77.83 1.02 3.01 76.81 2.21 7.45 74.90 5.80 17.10 74.39 6.72 19.40 73.55 8.38 23.31 72.85 10.57 28.05 72.24 11.80 30.52 71.58 13.18 33.13 69.72 17.65 41.00 68.95 19.62 43.66 68.58 20.71 45.43 67.35 23.86 49.77 65.89 28.54 55.09 64.87 32.35 58.48 63.88 36.07 61.27 63.23 39.34 64.50 62.61 41.38 65.49 62.17 44.41 67.57 61.48 49.97 71.11 61.00 53.92 72.91 60.49 54.76 73.57 60.35 59.65 74.68 60.30 61.60 75.53 60.20 63.04 76.12 60.09 64.48 76.69 59.97 66.90 77.74 59.54 72.01 79.33 59.32 79.07 82.62 59.26 82.99 83.59 59.28 84.97 84.69 59.31 85.96 85.24 59.46 89.92 87.93 59.72 91.10 88.73 59.70 92.44 89.79 59.84 93.90 91.02 59.91 95.26 92.56 60.18 96.13 93.58 60.88 98.89 97.93 61.13 100.00 100.00 == Spectral data == UV-Vis λmax ? nm Extinction coefficient, ε ? Expanding the van der Waals equation in \frac{1}{V_m} one finds that T_b = \frac{a}{Rb}.Verma, K.S. Cengage Physical Chemistry Part 1. Also at Boyle temperature the dip in a PV diagram tends to a straight line over a period of pressure. 100px 100px Two representations of chloroform. This page provides supplementary chemical data on chloroform. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. Several billion kilograms of chlorinated methanes are produced annually, mainly by chlorination of methane: :CH4 \+ x Cl2 → CH4−xClx \+ x HCl The most important is dichloromethane, which is mainly used as a solvent. Organochlorine chemistry is concerned with the properties of organochlorine compounds, or organochlorides, organic compounds containing at least one covalently bonded atom of chlorine. To convert from L^2bar/mol^2 to m^6 Pa/mol^2, divide by 10. a (L2bar/mol2) b (L/mol) Acetic acid 17.7098 0.1065 Acetic anhydride 20.158 0.1263 Acetone 16.02 0.1124 Acetonitrile 17.81 0.1168 Acetylene 4.516 0.0522 Ammonia 4.225 0.0371 Aniline 29.14 0.1486 Argon 1.355 0.03201 Benzene 18.24 0.1193 Bromobenzene 28.94 0.1539 Butane 14.66 0.1226 1-Butanol 20.94 0.1326 2-Butanone 19.97 0.1326 Carbon dioxide 3.640 0.04267 Carbon disulfide 11.77 0.07685 Carbon monoxide 1.505 0.0398500 Carbon tetrachloride 19.7483 0.1281 Chlorine 6.579 0.05622 Chlorobenzene 25.77 0.1453 Chloroethane 11.05 0.08651 Chloromethane 7.570 0.06483 Cyanogen 7.769 0.06901 Cyclohexane 23.11 0.1424 Cyclopropane 8.34 0.0747 Decane 52.74 0.3043 1-Decanol 59.51 0.3086 Diethyl ether 17.61 0.1344 Diethyl sulfide 19.00 0.1214 Dimethyl ether 8.180 0.07246 Dimethyl sulfide 13.04 0.09213 Dodecane 69.38 0.3758 1-Dodecanol 75.70 0.3750 Ethane 5.562 0.0638 Ethanethiol 11.39 0.08098 Ethanol 12.18 0.08407 Ethyl acetate 20.72 0.1412 Ethylamine 10.74 0.08409 Ethylene 4.612 0.0582 Fluorine 1.171 0.0290 Fluorobenzene 20.19 0.1286 Fluoromethane 4.692 0.05264 Freon 10.78 0.0998 Furan 12.74 0.0926 Germanium tetrachloride 22.90 0.1485 Helium 0.0346 0.0238 Heptane 31.06 0.2049 1-Heptanol 38.17 0.2150 Hexane 24.71 0.1735 1-Hexanol 31.79 0.1856 Hydrazine 8.46 0.0462 Hydrogen 0.2476 0.02661 Hydrogen bromide 4.510 0.04431 Hydrogen chloride 3.716 0.04081 Hydrogen cyanide 11.29 0.0881 Hydrogen fluoride 9.565 0.0739 Hydrogen iodide 6.309 0.0530 Hydrogen selenide 5.338 0.04637 Hydrogen sulfide 4.490 0.04287 Isobutane 13.32 0.1164 Iodobenzene 33.52 0.1656 Krypton 2.349 0.03978 Mercury 8.200 0.01696 Methane 2.283 0.04278 Methanol 9.649 0.06702 Methylamine 7.106 0.0588 Neon 0.2135 0.01709 Neopentane 17.17 0.1411 Nitric oxide 1.358 0.02789 Nitrogen 1.370 0.0387 Nitrogen dioxide 5.354 0.04424 Nitrogen trifluoride 3.58 0.0545 Nitrous oxide 3.832 0.04415 Octane 37.88 0.2374 1-Octanol 44.71 0.2442 Oxygen 1.382 0.03186 Ozone 3.570 0.0487 Pentane 19.26 0.146 1-Pentanol 25.88 0.1568 Phenol 22.93 0.1177 Phosphine 4.692 0.05156 Propane 8.779 0.08445 1-Propanol 16.26 0.1079 2-Propanol 15.82 0.1109 Propene 8.442 0.0824 Pyridine 19.77 0.1137 Pyrrole 18.82 0.1049 Radon 6.601 0.06239 Silane 4.377 0.05786 Silicon tetrafluoride 4.251 0.05571 Sulfur dioxide 6.803 0.05636 Sulfur hexafluoride 7.857 0.0879 Tetrachloromethane 20.01 0.1281 Tetrachlorosilane 20.96 0.1470 Tetrafluoroethylene 6.954 0.0809 Tetrafluoromethane 4.040 0.0633 Tetrafluorosilane 5.259 0.0724 Tetrahydrofuran 16.39 0.1082 Tin tetrachloride 27.27 0.1642 Thiophene 17.21 0.1058 Toluene 24.38 0.1463 1-1-1-Trichloroethane 20.15 0.1317 Trichloromethane 15.34 0.1019 Trifluoromethane 5.378 0.0640 Trimethylamine 13.37 0.1101 Water 5.536 0.03049 Xenon 4.250 0.05105 ==Units== 1 J·m3/mol2 = 1 m6·Pa/mol2 = 10 L2·bar/mol2 1 L2atm/mol2 = 0.101325 J·m3/mol2 = 0.101325 Pa·m6/mol2 1 dm3/mol = 1 L/mol = 1 m3/kmol (where kmol is kilomoles = 1000 moles) ==References== Category:Gas laws Constants (Data Page) J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −705.63 kJ/mol Standard molar entropy S ~~o~~ solid 109.29 J/(mol K) Heat capacity cp 91.12 J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid -674.80 kJ/mol Standard molar entropy S ~~o~~ liquid 172.91 J/(mol K) Heat capacity cp 125.5 J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas -584.59 kJ/mol Standard molar entropy S ~~o~~ gas 314.44 J/(mol K) Heat capacity cp 82.46 J/(mol K) == Spectral data == UV-Vis Spectrum Lambda-max nm Log Ε IR Spectrum NIST Major absorption bands cm−1 NMR Proton NMR ? K (? °C), ? The annual production in 1985 was around 13 million tons, almost all of which was converted into polyvinylchloride (PVC). ===Chloromethanes=== Most low molecular weight chlorinated hydrocarbons such as chloroform, dichloromethane, dichloroethene, and trichloroethane are useful solvents. Chlorine adds to the multiple bonds on alkenes and alkynes as well, giving di- or tetra-chloro compounds. ===Reaction with hydrogen chloride=== Alkenes react with hydrogen chloride (HCl) to give alkyl chlorides. Since higher order virial coefficients are generally much smaller than the second coefficient, the gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature (or when c = \frac{1}{V_m} or P are minimized). Supplementary data for aluminium chloride. == External MSDS == * Baker * Fisher * EM Science * Akzo Nobel (hexahydrate) * Science Stuff (hexahydrate) * External SDS == Thermodynamic properties == Phase behavior Triple point ? The Wurtz reaction reductively couples two alkyl halides to couple with sodium. ==Applications== ===Vinyl chloride=== The largest application of organochlorine chemistry is the production of vinyl chloride. Fourme, M. Renaud, C. R. Acad. Sci, Ser. C(Chim),1966, p. 69 C-Cl 1.75 Å Bond angle Cl-C-Cl 110.3° Dipole moment 1.08 D (gas) 1.04 DCRC Handbook of Chemistry and Physics. 89th ed./David R. Lide ed.-in- chief. Alternatively, the Appel reaction can be used: :250px ==Reactions== Alkyl chlorides are versatile building blocks in organic chemistry. – Close to that of Teflon Surface tension 28.5 dyn/cm at 10 °C 27.1 dyn/cm at 20 °C 26.67 dyn/cm at 25 °C 23.44 dyn/cm at 50 °C 21.7 dyn/cm at 60 °C 20.20 dyn/cm at 75 °C ViscosityLange's Handbook of Chemistry, 10th ed. pp. 1669–1674 0.786 mPa·s at –10 °C 0.699 mPa·s at 0 °C 0.563 mPa·s at 20 °C 0.542 mPa·s at 25 °C 0.464 mPa·s at 40 °C 0.389 mPa·s at 60 °C == Thermodynamic properties == Phase behavior Triple point 209.61 K (–63.54 °C), ? ",1410,3.8,313.0,7.82,144,A
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $\Delta T$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We assume the expansion occurs without exchange of heat (adiabatic expansion). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. Since the piston cannot move, the volume is constant. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. * Being a gaseous fuel, CNG mixes easily and evenly in air. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). The pressure inside is equal to atmospheric pressure. ",-0.347,257,3.42,0.925,4096,A
-Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. 3.17 Estimate the standard reaction Gibbs energy of $\mathrm{N}_2(\mathrm{~g})+3 \mathrm{H}_2(\mathrm{~g}) \rightarrow$ $2 \mathrm{NH}_3(\mathrm{~g})$ at $1000 \mathrm{~K}$ from their values at $298 \mathrm{~K}$.,"Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. * The heat capacity of the gas from the boiling point to room temperature. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. ",817.90,4.979,-191.2,+107,27,D
-"Could $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert a pressure of $20 \mathrm{atm}$ at $25^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert?","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. After repeating the experiment several times and using different amounts of mercury he found that under controlled conditions, the pressure of a gas is inversely proportional to the volume occupied by it. Pressure vessels for gas storage may also be classified by volume. The ship was to carry compressed natural gas in vertical pressure bottles; however, this design failed because of the high cost of the pressure vessels. Boyle's law has been stated as: > The absolute pressure exerted by a given mass of an ideal gas is inversely > proportional to the volume it occupies if the temperature and amount of gas > remain unchanged within a closed system.Levine, Ira. Most gases behave like ideal gases at moderate pressures and temperatures. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory.Levine, Ira. N. (1978), p. 11 notes that deviations occur with high pressures and temperatures. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. thumb|upright=1.3|An animation showing the relationship between pressure and volume when mass and temperature are held constant Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. That is, doubling the density of a quantity of air doubles its pressure. What the pressure should be according to the Hypothesis, that supposes the pressures and expansions to be in reciprocal relation."" High-pressure gas cylinders are also called bottles. Boyle's Law states that when the temperature of a given mass of confined gas is constant, the product of its pressure and volume is also constant. Conversely, reducing the volume of the gas increases the pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. VPT may refer to: * Valor por Tamaulipas * Vermont Public Television * Volume, temperature and pressure, the three parameters in the combined gas law ",0.4207,48.6,24.0,5.1,5.0,C
-"Could $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert a pressure of $20 \mathrm{atm}$ at $25^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert?","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. After repeating the experiment several times and using different amounts of mercury he found that under controlled conditions, the pressure of a gas is inversely proportional to the volume occupied by it. Pressure vessels for gas storage may also be classified by volume. * ISO 11439: Compressed natural gas (CNG) cylinders. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America, the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSA B51 in Canada, Australian Standards in Australia and other international standards like Lloyd's, Germanischer Lloyd, Det Norske Veritas, Société Générale de Surveillance (SGS S.A.), Lloyd’s Register Energy Nederland (formerly known as Stoomwezen) etc. Note that where the pressure-volume product is part of a safety standard, any incompressible liquid in the vessel can be excluded as it does not contribute to the potential energy stored in the vessel, so only the volume of the compressible part such as gas is used. ===List of standards=== * EN 13445: The current European Standard, harmonized with the Pressure Equipment Directive (Originally ""97/23/EC"", since 2014 ""2014/68/EU""). Further the volume of the gas is (4πr3)/3. Boyle's law has been stated as: > The absolute pressure exerted by a given mass of an ideal gas is inversely > proportional to the volume it occupies if the temperature and amount of gas > remain unchanged within a closed system.Levine, Ira. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory.Levine, Ira. N. (1978), p. 11 notes that deviations occur with high pressures and temperatures. Most gases behave like ideal gases at moderate pressures and temperatures. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The very small vessels used to make liquid butane fueled cigarette lighters are subjected to about 2 bar pressure, depending on ambient temperature. * EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. thumb|upright=1.3|An animation showing the relationship between pressure and volume when mass and temperature are held constant Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Other examples of pressure vessels are diving cylinders, recompression chambers, distillation towers, pressure reactors, autoclaves, and many other vessels in mining operations, oil refineries and petrochemical plants, nuclear reactor vessels, submarine and space ship habitats, atmospheric diving suits, pneumatic reservoirs, hydraulic reservoirs under pressure, rail vehicle airbrake reservoirs, road vehicle airbrake reservoirs, and storage vessels for high pressure permanent gases and liquified gases such as ammonia, chlorine, and LPG (propane, butane). That is, doubling the density of a quantity of air doubles its pressure. What the pressure should be according to the Hypothesis, that supposes the pressures and expansions to be in reciprocal relation."" Boyle's Law states that when the temperature of a given mass of confined gas is constant, the product of its pressure and volume is also constant. High-pressure gas cylinders are also called bottles. ",4.85,24,4.56,226,4.738,B
-The barometric formula relates the pressure of a gas of molar mass $M$ at an altitude $h$ to its pressure $p_0$ at sea level. Derive this relation by showing that the change in pressure $\mathrm{d} p$ for an infinitesimal change in altitude $\mathrm{d} h$ where the density is $\rho$ is $\mathrm{d} p=-\rho g \mathrm{~d} h$. Remember that $\rho$ depends on the pressure. Evaluate the external atmospheric pressure at a typical cruising altitude of an aircraft (11 km) when the pressure at ground level is 1.0 atm.,"The other two values (pressure P and density ρ) are computed by simultaneously solving the equations resulting from: * the vertical pressure gradient resulting from hydrostatic balance, which relates the rate of change of pressure with geopotential altitude: :: \frac{dP}{dh} = - \rho g , and * the ideal gas law in molar form, which relates pressure , density, and temperature: :: \ P = \rho R_{\rm specific}T at each geopotential altitude, where g is the standard acceleration of gravity, and Rspecific is the specific gas constant for dry air (287.0528J⋅kg−1⋅K−1). The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude. == Pressure equations == thumb|300px|Pressure as a function of the height above the sea level There are two equations for computing pressure as a function of height. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. Equation 1: \rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^* L_b}-1\right)} which is equivalent to the ratio of the relative pressure and temperature changes \rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} Equation 2: \rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where *{\rho} = mass density (kg/m3) *T_b = standard temperature (K) *L = standard temperature lapse rate (see table below) (K/m) in ISA *h = height above sea level (geopotential meters) *R^* = universal gas constant 8.3144598 N·m/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): *{\rho} = mass density (slug/ft3) *{T_b} = standard temperature (K) *{L} = standard temperature lapse rate (K/ft) *{h} = height above sea level (geopotential feet) *{R^*} = universal gas constant: 8.9494596×104 ft2/(s·K) *{g_0} = gravitational acceleration: 32.17405 ft/s2 *{M} = molar mass of Earth's air: 0.0289644 kg/mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. Pressure altitude is primarily used in aircraft-performance calculations and in high-altitude flight (i.e., above the transition altitude). == Inverse equation == Solving the equation for the pressure gives : p = 1013.25\left(1-\frac{h}{44307.694 m}\right)^{5.25530} hPa where are meter and hPa hecto-Pascal. Subscript b Height Above Sea Level (h) Mass Density (\rho) Standard Temperature (T') (K) Temperature Lapse Rate (L) (m) (ft) (kg/m3) (slug/ft3) (K/m) (K/ft) 0 0 0 1.2250 288.15 0.0065 0.0019812 1 11 000 36,089.24 0.36391 216.65 0.0 0.0 2 20 000 65,616.79 0.08803 216.65 -0.001 -0.0003048 3 32 000 104,986.87 0.01322 228.65 -0.0028 -0.00085344 4 47 000 154,199.48 0.00143 270.65 0.0 0.0 5 51 000 167,322.83 0.00086 270.65 0.0028 0.00085344 6 71 000 232,939.63 0.000064 214.65 0.002 0.0006096 ==Derivation== The barometric formula can be derived using the ideal gas law: P = \frac{\rho}{M} {R^*} T Assuming that all pressure is hydrostatic: dP = - \rho g\,dz and dividing the dP by the P expression we get: \frac{dP}{P} = - \frac{M g\,dz}{R^*T} Integrating this expression from the surface to the altitude z we get: P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}} Assuming linear temperature change T = T_0 - L z and constant molar mass and gravitational acceleration, we get the first barometric formula: P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}} Instead, assuming constant temperature, integrating gives the second barometric formula: P = P_0 e^{-M g z/R^*T} In this formulation, R* is the gas constant, and the term R*T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere). The National Oceanic and Atmospheric Administration (NOAA) published the following formula for directly converting atmospheric pressure in millibars ( \mathrm{mb} ) to pressure altitude in feet ( \mathrm{ft} ): : h = 145366.45 \left[ 1 - \left( \frac{\text{Station pressure in millibars}}{1013.25} \right)^{0.190284} \right]. Atmospheric pressure decreases following the Barometric formula with altitude while the O2 fraction remains constant to about , so pO2 decreases with altitude as well. The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of L_b: P = P_b \left[\frac{T_b - \left(h - h_b\right) L_b}{T_b}\right]^{\tfrac{g_0 M}{R^* L_b}} The second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude: P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/m) in ISA *h = height at which pressure is calculated (m) *h_b = height of reference level b (meters; e.g., hb = 11 000 m) *R^* = universal gas constant: 8.3144598 J/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol Or converted to imperial units:Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. The relationship between static pressure and pressure altitude is defined in terms of properties of the ISA. ==See also== * QNH * Flight level * Cabin altitude * Density altitude * Standard conditions for temperature and pressure * Barometric formula ==References== Category:Altitudes in aviation (The total air mass below a certain altitude is calculated by integrating over the density function.) Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. Pressure altitude is the altitude in the International Standard Atmosphere (ISA) with the same atmospheric pressure as that of the part of the atmosphere in question. Subscript b Height above sea level Static pressure Standard temperature (K) Temperature lapse rate Exponent g0 M / R L (m) (ft) (Pa) (inHg) (K/m) (K/ft) 0 0 0 101 325.00 29.92126 288.15 0.0065 0.0019812 5.2558 1 11 000 36,089 22 632.10 6.683245 216.65 0.0 0.0 -- 2 20 000 65,617 5474.89 1.616734 216.65 -0.001 -0.0003048 -34.1626 3 32 000 104,987 868.02 0.2563258 228.65 -0.0028 -0.00085344 -12.2009 4 47 000 154,199 110.91 0.0327506 270.65 0.0 0.0 -- 5 51 000 167,323 66.94 0.01976704 270.65 0.0028 0.00085344 12.2009 6 71 000 232,940 3.96 0.00116833 214.65 0.002 0.0006096 17.0813 ==Density equations== The expressions for calculating density are nearly identical to calculating pressure. The concentration of oxygen (O2) in sea-level air is 20.9%, so the partial pressure of O2 (pO2) is . A reference atmospheric model describes how the ideal gas properties (namely: pressure, temperature, density, and molecular weight) of an atmosphere change, primarily as a function of altitude, and sometimes also as a function of latitude, day of year, etc. For example, the U.S. Standard Atmosphere derives the values for air temperature, pressure, and mass density, as a function of altitude above sea level. For the isothermal-barotropic model, density and pressure turn out to be exponential functions of altitude. The increase in altitude necessary for P or ρ to drop to 1/e of its initial value is called the scale height: :H = \frac{R T}{M g_0} where R is the ideal gas constant, T is temperature, M is average molecular weight, and g0 is the gravitational acceleration at the planet's surface. Image:Liquid ocean atmosphere model.png ===Isothermal-barotropic approximation and scale height=== This atmospheric model assumes both molecular weight and temperature are constant over a wide range of altitude. For example, if the airfield elevation is 500 ~ \mathrm{ft} and the altimeter setting is 29.32 ~ \mathrm{inHg} , then : \begin{align} \text{PA} & = 500 + 1000 \times (29.92 - 29.32) \\\ & = 500 + 1000 \times 0.6 \\\ & = 500 + 600 \\\ & = 1100. \end{align} Alternatively, : \text{Pressure altitude (PA)} = \text{Elevation} + 30 \times (1013 - \text{QNH}). Other static atmospheric models may have other outputs, or depend on inputs besides altitude. ==Basic assumptions== The gas which comprises an atmosphere is usually assumed to be an ideal gas, which is to say: : \rho = \frac{M P}{R T} Where ρ is mass density, M is average molecular weight, P is pressure, T is temperature, and R is the ideal gas constant. ",344,0,399.0,0.72,0.66666666666,D
-A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What pressure indicates a temperature of $100.00^{\circ} \mathrm{C}$?,"By the 1940s, the triple point of water had been experimentally measured to be about 0.6% of standard atmospheric pressure and very close to 0.01 °C per the historical definition of Celsius then in use. The 13th CGPM also held in Resolution 4 that ""The kelvin, unit of thermodynamic temperature, is equal to the fraction of the thermodynamic temperature of the triple point of water."" The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. In 1954, with absolute zero having been experimentally determined to be about −273.15 °C per the definition of °C then in use, Resolution 3 of the 10th General Conference on Weights and Measures (CGPM) introduced a new internationally standardised Kelvin scale which defined the triple point as exactly 273.15 + 0.01 = 273.16 degrees Kelvin. The kelvin, symbol K, is a unit of measurement for temperature. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. In water, the critical point occurs at around Tc = , pc = and ρc = 356 kg/m3.The International Association for the Properties of Water and Steam ""Guideline on the Use of Fundamental Physical Constants and Basic Constants of Water"", 2001, p. 5 The existence of the liquid–gas critical point reveals a slight ambiguity in labelling the single phase regions. In the early decades of the 20th century, the Kelvin scale was often called the ""absolute Celsius"" scale, indicating Celsius degrees counted from absolute zero rather than the freezing point of water, and using the same symbol for regular Celsius degrees, °C.For example, Encyclopaedia Britannica editions from the 1920s and 1950s, one example being the article ""Planets"". === Triple point standard === In 1873, William Thomson's older brother James coined the term triple point to describe the combination of temperature and pressure at which the solid, liquid, and gas phases of a substance were capable of coexisting in thermodynamic equilibrium. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). In 1948, the Celsius scale was recalibrated by assigning the triple point temperature of water the value of 0.01 °C exactly and allowing the melting point at standard atmospheric pressure to have an empirically determined value (and the actual melting point at ambient pressure to have a fluctuating value) close to 0 °C. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. * The boiling point of water is 100 degrees. * The boiling point of water is 100 degrees. At constant pressure the maximum number of independent variables is three – the temperature and two concentration values. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. For practical purposes, the redefinition was unnoticed; water still freezes at 273.15 K (0 °C), and the triple point of water continues to be a commonly used laboratory reference temperature. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Historically, the Kelvin scale was developed from the Celsius scale, such that 273.15 K was 0 °C (the approximate melting point of ice) and a change of one kelvin was exactly equal to a change of one degree Celsius. The boiling point of water is the temperature at which the saturated vapour pressure equals the ambient pressure. English translation (extract). that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. ",1410,1.2,9.14,420,1.4,C
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. An average human produces about $10 \mathrm{MJ}$ of heat each day through metabolic activity. Human bodies are actually open systems, and the main mechanism of heat loss is through the evaporation of water. What mass of water should be evaporated each day to maintain constant temperature?","The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. The equation for evaporation given by Penman is: :E_{\mathrm{mass}}=\frac{m R_n + \rho_a c_p \left(\delta e \right) g_a }{\lambda_v \left(m + \gamma \right) } where: :m = Slope of the saturation vapor pressure curve (Pa K−1) :Rn = Net irradiance (W m−2) :ρa = density of air (kg m−3) :cp = heat capacity of air (J kg−1 K−1) :δe = vapor pressure deficit (Pa) :ga = momentum surface aerodynamic conductance (m s−1) :λv = latent heat of vaporization (J kg−1) :γ = psychrometric constant (Pa K−1) which (if the SI units in parentheses are used) will give the evaporation Emass in units of kg/(m2·s), kilograms of water evaporated every second for each square meter of area. Penman's equation requires daily mean temperature, wind speed, air pressure, and solar radiation to predict E. Simpler Hydrometeorological equations continue to be used where obtaining such data is impractical, to give comparable results within specific contexts, e.g. humid vs arid climates. ==Details== Numerous variations of the Penman equation are used to estimate evaporation from water, and land. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The normal human body temperature is often stated as . However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Human body temperature varies. Normal human body temperature varies slightly from person to person and by the time of day. Temperature, wind speed, relative humidity impact the values of m, g, cp, ρ, and δe. ==Shuttleworth (1993)== In 1993, W.Jim Shuttleworth modified and adapted the Penman equation to use SI, which made calculating evaporation simpler.Shuttleworth, J., Putting the vap' into evaporation http://www.hydrol-earth-syst-sci.net/11/210/2007/hess-11-210-2007.pdf The resultant equation is: :E_{\mathrm{mass}}=\frac{m R_n + \gamma * 6.43\left(1+0.536 * U_2 \right)\delta e}{\lambda_v \left(m + \gamma \right) } where: :Emass = Evaporation rate (mm day−1) :m = Slope of the saturation vapor pressure curve (kPa K−1) :Rn = Net irradiance (MJ m−2 day−1) :γ = psychrometric constant = \frac{0.0016286 * P_{kPa}} {\lambda_v} (kPa K−1) :U2 = wind speed (m s−1) :δe = vapor pressure deficit (kPa) :λv = latent heat of vaporization (MJ kg−1) Note: this formula implicitly includes the division of the numerator by the density of water (1000 kg m−3) to obtain evaporation in units of mm d−1 ==Some useful relationships== :δe = (es \- ea) = (1 – relative humidity) es :es = saturated vapor pressure of air, as is found inside plant stoma. :ea = vapor pressure of free flowing air. :es, mmHg = exp(21.07-5336/Ta), approximation by Merva, 1975Merva, G.E. 1975. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature In humans, the average internal temperature is widely accepted to be 37 °C (98.6 °F), a ""normal"" temperature established in the 1800s. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Therefore, the heat capacity ratio in this example is 1.4. This equation assumes a daily time step so that net heat exchange with the ground is insignificant, and a unit area surrounded by similar open water or vegetation so that net heat & vapor exchange with the surrounding area cancels out. While some people think of these averages as representing normal or ideal measurements, a wide range of temperatures has been found in healthy people. An individual's body temperature typically changes by about between its highest and lowest points each day. ",537,7,4.09,0.241,2.24,C
-A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What change of pressure indicates a change of $1.00 \mathrm{~K}$ at this temperature?,"Historically, the Kelvin scale was developed from the Celsius scale, such that 273.15 K was 0 °C (the approximate melting point of ice) and a change of one kelvin was exactly equal to a change of one degree Celsius. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. thumb|400px|Diagram showing pressure difference induced by a temperature difference. The kelvin, symbol K, is a unit of measurement for temperature. By the 1940s, the triple point of water had been experimentally measured to be about 0.6% of standard atmospheric pressure and very close to 0.01 °C per the historical definition of Celsius then in use. For practical purposes, the redefinition was unnoticed; water still freezes at 273.15 K (0 °C), and the triple point of water continues to be a commonly used laboratory reference temperature. In 1948, the Celsius scale was recalibrated by assigning the triple point temperature of water the value of 0.01 °C exactly and allowing the melting point at standard atmospheric pressure to have an empirically determined value (and the actual melting point at ambient pressure to have a fluctuating value) close to 0 °C. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). This value of ""−273"" was the negative reciprocal of 0.00366—the accepted coefficient of thermal expansion of an ideal gas per degree Celsius relative to the ice point, giving a remarkable consistency to the currently accepted value. In the early decades of the 20th century, the Kelvin scale was often called the ""absolute Celsius"" scale, indicating Celsius degrees counted from absolute zero rather than the freezing point of water, and using the same symbol for regular Celsius degrees, °C.For example, Encyclopaedia Britannica editions from the 1920s and 1950s, one example being the article ""Planets"". === Triple point standard === In 1873, William Thomson's older brother James coined the term triple point to describe the combination of temperature and pressure at which the solid, liquid, and gas phases of a substance were capable of coexisting in thermodynamic equilibrium. In 1954, with absolute zero having been experimentally determined to be about −273.15 °C per the definition of °C then in use, Resolution 3 of the 10th General Conference on Weights and Measures (CGPM) introduced a new internationally standardised Kelvin scale which defined the triple point as exactly 273.15 + 0.01 = 273.16 degrees Kelvin. The 13th CGPM also held in Resolution 4 that ""The kelvin, unit of thermodynamic temperature, is equal to the fraction of the thermodynamic temperature of the triple point of water."" This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. The Kelvin scale is an absolute scale, which is defined such that 0 K is absolute zero and a change of thermodynamic temperature by 1 kelvin corresponds to a change of thermal energy by . In 1967/1968, Resolution 3 of the 13th CGPM renamed the unit increment of thermodynamic temperature ""kelvin"", symbol K, replacing ""degree Kelvin"", symbol . English translation (extract). that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. In 2005, noting that the triple point could be influenced by the isotopic ratio of the hydrogen and oxygen making up a water sample and that this was ""now one of the major sources of the observed variability between different realizations of the water triple point"", the International Committee for Weights and Measures (CIPM), a committee of the CGPM, affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin would refer to water having the isotopic composition specified for Vienna Standard Mean Ocean Water. === 2019 redefinition === In 2005, the CIPM began a programme to redefine the kelvin (along with the other SI units) using a more experimentally rigorous method. At constant pressure the maximum number of independent variables is three – the temperature and two concentration values. ",-1270,0.0245,41.4,0.375,1.81,B
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $229 \mathrm{~J}$ of energy is supplied as heat to $3.0 \mathrm{~mol} \mathrm{Ar}(\mathrm{g})$ at constant pressure, the temperature of the sample increases by $2.55 \mathrm{~K}$. Calculate the molar heat capacities at constant pressure of the gas.","Then the molar heat capacity (at constant volume) would be :cV,m = fR where R is the ideal gas constant. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.Lange's Handbook of Chemistry, 10th ed. p. 1524 All methods for the measurement of specific heat apply to molar heat capacity as well. ==Units== The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . The following is a table of some constant- pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* H2 28.9 5.0 29.6 5.1 41.2 7.9 Not saturated.""Hydrogen"" NIST Chemistry WebBook, SRD 69, online. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where ""amol"" denotes an amount of the solid that contains the Avogadro number of atoms. Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. The last value corresponds almost exactly to the predicted value for f = 7. right|thumb|upright=1.25|Constant-volume specific heat capacity of diatomic gases (real gases) between about 200 K and 2000 K. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. At high enough temperature, its molar heat capacity then should be cP,m = 7.5 R = 62.63 J⋅K−1⋅mol−1. Therefore, the word ""molar"", not ""specific"", should always be used for this quantity. ==Definition== The molar heat capacity of a substance, which may be denoted by cm, is the heat capacity C of a sample of the substance, divided by the amount (moles) n of the substance in the sample: :cm{} \;=\; \frac{C}{n} \;=\; \frac{1}{n} \lim_{\Delta T \rightarrow 0}\frac{Q}{\Delta T} where Q is the amount of heat needed to raise the temperature of the sample by ΔT. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In other words, that theory predicts that the molar heat capacity at constant volume cV,m of all monatomic gases will be the same; specifically, :cV,m = R where R is the ideal gas constant, about 8.31446 J⋅K−1⋅mol−1 (which is the product of the Boltzmann constant kB and the Avogadro constant). According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). The same theory predicts that the molar heat capacity of a monatomic gas at constant pressure will be :cP,m = cV,m \+ R = R This prediction matches the experimental values, which, for helium through xenon, are 20.78, 20.79, 20.85, 20.95, and 21.01 J⋅K−1⋅mol−1, respectively;Shuen-Chen Hwang, Robert D. Lein, Daniel A. Morgan (2005). The molar heat capacity of the gas will then be determined only by the ""active"" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ",-20,+11,17.7,22,269,D
-Express the van der Waals parameters $b=0.0226 \mathrm{dm}^3 \mathrm{~mol}^{-1}$ in SI base units.,"* Analytical calculation of Van der Waals surfaces and volumes. Petitjean, On the Analytical Calculation of Van der Waals Surfaces and Volumes: Some Numerical Aspects, Journal of Computational Chemistry, Volume 15, Number 5, 1994, pp. 507–523. == External links == * VSAs for various molecules by Anton Antonov, The Wolfram Demonstrations Project, 2007. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. In Van der Waals' original derivation, given below, b' is four times the proper volume of the particle. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. Useful tabulated values of Van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will be seen to present different values for the Van der Waals radius of the same atom. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. ",−2,5,0.41887902047,0.000226,0.5,D
-A diving bell has an air space of $3.0 \mathrm{m}^3$ when on the deck of a boat. What is the volume of the air space when the bell has been lowered to a depth of $50 \mathrm{m}$? Take the mean density of sea water to be $1.025 \mathrm{g} \mathrm{cm}^{-3}$ and assume that the temperature is the same as on the surface.,"* Volume reduction of the air in an open bell due to increasing hydrostatic pressure as the bell is lowered is compensated. The bell is lowered into the water and to the working depth at a rate recommended by the decompression schedule, and which allows the divers to equalize comfortably. The bell is lowered through the water to working depth, so must be negatively buoyant. Each 10 metres (33 feet) of depth puts another atmosphere (1 bar, 14.7 psi, 101 kPa) of pressure on the hull, so at 300 metres (1,000 feet), the hull is withstanding thirty atmospheres (30 bar, 441 psi, 3,000 kPa) of water pressure. ===Test depth=== This is the maximum depth at which a submarine is permitted to operate under normal peacetime circumstances, and is tested during sea trials. Maximum Operating Depth (MOD) in metres sea water (msw) for pO2 1.2 to 1.6 MOD (msw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 390 190 123 90 70 66 57 47 40 34 30 22 17 13 10 8 6 1.5 365 178 115 84 65 61 53 44 37 32 28 20 15 11 9 7 5 1.4 340 165 107 78 60 57 48 40 34 29 25 18 13 10 8 6 4 1.3 315 153 98 71 55 52 44 36 31 26 23 16 12 9 6 4 3 1.2 290 140 90 65 50 47 40 33 28 23 20 14 10 7 5 3 2 These depths are rounded to the nearest metre. ==See also== * == Notes == == References == == Sources == * * * * Category:Dive planning Category:Underwater diving safety Category:Breathing gases Wet bells with an air space will have the air space topped up as the bell descends and the air is compressed by increasing hydrostatic pressure. The physics of the diving bell applies also to an underwater habitat equipped with a moon pool, which is like a diving bell enlarged to the size of a room or two, and with the water–air interface at the bottom confined to a section rather than forming the entire bottom of the structure. ===Wet bell=== thumb|upright|Open diving bell on a stern mounted launch and recovery system A wet bell is a platform for lowering and lifting divers to and from the underwater workplace, which has an air filled space, open at the bottom, where the divers can stand or sit with their heads out of the water. It transports this air to its diving bell to replenish the air supply in the bell. The pressure produced by depth in water, is converted to pressure in feet sea water (fsw) or metres sea water (msw) by multiplying with the appropriate conversion factor, 33 fsw per atm, or 10 msw per bar. Air is trapped inside the bell by pressure of the water at the interface. Of this total pressure which can be tolerated by the diver, 1 atmosphere is due to surface pressure of the Earth's air, and the rest is due to the depth in water. The bell is ballasted so as to remain upright in the water and to be negatively buoyant, so that it will sink even when full of air. If the divers are breathing from the bell airspace at the time, it may need to be vented with additional air to maintain a low carbon dioxide level. Adding pressurized gas ensures that the gas space within the bell remains at constant volume as the bell descends in the water. A diving bell is a rigid chamber used to transport divers from the surface to depth and back in open water, usually for the purpose of performing underwater work. The diving bell would be connected via the mating flange of an airlock to the deck decompression chamber or saturation system for transfer under pressure of the occupants. == Air-lock diving bells == thumb|Barge with air-lock diving bell for working on moorings Service vessel with diving bell which can be lowered to 10 m and accessed via airlock and a 2 m diameter access tube|thumb|right The air lock diving-bell plant was a purpose-built barge for the laying, examination and repair of moorings for battleships at Gibraltar harbour. So the 1 atmosphere or bar contributed by the air is subtracted to give the pressure due to the depth of water. In underwater diving activities such as saturation diving, technical diving and nitrox diving, the maximum operating depth (MOD) of a breathing gas is the depth below which the partial pressure of oxygen (pO2) of the gas mix exceeds an acceptable limit. For example, if a gas contains 36% oxygen and the maximum pO2 is 1.4 bar, the MOD (msw) is 10 msw/bar x [(1.4 bar / 0.36) − 1] = 28.9 msw. == Tables == Maximum Operating Depth (MOD) in feet sea water (fsw) for pO2 1.2 to 1.6 MOD (fsw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 1287 627 407 297 231 218 187 156 132 114 99 73 55 42 33 26 20 1.5 1205 586 380 276 215 203 173 144 122 105 91 66 50 38 29 22 17 1.4 1122 545 352 256 198 187 160 132 111 95 83 59 44 33 25 18 13 1.3 1040 503 325 235 182 171 146 120 101 86 74 53 39 28 21 15 10 1.2 957 462 297 215 165 156 132 108 91 77 66 46 33 24 17 11 7 These depths are rounded to the nearest foot. The internal pressure in the bell is usually kept at atmospheric pressure to minimise run time by eliminating the need for decompression, so the seal between the bell skirt and the submarine deck is critical to the safety of the operation. The 6 m × 4 m × 2.5 m bell is accessible through a 2 m diameter tube and an airlock. In 1689, Denis Papin suggested that the pressure and fresh air inside a diving bell could be maintained by a force pump or bellows. ",92, 35.91,0.05882352941,0.5,8.87,D
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The change in the Gibbs energy of a certain constant-pressure process was found to fit the expression $\Delta G / \text{J}=-85.40+36.5(T / \text{K})$. Calculate the value of $\Delta S$ for the process.,"The Gibbs energy of any system is and an infinitesimal change in G, at constant temperature and pressure, yields :dG=dU+pdV-TdS. The Gibbs free energy is expressed as : G(p,T) = U + pV - TS = H - TS where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy. Moreover, we also have \begin{align} K_\text{eq} &= e^{-\frac{\Delta_\text{r} G^\circ}{RT}}, \\\ \Delta_\text{r} G^\circ &= -RT\left(\ln K_\text{eq}\right) = -2.303\,RT\left(\log_{10} K_\text{eq}\right), \end{align} which relates the equilibrium constant with Gibbs free energy. The equation is:Physical chemistry, P. W. Atkins, Oxford University Press, 1978, where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all at constant pressure p. Integrating with respect to T (again p is constant) it becomes: : \frac{\Delta G^\ominus(T_2)}{T_2} - \frac{\Delta G^\ominus(T_1)}{T_1} = \Delta H^\ominus \left(\frac{1}{T_2} - \frac{1}{T_1}\right) This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T2 with knowledge of just the standard Gibbs free energy change of formation and the standard enthalpy change of formation for the individual components. This implies that at equilibrium Q_\text{r} = K_\text{eq} and \Delta_\text{r} G = 0. ==Standard Gibbs energy change of formation== Table of selected substancesCRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide. The equation reads:Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, :\left( \frac{\partial ( \Delta G^\ominus/T ) } {\partial T} \right)_p = - \frac {\Delta H^\ominus} {T^2} with ΔG as the change in Gibbs energy due to reaction, and ΔH as the enthalpy of reaction (often, but not necessarily, assumed to be independent of temperature). The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models The Gibbs free energy change , measured in joules in SI) is the maximum amount of non-volume expansion work that can be extracted from a closed system (one that can exchange heat and work with its surroundings, but not matter) at fixed temperature and pressure. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Substance (state) ΔfG° ΔfG° Substance (state) (kJ/mol) (kcal/mol) NO(g) 87.6 20.9 NO2(g) 51.3 12.3 N2O(g) 103.7 24.78 H2O(g) −228.6 −54.64 H2O(l) −237.1 −56.67 CO2(g) −394.4 −94.26 CO(g) −137.2 −32.79 CH4(g) −50.5 −12.1 C2H6(g) −32.0 −7.65 C3H8(g) −23.4 −5.59 C6H6(g) 129.7 29.76 C6H6(l) 124.5 31.00 The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). The equation states that the change in the G/T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor H/T2. ==Chemical reactions and work== The typical applications of this equation are to chemical reactions. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Also, using the reaction isotherm equation,Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, that is :\frac{\Delta G^\ominus}{T} = -R \ln K which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, Since the change in a system's Gibbs energy is equal to the maximum amount of non-expansion work that the system can do in a process, the Gibbs-Helmholtz equation may be used to estimate how much non-expansion work can be done by a chemical process as a function of temperature. As a necessary condition for the reaction to occur at constant temperature and pressure, ΔG must be smaller than the non- pressure-volume (non-pV, e.g. electrical) work, which is often equal to zero (then ΔG must be negative). One can think of ∆G as the amount of ""free"" or ""useful"" energy available to do non-pV work at constant temperature and pressure. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. ",1000,+107,0.70710678,-36.5,+116.0,D
+$\mathrm{Si}_2 \mathrm{H}_6(\mathrm{g}) \rightarrow \mathrm{SiH}_2(\mathrm{g})+\mathrm{SiH}_4(\mathrm{g})$","Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? \frac{1}{\sqrt{1 + \left(\frac{x-\xi}{\lambda}\right)^2}} e^{-\frac{1}{2}\left(\gamma+\delta \sinh^{-1} \left(\frac{x-\xi}{\lambda}\right)\right)^2} | cdf = \Phi \left(\gamma + \delta \sinh^{-1} \left( \frac{x-\xi}{\lambda} \right) \right) | mean = \xi - \lambda \exp \frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right) | median = \xi + \lambda \sinh \left( - \frac{\gamma}{\delta} \right) | mode = | variance = \frac{\lambda^2}{2} (\exp(\delta^{-2})-1) \left( \exp(\delta^{-2}) \cosh \left(\frac{2\gamma}{\delta} \right) +1 \right) | skewness = -\frac{\lambda^3\sqrt{e^{\delta^{-2}}}(e^{\delta^{-2}}-1)^{2}((e^{\delta^{-2}})(e^{\delta^{-2}}+2)\sinh(\frac{3\gamma}{\delta})+3\sinh(\frac{2\gamma}{\delta}))}{4(\operatorname{Variance}X)^{1.5}} | kurtosis = \frac{\lambda^4(e^{\delta^{-2}}-1)^2(K_{1}+K_2+K_3)}{8(\operatorname{Variance}X)^2} K_{1}=\left( e^{\delta^{-2}} \right)^{2}\left( \left( e^{\delta^{-2}} \right)^{4}+2\left( e^{\delta^{-2}} \right)^{3}+3\left( e^{\delta^{-2}} \right)^{2}-3 \right)\cosh\left( \frac{4\gamma}{\delta} \right) K_2=4\left( e^{\delta^{-2}} \right)^2 \left( \left( e^{\delta^{-2}} \right)+2 \right)\cosh\left( \frac{3\gamma}{\delta} \right) K_3=3\left( 2\left( e^{\delta^{-2}} \right)+1 \right) | entropy = | pgf = | mgf = | char = }} The Johnson's SU-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). K (? °C), ? K (? °C), ? K (? °C), ? K (? °C), ? ",+37,0.9522,"""3.2""",228,22.2036033112,D
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $\Delta S$.","In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We assume the expansion occurs without exchange of heat (adiabatic expansion). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. Since the piston cannot move, the volume is constant. The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. * Being a gaseous fuel, CNG mixes easily and evenly in air. ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). Especially, for the CNG Type 1 and Type 2 cylinders, many countries are able to make reliable and cost effective cylinders for conversion need. ==Energy density== CNG's energy density is the same as liquefied natural gas at 53.6 MJ/kg. ",30,+0.60,"""0.2553""",0.14,0.1792,B
+"The mass density of water vapour at $327.6 \mathrm{~atm}$ and $776.4 \mathrm{~K}$ is $133.2 \mathrm{~kg} \mathrm{~m}^{-3}$. Given that for water $T_{\mathrm{c}}=647.4 \mathrm{~K}, p_{\mathrm{c}}=218.3 \mathrm{~atm}, a=5.464 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}$, $b=0.03049 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and $M=18.02 \mathrm{~g} \mathrm{~mol}^{-1}$, calculate the compression factor from the virial expansion of the van der Waals equation.","Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated. ===Reduced form=== Although the material constant a and b in the usual form of the Van der Waals equation differs for every single fluid considered, the equation can be recast into an invariant form applicable to all fluids. Even with its acknowledged shortcomings, the pervasive use of the Van der Waals equation in standard university physical chemistry textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. This apparent discrepancy is resolved in the context of vapour–liquid equilibrium: at a particular temperature, there exist two points on the Van der Waals isotherm that have the same chemical potential, and thus a system in thermodynamic equilibrium will appear to traverse a straight line on the p–V diagram as the ratio of vapour to liquid changes. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. The corrected equation becomes : p = \frac{RT}{V_\mathrm{m}-b}. ",-167,0.7158,"""0.18""",9.8,0.123,B
+Express the van der Waals parameters $a=0.751 \mathrm{~atm} \mathrm{dm}^6 \mathrm{~mol}^{-2}$ in SI base units.,"* Analytical calculation of Van der Waals surfaces and volumes. Petitjean, On the Analytical Calculation of Van der Waals Surfaces and Volumes: Some Numerical Aspects, Journal of Computational Chemistry, Volume 15, Number 5, 1994, pp. 507–523. == External links == * VSAs for various molecules by Anton Antonov, The Wolfram Demonstrations Project, 2007. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. Useful tabulated values of Van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will be seen to present different values for the Van der Waals radius of the same atom. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Van der Waals volumes of a single atom or molecules are arrived at by dividing the macroscopically determined volumes by the Avogadro constant. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). ",0.0761,0.3359,"""2.3""",0.0625,9.30,A
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Estimate the change in the Gibbs energy of $1.0 \mathrm{dm}^3$ of benzene when the pressure acting on it is increased from $1.0 \mathrm{~atm}$ to $100 \mathrm{~atm}$.,"Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . thumb|2D model of a benzene molecule. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant is the constant of proportionality that relates the energy scale in physics to the temperature scale and the scale used for amount of substance. Uses formula: \scriptstyle \log_e P_{mmHg} =\scriptstyle \log_e (\frac {760} {101.325}) - 8.433613\log_e(T+273.15) - \frac {6281.040} {T+273.15} + 71.10718 + 6.198413 \times 10^{-06} (T+273.15)^2 obtained from CHERIC Note: yellow area is the region where the formula disagrees with tabulated data above. ==Distillation data== {| border=""1"" cellspacing=""0"" cellpadding=""6"" style=""margin: 0 0 0 0.5em; background: white; border- collapse: collapse; border-color: #C0C090;"" Vapor-liquid Equilibrium for Benzene/Ethanol P = 760 mm Hg BP Temp. °C % by mole ethanol liquid vapor 70.8 8.6 26.5 69.8 11.2 28.2 69.6 12.0 30.8 69.1 15.8 33.5 68.5 20.0 36.8 67.7 30.8 41.0 67.7 44.2 44.6 68.1 60.4 50.5 69.6 77.0 59.0 70.4 81.5 62.8 70.9 84.1 66.5 72.7 89.8 74.4 73.8 92.4 78.2 == Spectral data == UV-Vis Ionization potential 9.24 eV (74525.6 cm−1) S1 4.75 eV (38311.3 cm−1) S2 6.05 eV (48796.5 cm−1) λmax 255 nm Extinction coefficient, ε ? This page provides supplementary chemical data on benzene. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. (i) Indicates values calculated from ideal gas thermodynamic functions. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. On the elasticity of gases. 1875 (in Russian) Mendeleev also calculated it with high precision, within 0.3% of its modern value. It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per amount of substance, i.e. the pressure–volume product, rather than energy per temperature increment per particle. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of units of energy, temperature and amount of substance. ",0.1591549431,+10,"""1.33""",35.64, 0.0024,B
+"The mass density of water vapour at $327.6 \mathrm{~atm}$ and $776.4 \mathrm{~K}$ is $133.2 \mathrm{~kg} \mathrm{~m}^{-3}$. Given that for water $T_{\mathrm{c}}=647.4 \mathrm{~K}, p_{\mathrm{c}}=218.3 \mathrm{~atm}, a=5.464 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}$, $b=0.03049 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, and $M=18.02 \mathrm{~g} \mathrm{~mol}^{-1}$, calculate the compression factor from the data.","The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Regarding the measurement of upper-air humidity, this publication also reads (in Section 12.5.1): The saturation with respect to water cannot be measured much below –50 °C, so manufacturers should use one of the following expressions for calculating saturation vapour pressure relative to water at the lowest temperatures – Wexler (1976, 1977), reported by Flatau et al. (1992)., Hyland and Wexler (1983) or Sonntag (1994) – and not the Goff- Gratch equation recommended in earlier WMO publications. ==Experimental correlation== The original Goff–Gratch (1945) experimental correlation reads as follows: : \log\ e^*\ = -7.90298(T_\mathrm{st}/T-1)\ +\ 5.02808\ \log(T_\mathrm{st}/T) -\ 1.3816\times10^{-7}(10^{11.344(1-T/T_\mathrm{st})}-1) +\ 8.1328\times10^{-3}(10^{-3.49149(T_\mathrm{st}/T-1)}-1)\ +\ \log\ e^*_\mathrm{st} where: :log refers to the logarithm in base 10 :e* is the saturation water vapor pressure (hPa) :T is the absolute air temperature in kelvins :Tst is the steam-point (i.e. boiling point at 1 atm.) temperature (373.15 K) :e*st is e* at the steam-point pressure (1 atm = 1013.25 hPa) Similarly, the correlation for the saturation water vapor pressure over ice is: : \log\ e^*_i\ = -9.09718(T_0/T-1)\ -\ 3.56654\ \log(T_0/T) +\ 0.876793(1-T/T_0) +\ \log\ e^*_{i0} where: :log stands for the logarithm in base 10 :e*i is the saturation water vapor pressure over ice (hPa) :T is the air temperature (K) :T0 is the ice-point (triple point) temperature (273.16 K) :e*i0 is e* at the ice-point pressure (6.1173 hPa) ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Tetens equation *Lee–Kesler method ==References== * Goff, J. A., and Gratch, S. (1946) Low- pressure properties of water from −160 to 212 °F, in Transactions of the American Society of Heating and Ventilating Engineers, pp 95–122, presented at the 52nd annual meeting of the American Society of Heating and Ventilating Engineers, New York, 1946. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. * * ;Notes ==External links== * * Free Windows Program, Moisture Units Conversion Calculator w/Goff-Gratch equation — PhyMetrix Category:Atmospheric thermodynamics Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. For this use, air has a molecular weight of 28.97 atomic mass units, and all other gas and vapour molecular weights are divided by this number to derive their vapour density.HazMat Math: Calculating Vapor Density . Air is given a vapour density of one. Further g is acceleration due to gravity and ρ is the fluid density. The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. The Goff–Gratch equation is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the saturation water vapor pressure at a given temperature. thumb|The Mollier enthalpy–entropy diagram for water and steam. Further ρ is the (constant) fluid density and g is the gravitational acceleration. * Goff, J. A. (1957) Saturation pressure of water on the new Kelvin temperature scale, Transactions of the American Society of Heating and Ventilating Engineers, pp 347–354, presented at the semi-annual meeting of the American Society of Heating and Ventilating Engineers, Murray Bay, Que. Canada. * After correction, repeat this process until all data points have the same slope. ==See also== *Statistics ==Notes== ==Further reading== * Dubreuil P. (1974) Initiation à l'analyse hydrologique Masson& Cie et ORSTOM, Paris. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. Another similar equation based on more recent data is the Arden Buck equation. ==Historical note== This equation is named after the authors of the original scientific article who described how to calculate the saturation water vapor pressure above a flat free water surface as a function of temperature (Goff and Gratch, 1946). Double mass analysis is a simple graphical method to evaluate the consistency of hydrological data. The Mollier diagram coordinates are enthalpy h and humidity ratio x. The temperature-vapour pressure relation inversely describes the relation between the boiling point of water and the pressure. An understanding of vapour pressure is also relevant in explaining high altitude breathing and cavitation. ==Approximation formulas == There are many published approximations for calculating saturated vapour pressure over water and over ice. ",+93.4,2.3,"""1.0""",0.6957,311875200,D
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in Gibbs energy of $35 \mathrm{~g}$ of ethanol (mass density $0.789 \mathrm{~g} \mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from $1 \mathrm{~atm}$ to $3000 \mathrm{~atm}$.,"Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The partial pressures obey Dalton's law: P_i = y_i P, where is the total pressure and is the mole fraction of the component (so the partial pressures add up to the total pressure). It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to . *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., ) is given by d\mu = V_\mathrm{m}dP = RT \, \frac{dP}{P} = R T\, d \ln P,where ln p is the natural logarithm of p. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. The contribution of nonideality to the molar Gibbs energy of a real gas is equal to . The Reid vapor pressure (RVP) can differ substantially from the true vapor pressure (TVP) of a liquid mixture, since (1) RVP is the vapor pressure measured at 37.8 °C (100 °F) and the TVP is a function of the temperature; (2) RVP is defined as being measured at a vapor-to-liquid ratio of 4:1, whereas the TVP of mixtures can depend on the actual vapor-to-liquid ratio; (3) RVP will include the pressure associated with the presence of dissolved water and air in the sample (which is excluded by some but not all definitions of TVP); and (4) the RVP method is applied to a sample which has had the opportunity to volatilize somewhat prior to measurement: i.e., the sample container is required to be only 70-80% full of liquid ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Section 8.3(so that whatever volatilizes into the container headspace is lost prior to analysis); the sample then again volatilizes into the headspace of the D323 test chamber before it is heated to 37.8 degrees Celsius.Conversion between the two measures can be found here, from p. 7.1-54 onwards. ==See also== * Crude oil assay * Gasoline volatility * Vapor pressure ==External links== * ASTM D323 - 06 Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method) * Reid Vapor Pressure Requirements for Ethanol Congressional Research Service * USA's Environmental Protection Agency (EPA) publication AP-42, Compilation of Air Pollutant Emissions. Reid vapor pressure (RVP) is a common measure of the volatility of gasoline and other petroleum products.ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Section 1.1 It is defined as the absolute vapor pressure exerted by the vapor of the liquid and any dissolved gases/moisture at 37.8 °C (100 °F) as determined by the test method ASTM-D-323, which was first developed in 1930 ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), footnote 1 and has been revised several times (the latest version is ASTM D323-15a).ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method) The test method measures the vapor pressure of gasoline, volatile crude oil, jet fuels, naphtha, and other volatile petroleum products but is not applicable for liquefied petroleum gases.ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Sections 1.1 and 1.6 ASTM D323-15a requires that the sample be chilled to 0 to 1 degrees Celsius and then poured into the apparatus;ASTM D323-15a, Standard Test Method for Vapor Pressure of Petroleum Products (Reid Method), Sections 11.1 and 11.1.2 for any material that solidifies at this temperature, this step cannot be performed. ",5,0.68,"""152.67""",-167,12,E
+"The densities of air at $-85^{\circ} \mathrm{C}, 0^{\circ} \mathrm{C}$, and $100^{\circ} \mathrm{C}$ are $1.877 \mathrm{~g} \mathrm{dm}^{-3}, 1.294 \mathrm{~g}$ $\mathrm{dm}^{-3}$, and $0.946 \mathrm{~g} \mathrm{dm}^{-3}$, respectively. From these data, and assuming that air obeys Charles's law, determine a value for the absolute zero of temperature in degrees Celsius.","Rounding up 1.98°C to 2°C, this approximation simplifies to become :\begin{align} \text{DA} & \approx \text{PA} + 118.8 ~ \frac{\text{ft}}{^\circ \text{C}} \left[ T_\text{OA} + \frac{\text{PA}}{500 ~ \text{ft}} {^\circ \text{C}} - 15 ~ {^\circ \text{C}} \right] \\\\[3pt] & = 1.2376 \, \text{PA} + 118.8 ~ \frac{\text{ft}}{{}^\circ \text{C}} \, T_\text{OA} - 1782 ~ \text{ft}. \end{align} ==See also== *Outside air temperature *Barometric formula *Density of air *Hot and high *List of longest runways == Notes == ==References== * * * Advisory Circular AC 61-23C, Pilot's Handbook of Aeronautical Knowledge, U.S. Federal Aviation Administration, Revised 1997 * http://www.tpub.com/content/aerographer/14269/css/14269_74.htm * ==External links== *Density Altitude Calculator *Density Altitude influence on aircraft performance *NewByte Atmospheric Calculator Category:Altitudes in aviation Category:Atmospheric thermodynamics Unaware of the inaccuracies of mercury thermometers at the time, which were divided into equal portions between the fixed points, Dalton, after concluding in Essay II that in the case of vapours, “any elastic fluid expands nearly in a uniform manner into 1370 or 1380 parts by 180 degrees (Fahrenheit) of heat”, was unable to confirm it for gases. ==Relation to absolute zero== Charles's law appears to imply that the volume of a gas will descend to zero at a certain temperature (−266.66 °C according to Gay-Lussac's figures) or −273.15 °C. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In this formula, : \text{DA} , density altitude in meters (m); : P , (static) atmospheric pressure; : P_\text{SL} , standard sea-level atmospheric pressure, International Standard Atmosphere (ISA): 1013.25 hectopascals (hPa), or U.S. Standard Atmosphere: 29.92 inches of mercury (inHg); : T , outside air temperature in kelvins (K); : T_\text{SL} = 288.15K, ISA sea-level air temperature; : \Gamma = 0.0065K/m, ISA temperature lapse rate (below 11km); : R ≈ 8.3144598J/mol·K, ideal gas constant; : g ≈ 9.80665m/s, gravitational acceleration; : M ≈ 0.028964kg/mol, molar mass of dry air. ===The National Weather Service (NWS) formula=== The National Weather Service uses the following dry-air approximation to the formula for the density altitude above in its standard: : \text{DA}_\text{NWS} = 145442.16 ~ \text{ft} \left( 1 - \left[ 17.326 ~ \frac{^\circ \text{F}}{\text{inHg}} \ \frac{P}{459.67 ~ {{}^\circ \text{F}} + T} \right]^{0.235} \right). Subscript b Height above sea level Static pressure Standard temperature (K) Temperature lapse rate Exponent g0 M / R L (m) (ft) (Pa) (inHg) (K/m) (K/ft) 0 0 0 101 325.00 29.92126 288.15 0.0065 0.0019812 5.2558 1 11 000 36,089 22 632.10 6.683245 216.65 0.0 0.0 -- 2 20 000 65,617 5474.89 1.616734 216.65 -0.001 -0.0003048 -34.1626 3 32 000 104,987 868.02 0.2563258 228.65 -0.0028 -0.00085344 -12.2009 4 47 000 154,199 110.91 0.0327506 270.65 0.0 0.0 -- 5 51 000 167,323 66.94 0.01976704 270.65 0.0028 0.00085344 12.2009 6 71 000 232,940 3.96 0.00116833 214.65 0.002 0.0006096 17.0813 ==Density equations== The expressions for calculating density are nearly identical to calculating pressure. (Before going further, I should inform [you] that although I had recognized many times that the gases oxygen, nitrogen, hydrogen, and carbonic acid [i.e., carbon dioxide], and atmospheric air also expand from 0° to 80°, citizen Charles had noticed 15 years ago the same property in these gases; but having never published his results, it is by the merest chance that I knew of them.) although he credited the discovery to unpublished work from the 1780s by Jacques Charles. This equation does not contain the temperature and so is not what became known as Charles's Law. Equation 1: \rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^* L_b}-1\right)} which is equivalent to the ratio of the relative pressure and temperature changes \rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} Equation 2: \rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where *{\rho} = mass density (kg/m3) *T_b = standard temperature (K) *L = standard temperature lapse rate (see table below) (K/m) in ISA *h = height above sea level (geopotential meters) *R^* = universal gas constant 8.3144598 N·m/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): *{\rho} = mass density (slug/ft3) *{T_b} = standard temperature (K) *{L} = standard temperature lapse rate (K/ft) *{h} = height above sea level (geopotential feet) *{R^*} = universal gas constant: 8.9494596×104 ft2/(s·K) *{g_0} = gravitational acceleration: 32.17405 ft/s2 *{M} = molar mass of Earth's air: 0.0289644 kg/mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In this formula, : \text{PA} , pressure altitude in feet (ft) \approx \text{station elevation in feet} + 27 ~ \frac{\text{ft}}{\text{mb}} (1013 ~ \text{mb} - \text{QNH}) ; : \text{QNH} , atmospheric pressure in millibars (mb) adjusted to mean sea level; : T_\text{OA}, outside air temperature in degrees Celsius (°C); : T_\text{ISA} \approx 15 ~ {{}^\circ \text{C}} - 1.98 ~ {{}^\circ \text{C}} \, \frac{\text{PA}}{1000 ~ \text{ft}} , assuming that the outside air temperature falls at the rate of 1.98°C per 1,000ft of altitude until the tropopause (at ) is reached. The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of L_b: P = P_b \left[\frac{T_b - \left(h - h_b\right) L_b}{T_b}\right]^{\tfrac{g_0 M}{R^* L_b}} The second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude: P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/m) in ISA *h = height at which pressure is calculated (m) *h_b = height of reference level b (meters; e.g., hb = 11 000 m) *R^* = universal gas constant: 8.3144598 J/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol Or converted to imperial units:Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. In the absence of a firm record, the gas law relating volume to temperature cannot be attributed to Charles. A modern statement of Charles' law is: > When the pressure on a sample of a dry gas is held constant, the Kelvin > temperature and the volume will be in direct proportion.. Thomson did not assume that this was equal to the ""zero-volume point"" of Charles's law, merely that Charles's law provided the minimum temperature which could be attained. The first mention of a temperature at which the volume of a gas might descend to zero was by William Thomson (later known as Lord Kelvin) in 1848:. > This is what we might anticipate when we reflect that infinite cold must > correspond to a finite number of degrees of the air-thermometer below zero; > since if we push the strict principle of graduation, stated above, > sufficiently far, we should arrive at a point corresponding to the volume of > air being reduced to nothing, which would be marked as −273° of the scale > (−100/.366, if .366 be the coefficient of expansion); and therefore −273° of > the air-thermometer is a point which cannot be reached at any finite > temperature, however low. The values used for M, g0 and R* are in accordance with the U.S. Standard Atmosphere, 1976, and that the value for R* in particular does not agree with standard values for this constant. Subscript b Height Above Sea Level (h) Mass Density (\rho) Standard Temperature (T') (K) Temperature Lapse Rate (L) (m) (ft) (kg/m3) (slug/ft3) (K/m) (K/ft) 0 0 0 1.2250 288.15 0.0065 0.0019812 1 11 000 36,089.24 0.36391 216.65 0.0 0.0 2 20 000 65,616.79 0.08803 216.65 -0.001 -0.0003048 3 32 000 104,986.87 0.01322 228.65 -0.0028 -0.00085344 4 47 000 154,199.48 0.00143 270.65 0.0 0.0 5 51 000 167,322.83 0.00086 270.65 0.0028 0.00085344 6 71 000 232,939.63 0.000064 214.65 0.002 0.0006096 ==Derivation== The barometric formula can be derived using the ideal gas law: P = \frac{\rho}{M} {R^*} T Assuming that all pressure is hydrostatic: dP = - \rho g\,dz and dividing the dP by the P expression we get: \frac{dP}{P} = - \frac{M g\,dz}{R^*T} Integrating this expression from the surface to the altitude z we get: P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}} Assuming linear temperature change T = T_0 - L z and constant molar mass and gravitational acceleration, we get the first barometric formula: P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}} Instead, assuming constant temperature, integrating gives the second barometric formula: P = P_0 e^{-M g z/R^*T} In this formulation, R* is the gas constant, and the term R*T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere). The values used for M, g0, and R* are in accordance with the U.S. Standard Atmosphere, 1976, and the value for R* in particular does not agree with standard values for this constant.U.S. Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976. Note that the NWS standard specifies that the density altitude should be rounded to the nearest 100ft. ===Approximation formula for calculating the density altitude from the pressure altitude=== This is an easier formula to calculate (with great approximation) the density altitude from the pressure altitude and the ISA temperature deviation: : \text{DA} \approx \text{PA} + 118.8 ~ \frac{\text{ft}}{{^\circ \text{C}}} \left(T_\text{OA} - T_\text{ISA}\right). To derive Charles's law from kinetic theory, it is necessary to have a microscopic definition of temperature: this can be conveniently taken as the temperature being proportional to the average kinetic energy of the gas molecules, k: :T \propto \bar{E_{\rm k}}.\, Under this definition, the demonstration of Charles's law is almost trivial. This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. In this formula, : \text{DA}_\text{NWS} , National Weather Service density altitude in feet ( \text{ft} ); : P , station pressure (static atmospheric pressure) in inches of mercury (inHg); : T , station temperature (outside air temperature) in degrees Fahrenheit (°F). * Review of Amontons' findings: ""Sur une nouvelle proprieté de l'air, et une nouvelle construction de Thermométre"" (On a new property of the air and a new construction of thermometer), Histoire de l'Académie Royale des Sciences, 1–8 (submitted: 1702; published: 1743). and Francis Hauksbee* Englishman Francis Hauksbee (1660–1713) independently also discovered Charles's law: Francis Hauksbee (1708) ""An account of an experiment touching the different densities of air, from the greatest natural heat to the greatest natural cold in this climate,"" Philosophical Transactions of the Royal Society of London 26(315): 93–96. a century earlier. ",-100,-273,"""4.946""",-7.5,+65.49,B
+A certain gas obeys the van der Waals equation with $a=0.50 \mathrm{~m}^6 \mathrm{~Pa}$ $\mathrm{mol}^{-2}$. Its volume is found to be $5.00 \times 10^{-4} \mathrm{~m}^3 \mathrm{~mol}^{-1}$ at $273 \mathrm{~K}$ and $3.0 \mathrm{MPa}$. From this information calculate the van der Waals constant $b$. What is the compression factor for this gas at the prevailing temperature and pressure?,"The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density. ===Statistical thermodynamics derivation=== The canonical partition function Z of an ideal gas consisting of N = nNA identical (non-interacting) particles, is: : Z = \frac{z^N}{N!}\quad \hbox{with}\quad z = \frac{V}{\Lambda^3} where \Lambda is the thermal de Broglie wavelength, : \Lambda = \sqrt{\frac{h^2}{2\pi m k T}} with the usual definitions: h is the Planck constant, m the mass of a particle, k the Boltzmann constant and T the absolute temperature. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. For an ideal gas the compressibility factor is Z=1 per definition. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. Experimental values for the compressibility factor confirm this. According to van der Waals, the theorem of corresponding states (or principle/law of corresponding states) indicates that all fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility factor and all deviate from ideal gas behavior to about the same degree. page 141 Material constants that vary for each type of material are eliminated, in a recast reduced form of a constitutive equation. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. ",0.0761,7.136,"""4.0""",2,0.66,E
+Calculate the pressure exerted by $1.0 \mathrm{~mol} \mathrm{Xe}$ when it is confined to $1.0 \mathrm{dm}^3$ at $25^{\circ} \mathrm{C}$.,"The CGS unit of pressure is the barye (Ba), equal to 1 dyn·cm−2, or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre (g/cm2 or kg/cm2) and the like without properly identifying the force units. :P=\frac {2 \sigma_\theta s} {D}thumb|252x252px|Cylinder, where :P : internal pressure, :\sigma_\theta : allowable stress, :s : wall thickness, :D : outside diameter. Derivation of this equation This is derived from the definitions of pressure and weight density. Pressure is related to energy density and may be expressed in units such as joules per cubic metre (J/m3, which is equal to Pa). With the ""area"" in the numerator and the ""area"" in the denominator canceling each other out, we are left with :\text{pressure} = \text{weight density} \times \text{depth}. Presently or formerly popular pressure units include the following: *atmosphere (atm) *manometric units: **centimetre, inch, millimetre (torr) and micrometre (mTorr, micron) of mercury, **height of equivalent column of water, including millimetre (mm ), centimetre (cm ), metre, inch, and foot of water; *imperial and customary units: **kip, short ton-force, long ton-force, pound- force, ounce-force, and poundal per square inch, **short ton-force and long ton-force per square inch, **fsw (feet sea water) used in underwater diving, particularly in connection with diving pressure exposure and decompression; *non-SI metric units: **bar, decibar, millibar, ***msw (metres sea water), used in underwater diving, particularly in connection with diving pressure exposure and decompression, **kilogram-force, or kilopond, per square centimetre (technical atmosphere), **gram-force and tonne-force (metric ton- force) per square centimetre, **barye (dyne per square centimetre), **kilogram-force and tonne-force per square metre, **sthene per square metre (pieze). ===Examples=== 120px|thumbnail|right|The effects of an external pressure of 700 bar on an aluminum cylinder with wall thickness As an example of varying pressures, a finger can be pressed against a wall without making any lasting impression; however, the same finger pushing a thumbtack can easily damage the wall. The pressure exerted by a column of liquid of height h and density ρ is given by the hydrostatic pressure equation , where g is the gravitational acceleration. thumb|259x259px|English Parliament of General Election 1702 The 1702 English general election was the first to be held during the reign of Queen Anne, and was necessitated by the demise of William III. * P_0 is the pressure at the surface. * \rho(z) is the density of the material above the depth z. * g is the gravity acceleration in m/s^2 . Pressure is force magnitude applied over an area. Barlow's formula (called ""Kesselformel"" in German) relates the internal pressure that a pipeOr pressure vessel, or other cylindrical pressure containment structure. can withstand to its dimensions and the strength of its material. The molecular formula C23H21NO (molar mass: 327.42 g/mol, exact mass: 327.1623 u) may refer to: * JWH-015 * JWH-073 * JWH-120 Category:Molecular formulas The molecular formula C13H14O3 (molar mass: 218.248 g/mol, exact mass: 218.0943 u) may refer to: * NCS-382 * Toxol Category:Molecular formulas Pressure force acts in all directions at a point inside a gas. Then we have :\text{pressure} = \frac{\text{force}}{\text{area}} = \frac{\text{weight}}{\text{area}} = \frac{\text{weight density} \times \text{volume}}{\text{area}}, :\text{pressure} = \frac{\text{weight density} \times \text{(area} \times \text{depth)}}{\text{area}}. Pressure in open conditions usually can be approximated as the pressure in ""static"" or non-moving conditions (even in the ocean where there are waves and currents), because the motions create only negligible changes in the pressure. The negative gradient of pressure is called the force density. The pressure is the scalar proportionality constant that relates the two normal vectors: :d\mathbf{F}_n = -p\,d\mathbf{A} = -p\,\mathbf{n}\,dA. It is a fundamental parameter in thermodynamics, and it is conjugate to volume. ===Units=== thumb|right|Mercury column The SI unit for pressure is the pascal (Pa), equal to one newton per square metre (N/m2, or kg·m−1·s−2). In a stratigraphic layer that is in hydrostatic equilibrium; the overburden pressure at a depth z, assuming the magnitude of the gravity acceleration is approximately constant, is given by: P(z) = P_0 + g \int_{0}^{z} \rho(z) \, dz Where: * z is the depth in meters. This tensor may be expressed as the sum of the viscous stress tensor minus the hydrostatic pressure. The total pressure of a liquid, then, is ρgh plus the pressure of the atmosphere. ",21,4152,"""-20.0""",24,-0.347,A
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the total change in entropy, when a sample of nitrogen gas of mass $14 \mathrm{~g}$ at $298 \mathrm{~K}$ and $1.00 \mathrm{bar}$ doubles its volume in an isothermal reversible expansion.","The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. ISO 15971 is an ISO standard for calorific value measurement of natural gas and its substitutes. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. The limit of this sum as N \rightarrow \infty becomes an integral: :S^\circ = \sum_{k=1}^N \Delta S_k = \sum_{k=1}^N \frac{dQ_k}{T} \rightarrow \int _0 ^{T_2} \frac{dS}{dT} dT = \int _0 ^{T_2} \frac {C_{p_k}}{T} dT In this example, T_2 =298.15 K and C_{p_k} is the molar heat capacity at a constant pressure of the substance in the reversible process . However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. This quantity relates to the thermodynamic entropy as : \Delta S = 3/2 \ln K . In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. According to Trouton's rule, the entropy of vaporization (at standard pressure) of most liquids has similar values. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). * The heat capacity of the gas from the boiling point to room temperature. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. ",0.25,0,"""9.2e-06""",0.2307692308,2,B
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate $\Delta H$ when two copper blocks, each of mass $10.0 \mathrm{~kg}$, one at $100^{\circ} \mathrm{C}$ and the other at $0^{\circ} \mathrm{C}$, are placed in contact in an isolated container. The specific heat capacity of copper is $0.385 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$ and may be assumed constant over the temperature range involved.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The formula is: :\alpha = \frac{ k }{ \rho c_{p} } where * is thermal conductivity (W/(m·K)) * is specific heat capacity (J/(kg·K)) * is density (kg/m3) Together, can be considered the volumetric heat capacity (J/(m3·K)). Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. In heat transfer analysis, thermal diffusivity is the thermal conductivity divided by density and specific heat capacity at constant pressure. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). thumb|250px|The plot of the specific heat capacity versus temperature. Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, ), which reflects the relatively constant difference in work done during expansion for constant pressure vs. constant volume conditions. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities. The amount of energy added equals , with representing the change in temperature. Boca Raton, Florida, 2003; Section 4, Properties of the Elements and Inorganic Compounds; Vapor Pressure of the Metallic Elements :The equations reproduce the observed pressures to an accuracy of ±5% or better. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. ",2.3613,0,"""65.49""",14,0.086,B
+"A perfect gas undergoes isothermal compression, which reduces its volume by $2.20 \mathrm{dm}^3$. The final pressure and volume of the gas are $5.04 \mathrm{bar}$ and $4.65 \mathrm{dm}^3$, respectively. Calculate the original pressure of the gas in atm.","The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture (Dalton's Law). The partial pressures obey Dalton's law: P_i = y_i P, where is the total pressure and is the mole fraction of the component (so the partial pressures add up to the total pressure). It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules. That is, the mole fraction x_{\mathrm{i}} of an individual gas component in an ideal gas mixture can be expressed in terms of the component's partial pressure or the moles of the component: x_{\mathrm{i}} = \frac{p_{\mathrm{i}}}{p} = \frac{n_{\mathrm{i}}}{n} and the partial pressure of an individual gas component in an ideal gas can be obtained using this expression: p_{\mathrm{i}} = x_{\mathrm{i}} \cdot p where: x_{\mathrm{i}} = mole fraction of any individual gas component in a gas mixture p_{\mathrm{i}} = partial pressure of any individual gas component in a gas mixture n_{\mathrm{i}} = moles of any individual gas component in a gas mixture n = total moles of the gas mixture p = total pressure of the gas mixture The mole fraction of a gas component in a gas mixture is equal to the volumetric fraction of that component in a gas mixture.Frostberg State University's ""General Chemistry Online"" The ratio of partial pressures relies on the following isotherm relation: \frac{V_{\rm X}}{V_{\rm tot}} = \frac{p_{\rm X}}{p_{\rm tot}} = \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of any individual gas component (X) * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas (X) * ntot is the total amount of substance in gas mixture ==Partial volume (Amagat's law of additive volume)== The partial volume of a particular gas in a mixture is the volume of one component of the gas mixture. In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. The standard atmosphere (symbol: atm) is a unit of pressure defined as Pa. Using diving terms, partial pressure is calculated as: :partial pressure = (total absolute pressure) × (volume fraction of gas component) For the component gas ""i"": :pi = P × Fi For example, at underwater, the total absolute pressure is (i.e., 1 bar of atmospheric pressure + 5 bar of water pressure) and the partial pressures of the main components of air, oxygen 21% by volume and nitrogen approximately 79% by volume are: :pN2 = 6 bar × 0.79 = 4.7 bar absolute :pO2 = 6 bar × 0.21 = 1.3 bar absolute where: pi = partial pressure of gas component i = P_{\mathrm{i}} in the terms used in this article P = total pressure = P in the terms used in this article Fi = volume fraction of gas component i = mole fraction, x_{\mathrm{i}}, in the terms used in this article pN2 = partial pressure of nitrogen = P_\mathrm{N_2} in the terms used in this article pO2 = partial pressure of oxygen = P_\mathrm{O_2} in the terms used in this article The minimum safe lower limit for the partial pressures of oxygen in a breathing gas mixture for diving is absolute. V_{\rm X} = V_{\rm tot} \times \frac{p_{\rm X}}{p_{\rm tot}} = V_{\rm tot} \times \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of an individual gas component X in the mixture * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas X * ntot is the total amount of substance in the gas mixture ==Vapor pressure== thumb|right|A log-lin vapor pressure chart for various liquids Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases (i.e., liquid or solid). For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Typically, the maximum total partial pressure of narcotic gases used when planning for technical diving may be around 4.5 bar absolute, based on an equivalent narcotic depth of . In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . In chemistry and in various industries, the reference pressure referred to in standard temperature and pressure was commonly but standards have since diverged; in 1982, the International Union of Pure and Applied Chemistry recommended that for the purposes of specifying the physical properties of substances, standard pressure should be precisely .IUPAC.org, Gold Book, Standard Pressure ==Pressure units and equivalencies == A pressure of 1 atm can also be stated as: :≡ pascals (Pa) :≡ bar :≈ kgf/cm2 :≈ technical atmosphere :≈ m H2O, 4 °CThis is the customarily accepted value for cm–H2O, 4 °C. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. It is approximately equal to Earth's average atmospheric pressure at sea level. ==History== The standard atmosphere was originally defined as the pressure exerted by 760 mm of mercury at and standard gravity (gn = ). The partial pressure of oxygen also determines the maximum operating depth of a gas mixture. That is, at low pressures is the same as the pressure, so it has the same units as pressure. When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. ",7.00,3.38,"""0.4""",0.6296296296,+5.41,B
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the total change in entropy, when a sample of nitrogen gas of mass $14 \mathrm{~g}$ at $298 \mathrm{~K}$ and $1.00 \mathrm{bar}$ doubles its volume in an isothermal irreversible expansion against $p_{\mathrm{ex}}=0$.","However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. In astrophysics, what is referred to as ""entropy"" is actually the adiabatic constant derived as follows. The differential change of the chemical potential between two states of slightly different pressures but equal temperature (i.e., ) is given by d\mu = V_\mathrm{m}dP = RT \, \frac{dP}{P} = R T\, d \ln P,where ln p is the natural logarithm of p. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Numerical example: Nitrogen gas (N2) at 0 °C and a pressure of atmospheres (atm) has a fugacity of atm. ISO 15971 is an ISO standard for calorific value measurement of natural gas and its substitutes. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. * The heat capacity of the gas from the boiling point to room temperature. ",0,-6.9,"""3.07""",34,+2.9,E
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate the change in the molar Gibbs energy of hydrogen gas when its pressure is increased isothermally from $1.0 \mathrm{~atm}$ to 100.0 atm at $298 \mathrm{~K}$.,"The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. * The heat capacity of the gas from the boiling point to room temperature. When using the ISO value of R, the calculated pressure increases by only 0.62 pascal at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in). In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac{ \dfrac{\mathrm{force}}{\mathrm{area}} \times \mathrm{volume} } { \mathrm{amount} \times \mathrm{temperature} } Area and volume are (length)2 and (length)3 respectively. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. ", 4.56,0.24995,"""-3.8""",+11,0.7854,D
+"A perfect gas undergoes isothermal compression, which reduces its volume by $2.20 \mathrm{dm}^3$. The final pressure and volume of the gas are $5.04 \mathrm{bar}$ and $4.65 \mathrm{dm}^3$, respectively. Calculate the original pressure of the gas in bar.","The partial pressures obey Dalton's law: P_i = y_i P, where is the total pressure and is the mole fraction of the component (so the partial pressures add up to the total pressure). The total pressure of an ideal gas mixture is the sum of the partial pressures of the gases in the mixture (Dalton's Law). The partial pressure of a gas is a measure of thermodynamic activity of the gas's molecules. That is, the mole fraction x_{\mathrm{i}} of an individual gas component in an ideal gas mixture can be expressed in terms of the component's partial pressure or the moles of the component: x_{\mathrm{i}} = \frac{p_{\mathrm{i}}}{p} = \frac{n_{\mathrm{i}}}{n} and the partial pressure of an individual gas component in an ideal gas can be obtained using this expression: p_{\mathrm{i}} = x_{\mathrm{i}} \cdot p where: x_{\mathrm{i}} = mole fraction of any individual gas component in a gas mixture p_{\mathrm{i}} = partial pressure of any individual gas component in a gas mixture n_{\mathrm{i}} = moles of any individual gas component in a gas mixture n = total moles of the gas mixture p = total pressure of the gas mixture The mole fraction of a gas component in a gas mixture is equal to the volumetric fraction of that component in a gas mixture.Frostberg State University's ""General Chemistry Online"" The ratio of partial pressures relies on the following isotherm relation: \frac{V_{\rm X}}{V_{\rm tot}} = \frac{p_{\rm X}}{p_{\rm tot}} = \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of any individual gas component (X) * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas (X) * ntot is the total amount of substance in gas mixture ==Partial volume (Amagat's law of additive volume)== The partial volume of a particular gas in a mixture is the volume of one component of the gas mixture. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. In a mixture of gases, each constituent gas has a partial pressure which is the notional pressure of that constituent gas as if it alone occupied the entire volume of the original mixture at the same temperature. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. V_{\rm X} = V_{\rm tot} \times \frac{p_{\rm X}}{p_{\rm tot}} = V_{\rm tot} \times \frac{n_{\rm X}}{n_{\rm tot}} * VX is the partial volume of an individual gas component X in the mixture * Vtot is the total volume of the gas mixture * pX is the partial pressure of gas X * ptot is the total pressure of the gas mixture * nX is the amount of substance of gas X * ntot is the total amount of substance in the gas mixture ==Vapor pressure== thumb|right|A log-lin vapor pressure chart for various liquids Vapor pressure is the pressure of a vapor in equilibrium with its non-vapor phases (i.e., liquid or solid). Typically, the maximum total partial pressure of narcotic gases used when planning for technical diving may be around 4.5 bar absolute, based on an equivalent narcotic depth of . Using diving terms, partial pressure is calculated as: :partial pressure = (total absolute pressure) × (volume fraction of gas component) For the component gas ""i"": :pi = P × Fi For example, at underwater, the total absolute pressure is (i.e., 1 bar of atmospheric pressure + 5 bar of water pressure) and the partial pressures of the main components of air, oxygen 21% by volume and nitrogen approximately 79% by volume are: :pN2 = 6 bar × 0.79 = 4.7 bar absolute :pO2 = 6 bar × 0.21 = 1.3 bar absolute where: pi = partial pressure of gas component i = P_{\mathrm{i}} in the terms used in this article P = total pressure = P in the terms used in this article Fi = volume fraction of gas component i = mole fraction, x_{\mathrm{i}}, in the terms used in this article pN2 = partial pressure of nitrogen = P_\mathrm{N_2} in the terms used in this article pO2 = partial pressure of oxygen = P_\mathrm{O_2} in the terms used in this article The minimum safe lower limit for the partial pressures of oxygen in a breathing gas mixture for diving is absolute. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. VPT may refer to: * Valor por Tamaulipas * Vermont Public Television * Volume, temperature and pressure, the three parameters in the combined gas law The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude. == Pressure equations == thumb|300px|Pressure as a function of the height above the sea level There are two equations for computing pressure as a function of height. The partial pressure of oxygen also determines the maximum operating depth of a gas mixture. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. See real gas or perfect gas or gas for further understanding.) ==See also== *Hypsometric equation *NRLMSISE-00 *Vertical pressure variation == References == Category:Atmosphere Category:Vertical position Category:Pressure That is, at low pressures is the same as the pressure, so it has the same units as pressure. In aeronautical engineering, overall pressure ratio, or overall compression ratio, is the ratio of the stagnation pressure as measured at the front and rear of the compressor of a gas turbine engine. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. ", 10.7598,5654.86677646,"""1.43""",3.42,2,D
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate $\Delta S$ (for the system) when the state of $3.00 \mathrm{~mol}$ of perfect gas atoms, for which $C_{p, \mathrm{~m}}=\frac{5}{2} R$, is changed from $25^{\circ} \mathrm{C}$ and 1.00 atm to $125^{\circ} \mathrm{C}$ and $5.00 \mathrm{~atm}$. How do you rationalize the $\operatorname{sign}$ of $\Delta S$?","The δ34S (pronounced delta 34 S) value is a standardized method for reporting measurements of the ratio of two stable isotopes of sulfur, 34S:32S, in a sample against the equivalent ratio in a known reference standard. With VCDT as the reference standard, natural δ34S value variations have been recorded between -72‰ and +147‰. The δ34S value refers to a measure of the ratio of the two most common stable sulfur isotopes, 34S:32S, as measured in a sample against that same ratio as measured in a known reference standard. In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced ""delta fifteen n"") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. Thus the step is approximately 13.946 cents, and there are 86.049 steps per octave. :\begin{align} \frac{50\log_2{\left(\frac32\right)} + 28\log_2{\left(\frac54\right)} + 23\log_2{\left(\frac65\right)}}{50^2+28^2+23^2} = 0.011\,621\,2701 \\\ 0.011\,621\,2701 \times 1200 = 13.945\,524\,1627 \end{align} () The Bohlen–Pierce delta scale is based on the tritave and the 7:5:3 ""wide"" triad () and the 9:7:5 ""narrow"" triad () (rather than the conventional 4:5:6 triad). For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. That value can be calculated in per mil (‰, parts per thousand) as: :\delta \ce{^{34}S} = \left( \frac{\left( \frac\ce{^{34}S}\ce{^{32}S} \right)_\mathrm{sample}}{\left( \frac\ce{^{34}S}\ce{^{32}S} \right)_\mathrm{standard}} - 1 \right) \times 1000 ‰ Less commonly, if the appropriate isotope abundances are measured, similar formulae can be used to quantify ratio variations between 33S and 32S, and 36S and 32S, reported as δ33S and δ36S, respectively. ===Reference standard=== thumb|right|Troilite from the Canyon Diablo meteorite was the first reference standard for δ34S.|alt=A worn brown-red-gold space rock covered in smoothed pock-marks sits mounted in a museum. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Since the atmosphere is 99.6337% 14N and 0.3663% 15N, a is 0.003663 in the former case and 0.003663/0.996337 = 0.003676 in the latter. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. Both have the form 1000\frac{s-a}a ‰ (‰ = permil or parts per thousand) where s and a are the relative abundances of 15N in respectively the sample and the atmosphere. The lowercase delta character is used by convention, to be consistent with use in other areas of stable isotope chemistry. ISO 15971 is an ISO standard for calorific value measurement of natural gas and its substitutes. However s varies similarly; for example if in the sample 15N is 0.385% and 14N is 99.615%, s is 0.003850 in the former case and 0.00385/0.99615 = 0.003865 in the latter. From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. The value of 1000\frac{s-a}a is then 51.05‰ in the former case and 51.38‰ in the latter, an insignificant difference in practice given the typical range of -20 to 80 for . ==Applications== One use of 15N is as a tracer to determine the path taken by fertilizers applied to anything from pots to landscapes. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. Sometimes, other distinctions are made, such as between thermally perfect gas and calorically perfect gas, or between imperfect, semi-perfect, and perfect gases, and as well as the characteristics of ideal gases. The δ (delta) scale is a non-octave repeating musical scale. This temperature limit depends on the chemical composition of the gas and how accurate the calculations need to be, since molecular dissociation may be important at a higher or lower temperature which is intrinsically dependent on the molecular nature of the gas. ",1.154700538,4.3,"""-22.1""",169,2.567,C
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. A sample consisting of $3.00 \mathrm{~mol}$ of diatomic perfect gas molecules at $200 \mathrm{~K}$ is compressed reversibly and adiabatically until its temperature reaches $250 \mathrm{~K}$. Given that $C_{V, \mathrm{~m}}=27.5 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, calculate $\Delta H$.","Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. The temperature dependence at constant pressure is \left(\frac{\partial \ln f}{\partial T}\right)_P = \frac{\Delta H_\mathrm{m}}{R T^2}, where is the change in molar enthalpy as the gas expands, liquid vaporizes, or solid sublimates into a vacuum. Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Then, to a good approximation, most gases have the same value of for the same reduced temperature and pressure. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. This error is corrected in the above equation. ==Measurement== The fugacity can be deduced from measurements of volume as a function of pressure at constant temperature. When calculating the fugacity of the compressed phase, one can generally assume the volume is constant. ",+5.41,0.68,"""0.59""",3.8,58.2,A
+A sample of $255 \mathrm{mg}$ of neon occupies $3.00 \mathrm{dm}^3$ at $122 \mathrm{K}$. Use the perfect gas law to calculate the pressure of the gas.,"For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. Billion cubic meters of natural gas (non SI abbreviation: bcm) or cubic kilometer of natural gas is a measure of natural gas production and trade. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. That means that 1 billion cubic metres of natural gas by the International Energy Agency standard is equivalent to 1.017 billion cubic metres of natural gas by the Russian standard. ==Energy based definitions== Some other organizations use energy equivalent-based standards. Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. The pressure inside is equal to atmospheric pressure. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. 8) or of pressure P (p. 9). From both statistical mechanics and the simpler kinetic theory of gases, we expect the heat capacity of a monatomic ideal gas to be constant, since for such a gas only kinetic energy contributes to the internal energy and to within an arbitrary additive constant U = (3/2) n R T , and therefore C_V = (3/2) n R , a constant. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Thus we observe that for a monatomic gas, with 3 translational degrees of freedom per atom: : \gamma = \frac{5}{3} = 1.6666\ldots, As an example of this behavior, at 273 K (0 °C) the noble gases He, Ne, and Ar all have nearly the same value of , equal to 1.664. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. According to the Russian standard, the gas volume is measured at . Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Dalton's law is not strictly followed by real gases, with the deviation increasing with pressure. For example, terrestrial air is primarily made up of diatomic gases (around 78% nitrogen, N2, and 21% oxygen, O2), and at standard conditions it can be considered to be an ideal gas. Cedigaz uses a standard which is equivalent to per billion cubic metres. ==References== Category:Natural gas Category:Units of volume Category:Units of energy Category:Non-SI metric units Category:International Energy Agency Category:BP However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. ",1.2,0.042,"""152.67""",9.90,0.9984,B
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A chemical reaction takes place in a container of cross-sectional area $100 \mathrm{~cm}^2$. As a result of the reaction, a piston is pushed out through $10 \mathrm{~cm}$ against an external pressure of $1.0 \mathrm{~atm}$. Calculate the work done by the system.","In this case the work is given by (where is the pressure at the surface, is the increase of the volume of the system). When a system, for example, moles of a gas of volume at pressure and temperature , is created or brought to its present state from absolute zero, energy must be supplied equal to its internal energy plus , where is the work done in pushing against the ambient (atmospheric) pressure. The quantity of thermodynamic work is defined as work done by the system on its surroundings. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. In chemistry, experiments are often conducted at constant atmospheric pressure, and the pressure–volume work represents a small, well-defined energy exchange with the atmosphere, so that is the appropriate expression for the heat of reaction. The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). For systems at constant pressure, with no external work done other than the work, the change in enthalpy is the heat received by the system. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Such work done by compression is thermodynamic work as here defined. Thermodynamic work is defined for the purposes of thermodynamic calculations about bodies of material, known as thermodynamic systems. Real gases at common temperatures and pressures often closely approximate this behavior, which simplifies practical thermodynamic design and analysis. ==Definition== The enthalpy of a thermodynamic system is defined as the sum of its internal energy and the product of its pressure and volume: : , where is the internal energy, is pressure, and is the volume of the system; is sometimes referred to as the pressure energy . Thermodynamic work done by a thermodynamic system on its surroundings is defined so as to comply with this principle. The work is due to change of system volume by expansion or contraction of the system. To get an actual and precise physical measurement of a quantity of thermodynamic work, it is necessary to take account of the irreversibility by restoring the system to its initial condition by running a cycle, for example a Carnot cycle, that includes the target work as a step. As a result, the work done by the system also depends on the initial and final states. An Introduction to Thermal Physics, 2000, Addison Wesley Longman, San Francisco, CA, , p. 18 According to the first law of thermodynamics for a closed system, any net change in the internal energy U must be fully accounted for, in terms of heat Q entering the system and work W done by the system: :\Delta U = Q - W.\; Freedman, Roger A., and Young, Hugh D. (2008). 12th Edition. Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. Consequently, thermodynamic work is defined in terms of quantities that describe the states of materials, which appear as the usual thermodynamic state variables, such as volume, pressure, temperature, chemical composition, and electric polarization. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of mass into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of mass out as if it were driving a piston of fluid. ",-11.2,-100,"""6.2""",3.54,+93.4,B
+Use the van der Waals parameters for chlorine to calculate approximate values of the Boyle temperature of chlorine.,"J/(mol K) Enthalpy of combustion –473.2 kJ/mol ΔcH ~~o~~ Heat capacity, cp 114.25 J/(mol K) Gas properties Std enthalpy change of formation, ΔfH ~~o~~ gas –103.18 kJ/mol Standard molar entropy, S ~~o~~ gas 295.6 J/(mol K) at 25 °C Heat capacity, cp 65.33 J/(mol K) at 25 °C van der Waals' constantsLange's Handbook of Chemistry 10th ed, pp. 1522–1524 a = 1537 L2 kPa/mol2 b = 0.1022 liter per mole ==Vapor pressure of liquid== P in mm Hg 1 10 40 100 400 760 1520 3800 7600 15200 30400 45600 T in °C –58.0 –29.7 –7.1 10.4 42.7 61.3 83.9 120.0 152.3 191.8 237.5 — Table data obtained from CRC Handbook of Chemistry and Physics 44th ed. thumb|812px|left|log10 of Chloroform vapor pressure. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The molecular formula C3H4Cl2O2 (molar mass: 142.97 g/mol, exact mass: 141.9588 u) may refer to: * Chloroethyl chloroformate * Dalapon The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. Uses formula: \scriptstyle \log_e P_{mmHg} =\scriptstyle \log_e (\frac {760} {101.325}) - 10.07089 \log_e(T+273.15) - \frac {6351.140} {T+273.15} + 81.14393 + 9.127608 \times 10^{-6} (T+273.15)^2 obtained from CHERIC ==Distillation data== {| border=""1"" cellspacing=""0"" cellpadding=""6"" style=""margin: 0 0 0 0.5em; background: white; border-collapse: collapse; border-color: #C0C090;"" Vapor-liquid Equilibrium for Chloroform/Ethanol P = 101.325 kPa BP Temp. °C % by mole chloroform liquid vapor 78.15 0.0 0.0 78.07 0.52 1.59 77.83 1.02 3.01 76.81 2.21 7.45 74.90 5.80 17.10 74.39 6.72 19.40 73.55 8.38 23.31 72.85 10.57 28.05 72.24 11.80 30.52 71.58 13.18 33.13 69.72 17.65 41.00 68.95 19.62 43.66 68.58 20.71 45.43 67.35 23.86 49.77 65.89 28.54 55.09 64.87 32.35 58.48 63.88 36.07 61.27 63.23 39.34 64.50 62.61 41.38 65.49 62.17 44.41 67.57 61.48 49.97 71.11 61.00 53.92 72.91 60.49 54.76 73.57 60.35 59.65 74.68 60.30 61.60 75.53 60.20 63.04 76.12 60.09 64.48 76.69 59.97 66.90 77.74 59.54 72.01 79.33 59.32 79.07 82.62 59.26 82.99 83.59 59.28 84.97 84.69 59.31 85.96 85.24 59.46 89.92 87.93 59.72 91.10 88.73 59.70 92.44 89.79 59.84 93.90 91.02 59.91 95.26 92.56 60.18 96.13 93.58 60.88 98.89 97.93 61.13 100.00 100.00 == Spectral data == UV-Vis λmax ? nm Extinction coefficient, ε ? Expanding the van der Waals equation in \frac{1}{V_m} one finds that T_b = \frac{a}{Rb}.Verma, K.S. Cengage Physical Chemistry Part 1. Also at Boyle temperature the dip in a PV diagram tends to a straight line over a period of pressure. 100px 100px Two representations of chloroform. This page provides supplementary chemical data on chloroform. == Material Safety Data Sheet == The handling of this chemical may incur notable safety precautions. Several billion kilograms of chlorinated methanes are produced annually, mainly by chlorination of methane: :CH4 \+ x Cl2 → CH4−xClx \+ x HCl The most important is dichloromethane, which is mainly used as a solvent. Organochlorine chemistry is concerned with the properties of organochlorine compounds, or organochlorides, organic compounds containing at least one covalently bonded atom of chlorine. To convert from L^2bar/mol^2 to m^6 Pa/mol^2, divide by 10. a (L2bar/mol2) b (L/mol) Acetic acid 17.7098 0.1065 Acetic anhydride 20.158 0.1263 Acetone 16.02 0.1124 Acetonitrile 17.81 0.1168 Acetylene 4.516 0.0522 Ammonia 4.225 0.0371 Aniline 29.14 0.1486 Argon 1.355 0.03201 Benzene 18.24 0.1193 Bromobenzene 28.94 0.1539 Butane 14.66 0.1226 1-Butanol 20.94 0.1326 2-Butanone 19.97 0.1326 Carbon dioxide 3.640 0.04267 Carbon disulfide 11.77 0.07685 Carbon monoxide 1.505 0.0398500 Carbon tetrachloride 19.7483 0.1281 Chlorine 6.579 0.05622 Chlorobenzene 25.77 0.1453 Chloroethane 11.05 0.08651 Chloromethane 7.570 0.06483 Cyanogen 7.769 0.06901 Cyclohexane 23.11 0.1424 Cyclopropane 8.34 0.0747 Decane 52.74 0.3043 1-Decanol 59.51 0.3086 Diethyl ether 17.61 0.1344 Diethyl sulfide 19.00 0.1214 Dimethyl ether 8.180 0.07246 Dimethyl sulfide 13.04 0.09213 Dodecane 69.38 0.3758 1-Dodecanol 75.70 0.3750 Ethane 5.562 0.0638 Ethanethiol 11.39 0.08098 Ethanol 12.18 0.08407 Ethyl acetate 20.72 0.1412 Ethylamine 10.74 0.08409 Ethylene 4.612 0.0582 Fluorine 1.171 0.0290 Fluorobenzene 20.19 0.1286 Fluoromethane 4.692 0.05264 Freon 10.78 0.0998 Furan 12.74 0.0926 Germanium tetrachloride 22.90 0.1485 Helium 0.0346 0.0238 Heptane 31.06 0.2049 1-Heptanol 38.17 0.2150 Hexane 24.71 0.1735 1-Hexanol 31.79 0.1856 Hydrazine 8.46 0.0462 Hydrogen 0.2476 0.02661 Hydrogen bromide 4.510 0.04431 Hydrogen chloride 3.716 0.04081 Hydrogen cyanide 11.29 0.0881 Hydrogen fluoride 9.565 0.0739 Hydrogen iodide 6.309 0.0530 Hydrogen selenide 5.338 0.04637 Hydrogen sulfide 4.490 0.04287 Isobutane 13.32 0.1164 Iodobenzene 33.52 0.1656 Krypton 2.349 0.03978 Mercury 8.200 0.01696 Methane 2.283 0.04278 Methanol 9.649 0.06702 Methylamine 7.106 0.0588 Neon 0.2135 0.01709 Neopentane 17.17 0.1411 Nitric oxide 1.358 0.02789 Nitrogen 1.370 0.0387 Nitrogen dioxide 5.354 0.04424 Nitrogen trifluoride 3.58 0.0545 Nitrous oxide 3.832 0.04415 Octane 37.88 0.2374 1-Octanol 44.71 0.2442 Oxygen 1.382 0.03186 Ozone 3.570 0.0487 Pentane 19.26 0.146 1-Pentanol 25.88 0.1568 Phenol 22.93 0.1177 Phosphine 4.692 0.05156 Propane 8.779 0.08445 1-Propanol 16.26 0.1079 2-Propanol 15.82 0.1109 Propene 8.442 0.0824 Pyridine 19.77 0.1137 Pyrrole 18.82 0.1049 Radon 6.601 0.06239 Silane 4.377 0.05786 Silicon tetrafluoride 4.251 0.05571 Sulfur dioxide 6.803 0.05636 Sulfur hexafluoride 7.857 0.0879 Tetrachloromethane 20.01 0.1281 Tetrachlorosilane 20.96 0.1470 Tetrafluoroethylene 6.954 0.0809 Tetrafluoromethane 4.040 0.0633 Tetrafluorosilane 5.259 0.0724 Tetrahydrofuran 16.39 0.1082 Tin tetrachloride 27.27 0.1642 Thiophene 17.21 0.1058 Toluene 24.38 0.1463 1-1-1-Trichloroethane 20.15 0.1317 Trichloromethane 15.34 0.1019 Trifluoromethane 5.378 0.0640 Trimethylamine 13.37 0.1101 Water 5.536 0.03049 Xenon 4.250 0.05105 ==Units== 1 J·m3/mol2 = 1 m6·Pa/mol2 = 10 L2·bar/mol2 1 L2atm/mol2 = 0.101325 J·m3/mol2 = 0.101325 Pa·m6/mol2 1 dm3/mol = 1 L/mol = 1 m3/kmol (where kmol is kilomoles = 1000 moles) ==References== Category:Gas laws Constants (Data Page) J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −705.63 kJ/mol Standard molar entropy S ~~o~~ solid 109.29 J/(mol K) Heat capacity cp 91.12 J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid -674.80 kJ/mol Standard molar entropy S ~~o~~ liquid 172.91 J/(mol K) Heat capacity cp 125.5 J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas -584.59 kJ/mol Standard molar entropy S ~~o~~ gas 314.44 J/(mol K) Heat capacity cp 82.46 J/(mol K) == Spectral data == UV-Vis Spectrum Lambda-max nm Log Ε IR Spectrum NIST Major absorption bands cm−1 NMR Proton NMR ? K (? °C), ? The annual production in 1985 was around 13 million tons, almost all of which was converted into polyvinylchloride (PVC). ===Chloromethanes=== Most low molecular weight chlorinated hydrocarbons such as chloroform, dichloromethane, dichloroethene, and trichloroethane are useful solvents. Chlorine adds to the multiple bonds on alkenes and alkynes as well, giving di- or tetra-chloro compounds. ===Reaction with hydrogen chloride=== Alkenes react with hydrogen chloride (HCl) to give alkyl chlorides. Since higher order virial coefficients are generally much smaller than the second coefficient, the gas tends to behave as an ideal gas over a wider range of pressures when the temperature reaches the Boyle temperature (or when c = \frac{1}{V_m} or P are minimized). Supplementary data for aluminium chloride. == External MSDS == * Baker * Fisher * EM Science * Akzo Nobel (hexahydrate) * Science Stuff (hexahydrate) * External SDS == Thermodynamic properties == Phase behavior Triple point ? The Wurtz reaction reductively couples two alkyl halides to couple with sodium. ==Applications== ===Vinyl chloride=== The largest application of organochlorine chemistry is the production of vinyl chloride. Fourme, M. Renaud, C. R. Acad. Sci, Ser. C(Chim),1966, p. 69 C-Cl 1.75 Å Bond angle Cl-C-Cl 110.3° Dipole moment 1.08 D (gas) 1.04 DCRC Handbook of Chemistry and Physics. 89th ed./David R. Lide ed.-in- chief. Alternatively, the Appel reaction can be used: :250px ==Reactions== Alkyl chlorides are versatile building blocks in organic chemistry. – Close to that of Teflon Surface tension 28.5 dyn/cm at 10 °C 27.1 dyn/cm at 20 °C 26.67 dyn/cm at 25 °C 23.44 dyn/cm at 50 °C 21.7 dyn/cm at 60 °C 20.20 dyn/cm at 75 °C ViscosityLange's Handbook of Chemistry, 10th ed. pp. 1669–1674 0.786 mPa·s at –10 °C 0.699 mPa·s at 0 °C 0.563 mPa·s at 20 °C 0.542 mPa·s at 25 °C 0.464 mPa·s at 40 °C 0.389 mPa·s at 60 °C == Thermodynamic properties == Phase behavior Triple point 209.61 K (–63.54 °C), ? ",1410,3.8,"""313.0""",7.82,144,A
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Consider a system consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2(\mathrm{~g})$, initially at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~atm}$ and confined to a cylinder of cross-section $10.0 \mathrm{~cm}^2$. It is allowed to expand adiabatically against an external pressure of 1.0 atm until the piston has moved outwards through $20 \mathrm{~cm}$. Assume that carbon dioxide may be considered a perfect gas with $C_{V, \mathrm{~m}}=28.8 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ and calculate $\Delta T$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. Values can also be determined through finite-difference approximation. == Adiabatic process == This ratio gives the important relation for an isentropic (quasistatic, reversible, adiabatic process) process of a simple compressible calorically-perfect ideal gas: : PV^\gamma is constant Using the ideal gas law, PV = nRT: : P^{1-\gamma} T^\gamma is constant : TV^{\gamma-1} is constant where is the pressure of the gas, is the volume, and is the thermodynamic temperature. We assume the expansion occurs without exchange of heat (adiabatic expansion). To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. Since the piston cannot move, the volume is constant. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston. For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. * Being a gaseous fuel, CNG mixes easily and evenly in air. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). The pressure inside is equal to atmospheric pressure. ",-0.347,257,"""3.42""",0.925,4096,A
+Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. 3.17 Estimate the standard reaction Gibbs energy of $\mathrm{N}_2(\mathrm{~g})+3 \mathrm{H}_2(\mathrm{~g}) \rightarrow$ $2 \mathrm{NH}_3(\mathrm{~g})$ at $1000 \mathrm{~K}$ from their values at $298 \mathrm{~K}$.,"Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. * The heat capacity of the gas from the boiling point to room temperature. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically perfect gas and a thermally perfect gas. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB): R = N_{\rm A} k. The value of R is then obtained from the relation :c_\mathrm{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\mathrm{r}(\mathrm{Ar}) M_\mathrm{u}}}, where: *γ0 is the heat capacity ratio ( for monatomic gases such as argon); *T is the temperature, TTPW = 273.16 K by the definition of the kelvin at that time; *Ar(Ar) is the relative atomic mass of argon and Mu = as defined at the time. ",817.90,4.979,"""-191.2""",+107,27,D
+"Could $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert a pressure of $20 \mathrm{atm}$ at $25^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert?","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. After repeating the experiment several times and using different amounts of mercury he found that under controlled conditions, the pressure of a gas is inversely proportional to the volume occupied by it. Pressure vessels for gas storage may also be classified by volume. The ship was to carry compressed natural gas in vertical pressure bottles; however, this design failed because of the high cost of the pressure vessels. Boyle's law has been stated as: > The absolute pressure exerted by a given mass of an ideal gas is inversely > proportional to the volume it occupies if the temperature and amount of gas > remain unchanged within a closed system.Levine, Ira. Most gases behave like ideal gases at moderate pressures and temperatures. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory.Levine, Ira. N. (1978), p. 11 notes that deviations occur with high pressures and temperatures. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. thumb|upright=1.3|An animation showing the relationship between pressure and volume when mass and temperature are held constant Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. That is, doubling the density of a quantity of air doubles its pressure. What the pressure should be according to the Hypothesis, that supposes the pressures and expansions to be in reciprocal relation."" High-pressure gas cylinders are also called bottles. Boyle's Law states that when the temperature of a given mass of confined gas is constant, the product of its pressure and volume is also constant. Conversely, reducing the volume of the gas increases the pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. VPT may refer to: * Valor por Tamaulipas * Vermont Public Television * Volume, temperature and pressure, the three parameters in the combined gas law ",0.4207,48.6,"""24.0""",5.1,5.0,C
+"Could $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert a pressure of $20 \mathrm{atm}$ at $25^{\circ} \mathrm{C}$ if it behaved as a perfect gas? If not, what pressure would it exert?","If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. After repeating the experiment several times and using different amounts of mercury he found that under controlled conditions, the pressure of a gas is inversely proportional to the volume occupied by it. Pressure vessels for gas storage may also be classified by volume. * ISO 11439: Compressed natural gas (CNG) cylinders. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America, the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSA B51 in Canada, Australian Standards in Australia and other international standards like Lloyd's, Germanischer Lloyd, Det Norske Veritas, Société Générale de Surveillance (SGS S.A.), Lloyd’s Register Energy Nederland (formerly known as Stoomwezen) etc. Note that where the pressure-volume product is part of a safety standard, any incompressible liquid in the vessel can be excluded as it does not contribute to the potential energy stored in the vessel, so only the volume of the compressible part such as gas is used. ===List of standards=== * EN 13445: The current European Standard, harmonized with the Pressure Equipment Directive (Originally ""97/23/EC"", since 2014 ""2014/68/EU""). Further the volume of the gas is (4πr3)/3. Boyle's law has been stated as: > The absolute pressure exerted by a given mass of an ideal gas is inversely > proportional to the volume it occupies if the temperature and amount of gas > remain unchanged within a closed system.Levine, Ira. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. As improvements in technology permitted higher pressures and lower temperatures, deviations from the ideal gas behavior became noticeable, and the relationship between pressure and volume can only be accurately described employing real gas theory.Levine, Ira. N. (1978), p. 11 notes that deviations occur with high pressures and temperatures. Most gases behave like ideal gases at moderate pressures and temperatures. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. The very small vessels used to make liquid butane fueled cigarette lighters are subjected to about 2 bar pressure, depending on ambient temperature. * EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. thumb|upright=1.3|An animation showing the relationship between pressure and volume when mass and temperature are held constant Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Other examples of pressure vessels are diving cylinders, recompression chambers, distillation towers, pressure reactors, autoclaves, and many other vessels in mining operations, oil refineries and petrochemical plants, nuclear reactor vessels, submarine and space ship habitats, atmospheric diving suits, pneumatic reservoirs, hydraulic reservoirs under pressure, rail vehicle airbrake reservoirs, road vehicle airbrake reservoirs, and storage vessels for high pressure permanent gases and liquified gases such as ammonia, chlorine, and LPG (propane, butane). That is, doubling the density of a quantity of air doubles its pressure. What the pressure should be according to the Hypothesis, that supposes the pressures and expansions to be in reciprocal relation."" Boyle's Law states that when the temperature of a given mass of confined gas is constant, the product of its pressure and volume is also constant. High-pressure gas cylinders are also called bottles. ",4.85,24,"""4.56""",226,4.738,B
+The barometric formula relates the pressure of a gas of molar mass $M$ at an altitude $h$ to its pressure $p_0$ at sea level. Derive this relation by showing that the change in pressure $\mathrm{d} p$ for an infinitesimal change in altitude $\mathrm{d} h$ where the density is $\rho$ is $\mathrm{d} p=-\rho g \mathrm{~d} h$. Remember that $\rho$ depends on the pressure. Evaluate the external atmospheric pressure at a typical cruising altitude of an aircraft (11 km) when the pressure at ground level is 1.0 atm.,"The other two values (pressure P and density ρ) are computed by simultaneously solving the equations resulting from: * the vertical pressure gradient resulting from hydrostatic balance, which relates the rate of change of pressure with geopotential altitude: :: \frac{dP}{dh} = - \rho g , and * the ideal gas law in molar form, which relates pressure , density, and temperature: :: \ P = \rho R_{\rm specific}T at each geopotential altitude, where g is the standard acceleration of gravity, and Rspecific is the specific gas constant for dry air (287.0528J⋅kg−1⋅K−1). The barometric formula is a formula used to model how the pressure (or density) of the air changes with altitude. == Pressure equations == thumb|300px|Pressure as a function of the height above the sea level There are two equations for computing pressure as a function of height. NASA SP-7012, Second Revision, National Aeronautics and Space Administration, Washington, D.C. where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/ft) in ISA *h = height at which pressure is calculated (ft) *h_b = height of reference level b (feet; e.g., hb = 36,089 ft) *R^* = universal gas constant; using feet, kelvins, and (SI) moles: *g_0 = gravitational acceleration: 32.17405 ft/s2 *M = molar mass of Earth's air: 28.9644 lb/lb-mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. Equation 1: \rho = \rho_b \left[\frac{T_b - (h-h_b) L_b}{T_b}\right]^{\left(\frac{g_0 M}{R^* L_b}-1\right)} which is equivalent to the ratio of the relative pressure and temperature changes \rho = \rho_b \frac{P}{T} \frac{T_b}{P_b} Equation 2: \rho =\rho_b \exp\left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where *{\rho} = mass density (kg/m3) *T_b = standard temperature (K) *L = standard temperature lapse rate (see table below) (K/m) in ISA *h = height above sea level (geopotential meters) *R^* = universal gas constant 8.3144598 N·m/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol or, converted to U.S. gravitational foot-pound-second units (no longer used in U.K.): *{\rho} = mass density (slug/ft3) *{T_b} = standard temperature (K) *{L} = standard temperature lapse rate (K/ft) *{h} = height above sea level (geopotential feet) *{R^*} = universal gas constant: 8.9494596×104 ft2/(s·K) *{g_0} = gravitational acceleration: 32.17405 ft/s2 *{M} = molar mass of Earth's air: 0.0289644 kg/mol The value of subscript b ranges from 0 to 6 in accordance with each of seven successive layers of the atmosphere shown in the table below. Pressure altitude is primarily used in aircraft-performance calculations and in high-altitude flight (i.e., above the transition altitude). == Inverse equation == Solving the equation for the pressure gives : p = 1013.25\left(1-\frac{h}{44307.694 m}\right)^{5.25530} hPa where are meter and hPa hecto-Pascal. Subscript b Height Above Sea Level (h) Mass Density (\rho) Standard Temperature (T') (K) Temperature Lapse Rate (L) (m) (ft) (kg/m3) (slug/ft3) (K/m) (K/ft) 0 0 0 1.2250 288.15 0.0065 0.0019812 1 11 000 36,089.24 0.36391 216.65 0.0 0.0 2 20 000 65,616.79 0.08803 216.65 -0.001 -0.0003048 3 32 000 104,986.87 0.01322 228.65 -0.0028 -0.00085344 4 47 000 154,199.48 0.00143 270.65 0.0 0.0 5 51 000 167,322.83 0.00086 270.65 0.0028 0.00085344 6 71 000 232,939.63 0.000064 214.65 0.002 0.0006096 ==Derivation== The barometric formula can be derived using the ideal gas law: P = \frac{\rho}{M} {R^*} T Assuming that all pressure is hydrostatic: dP = - \rho g\,dz and dividing the dP by the P expression we get: \frac{dP}{P} = - \frac{M g\,dz}{R^*T} Integrating this expression from the surface to the altitude z we get: P = P_0 e^{-\int_{0}^{z}{M g dz/R^*T}} Assuming linear temperature change T = T_0 - L z and constant molar mass and gravitational acceleration, we get the first barometric formula: P = P_0 \cdot \left[\frac{T}{T_0}\right]^{\textstyle \frac{M g}{R^* L}} Instead, assuming constant temperature, integrating gives the second barometric formula: P = P_0 e^{-M g z/R^*T} In this formulation, R* is the gas constant, and the term R*T/Mg gives the scale height (approximately equal to 8.4 km for the troposphere). The National Oceanic and Atmospheric Administration (NOAA) published the following formula for directly converting atmospheric pressure in millibars ( \mathrm{mb} ) to pressure altitude in feet ( \mathrm{ft} ): : h = 145366.45 \left[ 1 - \left( \frac{\text{Station pressure in millibars}}{1013.25} \right)^{0.190284} \right]. Atmospheric pressure decreases following the Barometric formula with altitude while the O2 fraction remains constant to about , so pO2 decreases with altitude as well. The first equation is applicable to the standard model of the troposphere in which the temperature is assumed to vary with altitude at a lapse rate of L_b: P = P_b \left[\frac{T_b - \left(h - h_b\right) L_b}{T_b}\right]^{\tfrac{g_0 M}{R^* L_b}} The second equation is applicable to the standard model of the stratosphere in which the temperature is assumed not to vary with altitude: P = P_b \exp \left[\frac{-g_0 M \left(h-h_b\right)}{R^* T_b}\right] where: *P_b = reference pressure *T_b = reference temperature (K) *L_b = temperature lapse rate (K/m) in ISA *h = height at which pressure is calculated (m) *h_b = height of reference level b (meters; e.g., hb = 11 000 m) *R^* = universal gas constant: 8.3144598 J/(mol·K) *g_0 = gravitational acceleration: 9.80665 m/s2 *M = molar mass of Earth's air: 0.0289644 kg/mol Or converted to imperial units:Mechtly, E. A., 1973: The International System of Units, Physical Constants and Conversion Factors. The relationship between static pressure and pressure altitude is defined in terms of properties of the ISA. ==See also== * QNH * Flight level * Cabin altitude * Density altitude * Standard conditions for temperature and pressure * Barometric formula ==References== Category:Altitudes in aviation (The total air mass below a certain altitude is calculated by integrating over the density function.) Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. Pressure altitude is the altitude in the International Standard Atmosphere (ISA) with the same atmospheric pressure as that of the part of the atmosphere in question. Subscript b Height above sea level Static pressure Standard temperature (K) Temperature lapse rate Exponent g0 M / R L (m) (ft) (Pa) (inHg) (K/m) (K/ft) 0 0 0 101 325.00 29.92126 288.15 0.0065 0.0019812 5.2558 1 11 000 36,089 22 632.10 6.683245 216.65 0.0 0.0 -- 2 20 000 65,617 5474.89 1.616734 216.65 -0.001 -0.0003048 -34.1626 3 32 000 104,987 868.02 0.2563258 228.65 -0.0028 -0.00085344 -12.2009 4 47 000 154,199 110.91 0.0327506 270.65 0.0 0.0 -- 5 51 000 167,323 66.94 0.01976704 270.65 0.0028 0.00085344 12.2009 6 71 000 232,940 3.96 0.00116833 214.65 0.002 0.0006096 17.0813 ==Density equations== The expressions for calculating density are nearly identical to calculating pressure. The concentration of oxygen (O2) in sea-level air is 20.9%, so the partial pressure of O2 (pO2) is . A reference atmospheric model describes how the ideal gas properties (namely: pressure, temperature, density, and molecular weight) of an atmosphere change, primarily as a function of altitude, and sometimes also as a function of latitude, day of year, etc. For example, the U.S. Standard Atmosphere derives the values for air temperature, pressure, and mass density, as a function of altitude above sea level. For the isothermal-barotropic model, density and pressure turn out to be exponential functions of altitude. The increase in altitude necessary for P or ρ to drop to 1/e of its initial value is called the scale height: :H = \frac{R T}{M g_0} where R is the ideal gas constant, T is temperature, M is average molecular weight, and g0 is the gravitational acceleration at the planet's surface. Image:Liquid ocean atmosphere model.png ===Isothermal-barotropic approximation and scale height=== This atmospheric model assumes both molecular weight and temperature are constant over a wide range of altitude. For example, if the airfield elevation is 500 ~ \mathrm{ft} and the altimeter setting is 29.32 ~ \mathrm{inHg} , then : \begin{align} \text{PA} & = 500 + 1000 \times (29.92 - 29.32) \\\ & = 500 + 1000 \times 0.6 \\\ & = 500 + 600 \\\ & = 1100. \end{align} Alternatively, : \text{Pressure altitude (PA)} = \text{Elevation} + 30 \times (1013 - \text{QNH}). Other static atmospheric models may have other outputs, or depend on inputs besides altitude. ==Basic assumptions== The gas which comprises an atmosphere is usually assumed to be an ideal gas, which is to say: : \rho = \frac{M P}{R T} Where ρ is mass density, M is average molecular weight, P is pressure, T is temperature, and R is the ideal gas constant. ",344,0,"""399.0""",0.72,0.66666666666,D
+A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What pressure indicates a temperature of $100.00^{\circ} \mathrm{C}$?,"By the 1940s, the triple point of water had been experimentally measured to be about 0.6% of standard atmospheric pressure and very close to 0.01 °C per the historical definition of Celsius then in use. The 13th CGPM also held in Resolution 4 that ""The kelvin, unit of thermodynamic temperature, is equal to the fraction of the thermodynamic temperature of the triple point of water."" The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. In 1954, with absolute zero having been experimentally determined to be about −273.15 °C per the definition of °C then in use, Resolution 3 of the 10th General Conference on Weights and Measures (CGPM) introduced a new internationally standardised Kelvin scale which defined the triple point as exactly 273.15 + 0.01 = 273.16 degrees Kelvin. The kelvin, symbol K, is a unit of measurement for temperature. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. In water, the critical point occurs at around Tc = , pc = and ρc = 356 kg/m3.The International Association for the Properties of Water and Steam ""Guideline on the Use of Fundamental Physical Constants and Basic Constants of Water"", 2001, p. 5 The existence of the liquid–gas critical point reveals a slight ambiguity in labelling the single phase regions. In the early decades of the 20th century, the Kelvin scale was often called the ""absolute Celsius"" scale, indicating Celsius degrees counted from absolute zero rather than the freezing point of water, and using the same symbol for regular Celsius degrees, °C.For example, Encyclopaedia Britannica editions from the 1920s and 1950s, one example being the article ""Planets"". === Triple point standard === In 1873, William Thomson's older brother James coined the term triple point to describe the combination of temperature and pressure at which the solid, liquid, and gas phases of a substance were capable of coexisting in thermodynamic equilibrium. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). In 1948, the Celsius scale was recalibrated by assigning the triple point temperature of water the value of 0.01 °C exactly and allowing the melting point at standard atmospheric pressure to have an empirically determined value (and the actual melting point at ambient pressure to have a fluctuating value) close to 0 °C. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. * The boiling point of water is 100 degrees. * The boiling point of water is 100 degrees. At constant pressure the maximum number of independent variables is three – the temperature and two concentration values. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. For practical purposes, the redefinition was unnoticed; water still freezes at 273.15 K (0 °C), and the triple point of water continues to be a commonly used laboratory reference temperature. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Historically, the Kelvin scale was developed from the Celsius scale, such that 273.15 K was 0 °C (the approximate melting point of ice) and a change of one kelvin was exactly equal to a change of one degree Celsius. The boiling point of water is the temperature at which the saturated vapour pressure equals the ambient pressure. English translation (extract). that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. ",1410,1.2,"""9.14""",420,1.4,C
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. An average human produces about $10 \mathrm{MJ}$ of heat each day through metabolic activity. Human bodies are actually open systems, and the main mechanism of heat loss is through the evaporation of water. What mass of water should be evaporated each day to maintain constant temperature?","The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). This will depend on the latent heat release as: T_e \approx T + \frac{L_v}{c_{pd}} r where: * L_v : latent heat of evaporation (2400 kJ/kg at 25°C to 2600 kJ/kg at −40°C) * c_{pd} : specific heat at constant pressure for air (≈ 1004 J/(kg·K)) Tables exist for exact values of the last two coefficients. == See also == * Wet-bulb temperature * Potential temperature * Atmospheric thermodynamics * Equivalent potential temperature ==Bibliography== * M Robitzsch, Aequivalenttemperatur und Aequivalentthemometer, Meteorologische Zeitschrift, 1928, pp. 313-315. The equation for evaporation given by Penman is: :E_{\mathrm{mass}}=\frac{m R_n + \rho_a c_p \left(\delta e \right) g_a }{\lambda_v \left(m + \gamma \right) } where: :m = Slope of the saturation vapor pressure curve (Pa K−1) :Rn = Net irradiance (W m−2) :ρa = density of air (kg m−3) :cp = heat capacity of air (J kg−1 K−1) :δe = vapor pressure deficit (Pa) :ga = momentum surface aerodynamic conductance (m s−1) :λv = latent heat of vaporization (J kg−1) :γ = psychrometric constant (Pa K−1) which (if the SI units in parentheses are used) will give the evaporation Emass in units of kg/(m2·s), kilograms of water evaporated every second for each square meter of area. Penman's equation requires daily mean temperature, wind speed, air pressure, and solar radiation to predict E. Simpler Hydrometeorological equations continue to be used where obtaining such data is impractical, to give comparable results within specific contexts, e.g. humid vs arid climates. ==Details== Numerous variations of the Penman equation are used to estimate evaporation from water, and land. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The normal human body temperature is often stated as . However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Human body temperature varies. Normal human body temperature varies slightly from person to person and by the time of day. Temperature, wind speed, relative humidity impact the values of m, g, cp, ρ, and δe. ==Shuttleworth (1993)== In 1993, W.Jim Shuttleworth modified and adapted the Penman equation to use SI, which made calculating evaporation simpler.Shuttleworth, J., Putting the vap' into evaporation http://www.hydrol-earth-syst-sci.net/11/210/2007/hess-11-210-2007.pdf The resultant equation is: :E_{\mathrm{mass}}=\frac{m R_n + \gamma * 6.43\left(1+0.536 * U_2 \right)\delta e}{\lambda_v \left(m + \gamma \right) } where: :Emass = Evaporation rate (mm day−1) :m = Slope of the saturation vapor pressure curve (kPa K−1) :Rn = Net irradiance (MJ m−2 day−1) :γ = psychrometric constant = \frac{0.0016286 * P_{kPa}} {\lambda_v} (kPa K−1) :U2 = wind speed (m s−1) :δe = vapor pressure deficit (kPa) :λv = latent heat of vaporization (MJ kg−1) Note: this formula implicitly includes the division of the numerator by the density of water (1000 kg m−3) to obtain evaporation in units of mm d−1 ==Some useful relationships== :δe = (es \- ea) = (1 – relative humidity) es :es = saturated vapor pressure of air, as is found inside plant stoma. :ea = vapor pressure of free flowing air. :es, mmHg = exp(21.07-5336/Ta), approximation by Merva, 1975Merva, G.E. 1975. * J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages Category:Atmospheric thermodynamics Category:Atmospheric temperature In humans, the average internal temperature is widely accepted to be 37 °C (98.6 °F), a ""normal"" temperature established in the 1800s. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate. === Thermodynamic expressions === Values based on approximations (particularly ) are in many cases not sufficiently accurate for practical engineering calculations, such as flow rates through pipes and valves at moderate to high pressures. Therefore, the heat capacity ratio in this example is 1.4. This equation assumes a daily time step so that net heat exchange with the ground is insignificant, and a unit area surrounded by similar open water or vegetation so that net heat & vapor exchange with the surrounding area cancels out. While some people think of these averages as representing normal or ideal measurements, a wide range of temperatures has been found in healthy people. An individual's body temperature typically changes by about between its highest and lowest points each day. ",537,7,"""4.09""",0.241,2.24,C
+A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What change of pressure indicates a change of $1.00 \mathrm{~K}$ at this temperature?,"Historically, the Kelvin scale was developed from the Celsius scale, such that 273.15 K was 0 °C (the approximate melting point of ice) and a change of one kelvin was exactly equal to a change of one degree Celsius. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. thumb|400px|Diagram showing pressure difference induced by a temperature difference. The kelvin, symbol K, is a unit of measurement for temperature. By the 1940s, the triple point of water had been experimentally measured to be about 0.6% of standard atmospheric pressure and very close to 0.01 °C per the historical definition of Celsius then in use. For practical purposes, the redefinition was unnoticed; water still freezes at 273.15 K (0 °C), and the triple point of water continues to be a commonly used laboratory reference temperature. In 1948, the Celsius scale was recalibrated by assigning the triple point temperature of water the value of 0.01 °C exactly and allowing the melting point at standard atmospheric pressure to have an empirically determined value (and the actual melting point at ambient pressure to have a fluctuating value) close to 0 °C. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). This value of ""−273"" was the negative reciprocal of 0.00366—the accepted coefficient of thermal expansion of an ideal gas per degree Celsius relative to the ice point, giving a remarkable consistency to the currently accepted value. In the early decades of the 20th century, the Kelvin scale was often called the ""absolute Celsius"" scale, indicating Celsius degrees counted from absolute zero rather than the freezing point of water, and using the same symbol for regular Celsius degrees, °C.For example, Encyclopaedia Britannica editions from the 1920s and 1950s, one example being the article ""Planets"". === Triple point standard === In 1873, William Thomson's older brother James coined the term triple point to describe the combination of temperature and pressure at which the solid, liquid, and gas phases of a substance were capable of coexisting in thermodynamic equilibrium. In 1954, with absolute zero having been experimentally determined to be about −273.15 °C per the definition of °C then in use, Resolution 3 of the 10th General Conference on Weights and Measures (CGPM) introduced a new internationally standardised Kelvin scale which defined the triple point as exactly 273.15 + 0.01 = 273.16 degrees Kelvin. The 13th CGPM also held in Resolution 4 that ""The kelvin, unit of thermodynamic temperature, is equal to the fraction of the thermodynamic temperature of the triple point of water."" This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. The Kelvin scale is an absolute scale, which is defined such that 0 K is absolute zero and a change of thermodynamic temperature by 1 kelvin corresponds to a change of thermal energy by . In 1967/1968, Resolution 3 of the 13th CGPM renamed the unit increment of thermodynamic temperature ""kelvin"", symbol K, replacing ""degree Kelvin"", symbol . English translation (extract). that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. In 2005, noting that the triple point could be influenced by the isotopic ratio of the hydrogen and oxygen making up a water sample and that this was ""now one of the major sources of the observed variability between different realizations of the water triple point"", the International Committee for Weights and Measures (CIPM), a committee of the CGPM, affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin would refer to water having the isotopic composition specified for Vienna Standard Mean Ocean Water. === 2019 redefinition === In 2005, the CIPM began a programme to redefine the kelvin (along with the other SI units) using a more experimentally rigorous method. At constant pressure the maximum number of independent variables is three – the temperature and two concentration values. ",-1270,0.0245,"""41.4""",0.375,1.81,B
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $229 \mathrm{~J}$ of energy is supplied as heat to $3.0 \mathrm{~mol} \mathrm{Ar}(\mathrm{g})$ at constant pressure, the temperature of the sample increases by $2.55 \mathrm{~K}$. Calculate the molar heat capacities at constant pressure of the gas.","Then the molar heat capacity (at constant volume) would be :cV,m = fR where R is the ideal gas constant. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.Lange's Handbook of Chemistry, 10th ed. p. 1524 All methods for the measurement of specific heat apply to molar heat capacity as well. ==Units== The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . The following is a table of some constant- pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* H2 28.9 5.0 29.6 5.1 41.2 7.9 Not saturated.""Hydrogen"" NIST Chemistry WebBook, SRD 69, online. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where ""amol"" denotes an amount of the solid that contains the Avogadro number of atoms. Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. The last value corresponds almost exactly to the predicted value for f = 7. right|thumb|upright=1.25|Constant-volume specific heat capacity of diatomic gases (real gases) between about 200 K and 2000 K. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. At high enough temperature, its molar heat capacity then should be cP,m = 7.5 R = 62.63 J⋅K−1⋅mol−1. Therefore, the word ""molar"", not ""specific"", should always be used for this quantity. ==Definition== The molar heat capacity of a substance, which may be denoted by cm, is the heat capacity C of a sample of the substance, divided by the amount (moles) n of the substance in the sample: :cm{} \;=\; \frac{C}{n} \;=\; \frac{1}{n} \lim_{\Delta T \rightarrow 0}\frac{Q}{\Delta T} where Q is the amount of heat needed to raise the temperature of the sample by ΔT. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. In other words, that theory predicts that the molar heat capacity at constant volume cV,m of all monatomic gases will be the same; specifically, :cV,m = R where R is the ideal gas constant, about 8.31446 J⋅K−1⋅mol−1 (which is the product of the Boltzmann constant kB and the Avogadro constant). According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). The same theory predicts that the molar heat capacity of a monatomic gas at constant pressure will be :cP,m = cV,m \+ R = R This prediction matches the experimental values, which, for helium through xenon, are 20.78, 20.79, 20.85, 20.95, and 21.01 J⋅K−1⋅mol−1, respectively;Shuen-Chen Hwang, Robert D. Lein, Daniel A. Morgan (2005). The molar heat capacity of the gas will then be determined only by the ""active"" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). ",-20,+11,"""17.7""",22,269,D
+Express the van der Waals parameters $b=0.0226 \mathrm{dm}^3 \mathrm{~mol}^{-1}$ in SI base units.,"* Analytical calculation of Van der Waals surfaces and volumes. Petitjean, On the Analytical Calculation of Van der Waals Surfaces and Volumes: Some Numerical Aspects, Journal of Computational Chemistry, Volume 15, Number 5, 1994, pp. 507–523. == External links == * VSAs for various molecules by Anton Antonov, The Wolfram Demonstrations Project, 2007. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. In Van der Waals' original derivation, given below, b' is four times the proper volume of the particle. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. Useful tabulated values of Van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will be seen to present different values for the Van der Waals radius of the same atom. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Tabulated values of van der Waals radii are obtained by taking a weighted mean of a number of different experimental values, and, for this reason, different tables will often have different values for the van der Waals radius of the same atom. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. Blaney and G. E. Ewing, Annual Review of Physical Chemistry, Vol. 27, pp. 553-586 (1976): Van der Waals Molecules. ",−2,5,"""0.41887902047""",0.000226,0.5,D
+A diving bell has an air space of $3.0 \mathrm{m}^3$ when on the deck of a boat. What is the volume of the air space when the bell has been lowered to a depth of $50 \mathrm{m}$? Take the mean density of sea water to be $1.025 \mathrm{g} \mathrm{cm}^{-3}$ and assume that the temperature is the same as on the surface.,"* Volume reduction of the air in an open bell due to increasing hydrostatic pressure as the bell is lowered is compensated. The bell is lowered into the water and to the working depth at a rate recommended by the decompression schedule, and which allows the divers to equalize comfortably. The bell is lowered through the water to working depth, so must be negatively buoyant. Each 10 metres (33 feet) of depth puts another atmosphere (1 bar, 14.7 psi, 101 kPa) of pressure on the hull, so at 300 metres (1,000 feet), the hull is withstanding thirty atmospheres (30 bar, 441 psi, 3,000 kPa) of water pressure. ===Test depth=== This is the maximum depth at which a submarine is permitted to operate under normal peacetime circumstances, and is tested during sea trials. Maximum Operating Depth (MOD) in metres sea water (msw) for pO2 1.2 to 1.6 MOD (msw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 390 190 123 90 70 66 57 47 40 34 30 22 17 13 10 8 6 1.5 365 178 115 84 65 61 53 44 37 32 28 20 15 11 9 7 5 1.4 340 165 107 78 60 57 48 40 34 29 25 18 13 10 8 6 4 1.3 315 153 98 71 55 52 44 36 31 26 23 16 12 9 6 4 3 1.2 290 140 90 65 50 47 40 33 28 23 20 14 10 7 5 3 2 These depths are rounded to the nearest metre. ==See also== * == Notes == == References == == Sources == * * * * Category:Dive planning Category:Underwater diving safety Category:Breathing gases Wet bells with an air space will have the air space topped up as the bell descends and the air is compressed by increasing hydrostatic pressure. The physics of the diving bell applies also to an underwater habitat equipped with a moon pool, which is like a diving bell enlarged to the size of a room or two, and with the water–air interface at the bottom confined to a section rather than forming the entire bottom of the structure. ===Wet bell=== thumb|upright|Open diving bell on a stern mounted launch and recovery system A wet bell is a platform for lowering and lifting divers to and from the underwater workplace, which has an air filled space, open at the bottom, where the divers can stand or sit with their heads out of the water. It transports this air to its diving bell to replenish the air supply in the bell. The pressure produced by depth in water, is converted to pressure in feet sea water (fsw) or metres sea water (msw) by multiplying with the appropriate conversion factor, 33 fsw per atm, or 10 msw per bar. Air is trapped inside the bell by pressure of the water at the interface. Of this total pressure which can be tolerated by the diver, 1 atmosphere is due to surface pressure of the Earth's air, and the rest is due to the depth in water. The bell is ballasted so as to remain upright in the water and to be negatively buoyant, so that it will sink even when full of air. If the divers are breathing from the bell airspace at the time, it may need to be vented with additional air to maintain a low carbon dioxide level. Adding pressurized gas ensures that the gas space within the bell remains at constant volume as the bell descends in the water. A diving bell is a rigid chamber used to transport divers from the surface to depth and back in open water, usually for the purpose of performing underwater work. The diving bell would be connected via the mating flange of an airlock to the deck decompression chamber or saturation system for transfer under pressure of the occupants. == Air-lock diving bells == thumb|Barge with air-lock diving bell for working on moorings Service vessel with diving bell which can be lowered to 10 m and accessed via airlock and a 2 m diameter access tube|thumb|right The air lock diving-bell plant was a purpose-built barge for the laying, examination and repair of moorings for battleships at Gibraltar harbour. So the 1 atmosphere or bar contributed by the air is subtracted to give the pressure due to the depth of water. In underwater diving activities such as saturation diving, technical diving and nitrox diving, the maximum operating depth (MOD) of a breathing gas is the depth below which the partial pressure of oxygen (pO2) of the gas mix exceeds an acceptable limit. For example, if a gas contains 36% oxygen and the maximum pO2 is 1.4 bar, the MOD (msw) is 10 msw/bar x [(1.4 bar / 0.36) − 1] = 28.9 msw. == Tables == Maximum Operating Depth (MOD) in feet sea water (fsw) for pO2 1.2 to 1.6 MOD (fsw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 1287 627 407 297 231 218 187 156 132 114 99 73 55 42 33 26 20 1.5 1205 586 380 276 215 203 173 144 122 105 91 66 50 38 29 22 17 1.4 1122 545 352 256 198 187 160 132 111 95 83 59 44 33 25 18 13 1.3 1040 503 325 235 182 171 146 120 101 86 74 53 39 28 21 15 10 1.2 957 462 297 215 165 156 132 108 91 77 66 46 33 24 17 11 7 These depths are rounded to the nearest foot. The internal pressure in the bell is usually kept at atmospheric pressure to minimise run time by eliminating the need for decompression, so the seal between the bell skirt and the submarine deck is critical to the safety of the operation. The 6 m × 4 m × 2.5 m bell is accessible through a 2 m diameter tube and an airlock. In 1689, Denis Papin suggested that the pressure and fresh air inside a diving bell could be maintained by a force pump or bellows. ",92, 35.91,"""0.05882352941""",0.5,8.87,D
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The change in the Gibbs energy of a certain constant-pressure process was found to fit the expression $\Delta G / \text{J}=-85.40+36.5(T / \text{K})$. Calculate the value of $\Delta S$ for the process.,"The Gibbs energy of any system is and an infinitesimal change in G, at constant temperature and pressure, yields :dG=dU+pdV-TdS. The Gibbs free energy is expressed as : G(p,T) = U + pV - TS = H - TS where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy. Moreover, we also have \begin{align} K_\text{eq} &= e^{-\frac{\Delta_\text{r} G^\circ}{RT}}, \\\ \Delta_\text{r} G^\circ &= -RT\left(\ln K_\text{eq}\right) = -2.303\,RT\left(\log_{10} K_\text{eq}\right), \end{align} which relates the equilibrium constant with Gibbs free energy. The equation is:Physical chemistry, P. W. Atkins, Oxford University Press, 1978, where H is the enthalpy, T the absolute temperature and G the Gibbs free energy of the system, all at constant pressure p. Integrating with respect to T (again p is constant) it becomes: : \frac{\Delta G^\ominus(T_2)}{T_2} - \frac{\Delta G^\ominus(T_1)}{T_1} = \Delta H^\ominus \left(\frac{1}{T_2} - \frac{1}{T_1}\right) This equation quickly enables the calculation of the Gibbs free energy change for a chemical reaction at any temperature T2 with knowledge of just the standard Gibbs free energy change of formation and the standard enthalpy change of formation for the individual components. This implies that at equilibrium Q_\text{r} = K_\text{eq} and \Delta_\text{r} G = 0. ==Standard Gibbs energy change of formation== Table of selected substancesCRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide. The equation reads:Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, :\left( \frac{\partial ( \Delta G^\ominus/T ) } {\partial T} \right)_p = - \frac {\Delta H^\ominus} {T^2} with ΔG as the change in Gibbs energy due to reaction, and ΔH as the enthalpy of reaction (often, but not necessarily, assumed to be independent of temperature). The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models The Gibbs free energy change , measured in joules in SI) is the maximum amount of non-volume expansion work that can be extracted from a closed system (one that can exchange heat and work with its surroundings, but not matter) at fixed temperature and pressure. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Substance (state) ΔfG° ΔfG° Substance (state) (kJ/mol) (kcal/mol) NO(g) 87.6 20.9 NO2(g) 51.3 12.3 N2O(g) 103.7 24.78 H2O(g) −228.6 −54.64 H2O(l) −237.1 −56.67 CO2(g) −394.4 −94.26 CO(g) −137.2 −32.79 CH4(g) −50.5 −12.1 C2H6(g) −32.0 −7.65 C3H8(g) −23.4 −5.59 C6H6(g) 129.7 29.76 C6H6(l) 124.5 31.00 The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). The equation states that the change in the G/T ratio at constant pressure as a result of an infinitesimally small change in temperature is a factor H/T2. ==Chemical reactions and work== The typical applications of this equation are to chemical reactions. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Then the molar mass of air is computed by M0 = R/Rair = . ==U.S. Standard Atmosphere== The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R∗ as: Part 1, p. 3, (Linked file is 17 Meg) :R∗ = = . Also, using the reaction isotherm equation,Chemistry, Matter, and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, that is :\frac{\Delta G^\ominus}{T} = -R \ln K which relates the Gibbs energy to a chemical equilibrium constant, the van 't Hoff equation can be derived.Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, Since the change in a system's Gibbs energy is equal to the maximum amount of non-expansion work that the system can do in a process, the Gibbs-Helmholtz equation may be used to estimate how much non-expansion work can be done by a chemical process as a function of temperature. As a necessary condition for the reaction to occur at constant temperature and pressure, ΔG must be smaller than the non- pressure-volume (non-pV, e.g. electrical) work, which is often equal to zero (then ΔG must be negative). One can think of ∆G as the amount of ""free"" or ""useful"" energy available to do non-pV work at constant temperature and pressure. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. Moran and Shapiro, Fundamentals of Engineering Thermodynamics, Wiley, 4th Ed, 2000 In case of air, using the perfect gas law and the standard sea-level conditions (SSL) (air density ρ0 = 1.225 kg/m3, temperature T0 = 288.15 K and pressure p0 = ), we have that Rair = P0/(ρ0T0) = . However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. ",1000,+107,"""0.70710678""",-36.5,+116.0,D
 "Assume that all gases are perfect and that data refer to 298 K unless otherwise stated. At $298 \mathrm{~K}$ the standard enthalpy of combustion of sucrose is $-5797 \mathrm{~kJ}$ $\mathrm{mol}^{-1}$ and the standard Gibbs energy of the reaction is $-6333 \mathrm{~kJ} \mathrm{~mol}^{-1}$.
-Estimate the additional non-expansion work that may be obtained by raising the temperature to blood temperature, $37^{\circ} \mathrm{C}$.","The Sugden Award is an annual award for contributions to combustion research. Part V - Evaluation of Models for the chemical source term"" Combustion and Flame, 127 2023 (2001). * 2000. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. T.C. Chew, K. N.C. Bray, R. E. Britter, ""Spatially Resolved Flamelet Statistics for Reaction Rate Modelling"" Combustion and Flame 80 65-82 (1990). * 1989. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The prize is awarded by the British Section of The Combustion Institute for the published paper with at least one British Section member as author, which makes the most significant contribution to combustion research. Combustion Theory and Modelling is a bimonthly peer-reviewed scientific journal covering research on combustion. K.M. Leung and R.P. Lindstedt, ""Detailed modelling of C1-C3 alkane diffusion flames"" Combustion and Flame 102 129-160 (1995). * 1994. Progress in Energy and Combustion Science is a bimonthly peer-reviewed review journal published by Elsevier. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. The theoretical bases of indirect calorimetry: a review."" :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles F.C. Lockwood, M. Costa and P. Costen, ""Detailed Measurements in a heavy fuel oil-fired furnace"" Combustion Science and Technology 77 1-26 (1991). * 1990. Metabolism. 1988 Mar;37(3):287-301. ==Scientific background== Indirect calorimetry measures O2 consumption and CO2 production. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics ""Measuring RMR with Indirect Calorimetry (IC)."" R.S.M. Chrystie, I.S. Burns, C.F. Kaminski ""Temperature response of an acoustically- forced turbulent lean premixed flame: A quantitative experimental determination"", Combustion Science and Technology, vol. 185, pp. 180–199, (2013). * 2012. J.F.Griffiths and B.J.Whitaker, ""Thermokinetic Interactions Leading to Knock during Homogeneous Charge Compression Ignition"", Combustion and Flame 131 386-399 (2002). * 2001. Balthasar and M. Kraft, ""A stochastic approach to calculate the particle size distribution function of soot particles in laminar premixed flames"" Combustion and Flame 133 289 (2003). * 2002. ",-100,0.0029,4.0,-21,0.000216,D
-The composition of the atmosphere is approximately 80 per cent nitrogen and 20 per cent oxygen by mass. At what height above the surface of the Earth would the atmosphere become 90 per cent nitrogen and 10 per cent oxygen by mass? Assume that the temperature of the atmosphere is constant at $25^{\circ} \mathrm{C}$. What is the pressure of the atmosphere at that height?,"Total atmospheric mass is 5.1480×1018 kg (1.135×1019 lb), about 2.5% less than would be inferred from the average sea level pressure and Earth's area of 51007.2 megahectares, this portion being displaced by Earth's mountainous terrain. Hp is 8.4km, but for different gasses (measuring their partial pressure), it is again different and depends upon molar mass, giving 8.7 for nitrogen, 7.6 for oxygen and 5.6 for carbon dioxide. ====Total content==== Further note that since g, Earth's gravitational acceleration, is approximately constant with altitude in the atmosphere, the pressure at height h is proportional to the integral of the density in the column above h, and therefore to the mass in the atmosphere above height h. For an isothermal atmosphere, (1-\frac{1}{e}) or about 63% of the total mass of the atmosphere exists between the planet's surface and one scale height. In summary, the mass of Earth's atmosphere is distributed approximately as follows:Lutgens, Frederick K. and Edward J. Tarbuck (1995) The Atmosphere, Prentice Hall, 6th ed., pp. 14–17, * 50% is below . * 90% is below . * 99.99997% is below , the Kármán line. The concentration of oxygen (O2) in sea-level air is 20.9%, so the partial pressure of O2 (pO2) is . Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. The partial pressure of dry air p_\text{d} is found considering partial pressure, resulting in: p_\text{d} = p - p_\text{v} where p simply denotes the observed absolute pressure. ==Variation with altitude== thumb|upright=2.0|Standard atmosphere: , , ===Troposphere=== To calculate the density of air as a function of altitude, one requires additional parameters. For the troposphere, the lowest part (~10 km) of the atmosphere, they are listed below, along with their values according to the International Standard Atmosphere, using for calculation the universal gas constant instead of the air specific constant: *p_0, sea level standard atmospheric pressure, 101325Pa *T_0, sea level standard temperature, 288.15K *g, earth-surface gravitational acceleration, 9.80665m/s2 *L, temperature lapse rate, 0.0065K/m *R, ideal (universal) gas constant, 8.31446J/(mol·K) *M, molar mass of dry air, 0.0289652kg/mol Temperature at altitude h meters above sea level is approximated by the following formula (only valid inside the troposphere, no more than ~18km above Earth's surface (and lower away from Equator)): T = T_0 - L h The pressure at altitude h is given by: p = p_0 \left(1 - \frac{L h}{T_0}\right)^\frac{g M}{R L} Density can then be calculated according to a molar form of the ideal gas law: \rho = \frac{p M}{R T} = \frac{p M}{R T_0 \left(1 - \frac{Lh}{T_0}\right)} = \frac{p_0 M}{R T_0} \left(1 - \frac{L h}{T_0} \right)^{\frac{g M}{R L} - 1} where: *M, molar mass *R, ideal gas constant *T, absolute temperature *p, absolute pressure Note that the density close to the ground is \rho_0 = \frac{p_0 M}{R T_0} It can be easily verified that the hydrostatic equation holds: \frac{dp}{dh} = -g\rho . ====Exponential approximation==== As the temperature varies with height inside the troposphere by less than 25%, \frac{Lh}{T_0} < 0.25 and one may approximate: \rho = \rho_0 e^{\left(\frac{g M}{R L} - 1\right) \ln \left(1 - \frac{L h}{T_0}\right)} \approx \rho_0 e^{-\left(\frac{g M}{R L} - 1\right)\frac{L h}{T_0}} = \rho_0 e^{-\left(\frac{g M h}{R T_0} - \frac{L h}{T_0}\right)} Thus: \rho \approx \rho_0 e^{-h/H_n} Which is identical to the isothermal solution, except that Hn, the height scale of the exponential fall for density (as well as for number density n), is not equal to RT0/gM as one would expect for an isothermal atmosphere, but rather: \frac{1}{H_n} = \frac{g M}{R T_0} - \frac{L}{T_0} Which gives Hn = 10.4km. Air composition, temperature, and atmospheric pressure vary with altitude. The average temperature of the atmosphere at Earth's surface is or , depending on the reference. ==Physical properties== ===Pressure and thickness=== The average atmospheric pressure at sea level is defined by the International Standard Atmosphere as . Atmospheric pressure is the total weight of the air above unit area at the point where the pressure is measured. As of 2023, by mole fraction (i.e., by number of molecules), dry air contains 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, and small amounts of other gases. Pressure altitude is the altitude in the International Standard Atmosphere (ISA) with the same atmospheric pressure as that of the part of the atmosphere in question. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. According to the American National Center for Atmospheric Research, ""The total mean mass of the atmosphere is 5.1480 kg with an annual range due to water vapor of 1.2 or 1.5 kg, depending on whether surface pressure or water vapor data are used; somewhat smaller than the previous estimate. Therefore the mass fraction of the troposphere out of all the atmosphere is given using the approximated formula for p: 1 - \frac{p(h = 11\text{ km})}{p_0} = 1 - \left(\frac{T(11\text{ km})}{T_0} \right)^\frac{g M}{R L} \approx 76\% For nitrogen, it is 75%, while for oxygen this is 79%, and for carbon dioxide, 88%. ===Tropopause=== Higher than the troposphere, at the tropopause, the temperature is approximately constant with altitude (up to ~20km) and is 220K. The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. Air pressure actually decreases exponentially with altitude, dropping by half every or by a factor of 1/e (0.368) every , (this is called the scale height) -- for altitudes out to around . The atmospheric pressure at the top of the stratosphere is roughly 1/1000 the pressure at sea level. For example, the U.S. Standard Atmosphere derives the values for air temperature, pressure, and mass density, as a function of altitude above sea level. Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as U: \begin{align} p &= p(U) e^{-\frac{h - U}{H_\text{TP}}} = p_0 \left(1 - \frac{L U}{T_0}\right)^\frac{g M}{R L} e^{-\frac{h - U}{H_\text{TP}}} \\\ \rho &= \rho(U) e^{-\frac{h - U}{H_\text{TP}}} = \rho_0 \left(1 - \frac{L U}{T_0}\right)^{\frac{g M}{R L} - 1} e^{-\frac{h - U}{H_\text{TP}}} \end{align} ==Composition== ==See also== *Air *Atmospheric drag *Lighter than air *Density *Atmosphere of Earth *International Standard Atmosphere *U.S. Standard Atmosphere *NRLMSISE-00 ==Notes== ==References== ==External links== *Conversions of density units ρ by Sengpielaudio *Air density and density altitude calculations and by Richard Shelquist *Air density calculations by Sengpielaudio (section under Speed of sound in humid air) *Air density calculator by Engineering design encyclopedia *Atmospheric pressure calculator by wolfdynamics *Air iTools - Air density calculator for mobile by JSyA *Revised formula for the density of moist air (CIPM-2007) by NIST Category:Atmospheric thermodynamics A The average mass of the atmosphere is about 5 quadrillion (5) tonnes or 1/1,200,000 the mass of Earth. ",0.0029,14,1.16,1.2,2,A
-"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate $\Delta S_\text{tot}$ when two copper blocks, each of mass $10.0 \mathrm{~kg}$, one at $100^{\circ} \mathrm{C}$ and the other at $0^{\circ} \mathrm{C}$, are placed in contact in an isolated container. The specific heat capacity of copper is $0.385 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$ and may be assumed constant over the temperature range involved.","Copper has a thermal conductivity of 231 Btu/(hr-ft-F). However, the thermal conductivity of stainless steel is 1/30th times than that of copper. Copper heat exchangers for improving indoor ait quality: Cooling season at Ft. Jackson. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). As the solution circulates through the copper header, the temperature rises. Therefore, the heat capacity ratio in this example is 1.4. Copper has many desirable properties for thermally efficient and durable heat exchangers. This article focuses on beneficial properties and common applications of copper in heat exchangers. First and foremost, copper is an excellent conductor of heat. Thermal conductivity of some common metals Metal Thermal conductivity (Btu/(hr-ft-F)) (W/(m•K)) Silver 247.87 429 Copper 231 399 Gold 183 316 Aluminium 136 235 Yellow brass 69.33 120 Cast iron 46.33 80.1 Stainless steel 8.1 14.0 Further information about the thermal conductivity of selected metals is available. ===Corrosion resistance=== Corrosion resistance is essential in heat transfer applications where fluids are involved, such as in hot water tanks, radiators, etc. Copper heat exchangers are the preferred material in these units because of their high thermal conductivity and ease of fabrication. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The formula is: :\alpha = \frac{ k }{ \rho c_{p} } where * is thermal conductivity (W/(m·K)) * is specific heat capacity (J/(kg·K)) * is density (kg/m3) Together, can be considered the volumetric heat capacity (J/(m3·K)). thumb|250px|The plot of the specific heat capacity versus temperature. The copper heat pipe transfers thermal energy from within the solar tube into a copper header. This means that copper's high thermal conductivity allows heat to pass through it quickly. Non-copper heat exchangers are also available. Copper has a 60% better thermal conductivity rating than aluminum and has almost 30 times more thermal conductivity than stainless steel. Part 1: Feasibility of usage in a temperate zone; Part 2: Demonstration of usage in a cold zone; Final report to the International Copper Association Ltd. New copper heat exchanger technologies for specific applications are also introduced. ==History== Heat exchangers using copper and its alloys have evolved along with heat transfer technologies over the past several hundred years. During the same time period, antimicrobial copper was able to limit bacterial loads associated with the copper heat exchanger fins by 99.99% and fungal loads by 99.74%.Michels, H. (2011). ",11000,0.59,169.0,+93.4,+3.03,D
-"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\Delta U$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Therefore, the heat capacity ratio in this example is 1.4. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. ",1.92,-57.2,2.35,30,1.2,C
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of formation of ethylbenzene is $-12.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate its standard enthalpy of combustion.","For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). However the standard enthalpy of combustion is readily measurable using bomb calorimetry. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. The standard enthalpy of formation is then determined using Hess's law. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). The standard enthalpy of formation, which has been determined for a vast number of substances, is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements, with all substances in their standard states. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). * Standard enthalpy of formation is the enthalpy change when one mole of any compound is formed from its constituent elements in their standard states. Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. Most values of standard thermochemical data are tabulated at either (25°C, 1 bar) or (25°C, 1 atm). For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. This method is only approximate, however, because a reported bond energy is only an average value for different molecules with bonds between the same elements.Petrucci, Harwood and Herring, pages 422–423 ==References== Category:Enthalpy Category:Thermochemistry Category:Thermodynamics pl:Standardowe molowe ciepło tworzenia The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Progress in Energy and Combustion Science is a bimonthly peer-reviewed review journal published by Elsevier. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. ",0.63,-4564.7,-2.0,7.58,-50,B
+Estimate the additional non-expansion work that may be obtained by raising the temperature to blood temperature, $37^{\circ} \mathrm{C}$.","The Sugden Award is an annual award for contributions to combustion research. Part V - Evaluation of Models for the chemical source term"" Combustion and Flame, 127 2023 (2001). * 2000. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. T.C. Chew, K. N.C. Bray, R. E. Britter, ""Spatially Resolved Flamelet Statistics for Reaction Rate Modelling"" Combustion and Flame 80 65-82 (1990). * 1989. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The prize is awarded by the British Section of The Combustion Institute for the published paper with at least one British Section member as author, which makes the most significant contribution to combustion research. Combustion Theory and Modelling is a bimonthly peer-reviewed scientific journal covering research on combustion. K.M. Leung and R.P. Lindstedt, ""Detailed modelling of C1-C3 alkane diffusion flames"" Combustion and Flame 102 129-160 (1995). * 1994. Progress in Energy and Combustion Science is a bimonthly peer-reviewed review journal published by Elsevier. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. The theoretical bases of indirect calorimetry: a review."" :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles F.C. Lockwood, M. Costa and P. Costen, ""Detailed Measurements in a heavy fuel oil-fired furnace"" Combustion Science and Technology 77 1-26 (1991). * 1990. Metabolism. 1988 Mar;37(3):287-301. ==Scientific background== Indirect calorimetry measures O2 consumption and CO2 production. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics ""Measuring RMR with Indirect Calorimetry (IC)."" R.S.M. Chrystie, I.S. Burns, C.F. Kaminski ""Temperature response of an acoustically- forced turbulent lean premixed flame: A quantitative experimental determination"", Combustion Science and Technology, vol. 185, pp. 180–199, (2013). * 2012. J.F.Griffiths and B.J.Whitaker, ""Thermokinetic Interactions Leading to Knock during Homogeneous Charge Compression Ignition"", Combustion and Flame 131 386-399 (2002). * 2001. Balthasar and M. Kraft, ""A stochastic approach to calculate the particle size distribution function of soot particles in laminar premixed flames"" Combustion and Flame 133 289 (2003). * 2002. ",-100,0.0029,"""4.0""",-21,0.000216,D
+The composition of the atmosphere is approximately 80 per cent nitrogen and 20 per cent oxygen by mass. At what height above the surface of the Earth would the atmosphere become 90 per cent nitrogen and 10 per cent oxygen by mass? Assume that the temperature of the atmosphere is constant at $25^{\circ} \mathrm{C}$. What is the pressure of the atmosphere at that height?,"Total atmospheric mass is 5.1480×1018 kg (1.135×1019 lb), about 2.5% less than would be inferred from the average sea level pressure and Earth's area of 51007.2 megahectares, this portion being displaced by Earth's mountainous terrain. Hp is 8.4km, but for different gasses (measuring their partial pressure), it is again different and depends upon molar mass, giving 8.7 for nitrogen, 7.6 for oxygen and 5.6 for carbon dioxide. ====Total content==== Further note that since g, Earth's gravitational acceleration, is approximately constant with altitude in the atmosphere, the pressure at height h is proportional to the integral of the density in the column above h, and therefore to the mass in the atmosphere above height h. For an isothermal atmosphere, (1-\frac{1}{e}) or about 63% of the total mass of the atmosphere exists between the planet's surface and one scale height. In summary, the mass of Earth's atmosphere is distributed approximately as follows:Lutgens, Frederick K. and Edward J. Tarbuck (1995) The Atmosphere, Prentice Hall, 6th ed., pp. 14–17, * 50% is below . * 90% is below . * 99.99997% is below , the Kármán line. The concentration of oxygen (O2) in sea-level air is 20.9%, so the partial pressure of O2 (pO2) is . Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. The partial pressure of dry air p_\text{d} is found considering partial pressure, resulting in: p_\text{d} = p - p_\text{v} where p simply denotes the observed absolute pressure. ==Variation with altitude== thumb|upright=2.0|Standard atmosphere: , , ===Troposphere=== To calculate the density of air as a function of altitude, one requires additional parameters. For the troposphere, the lowest part (~10 km) of the atmosphere, they are listed below, along with their values according to the International Standard Atmosphere, using for calculation the universal gas constant instead of the air specific constant: *p_0, sea level standard atmospheric pressure, 101325Pa *T_0, sea level standard temperature, 288.15K *g, earth-surface gravitational acceleration, 9.80665m/s2 *L, temperature lapse rate, 0.0065K/m *R, ideal (universal) gas constant, 8.31446J/(mol·K) *M, molar mass of dry air, 0.0289652kg/mol Temperature at altitude h meters above sea level is approximated by the following formula (only valid inside the troposphere, no more than ~18km above Earth's surface (and lower away from Equator)): T = T_0 - L h The pressure at altitude h is given by: p = p_0 \left(1 - \frac{L h}{T_0}\right)^\frac{g M}{R L} Density can then be calculated according to a molar form of the ideal gas law: \rho = \frac{p M}{R T} = \frac{p M}{R T_0 \left(1 - \frac{Lh}{T_0}\right)} = \frac{p_0 M}{R T_0} \left(1 - \frac{L h}{T_0} \right)^{\frac{g M}{R L} - 1} where: *M, molar mass *R, ideal gas constant *T, absolute temperature *p, absolute pressure Note that the density close to the ground is \rho_0 = \frac{p_0 M}{R T_0} It can be easily verified that the hydrostatic equation holds: \frac{dp}{dh} = -g\rho . ====Exponential approximation==== As the temperature varies with height inside the troposphere by less than 25%, \frac{Lh}{T_0} < 0.25 and one may approximate: \rho = \rho_0 e^{\left(\frac{g M}{R L} - 1\right) \ln \left(1 - \frac{L h}{T_0}\right)} \approx \rho_0 e^{-\left(\frac{g M}{R L} - 1\right)\frac{L h}{T_0}} = \rho_0 e^{-\left(\frac{g M h}{R T_0} - \frac{L h}{T_0}\right)} Thus: \rho \approx \rho_0 e^{-h/H_n} Which is identical to the isothermal solution, except that Hn, the height scale of the exponential fall for density (as well as for number density n), is not equal to RT0/gM as one would expect for an isothermal atmosphere, but rather: \frac{1}{H_n} = \frac{g M}{R T_0} - \frac{L}{T_0} Which gives Hn = 10.4km. Air composition, temperature, and atmospheric pressure vary with altitude. The average temperature of the atmosphere at Earth's surface is or , depending on the reference. ==Physical properties== ===Pressure and thickness=== The average atmospheric pressure at sea level is defined by the International Standard Atmosphere as . Atmospheric pressure is the total weight of the air above unit area at the point where the pressure is measured. As of 2023, by mole fraction (i.e., by number of molecules), dry air contains 78.08% nitrogen, 20.95% oxygen, 0.93% argon, 0.04% carbon dioxide, and small amounts of other gases. Pressure altitude is the altitude in the International Standard Atmosphere (ISA) with the same atmospheric pressure as that of the part of the atmosphere in question. *R_\text{specific}, the specific gas constant for dry air, which using the values presented above would be approximately in J⋅kg−1⋅K−1. According to the American National Center for Atmospheric Research, ""The total mean mass of the atmosphere is 5.1480 kg with an annual range due to water vapor of 1.2 or 1.5 kg, depending on whether surface pressure or water vapor data are used; somewhat smaller than the previous estimate. Therefore the mass fraction of the troposphere out of all the atmosphere is given using the approximated formula for p: 1 - \frac{p(h = 11\text{ km})}{p_0} = 1 - \left(\frac{T(11\text{ km})}{T_0} \right)^\frac{g M}{R L} \approx 76\% For nitrogen, it is 75%, while for oxygen this is 79%, and for carbon dioxide, 88%. ===Tropopause=== Higher than the troposphere, at the tropopause, the temperature is approximately constant with altitude (up to ~20km) and is 220K. The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. Air pressure actually decreases exponentially with altitude, dropping by half every or by a factor of 1/e (0.368) every , (this is called the scale height) -- for altitudes out to around . The atmospheric pressure at the top of the stratosphere is roughly 1/1000 the pressure at sea level. For example, the U.S. Standard Atmosphere derives the values for air temperature, pressure, and mass density, as a function of altitude above sea level. Both the pressure and density obey this law, so, denoting the height of the border between the troposphere and the tropopause as U: \begin{align} p &= p(U) e^{-\frac{h - U}{H_\text{TP}}} = p_0 \left(1 - \frac{L U}{T_0}\right)^\frac{g M}{R L} e^{-\frac{h - U}{H_\text{TP}}} \\\ \rho &= \rho(U) e^{-\frac{h - U}{H_\text{TP}}} = \rho_0 \left(1 - \frac{L U}{T_0}\right)^{\frac{g M}{R L} - 1} e^{-\frac{h - U}{H_\text{TP}}} \end{align} ==Composition== ==See also== *Air *Atmospheric drag *Lighter than air *Density *Atmosphere of Earth *International Standard Atmosphere *U.S. Standard Atmosphere *NRLMSISE-00 ==Notes== ==References== ==External links== *Conversions of density units ρ by Sengpielaudio *Air density and density altitude calculations and by Richard Shelquist *Air density calculations by Sengpielaudio (section under Speed of sound in humid air) *Air density calculator by Engineering design encyclopedia *Atmospheric pressure calculator by wolfdynamics *Air iTools - Air density calculator for mobile by JSyA *Revised formula for the density of moist air (CIPM-2007) by NIST Category:Atmospheric thermodynamics A The average mass of the atmosphere is about 5 quadrillion (5) tonnes or 1/1,200,000 the mass of Earth. ",0.0029,14,"""1.16""",1.2,2,A
+"Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. Calculate $\Delta S_\text{tot}$ when two copper blocks, each of mass $10.0 \mathrm{~kg}$, one at $100^{\circ} \mathrm{C}$ and the other at $0^{\circ} \mathrm{C}$, are placed in contact in an isolated container. The specific heat capacity of copper is $0.385 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$ and may be assumed constant over the temperature range involved.","Copper has a thermal conductivity of 231 Btu/(hr-ft-F). However, the thermal conductivity of stainless steel is 1/30th times than that of copper. Copper heat exchangers for improving indoor ait quality: Cooling season at Ft. Jackson. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). As the solution circulates through the copper header, the temperature rises. Therefore, the heat capacity ratio in this example is 1.4. Copper has many desirable properties for thermally efficient and durable heat exchangers. This article focuses on beneficial properties and common applications of copper in heat exchangers. First and foremost, copper is an excellent conductor of heat. Thermal conductivity of some common metals Metal Thermal conductivity (Btu/(hr-ft-F)) (W/(m•K)) Silver 247.87 429 Copper 231 399 Gold 183 316 Aluminium 136 235 Yellow brass 69.33 120 Cast iron 46.33 80.1 Stainless steel 8.1 14.0 Further information about the thermal conductivity of selected metals is available. ===Corrosion resistance=== Corrosion resistance is essential in heat transfer applications where fluids are involved, such as in hot water tanks, radiators, etc. Copper heat exchangers are the preferred material in these units because of their high thermal conductivity and ease of fabrication. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The formula is: :\alpha = \frac{ k }{ \rho c_{p} } where * is thermal conductivity (W/(m·K)) * is specific heat capacity (J/(kg·K)) * is density (kg/m3) Together, can be considered the volumetric heat capacity (J/(m3·K)). thumb|250px|The plot of the specific heat capacity versus temperature. The copper heat pipe transfers thermal energy from within the solar tube into a copper header. This means that copper's high thermal conductivity allows heat to pass through it quickly. Non-copper heat exchangers are also available. Copper has a 60% better thermal conductivity rating than aluminum and has almost 30 times more thermal conductivity than stainless steel. Part 1: Feasibility of usage in a temperate zone; Part 2: Demonstration of usage in a cold zone; Final report to the International Copper Association Ltd. New copper heat exchanger technologies for specific applications are also introduced. ==History== Heat exchangers using copper and its alloys have evolved along with heat transfer technologies over the past several hundred years. During the same time period, antimicrobial copper was able to limit bacterial loads associated with the copper heat exchanger fins by 99.99% and fungal loads by 99.74%.Michels, H. (2011). ",11000,0.59,"""169.0""",+93.4,+3.03,D
+"Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\Delta U$.","Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. ===Cubic equation=== The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: :{v_R^3}-\frac{1}{3}\left({1+\frac{8T_R}{p_R}}\right){v_R^2} +\frac{3}{p_R}v_R- \frac{1}{p_R}= 0 At the critical temperature, where T_R=p_R=1 we get as expected :{v_R^3}-3v_R^2 +3v_R- 1=\left(v_R -1\right)^3 = 0 \quad \Longleftrightarrow \quad v_R=1 For TR < 1, there are 3 values for vR. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). Therefore, the heat capacity ratio in this example is 1.4. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. ",1.92,-57.2,"""2.35""",30,1.2,C
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of formation of ethylbenzene is $-12.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate its standard enthalpy of combustion.","For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). However the standard enthalpy of combustion is readily measurable using bomb calorimetry. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. The standard enthalpy of formation is then determined using Hess's law. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). The standard enthalpy of formation, which has been determined for a vast number of substances, is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements, with all substances in their standard states. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). * Standard enthalpy of formation is the enthalpy change when one mole of any compound is formed from its constituent elements in their standard states. Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. Most values of standard thermochemical data are tabulated at either (25°C, 1 bar) or (25°C, 1 atm). For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. This method is only approximate, however, because a reported bond energy is only an average value for different molecules with bonds between the same elements.Petrucci, Harwood and Herring, pages 422–423 ==References== Category:Enthalpy Category:Thermochemistry Category:Thermodynamics pl:Standardowe molowe ciepło tworzenia The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Progress in Energy and Combustion Science is a bimonthly peer-reviewed review journal published by Elsevier. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. ",0.63,-4564.7,"""-2.0""",7.58,-50,B
 "A scientist proposed the following equation of state:
 $$
 p=\frac{R T}{V_{\mathrm{m}}}-\frac{B}{V_{\mathrm{m}}^2}+\frac{C}{V_{\mathrm{m}}^3}
 $$
-Show that the equation leads to critical behaviour. Find the critical constants of the gas in terms of $B$ and $C$ and an expression for the critical compression factor.","When the equation expressed in reduced form, an identical equation is obtained for all gases: : P_\text{r} = \frac{3 T_\text{r}}{V_\text{r} - b'} - \frac{1}{b' \sqrt{T_\text{r}} V_\text{r} \left(V_\text{r}+b'\right)} where b' is: : b' = 2^{1/3}-1 \approx 0.25992 In addition, the compressibility factor at the critical point is the same for every substance: : Z_\text{c}=\frac{p_\text{c} V_\text{c}}{R T_\text{c}}=1/3 \approx 0.33333 This is an improvement over the van der Waals equation prediction of the critical compressibility factor, which is Z_\text{c} = 3/8 = 0.375 . It predicts a value of 3/8 = 0.375 that is found to be an overestimate when compared to real gases. ==Compressibility factor at the critical point== The compressibility factor at the critical point, which is defined as Z_c=\frac{P_c v_c \mu}{R T_c}, where the subscript c indicates physical quantities measured at the critical point, is predicted to be a constant independent of substance by many equations of state. Here T_c and P_c are known as the critical temperature and critical pressure of a gas. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. Because interactions between large numbers of molecules are rare, the virial equation is usually truncated after the third term. page73 When this truncation is assumed, the compressibility factor is linked to the intermolecular-force potential φ by: :Z = 1 + 2\pi \frac{N_\text{A}}{V_\text{m}} \int_0^\infty \left(1 - \exp \left(\frac{\varphi}{kT}\right)\right) r^2 dr The Real gas article features more theoretical methods to compute compressibility factors. ==Physical mechanism of temperature and pressure dependence== Deviations of the compressibility factor, Z, from unity are due to attractive and repulsive intermolecular forces. By substituting the variables in the reduced form and the compressibility factor at critical point : \\{p_\text{r}=p/P_\text{c}, T_\text{r}=T/T_\text{c}, V_\text{r}=V_\text{m}/V_\text{c}, Z_\text{c}=\frac{P_\text{c} V_\text{c}}{R T_\text{c}}\\} we obtain : p_\text{r} P_\text{c} = \frac{R\,T_\text{r} T_\text{c}}{V_\text{r} V_\text{c}-b} - \frac{a \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}\left(V_\text{r} V_\text{c+}b\right)} = \frac{R\,T_\text{r} T_\text{c}}{V_\text{r} V_\text{c}-\frac{\Omega_b\,R T_\text{c}}{P_\text{c}}} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}\left(V_\text{r} V_\text{c}+\frac{\Omega_b\,R T_\text{c}}{P_\text{c}}\right)} = : = \frac{R\,T_\text{r} T_\text{c}}{V_\text{c}\left(V_\text{r}-\frac{\Omega_b\,R T_\text{c}}{P_\text{c} V_\text{c}}\right)} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}^2\left(V_\text{r}+\frac{\Omega_b\,R T_\text{c}}{P_\text{c} V_\text{c}}\right)} = \frac{R\,T_\text{r} T_\text{c}}{V_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}^2\left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} thus leading to : p_\text{r} = \frac{R\,T_\text{r} T_\text{c}}{P_\text{c} V_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}^2} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}^2\left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} = \frac{T_\text{r}}{Z_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a}{Z_\text{c}^2} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} \left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} Thus, the Soave–Redlich–Kwong equation in reduced form only depends on ω and Z_\text{c} of the substance, contrary to both the VdW and RK equation which are consistent with the theorem of corresponding states and the reduced form is one for all substances: : p_\text{r} = \frac{T_\text{r}}{Z_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a}{Z_\text{c}^2} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} \left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} We can also write it in the polynomial form, with: : A = \frac{a \alpha P}{R^2 T^2} : B = \frac{bP}{RT} In terms of the compressibility factor, we have: : 0 = Z^3-Z^2+Z\left(A-B-B^2\right) - AB. The substance-specific constants a and b can be calculated from the critical properties p_\text{c} and V_\text{c} (noting that V_\text{c} is the molar volume at the critical point and p_\text{c} is the critical pressure) as: : a = 3 p_\text{c} V_\text{c}^2 : b = \frac{V_\text{c}}{3}. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. The unique relationship between the compressibility factor and the reduced temperature, T_r, and the reduced pressure, P_r, was first recognized by Johannes Diderik van der Waals in 1873 and is known as the two-parameter principle of corresponding states. The equation is given below, as are relationships between its parameters and the critical constants: : \begin{align} p &= \frac{R\,T}{V_\text{m} - b} - \frac{a}{\sqrt{T}\,V_\text{m}\left(V_\text{m} + b\right)} \\\\[3pt] a &= \frac{\Omega_a\,R^2 T_\text{c}^\frac{5}{2}}{p_\text{c}} \approx 0.42748\frac{R^2\,T_\text{c}^\frac{5}{2}}{P_\text{c}} \\\\[3pt] b &= \frac{\Omega_b\,R T_\text{c}}{P_\text{c}} \approx 0.08664\frac{R\,T_\text{c}}{p_\text{c}} \\\\[3pt] \Omega_a &= \left[9\left(2^{1/3}-1\right)\right]^{-1} \approx 0.42748 \\\\[3pt] \Omega_b &= \frac{2^{1/3}-1}{3} \approx 0.08664 \end{align} Another, equivalent form of the Redlich–Kwong equation is the expression of the model's compressibility factor: : Z=\frac{p V_\text{m}}{RT} = \frac{V_\text{m}}{V_\text{m} - b} - \frac{a}{R T^{3/2} \left(V_\text{m} + b\right)} The Redlich–Kwong equation is adequate for calculation of gas phase properties when the reduced pressure (defined in the previous section) is less than about one-half of the ratio of the temperature to the reduced temperature, : P_\text{r} < \frac{T}{2T_\text{c}}. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document) In general, deviation from ideal behaviour becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure. Above the Boyle temperature, the compressibility factor is always greater than unity and increases slowly but steadily as pressure increases. ==Experimental values== It is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomes important. Expressions for (a,b) written as functions of (T_\text{c},p_\text{c}) may also be obtained and are often used to parameterize the equation because the critical temperature and pressure are readily accessible to experiment. Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume. Experimental values for the compressibility factor confirm this. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document) *Theorem of corresponding states on SklogWiki. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound- specific empirical constants as input. # The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density. # Gases deviate from ideal-gas behavior the most in the vicinity of the critical point. page 139 ==Theoretical models== The virial equation is especially useful to describe the causes of non-ideality at a molecular level (very few gases are mono-atomic) as it is derived directly from statistical mechanics: :Z = 1 + \frac{B}{V_\mathrm{m}} + \frac{C}{V_\mathrm{m}^2} + \frac{D}{V_\mathrm{m}^3} + \dots Where the coefficients in the numerator are known as virial coefficients and are functions of temperature. Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. ",0.333333,4.86,-114.4,0.5,+0.60,A
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of decomposition of the yellow complex $\mathrm{H}_3 \mathrm{NSO}_2$ into $\mathrm{NH}_3$ and $\mathrm{SO}_2$ is $+40 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the standard enthalpy of formation of $\mathrm{H}_3 \mathrm{NSO}_2$.","The standard enthalpy of formation is then determined using Hess's law. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. This is true for all enthalpies of formation. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. The Sabatier reaction or Sabatier process produces methane and water from a reaction of hydrogen with carbon dioxide at elevated temperatures (optimally 300–400 °C) and pressures (perhaps 3 MPa ) in the presence of a nickel catalyst. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. All elements in their reference states (oxygen gas, solid carbon in the form of graphite, etc.) have a standard enthalpy of formation of zero, as there is no change involved in their formation. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. thumb|The Mollier enthalpy–entropy diagram for water and steam. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Vaporization ===GME=== *Kugler HK & Keller C (eds) 1985, Gmelin handbook of inorganic and organometallic chemistry, 8th ed., ",4, 0.0024,0.0,-383,7.00,D
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of carbon dioxide of mass $2.45 \mathrm{~g}$ at $27.0^{\circ} \mathrm{C}$ is allowed to expand reversibly and adiabatically from $500 \mathrm{~cm}^3$ to $3.00 \mathrm{dm}^3$. What is the work done by the gas?","Such work done by compression is thermodynamic work as here defined. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Carbon dioxide reforming (also known as dry reforming) is a method of producing synthesis gas (mixtures of hydrogen and carbon monoxide) from the reaction of carbon dioxide with hydrocarbons such as methane with the aid of noble metal catalysts (typically Ni or Ni alloys). For a quasi-static adiabatic process, the change in internal energy is equal to minus the integral amount of work done by the system, so the work also depends only on the initial and final states of the process and is one and the same for every intermediate path. The work is due to change of system volume by expansion or contraction of the system. Changes in the potential energy of a body as a whole with respect to forces in its surroundings, and in the kinetic energy of the body moving as a whole with respect to its surroundings, are by definition excluded from the body's cardinal energy (examples are internal energy and enthalpy). ===Nearly reversible transfer of energy by work in the surroundings=== In the surroundings of a thermodynamic system, external to it, all the various mechanical and non-mechanical macroscopic forms of work can be converted into each other with no limitation in principle due to the laws of thermodynamics, so that the energy conversion efficiency can approach 100% in some cases; such conversion is required to be frictionless, and consequently adiabatic.F.C.Andrews Thermodynamics: Principles and Applications (Wiley- Interscience 1971), , p.17-18. To get an actual and precise physical measurement of a quantity of thermodynamic work, it is necessary to take account of the irreversibility by restoring the system to its initial condition by running a cycle, for example a Carnot cycle, that includes the target work as a step. Adiabatic work is done without matter transfer and without heat transfer. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. Synthesis gas is conventionally produced via the steam reforming reaction or coal gasification. In a process of transfer of energy from or to a thermodynamic system, the change of internal energy of the system is defined in theory by the amount of adiabatic work that would have been necessary to reach the final from the initial state, such adiabatic work being measurable only through the externally measurable mechanical or deformation variables of the system, that provide full information about the forces exerted by the surroundings on the system during the process. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. The quantity of thermodynamic work is defined as work done by the system on its surroundings. # The gasification process occurs as the char reacts with steam and carbon dioxide to produce carbon monoxide and hydrogen, via the reactions C + H2O → H2 \+ CO and C + CO2 → 2CO. Gasification is a process that converts biomass- or fossil fuel-based carbonaceous materials into gases, including as the largest fractions: nitrogen (N2), carbon monoxide (CO), hydrogen (H2), and carbon dioxide (). In particular, in principle, all macroscopic forms of work can be converted into the mechanical work of lifting a weight, which was the original form of thermodynamic work considered by Carnot and Joule (see History section above). Such work is adiabatic for the surroundings, even though it is associated with friction within the system. Several kinds of thermodynamic work are especially important. The dry reforming reaction may be represented by: :CH4 + CO2 <=>[975^oC] 2CO + 2H2 Thus, two greenhouse gases are consumed and useful chemical building blocks, hydrogen and carbon monoxide, are produced. ",21,-32,0.36,3.2, -194,E
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \mathrm{~kg}$ person to climb through $4.0 \mathrm{~m}$ on the surface of the Earth.","Other definitions which roughly produce the same numbers have been devised, such as: : \text{1 MET}\ = 1 \, \frac{\text{kcal}}{\text{kg}\times\text{h}}\ = 4.184 \, \frac{\text{kJ}}{\text{kg}\times\text{h}} = 1.162 \, \frac{\text{W}}{\text{kg}} where * kcal = kilocalorie * kg = kilogram * h = hour * kJ = kilojoule * W = watt ===Based on watts produced and body surface area=== Still another definition is based on the body surface area, BSA, and energy itself, where the BSA is expressed in m2: : \text{1 MET}\ = 58.2 \, \frac{\text{J}}{\text{s}\times\text{BSA}}\ = 58.2 \, \frac{\text{W}}{\text{m}^2} = 18.4 \, \frac{\text{Btu}}{\text{h}\times\text{ft}^2} which is equal to the rate of energy produced per unit surface area of an average person seated at rest. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. Health and fitness studies often bracket cohort activity levels in MET⋅hours/week. ==Quantitative definitions== ===Based on oxygen utilization and body mass=== The original definition of metabolic equivalent of task is the oxygen used by a person in milliliters per minute per kilogram body mass divided by 3.5. The metabolic equivalent of task (MET) is the objective measure of the ratio of the rate at which a person expends energy, relative to the mass of that person, while performing some specific physical activity compared to a reference, currently set by convention at an absolute 3.5 mL of oxygen per kg per minute, which is the energy expended when sitting quietly by a reference individual, chosen to be roughly representative of the general population, and thereby suited to epidemiological surveys. A 4 MET activity expends 4 times the energy used by the body at rest. By convention, 1 MET is considered equivalent to the consumption of 3.5 ml O2·kg−1·min−1 (or 3.5 ml of oxygen per kilogram of body mass per minute) and is roughly equivalent to the expenditure of 1 kcal per kilogram of body weight per hour. In the winter of 1980-1981 Tasker was part of an eight-man team (with Alan Rouse, John Porter, Brian Hall, Adrian Burgess, Alan Burgess, Pete Thexton and Paul Nunn) attempting to make a difficult winter assault on the West Face of Mount Everest; this was unsuccessful but was recounted in Tasker's first book Everest the Cruel Way. A person could also achieve 120 MET-minutes by doing an 8 MET activity for 15 minutes. RMR measurements by calorimetry in medical surveys have shown that the conventional 1-MET value overestimates the actual resting O2 consumption and energy expenditures by about 20% to 30% on the average; body composition (ratio of body fat to lean body mass) accounted for most of the variance. ==Standardized definition for research== The Compendium of Physical Activities was developed for use in epidemiologic studies to standardize the assignment of MET intensities in physical activity questionnaires. If a person does a 4 MET activity for 30 minutes, he or she has done 4 x 30 = 120 MET-minutes (or 2.0 MET-hours) of physical activity. One MET is defined as 1 kcal/kg/hour and is roughly equivalent to the energy cost of sitting quietly. Since the RMR of a person depends mainly on lean body mass (and not total weight) and other physiological factors such as health status and age, actual RMR (and thus 1-MET energy equivalents) may vary significantly from the kcal/(kg·h) rule of thumb. Joe Tasker (12 May 1948 – 17 May 1982) was a British climber, active during the late 1970s and early 1980s. For example, 1 MET is the rate of energy expenditure while at rest. A small team consisting of Tasker, Boardman, and Doug Scott made an ascent of Kangchenjunga (at 8,598 m the third highest mountain in the world) by a new route from the North-West in 1979 (with Georges Bettembourg also on the team but not making the summit); this was also the first ascent of the mountain without the use of supplementary oxygen. Tasker had delivered his manuscript for his second book, Savage Arena, which recounted his climbing life from the 1960s-1980, on the eve of his departure for the British Everest expedition in 1982. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. The BSA of an average person is 1.8 m2 (19 ft2). *At a vertex where both climbers are at peaks or both climbers are at valleys, the degree is four: both climbers may choose independently of each other which direction to go. *At the vertex (0,0), the degree is one: the only possible direction for both climbers to go is onto the mountain. Although the RMR of any person may deviate from the reference value, MET can be thought of as an index of the intensity of activities: for example, an activity with a MET value of 2, such as walking at a slow pace (e.g., 3 km/h) would require twice the energy that an average person consumes at rest (e.g., sitting quietly). ==Use== MET: The ratio of the work metabolic rate to the resting metabolic rate. A MET is the ratio of the rate of energy expended during an activity to the rate of energy expended at rest. ",2.0,2600,0.082,12,+87.8,B
-What pressure would $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert at $25^{\circ} \mathrm{C}$ if it behaved as a van der Waals gas?,"* ISO 11439: Compressed natural gas (CNG) cylinders. Further the volume of the gas is (4πr3)/3. Pressure vessels for gas storage may also be classified by volume. * IS 2825–1969 (RE1977)_code_unfired_Pressure_vessels. * EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC. VPT may refer to: * Valor por Tamaulipas * Vermont Public Television * Volume, temperature and pressure, the three parameters in the combined gas law Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America, the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSA B51 in Canada, Australian Standards in Australia and other international standards like Lloyd's, Germanischer Lloyd, Det Norske Veritas, Société Générale de Surveillance (SGS S.A.), Lloyd’s Register Energy Nederland (formerly known as Stoomwezen) etc. Note that where the pressure-volume product is part of a safety standard, any incompressible liquid in the vessel can be excluded as it does not contribute to the potential energy stored in the vessel, so only the volume of the compressible part such as gas is used. ===List of standards=== * EN 13445: The current European Standard, harmonized with the Pressure Equipment Directive (Originally ""97/23/EC"", since 2014 ""2014/68/EU""). The very small vessels used to make liquid butane fueled cigarette lighters are subjected to about 2 bar pressure, depending on ambient temperature. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. EN 13445 - Unfired Pressure Vessels is a standard that provides rules for the design, fabrication, and inspection of pressure vessels EN 13445 consists of 8 parts: * EN 13445-1 : Unfired pressure vessels - Part 1: General * EN 13445-2 : Unfired pressure vessels - Part 2: Materials * EN 13445-3 : Unfired pressure vessels - Part 3: Design * EN 13445-4 : Unfired pressure vessels - Part 4: Fabrication * EN 13445-5 : Unfired pressure vessels - Part 5: Inspection and testing * EN 13445-6 : Unfired pressure vessels - Part 6: Requirements for the design and fabrication of pressure vessels and pressure parts constructed from spheroidal graphite cast iron * EN 13445-8 : Unfired pressure vessels - Part 8: Additional requirements for pressure vessels of aluminium and aluminium alloys * EN 13445-10:2015 : Unfired pressure vessels - Part 10: Additional requirements for pressure vessels of nickel and nickel alloys. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. From about 1975 until now, the standard pressure is . Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. ""Pressure Vessel Handbook, 14th Edition."" High-pressure gas cylinders are also called bottles. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus :M = {3 \over 2} nRT {\rho \over \sigma}. (see gas law) The other factors are constant for a given vessel shape and material. * ASME Boiler and Pressure Vessel Code Section VIII: Rules for Construction of Pressure Vessels. ",22,0.0182,1.25,3.07,260,A
-A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What change of pressure indicates a change of $1.00 \mathrm{~K}$ at the latter temperature?,":V \propto T\, or :\frac{V}{T}=k V is the volume, T is the thermodynamic temperature, k is the constant for the system. k is not a fixed constant across all systems and therefore needs to be found experimentally for a given system through testing with known temperature values. ==Pressure Thermometer and Absolute Zero== thumb|left|180px|Plots of pressure vs temperature for three different gas samples extrapolate to absolute zero. thumb|400px|Diagram showing pressure difference induced by a temperature difference. Historically, the Kelvin scale was developed from the Celsius scale, such that 273.15 K was 0 °C (the approximate melting point of ice) and a change of one kelvin was exactly equal to a change of one degree Celsius. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. The kelvin, symbol K, is a unit of measurement for temperature. By the 1940s, the triple point of water had been experimentally measured to be about 0.6% of standard atmospheric pressure and very close to 0.01 °C per the historical definition of Celsius then in use. For practical purposes, the redefinition was unnoticed; water still freezes at 273.15 K (0 °C), and the triple point of water continues to be a commonly used laboratory reference temperature. In 1948, the Celsius scale was recalibrated by assigning the triple point temperature of water the value of 0.01 °C exactly and allowing the melting point at standard atmospheric pressure to have an empirically determined value (and the actual melting point at ambient pressure to have a fluctuating value) close to 0 °C. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. In the early decades of the 20th century, the Kelvin scale was often called the ""absolute Celsius"" scale, indicating Celsius degrees counted from absolute zero rather than the freezing point of water, and using the same symbol for regular Celsius degrees, °C.For example, Encyclopaedia Britannica editions from the 1920s and 1950s, one example being the article ""Planets"". === Triple point standard === In 1873, William Thomson's older brother James coined the term triple point to describe the combination of temperature and pressure at which the solid, liquid, and gas phases of a substance were capable of coexisting in thermodynamic equilibrium. The 13th CGPM also held in Resolution 4 that ""The kelvin, unit of thermodynamic temperature, is equal to the fraction of the thermodynamic temperature of the triple point of water."" In 1967/1968, Resolution 3 of the 13th CGPM renamed the unit increment of thermodynamic temperature ""kelvin"", symbol K, replacing ""degree Kelvin"", symbol . English translation (extract). that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. In 2005, noting that the triple point could be influenced by the isotopic ratio of the hydrogen and oxygen making up a water sample and that this was ""now one of the major sources of the observed variability between different realizations of the water triple point"", the International Committee for Weights and Measures (CIPM), a committee of the CGPM, affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin would refer to water having the isotopic composition specified for Vienna Standard Mean Ocean Water. === 2019 redefinition === In 2005, the CIPM began a programme to redefine the kelvin (along with the other SI units) using a more experimentally rigorous method. Thus an increment of 1 °C equals of the temperature difference between the melting and boiling points. To the extent that the gas is ideal, the pressure depends linearly on temperature, and the extrapolation to zero pressure occurs at absolute zero. The Kelvin scale is an absolute scale, which is defined such that 0 K is absolute zero and a change of thermodynamic temperature by 1 kelvin corresponds to a change of thermal energy by . In 1954, with absolute zero having been experimentally determined to be about −273.15 °C per the definition of °C then in use, Resolution 3 of the 10th General Conference on Weights and Measures (CGPM) introduced a new internationally standardised Kelvin scale which defined the triple point as exactly 273.15 + 0.01 = 273.16 degrees Kelvin. This value of ""−273"" was the negative reciprocal of 0.00366—the accepted coefficient of thermal expansion of an ideal gas per degree Celsius relative to the ice point, giving a remarkable consistency to the currently accepted value. The pressure melting point of ice is the temperature at which ice melts at a given pressure. ",48,16,2.0,0.0182,0.0245,E
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \mathrm{~kg}$ person to climb through $4.0 \mathrm{~m}$ on the surface of the Moon $\left(g=1.60 \mathrm{~m} \mathrm{~s}^{-2}\right)$.","The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The mass of the Moon is M = 7.3458 × 1022 kg and the mean density is 3346 kg/m3. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. thumb|Apollo astronauts work on the Moon to collect samples and explore. The lunar GM = 4902.8001 km3/s2 from GRAIL analyses. Consequently, it is conventional to express the lunar mass M multiplied by the gravitational constant G. The atmosphere of the Moon is a very scant presence of gases surrounding the Moon. Another important source is the bombardment of the lunar surface by micrometeorites, the solar wind, and sunlight, in a process known as sputtering. == Escape velocity and atmospheric hold == Gases can: * be re-implanted into the regolith as a result of the Moon's gravity; * escape the Moon entirely if the particle is moving at or above the lunar escape velocity of , or ; * be lost to space either by solar radiation pressure or, if the gases are ionized, by being swept away in the solar wind's magnetic field. == Composition == What little atmosphere the Moon has consists of some unusual gases, including sodium and potassium, which are not found in the atmospheres of Earth, Mars, or Venus. Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% (= 1/6) of what they weigh on the Earth. ==Gravitational field== The gravitational field of the Moon has been measured by tracking the radio signals emitted by orbiting spacecraft. The ancient lunar atmosphere was eventually stripped away by solar winds and dissipated into space. == See also == * Atmosphere of Mercury * Exosphere * Lunar Atmosphere and Dust Environment Explorer (LADEE) * Orders of magnitude (pressure) * Sodium tail of the Moon == References == Category:Lunar science Moon Moon The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. thumb|upright=1.5|At the Swamp Works, a sculpture made of lunar soil simulant representing construction on the Moon by robots working together with humans. The average daytime abundances of the elements known to be present in the lunar atmosphere, in atoms per cubic centimeter, are as follows: *Argon: 20,000–100,000 *Helium: 5,000–30,000 *Neon: up to 20,000 *Sodium: 70 *Potassium: 17 *Hydrogen: fewer than 17 This yields approximately 80,000 total atoms per cubic centimeter, marginally higher than the quantity posited to exist in the atmosphere of Mercury. It returned approximately of Lunar surface material. The building is where Apollo astronauts practiced working with the Lunar Module for lunar landings and extravehicular activities. thumb|Students traverse a simulated crater in a moonbuggy they designed and built themselves. The NASA Human Exploration Rover Challenge, prior to 2014 referred to as the Great Moonbuggy Race, is an annual competition for high school and college students to design, build, and race human-powered, collapsible vehicles over simulated lunar/Martian terrain. The obstacles are constructed of discarded tires, plywood, some 20 tons of gravel and five tons of sand, all to simulate lunar craters, basins, and rilles. * The moonbuggy (pre-2014) must fit into a cube and be no more than 4 ft wide. Roger Joseph Boscovich was the first modern astronomer to argue for the lack of atmosphere around the Moon in his De lunae atmosphaera (1753). == Sources == One source of the lunar atmosphere is outgassing: the release of gases such as radon and helium resulting from radioactive decay within the crust and mantle. The C31 coefficient is large. ==Simulating lunar gravity== In January 2022 China was reported by the South China Morning Post to have built a small (60 centimeters in diameter) research facility to simulate low lunar gravity with the help of magnets. National Academies Press, Washington, DC (1997). ==Sample- return missions== ===First missions=== thumb|Lunar sample 60016 on display at Space Center Houston Lunar Samples Vault, at NASA's Johnson Space Center The Apollo program returned over of lunar rocks and regolith (including lunar 'soil') to the Lunar Receiving Laboratory in Houston.Orloff 2004, ""Extravehicular Activity"" Today, 75% of the samples are stored at the Lunar Sample Laboratory Facility built in 1979. ",1.22,0.245,420.0,4,0.264,C
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $120 \mathrm{mg}$ of naphthalene, $\mathrm{C}_{10} \mathrm{H}_8(\mathrm{~s})$, was burned in a bomb calorimeter the temperature rose by $3.05 \mathrm{~K}$. By how much will the temperature rise when $10 \mathrm{mg}$ of phenol, $\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}(\mathrm{s})$, is burned in the calorimeter under the same conditions?","The high heat values are conventionally measured with a bomb calorimeter. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. That is, the heat of combustion, ΔH°comb, is the heat of reaction of the following process: : (std.) + (c + - ) (g) → c (g) + (l) + (g) Chlorine and sulfur are not quite standardized; they are usually assumed to convert to hydrogen chloride gas and or gas, respectively, or to dilute aqueous hydrochloric and sulfuric acids, respectively, when the combustion is conducted in a bomb calorimeter containing some quantity of water. == Ways of determination == ===Gross and net=== Zwolinski and Wilhoit defined, in 1972, ""gross"" and ""net"" values for heats of combustion. * However, for true energy calculations in some specific cases, the higher heating value is correct. They may also be calculated as the difference between the heat of formation ΔH of the products and reactants (though this approach is somewhat artificial since most heats of formation are typically calculated from measured heats of combustion). Since the heat of combustion of these elements is known, the heating value can be calculated using Dulong's Formula: LHV [kJ/g]= 33.87mC \+ 122.3(mH \- mO ÷ 8) + 9.4mS where mC, mH, mO, mN, and mS are the contents of carbon, hydrogen, oxygen, nitrogen, and sulfur on any (wet, dry or ash free) basis, respectively. === Higher heating value === The higher heating value (HHV; gross energy, upper heating value, gross calorific value GCV, or higher calorific value; HCV) indicates the upper limit of the available thermal energy produced by a complete combustion of fuel. Among the variables affecting burn rate are pressure and temperature. The calorific value is the total energy released as heat when a substance undergoes complete combustion with oxygen under standard conditions. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics This value is important for fuels like wood or coal, which will usually contain some amount of water prior to burning. == Measuring heating values == The higher heating value is experimentally determined in a bomb calorimeter. Incomplete reaction at higher temperature further curtails the effect of a larger heat of combustion. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). The temperatures mentioned here are for a stoichiometric fuel-oxidizer mixture (i.e. equivalence ratio φ = 1). It may be expressed with the quantities: * energy/mole of fuel * energy/mass of fuel * energy/volume of the fuel There are two kinds of enthalpy of combustion, called high(er) and low(er) heat(ing) value, depending on how much the products are allowed to cool and whether compounds like are allowed to condense. This table is in Standard cubic metres (1atm, 15°C), to convert to values per Normal cubic metre (1atm, 0°C), multiply above table by 1.0549. == See also == * Adiabatic flame temperature * Cost of electricity by source * Electrical efficiency * Energy content of fuel * Energy conversion efficiency * Energy density * Energy value of coal * Exothermic reaction * Figure of merit * Fire * Food energy * Internal energy * ISO 15971 * Mechanical efficiency * Thermal efficiency * Wobbe index: heat density == References == ==Further reading== * == External links == * NIST Chemistry WebBook * Category:Engineering thermodynamics Category:Combustion Category:Fuels Category:Thermodynamic properties Category:Nuclear physics Category:Thermochemistry == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. thumb|upright|Ethanol burning with its spectrum depicted In the study of combustion, the adiabatic flame temperature is the temperature reached by a flame under ideal conditions. The definition in which the combustion products are all returned to the reference temperature is more easily calculated from the higher heating value than when using other definitions and will in fact give a slightly different answer. === Gross heating value === Gross heating value accounts for water in the exhaust leaving as vapor, as does LHV, but gross heating value also includes liquid water in the fuel prior to combustion. This result can be explained through Le Chatelier's principle. ==See also== *Flame speed ==References== == External links == === General information === * * Computation of adiabatic flame temperature * Adiabatic flame temperature === Tables === * adiabatic flame temperature of hydrogen, methane, propane and octane with oxygen or air as oxidizers * * Temperature of a blue flame and common materials === Calculators === * Online adiabatic flame temperature calculator using Cantera * Adiabatic flame temperature program * Gaseq, program for performing chemical equilibrium calculations. ",1.6,205,46.7,5.9,0.38,B
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from $57.4 \mathrm{kPa}$ and $1.0 \mathrm{dm}^3$ to a final volume of $2.0 \mathrm{dm}^3$. Take $\gamma=1.4$.","The solubility turned out to be very low: from 0.02 to 0.10 %. thumb|Carbon dioxide pressure-temperature phase diagram == Uses == thumb|219x219px|Fire extinguisher Uses of liquid carbon dioxide include the preservation of food, in fire extinguishers, and in commercial food processes. Liquid carbon dioxide is a type of liquid which is formed from highly compressed and cooled gaseous carbon dioxide. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. Accumulation of carbon dioxide is predominantly a result of hypoperfusion and not hypoxia. At this temperature, the pressure is measured in a range from 15 to 60 atmospheres. The solubility of water in liquid carbon dioxide is measured in a range of temperatures, ranging from to . thumb|Jets of liquid carbon dioxide Liquid carbon dioxide is the liquid state of carbon dioxide (), which cannot occur under atmospheric pressure. Normocapnia or normocarbia is a state of normal arterial carbon dioxide pressure, usually about 40 mmHg. == See also == * * * == References == *The Free Dictionary - Normocapnia Category:Diving medicine Category:Pulmonology Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. According to the model, each exhalatory segment of capnogram waveform follows the analytical expression: p_D(t) = p_A (1 - e ^{-\alpha}e^{\alpha e^{-t/\tau}}) where * p_D(t)represents the partial pressure of carbon dioxide measured by the capnogram as a function of time t since the beginning of exhalation. * p_Arepresents the alveolar partial pressure of carbon dioxide. * \alpharepresents the inverse of the dead space fraction (i.e. the ratio of tidal volume to dead space volume). * \taurepresents the pulmonary time constant (i.e. the product of pulmonary resistance and compliance) In particular, this model explains the rounded ""shark-fin"" shape of the capnogram observed in patients with obstructive lung disease. == See also == * Integrated pulmonary index * Medical equipment * Medical test * Respiratory monitoring * Colorimetric capnography == Citations == == External links == * CapnoBase.org: Respiratory signal database that contains clinical and simulated capnogram recordings Category:Anesthesia Category:Breath tests Category:Diagnostic emergency medicine Category:Diagnostic intensive care medicine Category:Diagnostic pulmonology However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). In healthy individuals, the difference between arterial blood and expired gas partial pressures is very small (normal difference 4-5 mmHg). Low-temperature carbon dioxide is commercially used in its solid form, commonly known as ""dry ice"". Gastric tonometry describes the measurement of the carbon dioxide level inside the stomach in order to assess the degree of blood flow to the stomach and bowel. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. Because the introduction of a nasogastric tube is almost routine in critically ill patients, the measurement of gastric carbon dioxide can be an easy method to monitor tissue perfusion. Capnography is the monitoring of the concentration or partial pressure of carbon dioxide () in the respiratory gases. For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). In this process known as high pressure ANG, a high pressure CNG tank is filled by absorbers such as activated carbon (which is an adsorbent with high surface area) and stores natural gas by both CNG and ANG mechanisms. ",0.22222222,1.1,4.86,22,49,D
-"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $229 \mathrm{~J}$ of energy is supplied as heat to $3.0 \mathrm{~mol} \mathrm{Ar}(\mathrm{g})$ at constant pressure, the temperature of the sample increases by $2.55 \mathrm{~K}$. Calculate the molar heat capacities at constant volume.","Then the molar heat capacity (at constant volume) would be :cV,m = fR where R is the ideal gas constant. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.Lange's Handbook of Chemistry, 10th ed. p. 1524 All methods for the measurement of specific heat apply to molar heat capacity as well. ==Units== The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where ""amol"" denotes an amount of the solid that contains the Avogadro number of atoms. The following is a table of some constant- pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* H2 28.9 5.0 29.6 5.1 41.2 7.9 Not saturated.""Hydrogen"" NIST Chemistry WebBook, SRD 69, online. Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. Therefore, the word ""molar"", not ""specific"", should always be used for this quantity. ==Definition== The molar heat capacity of a substance, which may be denoted by cm, is the heat capacity C of a sample of the substance, divided by the amount (moles) n of the substance in the sample: :cm{} \;=\; \frac{C}{n} \;=\; \frac{1}{n} \lim_{\Delta T \rightarrow 0}\frac{Q}{\Delta T} where Q is the amount of heat needed to raise the temperature of the sample by ΔT. The last value corresponds almost exactly to the predicted value for f = 7. right|thumb|upright=1.25|Constant-volume specific heat capacity of diatomic gases (real gases) between about 200 K and 2000 K. In other words, that theory predicts that the molar heat capacity at constant volume cV,m of all monatomic gases will be the same; specifically, :cV,m = R where R is the ideal gas constant, about 8.31446 J⋅K−1⋅mol−1 (which is the product of the Boltzmann constant kB and the Avogadro constant). At high enough temperature, its molar heat capacity then should be cP,m = 7.5 R = 62.63 J⋅K−1⋅mol−1. The molar heat capacity of the gas will then be determined only by the ""active"" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Therefore, the heat capacity of a sample of a solid substance is expected to be 3RNa, or (24.94 J/K)Na, where Na is the number of moles of atoms in the sample, not molecules. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). In modern terms, Dulong and Petit found that the heat capacity of a mole of many solid elements is about 3R, where R is the universal gas constant. The same theory predicts that the molar heat capacity of a monatomic gas at constant pressure will be :cP,m = cV,m \+ R = R This prediction matches the experimental values, which, for helium through xenon, are 20.78, 20.79, 20.85, 20.95, and 21.01 J⋅K−1⋅mol−1, respectively;Shuen-Chen Hwang, Robert D. Lein, Daniel A. Morgan (2005). ", 7.42,0.0526315789,30.0,+3.60,4,C
-Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The fugacity coefficient of a certain gas at $200 \mathrm{~K}$ and 50 bar is 0.72. Calculate the difference of its molar Gibbs energy from that of a perfect gas in the same state.,"Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to . The contribution of nonideality to the molar Gibbs energy of a real gas is equal to . For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The fugacity coefficient is . In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. In addition, there are natural reference states for fugacity (for example, an ideal gas makes a natural reference state for gas mixtures since the fugacity and pressure converge at low pressure). ===Gases=== In a mixture of gases, the fugacity of each component has a similar definition, with partial molar quantities instead of molar quantities (e.g., instead of and instead of ): dG_i = R T \,d \ln f_i and \lim_{P\to 0} \frac{f_i}{P_i} = 1, where is the partial pressure of component . The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. ",7.136,0.000216,0.0,3.8,-0.55,E
+Show that the equation leads to critical behaviour. Find the critical constants of the gas in terms of $B$ and $C$ and an expression for the critical compression factor.","When the equation expressed in reduced form, an identical equation is obtained for all gases: : P_\text{r} = \frac{3 T_\text{r}}{V_\text{r} - b'} - \frac{1}{b' \sqrt{T_\text{r}} V_\text{r} \left(V_\text{r}+b'\right)} where b' is: : b' = 2^{1/3}-1 \approx 0.25992 In addition, the compressibility factor at the critical point is the same for every substance: : Z_\text{c}=\frac{p_\text{c} V_\text{c}}{R T_\text{c}}=1/3 \approx 0.33333 This is an improvement over the van der Waals equation prediction of the critical compressibility factor, which is Z_\text{c} = 3/8 = 0.375 . It predicts a value of 3/8 = 0.375 that is found to be an overestimate when compared to real gases. ==Compressibility factor at the critical point== The compressibility factor at the critical point, which is defined as Z_c=\frac{P_c v_c \mu}{R T_c}, where the subscript c indicates physical quantities measured at the critical point, is predicted to be a constant independent of substance by many equations of state. Here T_c and P_c are known as the critical temperature and critical pressure of a gas. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. Because interactions between large numbers of molecules are rare, the virial equation is usually truncated after the third term. page73 When this truncation is assumed, the compressibility factor is linked to the intermolecular-force potential φ by: :Z = 1 + 2\pi \frac{N_\text{A}}{V_\text{m}} \int_0^\infty \left(1 - \exp \left(\frac{\varphi}{kT}\right)\right) r^2 dr The Real gas article features more theoretical methods to compute compressibility factors. ==Physical mechanism of temperature and pressure dependence== Deviations of the compressibility factor, Z, from unity are due to attractive and repulsive intermolecular forces. By substituting the variables in the reduced form and the compressibility factor at critical point : \\{p_\text{r}=p/P_\text{c}, T_\text{r}=T/T_\text{c}, V_\text{r}=V_\text{m}/V_\text{c}, Z_\text{c}=\frac{P_\text{c} V_\text{c}}{R T_\text{c}}\\} we obtain : p_\text{r} P_\text{c} = \frac{R\,T_\text{r} T_\text{c}}{V_\text{r} V_\text{c}-b} - \frac{a \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}\left(V_\text{r} V_\text{c+}b\right)} = \frac{R\,T_\text{r} T_\text{c}}{V_\text{r} V_\text{c}-\frac{\Omega_b\,R T_\text{c}}{P_\text{c}}} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}\left(V_\text{r} V_\text{c}+\frac{\Omega_b\,R T_\text{c}}{P_\text{c}}\right)} = : = \frac{R\,T_\text{r} T_\text{c}}{V_\text{c}\left(V_\text{r}-\frac{\Omega_b\,R T_\text{c}}{P_\text{c} V_\text{c}}\right)} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}^2\left(V_\text{r}+\frac{\Omega_b\,R T_\text{c}}{P_\text{c} V_\text{c}}\right)} = \frac{R\,T_\text{r} T_\text{c}}{V_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}^2\left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} thus leading to : p_\text{r} = \frac{R\,T_\text{r} T_\text{c}}{P_\text{c} V_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a\,R^2 T_\text{c}^2}{P_\text{c}^2} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} V_\text{c}^2\left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} = \frac{T_\text{r}}{Z_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a}{Z_\text{c}^2} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} \left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} Thus, the Soave–Redlich–Kwong equation in reduced form only depends on ω and Z_\text{c} of the substance, contrary to both the VdW and RK equation which are consistent with the theorem of corresponding states and the reduced form is one for all substances: : p_\text{r} = \frac{T_\text{r}}{Z_\text{c}\left(V_\text{r}-\frac{\Omega_b}{Z_\text{c}}\right)} - \frac{\frac{\Omega_a}{Z_\text{c}^2} \alpha\left(\omega, T_\text{r}\right)}{V_\text{r} \left(V_\text{r}+\frac{\Omega_b}{Z_\text{c}}\right)} We can also write it in the polynomial form, with: : A = \frac{a \alpha P}{R^2 T^2} : B = \frac{bP}{RT} In terms of the compressibility factor, we have: : 0 = Z^3-Z^2+Z\left(A-B-B^2\right) - AB. The substance-specific constants a and b can be calculated from the critical properties p_\text{c} and V_\text{c} (noting that V_\text{c} is the molar volume at the critical point and p_\text{c} is the critical pressure) as: : a = 3 p_\text{c} V_\text{c}^2 : b = \frac{V_\text{c}}{3}. The table below for a selection of gases uses the following conventions: * T_c: critical temperature [K] * P_c: critical pressure [Pa] * v_c: critical specific volume [m3⋅kg−1] * R: gas constant (8.314 J⋅K−1⋅mol−1) * \mu: Molar mass [kg⋅mol−1] Substance P_c [Pa] T_c [K] v_c [m3/kg] Z_c H2O 647.3 0.23 4He 5.2 0.31 He 5.2 0.30 H2 33.2 0.30 Ne 44.5 0.29 N2 126.2 0.29 Ar 150.7 0.29 Xe 289.7 0.29 O2 154.8 0.291 CO2 304.2 0.275 SO2 430.0 0.275 CH4 190.7 0.285 C3H8 370.0 0.267 ==See also== *Van der Waals equation *Equation of state *Compressibility factors *Johannes Diderik van der Waals equation *Noro- Frenkel law of corresponding states ==References== ==External links== * Properties of Natural Gases. The unique relationship between the compressibility factor and the reduced temperature, T_r, and the reduced pressure, P_r, was first recognized by Johannes Diderik van der Waals in 1873 and is known as the two-parameter principle of corresponding states. The equation is given below, as are relationships between its parameters and the critical constants: : \begin{align} p &= \frac{R\,T}{V_\text{m} - b} - \frac{a}{\sqrt{T}\,V_\text{m}\left(V_\text{m} + b\right)} \\\\[3pt] a &= \frac{\Omega_a\,R^2 T_\text{c}^\frac{5}{2}}{p_\text{c}} \approx 0.42748\frac{R^2\,T_\text{c}^\frac{5}{2}}{P_\text{c}} \\\\[3pt] b &= \frac{\Omega_b\,R T_\text{c}}{P_\text{c}} \approx 0.08664\frac{R\,T_\text{c}}{p_\text{c}} \\\\[3pt] \Omega_a &= \left[9\left(2^{1/3}-1\right)\right]^{-1} \approx 0.42748 \\\\[3pt] \Omega_b &= \frac{2^{1/3}-1}{3} \approx 0.08664 \end{align} Another, equivalent form of the Redlich–Kwong equation is the expression of the model's compressibility factor: : Z=\frac{p V_\text{m}}{RT} = \frac{V_\text{m}}{V_\text{m} - b} - \frac{a}{R T^{3/2} \left(V_\text{m} + b\right)} The Redlich–Kwong equation is adequate for calculation of gas phase properties when the reduced pressure (defined in the previous section) is less than about one-half of the ratio of the temperature to the reduced temperature, : P_\text{r} < \frac{T}{2T_\text{c}}. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document) In general, deviation from ideal behaviour becomes more significant the closer a gas is to a phase change, the lower the temperature or the larger the pressure. Above the Boyle temperature, the compressibility factor is always greater than unity and increases slowly but steadily as pressure increases. ==Experimental values== It is extremely difficult to generalize at what pressures or temperatures the deviation from the ideal gas becomes important. Expressions for (a,b) written as functions of (T_\text{c},p_\text{c}) may also be obtained and are often used to parameterize the equation because the critical temperature and pressure are readily accessible to experiment. Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume. Experimental values for the compressibility factor confirm this. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known. Includes a chart of compressibility factors versus reduced pressure and reduced temperature (on last page of the PDF document) *Theorem of corresponding states on SklogWiki. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Compressibility factor values are usually obtained by calculation from equations of state (EOS), such as the virial equation which take compound- specific empirical constants as input. # The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density. # Gases deviate from ideal-gas behavior the most in the vicinity of the critical point. page 139 ==Theoretical models== The virial equation is especially useful to describe the causes of non-ideality at a molecular level (very few gases are mono-atomic) as it is derived directly from statistical mechanics: :Z = 1 + \frac{B}{V_\mathrm{m}} + \frac{C}{V_\mathrm{m}^2} + \frac{D}{V_\mathrm{m}^3} + \dots Where the coefficients in the numerator are known as virial coefficients and are functions of temperature. Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. ",0.333333,4.86,"""-114.4""",0.5,+0.60,A
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of decomposition of the yellow complex $\mathrm{H}_3 \mathrm{NSO}_2$ into $\mathrm{NH}_3$ and $\mathrm{SO}_2$ is $+40 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the standard enthalpy of formation of $\mathrm{H}_3 \mathrm{NSO}_2$.","The standard enthalpy of formation is then determined using Hess's law. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. In chemistry and thermodynamics, the standard enthalpy of formation or standard heat of formation of a compound is the change of enthalpy during the formation of 1 mole of the substance from its constituent elements in their reference state, with all substances in their standard states. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. This is true for all enthalpies of formation. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. The Sabatier reaction or Sabatier process produces methane and water from a reaction of hydrogen with carbon dioxide at elevated temperatures (optimally 300–400 °C) and pressures (perhaps 3 MPa ) in the presence of a nickel catalyst. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. All elements in their reference states (oxygen gas, solid carbon in the form of graphite, etc.) have a standard enthalpy of formation of zero, as there is no change involved in their formation. However the standard enthalpy of combustion is readily measurable using bomb calorimetry. thumb|The Mollier enthalpy–entropy diagram for water and steam. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Vaporization ===GME=== *Kugler HK & Keller C (eds) 1985, Gmelin handbook of inorganic and organometallic chemistry, 8th ed., ",4, 0.0024,"""0.0""",-383,7.00,D
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of carbon dioxide of mass $2.45 \mathrm{~g}$ at $27.0^{\circ} \mathrm{C}$ is allowed to expand reversibly and adiabatically from $500 \mathrm{~cm}^3$ to $3.00 \mathrm{dm}^3$. What is the work done by the gas?","Such work done by compression is thermodynamic work as here defined. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. Carbon dioxide reforming (also known as dry reforming) is a method of producing synthesis gas (mixtures of hydrogen and carbon monoxide) from the reaction of carbon dioxide with hydrocarbons such as methane with the aid of noble metal catalysts (typically Ni or Ni alloys). For a quasi-static adiabatic process, the change in internal energy is equal to minus the integral amount of work done by the system, so the work also depends only on the initial and final states of the process and is one and the same for every intermediate path. The work is due to change of system volume by expansion or contraction of the system. Changes in the potential energy of a body as a whole with respect to forces in its surroundings, and in the kinetic energy of the body moving as a whole with respect to its surroundings, are by definition excluded from the body's cardinal energy (examples are internal energy and enthalpy). ===Nearly reversible transfer of energy by work in the surroundings=== In the surroundings of a thermodynamic system, external to it, all the various mechanical and non-mechanical macroscopic forms of work can be converted into each other with no limitation in principle due to the laws of thermodynamics, so that the energy conversion efficiency can approach 100% in some cases; such conversion is required to be frictionless, and consequently adiabatic.F.C.Andrews Thermodynamics: Principles and Applications (Wiley- Interscience 1971), , p.17-18. To get an actual and precise physical measurement of a quantity of thermodynamic work, it is necessary to take account of the irreversibility by restoring the system to its initial condition by running a cycle, for example a Carnot cycle, that includes the target work as a step. Adiabatic work is done without matter transfer and without heat transfer. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. Synthesis gas is conventionally produced via the steam reforming reaction or coal gasification. In a process of transfer of energy from or to a thermodynamic system, the change of internal energy of the system is defined in theory by the amount of adiabatic work that would have been necessary to reach the final from the initial state, such adiabatic work being measurable only through the externally measurable mechanical or deformation variables of the system, that provide full information about the forces exerted by the surroundings on the system during the process. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Thermodynamic work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. The quantity of thermodynamic work is defined as work done by the system on its surroundings. # The gasification process occurs as the char reacts with steam and carbon dioxide to produce carbon monoxide and hydrogen, via the reactions C + H2O → H2 \+ CO and C + CO2 → 2CO. Gasification is a process that converts biomass- or fossil fuel-based carbonaceous materials into gases, including as the largest fractions: nitrogen (N2), carbon monoxide (CO), hydrogen (H2), and carbon dioxide (). In particular, in principle, all macroscopic forms of work can be converted into the mechanical work of lifting a weight, which was the original form of thermodynamic work considered by Carnot and Joule (see History section above). Such work is adiabatic for the surroundings, even though it is associated with friction within the system. Several kinds of thermodynamic work are especially important. The dry reforming reaction may be represented by: :CH4 + CO2 <=>[975^oC] 2CO + 2H2 Thus, two greenhouse gases are consumed and useful chemical building blocks, hydrogen and carbon monoxide, are produced. ",21,-32,"""0.36""",3.2, -194,E
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \mathrm{~kg}$ person to climb through $4.0 \mathrm{~m}$ on the surface of the Earth.","Other definitions which roughly produce the same numbers have been devised, such as: : \text{1 MET}\ = 1 \, \frac{\text{kcal}}{\text{kg}\times\text{h}}\ = 4.184 \, \frac{\text{kJ}}{\text{kg}\times\text{h}} = 1.162 \, \frac{\text{W}}{\text{kg}} where * kcal = kilocalorie * kg = kilogram * h = hour * kJ = kilojoule * W = watt ===Based on watts produced and body surface area=== Still another definition is based on the body surface area, BSA, and energy itself, where the BSA is expressed in m2: : \text{1 MET}\ = 58.2 \, \frac{\text{J}}{\text{s}\times\text{BSA}}\ = 58.2 \, \frac{\text{W}}{\text{m}^2} = 18.4 \, \frac{\text{Btu}}{\text{h}\times\text{ft}^2} which is equal to the rate of energy produced per unit surface area of an average person seated at rest. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. Health and fitness studies often bracket cohort activity levels in MET⋅hours/week. ==Quantitative definitions== ===Based on oxygen utilization and body mass=== The original definition of metabolic equivalent of task is the oxygen used by a person in milliliters per minute per kilogram body mass divided by 3.5. The metabolic equivalent of task (MET) is the objective measure of the ratio of the rate at which a person expends energy, relative to the mass of that person, while performing some specific physical activity compared to a reference, currently set by convention at an absolute 3.5 mL of oxygen per kg per minute, which is the energy expended when sitting quietly by a reference individual, chosen to be roughly representative of the general population, and thereby suited to epidemiological surveys. A 4 MET activity expends 4 times the energy used by the body at rest. By convention, 1 MET is considered equivalent to the consumption of 3.5 ml O2·kg−1·min−1 (or 3.5 ml of oxygen per kilogram of body mass per minute) and is roughly equivalent to the expenditure of 1 kcal per kilogram of body weight per hour. In the winter of 1980-1981 Tasker was part of an eight-man team (with Alan Rouse, John Porter, Brian Hall, Adrian Burgess, Alan Burgess, Pete Thexton and Paul Nunn) attempting to make a difficult winter assault on the West Face of Mount Everest; this was unsuccessful but was recounted in Tasker's first book Everest the Cruel Way. A person could also achieve 120 MET-minutes by doing an 8 MET activity for 15 minutes. RMR measurements by calorimetry in medical surveys have shown that the conventional 1-MET value overestimates the actual resting O2 consumption and energy expenditures by about 20% to 30% on the average; body composition (ratio of body fat to lean body mass) accounted for most of the variance. ==Standardized definition for research== The Compendium of Physical Activities was developed for use in epidemiologic studies to standardize the assignment of MET intensities in physical activity questionnaires. If a person does a 4 MET activity for 30 minutes, he or she has done 4 x 30 = 120 MET-minutes (or 2.0 MET-hours) of physical activity. One MET is defined as 1 kcal/kg/hour and is roughly equivalent to the energy cost of sitting quietly. Since the RMR of a person depends mainly on lean body mass (and not total weight) and other physiological factors such as health status and age, actual RMR (and thus 1-MET energy equivalents) may vary significantly from the kcal/(kg·h) rule of thumb. Joe Tasker (12 May 1948 – 17 May 1982) was a British climber, active during the late 1970s and early 1980s. For example, 1 MET is the rate of energy expenditure while at rest. A small team consisting of Tasker, Boardman, and Doug Scott made an ascent of Kangchenjunga (at 8,598 m the third highest mountain in the world) by a new route from the North-West in 1979 (with Georges Bettembourg also on the team but not making the summit); this was also the first ascent of the mountain without the use of supplementary oxygen. Tasker had delivered his manuscript for his second book, Savage Arena, which recounted his climbing life from the 1960s-1980, on the eve of his departure for the British Everest expedition in 1982. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. The BSA of an average person is 1.8 m2 (19 ft2). *At a vertex where both climbers are at peaks or both climbers are at valleys, the degree is four: both climbers may choose independently of each other which direction to go. *At the vertex (0,0), the degree is one: the only possible direction for both climbers to go is onto the mountain. Although the RMR of any person may deviate from the reference value, MET can be thought of as an index of the intensity of activities: for example, an activity with a MET value of 2, such as walking at a slow pace (e.g., 3 km/h) would require twice the energy that an average person consumes at rest (e.g., sitting quietly). ==Use== MET: The ratio of the work metabolic rate to the resting metabolic rate. A MET is the ratio of the rate of energy expended during an activity to the rate of energy expended at rest. ",2.0,2600,"""0.082""",12,+87.8,B
+What pressure would $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert at $25^{\circ} \mathrm{C}$ if it behaved as a van der Waals gas?,"* ISO 11439: Compressed natural gas (CNG) cylinders. Further the volume of the gas is (4πr3)/3. Pressure vessels for gas storage may also be classified by volume. * IS 2825–1969 (RE1977)_code_unfired_Pressure_vessels. * EN 286 (Parts 1 to 4): European standard for simple pressure vessels (air tanks), harmonized with Council Directive 87/404/EEC. VPT may refer to: * Valor por Tamaulipas * Vermont Public Television * Volume, temperature and pressure, the three parameters in the combined gas law Because of that, the design and certification of pressure vessels is governed by design codes such as the ASME Boiler and Pressure Vessel Code in North America, the Pressure Equipment Directive of the EU (PED), Japanese Industrial Standard (JIS), CSA B51 in Canada, Australian Standards in Australia and other international standards like Lloyd's, Germanischer Lloyd, Det Norske Veritas, Société Générale de Surveillance (SGS S.A.), Lloyd’s Register Energy Nederland (formerly known as Stoomwezen) etc. Note that where the pressure-volume product is part of a safety standard, any incompressible liquid in the vessel can be excluded as it does not contribute to the potential energy stored in the vessel, so only the volume of the compressible part such as gas is used. ===List of standards=== * EN 13445: The current European Standard, harmonized with the Pressure Equipment Directive (Originally ""97/23/EC"", since 2014 ""2014/68/EU""). The very small vessels used to make liquid butane fueled cigarette lighters are subjected to about 2 bar pressure, depending on ambient temperature. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. EN 13445 - Unfired Pressure Vessels is a standard that provides rules for the design, fabrication, and inspection of pressure vessels EN 13445 consists of 8 parts: * EN 13445-1 : Unfired pressure vessels - Part 1: General * EN 13445-2 : Unfired pressure vessels - Part 2: Materials * EN 13445-3 : Unfired pressure vessels - Part 3: Design * EN 13445-4 : Unfired pressure vessels - Part 4: Fabrication * EN 13445-5 : Unfired pressure vessels - Part 5: Inspection and testing * EN 13445-6 : Unfired pressure vessels - Part 6: Requirements for the design and fabrication of pressure vessels and pressure parts constructed from spheroidal graphite cast iron * EN 13445-8 : Unfired pressure vessels - Part 8: Additional requirements for pressure vessels of aluminium and aluminium alloys * EN 13445-10:2015 : Unfired pressure vessels - Part 10: Additional requirements for pressure vessels of nickel and nickel alloys. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. From about 1975 until now, the standard pressure is . Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. ""Pressure Vessel Handbook, 14th Edition."" High-pressure gas cylinders are also called bottles. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus :M = {3 \over 2} nRT {\rho \over \sigma}. (see gas law) The other factors are constant for a given vessel shape and material. * ASME Boiler and Pressure Vessel Code Section VIII: Rules for Construction of Pressure Vessels. ",22,0.0182,"""1.25""",3.07,260,A
+A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What change of pressure indicates a change of $1.00 \mathrm{~K}$ at the latter temperature?,":V \propto T\, or :\frac{V}{T}=k V is the volume, T is the thermodynamic temperature, k is the constant for the system. k is not a fixed constant across all systems and therefore needs to be found experimentally for a given system through testing with known temperature values. ==Pressure Thermometer and Absolute Zero== thumb|left|180px|Plots of pressure vs temperature for three different gas samples extrapolate to absolute zero. thumb|400px|Diagram showing pressure difference induced by a temperature difference. Historically, the Kelvin scale was developed from the Celsius scale, such that 273.15 K was 0 °C (the approximate melting point of ice) and a change of one kelvin was exactly equal to a change of one degree Celsius. The numerical value of an absolute temperature, , on the 1848 scale is related to the absolute temperature of the melting point of water, , and the absolute temperature of the boiling point of water, , by * (1848 scale) = 100 () / () On this scale, an increase of 222 degrees always means an approximate doubling of absolute temperature regardless of the starting temperature. The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. The kelvin, symbol K, is a unit of measurement for temperature. By the 1940s, the triple point of water had been experimentally measured to be about 0.6% of standard atmospheric pressure and very close to 0.01 °C per the historical definition of Celsius then in use. For practical purposes, the redefinition was unnoticed; water still freezes at 273.15 K (0 °C), and the triple point of water continues to be a commonly used laboratory reference temperature. In 1948, the Celsius scale was recalibrated by assigning the triple point temperature of water the value of 0.01 °C exactly and allowing the melting point at standard atmospheric pressure to have an empirically determined value (and the actual melting point at ambient pressure to have a fluctuating value) close to 0 °C. This lets one interpolate actual vapor pressure from paired dry and wet thermometer bulb temperature readings. :: \gamma =\frac{ \left( c_p \right)_{air} * P }{ \lambda_v * MW_{ratio} } : \gamma = psychrometric constant [kPa °C−1], : P = atmospheric pressure [kPa], : \lambda_v = latent heat of water vaporization, 2.45 [MJ kg−1], : c_p = specific heat of air at constant pressure, [MJ kg−1 °C−1], : MW_{ratio} = ratio molecular weight of water vapor/dry air = 0.622. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. In the early decades of the 20th century, the Kelvin scale was often called the ""absolute Celsius"" scale, indicating Celsius degrees counted from absolute zero rather than the freezing point of water, and using the same symbol for regular Celsius degrees, °C.For example, Encyclopaedia Britannica editions from the 1920s and 1950s, one example being the article ""Planets"". === Triple point standard === In 1873, William Thomson's older brother James coined the term triple point to describe the combination of temperature and pressure at which the solid, liquid, and gas phases of a substance were capable of coexisting in thermodynamic equilibrium. The 13th CGPM also held in Resolution 4 that ""The kelvin, unit of thermodynamic temperature, is equal to the fraction of the thermodynamic temperature of the triple point of water."" In 1967/1968, Resolution 3 of the 13th CGPM renamed the unit increment of thermodynamic temperature ""kelvin"", symbol K, replacing ""degree Kelvin"", symbol . English translation (extract). that, at constant pressure, ideal gases expanded or contracted their volume linearly (Charles's law) by about 1/273 parts per degree Celsius of temperature's change up or down, between 0° and 100° C. In 2005, noting that the triple point could be influenced by the isotopic ratio of the hydrogen and oxygen making up a water sample and that this was ""now one of the major sources of the observed variability between different realizations of the water triple point"", the International Committee for Weights and Measures (CIPM), a committee of the CGPM, affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin would refer to water having the isotopic composition specified for Vienna Standard Mean Ocean Water. === 2019 redefinition === In 2005, the CIPM began a programme to redefine the kelvin (along with the other SI units) using a more experimentally rigorous method. Thus an increment of 1 °C equals of the temperature difference between the melting and boiling points. To the extent that the gas is ideal, the pressure depends linearly on temperature, and the extrapolation to zero pressure occurs at absolute zero. The Kelvin scale is an absolute scale, which is defined such that 0 K is absolute zero and a change of thermodynamic temperature by 1 kelvin corresponds to a change of thermal energy by . In 1954, with absolute zero having been experimentally determined to be about −273.15 °C per the definition of °C then in use, Resolution 3 of the 10th General Conference on Weights and Measures (CGPM) introduced a new internationally standardised Kelvin scale which defined the triple point as exactly 273.15 + 0.01 = 273.16 degrees Kelvin. This value of ""−273"" was the negative reciprocal of 0.00366—the accepted coefficient of thermal expansion of an ideal gas per degree Celsius relative to the ice point, giving a remarkable consistency to the currently accepted value. The pressure melting point of ice is the temperature at which ice melts at a given pressure. ",48,16,"""2.0""",0.0182,0.0245,E
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \mathrm{~kg}$ person to climb through $4.0 \mathrm{~m}$ on the surface of the Moon $\left(g=1.60 \mathrm{~m} \mathrm{~s}^{-2}\right)$.","The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The mass of the Moon is M = 7.3458 × 1022 kg and the mean density is 3346 kg/m3. Work in compressed air, compressed air work or hyperbaric work is occupational activity in an enclosed atmosphere at a controlled ambient pressure significantly higher than the adjacent normal atmospheric pressure. thumb|Apollo astronauts work on the Moon to collect samples and explore. The lunar GM = 4902.8001 km3/s2 from GRAIL analyses. Consequently, it is conventional to express the lunar mass M multiplied by the gravitational constant G. The atmosphere of the Moon is a very scant presence of gases surrounding the Moon. Another important source is the bombardment of the lunar surface by micrometeorites, the solar wind, and sunlight, in a process known as sputtering. == Escape velocity and atmospheric hold == Gases can: * be re-implanted into the regolith as a result of the Moon's gravity; * escape the Moon entirely if the particle is moving at or above the lunar escape velocity of , or ; * be lost to space either by solar radiation pressure or, if the gases are ionized, by being swept away in the solar wind's magnetic field. == Composition == What little atmosphere the Moon has consists of some unusual gases, including sodium and potassium, which are not found in the atmospheres of Earth, Mars, or Venus. Because weight is directly dependent upon gravitational acceleration, things on the Moon will weigh only 16.6% (= 1/6) of what they weigh on the Earth. ==Gravitational field== The gravitational field of the Moon has been measured by tracking the radio signals emitted by orbiting spacecraft. The ancient lunar atmosphere was eventually stripped away by solar winds and dissipated into space. == See also == * Atmosphere of Mercury * Exosphere * Lunar Atmosphere and Dust Environment Explorer (LADEE) * Orders of magnitude (pressure) * Sodium tail of the Moon == References == Category:Lunar science Moon Moon The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. thumb|upright=1.5|At the Swamp Works, a sculpture made of lunar soil simulant representing construction on the Moon by robots working together with humans. The average daytime abundances of the elements known to be present in the lunar atmosphere, in atoms per cubic centimeter, are as follows: *Argon: 20,000–100,000 *Helium: 5,000–30,000 *Neon: up to 20,000 *Sodium: 70 *Potassium: 17 *Hydrogen: fewer than 17 This yields approximately 80,000 total atoms per cubic centimeter, marginally higher than the quantity posited to exist in the atmosphere of Mercury. It returned approximately of Lunar surface material. The building is where Apollo astronauts practiced working with the Lunar Module for lunar landings and extravehicular activities. thumb|Students traverse a simulated crater in a moonbuggy they designed and built themselves. The NASA Human Exploration Rover Challenge, prior to 2014 referred to as the Great Moonbuggy Race, is an annual competition for high school and college students to design, build, and race human-powered, collapsible vehicles over simulated lunar/Martian terrain. The obstacles are constructed of discarded tires, plywood, some 20 tons of gravel and five tons of sand, all to simulate lunar craters, basins, and rilles. * The moonbuggy (pre-2014) must fit into a cube and be no more than 4 ft wide. Roger Joseph Boscovich was the first modern astronomer to argue for the lack of atmosphere around the Moon in his De lunae atmosphaera (1753). == Sources == One source of the lunar atmosphere is outgassing: the release of gases such as radon and helium resulting from radioactive decay within the crust and mantle. The C31 coefficient is large. ==Simulating lunar gravity== In January 2022 China was reported by the South China Morning Post to have built a small (60 centimeters in diameter) research facility to simulate low lunar gravity with the help of magnets. National Academies Press, Washington, DC (1997). ==Sample- return missions== ===First missions=== thumb|Lunar sample 60016 on display at Space Center Houston Lunar Samples Vault, at NASA's Johnson Space Center The Apollo program returned over of lunar rocks and regolith (including lunar 'soil') to the Lunar Receiving Laboratory in Houston.Orloff 2004, ""Extravehicular Activity"" Today, 75% of the samples are stored at the Lunar Sample Laboratory Facility built in 1979. ",1.22,0.245,"""420.0""",4,0.264,C
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $120 \mathrm{mg}$ of naphthalene, $\mathrm{C}_{10} \mathrm{H}_8(\mathrm{~s})$, was burned in a bomb calorimeter the temperature rose by $3.05 \mathrm{~K}$. By how much will the temperature rise when $10 \mathrm{mg}$ of phenol, $\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}(\mathrm{s})$, is burned in the calorimeter under the same conditions?","The high heat values are conventionally measured with a bomb calorimeter. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. That is, the heat of combustion, ΔH°comb, is the heat of reaction of the following process: : (std.) + (c + - ) (g) → c (g) + (l) + (g) Chlorine and sulfur are not quite standardized; they are usually assumed to convert to hydrogen chloride gas and or gas, respectively, or to dilute aqueous hydrochloric and sulfuric acids, respectively, when the combustion is conducted in a bomb calorimeter containing some quantity of water. == Ways of determination == ===Gross and net=== Zwolinski and Wilhoit defined, in 1972, ""gross"" and ""net"" values for heats of combustion. * However, for true energy calculations in some specific cases, the higher heating value is correct. They may also be calculated as the difference between the heat of formation ΔH of the products and reactants (though this approach is somewhat artificial since most heats of formation are typically calculated from measured heats of combustion). Since the heat of combustion of these elements is known, the heating value can be calculated using Dulong's Formula: LHV [kJ/g]= 33.87mC \+ 122.3(mH \- mO ÷ 8) + 9.4mS where mC, mH, mO, mN, and mS are the contents of carbon, hydrogen, oxygen, nitrogen, and sulfur on any (wet, dry or ash free) basis, respectively. === Higher heating value === The higher heating value (HHV; gross energy, upper heating value, gross calorific value GCV, or higher calorific value; HCV) indicates the upper limit of the available thermal energy produced by a complete combustion of fuel. Among the variables affecting burn rate are pressure and temperature. The calorific value is the total energy released as heat when a substance undergoes complete combustion with oxygen under standard conditions. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics This value is important for fuels like wood or coal, which will usually contain some amount of water prior to burning. == Measuring heating values == The higher heating value is experimentally determined in a bomb calorimeter. Incomplete reaction at higher temperature further curtails the effect of a larger heat of combustion. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). The temperatures mentioned here are for a stoichiometric fuel-oxidizer mixture (i.e. equivalence ratio φ = 1). It may be expressed with the quantities: * energy/mole of fuel * energy/mass of fuel * energy/volume of the fuel There are two kinds of enthalpy of combustion, called high(er) and low(er) heat(ing) value, depending on how much the products are allowed to cool and whether compounds like are allowed to condense. This table is in Standard cubic metres (1atm, 15°C), to convert to values per Normal cubic metre (1atm, 0°C), multiply above table by 1.0549. == See also == * Adiabatic flame temperature * Cost of electricity by source * Electrical efficiency * Energy content of fuel * Energy conversion efficiency * Energy density * Energy value of coal * Exothermic reaction * Figure of merit * Fire * Food energy * Internal energy * ISO 15971 * Mechanical efficiency * Thermal efficiency * Wobbe index: heat density == References == ==Further reading== * == External links == * NIST Chemistry WebBook * Category:Engineering thermodynamics Category:Combustion Category:Fuels Category:Thermodynamic properties Category:Nuclear physics Category:Thermochemistry == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. thumb|upright|Ethanol burning with its spectrum depicted In the study of combustion, the adiabatic flame temperature is the temperature reached by a flame under ideal conditions. The definition in which the combustion products are all returned to the reference temperature is more easily calculated from the higher heating value than when using other definitions and will in fact give a slightly different answer. === Gross heating value === Gross heating value accounts for water in the exhaust leaving as vapor, as does LHV, but gross heating value also includes liquid water in the fuel prior to combustion. This result can be explained through Le Chatelier's principle. ==See also== *Flame speed ==References== == External links == === General information === * * Computation of adiabatic flame temperature * Adiabatic flame temperature === Tables === * adiabatic flame temperature of hydrogen, methane, propane and octane with oxygen or air as oxidizers * * Temperature of a blue flame and common materials === Calculators === * Online adiabatic flame temperature calculator using Cantera * Adiabatic flame temperature program * Gaseq, program for performing chemical equilibrium calculations. ",1.6,205,"""46.7""",5.9,0.38,B
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from $57.4 \mathrm{kPa}$ and $1.0 \mathrm{dm}^3$ to a final volume of $2.0 \mathrm{dm}^3$. Take $\gamma=1.4$.","The solubility turned out to be very low: from 0.02 to 0.10 %. thumb|Carbon dioxide pressure-temperature phase diagram == Uses == thumb|219x219px|Fire extinguisher Uses of liquid carbon dioxide include the preservation of food, in fire extinguishers, and in commercial food processes. Liquid carbon dioxide is a type of liquid which is formed from highly compressed and cooled gaseous carbon dioxide. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Compressed natural gas (CNG) is a fuel gas mainly composed of methane (CH4), compressed to less than 1% of the volume it occupies at standard atmospheric pressure. Accumulation of carbon dioxide is predominantly a result of hypoperfusion and not hypoxia. At this temperature, the pressure is measured in a range from 15 to 60 atmospheres. The solubility of water in liquid carbon dioxide is measured in a range of temperatures, ranging from to . thumb|Jets of liquid carbon dioxide Liquid carbon dioxide is the liquid state of carbon dioxide (), which cannot occur under atmospheric pressure. Normocapnia or normocarbia is a state of normal arterial carbon dioxide pressure, usually about 40 mmHg. == See also == * * * == References == *The Free Dictionary - Normocapnia Category:Diving medicine Category:Pulmonology Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. According to the model, each exhalatory segment of capnogram waveform follows the analytical expression: p_D(t) = p_A (1 - e ^{-\alpha}e^{\alpha e^{-t/\tau}}) where * p_D(t)represents the partial pressure of carbon dioxide measured by the capnogram as a function of time t since the beginning of exhalation. * p_Arepresents the alveolar partial pressure of carbon dioxide. * \alpharepresents the inverse of the dead space fraction (i.e. the ratio of tidal volume to dead space volume). * \taurepresents the pulmonary time constant (i.e. the product of pulmonary resistance and compliance) In particular, this model explains the rounded ""shark-fin"" shape of the capnogram observed in patients with obstructive lung disease. == See also == * Integrated pulmonary index * Medical equipment * Medical test * Respiratory monitoring * Colorimetric capnography == Citations == == External links == * CapnoBase.org: Respiratory signal database that contains clinical and simulated capnogram recordings Category:Anesthesia Category:Breath tests Category:Diagnostic emergency medicine Category:Diagnostic intensive care medicine Category:Diagnostic pulmonology However, CNG requires a much larger volume to store the energy equivalent of petrol and the use of very high pressures (3000 to 4000 psi, or 205 to 275 bar). In healthy individuals, the difference between arterial blood and expired gas partial pressures is very small (normal difference 4-5 mmHg). Low-temperature carbon dioxide is commercially used in its solid form, commonly known as ""dry ice"". Gastric tonometry describes the measurement of the carbon dioxide level inside the stomach in order to assess the degree of blood flow to the stomach and bowel. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. Because the introduction of a nasogastric tube is almost routine in critically ill patients, the measurement of gastric carbon dioxide can be an easy method to monitor tissue perfusion. Capnography is the monitoring of the concentration or partial pressure of carbon dioxide () in the respiratory gases. For light-duty CNG cars to become a viable short-term climate strategy, methane leakage would need to be kept below 1.6% of total natural gas produced (approximately half the current amount for well to wheels – note difference from well to city)."" ** The lifecycle greenhouse gas emissions for CNG compressed from California's pipeline natural gas is given a value of 67.70 grams of -equivalent per megajoule (gCO2e/MJ) by CARB (the California Air Resources Board), approximately 28 percent lower than the average petrol fuel in that market (95.86 gCO2e/MJ). Natural gas is being experimentally stored at lower pressure in a form known as an ANG (adsorbed natural gas) cylinder, where it is adsorbed at 35 bar (500 psi, the pressure of gas in natural gas pipelines) in various sponge-like materials, such as carbon and MOFs (metal-organic frameworks). In this process known as high pressure ANG, a high pressure CNG tank is filled by absorbers such as activated carbon (which is an adsorbent with high surface area) and stores natural gas by both CNG and ANG mechanisms. ",0.22222222,1.1,"""4.86""",22,49,D
+"Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $229 \mathrm{~J}$ of energy is supplied as heat to $3.0 \mathrm{~mol} \mathrm{Ar}(\mathrm{g})$ at constant pressure, the temperature of the sample increases by $2.55 \mathrm{~K}$. Calculate the molar heat capacities at constant volume.","Then the molar heat capacity (at constant volume) would be :cV,m = fR where R is the ideal gas constant. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.Lange's Handbook of Chemistry, 10th ed. p. 1524 All methods for the measurement of specific heat apply to molar heat capacity as well. ==Units== The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where ""amol"" denotes an amount of the solid that contains the Avogadro number of atoms. The following is a table of some constant- pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* H2 28.9 5.0 29.6 5.1 41.2 7.9 Not saturated.""Hydrogen"" NIST Chemistry WebBook, SRD 69, online. Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. Therefore, the word ""molar"", not ""specific"", should always be used for this quantity. ==Definition== The molar heat capacity of a substance, which may be denoted by cm, is the heat capacity C of a sample of the substance, divided by the amount (moles) n of the substance in the sample: :cm{} \;=\; \frac{C}{n} \;=\; \frac{1}{n} \lim_{\Delta T \rightarrow 0}\frac{Q}{\Delta T} where Q is the amount of heat needed to raise the temperature of the sample by ΔT. The last value corresponds almost exactly to the predicted value for f = 7. right|thumb|upright=1.25|Constant-volume specific heat capacity of diatomic gases (real gases) between about 200 K and 2000 K. In other words, that theory predicts that the molar heat capacity at constant volume cV,m of all monatomic gases will be the same; specifically, :cV,m = R where R is the ideal gas constant, about 8.31446 J⋅K−1⋅mol−1 (which is the product of the Boltzmann constant kB and the Avogadro constant). At high enough temperature, its molar heat capacity then should be cP,m = 7.5 R = 62.63 J⋅K−1⋅mol−1. The molar heat capacity of the gas will then be determined only by the ""active"" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Therefore, the heat capacity of a sample of a solid substance is expected to be 3RNa, or (24.94 J/K)Na, where Na is the number of moles of atoms in the sample, not molecules. Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). In modern terms, Dulong and Petit found that the heat capacity of a mole of many solid elements is about 3R, where R is the universal gas constant. The same theory predicts that the molar heat capacity of a monatomic gas at constant pressure will be :cP,m = cV,m \+ R = R This prediction matches the experimental values, which, for helium through xenon, are 20.78, 20.79, 20.85, 20.95, and 21.01 J⋅K−1⋅mol−1, respectively;Shuen-Chen Hwang, Robert D. Lein, Daniel A. Morgan (2005). ", 7.42,0.0526315789,"""30.0""",+3.60,4,C
+Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The fugacity coefficient of a certain gas at $200 \mathrm{~K}$ and 50 bar is 0.72. Calculate the difference of its molar Gibbs energy from that of a perfect gas in the same state.,"Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to . The contribution of nonideality to the molar Gibbs energy of a real gas is equal to . For a gas obeying the van der Waals equation, the explicit formula for the fugacity coefficient is RT \ln \varphi = \frac{RTb}{V_\mathrm{m}-b} - \frac{2a}{V_\mathrm{m}} - RT \ln \left ( 1 - \frac{a(V_\mathrm{m}-b)}{RTV_\mathrm{m}^2}\right ) This formula is difficult to use, since the pressure depends on the molar volume through the equation of state; so one must choose a volume, calculate the pressure, and then use these two values on the right-hand side of the equation. ==History== The word fugacity is derived from the Latin fugere, to flee. This equation allows the fugacity to be calculated using tabulated values for saturated vapor pressure. This means that the molar Gibbs energy of real nitrogen at a pressure of 100 atm is equal to the molar Gibbs energy of nitrogen as an ideal gas at . Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . Often the pressure is low enough for the vapor phase to be considered an ideal gas, so the fugacity coefficient is approximately equal to 1. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy as the real gas. For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. In a pure substance, is equal to the Gibbs energy for a mole of the substance, and d\mu = dG_\mathrm{m} = -S_\mathrm{m}dT + V_\mathrm{m}dP, where and are the temperature and pressure, is the volume per mole and is the entropy per mole. ===Gas=== For an ideal gas the equation of state can be written as V_\mathrm{m}^\text{ideal} = \frac{RT}{P}, where is the ideal gas constant. The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The fugacity coefficient is . In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. A calorically perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, and enthalpy H that are functions of temperature only, i.e., U = U(T), H = H(T) * has heat capacities C_V, C_P that are constant, i.e., dU = C_V dT, dH = C_P dT and \Delta U = C_V \Delta T, \Delta H = C_P \Delta T, where \Delta is any finite (non-differential) change in each quantity. In addition, there are natural reference states for fugacity (for example, an ideal gas makes a natural reference state for gas mixtures since the fugacity and pressure converge at low pressure). ===Gases=== In a mixture of gases, the fugacity of each component has a similar definition, with partial molar quantities instead of molar quantities (e.g., instead of and instead of ): dG_i = R T \,d \ln f_i and \lim_{P\to 0} \frac{f_i}{P_i} = 1, where is the partial pressure of component . The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case. == Ideal- gas relations == For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U=U(n,T), where is the amount of substance in moles. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. ",7.136,0.000216,"""0.0""",3.8,-0.55,E
 "Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K.  Given the reactions (1) and (2) below, determine $\Delta_{\mathrm{r}} U^{\ominus}$ for reaction (3).
 
 (1) $\mathrm{H}_2(\mathrm{g})+\mathrm{Cl}_2(\mathrm{g}) \rightarrow 2 \mathrm{HCl}(\mathrm{g})$, $\Delta_{\mathrm{r}} H^{\ominus}=-184.62 \mathrm{~kJ} \mathrm{~mol}^{-1}$
 
 (2) $2 \mathrm{H}_2(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$, $\Delta_{\mathrm{r}} H^\ominus=-483.64 \mathrm{~kJ} \mathrm{~mol}^{-1}$
 
-(3) $4 \mathrm{HCl}(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{Cl}_2(\mathrm{g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. == Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). K (? °C), ? K (? °C), ? :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Ellis (Ed.) in Nuffield Advanced Science Book of Data, Longman, London, UK, 1972. == See also == Category:Thermodynamic properties Category:Chemical element data pages Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Bodek et al., Environmental Inorganic Chemistry, Pergamon Press, New York, (1988). ",91.17,0.7854,2.19,-0.28,-111.92,E
-Radiation from an X-ray source consists of two components of wavelengths $154.433 \mathrm{pm}$ and $154.051 \mathrm{pm}$. Calculate the difference in glancing angles $(2 \theta)$ of the diffraction lines arising from the two components in a diffraction pattern from planes of separation $77.8 \mathrm{pm}$.,"In the figure below, the line representing a ray makes an angle θ with the normal (dotted line). right|thumb|Grazing incidence diffraction geometry. * The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object. * The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object. The amount of diffraction depends on the size of the gap. When the incident angle \theta_\text{i} of the light onto the slit is non-zero (which causes a change in the path length), the intensity profile in the Fraunhofer regime (i.e. far field) becomes: :I(\theta) = I_0 \,\operatorname{sinc}^2 \left[ \frac{d \pi}{\lambda} (\sin\theta \pm \sin\theta_i)\right] The choice of plus/minus sign depends on the definition of the incident angle \theta_\text{i}.right|thumb|2-slit (top) and 5-slit diffraction of red laser light thumb|left|Diffraction of a red laser using a diffraction grating. right|thumb|A diffraction pattern of a 633 nm laser through a grid of 150 slits ===Diffraction grating=== thumb|Diffraction grating A diffraction grating is an optical component with a regular pattern. The beam is diffracted in the plane of the surface of the sample by the angle 2θ, and often also out of the plane. The main central beam, nulls, and phase reversals are apparent. right|thumb|300px|Graph and image of single-slit diffraction As an example, an exact equation can now be derived for the intensity of the diffraction pattern as a function of angle in the case of single-slit diffraction. Given that \delta_A-\delta_B\ll 1 and \alpha_A-\alpha_B\ll 1, at a second-order development it turns that \cos\delta_A\cos\delta_B \frac{(\alpha_A - \alpha_B)^2}{2} \approx \cos^2\delta_A \frac{(\alpha_A - \alpha_B)^2}{2}, so that :\theta \approx \sqrt{\left[(\alpha_A - \alpha_B)\cos\delta_A\right]^2 + (\delta_A-\delta_B)^2} === Small angular distance: planar approximation === thumb|Planar approximation of angular distance on sky If we consider a detector imaging a small sky field (dimension much less than one radian) with the y-axis pointing up, parallel to the meridian of right ascension \alpha, and the x-axis along the parallel of declination \delta, the angular separation can be written as: : \theta \approx \sqrt{\delta x^2 + \delta y^2} where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B. Note that the y-axis is equal to the declination, whereas the x-axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure). ==See also== * Milliradian * Gradian * Hour angle * Central angle * Angle of rotation * Angular diameter * Angular displacement * Great-circle distance * ==References== * CASTOR, author(s) unknown. Grazing incidence diffraction is used in X-ray spectroscopy and atom optics, where significant reflection can be achieved only at small values of the grazing angle. thumb|425x425px|The plane of incidence is defined by the incoming radiation's propagation vector and the normal vector of the surface. Determining the angle of reflection with respect to a planar surface is trivial, but the computation for almost any other surface is significantly more difficult. thumb|center|650px|Refraction of light at the interface between two media. ==Grazing angle or glancing angle== thumb|Focusing X-rays with glancing reflection When dealing with a beam that is nearly parallel to a surface, it is sometimes more useful to refer to the angle between the beam and the surface tangent, rather than that between the beam and the surface normal. Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. Diffraction is greatest when the size of the gap is similar to the wavelength of the wave. The 90-degree complement to the angle of incidence is called the grazing angle or glancing angle. Angular distance or angular separation, also known as apparent distance or apparent separation, denoted \theta, is the angle between the two sightlines, or between two point objects as viewed from an observer. The path difference is approximately \frac{d \sin(\theta)}{2} so that the minimum intensity occurs at an angle \theta_{min} given by :d\,\sin\theta_\text{min} = \lambda, where d is the width of the slit, \theta_\text{min} is the angle of incidence at which the minimum intensity occurs, and \lambda is the wavelength of the light. Now, substituting in \frac{2\pi}{\lambda} = k, the intensity (squared amplitude) I of the diffracted waves at an angle θ is given by: I(\theta) = I_0 {\left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right] }^2 ==Multiple slits== right|frame|Double-slit diffraction of red laser light right|frame|2-slit and 5-slit diffraction Let us again start with the mathematical representation of Huygens' principle. File:Two-Slit Diffraction.png|Generation of an interference pattern from two-slit diffraction. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different. thumb|A computer-generated image of an Airy disk. thumb| Computer-generated light diffraction pattern from a circular aperture of diameter 0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of 0.1 cm – 1 cm in steps of 0.1 cm. We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The main central beam, nulls, and phase reversals are apparent. right|thumb|Graph and image of single-slit diffraction. ",+37,3.38,0.14,2.14,0.020,D
-"A chemical reaction takes place in a container of cross-sectional area $50 \mathrm{~cm}^2$. As a result of the reaction, a piston is pushed out through $15 \mathrm{~cm}$ against an external pressure of $1.0 \mathrm{~atm}$. Calculate the work done by the system.","The work done is given by the dot product of the two vectors. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The work is due to change of system volume by expansion or contraction of the system. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. The work of the net force is calculated as the product of its magnitude and the particle displacement. The quantity of thermodynamic work is defined as work done by the system on its surroundings. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). work is represented by the following equation between differentials: \delta W = P \, dV where *\delta W (inexact differential) denotes an infinitesimal increment of work done by the system, transferring energy to the surroundings; *P denotes the pressure inside the system, that it exerts on the moving wall that transmits force to the surroundings.Borgnakke, C., Sontag, R. E. (2009). Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. As a result, the work done by the system also depends on the initial and final states. Then, for instance, to calculate the percent of the piston's stroke at which steam admission is cut off: *Calculate the angle whose cosine is twice the lap divided by the valve travel *Calculate the angle whose cosine is twice the (lap plus lead), divided by the valve travel Add the two angles and take the cosine of their sum; subtract 1 from that cosine and multiply the result by -50. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Therefore, work need only be computed for the gravitational forces acting on the bodies. Such work is adiabatic for the surroundings, even though it is associated with friction within the system. If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. Because it does not change the volume of the system it is not measured as pressure–volume work, and it is called isochoric work. ",-75,30,0.042,0.000216, 6.07,A
-"A mixture of water and ethanol is prepared with a mole fraction of water of 0.60 . If a small change in the mixture composition results in an increase in the chemical potential of water by $0.25 \mathrm{~J} \mathrm{~mol}^{-1}$, by how much will the chemical potential of ethanol change?","Specific heat = 2.44 kJ/(kg·K) === Acid-base chemistry === Ethanol is a neutral molecule and the pH of a solution of ethanol in water is nearly 7.00. Ethanol-water mixtures have less volume than the sum of their individual components at the given fractions. It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. Mixing equal volumes of ethanol and water results in only 1.92 volumes of mixture. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. The volume of alcohol in the solution can then be estimated. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Mixing ethanol and water is exothermic, with up to 777 J/mol being released at 298 K. Mixtures of ethanol and water form an azeotrope at about 89 mole-% ethanol and 11 mole-% water or a mixture of 95.6% ethanol by mass (or about 97% alcohol by volume) at normal pressure, which boils at 351 K (78 °C). Ethanol can be quantitatively converted to its conjugate base, the ethoxide ion (CH3CH2O−), by reaction with an alkali metal such as sodium: :2 CH3CH2OH + 2 Na → 2 CH3CH2ONa + H2 or a very strong base such as sodium hydride: :CH3CH2OH + NaH → CH3CH2ONa + H2 The acidities of water and ethanol are nearly the same, as indicated by their pKa of 15.7 and 16 respectively. As high as 30-50 kcal/mol changes in the potential energy surface (activation energies and relative stability) were calculated if the charge of the metal species was changed during the chemical transformation. ===Free radical syntheses=== Many free radical-based syntheses show large kinetic solvent effects that can reduce the rate of reaction and cause a planned reaction to follow an unwanted pathway. ==See also== * Cage effect ==References== Category:Physical chemistry Category:Reaction mechanisms A solution will have a lower and hence more negative water potential than that of pure water. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants A and B are related by : d\mu_\text{B} = -\frac{n_\text{A}}{n_\text{B}}\,d\mu_\text{A} where n_\text{A} is the number of moles of A and n_\text{B} is the number of moles of B. Water potential is the potential energy of water per unit volume relative to pure water in reference conditions. Ethanol is slightly more refractive than water, having a refractive index of 1.36242 (at λ=589.3 nm and ). From the above equation, the chemical potential is given by : \mu_i = \left(\frac{\partial U}{\partial N_i} \right)_{S,V, N_{j e i}}. It is defined as the number of millilitres (mL) of pure ethanol present in of solution at . Furthermore, the more solute molecules present, the more negative the solute potential is. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Under such nomenclature, the ethanol was mixed with 25% water to reduce the combustion chamber temperature. The addition of even a few percent of ethanol to water sharply reduces the surface tension of water. ",-0.38,131,1.81,2,8.7,A
-"The enthalpy of fusion of mercury is $2.292 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and its normal freezing point is $234.3 \mathrm{~K}$ with a change in molar volume of $+0.517 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$ on melting. At what temperature will the bottom of a column of mercury (density $13.6 \mathrm{~g} \mathrm{~cm}^{-3}$ ) of height $10.0 \mathrm{~m}$ be expected to freeze?","T_M(d) = T_{MB}(1-\frac{4\sigma\,_{sl}}{H_f\rho\,_sd}) Where: TMB = bulk melting temperature ::σsl = solid–liquid interface energy ::Hf = Bulk heat of fusion ::ρs = density of solid ::d = particle diameter ==Semiconductor/covalent nanoparticles== Equation 2 gives the general relation between the melting point of a metal nanoparticle and its diameter. T_M(d)=\frac{4T_{MB}}{H_fd}\left(\sigma\,_{sv}-\sigma\,_{lv}\left(\frac{\rho\,_s}{\rho\,_l}\right)^{2/3}\right) Where: σsv=solid-vapor interface energy ::σlv=liquid-vapor interface energy ::Hf=Bulk heat of fusion ::ρs=density of solid ::ρl=density of liquid ::d=diameter of nanoparticle ===Liquid shell nucleation model=== The liquid shell nucleation model (LSN) predicts that a surface layer of atoms melts prior to the bulk of the particle. :This article deals with melting/freezing point depression due to very small particle size. The theoretical size-dependent melting point of a material can be calculated through classical thermodynamic analysis. Equation 4 gives the normalized, size-dependent melting temperature of a material according to the liquid-drop model. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). T_M(d)=T_{MB}(1-(\frac{c}{d})^2) Where: TMB=bulk melting temperature ::c=materials constant ::d=particle diameter Equation 3 indicates that melting point depression is less pronounced in covalent nanoparticles due to the quadratic nature of particle size dependence in the melting Equation. ==Proposed mechanisms== The specific melting process for nanoparticles is currently unknown. K (? °C), ? K (? °C), ? K (? °C), ? K (? °C), ? T_M(d)=\frac{4T_{MB}}{H_fd}(\frac{\sigma\,_{sv}}{1-\frac{d_0}{d}}-\sigma\,_{lv}(1-\frac{\rho\,_s}{\rho\,_l})) Where: d0=atomic diameter ===Liquid nucleation and growth model=== The liquid nucleation and growth model (LNG) treats nanoparticle melting as a surface- initiated process. Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The decrease in melting temperature can be on the order of tens to hundreds of degrees for metals with nanometer dimensions. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? The melting temperature of a nanoparticle decreases sharply as the particle reaches critical diameter, usually < 50 nm for common engineering metals. The melting temperature of a nanoparticle is a function of its radius of curvature according to the LSN. The model calculates melting conditions as a function of two competing order parameters using Landau potentials. thumb|0 °C = 32 °F A degree of frost is a non-standard unit of measure for air temperature meaning degrees below melting point (also known as ""freezing point"") of water (0 degrees Celsius or 32 degrees Fahrenheit). The Mollier diagram coordinates are enthalpy h and humidity ratio x. More recently, researchers developed nanocalorimeters that directly measure the enthalpy and melting temperature of nanoparticles. ",0,1.154700538,0.6,16.3923,234.4,E
-"Suppose a nanostructure is modelled by an electron confined to a rectangular region with sides of lengths $L_1=1.0 \mathrm{~nm}$ and $L_2=2.0 \mathrm{~nm}$ and is subjected to thermal motion with a typical energy equal to $k T$, where $k$ is Boltzmann's constant. How low should the temperature be for the thermal energy to be comparable to the zero-point energy？","When L is comparable to or smaller than the mean free path (which is of the order 1 µm for carbon nanostructures ), the continuous energy model used for bulk materials no longer applies and nonlocal and nonequilibrium aspects to heat transfer also need to be considered. These issues can be addressed with phonon engineering, once nanoscale thermal behaviors have been studied and understood. ==The effect of the limited length of structure== In general two carrier types can contribute to thermal conductivity - electrons and phonons. Modeling of the low-temperature specific heat allows determination of the on-tube phonon velocity, the splitting of phonon subbands on a single tube, and the interaction between neighboring tubes in a bundle. ===Thermal conductivity measurements=== Measurements show a single-wall carbon nanotubes (SWNTs) room-temperature thermal conductivity about 3500 W/(m·K), and over 3000 W/(m·K) for individual multiwalled carbon nanotubes (MWNTs). Since the Fermi level in a metal at absolute zero is the energy of the highest occupied single particle state, then the Fermi energy in a metal is the energy difference between the Fermi level and lowest occupied single-particle state, at zero-temperature. ==Context== In quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons) obey the Pauli exclusion principle. The Fermi temperature is defined as T_\text{F} = \frac{E_\text{F}}{k_\text{B}}, where k_\text{B} is the Boltzmann constant, and E_\text{F} the Fermi energy. Only when the temperature exceeds the related Fermi temperature, do the particles begin to move significantly faster than at absolute zero. The Fermi temperature for a metal is a couple of orders of magnitude above room temperature. Junction temperature, short for transistor junction temperature, is the highest operating temperature of the actual semiconductor in an electronic device. In physics, the thermal conductance quantum g_0 describes the rate at which heat is transported through a single ballistic phonon channel with temperature T. For CNT, represented as 1-D ballistic electronic channel, the electronic conductance is quantized, with a universal value of :G_0 = \frac{2e^2}{h} Similarly, for a single ballistic 1-D channel, the thermal conductance is independent of materials parameters, and there exists a quantum of thermal conductance, which is linear in temperature: :G_{th} = \frac{\pi^2 {k_B}^2 T}{3h} Possible conditions for observation of this quantum were examined by Rego and Kirczenow. The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. These measurements employed suspended silicon nitride () nanostructures that exhibited a constant thermal conductance of 16 g_0 at temperatures below approximately 0.6 kelvin. == Relation to the quantum of electrical conductance == For ballistic electrical conductors, the electron contribution to the thermal conductance is also quantized as a result of the electrical conductance quantum and the Wiedemann–Franz law, which has been quantitatively measured at both cryogenic (~20 mK) and room temperature (~300K). Then : \kappa = L T \sigma = {\pi^2 k_{\rm B}^2 \over 3e^2}\times {n e^2 \over h} T = {\pi^2 k_{\rm B}^2 \over 3h} n T = g_0 n, where g_0 is the thermal conductance quantum defined above. ==References== ==See also== * Thermal properties of nanostructures Category:Mesoscopic physics Category:Nanotechnology Category:Quantum mechanics Category:Condensed matter physics Category:Physical quantities Category:Heat conduction It was shown that, using this formula and atomistically computed phonon dispersions (with interatomic potentials developed in ), it is possible to predictively calculate lattice thermal conductivity curves for nanowires, in good agreement with experiments. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. Berber et al. have calculated the phonon thermal conductivity of isolated nanotubes. In condensed matter physics, the recoil temperature is a fundamental lower limit of temperature attainable by some laser cooling schemes, and corresponds to the kinetic energy imparted in an atom initially at rest by the spontaneous emission of a photon. It may be that this weak coupling, which is problematic for mechanical applications of nanotubes, is an advantage for thermal applications. ====Phonon density of states for nanotubes==== The phonon density of states is to calculated through band structure of isolated nanotubes, which is studied in Saito et al. and Sanchez-Portal et al. Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics. As the devices continue to shrink further into the sub-100 nm range following the trend predicted by Moore’s law, the topic of thermal properties and transport in such nanoscale devices becomes increasingly important. Therefore, the phonon thermal conductivity displays a peak and decreases with increasing temperature. ",-8,1.5,5.5,0.7812,0.0761,C
-Calculate the change in Gibbs energy of $35 \mathrm{~g}$ of ethanol (mass density $0.789 \mathrm{~g} \mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from 1 atm to 3000 atm.,"The Gibbs energy of any system is and an infinitesimal change in G, at constant temperature and pressure, yields :dG=dU+pdV-TdS. The Gibbs free energy is expressed as : G(p,T) = U + pV - TS = H - TS where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy. This implies that at equilibrium Q_\text{r} = K_\text{eq} and \Delta_\text{r} G = 0. ==Standard Gibbs energy change of formation== Table of selected substancesCRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide. Alcoholic fermentation converts one mole of glucose into two moles of ethanol and two moles of carbon dioxide, producing two moles of ATP in the process. In isothermal, isobaric systems, Gibbs free energy can be thought of as a ""dynamic"" quantity, in that it is a representative measure of the competing effects of the enthalpic and entropic driving forces involved in a thermodynamic process. thumb|upright=1.9|Relation to other relevant parameters The temperature dependence of the Gibbs energy for an ideal gas is given by the Gibbs–Helmholtz equation, and its pressure dependence is given by \frac{G}{N} = \frac{G^\circ}{N} + kT\ln \frac{p}{p^\circ}. or more conveniently as its chemical potential: \frac{G}{N} = \mu = \mu^\circ + kT\ln \frac{p}{p^\circ}. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The Gibbs free energy change , measured in joules in SI) is the maximum amount of non-volume expansion work that can be extracted from a closed system (one that can exchange heat and work with its surroundings, but not matter) at fixed temperature and pressure. The Gibbs energy is the thermodynamic potential that is minimized when a system reaches chemical equilibrium at constant pressure and temperature when not driven by an applied electrolytic voltage. In an isobaric process, the pressure remains constant, so the heat interaction is the change in enthalpy. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. Substance (state) ΔfG° ΔfG° Substance (state) (kJ/mol) (kcal/mol) NO(g) 87.6 20.9 NO2(g) 51.3 12.3 N2O(g) 103.7 24.78 H2O(g) −228.6 −54.64 H2O(l) −237.1 −56.67 CO2(g) −394.4 −94.26 CO(g) −137.2 −32.79 CH4(g) −50.5 −12.1 C2H6(g) −32.0 −7.65 C3H8(g) −23.4 −5.59 C6H6(g) 129.7 29.76 C6H6(l) 124.5 31.00 The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). Moreover, we also have \begin{align} K_\text{eq} &= e^{-\frac{\Delta_\text{r} G^\circ}{RT}}, \\\ \Delta_\text{r} G^\circ &= -RT\left(\ln K_\text{eq}\right) = -2.303\,RT\left(\log_{10} K_\text{eq}\right), \end{align} which relates the equilibrium constant with Gibbs free energy. At constant pressure the above equation produces a Maxwell relation that links the change in open cell voltage with temperature T (a measurable quantity) to the change in entropy S when charge is passed isothermally and isobarically. Further, Gibbs stated: In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body... In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure-volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. When a system transforms reversibly from an initial state to a final state under these conditions, the decrease in Gibbs free energy equals the work done by the system to its surroundings, minus the work of the pressure forces. Approximately 2.8 gallons of ethanol are produced from one bushel of corn (0.42 liter per kilogram). However, simply substituting the above integrated result for U into the definition of G gives a standard expression for G: :\begin{align} G &= U + p V - TS\\\ &= \left(T S - p V + \sum_i \mu_i N_i \right) + p V - T S\\\ &= \sum_i \mu_i N_i. \end{align} This result shows that the chemical potential of a substance i is its (partial) mol(ecul)ar Gibbs free energy. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). thumb|400px|Diagram showing pressure difference induced by a temperature difference. The quantities on the right are all directly measurable. ==Useful identities to derive the Nernst equation== During a reversible electrochemical reaction at constant temperature and pressure, the following equations involving the Gibbs free energy hold: *\Delta_\text{r} G = \Delta_\text{r} G^\circ + R T \ln Q_\text{r} (see chemical equilibrium), *\Delta_\text{r} G^\circ = -R T \ln K_\text{eq} (for a system at chemical equilibrium), *\Delta_\text{r} G = w_\text{elec,rev} = -nF\mathcal{E} (for a reversible electrochemical process at constant temperature and pressure), *\Delta_\text{r} G^\circ = -nF\mathcal{E}^\circ (definition of \mathcal{E}^\circ), and rearranging gives \begin{align} nF\mathcal{E}^\circ &= RT \ln K_\text{eq}, \\\ nF\mathcal{E} &= nF\mathcal{E}^\circ - R T \ln Q_\text{r}, \\\ \mathcal{E} &= \mathcal{E}^\circ - \frac{R T}{n F} \ln Q_\text{r}, \end{align} which relates the cell potential resulting from the reaction to the equilibrium constant and reaction quotient for that reaction (Nernst equation), where * , Gibbs free energy change per mole of reaction, * , Gibbs free energy change per mole of reaction for unmixed reactants and products at standard conditions (i.e. 298K, 100kPa, 1M of each reactant and product), * , gas constant, * , absolute temperature, * , natural logarithm, * , reaction quotient (unitless), * , equilibrium constant (unitless), * , electrical work in a reversible process (chemistry sign convention), * , number of moles of electrons transferred in the reaction, * , Faraday constant (charge per mole of electrons), * \mathcal{E}, cell potential, * \mathcal{E}^\circ, standard cell potential. ",12,0.00539,6.0,14.5115,-17,A
-The promotion of an electron from the valence band into the conduction band in pure $\mathrm{TIO}_2$ by light absorption requires a wavelength of less than $350 \mathrm{~nm}$. Calculate the energy gap in electronvolts between the valence and conduction bands.,"Within the concept of bands, the energy gap between the valence band and the conduction band is the band gap. For materials with a direct band gap, valence electrons can be directly excited into the conduction band by a photon whose energy is larger than the bandgap. In graphs of the electronic band structure of solids, the band gap refers to the energy difference (often expressed in electronvolts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. This formula is valid only for light with photon energy larger, but not too much larger, than the band gap (more specifically, this formula assumes the bands are approximately parabolic), and ignores all other sources of absorption other than the band-to-band absorption in question, as well as the electrical attraction between the newly created electron and hole (see exciton). By plotting certain powers of the absorption coefficient against photon energy, one can normally tell both what value the band gap is, and whether or not it is direct. The term ""band gap"" refers to the energy difference between the top of the valence band and the bottom of the conduction band. A band gap is an energy range in a solid where no electron states can exist due to the quantization of energy. However, in order for a valence band electron to be promoted to the conduction band, it requires a specific minimum amount of energy for the transition. Especially in condensed-matter physics, an energy gap is often known more abstractly as a spectral gap, a term which need not be specific to electrons or solids. ==Band gap== If an energy gap exists in the band structure of a material, it is called band gap. The relationship between band gap energy and temperature can be described by Varshni's empirical expression (named after Y. P. Varshni), :E_g(T)=E_g(0)-\frac{\alpha T^2}{T+\beta}, where Eg(0), α and β are material constants. A semiconductor will not absorb photons of energy less than the band gap; and the energy of the electron-hole pair produced by a photon is equal to the bandgap energy. The band gap is called ""direct"" if the crystal momentum of electrons and holes is the same in both the conduction band and the valence band; an electron can directly emit a photon. On the other hand, for an indirect band gap, the formula is: :\alpha \propto \frac{(h u- E_{\text{g}}+E_{\text{p}})^2}{\exp(\frac{E_{\text{p}}}{kT})-1} + \frac{(h u- E_{\text{g}}-E_{\text{p}})^2}{1-\exp(-\frac{E_{\text{p}}}{kT})} where: *E_{\text{p}} is the energy of the phonon that assists in the transition *k is Boltzmann's constant *T is the thermodynamic temperature This formula involves the same approximations mentioned above. For the same reason as above, light with a photon energy close to the band gap can penetrate much farther before being absorbed in an indirect band gap material than a direct band gap one (at least insofar as the light absorption is due to exciting electrons across the band gap). In solid-state physics, the most important spectral gap is for the many-body system of electrons in a solid material, in which case it is often known as an energy gap. Since the mid-gap states do exist within some depth of the semiconductor, they must be a mixture (a Fourier series) of valence and conduction band states from the bulk. In solid-state physics, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. Group Material Symbol Band gap (eV) @ 302K Reference III–V Aluminium nitride AlN 6.0 IV Diamond C 5.5 IV Silicon Si 1.14 IV Germanium Ge 0.67 III–V Gallium nitride GaN 3.4 III–V Gallium phosphide GaP 2.26 III–V Gallium arsenide GaAs 1.43 IV–V Silicon nitride Si3N4 5 IV–VI Lead(II) sulfide PbS 0.37 IV–VI Silicon dioxide SiO2 9 Copper(I) oxide Cu2O 2.1 ==Optical versus electronic bandgap== In materials with a large exciton binding energy, it is possible for a photon to have just barely enough energy to create an exciton (bound electron–hole pair), but not enough energy to separate the electron and hole (which are electrically attracted to each other). In solid-state physics, an energy gap is an energy range in a solid where no electron states exist, i.e. an energy range where the density of states vanishes. In contrast, for materials with an indirect band gap, a photon and phonon must both be involved in a transition from the valence band top to the conduction band bottom, involving a momentum change. If there is a small band gap (Eg), then the flow of electrons from valence to conduction band is possible only if an external energy (thermal, etc.) is supplied; these groups with small Eg are called semiconductors. In an ""indirect"" gap, a photon cannot be emitted because the electron must pass through an intermediate state and transfer momentum to the crystal lattice. ",0.139,5.4,3.54,-0.0301,0.6321205588,C
-"Although the crystallization of large biological molecules may not be as readily accomplished as that of small molecules, their crystal lattices are no different. Tobacco seed globulin forms face-centred cubic crystals with unit cell dimension of $12.3 \mathrm{~nm}$ and a density of $1.287 \mathrm{~g} \mathrm{~cm}^{-3}$. Determine its molar mass.","The molecular formula C18H22O6 (molar mass: 334.36 g/mol, exact mass: 334.1416 u) may refer to: * Combretastatin * Combretastatin B-1 The molecular formula C3H9O6P (molar mass: 172.07 g/mol, exact mass: 172.0137 u) may refer to: * Glycerol 1-phosphate * Glycerol 2-phosphate (BGP) * Glycerol 3-phosphate Category:Molecular formulas A seed crystal is a small piece of single crystal or polycrystal material from which a large crystal of typically the same material is grown in a laboratory. The molecular formula C21H26O3 (molar mass: 326.42 g/mol, exact mass: 326.1882 u) may refer to: * Acitretin * Buparvaquone * Moxestrol * Octabenzone * RU-16117 * 11-Hydroxycannabinol Category:Molecular formulas The molecular formula C9H16N3O14P3 (molar mass: 483.16 g/mol) may refer to: * Cytidine triphosphate * Arabinofuranosylcytosine triphosphate Used to replicate material, the use of seed crystal to promote growth avoids the otherwise slow randomness of natural crystal growth and allows manufacture on a scale suitable for industry. ==Crystal enlargement== The large crystal can be grown by dipping the seed into a supersaturated solution, into molten material that is then cooled, or by growth on the seed face by passing vapor of the material to be grown over it. ==Theory== The theory behind this effect is thought to derive from the physical intermolecular interaction that occurs between compounds in a supersaturated solution (or possibly vapor). This was especially important in pharmaceutical applications where slight changes in molar mass (e.g. aggregation) or shape may result in different biological activity. The placement of a seed crystal into solution allows the recrystallization process to expedite by eliminating the need for random molecular collision or interaction. Absolute molar mass is a process used to determine the characteristics of molecules. == History == The first absolute measurements of molecular weights (i.e. made without reference to standards) were based on fundamental physical characteristics and their relation to the molar mass. In order to gain information about a polydisperse mixture of molar masses, a method for separating the different sizes was developed. To obtain molar mass, light scattering instruments need to measure the intensity of light scattered at zero angle. The purely mathematical root mean square radius is defined as the radii making up the molecule multiplied by the mass at that radius. == Bibliography == *A. Einstein, Ann. Phys. 33 (1910), 1275 *C.V. Raman, Indian J. Phys. 2 (1927), 1 *P.Debye, J. Appl. Phys. 15 (1944), 338 *B.H. Zimm, J. Chem. Phys. 13 (1945), 141 *B.H. Zimm, J. Chem. Phys. 16 (1948), 1093 *B.H. Zimm, R.S. Stein and P. Dotty, Pol. Bull. 1,(1945), 90 *M. Fixman, J. Chem. Phys. 23 (1955), 2074 *A.C. Ouano and W. Kaye J. Poly. Sci. A1(12) (1974), 1151 *Z. Grubisic, P. Rempp, and H. Benoit, J. Polym. The problem was that the system was calibrated according to the Vh characteristics of polymer standards that are not directly related to the molar mass. Also during the process of tempering chocolate, seed crystals can be used to promote the growth of favorable type V crystals ==See also== * Crystal structure * Crystallization * Laser heated pedestal growth * Micro-pulling-down * Polycrystal * Single crystal * Wafer (electronics) * Disappearing polymorphs ==References== Category:Crystals The next step is to convert the time at which the samples eluted into a measurement of molar mass. This information is the Root Mean Square radius of the molecule (RMS or Rg). As the demands on polymer properties increased, the necessity of getting absolute information on the molar mass and size also increased. Seeding is therefore said to decrease the necessary amount of time needed for nucleation to occur in a recrystallization process. ==Uses== One example where a seed crystal is used to grow large boules or ingots of a single crystal is the semiconductor industry where methods such as the Czochralski process or Bridgman technique are employed. This interaction can potentiate intermolecular forces between the separate molecules and form a basis for a crystal lattice. As previously noted, the MALS detector can also provide information about the size of the molecule. If the relationship between the molar mass and Vh of the standard is not the same as that of the unknown sample, then the calibration is invalid. A low angle light scattering system was developed in the early 1970s that allowed a single measurement to be used to calculate the molar mass. ",0.6296296296,0.5,22.0,1.8,3.61,E
-An electron is accelerated in an electron microscope from rest through a potential difference $\Delta \phi=100 \mathrm{kV}$ and acquires an energy of $e \Delta \phi$. What is its final speed?,"The kinetic energy Ke of an electron moving with velocity v is: :\displaystyle K_{\mathrm{e}} = (\gamma - 1)m_{\mathrm{e}} c^2, where me is the mass of electron. This wavelength, for example, is equal to 0.0037 nm for electrons accelerated across a 100,000-volt potential. The speed of an electron can approach, but never reach, the speed of light in vacuum, c. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. For an electron with rest mass m0, the rest energy is equal to: :\textstyle E_{\mathrm p} = m_0 c^2, where c is the speed of light in vacuum. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. The electron (symbol e) is on the left. For example, the Stanford linear accelerator can accelerate an electron to roughly 51 GeV. This energy is assumed to equal the electron's rest energy, defined by special relativity (E = mc2). The energy emission in turn causes a recoil of the electron, known as the Abraham–Lorentz–Dirac Force, which creates a friction that slows the electron. Electrons can be accelerated by suitable electric (or magnetic) fields, thereby acquiring kinetic energy. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. The required energy of the electrons is typically in the range 20–200 eV. The electron microscope directs a focused beam of electrons at a specimen. Electrons radiate or absorb energy in the form of photons when they are accelerated. For an electron, it has a value of . Relativistic electron beams are streams of electrons moving at relativistic speeds. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. The upper bound of the electron radius of 10−18 meters can be derived using the uncertainty relation in energy. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. When an excited electron falls back to a state of lower energy, it undergoes electron relaxation (deexcitationSakho, Ibrahima. ",1.88,8,0.28209479,1.51,0.333333,A
-The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature. By how much does the standard molar enthalpy of $\mathrm{SO}_2(\mathrm{~g})$ increase when the temperature is raised from $298.15 \mathrm{~K}$ to $1500 \mathrm{~K}$ ?,"The molar heat capacity generally increases with the molar mass, often varies with temperature and pressure, and is different for each state of matter. The following is a table of some constant- pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* H2 28.9 5.0 29.6 5.1 41.2 7.9 Not saturated.""Hydrogen"" NIST Chemistry WebBook, SRD 69, online. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.Lange's Handbook of Chemistry, 10th ed. p. 1524 All methods for the measurement of specific heat apply to molar heat capacity as well. ==Units== The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. Therefore, the word ""molar"", not ""specific"", should always be used for this quantity. ==Definition== The molar heat capacity of a substance, which may be denoted by cm, is the heat capacity C of a sample of the substance, divided by the amount (moles) n of the substance in the sample: :cm{} \;=\; \frac{C}{n} \;=\; \frac{1}{n} \lim_{\Delta T \rightarrow 0}\frac{Q}{\Delta T} where Q is the amount of heat needed to raise the temperature of the sample by ΔT. Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Then the molar heat capacity (at constant volume) would be :cV,m = fR where R is the ideal gas constant. On the other hand, the vibration modes only start to become active around 350 K (77 °C) Accordingly, the molar heat capacity cP,m is nearly constant at 29.1 J⋅K−1⋅mol−1 from 100 K to about 300 °C. According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). Since the molar heat capacity of a substance is the specific heat c times the molar mass of the substance M/N its numerical value is generally smaller than that of the specific heat. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). The value thus obtained is said to be the molar heat capacity at constant pressure (or isobaric), and is often denoted cP,m, cp,m, cP,m, etc. For example, ""H2O: 75.338 J⋅K−1⋅mol−1 (25 °C, 101.325 kPa)"" W. Wagner, J. R. Cooper, A. Dittmann, J. Kijima, H.-J. Kretzschmar, A. Kruse, R. Mare, K. Oguchi, H. Sato, I. Stöcker, O. Šifner, Y. Takaishi, I. Tanishita, J. Trübenbach and Th. Willkommen (2000): ""The IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam"", ASME J. Eng. Gas Turbines and Power, volume 122, pages 150–182 When not specified, published values of the molar heat capacity cm generally are valid for some standard conditions for temperature and pressure. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. These parameters are usually specified when giving the molar heat capacity of a substance. At high enough temperature, its molar heat capacity then should be cP,m = 7.5 R = 62.63 J⋅K−1⋅mol−1. ",1.94,1.07,1.2,62.2,1.06,D
-Suppose that the normalized wavefunction for an electron in a carbon nanotube of length $L=10.0 \mathrm{~nm}$ is: $\psi=(2 / L)^{1 / 2} \sin (\pi x / L)$. Calculate the probability that the electron is between $x=4.95 \mathrm{~nm}$ and $5.05 \mathrm{~nm}$.,"b) Linear dependence of the electron energy on the wave vector in CNTs; c) Dispersion relation near the Fermi energy for a semiconducting CNT; d) Dispersion relation near the Fermi energy for a metallic CNT Conduction in single-walled carbon nanotubes is quantized due to their one-dimensionality and the number of allowed electronic states is limited, if compared to bulk graphite. If a carbon nanotube is a ballistic conductor, but the contacts are nontransparent, the transmission probability, T, is reduced by back- scattering in the contacts. Scattering of electrons by optical phonons in carbon nanotube channels has two requirements: * The traveled length in the conduction channel between source and drain has to be greater than the optical phonon mean free path * The electron energy has to be greater than the critical optical phonon emission energy === Schottky barrier Ballistic conduction === thumb|400px|Figure 2: Example of the band structure of a ballistic CNT FET. In order to estimate the current in the carbon nanotube channel, the Landauer formula can be applied, which considers a one-dimensional channel, connected to two contacts – source and drain. In semiconducting CNTs at room temperature and for low energies, the mean free path is determined by the electron scattering from acoustic phonons, which results in lm ≈ 0.5μm. Single-walled carbon nanotubes in the fields of quantum mechanics and nanoelectronics, have the ability to conduct electricity. When ballistically conducted, the electrons travel through the nanotubes channel without experiencing scattering due to impurities, local defects or lattice vibrations. ""Carbon Nanotube and Graphene Device Physics"", Cambridge UP, 2011. Another way to make carbon nanotube transistors has been to use random networks of them. Carbon nanotube transistors as logic-gate circuits with densities comparable to modern CMOS technology has not yet been demonstrated. The potential of carbon nanotubes was demonstrated in 2003 when room- temperature ballistic transistors with ohmic metal contacts and high-k gate dielectric were reported, showing 20–30x higher ON current than state-of-the- art Si MOSFETs. Carbon nanotube chemistry involves chemical reactions, which are used to modify the properties of carbon nanotubes (CNTs). In order to derive the current-voltage (I-V) characteristics for a ballistic CNT FET, one can start with Planck's postulate, which relates the energy of the i-th state to its frequency: E_i=h u_i=\frac{h}{2e}\frac{2e}{T_i}=\frac{h}{2e}I_i The total current for a many-state system is then the sum over the energy of each state multiplied by the occupation probability function, in this case the Fermi–Dirac statistics: I_i=\frac{2e}{h}\sum_{i}E_i\frac{1}{1+e^{\frac{E-E_f}{k_BT}}} For a system with dense states, the discrete sum can be approximated by an integral: I_i=\frac{2e}{h}\int \frac{1}{1+e^{\frac{E-E_f}{k_BT}}}dE In CNT FETs, the charge carriers move either left (negative velocity) or right (positive velocity) and the resulting net current is called drain current. When ballistic in nature conductance can be treated as if the electrons experience no scattering. == Conductance quantization and Landauer formula == thumb|400px|Figure 1: a) Energy contour plot of the electronic band structure in CNTs.; thumbnail|right|Plot of probit function In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. The exceptional electrical and mechanical properties of carbon nanotubes have made them alternatives to the traditional electrical actuators for both microscopic and macroscopic applications. These processes were essential for low yield production of carbon nanotubes where carbon particles, amorphous carbon particles and coatings comprised a significant percentage of the overall material and are still important for the introduction of surface functional groups. A carbon nanotube quantum dot (CNT QD) is a small region of a carbon nanotube in which electrons are confined. ==Formation== A CNT QD is formed when electrons are confined to a small region within a carbon nanotube. The CNT QD is modelled as an Anderson-type model, which can be reduced by Schrieffer-Wolff transformation to an effective Kondo-type model at low temperature. ==Other nanotube system== Similar mesoscopic devices have been constructed from elements other than carbon. ""Nanowelded Carbon Nanotubes from Field-effect Transistors to Solar Microcells"", Heidelberg: Springer, 2009. Major obstacles to nanotube-based microelectronics include the absence of technology for mass production, circuit density, positioning of individual electrical contacts, sample purity, control over length, chirality and desired alignment, thermal budget and contact resistance. Carbon nanotubes (CNTs) are cylinders of one or more layers of graphene (lattice). ",0.08,226,1.0,0.020,0.011,D
-"A sample of the sugar D-ribose of mass $0.727 \mathrm{~g}$ was placed in a calorimeter and then ignited in the presence of excess oxygen. The temperature rose by $0.910 \mathrm{~K}$. In a separate experiment in the same calorimeter, the combustion of $0.825 \mathrm{~g}$ of benzoic acid, for which the internal energy of combustion is $-3251 \mathrm{~kJ} \mathrm{~mol}^{-1}$, gave a temperature rise of $1.940 \mathrm{~K}$. Calculate the enthalpy of formation of D-ribose.","However the standard enthalpy of combustion is readily measurable using bomb calorimetry. The standard enthalpy of formation is then determined using Hess's law. That is, the heat of combustion, ΔH°comb, is the heat of reaction of the following process: : (std.) + (c + - ) (g) → c (g) + (l) + (g) Chlorine and sulfur are not quite standardized; they are usually assumed to convert to hydrogen chloride gas and or gas, respectively, or to dilute aqueous hydrochloric and sulfuric acids, respectively, when the combustion is conducted in a bomb calorimeter containing some quantity of water. == Ways of determination == ===Gross and net=== Zwolinski and Wilhoit defined, in 1972, ""gross"" and ""net"" values for heats of combustion. The calorific value is the total energy released as heat when a substance undergoes complete combustion with oxygen under standard conditions. They may also be calculated as the difference between the heat of formation ΔH of the products and reactants (though this approach is somewhat artificial since most heats of formation are typically calculated from measured heats of combustion). For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. The experimental heat of formation of ethane is -20.03 kcal/mol and ethane consists of 2 P groups. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. It may be expressed with the quantities: * energy/mole of fuel * energy/mass of fuel * energy/volume of the fuel There are two kinds of enthalpy of combustion, called high(er) and low(er) heat(ing) value, depending on how much the products are allowed to cool and whether compounds like are allowed to condense. The temperatures mentioned here are for a stoichiometric fuel-oxidizer mixture (i.e. equivalence ratio φ = 1). alt=|thumb| Fischer Projection Ribose is a simple sugar and carbohydrate with molecular formula C5H10O5 and the linear-form composition H−(C=O)−(CHOH)4−H. 150px 150px Comparison of the chemical structures of ribose (top) and deoxyribose (bottom). This is the same as the thermodynamic heat of combustion since the enthalpy change for the reaction assumes a common temperature of the compounds before and after combustion, in which case the water produced by combustion is condensed to a liquid. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. Since the heat of combustion of these elements is known, the heating value can be calculated using Dulong's Formula: LHV [kJ/g]= 33.87mC \+ 122.3(mH \- mO ÷ 8) + 9.4mS where mC, mH, mO, mN, and mS are the contents of carbon, hydrogen, oxygen, nitrogen, and sulfur on any (wet, dry or ash free) basis, respectively. === Higher heating value === The higher heating value (HHV; gross energy, upper heating value, gross calorific value GCV, or higher calorific value; HCV) indicates the upper limit of the available thermal energy produced by a complete combustion of fuel. The enthalpy of reaction can then be analyzed by applying Hess's Law, which states that the sum of the enthalpy changes for a number of individual reaction steps equals the enthalpy change of the overall reaction. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. The following figure illustrates that the effects of dissociation tend to lower the adiabatic flame temperature. Incomplete reaction at higher temperature further curtails the effect of a larger heat of combustion. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). ",-0.347,-1270,0.4772,1.3,7.136,B
-"An electron confined to a metallic nanoparticle is modelled as a particle in a one-dimensional box of length $L$. If the electron is in the state $n=1$, calculate the probability of finding it in the following regions: $0 \leq x \leq \frac{1}{2} L$.","To a first approximation (i.e. assuming that the charges are distributed randomly), the molar configurational electronic entropy is given by: :S \approx n_\text{sites} \left [ x \ln x + (1-x) \ln (1-x) \right ] where is the fraction of sites on which a localized electron/hole could reside (typically a transition metal site), and is the concentration of localized electrons/holes. Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. The probability of occupation of each eigenstate is given by the Fermi function, : :p(E)=f=\frac{1}{e^{(E-E_{\rm F}) / k_{\rm B} T} + 1} where is the Fermi energy and is the absolute temperature. For one-electron systems, the electron density at any point is proportional to the square magnitude of the wavefunction. == Definition == The electronic density corresponding to a normalised N-electron wavefunction \Psi (with \textbf r and s denoting spatial and spin variables respectively) is defined as : \rho(\mathbf{r}) = \langle\Psi|\hat{\rho}(\mathbf{r})|\Psi\rangle, where the operator corresponding to the density observable is :\hat{\rho}(\mathbf{r}) = \sum_{i=1}^{N}\ \delta(\mathbf{r}-\mathbf{r}_{i}). The density is determined, through definition, by the normalised N-electron wavefunction which itself depends upon 4N variables (3N spatial and N spin coordinates). Some softwareor example, the Spartan program from Wavefunction, Inc. also allows for specification of the electron density in terms of percentage of total electrons enclosed. For every possible transfer of an electron from an occupied site i to an unoccupied site j , the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e., \Delta E>=0 . In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The Fermi–Dirac distribution implies that each eigenstate of a system, , is occupied with a certain probability, . Of course, the localized charges are not distributed randomly, as the charges will interact electrostatically with one another, and so the above formula should only be regarded as an approximation to the configurational atomic entropy. We will solve for each independently: Let E be an energy value above the well (E>0) * For 0 < x < (a-b): \begin{align} \frac{-\hbar^2}{2m} \psi_{xx} &= E \psi \\\ \Rightarrow \psi &= A e^{i \alpha x} + A' e^{-i \alpha x} & \left( \alpha^2 = {2mE \over \hbar^2} \right) \end{align} *For -b : \begin{align} \frac{-\hbar^2}{2m} \psi_{xx} &= (E+V_0)\psi \\\ \Rightarrow \psi &= B e^{i \beta x} + B' e^{-i \beta x} & \left( \beta^2 = {2m(E+V_0) \over \hbar^2} \right). \end{align} To find u(x) in each region, we need to manipulate the electron's wavefunction: \begin{align} \psi(0 And in the same manner: u(-b To complete the solution we need to make sure the probability function is continuous and smooth, i.e.: \psi(0^{-})=\psi(0^{+}) \qquad \psi'(0^{-})=\psi'(0^{+}). In these situations, the density simplifies to :\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2. == General properties == From its definition, the electron density is a non-negative function integrating to the total number of electrons. Moreover, in condensed matter and molecules, the electron clouds of the atoms usually overlap to some extent, and some of the electrons may roam over a large region encompassing two or more atoms. Switching from summing over individual states to integrating over energy levels, the entropy can be written as: :S=-k_{\rm B} \int n(E) \left [ p(E) \ln p(E) +(1- p(E)) \ln \left ( 1- p(E)\right ) \right ]dE where is the density of states of the solid. Metals have non-zero density of states at the Fermi level. As the density of states at the Fermi level varies widely between systems, this approximation is a reasonable heuristic for inferring when it may be necessary to include electronic entropy in the thermodynamic description of a system; only systems with large densities of states at the Fermi level should exhibit non-negligible electronic entropy (where large may be approximately defined as ). == Application to different materials classes == Insulators have zero density of states at the Fermi level due to their band gaps. The occupation numbers are not limited to the range of zero to two, and therefore sometimes even the response density can be negative in certain regions of space. ==Overview== In molecules, regions of large electron density are usually found around the atom, and its bonds. Electrons and Holes in Semiconductors: With Applications to Transistor Electronics, Bell Telephone Laboratories series, Van Nostrand. The observation of this is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the Coulomb gap. An earlier version containing some of the material appeared in the Bell Labs Technical Journal in 1949. ==Contents== The arrangement of chapters is as follows: Part I INTRODUCTION TO TRANSISTOR ELECTRONICS *Chapter 1: THE BULK PROPERTIES OF SEMICONDUCTORS *Chapter 2: THE TRANSISTOR AS A CIRCUIT ELEMENT *Chapter 3: QUANTITATIVE STUDIES OF INJECTION OF HOLES AND ELECTRONS *Chapter 4: ON THE PHYSICAL THEORY OF TRANSISTORS *Chapter 5: QUANTUM STATES, ENERGY BANDS, AND BRILLOUIN ZONES PART II DESCRIPTIVE THEORY OF SEMICONDUCTORS *Chapter 6: VELOCITIES AND CURRENTS FOR ELECTRONS I N CRYSTALS *Chapter 7: ELECTRONS AND HOLES IN ELECTRIC AND MAGNETIC FIELDS *Chapter 8: INTRODUCTORY THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 9: DISTRIBUTIONS OF QUANTUM STATES IN ENERGY *Chapter 10: FERMI-DIRAC STATISTICS FOR SEMICONDUCTORS *Chapter 11: MATHEMATICAL THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 12: APPLICATIONS TO TRANSISTOR ELECTRONICS PART III QUANTUM MECHANICAL FOUNDATIONS *Chapter 13: INTRODUCTION TO PART III *Chapter 14: ELEMENTARY QUANTUM MECHANICS WITH CIRCUIT THEORY ANALOGUES *Chapter 15: THEORY OF ELECTRON AND HOLE VELOCITIES, CURRENTS AND ACCELERATIONS *Chapter 16: STATISTICAL MECHANICS FOR SEMICONDUCTORS *Chapter 17: THE THEORY OF TRANSITION PROBABILITIES FOR HOLES AND ELECTRONS *APPENDIX A Units *APPENDIX B Periodic Table *APPENDIX C Values of the physical constants *APPENDIX D Energy conversion chart *NAME INDEX *SUBJECT INDEX Category:1950 non-fiction books Category:1950 in science Category:Physics textbooks Together with the normalization property places acceptable densities within the intersection of L1 and L3 – a superset of \mathcal{J}_{N}. == Topology == The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus. === Nuclear cusp condition === The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron- nucleus Coulomb potential. ",6,0.829,-20.0,4500,0.5,E
-"The carbon-carbon bond length in diamond is $154.45 \mathrm{pm}$. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is $3.516 \mathrm{~g} \mathrm{~cm}^{-3}$.","The atomic packing factor of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the vertices of the structure and are as large as possible without overlapping) is ≈ 0.34,. significantly smaller (indicating a less dense structure) than the packing factors for the face- centered and body-centered cubic lattices.. The toughness of natural diamond has been measured as 7.5–10 MPa·m1/2. It also has a high density, ranging from 3150 to 3530 kilograms per cubic metre (over three times the density of water) in natural diamonds and 3520 kg/m in pure diamond. St Edmundsbury Press Ltd, Bury St Edwards. ==External links== *Properties of diamond *Properties of diamond (S. Sque, PhD thesis, 2005, University of Exeter, UK) Category:Diamond Category:Allotropes of carbon Diamond Category:Superhard materials Diamond is the allotrope of carbon in which the carbon atoms are arranged in the specific type of cubic lattice called diamond cubic. Diamond is extremely strong owing to its crystal structure, known as diamond cubic, in which each carbon atom has four neighbors covalently bonded to it. thumb|upright=1.25|Main diamond producing countries Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. A lattice of 3×3×3 unit cells thumb|245px|Molar volume vs. pressure at room temperature. thumb|3D ball-and- stick model of a diamond lattice The precise tensile strength of diamond is unknown, though strength up to has been observed, and theoretically it could be as high as depending on the sample volume/size, the perfection of diamond lattice and on its orientation: Tensile strength is the highest for the [100] crystal direction (normal to the cubic face), smaller for the [110] and the smallest for the [111] axis (along the longest cube diagonal). thumb|250px|3D ball-and-stick model of a diamond lattice The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. The following issues are considered: * The hardness of diamond and its ability to cleave strongly depend on the crystal orientation. A similar proportion of diamonds comes from the lower mantle at depths between 660 and 800 km. Diamond is thermodynamically stable at high pressures and temperatures, with the phase transition from graphite occurring at greater temperatures as the pressure increases. The Allnatt Diamond is a diamond measuring 101.29 carats (20.258 g) with a cushion cut, rated in color as Fancy Vivid Yellow by the Gemological Institute of America. Thermal conductivity of natural diamond was measured to be about 2200 W/(m·K), which is five times more than silver, the most thermally conductive metal. Diamonds are carbon crystals that form under high temperatures and extreme pressures such as deep within the Earth. Diamonds crystallize in the diamond cubic crystal system (space group Fdm) and consist of tetrahedrally, covalently bonded carbon atoms. The diamond crystal lattice is exceptionally strong, and only atoms of nitrogen, boron, and hydrogen can be introduced into diamond during the growth at significant concentrations (up to atomic percents). Some extrasolar planets may be almost entirely composed of diamond. Thus, graphite is much softer than diamond. Unlike many other minerals, the specific gravity of diamond crystals (3.52) has rather small variation from diamond to diamond. ==Hardness and crystal structure== Known to the ancient Greeks as (, 'proper, unalterable, unbreakable') and sometimes called adamant, diamond is the hardest known naturally occurring material, and serves as the definition of 10 on the Mohs scale of mineral hardness. At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. The Amsterdam Diamond is a black diamond weighing , and has 145 facets. This was determined by comparing the ratios of carbon isotopes present. == Optical and electronic properties == The optical absorption for all diamondoids lies deep in the ultraviolet spectral region with optical band gaps around 6 electronvolts and higher. ",7.654,0.0625,9.13,2,90,A
-"A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of seawater in the visible region is $6.2 \times 10^{-3} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~cm}^{-1}$, calculate the depth at which a diver will experience half the surface intensity of light.","To calculate this coefficient, light energy is measured at a series of depths from the surface to the depth of 1% illumination. As the depth increases, more light is absorbed by the water. By using artificial light, it is possible to view an object in full color at greater depths. ==Need== Water attenuates light by absorption, so use of a dive light will improve a diver's underwater vision at depth. Maximum Operating Depth (MOD) in metres sea water (msw) for pO2 1.2 to 1.6 MOD (msw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 390 190 123 90 70 66 57 47 40 34 30 22 17 13 10 8 6 1.5 365 178 115 84 65 61 53 44 37 32 28 20 15 11 9 7 5 1.4 340 165 107 78 60 57 48 40 34 29 25 18 13 10 8 6 4 1.3 315 153 98 71 55 52 44 36 31 26 23 16 12 9 6 4 3 1.2 290 140 90 65 50 47 40 33 28 23 20 14 10 7 5 3 2 These depths are rounded to the nearest metre. ==See also== * == Notes == == References == == Sources == * * * * Category:Dive planning Category:Underwater diving safety Category:Breathing gases By directing the same type of beam downwards, the depth to the bottom of the ocean could be calculated. A dive light is a light source carried by an underwater diver to illuminate the underwater environment. The United States Navy Experimental Diving Unit continues to evaluate dive lights for wet and dry illumination output, battery duration, watertight integrity, as well as maximum operating depth. Then, the exponential decline in light is calculated using Beer’s Law with the equation: {I_z \over I_0}= e^{-kz} where k is the light attenuation coefficient, Iz is the intensity of light at depth z, and I0 is the intensity of light at the ocean surface.Idso, Sherwood B. and Gilbert, R. Gene (1974) On the Universality of the Poole and Atkins Secchi Disk: Light Extinction Equation British Ecological Society. Of this total pressure which can be tolerated by the diver, 1 atmosphere is due to surface pressure of the Earth's air, and the rest is due to the depth in water. For example, if a gas contains 36% oxygen and the maximum pO2 is 1.4 bar, the MOD (msw) is 10 msw/bar x [(1.4 bar / 0.36) − 1] = 28.9 msw. == Tables == Maximum Operating Depth (MOD) in feet sea water (fsw) for pO2 1.2 to 1.6 MOD (fsw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 1287 627 407 297 231 218 187 156 132 114 99 73 55 42 33 26 20 1.5 1205 586 380 276 215 203 173 144 122 105 91 66 50 38 29 22 17 1.4 1122 545 352 256 198 187 160 132 111 95 83 59 44 33 25 18 13 1.3 1040 503 325 235 182 171 146 120 101 86 74 53 39 28 21 15 10 1.2 957 462 297 215 165 156 132 108 91 77 66 46 33 24 17 11 7 These depths are rounded to the nearest foot. 300px|thumb|right|A canister style dive lightNight diving is underwater diving done during the hours of darkness. In addition to light penetration, the term water clarity is also often used to describe underwater visibility. In underwater diving activities such as saturation diving, technical diving and nitrox diving, the maximum operating depth (MOD) of a breathing gas is the depth below which the partial pressure of oxygen (pO2) of the gas mix exceeds an acceptable limit. For example, if a gas contains 36% oxygen (FO2 = 0.36) and the limiting maximum pO2 is chosen at 1.4 atmospheres absolute, the MOD in feet of seawater (fsw) is 33 fsw/atm x [(1.4 ata / 0.36) − 1] = 95.3 fsw. The diver can experience a different underwater environment at night, because many marine animals are nocturnal. The pressure produced by depth in water, is converted to pressure in feet sea water (fsw) or metres sea water (msw) by multiplying with the appropriate conversion factor, 33 fsw per atm, or 10 msw per bar. thumb|upright=1.5|A diver enters crystal clear water in Lake Huron. The depth at which the disk is no longer visible is taken as a measure of the transparency of the water. A dive light is routinely used during night dives and cave dives, when there is little or no natural light, but also has a useful function during the day, as water absorbs the longer (red) wavelengths first then the yellow and green with increasing depth. Bright dive lights have values from about 2500 lumens. This corresponds to a sound intensity 5.4 dB, or 3.5 times, higher than the threshold in air (see Measurements above). ====Safety thresholds==== High levels of underwater sound create a potential hazard to human divers. A modern dive light usually has an output of at least about 100 lumens. ",556,26.9,4.85,0.064,0.87,E
-Calculate the molar energy required to reverse the direction of an $\mathrm{H}_2 \mathrm{O}$ molecule located $100 \mathrm{pm}$ from a $\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \mathrm{D}$.,"thumb|Lewis Structure of H2O indicating bond angle and bond length Water () is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. 150px|thumb|Shape of water molecule showing that the real bond angle 104.5° deviates from the ideal sp3 angle of 109.5°. The bond angle for water is 104.5°. These tables list values of molar ionization energies, measured in kJ⋅mol−1. In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the lone pairs, while that leaves orbitals with more p character directed towards the hydrogens, resulting in deviation from idealized O(sp3) hybrid orbitals with 25% s character and 75% p character. The first molar ionization energy applies to the neutral atoms. The bond angles in those molecules are 104.5° and 107° respectively, which are below the expected tetrahedral angle of 109.5°. In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the very electropositive lone pairs, while that leaves orbitals with more p character directed towards the hydrogens. thumb|200px|Model of the hydrogen molecule and its axial projection In addition to the model of the atom, Niels Bohr also proposed a model of the chemical bond. A particularly well known example is water, where the angle between the two O-H bonds is only 104.5°. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. thumb|Example of bent electron arrangement (water molecule). However, the bond angle between the two O–H bonds is only 104.5°, rather than the 109.5° of a regular tetrahedron, because the two lone pairs (whose density or probability envelopes lie closer to the oxygen nucleus) exert a greater mutual repulsion than the two bond pairs. thumb|300px|Calculated structure of a (H2O)100 icosahedral water cluster. In the case of water, with its 104.5° HOH angle, the OH bonding orbitals are constructed from O(~sp4.0) orbitals (~20% s, ~80% p), while the lone pairs consist of O(~sp2.3) orbitals (~30% s, ~70% p). The four hydrogen atoms are positioned at the vertices of a tetrahedron, and the bond angle is cos−1(−) ≈ 109° 28′.""Angle Between 2 Legs of a Tetrahedron"" – Maze5.net This is referred to as an AX4 type of molecule. Oxygen difluoride 103.8° As one moves down the table, the substituents become more electronegative and the bond angle between them decreases. For simple molecules, pictorially generating their MO diagram can be achieved without extensive knowledge of point group theory and using reducible and irreducible representations.thumb|330x330px|Hybridized MO of H2O Note that the size of the atomic orbitals in the final molecular orbital are different from the size of the original atomic orbitals, this is due to different mixing proportions between the oxygen and hydrogen orbitals since their initial atomic orbital energies are different. By directing hybrid orbitals of more p character towards the fluorine, the energy of that bond is not increased very much. This is in open agreement with the true bond angle of 104.45°. In other words, if water was formed from two identical O-H bonds and two identical sp3 lone pairs on the oxygen atom as predicted by valence bond theory, then its photoelectron spectrum (PES) would have two (degenerate) peaks and energy, one for the two O-H bonds and the other for the two sp3 lone pairs. The bond angles between substituents are ~109.5°, ~120°, and 180°. ",1.07,4.49,205.0,4,2.14,A
-"In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ at a constant volume of $1.000 \mathrm{~m}^3$. The gas enters the container at $300 \mathrm{~K}$ and $100 \mathrm{~atm}$. The mass of the gas is $92.4 \mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \mathrm{~K}$. For nitrogen, $a=1.39 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}, b=0.0391 \mathrm{dm}^3 \mathrm{~mol}^{-1}$.","Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. ",+3.60,0.2553,140.0, 4.56,0.9966,C
-The chemical shift of the $\mathrm{CH}_3$ protons in acetaldehyde (ethanal) is $\delta=2.20$ and that of the $\mathrm{CHO}$ proton is 9.80 . What is the difference in local magnetic field between the two regions of the molecule when the applied field is $1.5 \mathrm{~T}$,"In this way the acetylenic protons are located in the cone-shaped shielding zone hence the upfield shift. :thumb|none|400px|Induced magnetic field of alkynes in external magnetic fields, field lines in grey. ==Magnetic properties of most common nuclei== 1H and 13C are not the only nuclei susceptible to NMR experiments. When a signal is found with a higher chemical shift: * the applied effective magnetic field is lower, if the resonance frequency is fixed (as in old traditional CW spectrometers) * the frequency is higher, when the applied magnetic field is static (normal case in FT spectrometers) * the nucleus is more deshielded * the signal or shift is downfield or at low field or paramagnetic Conversely a lower chemical shift is called a diamagnetic shift, and is upfield and more shielded. ==Diamagnetic shielding== In real molecules protons are surrounded by a cloud of charge due to adjacent bonds and atoms. The induced magnetic field lines are parallel to the external field at the location of the alkene protons which therefore shift downfield to a 4.5 ppm to 7.5 ppm range. Although the absolute resonance frequency depends on the applied magnetic field, the chemical shift is independent of external magnetic field strength. Acetaldehyde (IUPAC systematic name ethanal) is an organic chemical compound with the formula CH3CHO, sometimes abbreviated by chemists as MeCHO (Me = methyl). The nucleus is said to be experiencing a diamagnetic shielding. ==Factors causing chemical shifts== Important factors influencing chemical shift are electron density, electronegativity of neighboring groups and anisotropic induced magnetic field effects. Field effects are relatively weak, and diminish rapidly with distance, but have still been found to alter molecular properties such as acidity.thumb|160x160px|Field effect on a carbonyl arising from the dipole in a C-F bond. == Field sources == thumb|A bicycloheptane acid with an electron- withdrawing substituent, X, at the 4-position experiences a field effect on the acidic proton from the C-X bond dipole.|left|180x180pxleft|thumb|180x180px|A bicyclooctance acid with an electron-withdrawing substituent, X, at the 4-position experiences the same field effect on the acidic proton from the C-X bond dipole as the related bicylcoheptane. In nuclear magnetic resonance (NMR) spectroscopy, the chemical shift is the resonant frequency of an atomic nucleus relative to a standard in a magnetic field. As noted above, a consensus CSI method that filters upfield/downfield chemical shift changes in 13Cα, 13Cβ, and 13C' atoms in a similar manner to 1Hα shifts has also been developed. The protons in aromatic compounds are shifted downfield even further with a signal for benzene at 7.73 ppm as a consequence of a diamagnetic ring current. Field effects have also been shown in substituted arenes to dominate the electrostatic potential maps, which are maps of electron density used to explain intermolecular interactions. == Evidence for field effects == thumb|This octane derivative has only a single linker between the electron-withdrawing substituent and the acidic group.|180x180px Localized electronic effects are a combination of inductive and field effects. While chemical shift is referenced in order that the units are equivalent across different field strengths, the actual frequency separation in Hertz scales with field strength (). The variations of nuclear magnetic resonance frequencies of the same kind of nucleus, due to variations in the electron distribution, is called the chemical shift. This can be attributed to a field effect because in the same compound with the chlorines pointed away from the acidic group the pKa is lower, and if the effect were inductive the conformational position would not matter.thumb|The dichloroethano-bridged anthroic acid isomer with the C-Cl bond dipole oriented over the carboxylic acid has pKa of 6.07.|left|200x200px thumb|The isomer of dichloroethano- bridged anthroic acid in which the C-Cl dipole points away from the carboxylic acid has a pKa of 5.67.|left|200x200px == References == Category:Chemical properties Category:Chemistry Category:Electrostatics Category:Electromagnetism Category:Molecular physics Category:Molecules Category:Physical chemistry In carbon NMR the chemical shift of the carbon nuclei increase in the same order from around −10 ppm to 70 ppm. The total magnetic field experienced by a nucleus includes local magnetic fields induced by currents of electrons in the molecular orbitals (electrons have a magnetic moment themselves). In proton NMR of methyl halides (CH3X) the chemical shift of the methyl protons increase in the order from 2.16 ppm to 4.26 ppm reflecting this trend. Anisotropic induced magnetic field effects are the result of a local induced magnetic field experienced by a nucleus resulting from circulating electrons that can either be paramagnetic when it is parallel to the applied field or diamagnetic when it is opposed to it. However, the experimental data shows that effect on acidity in related octanes and cubanes is very similar, and therefore the dominant effect must be through space.thumb|This octane derivative has two linkers between the electron- withdrawing substituent and the acidic group.|179x179px thumb|This cubane derivative has four linkers but the acidic proton still feels the same effect from the C-X dipole because the interaction is a field effect.|180x180pxIn the cis-11,12-dichloro-9,10-dihydro-9,10-ethano-2-anthroic acid syn and anti isomers seen below and to the left, the chlorines provide a field effect. As is the case for NMR the chemical shift reflects the electron density at the atomic nucleus. ==See also== * EuFOD, a shift agent * MRI * Nuclear magnetic resonance * Nuclear magnetic resonance spectroscopy of carbohydrates * Nuclear magnetic resonance spectroscopy of nucleic acids * Nuclear magnetic resonance spectroscopy of proteins * Protein NMR * Random coil index * Relaxation (NMR) * Solid-state NMR * TRISPHAT, a chiral shift reagent for cations * Zeeman effect ==References== ==External links== *chem.wisc.edu *BioMagResBank *NMR Table *Proton chemical shifts *Carbon chemical shifts * Online tutorials (these generally involve combined use of IR, 1H NMR, 13C NMR and mass spectrometry) **Problem set 1 (see also this link for more background information on spin-spin coupling) **Problem set 2 **Problem set 4 **Problem set 5 **Combined solutions to problem set 5 (Problems 1-32) and (Problems 33-64) Category:Nuclear chemistry Category:Nuclear physics Category:Nuclear magnetic resonance spectroscopy pl:Spektroskopia NMR#Przesunięcie chemiczne The size of the chemical shift is given with respect to a reference frequency or reference sample (see also chemical shift referencing), usually a molecule with a barely distorted electron distribution. ==Operating frequency== The operating (or Larmor) frequency of a magnet is calculated from the Larmor equation : \omega_{0} = \gamma B_0\,, where is the actual strength of the magnet in units like Teslas or Gauss, and is the gyromagnetic ratio of the nucleus being tested which is in turn calculated from its magnetic moment and spin number with the nuclear magneton and the Planck constant : : \gamma = \frac{\mu\,\mu_\mathrm{N}}{hI}\,. SHIFTCOR identifies potential chemical shift referencing problems by comparing the difference between the average value of each set of observed backbone (1Hα, 13Cα, 13Cβ, 13CO, 15N and 1HN) shifts and their corresponding predicted chemical shifts. ",+107,11,7.136,0.4772,7200,B
-Suppose that the junction between two semiconductors can be represented by a barrier of height $2.0 \mathrm{eV}$ and length $100 \mathrm{pm}$. Calculate the transmission probability of an electron with energy $1.5 \mathrm{eV}$.,"Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height. === E = V0 === The transmission probability at E=V_0 is T=\frac{1}{1+ma^2V_0/2\hbar^2}. ==Remarks and applications== The calculation presented above may at first seem unrealistic and hardly useful. However, according to quantum mechanics, the electron has a non-zero wave amplitude in the barrier, and hence it has some probability of passing through the barrier. Note that these expressions hold for any energy If then so there is a singularity in both of these expressions. ==Analysis of the obtained expressions== ===E < V0=== thumb|350x350px|Transmission probability through a finite potential barrier for \sqrt{2m V_0} a / \hbar = 1, 3, and 7. The surprising result is that for energies less than the barrier height, E < V_0 there is a non-zero probability T=|t|^2= \frac{1}{1+\frac{V_0^2\sinh^2(k_1 a)}{4E(V_0-E)}} for the particle to be transmitted through the barrier, with .}} Classically, the electron has zero probability of passing through the barrier. For electrons, the barrier height \Phi_{B_{n}}can be easily calculated as the difference between the metal work function and the electron affinity of the semiconductor: \Phi_{B_n}=\Phi_M-\chi While the barrier height for holes is equal to the difference between the energy gap of the semiconductor and the energy barrier for electrons: \Phi_{B_p}=E_\text{gap}-\Phi_{B_n} In reality, what can happen is that charged interface states can pin the Fermi level at a certain energy value no matter the work function values, influencing the barrier height for both carriers. thumb|right|Schematic representation of an electron tunneling through a barrier In electronics/spintronics, a tunnel junction is a barrier, such as a thin insulating layer or electric potential, between two electrically conducting materials. To a first approximation, the barrier between a metal and a semiconductor is predicted by the Schottky–Mott rule to be proportional to the difference of the metal- vacuum work function and the semiconductor-vacuum electron affinity. The current-voltage relationship is qualitatively the same as with a p-n junction, however the physical process is somewhat different. === Conduction values === The thermionic emission can be formulated as following: J_{th}= A^{**}T^2e^{-\frac{\Phi_{B_{n,p}}}{k_bT}}\biggl(e^{\frac{qV}{k_bT}}-1\biggr) While the tunneling current density can be expressed, for a triangular shaped barrier (considering WKB approximation) as: J_{T_{n,p}}= \frac{q^3E^2}{16\pi^2\hbar\Phi_{B_{n,p}}} e^{\frac{-4\Phi_{B_{n,p}}^{3/2}\sqrt{2m^*_{n,p}}}{3q\hbar E}} From both formulae it is clear that the current contributions are related to the barrier height for both electrons and holes. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. The Monte Carlo method for electron transport is a semiclassical Monte Carlo (MC) approach of modeling semiconductor transport. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector whereas within the barrier it is exponentially damped over a distance If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed. === E > V0 === In this case T=|t|^2= \frac{1}{1+\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}}, where Equally surprising is that for energies larger than the barrier height, E > V_0, the particle may be reflected from the barrier with a non-zero probability R=|r|^2=1-T. Electrons (or quasiparticles) pass through the barrier by the process of quantum tunnelling. A classical particle with energy E larger than the barrier height V_0 would always pass the barrier, and a classical particle with E < V_0 incident on the barrier would always get reflected. An earlier version containing some of the material appeared in the Bell Labs Technical Journal in 1949. ==Contents== The arrangement of chapters is as follows: Part I INTRODUCTION TO TRANSISTOR ELECTRONICS *Chapter 1: THE BULK PROPERTIES OF SEMICONDUCTORS *Chapter 2: THE TRANSISTOR AS A CIRCUIT ELEMENT *Chapter 3: QUANTITATIVE STUDIES OF INJECTION OF HOLES AND ELECTRONS *Chapter 4: ON THE PHYSICAL THEORY OF TRANSISTORS *Chapter 5: QUANTUM STATES, ENERGY BANDS, AND BRILLOUIN ZONES PART II DESCRIPTIVE THEORY OF SEMICONDUCTORS *Chapter 6: VELOCITIES AND CURRENTS FOR ELECTRONS I N CRYSTALS *Chapter 7: ELECTRONS AND HOLES IN ELECTRIC AND MAGNETIC FIELDS *Chapter 8: INTRODUCTORY THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 9: DISTRIBUTIONS OF QUANTUM STATES IN ENERGY *Chapter 10: FERMI-DIRAC STATISTICS FOR SEMICONDUCTORS *Chapter 11: MATHEMATICAL THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 12: APPLICATIONS TO TRANSISTOR ELECTRONICS PART III QUANTUM MECHANICAL FOUNDATIONS *Chapter 13: INTRODUCTION TO PART III *Chapter 14: ELEMENTARY QUANTUM MECHANICS WITH CIRCUIT THEORY ANALOGUES *Chapter 15: THEORY OF ELECTRON AND HOLE VELOCITIES, CURRENTS AND ACCELERATIONS *Chapter 16: STATISTICAL MECHANICS FOR SEMICONDUCTORS *Chapter 17: THE THEORY OF TRANSITION PROBABILITIES FOR HOLES AND ELECTRONS *APPENDIX A Units *APPENDIX B Periodic Table *APPENDIX C Values of the physical constants *APPENDIX D Energy conversion chart *NAME INDEX *SUBJECT INDEX Category:1950 non-fiction books Category:1950 in science Category:Physics textbooks Under large voltage bias, the electric current flowing through the barrier is essentially governed by the laws of thermionic emission, combined with the fact that the Schottky barrier is fixed relative to the metal's Fermi level.This interpretation is due to Hans Bethe, after the incorrect theory of Schottky, see * Under forward bias, there are many thermally excited electrons in the semiconductor that are able to pass over the barrier. The scattering events and the duration of particle flight is determined through the use of random numbers. == Background == === Boltzmann transport equation === The Boltzmann transport equation model has been the main tool used in the analysis of transport in semiconductors. Within these approximations, Fermi's Golden Rule gives, to the first order, the transition probability per unit time for a scattering mechanism from a state |k \rangle to a state |k' \rangle: : S(k,k') = \frac{2\pi}{\hbar} \left | \langle k|H'|k' \rangle \right |^2 \cdot \delta(E - E') where H' is the perturbation Hamiltonian representing the collision and E and E′ are respectively the initial and final energies of the system constituted of both the carrier and the electron and phonon gas. To study the quantum case, consider the following situation: a particle incident on the barrier from the left side It may be reflected or transmitted To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations A_r = 1 (incoming particle), A_l = r (reflection), C_l = 0 (no incoming particle from the right), and C_r = t (transmission). 350px|right|thumb In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called ""quantum tunneling"") and wave-mechanical reflection. A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal–semiconductor junction. The value of ΦB depends on the combination of metal and semiconductor.Schottky barrier tutorial. ",54.7, 9.73,0.8,3,1.39,C
-"The diffusion coefficient of a particular kind of t-RNA molecule is $D=1.0 \times 10^{-11} \mathrm{~m}^2 \mathrm{~s}^{-1}$ in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance $1.0 \mu \mathrm{m}$, corresponding to the radius of the cell?","The rate of diffusion NA, is usually expressed as the number of moles diffusing across unit area in unit time. Of mass transport mechanisms, molecular diffusion is known as a slower one. === Biology === In cell biology, diffusion is a main form of transport for necessary materials such as amino acids within cells. The rate of diffusion of A, NA, depend on concentration gradient and the average velocity with which the molecules of A moves in the x direction. The source term in the diffusion equation becomes S(\vec{r},t, \vec{r'},t')=\delta(\vec{r}-\vec{r'})\delta(t-t'), where \vec{r} is the position at which fluence rate is measured and \vec{r'} is the position of the source. For the vector form, the RTE is multiplied by direction \hat{s} before evaluation.): : \frac{\partial \Phi(\vec{r},t)}{c\partial t} + \mu_a\Phi(\vec{r},t) + abla \cdot \vec{J}(\vec{r},t) = S(\vec{r},t) : \frac{\partial \vec{J}(\vec{r},t)}{c\partial t} + (\mu_a+\mu_s')\vec{J}(\vec{r},t) + \frac{1}{3} abla \Phi(\vec{r},t) = 0 The diffusion approximation is limited to systems where reduced scattering coefficients are much larger than their absorption coefficients and having a minimum layer thickness of the order of a few transport mean free path. ===The diffusion equation=== Using the second assumption of diffusion theory, we note that the fractional change in current density \vec{J}(\vec{r},t) over one transport mean free path is negligible. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. In practice, Knudsen diffusion applies only to gases because the mean free path for molecules in the liquid state is very small, typically near the diameter of the molecule itself. == Mathematical description == The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the kinetic theory of gases: :{D_{AA*}} = {{\lambda u} \over {3}} = {{\lambda}\over{3}} \sqrt{{8R T}\over {\pi M_{A}}} For Knudsen diffusion, path length λ is replaced with pore diameter d, as species A is now more likely to collide with the pore wall as opposed with another molecule. A common approximation summarized here is the diffusion approximation. The nuclear lamina is a dense (~30 to 100 nm thick) fibrillar network inside the nucleus of eukaryote cells. The result of diffusion is a gradual mixing of material such that the distribution of molecules is uniform. Nuclear collision length is the mean free path of a particle before undergoing a nuclear reaction, for a given particle in a given medium. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale. Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. Molecular diffusion is typically described mathematically using Fick's laws of diffusion. == Applications == Diffusion is of fundamental importance in many disciplines of physics, chemistry, and biology. For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases. In physiology, transport maximum (alternatively Tm or Tmax) refers to the point at which increase in concentration of a substance does not result in an increase in movement of a substance across a cell membrane. For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases. Substituting Fick's law into the scalar representation of the RTE gives the diffusion equation: : \frac{1}{c}\frac{\partial \Phi(\vec{r},t)}{\partial t} + \mu_a\Phi(\vec{r},t) - abla \cdot [D abla\Phi(\vec{r},t)] = S(\vec{r},t) D=\frac{1}{3(\mu_a+\mu_s')} is the diffusion coefficient and μ's=(1-g)μs is the reduced scattering coefficient. * Chemical diffusion occurs in a presence of concentration (or chemical potential) gradient and it results in net transport of mass. This is the process described by the diffusion equation. thumb|300px|Schematic drawing of a molecule in a cylindrical pore in the case of Knudsen diffusion; are indicated the pore diameter () and the free path of the particle (). Generally, the diffusion approximation is less accurate as the absorption coefficient μa increases and the scattering coefficient μs decreases. ",0.000216,1.7,-273.0,1.4,3.42,B
-At what pressure does the mean free path of argon at $20^{\circ} \mathrm{C}$ become comparable to the diameter of a $100 \mathrm{~cm}^3$ vessel that contains it? Take $\sigma=0.36 \mathrm{~nm}^2$,"New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. The formulae of pressure vessel design standards are extension of Lamé's theorem by putting some limit on ratio of inner radius and thickness. Stress in a thin-walled pressure vessel in the shape of a cylinder is :\sigma_\theta = \frac{pr}{t}, :\sigma_{\rm long} = \frac{pr}{2t}, where: * \sigma_\theta is hoop stress, or stress in the circumferential direction * \sigma_{long} is stress in the longitudinal direction * p is internal gauge pressure * r is the inner radius of the cylinder * t is thickness of the cylinder wall. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. Therefore, pressure vessels are designed to have a thickness proportional to the radius of tank and the pressure of the tank and inversely proportional to the maximum allowed normal stress of the particular material used in the walls of the container. The normal (tensile) stress in the walls of the container is proportional to the pressure and radius of the vessel and inversely proportional to the thickness of the walls. (Pa)For a sphere the thickness d = rP/2σ, where r is the radius of the tank. thumb|upright=1.5|In inverse depth parametrization, a point is identified by its inverse depth \rho = \frac{1}{\left\Vert \mathbf{p} - \mathbf{c}_0\right\Vert} along the ray, with direction v = (\cos \phi \sin \theta, -\sin \phi, \cos \phi \cos \theta), from which it was first observed. Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. Construction methods and materials may be chosen to suit the pressure application, and will depend on the size of the vessel, the contents, working pressure, mass constraints, and the number of items required. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. ""Pressure Vessel Handbook, 14th Edition."" File:Ресивер хладагента FP-LR-100.png|Cylindrical pressure vessel. For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: Thickness has to be less than 0.356 times inner radius :\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE} Cylindrical shells: Thickness has to be less than 0.5 times inner radius :\sigma_\theta = \frac{p(r + 0.6t)}{tE} :\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE} where E is the joint efficiency, and all others variables as stated above. However, a spherical shape is difficult to manufacture, and therefore more expensive, so most pressure vessels are cylindrical with 2:1 semi-elliptical heads or end caps on each end. For cylindrical vessels with a diameter up to 600 mm (NPS of 24 in), it is possible to use seamless pipe for the shell, thus avoiding many inspection and testing issues, mainly the nondestructive examination of radiography for the long seam if required. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. A disadvantage of these vessels is that greater diameters are more expensive, so that for example the most economic shape of a , pressure vessel might be a diameter of and a length of including the 2:1 semi-elliptical domed end caps. ===Construction materials=== thumb|200px|Composite overwrapped pressure vessel with titanium liner. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus :M = {3 \over 2} nRT {\rho \over \sigma}. (see gas law) The other factors are constant for a given vessel shape and material. A vessel can be considered ""thin-walled"" if the diameter is at least 10 times (sometimes cited as 20 times) greater than the wall thickness.Richard Budynas, J. Nisbett, Shigley's Mechanical Engineering Design, 8th ed., For these reasons, the definition of a pressure vessel varies from country to country. ",0.195,4152,170.0,0.22222222,58.2,A
-"The equilibrium pressure of $\mathrm{O}_2$ over solid silver and silver oxide, $\mathrm{Ag}_2 \mathrm{O}$, at $298 \mathrm{~K}$ is $11.85 \mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\mathrm{Ag}_2 \mathrm{O}(\mathrm{s})$ at $298 \mathrm{~K}$.","Silver oxide is the chemical compound with the formula Ag2O. Silver sulfate is the inorganic compound with the formula Ag2SO4. Silver sulfite is the chemical compound with the formula Ag2SO3. In 1993, AgF2 cost between 1000-1400 US dollars per kg. ==Composition and structure== AgF2 is a white crystalline powder, but it is usually black/brown due to impurities. The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. For some time, it was doubted that silver was actually in the +2 oxidation state, rather than some combination of states such as AgI[AgIIIF4], which would be similar to silver(I,III) oxide. Such reactions often work best when the silver oxide is prepared in situ from silver nitrate and alkali hydroxide. ==References== ==External links== * Annealing of Silver Oxide – Demonstration experiment: Instruction and video Category:Silver compounds Category:Transition metal oxides The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). This reaction does not afford appreciable amounts of silver hydroxide due to the favorable energetics for the following reaction:Holleman, A. F.; Wiberg, E. ""Inorganic Chemistry"" Academic Press: San Diego, 2001. . :2  AgOH -> Ag2O + H2O (pK = 2.875) With suitably controlled conditions, this reaction can be used to prepare Ag2O powder with properties suitable for several uses including as a fine grained conductive paste filler. ==Structure and properties== Ag2O features linear, two-coordinate Ag centers linked by tetrahedral oxides. Silver(II) fluoride is a chemical compound with the formula AgF2. Silver sulfate and anhydrous sodium sulfate adopt the same structure. ==Silver(II) sulfate== The synthesis of silver(II) sulfate (AgSO4) with a divalent silver ion instead of a monovalent silver ion was first reported in 2010 by adding sulfuric acid to silver(II) fluoride (HF escapes). :AgNO3 + NaCl -> AgCl(v) + NaNO3 :2 AgNO3 + CoCl2 -> 2 AgCl(v) + Co(NO3)2 It can also be produced by the reaction of silver metal and aqua regia, however, the insolubility of silver chloride decelerates the reaction. Silver chloride is a chemical compound with the chemical formula AgCl. AgF and AgBr crystallize similarly.Wells, A.F. (1984) Structural Inorganic Chemistry, Oxford: Clarendon Press. . The AgI[AgIIIF4] was found to be present at high temperatures, but it was unstable with respect to AgF2. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. In the gas phase, AgF2 is believed to have D∞h symmetry. This unstable silver compound when heated and/or in light it decomposes to silver dithionate and silver sulfate. ==Preparation== Silver sulfite can be prepared by dissolving silver nitrate with the stoichiometric quantity of sodium sulfite solution, yielding a precipitation of silver sulfite by the following reaction: :2 AgNO3 \+ Na2SO3 Ag2SO3 \+ 2 NaNO3 After precipitation then filtering silver sulfite, washing it using well-boiled water, and drying it in vacuum. ==References== Category:Silver compounds Category:Sulfites Silver usually exists in its +1 oxidation state. It is formed as an intermediate in the catalysis of gaseous reactions with fluorine by silver. It is a rare example of a silver(II) compound. ",152.67,0,0.36,-11.2,6.9,D
-"When alkali metals dissolve in liquid ammonia, their atoms each lose an electron and give rise to a deep-blue solution that contains unpaired electrons occupying cavities in the solvent. These 'metal-ammonia solutions' have a maximum absorption at $1500 \mathrm{~nm}$. Supposing that the absorption is due to the excitation of an electron in a spherical square well from its ground state to the next-higher state (see the preceding problem for information), what is the radius of the cavity?","The problems stems from the fact that a cavity is an open non- Hermitian system with leakage and absorption. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), it follows that cavity length must be an integer multiple of half- wavelength at resonance.David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics , and Waldron in the radio frequency domain. Because the ions overlap, their separation in the crystal will be less than the sum of their soft-sphere radii. That is, the distance between two neighboring iodides in the crystal is assumed to be twice the radius of the iodide ion, which was deduced to be 214 pm. Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. The Q factor of a resonant cavity can be calculated using cavity perturbation theory and expressions for stored electric and magnetic energy. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. The microwaves bounce back and forth between the walls of the cavity. Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.Montgomery, C. G. & Dicke, Robert H. & Edward M. Purcell, Principles of microwave circuits / edited by C.G. Montgomery, R.H. Dicke, E.M. Purcell, Peter Peregrinus on behalf of the Institution of Electrical Engineers, London, U.K., 1987. === Resonant frequencies === The resonant frequencies of a cavity are a function of its geometry. ==== Rectangular cavity ==== thumb|Rectangular cavity Resonance frequencies of a rectangular microwave cavity for any \scriptstyle TE_{mnl} or \scriptstyle TM_{mnl} resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. The process of gas or liquid which penetrate into the body of adsorbent is commonly known as absorption. 550px|link=https://doi.org/10.1351/goldbook.A00036|thumb|right|alt=IUPAC definition for absorption|[https://doi.org/10.1351/goldbook.A00036 https://doi.org/10.1351/goldbook.A00036]. ==Equation== If absorption is a physical process not accompanied by any other physical or chemical process, it usually follows the Nernst distribution law: :""the ratio of concentrations of some solute species in two bulk phases when it is equilibrium and in contact is constant for a given solute and bulk phases"": :: \frac{[x]_{1}}{[x]_{2}} = \text{constant} = K_{N(x,12)} The value of constant KN depends on temperature and is called partition coefficient. A microwave cavity or radio frequency cavity (RF cavity) is a special type of resonator, consisting of a closed (or largely closed) metal structure that confines electromagnetic fields in the microwave or RF region of the spectrum. Cavity perturbation measurement techniques for material characterization are used in many fields ranging from physics and material science to medicine and biology.Wenquan Che; Zhanxian Wang; Yumei Chang; Russer, P.; ""Permittivity Measurement of Biological Materials with Improved Microwave Cavity Perturbation Technique,"" Microwave Conference, 2008. In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. ;TM modes:T. Wangler, RF linear accelerators, Wiley (2008) f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2} ;TE modes: f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X'_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2} Here, \scriptstyle X_{mn} denotes the \scriptstyle n-th zero of the \scriptstyle m-th Bessel function, and \scriptstyle X'_{mn} denotes the \scriptstyle n-th zero of the derivative of the \scriptstyle m-th Bessel function. === Quality factor === The quality factor \scriptstyle Q of a cavity can be decomposed into three parts, representing different power loss mechanisms. *\scriptstyle Q_c, resulting from the power loss in the walls which have finite conductivity *\scriptstyle Q_d, resulting from the power loss in the lossy dielectric material filling the cavity. *\scriptstyle Q_{ext}, resulting from power loss through unclosed surfaces (holes) of the cavity geometry. Ideally, the material to be measured is introduced into the cavity at the position of maximum electric or magnetic field. This frequency is given by \cdot k_{mnl}\\\ &= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 + \left(\frac{l\pi}{d}\right)^2}\\\ &= \frac{c}{2\sqrt{\mu_r\epsilon_r}}\sqrt{\left( \frac{m}{a}\right) ^2+\left(\frac{n}{b}\right) ^2 + \left(\frac{l}{d}\right) ^2} \end{align}|}} where \scriptstyle k_{mnl} is the wavenumber, with \scriptstyle m, \scriptstyle n, \scriptstyle l being the mode numbers and \scriptstyle a, \scriptstyle b, \scriptstyle d being the corresponding dimensions; c is the speed of light in vacuum; and \scriptstyle \mu_r and \scriptstyle \epsilon_r are relative permeability and permittivity of the cavity filling respectively. ==== Cylindrical cavity ==== thumb|Cylindrical cavity The field solutions of a cylindrical cavity of length \scriptstyle L and radius \scriptstyle R follow from the solutions of a cylindrical waveguide with additional electric boundary conditions at the position of the enclosing plates. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The concept can be extended to solvated ions in liquid solutions taking into consideration the solvation shell. == Trends == X− NaX AgX F 464 492 Cl 564 555 Br 598 577 Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. ",0,-32,2500.0,131,0.69,E
-Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ? ,"Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. Electron diffraction occurs due to elastic scattering, when there is no change in the energy of the electrons during their interactions with atoms. The electron pulse undergoes diffraction as a result of interacting with the sample. For electron diffraction the electrons behave as if they are non-relativistic particles of mass m^* in terms of how they interact with the atoms. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. *The development of new approaches to reduce dynamical effects such as precession electron diffraction and three-dimensional diffraction methods. A fast electron beam is generated in an electron gun, enters a diffraction chamber typically at a vacuum of 10−7 mbar. *Fast and accurate methods to calculate intensities for low-energy electron diffraction so it could be used to determine atomic positions, for instance references. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. Principles of Electron Optics. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. framed|Geometry of electron beam in precession electron diffraction. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. Strictly, the term electron diffraction refers to how electrons are scattered by atoms, a process that is mathematically modelled by solving forms of Schrödinger equation. # The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called kinematical diffraction and is also a far-field or Fraunhofer approach. The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. == Historical development == The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. Scheme 1 shows the schematic procedure of an electron diffraction experiment. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments, with the information available in real time rather than requiring photographic processing after the experiment. ==Basics== === Geometrical considerations === What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. Electron diffraction also plays a major role in the contrast of images in electron microscopes. UED can provide a wealth of dynamics on charge carriers, atoms, and molecules. ==History== The design of early ultrafast electron diffraction instruments was based on x-ray streak cameras, the first reported UED experiment demonstrating an electron pulse length of 100 ps. ==Electron Pulse Production== The electron pulses are typically produced by the process of photoemission in which a fs optical pulse is directed toward a photocathode. However, with Cs corrected microscopes, the probe can be made much smaller. === Practical considerations === Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV. ",+11,5275,7.27,0.312,0.1792,C
-Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ? ,"Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. For electron diffraction the electrons behave as if they are non-relativistic particles of mass m^* in terms of how they interact with the atoms. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. Electron diffraction occurs due to elastic scattering, when there is no change in the energy of the electrons during their interactions with atoms. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. Principles of Electron Optics. *Fast and accurate methods to calculate intensities for low-energy electron diffraction so it could be used to determine atomic positions, for instance references. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. *The development of new approaches to reduce dynamical effects such as precession electron diffraction and three-dimensional diffraction methods. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. framed|Geometry of electron beam in precession electron diffraction. The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. Strictly, the term electron diffraction refers to how electrons are scattered by atoms, a process that is mathematically modelled by solving forms of Schrödinger equation. # The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called kinematical diffraction and is also a far-field or Fraunhofer approach. However, with Cs corrected microscopes, the probe can be made much smaller. === Practical considerations === Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. == Historical development == The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments, with the information available in real time rather than requiring photographic processing after the experiment. ==Basics== === Geometrical considerations === What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. These were rapidly followed by the first non-relativistic diffraction model for electrons by Hans Bethe based upon the Schrödinger equation, which is very close to how electron diffraction is now described. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. While the wavevector increases as the energy increases, the change in the effective mass compensates this so even at the very high energies used in electron diffraction there are still significant interactions. Relativistic electron beams are streams of electrons moving at relativistic speeds. ",7.27,10.065778,0.9,-1.0,4.16,A
-"Nelson, et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the $\mathrm{H}$ atoms in $\mathrm{NH}_3$ formed hydrogen bonds, but found none. For example, they found that the complex of $\mathrm{NH}_3$ and $\mathrm{CO}_2$ has the carbon atom nearest the nitrogen (299 pm away): the $\mathrm{CO}_2$ molecule is at right angles to the $\mathrm{C}-\mathrm{N}$ 'bond', and the $\mathrm{H}$ atoms of $\mathrm{NH}_3$ are pointing away from the $\mathrm{CO}_2$. The magnitude of the permanent dipole moment of this complex is reported as $1.77 \mathrm{D}$. If the $\mathrm{N}$ and $\mathrm{C}$ atoms are the centres of the negative and positive charge distributions, respectively, what is the magnitude of those partial charges (as multiples of $e$ )?","Theoretically, the bond strength of the hydrogen bonds can be assessed using NCI index, non-covalent interactions index, which allows a visualization of these non-covalent interactions, as its name indicates, using the electron density of the system. Another study found a much smaller number of hydrogen bonds: 2.357 at 25 °C. Quantum chemical calculations of the relevant interresidue potential constants (compliance constants) revealed large differences between individual H bonds of the same type. van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 Van der Waals radii taken from Bondi's compilation (1964). Defining and counting the hydrogen bonds is not straightforward however. Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09 Å for the van der Waals radius of hydrogen as opposed to Bondi's 1.20 Å. As well, it has been argued that the Van der Waals radius is not a fixed property of an atom in all circumstances, rather, that it will vary with the chemical environment of the atom. == Gallery == File:3D ammoniac.PNG|alt=Ammonia, van-der-Waals-based model|Ammonia, NH3, space- filling, Van der Waal's-based representation, nitrogen (N) in blue, hydrogen (H) in white. This is slightly different from the intramolecular bound states of, for example, covalent or ionic bonds; however, hydrogen bonding is generally still a bound state phenomenon, since the interaction energy has a net negative sum. This description of the hydrogen bond has been proposed to describe unusually short distances generally observed between or . ===Structural details=== The distance is typically ≈110 pm, whereas the distance is ≈160 to 200 pm. In weaker hydrogen bonds,Desiraju, G. R. and Steiner, T. Some of Bondi's figures are given in the table at the top of this article, and they remain the most widely used ""consensus"" values for the van der Waals radii of the elements. ==Hydrogen bonds in small molecules== thumb|right|Crystal structure of hexagonal ice. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The coefficient E_{0} is positive and of the order V\alpha^{3}, where V is the ionization energy and \alpha is the mean atomic polarizability; the exact value of E_{0} depends on the magnitudes of the dipole matrix elements and on the energies of the p orbitals. ==References== * Category:Chemical bonding Category:Quantum mechanical potentials The ideal bond angle depends on the nature of the hydrogen bond donor. Structural details, in particular distances between donor and acceptor which are smaller than the sum of the van der Waals radii can be taken as indication of the hydrogen bond strength. 100px 120px 100px From top to bottom, azides, nitrones, and nitro compounds are examples of 1,3-dipoles. The Hydrogen Bond Franklin Classics, 2018), Jeffrey, G. A.; In organic chemistry, a 1,3-dipolar compound or 1,3-dipole is a dipolar compound with delocalized electrons and a separation of charge over three atoms. Generally, the hydrogen bond is characterized by a proton acceptor that is a lone pair of electrons in nonmetallic atoms (most notably in the nitrogen, and chalcogen groups). The strength of intramolecular hydrogen bonds can be studied with equilibria between conformers with and without hydrogen bonds. ",0.36,3.52,0.123,0.118,29.9,C
-"The NOF molecule is an asymmetric rotor with rotational constants $3.1752 \mathrm{~cm}^{-1}, 0.3951 \mathrm{~cm}^{-1}$, and $0.3505 \mathrm{~cm}^{-1}$. Calculate the rotational partition function of the molecule at $25^{\circ} \mathrm{C}$.","Nitrosyl fluoride (NOF) is a covalently bonded nitrosyl compound. ==Reactions== NOF is a highly reactive fluorinating agent that converts many metals to their fluorides, releasing nitric oxide in the process: :n NOF + M → MFn \+ n NO NOF also fluorinates fluorides to form adducts that have a salt- like character, such as NOBF4. In terms of these constants, the rotational partition function can be written in the high temperature limit as G. Herzberg, ibid, Equation (V,29) \zeta^\text{rot} \approx \frac{\sqrt{\pi} }{\sigma} \sqrt{ \frac{ (k_\text{B} T)^3 }{ A B C }} with \sigma again known as the rotational symmetry number G. Herzberg, ibid; see Table 140 for values for common molecular point groups which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. NF4F.jpg|NF4+F− R3m structure NF4-2NF6F.jpg|(NF4+)2NF6−F− I4/m structure NF4NF6.jpg|NF4+NF6− P4/n structure ==Covalent molecule== thumb|left|upright=1.55|Possible structures of NF5 and analogous fluorohydrides For a NF5 molecule to form, five fluorine atoms have to be arranged around a nitrogen atom. In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor \sigma = 2 with \sigma known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without. ==Nonlinear molecules== A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A, B, and C , which can often be determined by rotational spectroscopy. Nitrogen pentafluoride (NF5) is a theoretical compound of nitrogen and fluorine that is hypothesized to exist based on the existence of the pentafluorides of the atoms below nitrogen in the periodic table, such as phosphorus pentafluoride. Calculations show that the NF5 molecule is thermodynamically favourably inclined to form NF4 and F radicals with energy 36 kJ/mol and a transition barrier around 67–84 kJ/mol. Nitrogen pentafluoride also violates the octet rule in which compounds with eight outer shell electrons are particularly stable. ==References== Category:Nitrogen fluorides Category:Hypothetical chemical compounds Category:Nitrogen(V) compounds For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E_J^\text{rot} = \frac{\mathbf{J}^2}{2I} = \frac{J(J+1)\hbar^2}{2I} = J(J+1)B. J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0,1,2, \ldots, B = \frac{\hbar^2}{2I} is the rotational constant, and I is the moment of inertia. N-Fluoropyridinium triflate is an organofluorine compound with the formula [C5H5NF]O3SCF3. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k_\text{B} T . == Quantum symmetry effects == For a diatomic molecule with a center of symmetry, such as \rm H_2, N_2, CO_2, or \mathrm{ H_2 C_2} (i.e. D_{\infty h} point group), rotation of a molecule by \pi radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. Theoretical models of the nitrogen pentafluoride molecule are either a trigonal bipyramidal covalently bound molecule with symmetry group D3h, or NFF−, which would be an ionic solid. ==Ionic solid== A variety of other tetrafluoroammonium salts are known (NFX−), as are fluoride salts of other ammonium cations ). Let each rotating molecule be associated with a unit vector \hat{n}; for example, \hat{n} might represent the orientation of an electric or magnetic dipole moment. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: \zeta^\text{rot} \approx \frac{5.34 \times 10^6}{\sigma} \sqrt{ \frac{T^3}{A B C}} where the prefactor comes from \sqrt{\frac{(\pi k_\text{B}) ^3}{h^3}} = 5.34 \times 10^6 when A, B, and C are expressed in units of MHz. The expressions for \zeta^\text{rot} works for asymmetric, symmetric and spherical top rotors. ==References== ==See also== * Translational partition function * Vibrational partition function * Partition function (mathematics) Category:Equations of physics Category:Partition functions The mean thermal rotational energy per molecule can now be computed by taking the derivative of \zeta^\text{rot} with respect to temperature T. Molecule \theta_{\mathrm{R}} (K)P. Atkins and J. de Paula ""Physical Chemistry"", 9th edition (W.H. Freeman 2010), Table 13.2, Data section in appendix H2 87.6 N2 2.88 O2 2.08 F2 1.27 HF 30.2 HCl 15.2 CO2 0.561P. Dominik Kurzydłowski and Patryk Zaleski-Ejgierd predict that a mixture of fluorine and nitrogen trifluoride under pressure between 10 and 33 GPa forms NFF− with space group R3m. This has CAS number 71485-49-9.Tetrafluoroammonium bifluoride I. J. Solomon believed that nitrogen pentafluoride was produced by the thermal decomposition of NF4AsF6, but experimental results were not reproduced. thumb|A molecule with a red cross on its front undergoing 3 dimensional rotational diffusion. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants. It is a salt, consisting of the N-fluoropyridinium cation ([C5H5NF]+) and the triflate anion. The characteristic rotational temperature ( or ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. ",-111.92,72,0.38,-0.16,7.97,E
-Suppose that $2.5 \mathrm{mmol} \mathrm{N}_2$ (g) occupies $42 \mathrm{~cm}^3$ at $300 \mathrm{~K}$ and expands isothermally to $600 \mathrm{~cm}^3$. Calculate $\Delta G$ for the process.,"For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. For the above defined path we have that and thus , and hence the increase in entropy for the Joule expansion is \Delta S=\int_i^f\mathrm{d}S=\int_{V_0}^{2V_0} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2. As a result, the change in internal energy, \Delta U, is zero. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. Negative and positive thermal expansion hereby compensate each other to a certain amount if the temperature is changed. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034 k_B T. == References == Category:SI derived units In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced ""delta fifteen n"") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. Thermodynamics, p. 414. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work given by W = -\int_{2V_0}^{V_0} P\,\mathrm{d}V = - \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V = nRT\ln 2 = T \Delta S_\text{gas}. From these initial measurements, Gibbs free energy changes (\Delta G) and entropy changes (\Delta S) can be determined using the relationship: ::: \Delta G = -RT\ln{K_a} = \Delta H -T\Delta S (where R is the gas constant and T is the absolute temperature). In thermodynamics, Stefan's formula says that the specific surface energy at a given interface is determined by the respective enthalpy difference \scriptstyle \Delta H^*. : \sigma = \gamma_0 \left( \frac{\Delta H^*}{N_\text{A}^{1/3}V_\text{m}^{2/3}}\right), where σ is the specific surface energy, NA is the Avogadro constant, \gamma_0 is a steric dimensionless coefficient, and Vm is the molar volume. ==References== Category:Thermodynamic equations Category:Chemical thermodynamics For example, the Gibbs free energy of a compound in the area of thermochemistry is often quantified in units of kilojoules per mole (symbol: kJ·mol−1 or kJ/mol), with 1 kilojoule = 1000 joules. Since 1 mole = 6.02214076 particles (atoms, molecules, ions etc.), 1 joule per mole is equal to 1 joule divided by 6.02214076 particles, ≈1.660539 joule per particle. The joule per mole (symbol: J·mol−1 or J/mol) is the unit of energy per amount of substance in the International System of Units (SI), such that energy is measured in joules, and the amount of substance is measured in moles. Negative thermal expansion (NTE) is an unusual physicochemical process in which some materials contract upon heating, rather than expand as most other materials do. thumb|240px|The Joule expansion, in which a volume is expanded to a volume in a thermally isolated chamber. thumb|180px|A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas. It is also an SI derived unit of molar thermodynamic energy defined as the energy equal to one joule in one mole of substance. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. ===Ideal gases=== If the gas is ideal, both the initial (T_{\mathrm{i}}, P_{\mathrm{i}}, V_{\mathrm{i}}) and final (T_{\mathrm{f}}, P_{\mathrm{f}}, V_{\mathrm{f}}) conditions follow the Ideal Gas Law, so that initially P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}} and then, after the tap is opened, P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}. Thus in 2D and 3D negative thermal expansion in close-packed systems with pair interactions is realized even when the third derivative of the potential is zero or even negative. For a monatomic ideal gas , with the molar heat capacity at constant volume. Isothermal titration calorimetry for chiral chemistry. Therefore, the sign of thermal expansion coefficient is determined by the sign of the third derivative of the potential. ",-1.0, 35.91,1110.0,0.00539,-17,E
+(3) $4 \mathrm{HCl}(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{Cl}_2(\mathrm{g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. == Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m��K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). K (? °C), ? K (? °C), ? :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Ellis (Ed.) in Nuffield Advanced Science Book of Data, Longman, London, UK, 1972. == See also == Category:Thermodynamic properties Category:Chemical element data pages Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Bodek et al., Environmental Inorganic Chemistry, Pergamon Press, New York, (1988). ",91.17,0.7854,"""2.19""",-0.28,-111.92,E
+Radiation from an X-ray source consists of two components of wavelengths $154.433 \mathrm{pm}$ and $154.051 \mathrm{pm}$. Calculate the difference in glancing angles $(2 \theta)$ of the diffraction lines arising from the two components in a diffraction pattern from planes of separation $77.8 \mathrm{pm}$.,"In the figure below, the line representing a ray makes an angle θ with the normal (dotted line). right|thumb|Grazing incidence diffraction geometry. * The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object. * The diffraction angles are invariant under scaling; that is, they depend only on the ratio of the wavelength to the size of the diffracting object. The amount of diffraction depends on the size of the gap. When the incident angle \theta_\text{i} of the light onto the slit is non-zero (which causes a change in the path length), the intensity profile in the Fraunhofer regime (i.e. far field) becomes: :I(\theta) = I_0 \,\operatorname{sinc}^2 \left[ \frac{d \pi}{\lambda} (\sin\theta \pm \sin\theta_i)\right] The choice of plus/minus sign depends on the definition of the incident angle \theta_\text{i}.right|thumb|2-slit (top) and 5-slit diffraction of red laser light thumb|left|Diffraction of a red laser using a diffraction grating. right|thumb|A diffraction pattern of a 633 nm laser through a grid of 150 slits ===Diffraction grating=== thumb|Diffraction grating A diffraction grating is an optical component with a regular pattern. The beam is diffracted in the plane of the surface of the sample by the angle 2θ, and often also out of the plane. The main central beam, nulls, and phase reversals are apparent. right|thumb|300px|Graph and image of single-slit diffraction As an example, an exact equation can now be derived for the intensity of the diffraction pattern as a function of angle in the case of single-slit diffraction. Given that \delta_A-\delta_B\ll 1 and \alpha_A-\alpha_B\ll 1, at a second-order development it turns that \cos\delta_A\cos\delta_B \frac{(\alpha_A - \alpha_B)^2}{2} \approx \cos^2\delta_A \frac{(\alpha_A - \alpha_B)^2}{2}, so that :\theta \approx \sqrt{\left[(\alpha_A - \alpha_B)\cos\delta_A\right]^2 + (\delta_A-\delta_B)^2} === Small angular distance: planar approximation === thumb|Planar approximation of angular distance on sky If we consider a detector imaging a small sky field (dimension much less than one radian) with the y-axis pointing up, parallel to the meridian of right ascension \alpha, and the x-axis along the parallel of declination \delta, the angular separation can be written as: : \theta \approx \sqrt{\delta x^2 + \delta y^2} where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B. Note that the y-axis is equal to the declination, whereas the x-axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure). ==See also== * Milliradian * Gradian * Hour angle * Central angle * Angle of rotation * Angular diameter * Angular displacement * Great-circle distance * ==References== * CASTOR, author(s) unknown. Grazing incidence diffraction is used in X-ray spectroscopy and atom optics, where significant reflection can be achieved only at small values of the grazing angle. thumb|425x425px|The plane of incidence is defined by the incoming radiation's propagation vector and the normal vector of the surface. Determining the angle of reflection with respect to a planar surface is trivial, but the computation for almost any other surface is significantly more difficult. thumb|center|650px|Refraction of light at the interface between two media. ==Grazing angle or glancing angle== thumb|Focusing X-rays with glancing reflection When dealing with a beam that is nearly parallel to a surface, it is sometimes more useful to refer to the angle between the beam and the surface tangent, rather than that between the beam and the surface normal. Diffraction is the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. Diffraction is greatest when the size of the gap is similar to the wavelength of the wave. The 90-degree complement to the angle of incidence is called the grazing angle or glancing angle. Angular distance or angular separation, also known as apparent distance or apparent separation, denoted \theta, is the angle between the two sightlines, or between two point objects as viewed from an observer. The path difference is approximately \frac{d \sin(\theta)}{2} so that the minimum intensity occurs at an angle \theta_{min} given by :d\,\sin\theta_\text{min} = \lambda, where d is the width of the slit, \theta_\text{min} is the angle of incidence at which the minimum intensity occurs, and \lambda is the wavelength of the light. Now, substituting in \frac{2\pi}{\lambda} = k, the intensity (squared amplitude) I of the diffracted waves at an angle θ is given by: I(\theta) = I_0 {\left[ \operatorname{sinc} \left( \frac{\pi a}{\lambda} \sin \theta \right) \right] }^2 ==Multiple slits== right|frame|Double-slit diffraction of red laser light right|frame|2-slit and 5-slit diffraction Let us again start with the mathematical representation of Huygens' principle. File:Two-Slit Diffraction.png|Generation of an interference pattern from two-slit diffraction. The figure shows the light diffracted by 2-element and 5-element gratings where the grating spacings are the same; it can be seen that the maxima are in the same position, but the detailed structures of the intensities are different. thumb|A computer-generated image of an Airy disk. thumb| Computer-generated light diffraction pattern from a circular aperture of diameter 0.5 micrometre at a wavelength of 0.6 micrometre (red-light) at distances of 0.1 cm – 1 cm in steps of 0.1 cm. We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. The main central beam, nulls, and phase reversals are apparent. right|thumb|Graph and image of single-slit diffraction. ",+37,3.38,"""0.14""",2.14,0.020,D
+"A chemical reaction takes place in a container of cross-sectional area $50 \mathrm{~cm}^2$. As a result of the reaction, a piston is pushed out through $15 \mathrm{~cm}$ against an external pressure of $1.0 \mathrm{~atm}$. Calculate the work done by the system.","The work done is given by the dot product of the two vectors. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The work is due to change of system volume by expansion or contraction of the system. Then for a given amount of work transferred, the exchange of volumes involves different pressures, inversely with the piston areas, for mechanical equilibrium. The work of the net force is calculated as the product of its magnitude and the particle displacement. The quantity of thermodynamic work is defined as work done by the system on its surroundings. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . The work done by the system on its surroundings is calculated from the quantities that constitute the whole cycle.Lavenda, B.H. (2010). work is represented by the following equation between differentials: \delta W = P \, dV where *\delta W (inexact differential) denotes an infinitesimal increment of work done by the system, transferring energy to the surroundings; *P denotes the pressure inside the system, that it exerts on the moving wall that transmits force to the surroundings.Borgnakke, C., Sontag, R. E. (2009). Pressure–volume work is a kind of contact work, because it occurs through direct material contact with the surrounding wall or matter at the boundary of the system. When work, for example pressure–volume work, is done on its surroundings by a closed system that cannot pass heat in or out because it is confined by an adiabatic wall, the work is said to be adiabatic for the system as well as for the surroundings. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Given only the initial state and the final state of the system, one can only say what the total change in internal energy was, not how much of the energy went out as heat, and how much as work. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. As a result, the work done by the system also depends on the initial and final states. Then, for instance, to calculate the percent of the piston's stroke at which steam admission is cut off: *Calculate the angle whose cosine is twice the lap divided by the valve travel *Calculate the angle whose cosine is twice the (lap plus lead), divided by the valve travel Add the two angles and take the cosine of their sum; subtract 1 from that cosine and multiply the result by -50. Work done on a thermodynamic system, by devices or systems in the surroundings, is performed by actions such as compression, and includes shaft work, stirring, and rubbing. Therefore, work need only be computed for the gravitational forces acting on the bodies. Such work is adiabatic for the surroundings, even though it is associated with friction within the system. If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. Because it does not change the volume of the system it is not measured as pressure–volume work, and it is called isochoric work. ",-75,30,"""0.042""",0.000216, 6.07,A
+"A mixture of water and ethanol is prepared with a mole fraction of water of 0.60 . If a small change in the mixture composition results in an increase in the chemical potential of water by $0.25 \mathrm{~J} \mathrm{~mol}^{-1}$, by how much will the chemical potential of ethanol change?","Specific heat = 2.44 kJ/(kg·K) === Acid-base chemistry === Ethanol is a neutral molecule and the pH of a solution of ethanol in water is nearly 7.00. Ethanol-water mixtures have less volume than the sum of their individual components at the given fractions. It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. Mixing equal volumes of ethanol and water results in only 1.92 volumes of mixture. The chemical potential of a species in a mixture is defined as the rate of change of free energy of a thermodynamic system with respect to the change in the number of atoms or molecules of the species that are added to the system. The volume of alcohol in the solution can then be estimated. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Mixing ethanol and water is exothermic, with up to 777 J/mol being released at 298 K. Mixtures of ethanol and water form an azeotrope at about 89 mole-% ethanol and 11 mole-% water or a mixture of 95.6% ethanol by mass (or about 97% alcohol by volume) at normal pressure, which boils at 351 K (78 °C). Ethanol can be quantitatively converted to its conjugate base, the ethoxide ion (CH3CH2O−), by reaction with an alkali metal such as sodium: :2 CH3CH2OH + 2 Na → 2 CH3CH2ONa + H2 or a very strong base such as sodium hydride: :CH3CH2OH + NaH → CH3CH2ONa + H2 The acidities of water and ethanol are nearly the same, as indicated by their pKa of 15.7 and 16 respectively. As high as 30-50 kcal/mol changes in the potential energy surface (activation energies and relative stability) were calculated if the charge of the metal species was changed during the chemical transformation. ===Free radical syntheses=== Many free radical-based syntheses show large kinetic solvent effects that can reduce the rate of reaction and cause a planned reaction to follow an unwanted pathway. ==See also== * Cage effect ==References== Category:Physical chemistry Category:Reaction mechanisms A solution will have a lower and hence more negative water potential than that of pure water. For example, in a binary mixture, at constant temperature and pressure, the chemical potentials of the two participants A and B are related by : d\mu_\text{B} = -\frac{n_\text{A}}{n_\text{B}}\,d\mu_\text{A} where n_\text{A} is the number of moles of A and n_\text{B} is the number of moles of B. Water potential is the potential energy of water per unit volume relative to pure water in reference conditions. Ethanol is slightly more refractive than water, having a refractive index of 1.36242 (at λ=589.3 nm and ). From the above equation, the chemical potential is given by : \mu_i = \left(\frac{\partial U}{\partial N_i} \right)_{S,V, N_{j e i}}. It is defined as the number of millilitres (mL) of pure ethanol present in of solution at . Furthermore, the more solute molecules present, the more negative the solute potential is. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Under such nomenclature, the ethanol was mixed with 25% water to reduce the combustion chamber temperature. The addition of even a few percent of ethanol to water sharply reduces the surface tension of water. ",-0.38,131,"""1.81""",2,8.7,A
+"The enthalpy of fusion of mercury is $2.292 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and its normal freezing point is $234.3 \mathrm{~K}$ with a change in molar volume of $+0.517 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$ on melting. At what temperature will the bottom of a column of mercury (density $13.6 \mathrm{~g} \mathrm{~cm}^{-3}$ ) of height $10.0 \mathrm{~m}$ be expected to freeze?","T_M(d) = T_{MB}(1-\frac{4\sigma\,_{sl}}{H_f\rho\,_sd}) Where: TMB = bulk melting temperature ::σsl = solid–liquid interface energy ::Hf = Bulk heat of fusion ::ρs = density of solid ::d = particle diameter ==Semiconductor/covalent nanoparticles== Equation 2 gives the general relation between the melting point of a metal nanoparticle and its diameter. T_M(d)=\frac{4T_{MB}}{H_fd}\left(\sigma\,_{sv}-\sigma\,_{lv}\left(\frac{\rho\,_s}{\rho\,_l}\right)^{2/3}\right) Where: σsv=solid-vapor interface energy ::σlv=liquid-vapor interface energy ::Hf=Bulk heat of fusion ::ρs=density of solid ::ρl=density of liquid ::d=diameter of nanoparticle ===Liquid shell nucleation model=== The liquid shell nucleation model (LSN) predicts that a surface layer of atoms melts prior to the bulk of the particle. :This article deals with melting/freezing point depression due to very small particle size. The theoretical size-dependent melting point of a material can be calculated through classical thermodynamic analysis. Equation 4 gives the normalized, size-dependent melting temperature of a material according to the liquid-drop model. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). T_M(d)=T_{MB}(1-(\frac{c}{d})^2) Where: TMB=bulk melting temperature ::c=materials constant ::d=particle diameter Equation 3 indicates that melting point depression is less pronounced in covalent nanoparticles due to the quadratic nature of particle size dependence in the melting Equation. ==Proposed mechanisms== The specific melting process for nanoparticles is currently unknown. K (? °C), ? K (? °C), ? K (? °C), ? K (? °C), ? T_M(d)=\frac{4T_{MB}}{H_fd}(\frac{\sigma\,_{sv}}{1-\frac{d_0}{d}}-\sigma\,_{lv}(1-\frac{\rho\,_s}{\rho\,_l})) Where: d0=atomic diameter ===Liquid nucleation and growth model=== The liquid nucleation and growth model (LNG) treats nanoparticle melting as a surface- initiated process. Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The decrease in melting temperature can be on the order of tens to hundreds of degrees for metals with nanometer dimensions. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? The melting temperature of a nanoparticle decreases sharply as the particle reaches critical diameter, usually < 50 nm for common engineering metals. The melting temperature of a nanoparticle is a function of its radius of curvature according to the LSN. The model calculates melting conditions as a function of two competing order parameters using Landau potentials. thumb|0 °C = 32 °F A degree of frost is a non-standard unit of measure for air temperature meaning degrees below melting point (also known as ""freezing point"") of water (0 degrees Celsius or 32 degrees Fahrenheit). The Mollier diagram coordinates are enthalpy h and humidity ratio x. More recently, researchers developed nanocalorimeters that directly measure the enthalpy and melting temperature of nanoparticles. ",0,1.154700538,"""0.6""",16.3923,234.4,E
+"Suppose a nanostructure is modelled by an electron confined to a rectangular region with sides of lengths $L_1=1.0 \mathrm{~nm}$ and $L_2=2.0 \mathrm{~nm}$ and is subjected to thermal motion with a typical energy equal to $k T$, where $k$ is Boltzmann's constant. How low should the temperature be for the thermal energy to be comparable to the zero-point energy？","When L is comparable to or smaller than the mean free path (which is of the order 1 µm for carbon nanostructures ), the continuous energy model used for bulk materials no longer applies and nonlocal and nonequilibrium aspects to heat transfer also need to be considered. These issues can be addressed with phonon engineering, once nanoscale thermal behaviors have been studied and understood. ==The effect of the limited length of structure== In general two carrier types can contribute to thermal conductivity - electrons and phonons. Modeling of the low-temperature specific heat allows determination of the on-tube phonon velocity, the splitting of phonon subbands on a single tube, and the interaction between neighboring tubes in a bundle. ===Thermal conductivity measurements=== Measurements show a single-wall carbon nanotubes (SWNTs) room-temperature thermal conductivity about 3500 W/(m·K), and over 3000 W/(m·K) for individual multiwalled carbon nanotubes (MWNTs). Since the Fermi level in a metal at absolute zero is the energy of the highest occupied single particle state, then the Fermi energy in a metal is the energy difference between the Fermi level and lowest occupied single-particle state, at zero-temperature. ==Context== In quantum mechanics, a group of particles known as fermions (for example, electrons, protons and neutrons) obey the Pauli exclusion principle. The Fermi temperature is defined as T_\text{F} = \frac{E_\text{F}}{k_\text{B}}, where k_\text{B} is the Boltzmann constant, and E_\text{F} the Fermi energy. Only when the temperature exceeds the related Fermi temperature, do the particles begin to move significantly faster than at absolute zero. The Fermi temperature for a metal is a couple of orders of magnitude above room temperature. Junction temperature, short for transistor junction temperature, is the highest operating temperature of the actual semiconductor in an electronic device. In physics, the thermal conductance quantum g_0 describes the rate at which heat is transported through a single ballistic phonon channel with temperature T. For CNT, represented as 1-D ballistic electronic channel, the electronic conductance is quantized, with a universal value of :G_0 = \frac{2e^2}{h} Similarly, for a single ballistic 1-D channel, the thermal conductance is independent of materials parameters, and there exists a quantum of thermal conductance, which is linear in temperature: :G_{th} = \frac{\pi^2 {k_B}^2 T}{3h} Possible conditions for observation of this quantum were examined by Rego and Kirczenow. The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature. These measurements employed suspended silicon nitride () nanostructures that exhibited a constant thermal conductance of 16 g_0 at temperatures below approximately 0.6 kelvin. == Relation to the quantum of electrical conductance == For ballistic electrical conductors, the electron contribution to the thermal conductance is also quantized as a result of the electrical conductance quantum and the Wiedemann–Franz law, which has been quantitatively measured at both cryogenic (~20 mK) and room temperature (~300K). Then : \kappa = L T \sigma = {\pi^2 k_{\rm B}^2 \over 3e^2}\times {n e^2 \over h} T = {\pi^2 k_{\rm B}^2 \over 3h} n T = g_0 n, where g_0 is the thermal conductance quantum defined above. ==References== ==See also== * Thermal properties of nanostructures Category:Mesoscopic physics Category:Nanotechnology Category:Quantum mechanics Category:Condensed matter physics Category:Physical quantities Category:Heat conduction It was shown that, using this formula and atomistically computed phonon dispersions (with interatomic potentials developed in ), it is possible to predictively calculate lattice thermal conductivity curves for nanowires, in good agreement with experiments. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. Berber et al. have calculated the phonon thermal conductivity of isolated nanotubes. In condensed matter physics, the recoil temperature is a fundamental lower limit of temperature attainable by some laser cooling schemes, and corresponds to the kinetic energy imparted in an atom initially at rest by the spontaneous emission of a photon. It may be that this weak coupling, which is problematic for mechanical applications of nanotubes, is an advantage for thermal applications. ====Phonon density of states for nanotubes==== The phonon density of states is to calculated through band structure of isolated nanotubes, which is studied in Saito et al. and Sanchez-Portal et al. Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. The Fermi temperature can be thought of as the temperature at which thermal effects are comparable to quantum effects associated with Fermi statistics. As the devices continue to shrink further into the sub-100 nm range following the trend predicted by Moore’s law, the topic of thermal properties and transport in such nanoscale devices becomes increasingly important. Therefore, the phonon thermal conductivity displays a peak and decreases with increasing temperature. ",-8,1.5,"""5.5""",0.7812,0.0761,C
+Calculate the change in Gibbs energy of $35 \mathrm{~g}$ of ethanol (mass density $0.789 \mathrm{~g} \mathrm{~cm}^{-3}$ ) when the pressure is increased isothermally from 1 atm to 3000 atm.,"The Gibbs energy of any system is and an infinitesimal change in G, at constant temperature and pressure, yields :dG=dU+pdV-TdS. The Gibbs free energy is expressed as : G(p,T) = U + pV - TS = H - TS where p is pressure, T is the temperature, U is the internal energy, V is volume, H is the enthalpy, and S is the entropy. This implies that at equilibrium Q_\text{r} = K_\text{eq} and \Delta_\text{r} G = 0. ==Standard Gibbs energy change of formation== Table of selected substancesCRC Handbook of Chemistry and Physics, 2009, pp. 5-4–5-42, 90th ed., Lide. Alcoholic fermentation converts one mole of glucose into two moles of ethanol and two moles of carbon dioxide, producing two moles of ATP in the process. In isothermal, isobaric systems, Gibbs free energy can be thought of as a ""dynamic"" quantity, in that it is a representative measure of the competing effects of the enthalpic and entropic driving forces involved in a thermodynamic process. thumb|upright=1.9|Relation to other relevant parameters The temperature dependence of the Gibbs energy for an ideal gas is given by the Gibbs–Helmholtz equation, and its pressure dependence is given by \frac{G}{N} = \frac{G^\circ}{N} + kT\ln \frac{p}{p^\circ}. or more conveniently as its chemical potential: \frac{G}{N} = \mu = \mu^\circ + kT\ln \frac{p}{p^\circ}. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The Gibbs free energy change , measured in joules in SI) is the maximum amount of non-volume expansion work that can be extracted from a closed system (one that can exchange heat and work with its surroundings, but not matter) at fixed temperature and pressure. The Gibbs energy is the thermodynamic potential that is minimized when a system reaches chemical equilibrium at constant pressure and temperature when not driven by an applied electrolytic voltage. In an isobaric process, the pressure remains constant, so the heat interaction is the change in enthalpy. Each quantity in the equations above can be divided by the amount of substance, measured in moles, to form molar Gibbs free energy. Substance (state) ΔfG° ΔfG° Substance (state) (kJ/mol) (kcal/mol) NO(g) 87.6 20.9 NO2(g) 51.3 12.3 N2O(g) 103.7 24.78 H2O(g) −228.6 −54.64 H2O(l) −237.1 −56.67 CO2(g) −394.4 −94.26 CO(g) −137.2 −32.79 CH4(g) −50.5 −12.1 C2H6(g) −32.0 −7.65 C3H8(g) −23.4 −5.59 C6H6(g) 129.7 29.76 C6H6(l) 124.5 31.00 The standard Gibbs free energy of formation of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of that substance from its component elements, in their standard states (the most stable form of the element at 25 °C and 100 kPa). Moreover, we also have \begin{align} K_\text{eq} &= e^{-\frac{\Delta_\text{r} G^\circ}{RT}}, \\\ \Delta_\text{r} G^\circ &= -RT\left(\ln K_\text{eq}\right) = -2.303\,RT\left(\log_{10} K_\text{eq}\right), \end{align} which relates the equilibrium constant with Gibbs free energy. At constant pressure the above equation produces a Maxwell relation that links the change in open cell voltage with temperature T (a measurable quantity) to the change in entropy S when charge is passed isothermally and isobarically. Further, Gibbs stated: In this description, as used by Gibbs, ε refers to the internal energy of the body, η refers to the entropy of the body, and ν is the volume of the body... In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work, other than pressure-volume work, that may be performed by a thermodynamically closed system at constant temperature and pressure. The entropy of vaporization is then equal to the heat of vaporization divided by the boiling point: : \Delta S_\text{vap} = \frac{\Delta H_\text{vap}}{T_\text{vap}}. When a system transforms reversibly from an initial state to a final state under these conditions, the decrease in Gibbs free energy equals the work done by the system to its surroundings, minus the work of the pressure forces. Approximately 2.8 gallons of ethanol are produced from one bushel of corn (0.42 liter per kilogram). However, simply substituting the above integrated result for U into the definition of G gives a standard expression for G: :\begin{align} G &= U + p V - TS\\\ &= \left(T S - p V + \sum_i \mu_i N_i \right) + p V - T S\\\ &= \sum_i \mu_i N_i. \end{align} This result shows that the chemical potential of a substance i is its (partial) mol(ecul)ar Gibbs free energy. At standard pressure , the value is denoted as and normally expressed in joules per mole-kelvin, J/(mol·K). thumb|400px|Diagram showing pressure difference induced by a temperature difference. The quantities on the right are all directly measurable. ==Useful identities to derive the Nernst equation== During a reversible electrochemical reaction at constant temperature and pressure, the following equations involving the Gibbs free energy hold: *\Delta_\text{r} G = \Delta_\text{r} G^\circ + R T \ln Q_\text{r} (see chemical equilibrium), *\Delta_\text{r} G^\circ = -R T \ln K_\text{eq} (for a system at chemical equilibrium), *\Delta_\text{r} G = w_\text{elec,rev} = -nF\mathcal{E} (for a reversible electrochemical process at constant temperature and pressure), *\Delta_\text{r} G^\circ = -nF\mathcal{E}^\circ (definition of \mathcal{E}^\circ), and rearranging gives \begin{align} nF\mathcal{E}^\circ &= RT \ln K_\text{eq}, \\\ nF\mathcal{E} &= nF\mathcal{E}^\circ - R T \ln Q_\text{r}, \\\ \mathcal{E} &= \mathcal{E}^\circ - \frac{R T}{n F} \ln Q_\text{r}, \end{align} which relates the cell potential resulting from the reaction to the equilibrium constant and reaction quotient for that reaction (Nernst equation), where * , Gibbs free energy change per mole of reaction, * , Gibbs free energy change per mole of reaction for unmixed reactants and products at standard conditions (i.e. 298K, 100kPa, 1M of each reactant and product), * , gas constant, * , absolute temperature, * , natural logarithm, * , reaction quotient (unitless), * , equilibrium constant (unitless), * , electrical work in a reversible process (chemistry sign convention), * , number of moles of electrons transferred in the reaction, * , Faraday constant (charge per mole of electrons), * \mathcal{E}, cell potential, * \mathcal{E}^\circ, standard cell potential. ",12,0.00539,"""6.0""",14.5115,-17,A
+The promotion of an electron from the valence band into the conduction band in pure $\mathrm{TIO}_2$ by light absorption requires a wavelength of less than $350 \mathrm{~nm}$. Calculate the energy gap in electronvolts between the valence and conduction bands.,"Within the concept of bands, the energy gap between the valence band and the conduction band is the band gap. For materials with a direct band gap, valence electrons can be directly excited into the conduction band by a photon whose energy is larger than the bandgap. In graphs of the electronic band structure of solids, the band gap refers to the energy difference (often expressed in electronvolts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. This formula is valid only for light with photon energy larger, but not too much larger, than the band gap (more specifically, this formula assumes the bands are approximately parabolic), and ignores all other sources of absorption other than the band-to-band absorption in question, as well as the electrical attraction between the newly created electron and hole (see exciton). By plotting certain powers of the absorption coefficient against photon energy, one can normally tell both what value the band gap is, and whether or not it is direct. The term ""band gap"" refers to the energy difference between the top of the valence band and the bottom of the conduction band. A band gap is an energy range in a solid where no electron states can exist due to the quantization of energy. However, in order for a valence band electron to be promoted to the conduction band, it requires a specific minimum amount of energy for the transition. Especially in condensed-matter physics, an energy gap is often known more abstractly as a spectral gap, a term which need not be specific to electrons or solids. ==Band gap== If an energy gap exists in the band structure of a material, it is called band gap. The relationship between band gap energy and temperature can be described by Varshni's empirical expression (named after Y. P. Varshni), :E_g(T)=E_g(0)-\frac{\alpha T^2}{T+\beta}, where Eg(0), α and β are material constants. A semiconductor will not absorb photons of energy less than the band gap; and the energy of the electron-hole pair produced by a photon is equal to the bandgap energy. The band gap is called ""direct"" if the crystal momentum of electrons and holes is the same in both the conduction band and the valence band; an electron can directly emit a photon. On the other hand, for an indirect band gap, the formula is: :\alpha \propto \frac{(h u- E_{\text{g}}+E_{\text{p}})^2}{\exp(\frac{E_{\text{p}}}{kT})-1} + \frac{(h u- E_{\text{g}}-E_{\text{p}})^2}{1-\exp(-\frac{E_{\text{p}}}{kT})} where: *E_{\text{p}} is the energy of the phonon that assists in the transition *k is Boltzmann's constant *T is the thermodynamic temperature This formula involves the same approximations mentioned above. For the same reason as above, light with a photon energy close to the band gap can penetrate much farther before being absorbed in an indirect band gap material than a direct band gap one (at least insofar as the light absorption is due to exciting electrons across the band gap). In solid-state physics, the most important spectral gap is for the many-body system of electrons in a solid material, in which case it is often known as an energy gap. Since the mid-gap states do exist within some depth of the semiconductor, they must be a mixture (a Fourier series) of valence and conduction band states from the bulk. In solid-state physics, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. Group Material Symbol Band gap (eV) @ 302K Reference III–V Aluminium nitride AlN 6.0 IV Diamond C 5.5 IV Silicon Si 1.14 IV Germanium Ge 0.67 III–V Gallium nitride GaN 3.4 III–V Gallium phosphide GaP 2.26 III–V Gallium arsenide GaAs 1.43 IV–V Silicon nitride Si3N4 5 IV–VI Lead(II) sulfide PbS 0.37 IV–VI Silicon dioxide SiO2 9 Copper(I) oxide Cu2O 2.1 ==Optical versus electronic bandgap== In materials with a large exciton binding energy, it is possible for a photon to have just barely enough energy to create an exciton (bound electron–hole pair), but not enough energy to separate the electron and hole (which are electrically attracted to each other). In solid-state physics, an energy gap is an energy range in a solid where no electron states exist, i.e. an energy range where the density of states vanishes. In contrast, for materials with an indirect band gap, a photon and phonon must both be involved in a transition from the valence band top to the conduction band bottom, involving a momentum change. If there is a small band gap (Eg), then the flow of electrons from valence to conduction band is possible only if an external energy (thermal, etc.) is supplied; these groups with small Eg are called semiconductors. In an ""indirect"" gap, a photon cannot be emitted because the electron must pass through an intermediate state and transfer momentum to the crystal lattice. ",0.139,5.4,"""3.54""",-0.0301,0.6321205588,C
+"Although the crystallization of large biological molecules may not be as readily accomplished as that of small molecules, their crystal lattices are no different. Tobacco seed globulin forms face-centred cubic crystals with unit cell dimension of $12.3 \mathrm{~nm}$ and a density of $1.287 \mathrm{~g} \mathrm{~cm}^{-3}$. Determine its molar mass.","The molecular formula C18H22O6 (molar mass: 334.36 g/mol, exact mass: 334.1416 u) may refer to: * Combretastatin * Combretastatin B-1 The molecular formula C3H9O6P (molar mass: 172.07 g/mol, exact mass: 172.0137 u) may refer to: * Glycerol 1-phosphate * Glycerol 2-phosphate (BGP) * Glycerol 3-phosphate Category:Molecular formulas A seed crystal is a small piece of single crystal or polycrystal material from which a large crystal of typically the same material is grown in a laboratory. The molecular formula C21H26O3 (molar mass: 326.42 g/mol, exact mass: 326.1882 u) may refer to: * Acitretin * Buparvaquone * Moxestrol * Octabenzone * RU-16117 * 11-Hydroxycannabinol Category:Molecular formulas The molecular formula C9H16N3O14P3 (molar mass: 483.16 g/mol) may refer to: * Cytidine triphosphate * Arabinofuranosylcytosine triphosphate Used to replicate material, the use of seed crystal to promote growth avoids the otherwise slow randomness of natural crystal growth and allows manufacture on a scale suitable for industry. ==Crystal enlargement== The large crystal can be grown by dipping the seed into a supersaturated solution, into molten material that is then cooled, or by growth on the seed face by passing vapor of the material to be grown over it. ==Theory== The theory behind this effect is thought to derive from the physical intermolecular interaction that occurs between compounds in a supersaturated solution (or possibly vapor). This was especially important in pharmaceutical applications where slight changes in molar mass (e.g. aggregation) or shape may result in different biological activity. The placement of a seed crystal into solution allows the recrystallization process to expedite by eliminating the need for random molecular collision or interaction. Absolute molar mass is a process used to determine the characteristics of molecules. == History == The first absolute measurements of molecular weights (i.e. made without reference to standards) were based on fundamental physical characteristics and their relation to the molar mass. In order to gain information about a polydisperse mixture of molar masses, a method for separating the different sizes was developed. To obtain molar mass, light scattering instruments need to measure the intensity of light scattered at zero angle. The purely mathematical root mean square radius is defined as the radii making up the molecule multiplied by the mass at that radius. == Bibliography == *A. Einstein, Ann. Phys. 33 (1910), 1275 *C.V. Raman, Indian J. Phys. 2 (1927), 1 *P.Debye, J. Appl. Phys. 15 (1944), 338 *B.H. Zimm, J. Chem. Phys. 13 (1945), 141 *B.H. Zimm, J. Chem. Phys. 16 (1948), 1093 *B.H. Zimm, R.S. Stein and P. Dotty, Pol. Bull. 1,(1945), 90 *M. Fixman, J. Chem. Phys. 23 (1955), 2074 *A.C. Ouano and W. Kaye J. Poly. Sci. A1(12) (1974), 1151 *Z. Grubisic, P. Rempp, and H. Benoit, J. Polym. The problem was that the system was calibrated according to the Vh characteristics of polymer standards that are not directly related to the molar mass. Also during the process of tempering chocolate, seed crystals can be used to promote the growth of favorable type V crystals ==See also== * Crystal structure * Crystallization * Laser heated pedestal growth * Micro-pulling-down * Polycrystal * Single crystal * Wafer (electronics) * Disappearing polymorphs ==References== Category:Crystals The next step is to convert the time at which the samples eluted into a measurement of molar mass. This information is the Root Mean Square radius of the molecule (RMS or Rg). As the demands on polymer properties increased, the necessity of getting absolute information on the molar mass and size also increased. Seeding is therefore said to decrease the necessary amount of time needed for nucleation to occur in a recrystallization process. ==Uses== One example where a seed crystal is used to grow large boules or ingots of a single crystal is the semiconductor industry where methods such as the Czochralski process or Bridgman technique are employed. This interaction can potentiate intermolecular forces between the separate molecules and form a basis for a crystal lattice. As previously noted, the MALS detector can also provide information about the size of the molecule. If the relationship between the molar mass and Vh of the standard is not the same as that of the unknown sample, then the calibration is invalid. A low angle light scattering system was developed in the early 1970s that allowed a single measurement to be used to calculate the molar mass. ",0.6296296296,0.5,"""22.0""",1.8,3.61,E
+An electron is accelerated in an electron microscope from rest through a potential difference $\Delta \phi=100 \mathrm{kV}$ and acquires an energy of $e \Delta \phi$. What is its final speed?,"The kinetic energy Ke of an electron moving with velocity v is: :\displaystyle K_{\mathrm{e}} = (\gamma - 1)m_{\mathrm{e}} c^2, where me is the mass of electron. This wavelength, for example, is equal to 0.0037 nm for electrons accelerated across a 100,000-volt potential. The speed of an electron can approach, but never reach, the speed of light in vacuum, c. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. For an electron with rest mass m0, the rest energy is equal to: :\textstyle E_{\mathrm p} = m_0 c^2, where c is the speed of light in vacuum. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. The electron (symbol e) is on the left. For example, the Stanford linear accelerator can accelerate an electron to roughly 51 GeV. This energy is assumed to equal the electron's rest energy, defined by special relativity (E = mc2). The energy emission in turn causes a recoil of the electron, known as the Abraham–Lorentz–Dirac Force, which creates a friction that slows the electron. Electrons can be accelerated by suitable electric (or magnetic) fields, thereby acquiring kinetic energy. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. The required energy of the electrons is typically in the range 20–200 eV. The electron microscope directs a focused beam of electrons at a specimen. Electrons radiate or absorb energy in the form of photons when they are accelerated. For an electron, it has a value of . Relativistic electron beams are streams of electrons moving at relativistic speeds. For the special case of an electrostatic field that is surpassed by a particle, the acceleration voltage is directly given by integrating the electric field along its path. The upper bound of the electron radius of 10−18 meters can be derived using the uncertainty relation in energy. In accelerator physics, the term acceleration voltage means the effective voltage surpassed by a charged particle along a defined straight line. When an excited electron falls back to a state of lower energy, it undergoes electron relaxation (deexcitationSakho, Ibrahima. ",1.88,8,"""0.28209479""",1.51,0.333333,A
+The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature. By how much does the standard molar enthalpy of $\mathrm{SO}_2(\mathrm{~g})$ increase when the temperature is raised from $298.15 \mathrm{~K}$ to $1500 \mathrm{~K}$ ?,"The molar heat capacity generally increases with the molar mass, often varies with temperature and pressure, and is different for each state of matter. The following is a table of some constant- pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* H2 28.9 5.0 29.6 5.1 41.2 7.9 Not saturated.""Hydrogen"" NIST Chemistry WebBook, SRD 69, online. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume.Lange's Handbook of Chemistry, 10th ed. p. 1524 All methods for the measurement of specific heat apply to molar heat capacity as well. ==Units== The SI unit of molar heat capacity heat is joule per kelvin per mole (J/(K⋅mol), J/(K mol), J K−1 mol−1, etc.). On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. Therefore, the word ""molar"", not ""specific"", should always be used for this quantity. ==Definition== The molar heat capacity of a substance, which may be denoted by cm, is the heat capacity C of a sample of the substance, divided by the amount (moles) n of the substance in the sample: :cm{} \;=\; \frac{C}{n} \;=\; \frac{1}{n} \lim_{\Delta T \rightarrow 0}\frac{Q}{\Delta T} where Q is the amount of heat needed to raise the temperature of the sample by ΔT. Like the heat capacity of an object, the molar heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T of the sample and the pressure P applied to it. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Then the molar heat capacity (at constant volume) would be :cV,m = fR where R is the ideal gas constant. On the other hand, the vibration modes only start to become active around 350 K (77 °C) Accordingly, the molar heat capacity cP,m is nearly constant at 29.1 J⋅K−1⋅mol−1 from 100 K to about 300 °C. According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). Since the molar heat capacity of a substance is the specific heat c times the molar mass of the substance M/N its numerical value is generally smaller than that of the specific heat. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Like the specific heat, the measured molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated (at constant pressure, or isobaric) than when it is heated in a closed vessel that prevents expansion (at constant volume, or isochoric). The value thus obtained is said to be the molar heat capacity at constant pressure (or isobaric), and is often denoted cP,m, cp,m, cP,m, etc. For example, ""H2O: 75.338 J⋅K−1⋅mol−1 (25 °C, 101.325 kPa)"" W. Wagner, J. R. Cooper, A. Dittmann, J. Kijima, H.-J. Kretzschmar, A. Kruse, R. Mare, K. Oguchi, H. Sato, I. Stöcker, O. Šifner, Y. Takaishi, I. Tanishita, J. Trübenbach and Th. Willkommen (2000): ""The IAPWS industrial formulation 1997 for the thermodynamic properties of water and steam"", ASME J. Eng. Gas Turbines and Power, volume 122, pages 150–182 When not specified, published values of the molar heat capacity cm generally are valid for some standard conditions for temperature and pressure. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. These parameters are usually specified when giving the molar heat capacity of a substance. At high enough temperature, its molar heat capacity then should be cP,m = 7.5 R = 62.63 J⋅K−1⋅mol−1. ",1.94,1.07,"""1.2""",62.2,1.06,D
+Suppose that the normalized wavefunction for an electron in a carbon nanotube of length $L=10.0 \mathrm{~nm}$ is: $\psi=(2 / L)^{1 / 2} \sin (\pi x / L)$. Calculate the probability that the electron is between $x=4.95 \mathrm{~nm}$ and $5.05 \mathrm{~nm}$.,"b) Linear dependence of the electron energy on the wave vector in CNTs; c) Dispersion relation near the Fermi energy for a semiconducting CNT; d) Dispersion relation near the Fermi energy for a metallic CNT Conduction in single-walled carbon nanotubes is quantized due to their one-dimensionality and the number of allowed electronic states is limited, if compared to bulk graphite. If a carbon nanotube is a ballistic conductor, but the contacts are nontransparent, the transmission probability, T, is reduced by back- scattering in the contacts. Scattering of electrons by optical phonons in carbon nanotube channels has two requirements: * The traveled length in the conduction channel between source and drain has to be greater than the optical phonon mean free path * The electron energy has to be greater than the critical optical phonon emission energy === Schottky barrier Ballistic conduction === thumb|400px|Figure 2: Example of the band structure of a ballistic CNT FET. In order to estimate the current in the carbon nanotube channel, the Landauer formula can be applied, which considers a one-dimensional channel, connected to two contacts – source and drain. In semiconducting CNTs at room temperature and for low energies, the mean free path is determined by the electron scattering from acoustic phonons, which results in lm ≈ 0.5μm. Single-walled carbon nanotubes in the fields of quantum mechanics and nanoelectronics, have the ability to conduct electricity. When ballistically conducted, the electrons travel through the nanotubes channel without experiencing scattering due to impurities, local defects or lattice vibrations. ""Carbon Nanotube and Graphene Device Physics"", Cambridge UP, 2011. Another way to make carbon nanotube transistors has been to use random networks of them. Carbon nanotube transistors as logic-gate circuits with densities comparable to modern CMOS technology has not yet been demonstrated. The potential of carbon nanotubes was demonstrated in 2003 when room- temperature ballistic transistors with ohmic metal contacts and high-k gate dielectric were reported, showing 20–30x higher ON current than state-of-the- art Si MOSFETs. Carbon nanotube chemistry involves chemical reactions, which are used to modify the properties of carbon nanotubes (CNTs). In order to derive the current-voltage (I-V) characteristics for a ballistic CNT FET, one can start with Planck's postulate, which relates the energy of the i-th state to its frequency: E_i=h u_i=\frac{h}{2e}\frac{2e}{T_i}=\frac{h}{2e}I_i The total current for a many-state system is then the sum over the energy of each state multiplied by the occupation probability function, in this case the Fermi–Dirac statistics: I_i=\frac{2e}{h}\sum_{i}E_i\frac{1}{1+e^{\frac{E-E_f}{k_BT}}} For a system with dense states, the discrete sum can be approximated by an integral: I_i=\frac{2e}{h}\int \frac{1}{1+e^{\frac{E-E_f}{k_BT}}}dE In CNT FETs, the charge carriers move either left (negative velocity) or right (positive velocity) and the resulting net current is called drain current. When ballistic in nature conductance can be treated as if the electrons experience no scattering. == Conductance quantization and Landauer formula == thumb|400px|Figure 1: a) Energy contour plot of the electronic band structure in CNTs.; thumbnail|right|Plot of probit function In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. The exceptional electrical and mechanical properties of carbon nanotubes have made them alternatives to the traditional electrical actuators for both microscopic and macroscopic applications. These processes were essential for low yield production of carbon nanotubes where carbon particles, amorphous carbon particles and coatings comprised a significant percentage of the overall material and are still important for the introduction of surface functional groups. A carbon nanotube quantum dot (CNT QD) is a small region of a carbon nanotube in which electrons are confined. ==Formation== A CNT QD is formed when electrons are confined to a small region within a carbon nanotube. The CNT QD is modelled as an Anderson-type model, which can be reduced by Schrieffer-Wolff transformation to an effective Kondo-type model at low temperature. ==Other nanotube system== Similar mesoscopic devices have been constructed from elements other than carbon. ""Nanowelded Carbon Nanotubes from Field-effect Transistors to Solar Microcells"", Heidelberg: Springer, 2009. Major obstacles to nanotube-based microelectronics include the absence of technology for mass production, circuit density, positioning of individual electrical contacts, sample purity, control over length, chirality and desired alignment, thermal budget and contact resistance. Carbon nanotubes (CNTs) are cylinders of one or more layers of graphene (lattice). ",0.08,226,"""1.0""",0.020,0.011,D
+"A sample of the sugar D-ribose of mass $0.727 \mathrm{~g}$ was placed in a calorimeter and then ignited in the presence of excess oxygen. The temperature rose by $0.910 \mathrm{~K}$. In a separate experiment in the same calorimeter, the combustion of $0.825 \mathrm{~g}$ of benzoic acid, for which the internal energy of combustion is $-3251 \mathrm{~kJ} \mathrm{~mol}^{-1}$, gave a temperature rise of $1.940 \mathrm{~K}$. Calculate the enthalpy of formation of D-ribose.","However the standard enthalpy of combustion is readily measurable using bomb calorimetry. The standard enthalpy of formation is then determined using Hess's law. That is, the heat of combustion, ΔH°comb, is the heat of reaction of the following process: : (std.) + (c + - ) (g) → c (g) + (l) + (g) Chlorine and sulfur are not quite standardized; they are usually assumed to convert to hydrogen chloride gas and or gas, respectively, or to dilute aqueous hydrochloric and sulfuric acids, respectively, when the combustion is conducted in a bomb calorimeter containing some quantity of water. == Ways of determination == ===Gross and net=== Zwolinski and Wilhoit defined, in 1972, ""gross"" and ""net"" values for heats of combustion. The calorific value is the total energy released as heat when a substance undergoes complete combustion with oxygen under standard conditions. They may also be calculated as the difference between the heat of formation ΔH of the products and reactants (though this approach is somewhat artificial since most heats of formation are typically calculated from measured heats of combustion). For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of formation is measured in units of energy per amount of substance, usually stated in kilojoule per mole (kJ mol−1), but also in kilocalorie per mole, joule per mole or kilocalorie per gram (any combination of these units conforming to the energy per mass or amount guideline). The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. The experimental heat of formation of ethane is -20.03 kcal/mol and ethane consists of 2 P groups. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. It may be expressed with the quantities: * energy/mole of fuel * energy/mass of fuel * energy/volume of the fuel There are two kinds of enthalpy of combustion, called high(er) and low(er) heat(ing) value, depending on how much the products are allowed to cool and whether compounds like are allowed to condense. The temperatures mentioned here are for a stoichiometric fuel-oxidizer mixture (i.e. equivalence ratio φ = 1). alt=|thumb| Fischer Projection Ribose is a simple sugar and carbohydrate with molecular formula C5H10O5 and the linear-form composition H−(C=O)−(CHOH)4−H. 150px 150px Comparison of the chemical structures of ribose (top) and deoxyribose (bottom). This is the same as the thermodynamic heat of combustion since the enthalpy change for the reaction assumes a common temperature of the compounds before and after combustion, in which case the water produced by combustion is condensed to a liquid. It is possible to predict heats of formation for simple unstrained organic compounds with the heat of formation group additivity method. == Use in calculation for other reactions == The standard enthalpy change of any reaction can be calculated from the standard enthalpies of formation of reactants and products using Hess's law. Since the heat of combustion of these elements is known, the heating value can be calculated using Dulong's Formula: LHV [kJ/g]= 33.87mC \+ 122.3(mH \- mO ÷ 8) + 9.4mS where mC, mH, mO, mN, and mS are the contents of carbon, hydrogen, oxygen, nitrogen, and sulfur on any (wet, dry or ash free) basis, respectively. === Higher heating value === The higher heating value (HHV; gross energy, upper heating value, gross calorific value GCV, or higher calorific value; HCV) indicates the upper limit of the available thermal energy produced by a complete combustion of fuel. The enthalpy of reaction can then be analyzed by applying Hess's Law, which states that the sum of the enthalpy changes for a number of individual reaction steps equals the enthalpy change of the overall reaction. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. The following figure illustrates that the effects of dissociation tend to lower the adiabatic flame temperature. Incomplete reaction at higher temperature further curtails the effect of a larger heat of combustion. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). ",-0.347,-1270,"""0.4772""",1.3,7.136,B
+"An electron confined to a metallic nanoparticle is modelled as a particle in a one-dimensional box of length $L$. If the electron is in the state $n=1$, calculate the probability of finding it in the following regions: $0 \leq x \leq \frac{1}{2} L$.","To a first approximation (i.e. assuming that the charges are distributed randomly), the molar configurational electronic entropy is given by: :S \approx n_\text{sites} \left [ x \ln x + (1-x) \ln (1-x) \right ] where is the fraction of sites on which a localized electron/hole could reside (typically a transition metal site), and is the concentration of localized electrons/holes. Electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. The probability of occupation of each eigenstate is given by the Fermi function, : :p(E)=f=\frac{1}{e^{(E-E_{\rm F}) / k_{\rm B} T} + 1} where is the Fermi energy and is the absolute temperature. For one-electron systems, the electron density at any point is proportional to the square magnitude of the wavefunction. == Definition == The electronic density corresponding to a normalised N-electron wavefunction \Psi (with \textbf r and s denoting spatial and spin variables respectively) is defined as : \rho(\mathbf{r}) = \langle\Psi|\hat{\rho}(\mathbf{r})|\Psi\rangle, where the operator corresponding to the density observable is :\hat{\rho}(\mathbf{r}) = \sum_{i=1}^{N}\ \delta(\mathbf{r}-\mathbf{r}_{i}). The density is determined, through definition, by the normalised N-electron wavefunction which itself depends upon 4N variables (3N spatial and N spin coordinates). Some softwareor example, the Spartan program from Wavefunction, Inc. also allows for specification of the electron density in terms of percentage of total electrons enclosed. For every possible transfer of an electron from an occupied site i to an unoccupied site j , the energy invested should be positive, since we are assuming we are in the ground state of the system, i.e., \Delta E>=0 . In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The Fermi–Dirac distribution implies that each eigenstate of a system, , is occupied with a certain probability, . Of course, the localized charges are not distributed randomly, as the charges will interact electrostatically with one another, and so the above formula should only be regarded as an approximation to the configurational atomic entropy. We will solve for each independently: Let E be an energy value above the well (E>0) * For 0 < x < (a-b): \begin{align} \frac{-\hbar^2}{2m} \psi_{xx} &= E \psi \\\ \Rightarrow \psi &= A e^{i \alpha x} + A' e^{-i \alpha x} & \left( \alpha^2 = {2mE \over \hbar^2} \right) \end{align} *For -b : \begin{align} \frac{-\hbar^2}{2m} \psi_{xx} &= (E+V_0)\psi \\\ \Rightarrow \psi &= B e^{i \beta x} + B' e^{-i \beta x} & \left( \beta^2 = {2m(E+V_0) \over \hbar^2} \right). \end{align} To find u(x) in each region, we need to manipulate the electron's wavefunction: \begin{align} \psi(0 And in the same manner: u(-b To complete the solution we need to make sure the probability function is continuous and smooth, i.e.: \psi(0^{-})=\psi(0^{+}) \qquad \psi'(0^{-})=\psi'(0^{+}). In these situations, the density simplifies to :\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2. == General properties == From its definition, the electron density is a non-negative function integrating to the total number of electrons. Moreover, in condensed matter and molecules, the electron clouds of the atoms usually overlap to some extent, and some of the electrons may roam over a large region encompassing two or more atoms. Switching from summing over individual states to integrating over energy levels, the entropy can be written as: :S=-k_{\rm B} \int n(E) \left [ p(E) \ln p(E) +(1- p(E)) \ln \left ( 1- p(E)\right ) \right ]dE where is the density of states of the solid. Metals have non-zero density of states at the Fermi level. As the density of states at the Fermi level varies widely between systems, this approximation is a reasonable heuristic for inferring when it may be necessary to include electronic entropy in the thermodynamic description of a system; only systems with large densities of states at the Fermi level should exhibit non-negligible electronic entropy (where large may be approximately defined as ). == Application to different materials classes == Insulators have zero density of states at the Fermi level due to their band gaps. The occupation numbers are not limited to the range of zero to two, and therefore sometimes even the response density can be negative in certain regions of space. ==Overview== In molecules, regions of large electron density are usually found around the atom, and its bonds. Electrons and Holes in Semiconductors: With Applications to Transistor Electronics, Bell Telephone Laboratories series, Van Nostrand. The observation of this is expected to occur below a certain temperature, such that the optimal energy of hopping would be smaller than the width of the Coulomb gap. An earlier version containing some of the material appeared in the Bell Labs Technical Journal in 1949. ==Contents== The arrangement of chapters is as follows: Part I INTRODUCTION TO TRANSISTOR ELECTRONICS *Chapter 1: THE BULK PROPERTIES OF SEMICONDUCTORS *Chapter 2: THE TRANSISTOR AS A CIRCUIT ELEMENT *Chapter 3: QUANTITATIVE STUDIES OF INJECTION OF HOLES AND ELECTRONS *Chapter 4: ON THE PHYSICAL THEORY OF TRANSISTORS *Chapter 5: QUANTUM STATES, ENERGY BANDS, AND BRILLOUIN ZONES PART II DESCRIPTIVE THEORY OF SEMICONDUCTORS *Chapter 6: VELOCITIES AND CURRENTS FOR ELECTRONS I N CRYSTALS *Chapter 7: ELECTRONS AND HOLES IN ELECTRIC AND MAGNETIC FIELDS *Chapter 8: INTRODUCTORY THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 9: DISTRIBUTIONS OF QUANTUM STATES IN ENERGY *Chapter 10: FERMI-DIRAC STATISTICS FOR SEMICONDUCTORS *Chapter 11: MATHEMATICAL THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 12: APPLICATIONS TO TRANSISTOR ELECTRONICS PART III QUANTUM MECHANICAL FOUNDATIONS *Chapter 13: INTRODUCTION TO PART III *Chapter 14: ELEMENTARY QUANTUM MECHANICS WITH CIRCUIT THEORY ANALOGUES *Chapter 15: THEORY OF ELECTRON AND HOLE VELOCITIES, CURRENTS AND ACCELERATIONS *Chapter 16: STATISTICAL MECHANICS FOR SEMICONDUCTORS *Chapter 17: THE THEORY OF TRANSITION PROBABILITIES FOR HOLES AND ELECTRONS *APPENDIX A Units *APPENDIX B Periodic Table *APPENDIX C Values of the physical constants *APPENDIX D Energy conversion chart *NAME INDEX *SUBJECT INDEX Category:1950 non-fiction books Category:1950 in science Category:Physics textbooks Together with the normalization property places acceptable densities within the intersection of L1 and L3 – a superset of \mathcal{J}_{N}. == Topology == The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus. === Nuclear cusp condition === The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron- nucleus Coulomb potential. ",6,0.829,"""-20.0""",4500,0.5,E
+"The carbon-carbon bond length in diamond is $154.45 \mathrm{pm}$. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is $3.516 \mathrm{~g} \mathrm{~cm}^{-3}$.","The atomic packing factor of the diamond cubic structure (the proportion of space that would be filled by spheres that are centered on the vertices of the structure and are as large as possible without overlapping) is ≈ 0.34,. significantly smaller (indicating a less dense structure) than the packing factors for the face- centered and body-centered cubic lattices.. The toughness of natural diamond has been measured as 7.5–10 MPa·m1/2. It also has a high density, ranging from 3150 to 3530 kilograms per cubic metre (over three times the density of water) in natural diamonds and 3520 kg/m in pure diamond. St Edmundsbury Press Ltd, Bury St Edwards. ==External links== *Properties of diamond *Properties of diamond (S. Sque, PhD thesis, 2005, University of Exeter, UK) Category:Diamond Category:Allotropes of carbon Diamond Category:Superhard materials Diamond is the allotrope of carbon in which the carbon atoms are arranged in the specific type of cubic lattice called diamond cubic. Diamond is extremely strong owing to its crystal structure, known as diamond cubic, in which each carbon atom has four neighbors covalently bonded to it. thumb|upright=1.25|Main diamond producing countries Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure called diamond cubic. A lattice of 3×3×3 unit cells thumb|245px|Molar volume vs. pressure at room temperature. thumb|3D ball-and- stick model of a diamond lattice The precise tensile strength of diamond is unknown, though strength up to has been observed, and theoretically it could be as high as depending on the sample volume/size, the perfection of diamond lattice and on its orientation: Tensile strength is the highest for the [100] crystal direction (normal to the cubic face), smaller for the [110] and the smallest for the [111] axis (along the longest cube diagonal). thumb|250px|3D ball-and-stick model of a diamond lattice The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. The following issues are considered: * The hardness of diamond and its ability to cleave strongly depend on the crystal orientation. A similar proportion of diamonds comes from the lower mantle at depths between 660 and 800 km. Diamond is thermodynamically stable at high pressures and temperatures, with the phase transition from graphite occurring at greater temperatures as the pressure increases. The Allnatt Diamond is a diamond measuring 101.29 carats (20.258 g) with a cushion cut, rated in color as Fancy Vivid Yellow by the Gemological Institute of America. Thermal conductivity of natural diamond was measured to be about 2200 W/(m·K), which is five times more than silver, the most thermally conductive metal. Diamonds are carbon crystals that form under high temperatures and extreme pressures such as deep within the Earth. Diamonds crystallize in the diamond cubic crystal system (space group Fdm) and consist of tetrahedrally, covalently bonded carbon atoms. The diamond crystal lattice is exceptionally strong, and only atoms of nitrogen, boron, and hydrogen can be introduced into diamond during the growth at significant concentrations (up to atomic percents). Some extrasolar planets may be almost entirely composed of diamond. Thus, graphite is much softer than diamond. Unlike many other minerals, the specific gravity of diamond crystals (3.52) has rather small variation from diamond to diamond. ==Hardness and crystal structure== Known to the ancient Greeks as (, 'proper, unalterable, unbreakable') and sometimes called adamant, diamond is the hardest known naturally occurring material, and serves as the definition of 10 on the Mohs scale of mineral hardness. At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. The Amsterdam Diamond is a black diamond weighing , and has 145 facets. This was determined by comparing the ratios of carbon isotopes present. == Optical and electronic properties == The optical absorption for all diamondoids lies deep in the ultraviolet spectral region with optical band gaps around 6 electronvolts and higher. ",7.654,0.0625,"""9.13""",2,90,A
+"A swimmer enters a gloomier world (in one sense) on diving to greater depths. Given that the mean molar absorption coefficient of seawater in the visible region is $6.2 \times 10^{-3} \mathrm{dm}^3 \mathrm{~mol}^{-1} \mathrm{~cm}^{-1}$, calculate the depth at which a diver will experience half the surface intensity of light.","To calculate this coefficient, light energy is measured at a series of depths from the surface to the depth of 1% illumination. As the depth increases, more light is absorbed by the water. By using artificial light, it is possible to view an object in full color at greater depths. ==Need== Water attenuates light by absorption, so use of a dive light will improve a diver's underwater vision at depth. Maximum Operating Depth (MOD) in metres sea water (msw) for pO2 1.2 to 1.6 MOD (msw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 390 190 123 90 70 66 57 47 40 34 30 22 17 13 10 8 6 1.5 365 178 115 84 65 61 53 44 37 32 28 20 15 11 9 7 5 1.4 340 165 107 78 60 57 48 40 34 29 25 18 13 10 8 6 4 1.3 315 153 98 71 55 52 44 36 31 26 23 16 12 9 6 4 3 1.2 290 140 90 65 50 47 40 33 28 23 20 14 10 7 5 3 2 These depths are rounded to the nearest metre. ==See also== * == Notes == == References == == Sources == * * * * Category:Dive planning Category:Underwater diving safety Category:Breathing gases By directing the same type of beam downwards, the depth to the bottom of the ocean could be calculated. A dive light is a light source carried by an underwater diver to illuminate the underwater environment. The United States Navy Experimental Diving Unit continues to evaluate dive lights for wet and dry illumination output, battery duration, watertight integrity, as well as maximum operating depth. Then, the exponential decline in light is calculated using Beer’s Law with the equation: {I_z \over I_0}= e^{-kz} where k is the light attenuation coefficient, Iz is the intensity of light at depth z, and I0 is the intensity of light at the ocean surface.Idso, Sherwood B. and Gilbert, R. Gene (1974) On the Universality of the Poole and Atkins Secchi Disk: Light Extinction Equation British Ecological Society. Of this total pressure which can be tolerated by the diver, 1 atmosphere is due to surface pressure of the Earth's air, and the rest is due to the depth in water. For example, if a gas contains 36% oxygen and the maximum pO2 is 1.4 bar, the MOD (msw) is 10 msw/bar x [(1.4 bar / 0.36) − 1] = 28.9 msw. == Tables == Maximum Operating Depth (MOD) in feet sea water (fsw) for pO2 1.2 to 1.6 MOD (fsw) % oxygen 4 8 12 16 20 21 24 28 32 36 40 50 60 70 80 90 100 Maximum pO2 (bar) 1.6 1287 627 407 297 231 218 187 156 132 114 99 73 55 42 33 26 20 1.5 1205 586 380 276 215 203 173 144 122 105 91 66 50 38 29 22 17 1.4 1122 545 352 256 198 187 160 132 111 95 83 59 44 33 25 18 13 1.3 1040 503 325 235 182 171 146 120 101 86 74 53 39 28 21 15 10 1.2 957 462 297 215 165 156 132 108 91 77 66 46 33 24 17 11 7 These depths are rounded to the nearest foot. 300px|thumb|right|A canister style dive lightNight diving is underwater diving done during the hours of darkness. In addition to light penetration, the term water clarity is also often used to describe underwater visibility. In underwater diving activities such as saturation diving, technical diving and nitrox diving, the maximum operating depth (MOD) of a breathing gas is the depth below which the partial pressure of oxygen (pO2) of the gas mix exceeds an acceptable limit. For example, if a gas contains 36% oxygen (FO2 = 0.36) and the limiting maximum pO2 is chosen at 1.4 atmospheres absolute, the MOD in feet of seawater (fsw) is 33 fsw/atm x [(1.4 ata / 0.36) − 1] = 95.3 fsw. The diver can experience a different underwater environment at night, because many marine animals are nocturnal. The pressure produced by depth in water, is converted to pressure in feet sea water (fsw) or metres sea water (msw) by multiplying with the appropriate conversion factor, 33 fsw per atm, or 10 msw per bar. thumb|upright=1.5|A diver enters crystal clear water in Lake Huron. The depth at which the disk is no longer visible is taken as a measure of the transparency of the water. A dive light is routinely used during night dives and cave dives, when there is little or no natural light, but also has a useful function during the day, as water absorbs the longer (red) wavelengths first then the yellow and green with increasing depth. Bright dive lights have values from about 2500 lumens. This corresponds to a sound intensity 5.4 dB, or 3.5 times, higher than the threshold in air (see Measurements above). ====Safety thresholds==== High levels of underwater sound create a potential hazard to human divers. A modern dive light usually has an output of at least about 100 lumens. ",556,26.9,"""4.85""",0.064,0.87,E
+Calculate the molar energy required to reverse the direction of an $\mathrm{H}_2 \mathrm{O}$ molecule located $100 \mathrm{pm}$ from a $\mathrm{Li}^{+}$ ion. Take the magnitude of the dipole moment of water as $1.85 \mathrm{D}$.,"thumb|Lewis Structure of H2O indicating bond angle and bond length Water () is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. 150px|thumb|Shape of water molecule showing that the real bond angle 104.5° deviates from the ideal sp3 angle of 109.5°. The bond angle for water is 104.5°. These tables list values of molar ionization energies, measured in kJ⋅mol−1. In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the lone pairs, while that leaves orbitals with more p character directed towards the hydrogens, resulting in deviation from idealized O(sp3) hybrid orbitals with 25% s character and 75% p character. The first molar ionization energy applies to the neutral atoms. The bond angles in those molecules are 104.5° and 107° respectively, which are below the expected tetrahedral angle of 109.5°. In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the very electropositive lone pairs, while that leaves orbitals with more p character directed towards the hydrogens. thumb|200px|Model of the hydrogen molecule and its axial projection In addition to the model of the atom, Niels Bohr also proposed a model of the chemical bond. A particularly well known example is water, where the angle between the two O-H bonds is only 104.5°. The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. thumb|Example of bent electron arrangement (water molecule). However, the bond angle between the two O–H bonds is only 104.5°, rather than the 109.5° of a regular tetrahedron, because the two lone pairs (whose density or probability envelopes lie closer to the oxygen nucleus) exert a greater mutual repulsion than the two bond pairs. thumb|300px|Calculated structure of a (H2O)100 icosahedral water cluster. In the case of water, with its 104.5° HOH angle, the OH bonding orbitals are constructed from O(~sp4.0) orbitals (~20% s, ~80% p), while the lone pairs consist of O(~sp2.3) orbitals (~30% s, ~70% p). The four hydrogen atoms are positioned at the vertices of a tetrahedron, and the bond angle is cos−1(−) ≈ 109° 28′.""Angle Between 2 Legs of a Tetrahedron"" – Maze5.net This is referred to as an AX4 type of molecule. Oxygen difluoride 103.8° As one moves down the table, the substituents become more electronegative and the bond angle between them decreases. For simple molecules, pictorially generating their MO diagram can be achieved without extensive knowledge of point group theory and using reducible and irreducible representations.thumb|330x330px|Hybridized MO of H2O Note that the size of the atomic orbitals in the final molecular orbital are different from the size of the original atomic orbitals, this is due to different mixing proportions between the oxygen and hydrogen orbitals since their initial atomic orbital energies are different. By directing hybrid orbitals of more p character towards the fluorine, the energy of that bond is not increased very much. This is in open agreement with the true bond angle of 104.45°. In other words, if water was formed from two identical O-H bonds and two identical sp3 lone pairs on the oxygen atom as predicted by valence bond theory, then its photoelectron spectrum (PES) would have two (degenerate) peaks and energy, one for the two O-H bonds and the other for the two sp3 lone pairs. The bond angles between substituents are ~109.5°, ~120°, and 180°. ",1.07,4.49,"""205.0""",4,2.14,A
+"In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ at a constant volume of $1.000 \mathrm{~m}^3$. The gas enters the container at $300 \mathrm{~K}$ and $100 \mathrm{~atm}$. The mass of the gas is $92.4 \mathrm{~kg}$. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of $500 \mathrm{~K}$. For nitrogen, $a=1.39 \mathrm{dm}^6 \mathrm{~atm} \mathrm{~mol}^{-2}, b=0.0391 \mathrm{dm}^3 \mathrm{~mol}^{-1}$.","Observe further that the pressure p goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when V = nb. ===Gas mixture=== If a mixture of n gases is being considered, and each gas has its own a (attraction between molecules) and b (volume occupied by molecules) values, then a and b for the mixture can be calculated as :m = total number of moles of gas present, :for each i, m_i = number of moles of gas i present, and x_i = \frac{m_i}{m} :a = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{a_i a_j}) :b = \sum_{i=1}^{i=n} \sum_{j=1}^{i=n} ( x_i x_j \sqrt{b_i b_j}) and the rule of adding partial pressures becomes invalid if the numerical result of the equation \left(p + ({n^2 a}/{V^2})\right)\left(V-nb\right) = nRT is significantly different from the ideal gas equation pV = nRT . ===Reduced form=== The Van der Waals equation can also be expressed in terms of reduced properties: :\left(P_r + \frac{3}{V_r^2}\right)\left(V_r-\frac{1}{3}\right) = \frac{8}{3}T_r The equation in reduced form is exactly the same for every gas, this is consistent with the Theorem of corresponding states. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made. ===Compressibility factor=== The compressibility factor for the Van der Waals equation is: : Z = \frac{P V_m}{RT} = \frac{V_m}{V_m-b} - \frac{a}{V_m RT} Or in reduced form by substitution of P_r=P/P_c, T_r=T/T_c, V_r=V_m/V_c: : Z = \frac{V_r}{V_r-\frac{1}{3}} - \frac{9}{8 V_r T_r} At the critical point: : Z_c = \frac{P_c V_c}{RT_c} = \frac{3}{8} = 0.375 ==Validity== The Van der Waals equation is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour.. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The ideal gas law states that the volume V occupied by n moles of any gas has a pressure P at temperature T given by the following relationship, where R is the gas constant: :PV=nRT To account for the volume occupied by real gas molecules, the Van der Waals equation replaces V/n in the ideal gas law with (V_m-b), where Vm is the molar volume of the gas and b is the volume occupied by the molecules of one mole: :P(V_m - b)=R T thumb|Van der Waals equation on a wall in Leiden The second modification made to the ideal gas law accounts for interaction between molecules of the gas. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Above the critical temperature, TC, the Van der Waals equation is an improvement over the ideal gas law, and for lower temperatures, i.e., T < TC, the equation is also qualitatively reasonable for the liquid and low-pressure gaseous states; however, with respect to the first-order phase transition, i.e., the range of (p, V, T) where a liquid phase and a gas phase would be in equilibrium, the equation appears to fail to predict observed experimental behaviour, in the sense that p is typically observed to be constant as a function of V for a given temperature in the two-phase region. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. Since: :p=-\left(\frac{\partial A}{\partial V}\right)_{T,N} where A is the Helmholtz free energy, it follows that the equal area rule can be expressed as: :p_V=\frac{A(V_L,T,N)-A(V_G,T,N)}{V_G-V_L}is Since the gas and liquid volumes are functions of pV and T only, this equation is then solved numerically to obtain pV as a function of temperature (and number of particles N), which may then be used to determine the gas and liquid volumes. thumb|800px|Locus of coexistence for two phases of Van der Waals fluid A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown in the accompanying figure. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. In this case specific volume changes are small and it can be written in a simplified form: :(p+A)(V-B)=CT, \, where p is the pressure, V is specific volume, T is the temperature and A, B, C are parameters. == See also == * Gas laws * Ideal gas * Inversion temperature * Iteration * Maxwell construction * Real gas * Theorem of corresponding states * Van der Waals constants (data page) * Redlich–Kwong equation of state ==References== ==Further reading== * * . The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. ",+3.60,0.2553,"""140.0""", 4.56,0.9966,C
+The chemical shift of the $\mathrm{CH}_3$ protons in acetaldehyde (ethanal) is $\delta=2.20$ and that of the $\mathrm{CHO}$ proton is 9.80 . What is the difference in local magnetic field between the two regions of the molecule when the applied field is $1.5 \mathrm{~T}$,"In this way the acetylenic protons are located in the cone-shaped shielding zone hence the upfield shift. :thumb|none|400px|Induced magnetic field of alkynes in external magnetic fields, field lines in grey. ==Magnetic properties of most common nuclei== 1H and 13C are not the only nuclei susceptible to NMR experiments. When a signal is found with a higher chemical shift: * the applied effective magnetic field is lower, if the resonance frequency is fixed (as in old traditional CW spectrometers) * the frequency is higher, when the applied magnetic field is static (normal case in FT spectrometers) * the nucleus is more deshielded * the signal or shift is downfield or at low field or paramagnetic Conversely a lower chemical shift is called a diamagnetic shift, and is upfield and more shielded. ==Diamagnetic shielding== In real molecules protons are surrounded by a cloud of charge due to adjacent bonds and atoms. The induced magnetic field lines are parallel to the external field at the location of the alkene protons which therefore shift downfield to a 4.5 ppm to 7.5 ppm range. Although the absolute resonance frequency depends on the applied magnetic field, the chemical shift is independent of external magnetic field strength. Acetaldehyde (IUPAC systematic name ethanal) is an organic chemical compound with the formula CH3CHO, sometimes abbreviated by chemists as MeCHO (Me = methyl). The nucleus is said to be experiencing a diamagnetic shielding. ==Factors causing chemical shifts== Important factors influencing chemical shift are electron density, electronegativity of neighboring groups and anisotropic induced magnetic field effects. Field effects are relatively weak, and diminish rapidly with distance, but have still been found to alter molecular properties such as acidity.thumb|160x160px|Field effect on a carbonyl arising from the dipole in a C-F bond. == Field sources == thumb|A bicycloheptane acid with an electron- withdrawing substituent, X, at the 4-position experiences a field effect on the acidic proton from the C-X bond dipole.|left|180x180pxleft|thumb|180x180px|A bicyclooctance acid with an electron-withdrawing substituent, X, at the 4-position experiences the same field effect on the acidic proton from the C-X bond dipole as the related bicylcoheptane. In nuclear magnetic resonance (NMR) spectroscopy, the chemical shift is the resonant frequency of an atomic nucleus relative to a standard in a magnetic field. As noted above, a consensus CSI method that filters upfield/downfield chemical shift changes in 13Cα, 13Cβ, and 13C' atoms in a similar manner to 1Hα shifts has also been developed. The protons in aromatic compounds are shifted downfield even further with a signal for benzene at 7.73 ppm as a consequence of a diamagnetic ring current. Field effects have also been shown in substituted arenes to dominate the electrostatic potential maps, which are maps of electron density used to explain intermolecular interactions. == Evidence for field effects == thumb|This octane derivative has only a single linker between the electron-withdrawing substituent and the acidic group.|180x180px Localized electronic effects are a combination of inductive and field effects. While chemical shift is referenced in order that the units are equivalent across different field strengths, the actual frequency separation in Hertz scales with field strength (). The variations of nuclear magnetic resonance frequencies of the same kind of nucleus, due to variations in the electron distribution, is called the chemical shift. This can be attributed to a field effect because in the same compound with the chlorines pointed away from the acidic group the pKa is lower, and if the effect were inductive the conformational position would not matter.thumb|The dichloroethano-bridged anthroic acid isomer with the C-Cl bond dipole oriented over the carboxylic acid has pKa of 6.07.|left|200x200px thumb|The isomer of dichloroethano- bridged anthroic acid in which the C-Cl dipole points away from the carboxylic acid has a pKa of 5.67.|left|200x200px == References == Category:Chemical properties Category:Chemistry Category:Electrostatics Category:Electromagnetism Category:Molecular physics Category:Molecules Category:Physical chemistry In carbon NMR the chemical shift of the carbon nuclei increase in the same order from around −10 ppm to 70 ppm. The total magnetic field experienced by a nucleus includes local magnetic fields induced by currents of electrons in the molecular orbitals (electrons have a magnetic moment themselves). In proton NMR of methyl halides (CH3X) the chemical shift of the methyl protons increase in the order from 2.16 ppm to 4.26 ppm reflecting this trend. Anisotropic induced magnetic field effects are the result of a local induced magnetic field experienced by a nucleus resulting from circulating electrons that can either be paramagnetic when it is parallel to the applied field or diamagnetic when it is opposed to it. However, the experimental data shows that effect on acidity in related octanes and cubanes is very similar, and therefore the dominant effect must be through space.thumb|This octane derivative has two linkers between the electron- withdrawing substituent and the acidic group.|179x179px thumb|This cubane derivative has four linkers but the acidic proton still feels the same effect from the C-X dipole because the interaction is a field effect.|180x180pxIn the cis-11,12-dichloro-9,10-dihydro-9,10-ethano-2-anthroic acid syn and anti isomers seen below and to the left, the chlorines provide a field effect. As is the case for NMR the chemical shift reflects the electron density at the atomic nucleus. ==See also== * EuFOD, a shift agent * MRI * Nuclear magnetic resonance * Nuclear magnetic resonance spectroscopy of carbohydrates * Nuclear magnetic resonance spectroscopy of nucleic acids * Nuclear magnetic resonance spectroscopy of proteins * Protein NMR * Random coil index * Relaxation (NMR) * Solid-state NMR * TRISPHAT, a chiral shift reagent for cations * Zeeman effect ==References== ==External links== *chem.wisc.edu *BioMagResBank *NMR Table *Proton chemical shifts *Carbon chemical shifts * Online tutorials (these generally involve combined use of IR, 1H NMR, 13C NMR and mass spectrometry) **Problem set 1 (see also this link for more background information on spin-spin coupling) **Problem set 2 **Problem set 4 **Problem set 5 **Combined solutions to problem set 5 (Problems 1-32) and (Problems 33-64) Category:Nuclear chemistry Category:Nuclear physics Category:Nuclear magnetic resonance spectroscopy pl:Spektroskopia NMR#Przesunięcie chemiczne The size of the chemical shift is given with respect to a reference frequency or reference sample (see also chemical shift referencing), usually a molecule with a barely distorted electron distribution. ==Operating frequency== The operating (or Larmor) frequency of a magnet is calculated from the Larmor equation : \omega_{0} = \gamma B_0\,, where is the actual strength of the magnet in units like Teslas or Gauss, and is the gyromagnetic ratio of the nucleus being tested which is in turn calculated from its magnetic moment and spin number with the nuclear magneton and the Planck constant : : \gamma = \frac{\mu\,\mu_\mathrm{N}}{hI}\,. SHIFTCOR identifies potential chemical shift referencing problems by comparing the difference between the average value of each set of observed backbone (1Hα, 13Cα, 13Cβ, 13CO, 15N and 1HN) shifts and their corresponding predicted chemical shifts. ",+107,11,"""7.136""",0.4772,7200,B
+Suppose that the junction between two semiconductors can be represented by a barrier of height $2.0 \mathrm{eV}$ and length $100 \mathrm{pm}$. Calculate the transmission probability of an electron with energy $1.5 \mathrm{eV}$.,"Note that the probabilities and amplitudes as written are for any energy (above/below) the barrier height. === E = V0 === The transmission probability at E=V_0 is T=\frac{1}{1+ma^2V_0/2\hbar^2}. ==Remarks and applications== The calculation presented above may at first seem unrealistic and hardly useful. However, according to quantum mechanics, the electron has a non-zero wave amplitude in the barrier, and hence it has some probability of passing through the barrier. Note that these expressions hold for any energy If then so there is a singularity in both of these expressions. ==Analysis of the obtained expressions== ===E < V0=== thumb|350x350px|Transmission probability through a finite potential barrier for \sqrt{2m V_0} a / \hbar = 1, 3, and 7. The surprising result is that for energies less than the barrier height, E < V_0 there is a non-zero probability T=|t|^2= \frac{1}{1+\frac{V_0^2\sinh^2(k_1 a)}{4E(V_0-E)}} for the particle to be transmitted through the barrier, with .}} Classically, the electron has zero probability of passing through the barrier. For electrons, the barrier height \Phi_{B_{n}}can be easily calculated as the difference between the metal work function and the electron affinity of the semiconductor: \Phi_{B_n}=\Phi_M-\chi While the barrier height for holes is equal to the difference between the energy gap of the semiconductor and the energy barrier for electrons: \Phi_{B_p}=E_\text{gap}-\Phi_{B_n} In reality, what can happen is that charged interface states can pin the Fermi level at a certain energy value no matter the work function values, influencing the barrier height for both carriers. thumb|right|Schematic representation of an electron tunneling through a barrier In electronics/spintronics, a tunnel junction is a barrier, such as a thin insulating layer or electric potential, between two electrically conducting materials. To a first approximation, the barrier between a metal and a semiconductor is predicted by the Schottky–Mott rule to be proportional to the difference of the metal- vacuum work function and the semiconductor-vacuum electron affinity. The current-voltage relationship is qualitatively the same as with a p-n junction, however the physical process is somewhat different. === Conduction values === The thermionic emission can be formulated as following: J_{th}= A^{**}T^2e^{-\frac{\Phi_{B_{n,p}}}{k_bT}}\biggl(e^{\frac{qV}{k_bT}}-1\biggr) While the tunneling current density can be expressed, for a triangular shaped barrier (considering WKB approximation) as: J_{T_{n,p}}= \frac{q^3E^2}{16\pi^2\hbar\Phi_{B_{n,p}}} e^{\frac{-4\Phi_{B_{n,p}}^{3/2}\sqrt{2m^*_{n,p}}}{3q\hbar E}} From both formulae it is clear that the current contributions are related to the barrier height for both electrons and holes. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. The Monte Carlo method for electron transport is a semiclassical Monte Carlo (MC) approach of modeling semiconductor transport. The transmission is exponentially suppressed with the barrier width, which can be understood from the functional form of the wave function: Outside of the barrier it oscillates with wave vector whereas within the barrier it is exponentially damped over a distance If the barrier is much wider than this decay length, the left and right part are virtually independent and tunneling as a consequence is suppressed. === E > V0 === In this case T=|t|^2= \frac{1}{1+\frac{V_0^2\sin^2(k_1 a)}{4E(E-V_0)}}, where Equally surprising is that for energies larger than the barrier height, E > V_0, the particle may be reflected from the barrier with a non-zero probability R=|r|^2=1-T. Electrons (or quasiparticles) pass through the barrier by the process of quantum tunnelling. A classical particle with energy E larger than the barrier height V_0 would always pass the barrier, and a classical particle with E < V_0 incident on the barrier would always get reflected. An earlier version containing some of the material appeared in the Bell Labs Technical Journal in 1949. ==Contents== The arrangement of chapters is as follows: Part I INTRODUCTION TO TRANSISTOR ELECTRONICS *Chapter 1: THE BULK PROPERTIES OF SEMICONDUCTORS *Chapter 2: THE TRANSISTOR AS A CIRCUIT ELEMENT *Chapter 3: QUANTITATIVE STUDIES OF INJECTION OF HOLES AND ELECTRONS *Chapter 4: ON THE PHYSICAL THEORY OF TRANSISTORS *Chapter 5: QUANTUM STATES, ENERGY BANDS, AND BRILLOUIN ZONES PART II DESCRIPTIVE THEORY OF SEMICONDUCTORS *Chapter 6: VELOCITIES AND CURRENTS FOR ELECTRONS I N CRYSTALS *Chapter 7: ELECTRONS AND HOLES IN ELECTRIC AND MAGNETIC FIELDS *Chapter 8: INTRODUCTORY THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 9: DISTRIBUTIONS OF QUANTUM STATES IN ENERGY *Chapter 10: FERMI-DIRAC STATISTICS FOR SEMICONDUCTORS *Chapter 11: MATHEMATICAL THEORY OF CONDUCTIVITY AND HALL EFFECT *Chapter 12: APPLICATIONS TO TRANSISTOR ELECTRONICS PART III QUANTUM MECHANICAL FOUNDATIONS *Chapter 13: INTRODUCTION TO PART III *Chapter 14: ELEMENTARY QUANTUM MECHANICS WITH CIRCUIT THEORY ANALOGUES *Chapter 15: THEORY OF ELECTRON AND HOLE VELOCITIES, CURRENTS AND ACCELERATIONS *Chapter 16: STATISTICAL MECHANICS FOR SEMICONDUCTORS *Chapter 17: THE THEORY OF TRANSITION PROBABILITIES FOR HOLES AND ELECTRONS *APPENDIX A Units *APPENDIX B Periodic Table *APPENDIX C Values of the physical constants *APPENDIX D Energy conversion chart *NAME INDEX *SUBJECT INDEX Category:1950 non-fiction books Category:1950 in science Category:Physics textbooks Under large voltage bias, the electric current flowing through the barrier is essentially governed by the laws of thermionic emission, combined with the fact that the Schottky barrier is fixed relative to the metal's Fermi level.This interpretation is due to Hans Bethe, after the incorrect theory of Schottky, see * Under forward bias, there are many thermally excited electrons in the semiconductor that are able to pass over the barrier. The scattering events and the duration of particle flight is determined through the use of random numbers. == Background == === Boltzmann transport equation === The Boltzmann transport equation model has been the main tool used in the analysis of transport in semiconductors. Within these approximations, Fermi's Golden Rule gives, to the first order, the transition probability per unit time for a scattering mechanism from a state |k \rangle to a state |k' \rangle: : S(k,k') = \frac{2\pi}{\hbar} \left | \langle k|H'|k' \rangle \right |^2 \cdot \delta(E - E') where H' is the perturbation Hamiltonian representing the collision and E and E′ are respectively the initial and final energies of the system constituted of both the carrier and the electron and phonon gas. To study the quantum case, consider the following situation: a particle incident on the barrier from the left side It may be reflected or transmitted To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations A_r = 1 (incoming particle), A_l = r (reflection), C_l = 0 (no incoming particle from the right), and C_r = t (transmission). 350px|right|thumb In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called ""quantum tunneling"") and wave-mechanical reflection. A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal–semiconductor junction. The value of ΦB depends on the combination of metal and semiconductor.Schottky barrier tutorial. ",54.7, 9.73,"""0.8""",3,1.39,C
+"The diffusion coefficient of a particular kind of t-RNA molecule is $D=1.0 \times 10^{-11} \mathrm{~m}^2 \mathrm{~s}^{-1}$ in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance $1.0 \mu \mathrm{m}$, corresponding to the radius of the cell?","The rate of diffusion NA, is usually expressed as the number of moles diffusing across unit area in unit time. Of mass transport mechanisms, molecular diffusion is known as a slower one. === Biology === In cell biology, diffusion is a main form of transport for necessary materials such as amino acids within cells. The rate of diffusion of A, NA, depend on concentration gradient and the average velocity with which the molecules of A moves in the x direction. The source term in the diffusion equation becomes S(\vec{r},t, \vec{r'},t')=\delta(\vec{r}-\vec{r'})\delta(t-t'), where \vec{r} is the position at which fluence rate is measured and \vec{r'} is the position of the source. For the vector form, the RTE is multiplied by direction \hat{s} before evaluation.): : \frac{\partial \Phi(\vec{r},t)}{c\partial t} + \mu_a\Phi(\vec{r},t) + abla \cdot \vec{J}(\vec{r},t) = S(\vec{r},t) : \frac{\partial \vec{J}(\vec{r},t)}{c\partial t} + (\mu_a+\mu_s')\vec{J}(\vec{r},t) + \frac{1}{3} abla \Phi(\vec{r},t) = 0 The diffusion approximation is limited to systems where reduced scattering coefficients are much larger than their absorption coefficients and having a minimum layer thickness of the order of a few transport mean free path. ===The diffusion equation=== Using the second assumption of diffusion theory, we note that the fractional change in current density \vec{J}(\vec{r},t) over one transport mean free path is negligible. Diffusion explains the net flux of molecules from a region of higher concentration to one of lower concentration. In practice, Knudsen diffusion applies only to gases because the mean free path for molecules in the liquid state is very small, typically near the diameter of the molecule itself. == Mathematical description == The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the kinetic theory of gases: :{D_{AA*}} = {{\lambda u} \over {3}} = {{\lambda}\over{3}} \sqrt{{8R T}\over {\pi M_{A}}} For Knudsen diffusion, path length λ is replaced with pore diameter d, as species A is now more likely to collide with the pore wall as opposed with another molecule. A common approximation summarized here is the diffusion approximation. The nuclear lamina is a dense (~30 to 100 nm thick) fibrillar network inside the nucleus of eukaryote cells. The result of diffusion is a gradual mixing of material such that the distribution of molecules is uniform. Nuclear collision length is the mean free path of a particle before undergoing a nuclear reaction, for a given particle in a given medium. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale. Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. Molecular diffusion is typically described mathematically using Fick's laws of diffusion. == Applications == Diffusion is of fundamental importance in many disciplines of physics, chemistry, and biology. For an attractive interaction between particles, the diffusion coefficient tends to decrease as concentration increases. In physiology, transport maximum (alternatively Tm or Tmax) refers to the point at which increase in concentration of a substance does not result in an increase in movement of a substance across a cell membrane. For a repulsive interaction between particles, the diffusion coefficient tends to increase as concentration increases. Substituting Fick's law into the scalar representation of the RTE gives the diffusion equation: : \frac{1}{c}\frac{\partial \Phi(\vec{r},t)}{\partial t} + \mu_a\Phi(\vec{r},t) - abla \cdot [D abla\Phi(\vec{r},t)] = S(\vec{r},t) D=\frac{1}{3(\mu_a+\mu_s')} is the diffusion coefficient and μ's=(1-g)μs is the reduced scattering coefficient. * Chemical diffusion occurs in a presence of concentration (or chemical potential) gradient and it results in net transport of mass. This is the process described by the diffusion equation. thumb|300px|Schematic drawing of a molecule in a cylindrical pore in the case of Knudsen diffusion; are indicated the pore diameter () and the free path of the particle (). Generally, the diffusion approximation is less accurate as the absorption coefficient μa increases and the scattering coefficient μs decreases. ",0.000216,1.7,"""-273.0""",1.4,3.42,B
+At what pressure does the mean free path of argon at $20^{\circ} \mathrm{C}$ become comparable to the diameter of a $100 \mathrm{~cm}^3$ vessel that contains it? Take $\sigma=0.36 \mathrm{~nm}^2$,"New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. The formulae of pressure vessel design standards are extension of Lamé's theorem by putting some limit on ratio of inner radius and thickness. Stress in a thin-walled pressure vessel in the shape of a cylinder is :\sigma_\theta = \frac{pr}{t}, :\sigma_{\rm long} = \frac{pr}{2t}, where: * \sigma_\theta is hoop stress, or stress in the circumferential direction * \sigma_{long} is stress in the longitudinal direction * p is internal gauge pressure * r is the inner radius of the cylinder * t is thickness of the cylinder wall. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. PV Publishing, Inc. Oklahoma City, OK ==Further reading== * Megyesy, Eugene F. (2008, 14th ed.) Pressure Vessel Handbook. Therefore, pressure vessels are designed to have a thickness proportional to the radius of tank and the pressure of the tank and inversely proportional to the maximum allowed normal stress of the particular material used in the walls of the container. The normal (tensile) stress in the walls of the container is proportional to the pressure and radius of the vessel and inversely proportional to the thickness of the walls. (Pa)For a sphere the thickness d = rP/2σ, where r is the radius of the tank. thumb|upright=1.5|In inverse depth parametrization, a point is identified by its inverse depth \rho = \frac{1}{\left\Vert \mathbf{p} - \mathbf{c}_0\right\Vert} along the ray, with direction v = (\cos \phi \sin \theta, -\sin \phi, \cos \phi \cos \theta), from which it was first observed. Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. Construction methods and materials may be chosen to suit the pressure application, and will depend on the size of the vessel, the contents, working pressure, mass constraints, and the number of items required. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. ""Pressure Vessel Handbook, 14th Edition."" File:Ресивер хладагента FP-LR-100.png|Cylindrical pressure vessel. For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: Thickness has to be less than 0.356 times inner radius :\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE} Cylindrical shells: Thickness has to be less than 0.5 times inner radius :\sigma_\theta = \frac{p(r + 0.6t)}{tE} :\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE} where E is the joint efficiency, and all others variables as stated above. However, a spherical shape is difficult to manufacture, and therefore more expensive, so most pressure vessels are cylindrical with 2:1 semi-elliptical heads or end caps on each end. For cylindrical vessels with a diameter up to 600 mm (NPS of 24 in), it is possible to use seamless pipe for the shell, thus avoiding many inspection and testing issues, mainly the nondestructive examination of radiography for the long seam if required. A pressure vessel is a container designed to hold gases or liquids at a pressure substantially different from the ambient pressure. A disadvantage of these vessels is that greater diameters are more expensive, so that for example the most economic shape of a , pressure vessel might be a diameter of and a length of including the 2:1 semi-elliptical domed end caps. ===Construction materials=== thumb|200px|Composite overwrapped pressure vessel with titanium liner. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus :M = {3 \over 2} nRT {\rho \over \sigma}. (see gas law) The other factors are constant for a given vessel shape and material. A vessel can be considered ""thin-walled"" if the diameter is at least 10 times (sometimes cited as 20 times) greater than the wall thickness.Richard Budynas, J. Nisbett, Shigley's Mechanical Engineering Design, 8th ed., For these reasons, the definition of a pressure vessel varies from country to country. ",0.195,4152,"""170.0""",0.22222222,58.2,A
+"The equilibrium pressure of $\mathrm{O}_2$ over solid silver and silver oxide, $\mathrm{Ag}_2 \mathrm{O}$, at $298 \mathrm{~K}$ is $11.85 \mathrm{~Pa}$. Calculate the standard Gibbs energy of formation of $\mathrm{Ag}_2 \mathrm{O}(\mathrm{s})$ at $298 \mathrm{~K}$.","Silver oxide is the chemical compound with the formula Ag2O. Silver sulfate is the inorganic compound with the formula Ag2SO4. Silver sulfite is the chemical compound with the formula Ag2SO3. In 1993, AgF2 cost between 1000-1400 US dollars per kg. ==Composition and structure== AgF2 is a white crystalline powder, but it is usually black/brown due to impurities. The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. For some time, it was doubted that silver was actually in the +2 oxidation state, rather than some combination of states such as AgI[AgIIIF4], which would be similar to silver(I,III) oxide. Such reactions often work best when the silver oxide is prepared in situ from silver nitrate and alkali hydroxide. ==References== ==External links== * Annealing of Silver Oxide – Demonstration experiment: Instruction and video Category:Silver compounds Category:Transition metal oxides The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). This reaction does not afford appreciable amounts of silver hydroxide due to the favorable energetics for the following reaction:Holleman, A. F.; Wiberg, E. ""Inorganic Chemistry"" Academic Press: San Diego, 2001. . :2  AgOH -> Ag2O + H2O (pK = 2.875) With suitably controlled conditions, this reaction can be used to prepare Ag2O powder with properties suitable for several uses including as a fine grained conductive paste filler. ==Structure and properties== Ag2O features linear, two-coordinate Ag centers linked by tetrahedral oxides. Silver(II) fluoride is a chemical compound with the formula AgF2. Silver sulfate and anhydrous sodium sulfate adopt the same structure. ==Silver(II) sulfate== The synthesis of silver(II) sulfate (AgSO4) with a divalent silver ion instead of a monovalent silver ion was first reported in 2010 by adding sulfuric acid to silver(II) fluoride (HF escapes). :AgNO3 + NaCl -> AgCl(v) + NaNO3 :2 AgNO3 + CoCl2 -> 2 AgCl(v) + Co(NO3)2 It can also be produced by the reaction of silver metal and aqua regia, however, the insolubility of silver chloride decelerates the reaction. Silver chloride is a chemical compound with the chemical formula AgCl. AgF and AgBr crystallize similarly.Wells, A.F. (1984) Structural Inorganic Chemistry, Oxford: Clarendon Press. . The AgI[AgIIIF4] was found to be present at high temperatures, but it was unstable with respect to AgF2. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. In the gas phase, AgF2 is believed to have D∞h symmetry. This unstable silver compound when heated and/or in light it decomposes to silver dithionate and silver sulfate. ==Preparation== Silver sulfite can be prepared by dissolving silver nitrate with the stoichiometric quantity of sodium sulfite solution, yielding a precipitation of silver sulfite by the following reaction: :2 AgNO3 \+ Na2SO3 Ag2SO3 \+ 2 NaNO3 After precipitation then filtering silver sulfite, washing it using well-boiled water, and drying it in vacuum. ==References== Category:Silver compounds Category:Sulfites Silver usually exists in its +1 oxidation state. It is formed as an intermediate in the catalysis of gaseous reactions with fluorine by silver. It is a rare example of a silver(II) compound. ",152.67,0,"""0.36""",-11.2,6.9,D
+"When alkali metals dissolve in liquid ammonia, their atoms each lose an electron and give rise to a deep-blue solution that contains unpaired electrons occupying cavities in the solvent. These 'metal-ammonia solutions' have a maximum absorption at $1500 \mathrm{~nm}$. Supposing that the absorption is due to the excitation of an electron in a spherical square well from its ground state to the next-higher state (see the preceding problem for information), what is the radius of the cavity?","The problems stems from the fact that a cavity is an open non- Hermitian system with leakage and absorption. Because of these boundary conditions that must be satisfied at resonance (tangential electric fields must be zero at cavity walls), it follows that cavity length must be an integer multiple of half- wavelength at resonance.David Pozar, Microwave Engineering, 2nd edition, Wiley, New York, NY, 1998. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. Cavity perturbation theory has been initially proposed by Bethe-Schwinger in optics , and Waldron in the radio frequency domain. Because the ions overlap, their separation in the crystal will be less than the sum of their soft-sphere radii. That is, the distance between two neighboring iodides in the crystal is assumed to be twice the radius of the iodide ion, which was deduced to be 214 pm. Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. The Q factor of a resonant cavity can be calculated using cavity perturbation theory and expressions for stored electric and magnetic energy. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. The microwaves bounce back and forth between the walls of the cavity. Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. External coupling structure has an effect on cavity performance and needs to be considered in the overall analysis.Montgomery, C. G. & Dicke, Robert H. & Edward M. Purcell, Principles of microwave circuits / edited by C.G. Montgomery, R.H. Dicke, E.M. Purcell, Peter Peregrinus on behalf of the Institution of Electrical Engineers, London, U.K., 1987. === Resonant frequencies === The resonant frequencies of a cavity are a function of its geometry. ==== Rectangular cavity ==== thumb|Rectangular cavity Resonance frequencies of a rectangular microwave cavity for any \scriptstyle TE_{mnl} or \scriptstyle TM_{mnl} resonant mode can be found by imposing boundary conditions on electromagnetic field expressions. The process of gas or liquid which penetrate into the body of adsorbent is commonly known as absorption. 550px|link=https://doi.org/10.1351/goldbook.A00036|thumb|right|alt=IUPAC definition for absorption|[https://doi.org/10.1351/goldbook.A00036 https://doi.org/10.1351/goldbook.A00036]. ==Equation== If absorption is a physical process not accompanied by any other physical or chemical process, it usually follows the Nernst distribution law: :""the ratio of concentrations of some solute species in two bulk phases when it is equilibrium and in contact is constant for a given solute and bulk phases"": :: \frac{[x]_{1}}{[x]_{2}} = \text{constant} = K_{N(x,12)} The value of constant KN depends on temperature and is called partition coefficient. A microwave cavity or radio frequency cavity (RF cavity) is a special type of resonator, consisting of a closed (or largely closed) metal structure that confines electromagnetic fields in the microwave or RF region of the spectrum. Cavity perturbation measurement techniques for material characterization are used in many fields ranging from physics and material science to medicine and biology.Wenquan Che; Zhanxian Wang; Yumei Chang; Russer, P.; ""Permittivity Measurement of Biological Materials with Improved Microwave Cavity Perturbation Technique,"" Microwave Conference, 2008. In Schrödinger's quantum-mechanical theory of the hydrogen atom, the Bohr radius is the value of the radial coordinate for which the radial probability density of the electron position is highest. ;TM modes:T. Wangler, RF linear accelerators, Wiley (2008) f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2} ;TE modes: f_{mnp}=\frac{c}{2\pi\sqrt{\mu_r\epsilon_r}} \sqrt{\left(\frac{X'_{mn}}{R}\right)^2 + \left(\frac{p \pi}{L}\right)^2} Here, \scriptstyle X_{mn} denotes the \scriptstyle n-th zero of the \scriptstyle m-th Bessel function, and \scriptstyle X'_{mn} denotes the \scriptstyle n-th zero of the derivative of the \scriptstyle m-th Bessel function. === Quality factor === The quality factor \scriptstyle Q of a cavity can be decomposed into three parts, representing different power loss mechanisms. *\scriptstyle Q_c, resulting from the power loss in the walls which have finite conductivity *\scriptstyle Q_d, resulting from the power loss in the lossy dielectric material filling the cavity. *\scriptstyle Q_{ext}, resulting from power loss through unclosed surfaces (holes) of the cavity geometry. Ideally, the material to be measured is introduced into the cavity at the position of maximum electric or magnetic field. This frequency is given by \cdot k_{mnl}\\\ &= \frac{c}{2\pi\sqrt{\mu_r\epsilon_r}}\sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2 + \left(\frac{l\pi}{d}\right)^2}\\\ &= \frac{c}{2\sqrt{\mu_r\epsilon_r}}\sqrt{\left( \frac{m}{a}\right) ^2+\left(\frac{n}{b}\right) ^2 + \left(\frac{l}{d}\right) ^2} \end{align}|}} where \scriptstyle k_{mnl} is the wavenumber, with \scriptstyle m, \scriptstyle n, \scriptstyle l being the mode numbers and \scriptstyle a, \scriptstyle b, \scriptstyle d being the corresponding dimensions; c is the speed of light in vacuum; and \scriptstyle \mu_r and \scriptstyle \epsilon_r are relative permeability and permittivity of the cavity filling respectively. ==== Cylindrical cavity ==== thumb|Cylindrical cavity The field solutions of a cylindrical cavity of length \scriptstyle L and radius \scriptstyle R follow from the solutions of a cylindrical waveguide with additional electric boundary conditions at the position of the enclosing plates. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The concept can be extended to solvated ions in liquid solutions taking into consideration the solvation shell. == Trends == X− NaX AgX F 464 492 Cl 564 555 Br 598 577 Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. ",0,-32,"""2500.0""",131,0.69,E
+Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ? ,"Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. Electron diffraction occurs due to elastic scattering, when there is no change in the energy of the electrons during their interactions with atoms. The electron pulse undergoes diffraction as a result of interacting with the sample. For electron diffraction the electrons behave as if they are non-relativistic particles of mass m^* in terms of how they interact with the atoms. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. *The development of new approaches to reduce dynamical effects such as precession electron diffraction and three-dimensional diffraction methods. A fast electron beam is generated in an electron gun, enters a diffraction chamber typically at a vacuum of 10−7 mbar. *Fast and accurate methods to calculate intensities for low-energy electron diffraction so it could be used to determine atomic positions, for instance references. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. Principles of Electron Optics. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. framed|Geometry of electron beam in precession electron diffraction. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. Strictly, the term electron diffraction refers to how electrons are scattered by atoms, a process that is mathematically modelled by solving forms of Schrödinger equation. # The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called kinematical diffraction and is also a far-field or Fraunhofer approach. The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. == Historical development == The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. Scheme 1 shows the schematic procedure of an electron diffraction experiment. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments, with the information available in real time rather than requiring photographic processing after the experiment. ==Basics== === Geometrical considerations === What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. Electron diffraction also plays a major role in the contrast of images in electron microscopes. UED can provide a wealth of dynamics on charge carriers, atoms, and molecules. ==History== The design of early ultrafast electron diffraction instruments was based on x-ray streak cameras, the first reported UED experiment demonstrating an electron pulse length of 100 ps. ==Electron Pulse Production== The electron pulses are typically produced by the process of photoemission in which a fs optical pulse is directed toward a photocathode. However, with Cs corrected microscopes, the probe can be made much smaller. === Practical considerations === Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV. ",+11,5275,"""7.27""",0.312,0.1792,C
+Electron diffraction makes use of electrons with wavelengths comparable to bond lengths. To what speed must an electron be accelerated for it to have a wavelength of $100 \mathrm{pm}$ ? ,"Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. For electron diffraction the electrons behave as if they are non-relativistic particles of mass m^* in terms of how they interact with the atoms. The wavelength of the electrons \lambda in vacuum is : \lambda = \frac1k = \frac{h}{\sqrt{2m^* E}}=\frac{h c}{\sqrt{E(2 m_0 c^2 + E)}}, and can range from about 0.1 nanometers, roughly the size of an atom, down to a thousandth of that. Electron diffraction occurs due to elastic scattering, when there is no change in the energy of the electrons during their interactions with atoms. For context, the typical energy of a chemical bond is a few eV; electron diffraction involves electrons up to 5,000,000 eV. Principles of Electron Optics. *Fast and accurate methods to calculate intensities for low-energy electron diffraction so it could be used to determine atomic positions, for instance references. Electron diffraction refers to changes in the direction of electron beams due to interactions with atoms. *The development of new approaches to reduce dynamical effects such as precession electron diffraction and three-dimensional diffraction methods. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. framed|Geometry of electron beam in precession electron diffraction. The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. Strictly, the term electron diffraction refers to how electrons are scattered by atoms, a process that is mathematically modelled by solving forms of Schrödinger equation. # The first level of more accuracy where it is approximated that the electrons are only scattered once, which is called kinematical diffraction and is also a far-field or Fraunhofer approach. However, with Cs corrected microscopes, the probe can be made much smaller. === Practical considerations === Precession electron diffraction is typically conducted using accelerating voltages between 100-400 kV. The theory of precession electron diffraction is still an active area of research, and efforts to improve on the ability to correct measured intensities without a priori knowledge are ongoing. == Historical development == The first precession electron diffraction system was developed by Vincent and Midgley in Bristol, UK and published in 1994. These have efficiencies and accuracies that can be a thousand or more times that of the photographic film used in the earliest experiments, with the information available in real time rather than requiring photographic processing after the experiment. ==Basics== === Geometrical considerations === What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. These were rapidly followed by the first non-relativistic diffraction model for electrons by Hans Bethe based upon the Schrödinger equation, which is very close to how electron diffraction is now described. Given sufficient voltage, the electron can be accelerated sufficiently fast to exhibit measurable relativistic effects. Then, the rule is that the amount of energy absorbed by an electron may allow for the electron to be promoted from a vibrational and electronic ground state to a vibrational and electronic excited state. While the wavevector increases as the energy increases, the change in the effective mass compensates this so even at the very high energies used in electron diffraction there are still significant interactions. Relativistic electron beams are streams of electrons moving at relativistic speeds. ",7.27,10.065778,"""0.9""",-1.0,4.16,A
+"Nelson, et al. (Science 238, 1670 (1987)) examined several weakly bound gas-phase complexes of ammonia in search of examples in which the $\mathrm{H}$ atoms in $\mathrm{NH}_3$ formed hydrogen bonds, but found none. For example, they found that the complex of $\mathrm{NH}_3$ and $\mathrm{CO}_2$ has the carbon atom nearest the nitrogen (299 pm away): the $\mathrm{CO}_2$ molecule is at right angles to the $\mathrm{C}-\mathrm{N}$ 'bond', and the $\mathrm{H}$ atoms of $\mathrm{NH}_3$ are pointing away from the $\mathrm{CO}_2$. The magnitude of the permanent dipole moment of this complex is reported as $1.77 \mathrm{D}$. If the $\mathrm{N}$ and $\mathrm{C}$ atoms are the centres of the negative and positive charge distributions, respectively, what is the magnitude of those partial charges (as multiples of $e$ )?","Theoretically, the bond strength of the hydrogen bonds can be assessed using NCI index, non-covalent interactions index, which allows a visualization of these non-covalent interactions, as its name indicates, using the electron density of the system. Another study found a much smaller number of hydrogen bonds: 2.357 at 25 °C. Quantum chemical calculations of the relevant interresidue potential constants (compliance constants) revealed large differences between individual H bonds of the same type. van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 Van der Waals radii taken from Bondi's compilation (1964). Defining and counting the hydrogen bonds is not straightforward however. Scott Rowland and Robin Taylor re-examined these 1964 figures in the light of more recent crystallographic data: on the whole, the agreement was very good, although they recommend a value of 1.09 Å for the van der Waals radius of hydrogen as opposed to Bondi's 1.20 Å. As well, it has been argued that the Van der Waals radius is not a fixed property of an atom in all circumstances, rather, that it will vary with the chemical environment of the atom. == Gallery == File:3D ammoniac.PNG|alt=Ammonia, van-der-Waals-based model|Ammonia, NH3, space- filling, Van der Waal's-based representation, nitrogen (N) in blue, hydrogen (H) in white. This is slightly different from the intramolecular bound states of, for example, covalent or ionic bonds; however, hydrogen bonding is generally still a bound state phenomenon, since the interaction energy has a net negative sum. This description of the hydrogen bond has been proposed to describe unusually short distances generally observed between or . ===Structural details=== The distance is typically ≈110 pm, whereas the distance is ≈160 to 200 pm. In weaker hydrogen bonds,Desiraju, G. R. and Steiner, T. Some of Bondi's figures are given in the table at the top of this article, and they remain the most widely used ""consensus"" values for the van der Waals radii of the elements. ==Hydrogen bonds in small molecules== thumb|right|Crystal structure of hexagonal ice. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The coefficient E_{0} is positive and of the order V\alpha^{3}, where V is the ionization energy and \alpha is the mean atomic polarizability; the exact value of E_{0} depends on the magnitudes of the dipole matrix elements and on the energies of the p orbitals. ==References== * Category:Chemical bonding Category:Quantum mechanical potentials The ideal bond angle depends on the nature of the hydrogen bond donor. Structural details, in particular distances between donor and acceptor which are smaller than the sum of the van der Waals radii can be taken as indication of the hydrogen bond strength. 100px 120px 100px From top to bottom, azides, nitrones, and nitro compounds are examples of 1,3-dipoles. The Hydrogen Bond Franklin Classics, 2018), Jeffrey, G. A.; In organic chemistry, a 1,3-dipolar compound or 1,3-dipole is a dipolar compound with delocalized electrons and a separation of charge over three atoms. Generally, the hydrogen bond is characterized by a proton acceptor that is a lone pair of electrons in nonmetallic atoms (most notably in the nitrogen, and chalcogen groups). The strength of intramolecular hydrogen bonds can be studied with equilibria between conformers with and without hydrogen bonds. ",0.36,3.52,"""0.123""",0.118,29.9,C
+"The NOF molecule is an asymmetric rotor with rotational constants $3.1752 \mathrm{~cm}^{-1}, 0.3951 \mathrm{~cm}^{-1}$, and $0.3505 \mathrm{~cm}^{-1}$. Calculate the rotational partition function of the molecule at $25^{\circ} \mathrm{C}$.","Nitrosyl fluoride (NOF) is a covalently bonded nitrosyl compound. ==Reactions== NOF is a highly reactive fluorinating agent that converts many metals to their fluorides, releasing nitric oxide in the process: :n NOF + M → MFn \+ n NO NOF also fluorinates fluorides to form adducts that have a salt- like character, such as NOBF4. In terms of these constants, the rotational partition function can be written in the high temperature limit as G. Herzberg, ibid, Equation (V,29) \zeta^\text{rot} \approx \frac{\sqrt{\pi} }{\sigma} \sqrt{ \frac{ (k_\text{B} T)^3 }{ A B C }} with \sigma again known as the rotational symmetry number G. Herzberg, ibid; see Table 140 for values for common molecular point groups which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. NF4F.jpg|NF4+F− R3m structure NF4-2NF6F.jpg|(NF4+)2NF6−F− I4/m structure NF4NF6.jpg|NF4+NF6− P4/n structure ==Covalent molecule== thumb|left|upright=1.55|Possible structures of NF5 and analogous fluorohydrides For a NF5 molecule to form, five fluorine atoms have to be arranged around a nitrogen atom. In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor \sigma = 2 with \sigma known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without. ==Nonlinear molecules== A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A, B, and C , which can often be determined by rotational spectroscopy. Nitrogen pentafluoride (NF5) is a theoretical compound of nitrogen and fluorine that is hypothesized to exist based on the existence of the pentafluorides of the atoms below nitrogen in the periodic table, such as phosphorus pentafluoride. Calculations show that the NF5 molecule is thermodynamically favourably inclined to form NF4 and F radicals with energy 36 kJ/mol and a transition barrier around 67–84 kJ/mol. Nitrogen pentafluoride also violates the octet rule in which compounds with eight outer shell electrons are particularly stable. ==References== Category:Nitrogen fluorides Category:Hypothetical chemical compounds Category:Nitrogen(V) compounds For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E_J^\text{rot} = \frac{\mathbf{J}^2}{2I} = \frac{J(J+1)\hbar^2}{2I} = J(J+1)B. J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0,1,2, \ldots, B = \frac{\hbar^2}{2I} is the rotational constant, and I is the moment of inertia. N-Fluoropyridinium triflate is an organofluorine compound with the formula [C5H5NF]O3SCF3. In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k_\text{B} T . == Quantum symmetry effects == For a diatomic molecule with a center of symmetry, such as \rm H_2, N_2, CO_2, or \mathrm{ H_2 C_2} (i.e. D_{\infty h} point group), rotation of a molecule by \pi radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. Theoretical models of the nitrogen pentafluoride molecule are either a trigonal bipyramidal covalently bound molecule with symmetry group D3h, or NFF−, which would be an ionic solid. ==Ionic solid== A variety of other tetrafluoroammonium salts are known (NFX−), as are fluoride salts of other ammonium cations ). Let each rotating molecule be associated with a unit vector \hat{n}; for example, \hat{n} might represent the orientation of an electric or magnetic dipole moment. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: \zeta^\text{rot} \approx \frac{5.34 \times 10^6}{\sigma} \sqrt{ \frac{T^3}{A B C}} where the prefactor comes from \sqrt{\frac{(\pi k_\text{B}) ^3}{h^3}} = 5.34 \times 10^6 when A, B, and C are expressed in units of MHz. The expressions for \zeta^\text{rot} works for asymmetric, symmetric and spherical top rotors. ==References== ==See also== * Translational partition function * Vibrational partition function * Partition function (mathematics) Category:Equations of physics Category:Partition functions The mean thermal rotational energy per molecule can now be computed by taking the derivative of \zeta^\text{rot} with respect to temperature T. Molecule \theta_{\mathrm{R}} (K)P. Atkins and J. de Paula ""Physical Chemistry"", 9th edition (W.H. Freeman 2010), Table 13.2, Data section in appendix H2 87.6 N2 2.88 O2 2.08 F2 1.27 HF 30.2 HCl 15.2 CO2 0.561P. Dominik Kurzydłowski and Patryk Zaleski-Ejgierd predict that a mixture of fluorine and nitrogen trifluoride under pressure between 10 and 33 GPa forms NFF− with space group R3m. This has CAS number 71485-49-9.Tetrafluoroammonium bifluoride I. J. Solomon believed that nitrogen pentafluoride was produced by the thermal decomposition of NF4AsF6, but experimental results were not reproduced. thumb|A molecule with a red cross on its front undergoing 3 dimensional rotational diffusion. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants. It is a salt, consisting of the N-fluoropyridinium cation ([C5H5NF]+) and the triflate anion. The characteristic rotational temperature ( or ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. ",-111.92,72,"""0.38""",-0.16,7.97,E
+Suppose that $2.5 \mathrm{mmol} \mathrm{N}_2$ (g) occupies $42 \mathrm{~cm}^3$ at $300 \mathrm{~K}$ and expands isothermally to $600 \mathrm{~cm}^3$. Calculate $\Delta G$ for the process.,"For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. For the above defined path we have that and thus , and hence the increase in entropy for the Joule expansion is \Delta S=\int_i^f\mathrm{d}S=\int_{V_0}^{2V_0} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2. As a result, the change in internal energy, \Delta U, is zero. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. Negative and positive thermal expansion hereby compensate each other to a certain amount if the temperature is changed. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034 k_B T. == References == Category:SI derived units In geochemistry, hydrology, paleoclimatology and paleoceanography, δ15N (pronounced ""delta fifteen n"") or delta-N-15 is a measure of the ratio of the two stable isotopes of nitrogen, 15N:14N. ==Formulas== Two very similar expressions for are in wide use in hydrology. Thermodynamics, p. 414. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work given by W = -\int_{2V_0}^{V_0} P\,\mathrm{d}V = - \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V = nRT\ln 2 = T \Delta S_\text{gas}. From these initial measurements, Gibbs free energy changes (\Delta G) and entropy changes (\Delta S) can be determined using the relationship: ::: \Delta G = -RT\ln{K_a} = \Delta H -T\Delta S (where R is the gas constant and T is the absolute temperature). In thermodynamics, Stefan's formula says that the specific surface energy at a given interface is determined by the respective enthalpy difference \scriptstyle \Delta H^*. : \sigma = \gamma_0 \left( \frac{\Delta H^*}{N_\text{A}^{1/3}V_\text{m}^{2/3}}\right), where σ is the specific surface energy, NA is the Avogadro constant, \gamma_0 is a steric dimensionless coefficient, and Vm is the molar volume. ==References== Category:Thermodynamic equations Category:Chemical thermodynamics For example, the Gibbs free energy of a compound in the area of thermochemistry is often quantified in units of kilojoules per mole (symbol: kJ·mol−1 or kJ/mol), with 1 kilojoule = 1000 joules. Since 1 mole = 6.02214076 particles (atoms, molecules, ions etc.), 1 joule per mole is equal to 1 joule divided by 6.02214076 particles, ≈1.660539 joule per particle. The joule per mole (symbol: J·mol−1 or J/mol) is the unit of energy per amount of substance in the International System of Units (SI), such that energy is measured in joules, and the amount of substance is measured in moles. Negative thermal expansion (NTE) is an unusual physicochemical process in which some materials contract upon heating, rather than expand as most other materials do. thumb|240px|The Joule expansion, in which a volume is expanded to a volume in a thermally isolated chamber. thumb|180px|A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas. It is also an SI derived unit of molar thermodynamic energy defined as the energy equal to one joule in one mole of substance. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. ===Ideal gases=== If the gas is ideal, both the initial (T_{\mathrm{i}}, P_{\mathrm{i}}, V_{\mathrm{i}}) and final (T_{\mathrm{f}}, P_{\mathrm{f}}, V_{\mathrm{f}}) conditions follow the Ideal Gas Law, so that initially P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}} and then, after the tap is opened, P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}. Thus in 2D and 3D negative thermal expansion in close-packed systems with pair interactions is realized even when the third derivative of the potential is zero or even negative. For a monatomic ideal gas , with the molar heat capacity at constant volume. Isothermal titration calorimetry for chiral chemistry. Therefore, the sign of thermal expansion coefficient is determined by the sign of the third derivative of the potential. ",-1.0, 35.91,"""1110.0""",0.00539,-17,E
 "Calculate the standard potential of the $\mathrm{Ce}^{4+} / \mathrm{Ce}$ couple from the values for the $\mathrm{Ce}^{3+} / \mathrm{Ce}$ and $\mathrm{Ce}^{4+} / \mathrm{Ce}^{3+}$ couples.
-","Cerium(III) sulfate, also called cerous sulfate, is an inorganic compound with the formula Ce2(SO4)3. 94 Ceti (HD 19994) is a trinary star system approximately 73 light-years away in the constellation Cetus. 94 Ceti A is a yellow-white dwarf star with about 1.3 times the mass of the Sun while 94 Ceti B and C are red dwarf stars. Cerianite-(Ce) is a relatively rare oxide mineral, belonging to uraninite group with the formula .Graham, A.R., 1955. The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . * After dividing by the number of electrons, the standard potential E° is related to the standard Gibbs free energy of formation ΔGf° by: E = \frac{\sum \Delta G_\text{left}-\sum \Delta G_\text{right}}{F} where F is the Faraday constant. thumb|upright=0.4|right|Ceres-1 Galactic Energy () is a Chinese private space launch enterprise flying the Ceres-1 and developing the Pallas-1 and 2 orbital rockets. {{Chembox | ImageFile = | ImageSize = | ImageAlt = | IUPACName = Triiodocerium | OtherNames = Cerous triiodide, Cerium triiodide | Section1 = | Section2 = | Section3 = | Section4 = }} Cerium(III) iodide (CeI3) is the compound formed by cerium(III) cations and iodide anions. == Preparation == Cerium metal reacts with iodine when heated to form cerium(III) iodide: : It is also formed when cerium reacts with mercury(II) iodide at high temperatures: : == Structure == Cerium(III) iodide adopts the plutonium(III) bromide crystal structure. For example: : \+ Cu(s) ( = +0.520 V) Cu + 2 Cu(s) ( = +0.337 V) Cu + ( = +0.159 V) :Calculating the potential using Gibbs free energy ( = 2 – ) gives the potential for as 0.154 V, not the experimental value of 0.159 V. : __TOC__ ==Table of standard electrode potentials== Legend: (s) - solid; (l) - liquid; (g) - gas; (aq) - aqueous (default for all charged species); (Hg) - amalgam; bold - water electrolysis equations. Cerianite-(Ce) associates with minerals of the apatite group, bastnäsite-group minerals, calcite, feldspar, ""fluocerite"", ""hydromica"", ilmenite, nepheline, magnetite, ""törnebohmite"" and tremolite. It is most stable if its inclination is either 65 or 115, ± 3. ==See also== * 79 Ceti * 81 Ceti * Lists of exoplanets ==References== ==External links== * SolStation: 94 Ceti 2 + orbits * 94 Ceti by Professor Jim Kaler. It contains 8-coordinate bicapped trigonal prismatic Ce3+ ions. == Uses == Cerium(III) iodide is used as a pharmaceutical intermediate and as a starting material for organocerium compounds. ==References== Category:Cerium(III) compounds Category:Iodides Category:Lanthanide halides The temperature of this dust is 40 K. ==Stellar system== This system is a hierarchical triple star system with 94 Ceti A being orbited by 94 Ceti BC, a pair of M dwarfs, in 2000 years. 94 Ceti B and C meanwhile orbit each other in a 1-year orbit. ==Planetary system== On 7 August 2000, a planet was announced by the Geneva Extrasolar Planet Search team as a result of radial velocity measurements taken with the Swiss 1.2-metre Leonhard Euler Telescope at La Silla Observatory in Chile. Cerianite CeO2: a new rare-earth oxide mineral. It is the most simple cerium mineral known. ==Notes on chemistry== Beside thorium cerianite-(Ce) may contain trace niobium, yttrium, lanthanum, ytterbium, zirconium and tantalum. ==Crystal structure== For details on crystal structure see cerium(IV) oxide. American Mineralogist 40, 560-564 It is one of a few currently known minerals containing essential tetravalent cerium, the other examples being stetindite and dyrnaesite-(La). ==Occurrence and association== Cerianite-(Ce) is associated with alkaline rocks, mostly nepheline syenites. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. The first launch of Ceres-1 took place at 7 November 2020, successfully placing the Tianqi 11 (also transcribed Tiange, also known as TQ 11, and Scorpio 1, COSPAR 2020-080A) satellite in orbit. On 6 December 2021, Galactic Energy launched its second Ceres-1 rocket, becoming the first Chinese private firm to reach orbit twice. All of the reactions should be divided by the stoichiometric coefficient for the electron to get the corresponding corrected reaction equation. The relation in electrode potential of metals in saltwater (as electrolyte) is given in the galvanic series. Using three Pallas-1 booster cores as its first stage, Pallas-2 will be capable of putting a 14-tonne payload into low Earth orbit. == Marketplace == Galactic Space is in competition with several other Chinese space rocket startups, being LandSpace, LinkSpace, ExPace, i-Space, OneSpace and Deep Blue Aerospace. == Launches == Rocket & Serial Date Payload Orbit Launch Site Outcome Notes Ceres-1 Y1 7 November 2020, 07:12 UTC Tianqi-11 (Scorpio-1) SSO Jiuquan First flight of Ceres-1. Cerium (III) sulfate tetrahydrate is a white solid that releases its water of crystallisation at 220 °C. ",-1.46,-87.8,135.36,1.11,0.15,A
-"An effusion cell has a circular hole of diameter $1.50 \mathrm{~mm}$. If the molar mass of the solid in the cell is $300 \mathrm{~g} \mathrm{~mol}^{-1}$ and its vapour pressure is $0.735 \mathrm{~Pa}$ at $500 \mathrm{~K}$, by how much will the mass of the solid decrease in a period of $1.00 \mathrm{~h}$ ?","If a small area A on the container is punched to become a small hole, the effusive flow rate will be \begin{align} Q_\text{effusion} &= J_\text{impingement} \times A \\\ &= \frac{P A}{\sqrt{2 \pi m k_{B} T}} \\\ &= \frac{P A N_A}{\sqrt{2 \pi M R T}} \end{align} where M is the molar mass, N_A is the Avogadro constant, and R = N_A k_B is the gas constant. The vapor slowly effuses through a pinhole, and the loss of mass is proportional to the vapor pressure and can be used to determine this pressure. Assuming the pressure difference between the two sides of the barrier is much smaller than P_{\rm avg}, the average absolute pressure in the system (i.e. \Delta P\ll P_{\rm avg}), it is possible to express effusion flow as a volumetric flow rate as follows: :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi k_BT}{32m}} or :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi RT}{32M}} where \Phi_V is the volumetric flow rate of the gas, P_{\rm avg} is the average pressure on either side of the orifice, and d is the hole diameter. ==Effect of molecular weight== At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. The rate \Phi_N at which a gas of molar mass M effuses (typically expressed as the number of molecules passing through the hole per second) is thenPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W.H.Freeman 2006) p.756 : \Phi_N = \frac{\Delta PAN_A}{\sqrt{2\pi MRT}}. The change in mass is the amount that flows after crossing the boundary for some time duration, not the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for steady flow. ==Alternative equations== thumb|245x245px|Illustration of volume flow rate. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice. ==Knudsen effusion cell== The Knudsen effusion cell is used to measure the vapor pressures of a solid with very low vapor pressure. The effusion rate for a gas depends directly on the average velocity of its particles. Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. Hence, a relation exists between the Spalding mass transfer number and the Spalding heat transfer number and writes: :B_T=\left( 1+B_M\right)^{\frac{1}{Le}\frac{C_{p,F}}{C_{p,g}}}-1 where: * Le is the gas film Lewis number (-) * C_{p,g} is the gas film specific heat at constant pressure (J.Kg−1.K−1) The droplet vaporization rate can be expressed as a function of the Sherwood number. As a consequence, the vaporization rate increases with the droplet Reynolds number. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules.K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18. Such a solid forms a vapor at low pressure by sublimation. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. ===Graham's law=== Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. ==Effusion into vacuum== Effusion from an equilibrated container into outside vacuum can be calculated based on kinetic theory. Droplet vaporization model for spray combustion calculations, Int. J. Heat Mass Transfer, Vol. 32, No. 9, pp. 1605-1618. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. : {\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1} where M_1 and M_2 represent the molar masses of the gases. Here \Delta P is the gas pressure difference across the barrier, A is the area of the hole, N_A is the Avogadro constant, R is the gas constant and T is the absolute temperature. The average velocity of effused particles is \begin{align} \overline{v_x}&=\overline{v_y}=0\\\ \overline{v_z}&=\sqrt{\frac{\pi k_BT}{2m}}. \end{align} Combined with the effusive flow rate, the recoil/thrust force on the system itself is F=m\overline{v_z}{\times}Q_\text{effusion}=\frac{PA}{2}. As the radius R of the sphere shrinks, the diameter of the cylinder must also shrink in order that h can remain the same. The conservation equation of mass simplifies to: :\rho_g r^2 u = cte = \left( \rho_g r^2 u\right)_s = \frac{\dot{m}_F}{4\pi} Combining the conservation equations for mass and fuel vapor mass fraction the following differential equation for the fuel vapor mass fraction Y_F(r) is obtained: :4 \pi r^2 \rho_g \mathcal{D} \frac{\mathrm{d}Y_F(r)}{\mathrm{d}r} = \dot{m}_F \left( Y_F(r)-1\right) Integrating this equation between r and the ambient gas phase region r = \infty and applying the boundary condition at r=r_d gives the expression for the droplet vaporization rate: :\dot{m}_F = 4 \pi \rho_g \mathcal{D} r_d \ln \left(1+ B_M \right) and :B_M=\frac{Y_{F,\infty}-Y_{F,s}}{Y_{F,s}-1} where: * B_M is the Spalding mass transfer number Phase equilibrium is assumed at the droplet surface and the mole fraction of fuel vapor at the droplet surface is obtained via the use of the Clapeyron's equation. A comparison of vaporization models in spray calculations, AIAA Journal, Vol. 22, No 10, p. 1448. for its balance between computational costs and accuracy. The vaporizing droplet (droplet vaporization) problem is a challenging issue in fluid dynamics. ",16,0.466,243.0,30,+17.7,A
-"The speed of a certain proton is $6.1 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated?","However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. One way to quantify the precision of the position and momentum is the standard deviation σ. H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. H_p = -\sum_{j=-\infty}^\infty \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53 The entropic uncertainty is indeed larger than the limiting value. For all n=1, \, 2, \, 3,\, \ldots, the quantity \sqrt{\frac{n^2\pi^2}{3}-2} is greater than 1, so the uncertainty principle is never violated. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. Uncertainty of measurement results. For numerical concreteness, the smallest value occurs when n = 1, in which case \sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{\pi^2}{3}-2} \approx 0.568 \hbar > \frac{\hbar}{2}. ===Constant momentum=== 360 px|thumb|right|Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p0 according to \varphi(p) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}\right), where we have introduced a reference scale x_0=\sqrt{\hbar/m\omega_0}, with \omega_0>0 describing the width of the distribution—cf. nondimensionalization. * Grabe, M ., Measurement Uncertainties in Science and Technology, Springer 2005. Evaluating the Uncertainty of Measurement. In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Under the above definition, the entropic uncertainty relation is H_x + H_p > \ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h} \right). Entropic uncertainty of the normal distribution We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. In doing so, we find the momentum of the particle to > arbitrary accuracy by conservation of momentum. In physics, the proton-to-electron mass ratio, μ or β, is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely: :μ = The number in parentheses is the measurement uncertainty on the last two digits, corresponding to a relative standard uncertainty of ==Discussion== μ is an important fundamental physical constant because: * Baryonic matter consists of quarks and particles made from quarks, like protons and neutrons. * Introduction to evaluating uncertainty of measurement * JCGM 200:2008. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. The measurement uncertainty strongly depends on the size of the measured value itself, e.g. amplitude-proportional. == See also == * measurement uncertainty * uncertainty * accuracy and precision * confidence Interval Category:Measurement The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The formula for cutting off the calculated value of is :U=\min \left\\{ ~9, ~ \max \Bigl\\{ \; 0, \; \left\lfloor 9\cdot\frac{\log r}{\;\log 648{,}000\;} \right\rfloor + 1 \; \Bigr\\} ~ \right\\} For instance: As of 10 September 2016, Ceres technically has an uncertainty of around −2.6, but is instead displayed as the minimal 0. The U value should not be used as a predictor for the uncertainty in the future motion of near-Earth objects. == Orbital uncertainty == Classical Kuiper belt objects 40–50 AU from the Sun JPL SBDB Uncertainty parameter Horizons January 2018 Uncertainty in distance from the Sun (millions of kilometers) Object Reference Ephemeris Location: @sun Table setting: 39 0 ±0.01 (134340) Pluto E2022-J69 1 ±0.04 2 ±0.14 20000 Varuna 3 ±0.84 19521 Chaos 4 ±1.4 5 ±8.2 6 ±70. 7 ±190. 8 ±590. 9 ±1,600. 1995 GJ ‘D’ Data insufficient for orbit determination. ",52,1.6,0.0,-8,76,A
-It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of $1.0 \mathrm{kT}$ ?,"More fundamental to the radical-pair mechanism, however, is the fact that radical-pair electrons both have spin, short for spin angular momentum, which gives each separate radical a magnetic moment. Magnetic isotope effects arise when a chemical reaction involves spin- selective processes, such as the radical pair mechanism. ""Normal"" high field NMR relies on the detection of spin- precession with inductive detection with a simple coil. Most commonly demonstrated in reactions of organic compounds involving radical intermediates, a magnetic field can speed up a reaction by decreasing the frequency of reverse reactions. === History === The radical-pair mechanism emerged as an explanation to CIDNP and CIDEP and was proposed in 1969 by Closs; Kaptein and Oosterhoff. === Radicals and radical- pairs === thumb|Example radical: Structure of Hydrocarboxyl radical, lone electron indicated as single black dot |145x145px A radical is a molecule with an odd number of electrons, and is induced in a variety of ways, including ultra-violet radiation. Therefore, spin states can be altered by magnetic fields. === Singlet and triplet spin states === The radical-pair is characterized as triplet or singlet by the spin state of the two lone electrons, paired together. The radical-pair mechanism explains how external magnetic fields can prevent radical-pair recombination with Zeeman interactions, the interaction between spin and an external magnetic field, and shows how a higher occurrence of the triplet state accelerates radical reactions because triplets can proceed only to products, and singlets are in equilibrium with the reactants as well as with the products. Spin chemistry is a sub-field of chemistry and physics, positioned at the intersection of chemical kinetics, photochemistry, magnetic resonance and free radical chemistry, that deals with magnetic and spin effects in chemical reactions. Low field NMR spans a range of different nuclear magnetic resonance (NMR) modalities, going from NMR conducted in permanent magnets, supporting magnetic fields of a few tesla (T), all the way down to zero field NMR, where the Earth's field is carefully shielded such that magnetic fields of nanotesla (nT) are achieved where nuclear spin precession is close to zero. Spinhenge@home was a volunteer computing project on the BOINC platform, which performs extensive numerical simulations concerning the physical characteristics of magnetic molecules. Spin chemistry concerns phenomena such as chemically induced dynamic nuclear polarization (CIDNP), chemically induced electron polarization (CIDEP), magnetic isotope effects in chemical reactions, and it is hypothesized to be key in the underlying mechanism for avian magnetoreception and consciousness. == Radical-pair mechanism == The radical-pair mechanism explains how a magnetic field can affect reaction kinetics by affecting electron spin dynamics. {{Chembox | ImageFile = File:Kölsch Radical V.1.svg | ImageCaption = Two resonance forms showing the predominant locations of the unpaired electron at the 1 and 3 positions | ImageSize = | ImageAlt = | PIN = 9-[(9H-Fluoren-9-ylidene)(phenyl)methyl]-9H-fluoren-9-yl | OtherNames = | Section1 = | Section2 = | Section3 = }} The Koelsch radical (also known as Koelsch's radical and 1,3-bisdiphenylene-2-phenylallyl or α,γ-bisdiphenylene- β-phenylallyl, abbreviated BDPA) is a chemical compound that is an unusually stable carbon-centered radical, due to its resonance structures. ==Properties== BDPA is an unusually stable radical compound due to the extent to which its electrons are delocalized through resonance structures. Zeeman interactions can “flip” only one of the radical's electron's spin if the radical-pair is anisotropic, thereby converting singlet radical-pairs to triplets. thumb|Typical Reaction Scheme of the Radical-pair Mechanism, which shows the effect of alternative product formation from singlet versus triplet radical-pairs. The spin relationship is such that the two unpaired electrons, one in each radical molecule, may have opposite spin (singlet; anticorrelated), or the same spin (triplet; correlated). Electron spin resonance can be employed to quantify the probe's concentration. ==References== Category:Molecular physics A spin probe is a molecule with stable free radical character that carries a functional group. Furthermore, the spin of each electron previously involved in the bond is conserved, which means that the radical- pair now formed is a singlet (each electron has opposite spin, as in the origin bond). Low field NMR also includes Earth's field NMR where simply the Earth's magnetic field is exploited to cause nuclear spin-precession which is detected. The Zeeman and Hyperfine Interactions take effect in the yellow box, denoted as step 4 in the process|367x367px The Zeeman interaction is an interaction between spin and external magnetic field, and is given by the equation :\Delta E=h u_L=g\mu_BB, where \Delta E is the energy of the Zeeman interaction, u_L is the Larmor frequency, B is the external magnetic field, \mu_B is the Bohr magneton, h is Planck's constant, and g is the g-factor of a free electron, 2.002319, which is slightly different in different radicals. In a broad sense, Low-field NMR is the branch of NMR that is not conducted in superconducting high-field magnets. It has been observed that migratory birds lose their navigational abilities in such conditions where the Zeeman interaction is obstructed in radical-pairs. == External links == * Spin chemistry portal ==References== Category:Physical chemistry Category:Nuclear magnetic resonance It is common to see the Zeeman interaction formulated in other ways. === Hyperfine interactions === Hyperfine interactions, the internal magnetic fields of local magnetic isotopes, play a significant role as well in the spin dynamics of radical-pairs. === Zeeman interactions and magnetoreception === Because the Zeeman interaction is a function of magnetic field and Larmor frequency, it can be obstructed or amplified by altering the external magnetic or the Larmor frequency with experimental instruments that generate oscillating fields. The project began beta testing on September 1, 2006 and used the Metropolis Monte Carlo algorithm to calculate and simulate spin dynamics in nanoscale molecular magnets. ",-1.00,2.8,-21.2,3.51,537,B
-"A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \mathrm{~s}$. Ignore air resistance?","The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. For example, for a speed of the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg). Kinetic energy is the movement energy of an object. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. Factoring out the rest energy gives: :E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is \frac{1}{2}mv^2. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: :E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) - m c^2 = \frac{1}{2} m v^2 So, the total energy E_k can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. ""Falling"" is a song by industrial rock band Gravity Kills from the album Perversion, released by TVT Records in 1998. ==Release== ""Falling"" reached No. 35 on Billboard's Mainstream Rock chart on July 4, 1998. The energy is not destroyed; it has only been converted to another form by friction. Notice that this can be obtained by replacing p by \hat p in the classical expression for kinetic energy in terms of momentum, :E_\text{k} = \frac{p^2}{2m}. The mathematical by- product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content :E_\text{rest} = E_0 = m c^2 At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Thus for a stationary observer (v = 0) :u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{tt}}} and thus the kinetic energy takes the form :E_\text{k} = -m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,. Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual. ==Overview== Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. Then kinetic energy is the total relativistic energy minus the rest energy: E_K = E - m_{0}c^2 = \sqrt{(p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2}- m_{0}c^2 At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. thumb|450px|center| Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as :E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12,960 \,\text{J} = 12.96 \,\text{kJ} When a person throws a ball, the person does work on it to give it speed as it leaves the hand. Substituting, we get:Physics notes - Kinetic energy in the CM frame . ",475,35.2,48.0,57.2, -194,C
-"A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \mathrm{~s}$. Ignore air resistance?","The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. For example, for a speed of the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg). Kinetic energy is the movement energy of an object. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is \frac{1}{2}mv^2. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. Factoring out the rest energy gives: :E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: :E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) - m c^2 = \frac{1}{2} m v^2 So, the total energy E_k can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds. The energy is not destroyed; it has only been converted to another form by friction. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. ""Falling"" is a song by industrial rock band Gravity Kills from the album Perversion, released by TVT Records in 1998. ==Release== ""Falling"" reached No. 35 on Billboard's Mainstream Rock chart on July 4, 1998. Notice that this can be obtained by replacing p by \hat p in the classical expression for kinetic energy in terms of momentum, :E_\text{k} = \frac{p^2}{2m}. The mathematical by- product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content :E_\text{rest} = E_0 = m c^2 At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Thus for a stationary observer (v = 0) :u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{tt}}} and thus the kinetic energy takes the form :E_\text{k} = -m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,. Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual. ==Overview== Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. Then kinetic energy is the total relativistic energy minus the rest energy: E_K = E - m_{0}c^2 = \sqrt{(p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2}- m_{0}c^2 At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. thumb|450px|center| Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as :E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12,960 \,\text{J} = 12.96 \,\text{kJ} When a person throws a ball, the person does work on it to give it speed as it leaves the hand. Acceleration due to gravity, acceleration of gravity or gravity acceleration may refer to: *Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general *Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth *Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth ==See also== *g-force, the acceleration of a body relative to free-fall ",47,1.07,48.0,0.9984,358800,C
-"The flux of visible photons reaching Earth from the North Star is about $4 \times 10^3 \mathrm{~mm}^{-2} \mathrm{~s}^{-1}$. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about $40 \mathrm{~mm}^2$ and the response time of the eye is about $0.1 \mathrm{~s}$. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused onto the retina in $0.1 \mathrm{~s}$ ?","Choosing parameter values thought typical of normal dark-site observations (e.g. eye pupil 0.7cm and F=2) he found N=7.69.Crumey, op. cit., Eq. Various authorsCited in Crumey, op. cit., Sec. 3.2. have stated the limiting magnitude of a telescope with entrance pupil D centimetres to be of the form : m = 5 logD \+ N with suggested values for the constant N ranging from 6.8 to 8.7. The astronomer H.D. Curtis reported his naked-eye limit as 6.53, but by looking at stars through a hole in a black screen (i.e. against a totally dark background) was able to see one of magnitude 8.3, and possibly one of 8.9.Section=1.6.5 of Naked-eye magnitude limits can be modelled theoretically using laboratory data on human contrast thresholds at various background brightness levels. More generally, for situations where it is possible to raise a telescope's magnification high enough to make the sky background effectively black, the limiting magnitude is approximated by :m = 5 logD \+ 8 – 2.5 log (p^2F/T) where D and F are as stated above, p is the observer's pupil diameter in centimetres, and T is the telescope transmittance (e.g. 0.75 for a typical reflector).Crumey, A. Modelling the Visibility of Deep-Sky Objects. upright=1.6|thumb|Visual effect of night sky's brightness. The pupil magnification of an optical system is the ratio of the diameter of the exit pupil to the diameter of the entrance pupil. In the dark it will be the same at first, but will approach the maximum distance for a wide pupil 3 to 8 mm. thumb|The apparent position of a star viewed from the Earth depends on the Earth's velocity. The very darkest skies have a zenith surface brightness of approximately 22 mag arcsec−2, so it can be seen from the equation that such a sky would be expected to show stars approximately 0.4 mag fainter than one with a surface brightness of 21 mag arcsec−2. Crumey obtained a formula for N as a function of the sky surface brightness, telescope magnification, observer's eye pupil diameter and other parameters including the personal factor F discussed above. For example, the cat's slit pupil can change the light intensity on the retina 135-fold compared to 10-fold in humans. However, the limiting visibility is 7th magnitude for faint stars visible from dark rural areas located 200 kilometers from major cities. Crumey showed that for a sky background with surface brightness \mu_{sky} > 21 mag arcsec−2, the visual limit m could be expressed as: :m=0.4260\mu_{sky} –2.3650–2.5logF where F is a ""field factor"" specific to the observer and viewing situation.Crumey, op. cit., Eq. The image of the pupil as seen from outside the eye is the entrance pupil, which does not exactly correspond to the location and size of the physical pupil because it is magnified by the cornea. Peripheral Light Focusing (PLF) can be described as the focusing of Solar Ultraviolet Radiation (SUVR) at the nasal limbus of the cornea. Bowen did not record parameters such as his eye pupil diameter, naked-eye magnitude limit, or the extent of light loss in his telescopes; but because he made observations at a range of magnifications using three telescopes (with apertures 0.33 inch, 6 inch and 60 inch), Crumey was able to construct a system of simultaneous equations from which the remaining parameters could be deduced. A star's brightness (more precisely its illuminance) must exceed the sky's surface brightness (i.e. luminance) by a sufficient amount. In addition to dilation and contraction caused by light and darkness, it has been shown that solving simple multiplication problems affects the size of the pupil. The limiting magnitude will depend on the observer, and will increase with the eye's dark adaptation. From brightly lit Midtown Manhattan, the limiting magnitude is possibly 2.0, meaning that from the heart of New York City only approximately 15 stars will be visible at any given time. For example, at the peak age of 15, the dark- adapted pupil can vary from 4 mm to 9 mm with different individuals. This corresponds to roughly 250 visible stars, or one-tenth the number that can be perceived under perfectly dark skies. ",-1.78,30,4.4,0.7812,0.54,C
+","Cerium(III) sulfate, also called cerous sulfate, is an inorganic compound with the formula Ce2(SO4)3. 94 Ceti (HD 19994) is a trinary star system approximately 73 light-years away in the constellation Cetus. 94 Ceti A is a yellow-white dwarf star with about 1.3 times the mass of the Sun while 94 Ceti B and C are red dwarf stars. Cerianite-(Ce) is a relatively rare oxide mineral, belonging to uraninite group with the formula .Graham, A.R., 1955. The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . * After dividing by the number of electrons, the standard potential E° is related to the standard Gibbs free energy of formation ΔGf° by: E = \frac{\sum \Delta G_\text{left}-\sum \Delta G_\text{right}}{F} where F is the Faraday constant. thumb|upright=0.4|right|Ceres-1 Galactic Energy () is a Chinese private space launch enterprise flying the Ceres-1 and developing the Pallas-1 and 2 orbital rockets. {{Chembox | ImageFile = | ImageSize = | ImageAlt = | IUPACName = Triiodocerium | OtherNames = Cerous triiodide, Cerium triiodide | Section1 = | Section2 = | Section3 = | Section4 = }} Cerium(III) iodide (CeI3) is the compound formed by cerium(III) cations and iodide anions. == Preparation == Cerium metal reacts with iodine when heated to form cerium(III) iodide: : It is also formed when cerium reacts with mercury(II) iodide at high temperatures: : == Structure == Cerium(III) iodide adopts the plutonium(III) bromide crystal structure. For example: : \+ Cu(s) ( = +0.520 V) Cu + 2 Cu(s) ( = +0.337 V) Cu + ( = +0.159 V) :Calculating the potential using Gibbs free energy ( = 2 – ) gives the potential for as 0.154 V, not the experimental value of 0.159 V. : __TOC__ ==Table of standard electrode potentials== Legend: (s) - solid; (l) - liquid; (g) - gas; (aq) - aqueous (default for all charged species); (Hg) - amalgam; bold - water electrolysis equations. Cerianite-(Ce) associates with minerals of the apatite group, bastnäsite-group minerals, calcite, feldspar, ""fluocerite"", ""hydromica"", ilmenite, nepheline, magnetite, ""törnebohmite"" and tremolite. It is most stable if its inclination is either 65 or 115, ± 3. ==See also== * 79 Ceti * 81 Ceti * Lists of exoplanets ==References== ==External links== * SolStation: 94 Ceti 2 + orbits * 94 Ceti by Professor Jim Kaler. It contains 8-coordinate bicapped trigonal prismatic Ce3+ ions. == Uses == Cerium(III) iodide is used as a pharmaceutical intermediate and as a starting material for organocerium compounds. ==References== Category:Cerium(III) compounds Category:Iodides Category:Lanthanide halides The temperature of this dust is 40 K. ==Stellar system== This system is a hierarchical triple star system with 94 Ceti A being orbited by 94 Ceti BC, a pair of M dwarfs, in 2000 years. 94 Ceti B and C meanwhile orbit each other in a 1-year orbit. ==Planetary system== On 7 August 2000, a planet was announced by the Geneva Extrasolar Planet Search team as a result of radial velocity measurements taken with the Swiss 1.2-metre Leonhard Euler Telescope at La Silla Observatory in Chile. Cerianite CeO2: a new rare-earth oxide mineral. It is the most simple cerium mineral known. ==Notes on chemistry== Beside thorium cerianite-(Ce) may contain trace niobium, yttrium, lanthanum, ytterbium, zirconium and tantalum. ==Crystal structure== For details on crystal structure see cerium(IV) oxide. American Mineralogist 40, 560-564 It is one of a few currently known minerals containing essential tetravalent cerium, the other examples being stetindite and dyrnaesite-(La). ==Occurrence and association== Cerianite-(Ce) is associated with alkaline rocks, mostly nepheline syenites. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. The first launch of Ceres-1 took place at 7 November 2020, successfully placing the Tianqi 11 (also transcribed Tiange, also known as TQ 11, and Scorpio 1, COSPAR 2020-080A) satellite in orbit. On 6 December 2021, Galactic Energy launched its second Ceres-1 rocket, becoming the first Chinese private firm to reach orbit twice. All of the reactions should be divided by the stoichiometric coefficient for the electron to get the corresponding corrected reaction equation. The relation in electrode potential of metals in saltwater (as electrolyte) is given in the galvanic series. Using three Pallas-1 booster cores as its first stage, Pallas-2 will be capable of putting a 14-tonne payload into low Earth orbit. == Marketplace == Galactic Space is in competition with several other Chinese space rocket startups, being LandSpace, LinkSpace, ExPace, i-Space, OneSpace and Deep Blue Aerospace. == Launches == Rocket & Serial Date Payload Orbit Launch Site Outcome Notes Ceres-1 Y1 7 November 2020, 07:12 UTC Tianqi-11 (Scorpio-1) SSO Jiuquan First flight of Ceres-1. Cerium (III) sulfate tetrahydrate is a white solid that releases its water of crystallisation at 220 °C. ",-1.46,-87.8,"""135.36""",1.11,0.15,A
+"An effusion cell has a circular hole of diameter $1.50 \mathrm{~mm}$. If the molar mass of the solid in the cell is $300 \mathrm{~g} \mathrm{~mol}^{-1}$ and its vapour pressure is $0.735 \mathrm{~Pa}$ at $500 \mathrm{~K}$, by how much will the mass of the solid decrease in a period of $1.00 \mathrm{~h}$ ?","If a small area A on the container is punched to become a small hole, the effusive flow rate will be \begin{align} Q_\text{effusion} &= J_\text{impingement} \times A \\\ &= \frac{P A}{\sqrt{2 \pi m k_{B} T}} \\\ &= \frac{P A N_A}{\sqrt{2 \pi M R T}} \end{align} where M is the molar mass, N_A is the Avogadro constant, and R = N_A k_B is the gas constant. The vapor slowly effuses through a pinhole, and the loss of mass is proportional to the vapor pressure and can be used to determine this pressure. Assuming the pressure difference between the two sides of the barrier is much smaller than P_{\rm avg}, the average absolute pressure in the system (i.e. \Delta P\ll P_{\rm avg}), it is possible to express effusion flow as a volumetric flow rate as follows: :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi k_BT}{32m}} or :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi RT}{32M}} where \Phi_V is the volumetric flow rate of the gas, P_{\rm avg} is the average pressure on either side of the orifice, and d is the hole diameter. ==Effect of molecular weight== At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. The rate \Phi_N at which a gas of molar mass M effuses (typically expressed as the number of molecules passing through the hole per second) is thenPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W.H.Freeman 2006) p.756 : \Phi_N = \frac{\Delta PAN_A}{\sqrt{2\pi MRT}}. The change in mass is the amount that flows after crossing the boundary for some time duration, not the initial amount of mass at the boundary minus the final amount at the boundary, since the change in mass flowing through the area would be zero for steady flow. ==Alternative equations== thumb|245x245px|Illustration of volume flow rate. Thus, the faster the gas particles are moving, the more likely they are to pass through the effusion orifice. ==Knudsen effusion cell== The Knudsen effusion cell is used to measure the vapor pressures of a solid with very low vapor pressure. The effusion rate for a gas depends directly on the average velocity of its particles. Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. Hence, a relation exists between the Spalding mass transfer number and the Spalding heat transfer number and writes: :B_T=\left( 1+B_M\right)^{\frac{1}{Le}\frac{C_{p,F}}{C_{p,g}}}-1 where: * Le is the gas film Lewis number (-) * C_{p,g} is the gas film specific heat at constant pressure (J.Kg−1.K−1) The droplet vaporization rate can be expressed as a function of the Sherwood number. As a consequence, the vaporization rate increases with the droplet Reynolds number. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules.K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18. Such a solid forms a vapor at low pressure by sublimation. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. ===Graham's law=== Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. ==Effusion into vacuum== Effusion from an equilibrated container into outside vacuum can be calculated based on kinetic theory. Droplet vaporization model for spray combustion calculations, Int. J. Heat Mass Transfer, Vol. 32, No. 9, pp. 1605-1618. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. : {\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1} where M_1 and M_2 represent the molar masses of the gases. Here \Delta P is the gas pressure difference across the barrier, A is the area of the hole, N_A is the Avogadro constant, R is the gas constant and T is the absolute temperature. The average velocity of effused particles is \begin{align} \overline{v_x}&=\overline{v_y}=0\\\ \overline{v_z}&=\sqrt{\frac{\pi k_BT}{2m}}. \end{align} Combined with the effusive flow rate, the recoil/thrust force on the system itself is F=m\overline{v_z}{\times}Q_\text{effusion}=\frac{PA}{2}. As the radius R of the sphere shrinks, the diameter of the cylinder must also shrink in order that h can remain the same. The conservation equation of mass simplifies to: :\rho_g r^2 u = cte = \left( \rho_g r^2 u\right)_s = \frac{\dot{m}_F}{4\pi} Combining the conservation equations for mass and fuel vapor mass fraction the following differential equation for the fuel vapor mass fraction Y_F(r) is obtained: :4 \pi r^2 \rho_g \mathcal{D} \frac{\mathrm{d}Y_F(r)}{\mathrm{d}r} = \dot{m}_F \left( Y_F(r)-1\right) Integrating this equation between r and the ambient gas phase region r = \infty and applying the boundary condition at r=r_d gives the expression for the droplet vaporization rate: :\dot{m}_F = 4 \pi \rho_g \mathcal{D} r_d \ln \left(1+ B_M \right) and :B_M=\frac{Y_{F,\infty}-Y_{F,s}}{Y_{F,s}-1} where: * B_M is the Spalding mass transfer number Phase equilibrium is assumed at the droplet surface and the mole fraction of fuel vapor at the droplet surface is obtained via the use of the Clapeyron's equation. A comparison of vaporization models in spray calculations, AIAA Journal, Vol. 22, No 10, p. 1448. for its balance between computational costs and accuracy. The vaporizing droplet (droplet vaporization) problem is a challenging issue in fluid dynamics. ",16,0.466,"""243.0""",30,+17.7,A
+"The speed of a certain proton is $6.1 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. If the uncertainty in its momentum is to be reduced to 0.0100 per cent, what uncertainty in its location must be tolerated?","However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. One way to quantify the precision of the position and momentum is the standard deviation σ. H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. H_p = -\sum_{j=-\infty}^\infty \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53 The entropic uncertainty is indeed larger than the limiting value. For all n=1, \, 2, \, 3,\, \ldots, the quantity \sqrt{\frac{n^2\pi^2}{3}-2} is greater than 1, so the uncertainty principle is never violated. The uncertainty principle also says that eliminating uncertainty about position maximizes uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. Uncertainty of measurement results. For numerical concreteness, the smallest value occurs when n = 1, in which case \sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{\pi^2}{3}-2} \approx 0.568 \hbar > \frac{\hbar}{2}. ===Constant momentum=== 360 px|thumb|right|Position space probability density of an initially Gaussian state moving at minimally uncertain, constant momentum in free space Assume a particle initially has a momentum space wave function described by a normal distribution around some constant momentum p0 according to \varphi(p) = \left(\frac{x_0}{\hbar \sqrt{\pi}} \right)^{1/2} \exp\left(\frac{-x_0^2 (p-p_0)^2}{2\hbar^2}\right), where we have introduced a reference scale x_0=\sqrt{\hbar/m\omega_0}, with \omega_0>0 describing the width of the distribution—cf. nondimensionalization. * Grabe, M ., Measurement Uncertainties in Science and Technology, Springer 2005. Evaluating the Uncertainty of Measurement. In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. Under the above definition, the entropic uncertainty relation is H_x + H_p > \ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h} \right). Entropic uncertainty of the normal distribution We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. In doing so, we find the momentum of the particle to > arbitrary accuracy by conservation of momentum. In physics, the proton-to-electron mass ratio, μ or β, is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely: :μ = The number in parentheses is the measurement uncertainty on the last two digits, corresponding to a relative standard uncertainty of ==Discussion== μ is an important fundamental physical constant because: * Baryonic matter consists of quarks and particles made from quarks, like protons and neutrons. * Introduction to evaluating uncertainty of measurement * JCGM 200:2008. In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. The measurement uncertainty strongly depends on the size of the measured value itself, e.g. amplitude-proportional. == See also == * measurement uncertainty * uncertainty * accuracy and precision * confidence Interval Category:Measurement The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant. The formula for cutting off the calculated value of is :U=\min \left\\{ ~9, ~ \max \Bigl\\{ \; 0, \; \left\lfloor 9\cdot\frac{\log r}{\;\log 648{,}000\;} \right\rfloor + 1 \; \Bigr\\} ~ \right\\} For instance: As of 10 September 2016, Ceres technically has an uncertainty of around −2.6, but is instead displayed as the minimal 0. The U value should not be used as a predictor for the uncertainty in the future motion of near-Earth objects. == Orbital uncertainty == Classical Kuiper belt objects 40–50 AU from the Sun JPL SBDB Uncertainty parameter Horizons January 2018 Uncertainty in distance from the Sun (millions of kilometers) Object Reference Ephemeris Location: @sun Table setting: 39 0 ±0.01 (134340) Pluto E2022-J69 1 ±0.04 2 ±0.14 20000 Varuna 3 ±0.84 19521 Chaos 4 ±1.4 5 ±8.2 6 ±70. 7 ±190. 8 ±590. 9 ±1,600. 1995 GJ ‘D’ Data insufficient for orbit determination. ",52,1.6,"""0.0""",-8,76,A
+It is possible to produce very high magnetic fields over small volumes by special techniques. What would be the resonance frequency of an electron spin in an organic radical in a field of $1.0 \mathrm{kT}$ ?,"More fundamental to the radical-pair mechanism, however, is the fact that radical-pair electrons both have spin, short for spin angular momentum, which gives each separate radical a magnetic moment. Magnetic isotope effects arise when a chemical reaction involves spin- selective processes, such as the radical pair mechanism. ""Normal"" high field NMR relies on the detection of spin- precession with inductive detection with a simple coil. Most commonly demonstrated in reactions of organic compounds involving radical intermediates, a magnetic field can speed up a reaction by decreasing the frequency of reverse reactions. === History === The radical-pair mechanism emerged as an explanation to CIDNP and CIDEP and was proposed in 1969 by Closs; Kaptein and Oosterhoff. === Radicals and radical- pairs === thumb|Example radical: Structure of Hydrocarboxyl radical, lone electron indicated as single black dot |145x145px A radical is a molecule with an odd number of electrons, and is induced in a variety of ways, including ultra-violet radiation. Therefore, spin states can be altered by magnetic fields. === Singlet and triplet spin states === The radical-pair is characterized as triplet or singlet by the spin state of the two lone electrons, paired together. The radical-pair mechanism explains how external magnetic fields can prevent radical-pair recombination with Zeeman interactions, the interaction between spin and an external magnetic field, and shows how a higher occurrence of the triplet state accelerates radical reactions because triplets can proceed only to products, and singlets are in equilibrium with the reactants as well as with the products. Spin chemistry is a sub-field of chemistry and physics, positioned at the intersection of chemical kinetics, photochemistry, magnetic resonance and free radical chemistry, that deals with magnetic and spin effects in chemical reactions. Low field NMR spans a range of different nuclear magnetic resonance (NMR) modalities, going from NMR conducted in permanent magnets, supporting magnetic fields of a few tesla (T), all the way down to zero field NMR, where the Earth's field is carefully shielded such that magnetic fields of nanotesla (nT) are achieved where nuclear spin precession is close to zero. Spinhenge@home was a volunteer computing project on the BOINC platform, which performs extensive numerical simulations concerning the physical characteristics of magnetic molecules. Spin chemistry concerns phenomena such as chemically induced dynamic nuclear polarization (CIDNP), chemically induced electron polarization (CIDEP), magnetic isotope effects in chemical reactions, and it is hypothesized to be key in the underlying mechanism for avian magnetoreception and consciousness. == Radical-pair mechanism == The radical-pair mechanism explains how a magnetic field can affect reaction kinetics by affecting electron spin dynamics. {{Chembox | ImageFile = File:Kölsch Radical V.1.svg | ImageCaption = Two resonance forms showing the predominant locations of the unpaired electron at the 1 and 3 positions | ImageSize = | ImageAlt = | PIN = 9-[(9H-Fluoren-9-ylidene)(phenyl)methyl]-9H-fluoren-9-yl | OtherNames = | Section1 = | Section2 = | Section3 = }} The Koelsch radical (also known as Koelsch's radical and 1,3-bisdiphenylene-2-phenylallyl or α,γ-bisdiphenylene- β-phenylallyl, abbreviated BDPA) is a chemical compound that is an unusually stable carbon-centered radical, due to its resonance structures. ==Properties== BDPA is an unusually stable radical compound due to the extent to which its electrons are delocalized through resonance structures. Zeeman interactions can “flip” only one of the radical's electron's spin if the radical-pair is anisotropic, thereby converting singlet radical-pairs to triplets. thumb|Typical Reaction Scheme of the Radical-pair Mechanism, which shows the effect of alternative product formation from singlet versus triplet radical-pairs. The spin relationship is such that the two unpaired electrons, one in each radical molecule, may have opposite spin (singlet; anticorrelated), or the same spin (triplet; correlated). Electron spin resonance can be employed to quantify the probe's concentration. ==References== Category:Molecular physics A spin probe is a molecule with stable free radical character that carries a functional group. Furthermore, the spin of each electron previously involved in the bond is conserved, which means that the radical- pair now formed is a singlet (each electron has opposite spin, as in the origin bond). Low field NMR also includes Earth's field NMR where simply the Earth's magnetic field is exploited to cause nuclear spin-precession which is detected. The Zeeman and Hyperfine Interactions take effect in the yellow box, denoted as step 4 in the process|367x367px The Zeeman interaction is an interaction between spin and external magnetic field, and is given by the equation :\Delta E=h u_L=g\mu_BB, where \Delta E is the energy of the Zeeman interaction, u_L is the Larmor frequency, B is the external magnetic field, \mu_B is the Bohr magneton, h is Planck's constant, and g is the g-factor of a free electron, 2.002319, which is slightly different in different radicals. In a broad sense, Low-field NMR is the branch of NMR that is not conducted in superconducting high-field magnets. It has been observed that migratory birds lose their navigational abilities in such conditions where the Zeeman interaction is obstructed in radical-pairs. == External links == * Spin chemistry portal ==References== Category:Physical chemistry Category:Nuclear magnetic resonance It is common to see the Zeeman interaction formulated in other ways. === Hyperfine interactions === Hyperfine interactions, the internal magnetic fields of local magnetic isotopes, play a significant role as well in the spin dynamics of radical-pairs. === Zeeman interactions and magnetoreception === Because the Zeeman interaction is a function of magnetic field and Larmor frequency, it can be obstructed or amplified by altering the external magnetic or the Larmor frequency with experimental instruments that generate oscillating fields. The project began beta testing on September 1, 2006 and used the Metropolis Monte Carlo algorithm to calculate and simulate spin dynamics in nanoscale molecular magnets. ",-1.00,2.8,"""-21.2""",3.51,537,B
+"A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \mathrm{~s}$. Ignore air resistance?","The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. For example, for a speed of the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg). Kinetic energy is the movement energy of an object. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. Factoring out the rest energy gives: :E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is \frac{1}{2}mv^2. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: :E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) - m c^2 = \frac{1}{2} m v^2 So, the total energy E_k can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. ""Falling"" is a song by industrial rock band Gravity Kills from the album Perversion, released by TVT Records in 1998. ==Release== ""Falling"" reached No. 35 on Billboard's Mainstream Rock chart on July 4, 1998. The energy is not destroyed; it has only been converted to another form by friction. Notice that this can be obtained by replacing p by \hat p in the classical expression for kinetic energy in terms of momentum, :E_\text{k} = \frac{p^2}{2m}. The mathematical by- product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content :E_\text{rest} = E_0 = m c^2 At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Thus for a stationary observer (v = 0) :u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{tt}}} and thus the kinetic energy takes the form :E_\text{k} = -m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,. Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual. ==Overview== Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. Then kinetic energy is the total relativistic energy minus the rest energy: E_K = E - m_{0}c^2 = \sqrt{(p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2}- m_{0}c^2 At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. thumb|450px|center| Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as :E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12,960 \,\text{J} = 12.96 \,\text{kJ} When a person throws a ball, the person does work on it to give it speed as it leaves the hand. Substituting, we get:Physics notes - Kinetic energy in the CM frame . ",475,35.2,"""48.0""",57.2, -194,C
+"A particle of mass $1.0 \mathrm{~g}$ is released near the surface of the Earth, where the acceleration of free fall is $g=8.91 \mathrm{~m} \mathrm{~s}^{-2}$. What will be its kinetic energy after  $1.0 \mathrm{~s}$. Ignore air resistance?","The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. For example, for a speed of the correction to the Newtonian kinetic energy is 0.0417 J/kg (on a Newtonian kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a Newtonian kinetic energy of 5 GJ/kg). Kinetic energy is the movement energy of an object. In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion., Chapter 1, p. 9 It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. The kinetic energy of an object is related to its momentum by the equation: :E_\text{k} = \frac{p^2}{2m} where: *p is momentum *m is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass m, whose center of mass is moving in a straight line with speed v, as seen above is equal to : E_\text{t} = \frac{1}{2} mv^2 where: *m is the mass of the body *v is the speed of the center of mass of the body. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is \frac{1}{2}mv^2. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., :Fs = \frac{1}{2} mv^2 Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. Factoring out the rest energy gives: :E_\text{k} = m c^2 \left( \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - 1 \right) \,. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: :E_\text{k} \approx m c^2 \left(1 + \frac{1}{2} \frac{v^2}{c^2}\right) - m c^2 = \frac{1}{2} m v^2 So, the total energy E_k can be partitioned into the rest mass energy plus the Newtonian kinetic energy at low speeds. The energy is not destroyed; it has only been converted to another form by friction. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. ""Falling"" is a song by industrial rock band Gravity Kills from the album Perversion, released by TVT Records in 1998. ==Release== ""Falling"" reached No. 35 on Billboard's Mainstream Rock chart on July 4, 1998. Notice that this can be obtained by replacing p by \hat p in the classical expression for kinetic energy in terms of momentum, :E_\text{k} = \frac{p^2}{2m}. The mathematical by- product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content :E_\text{rest} = E_0 = m c^2 At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy. Thus for a stationary observer (v = 0) :u_{\text{obs}}^{t} = c \sqrt{\frac{-1}{g_{tt}}} and thus the kinetic energy takes the form :E_\text{k} = -m g_{tt} u^t u_{\text{obs}}^t - m c^2 = m c^2 \sqrt{\frac{g_{tt}}{g_{tt} + g_{ss} v^2}} - m c^2\,. Sir William Thomson and Professor Tait have lately substituted the word 'kinetic' for 'actual. ==Overview== Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. Then kinetic energy is the total relativistic energy minus the rest energy: E_K = E - m_{0}c^2 = \sqrt{(p \textrm c)^2 + \left(m_0 \textrm c^2\right)^2}- m_{0}c^2 At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. thumb|450px|center| Log of relativistic kinetic energy versus log relativistic momentum, for many objects of vastly different scales. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as :E_\text{k} = \frac{1}{2} \cdot 80 \,\text{kg} \cdot \left(18 \,\text{m/s}\right)^2 = 12,960 \,\text{J} = 12.96 \,\text{kJ} When a person throws a ball, the person does work on it to give it speed as it leaves the hand. Acceleration due to gravity, acceleration of gravity or gravity acceleration may refer to: *Gravitational acceleration, the acceleration caused by the gravitational attraction of massive bodies in general *Gravity of Earth, the acceleration caused by the combination of gravitational attraction and centrifugal force of the Earth *Standard gravity, or g, the standard value of gravitational acceleration at sea level on Earth ==See also== *g-force, the acceleration of a body relative to free-fall ",47,1.07,"""48.0""",0.9984,358800,C
+"The flux of visible photons reaching Earth from the North Star is about $4 \times 10^3 \mathrm{~mm}^{-2} \mathrm{~s}^{-1}$. Of these photons, 30 per cent are absorbed or scattered by the atmosphere and 25 per cent of the surviving photons are scattered by the surface of the cornea of the eye. A further 9 per cent are absorbed inside the cornea. The area of the pupil at night is about $40 \mathrm{~mm}^2$ and the response time of the eye is about $0.1 \mathrm{~s}$. Of the photons passing through the pupil, about 43 per cent are absorbed in the ocular medium. How many photons from the North Star are focused onto the retina in $0.1 \mathrm{~s}$ ?","Choosing parameter values thought typical of normal dark-site observations (e.g. eye pupil 0.7cm and F=2) he found N=7.69.Crumey, op. cit., Eq. Various authorsCited in Crumey, op. cit., Sec. 3.2. have stated the limiting magnitude of a telescope with entrance pupil D centimetres to be of the form : m = 5 logD \+ N with suggested values for the constant N ranging from 6.8 to 8.7. The astronomer H.D. Curtis reported his naked-eye limit as 6.53, but by looking at stars through a hole in a black screen (i.e. against a totally dark background) was able to see one of magnitude 8.3, and possibly one of 8.9.Section=1.6.5 of Naked-eye magnitude limits can be modelled theoretically using laboratory data on human contrast thresholds at various background brightness levels. More generally, for situations where it is possible to raise a telescope's magnification high enough to make the sky background effectively black, the limiting magnitude is approximated by :m = 5 logD \+ 8 – 2.5 log (p^2F/T) where D and F are as stated above, p is the observer's pupil diameter in centimetres, and T is the telescope transmittance (e.g. 0.75 for a typical reflector).Crumey, A. Modelling the Visibility of Deep-Sky Objects. upright=1.6|thumb|Visual effect of night sky's brightness. The pupil magnification of an optical system is the ratio of the diameter of the exit pupil to the diameter of the entrance pupil. In the dark it will be the same at first, but will approach the maximum distance for a wide pupil 3 to 8 mm. thumb|The apparent position of a star viewed from the Earth depends on the Earth's velocity. The very darkest skies have a zenith surface brightness of approximately 22 mag arcsec−2, so it can be seen from the equation that such a sky would be expected to show stars approximately 0.4 mag fainter than one with a surface brightness of 21 mag arcsec−2. Crumey obtained a formula for N as a function of the sky surface brightness, telescope magnification, observer's eye pupil diameter and other parameters including the personal factor F discussed above. For example, the cat's slit pupil can change the light intensity on the retina 135-fold compared to 10-fold in humans. However, the limiting visibility is 7th magnitude for faint stars visible from dark rural areas located 200 kilometers from major cities. Crumey showed that for a sky background with surface brightness \mu_{sky} > 21 mag arcsec−2, the visual limit m could be expressed as: :m=0.4260\mu_{sky} –2.3650–2.5logF where F is a ""field factor"" specific to the observer and viewing situation.Crumey, op. cit., Eq. The image of the pupil as seen from outside the eye is the entrance pupil, which does not exactly correspond to the location and size of the physical pupil because it is magnified by the cornea. Peripheral Light Focusing (PLF) can be described as the focusing of Solar Ultraviolet Radiation (SUVR) at the nasal limbus of the cornea. Bowen did not record parameters such as his eye pupil diameter, naked-eye magnitude limit, or the extent of light loss in his telescopes; but because he made observations at a range of magnifications using three telescopes (with apertures 0.33 inch, 6 inch and 60 inch), Crumey was able to construct a system of simultaneous equations from which the remaining parameters could be deduced. A star's brightness (more precisely its illuminance) must exceed the sky's surface brightness (i.e. luminance) by a sufficient amount. In addition to dilation and contraction caused by light and darkness, it has been shown that solving simple multiplication problems affects the size of the pupil. The limiting magnitude will depend on the observer, and will increase with the eye's dark adaptation. From brightly lit Midtown Manhattan, the limiting magnitude is possibly 2.0, meaning that from the heart of New York City only approximately 15 stars will be visible at any given time. For example, at the peak age of 15, the dark- adapted pupil can vary from 4 mm to 9 mm with different individuals. This corresponds to roughly 250 visible stars, or one-tenth the number that can be perceived under perfectly dark skies. ",-1.78,30,"""4.4""",0.7812,0.54,C
 "When ultraviolet radiation of wavelength $58.4 \mathrm{~nm}$ from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of $1.59 \times 10^6 \mathrm{~m} \mathrm{~s}^{-1}$. Calculate the ionization energy of krypton.
-","The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. Krypton light has many spectral lines, and krypton plasma is useful in bright, high-powered gas lasers (krypton ion and excimer lasers), each of which resonates and amplifies a single spectral line. The average atmospheric concentration of krypton-85 was approximately 0.6 Bq/m3 in 1976, and has increased to approximately 1.3 Bq/m3 as of 2005. Krypton's concentration in the atmosphere is about 1 ppm. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The first measurements suggest an abundance of krypton in space. ==Applications== left|thumb|Krypton gas discharge tube Krypton's multiple emission lines make ionized krypton gas discharges appear whitish, which in turn makes krypton-based bulbs useful in photography as a white light source. In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. Krypton-85 (85Kr) is a radioisotope of krypton. The nth ionization energy refers to the amount of energy required to remove the most loosely bound electron from the species having a positive charge of (n − 1). There are two main ways in which ionization energy is calculated. The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or ""HOMO"" and the lowest unoccupied molecular orbital or ""LUMO"", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. The energy of these electrons that gives rise to a sharp onset of the current of ions and freed electrons through the tube will match the ionization energy of the atoms. == Atoms: values and trends == Generally, the (N+1)th ionization energy of a particular element is larger than the Nth ionization energy (it may also be noted that the ionization energy of an anion is generally less than that of cations and neutral atom for the same element). In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. In 1960, the International Bureau of Weights and Measures defined the meter as 1,650,763.73 wavelengths of light emitted in the vacuum corresponding to the transition between the 2p10 and 5d5 levels in the isotope krypton-86. Krypton-85 has a half-life of 10.756 years and a maximum decay energy of 687 keV. Most or all of this krypton-85 is retained in the spent nuclear fuel rods; spent fuel on discharge from a reactor contains between 0.13–1.8 PBq/Mg of krypton-85. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. This in turn makes its ionization energies increase by 18 kJ/mol−1. That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. The krypton-86 definition lasted until the October 1983 conference, which redefined the meter as the distance that light travels in vacuum during 1/299,792,458 s.Unit of length (meter), NIST ==Characteristics== Krypton is characterized by several sharp emission lines (spectral signatures) the strongest being green and yellow. Krypton, like the other noble gases, is used in lighting and photography. ",0.68,14,0.0245,-1.0,7,B
-" If $125 \mathrm{~cm}^3$ of hydrogen gas effuses through a small hole in 135 seconds, how long will it take the same volume of oxygen gas to effuse under the same temperature and pressure?","Here \Delta P is the gas pressure difference across the barrier, A is the area of the hole, N_A is the Avogadro constant, R is the gas constant and T is the absolute temperature. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. : {\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1} where M_1 and M_2 represent the molar masses of the gases. The effusion rate for a gas depends directly on the average velocity of its particles. The rate \Phi_N at which a gas of molar mass M effuses (typically expressed as the number of molecules passing through the hole per second) is thenPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W.H.Freeman 2006) p.756 : \Phi_N = \frac{\Delta PAN_A}{\sqrt{2\pi MRT}}. If a small area A on the container is punched to become a small hole, the effusive flow rate will be \begin{align} Q_\text{effusion} &= J_\text{impingement} \times A \\\ &= \frac{P A}{\sqrt{2 \pi m k_{B} T}} \\\ &= \frac{P A N_A}{\sqrt{2 \pi M R T}} \end{align} where M is the molar mass, N_A is the Avogadro constant, and R = N_A k_B is the gas constant. Assuming the pressure difference between the two sides of the barrier is much smaller than P_{\rm avg}, the average absolute pressure in the system (i.e. \Delta P\ll P_{\rm avg}), it is possible to express effusion flow as a volumetric flow rate as follows: :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi k_BT}{32m}} or :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi RT}{32M}} where \Phi_V is the volumetric flow rate of the gas, P_{\rm avg} is the average pressure on either side of the orifice, and d is the hole diameter. ==Effect of molecular weight== At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. ===Graham's law=== Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. The Lockman Hole is an area of the sky in which minimal amounts of neutral hydrogen gas are observed. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus Such a hole is often described as a pinhole and the escape of the gas is due to the pressure difference between the container and the exterior. In this region, the typical column density of neutral hydrogen is NH = 0.6 x 1020 cm−2. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules.K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18. Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. The number of atomic or molecular collisions with a wall of a container per unit area per unit time (impingement rate) is given by: J_\text{impingement} = \frac{P}{\sqrt{2 \pi m k_{B} T}}. assuming mean free path is much greater than pinhole diameter and the gas can be treated as an ideal gas. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. Pressure vessels for gas storage may also be classified by volume. Conversely, when the diameter is larger than the mean free path of the gas, flow obeys the Sampson flow law. Forming gas is used as an atmosphere for processes that need the properties of hydrogen gas. Under these conditions, essentially all molecules which arrive at the hole continue and pass through the hole, since collisions between molecules in the region of the hole are negligible. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. ", 258.14,0.0029,15.0,537,52,D
-The vibrational wavenumber of $\mathrm{Br}_2$ is $323.2 \mathrm{~cm}^{-1}$. Evaluate the vibrational partition function explicitly (without approximation) and plot its value as a function of temperature. At what temperature is the value within 5 per cent of the value calculated from the approximate formula?,"The vibrational temperature is used commonly when finding the vibrational partition function. In terms of the vibrational wavenumbers we can write the partition function as Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{ u}_j}{k_\text{B} T}} } It is convenient to define a characteristic vibrational temperature \Theta_{i,\text{vib}} = \frac{h u_i}{k_\text{B}} where u is experimentally determined for each vibrational mode by taking a spectrum or by calculation. Using this approximation we can derive a closed form expression for the vibrational partition function. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. ==Definition== For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_{j,n} is the energy of j-th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The vibrational partition functionDonald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. In the Central England Temperature series, dating back to 1659, at the time it was the 2nd hottest July on record, the hottest since 1783. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics For an isolated molecule of n atoms, the number of vibrational modes (i.e. values of j) is 3n − 5 for linear molecules and 3n − 6 for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons. ==Approximations== ===Quantum harmonic oscillator=== The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } where E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j is total vibrational zero point energy of the system. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of . By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} == References == ==See also== * Partition function (mathematics) Category:Partition functions The specific heat capacity has a sharp peak as the temperature approaches the lambda point. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. Often the wavenumber, \tilde{ u} with units of cm−1 is given instead of the angular frequency of a vibrational mode and also often misnamed frequency. The 1808 United Kingdom heat wave was a period of exceptionally high temperatures during July 1808. Given a solution of the heat equation, the value of for a small positive value of may be approximated as times the average value of the function over a sphere of very small radius centered at . ===Character of the solutions=== right|frame|Solution of a 1D heat partial differential equation. A quantum harmonic oscillator has an energy spectrum characterized by: E_{j,n} = \hbar\omega_j\left(n_j + \frac{1}{2}\right) where j runs over vibrational modes and n_j is the vibrational quantum number in the j-th mode, \hbar is Planck's constant, h, divided by 2 \pi and \omega_j is the angular frequency of the j'th mode. ",+37,432,-5.0,4500,-0.75,D
+","The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. Krypton light has many spectral lines, and krypton plasma is useful in bright, high-powered gas lasers (krypton ion and excimer lasers), each of which resonates and amplifies a single spectral line. The average atmospheric concentration of krypton-85 was approximately 0.6 Bq/m3 in 1976, and has increased to approximately 1.3 Bq/m3 as of 2005. Krypton's concentration in the atmosphere is about 1 ppm. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The first measurements suggest an abundance of krypton in space. ==Applications== left|thumb|Krypton gas discharge tube Krypton's multiple emission lines make ionized krypton gas discharges appear whitish, which in turn makes krypton-based bulbs useful in photography as a white light source. In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. Krypton-85 (85Kr) is a radioisotope of krypton. The nth ionization energy refers to the amount of energy required to remove the most loosely bound electron from the species having a positive charge of (n − 1). There are two main ways in which ionization energy is calculated. The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or ""HOMO"" and the lowest unoccupied molecular orbital or ""LUMO"", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. The energy of these electrons that gives rise to a sharp onset of the current of ions and freed electrons through the tube will match the ionization energy of the atoms. == Atoms: values and trends == Generally, the (N+1)th ionization energy of a particular element is larger than the Nth ionization energy (it may also be noted that the ionization energy of an anion is generally less than that of cations and neutral atom for the same element). In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. In 1960, the International Bureau of Weights and Measures defined the meter as 1,650,763.73 wavelengths of light emitted in the vacuum corresponding to the transition between the 2p10 and 5d5 levels in the isotope krypton-86. Krypton-85 has a half-life of 10.756 years and a maximum decay energy of 687 keV. Most or all of this krypton-85 is retained in the spent nuclear fuel rods; spent fuel on discharge from a reactor contains between 0.13–1.8 PBq/Mg of krypton-85. thumb|right|Final amplifier of the Nike laser where laser beam energy is increased from 150 J to ~5 kJ by passing through a krypton/fluorine/argon gas mixture excited by irradiation with two opposing 670,000 volt electron beams. This in turn makes its ionization energies increase by 18 kJ/mol−1. That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. The krypton-86 definition lasted until the October 1983 conference, which redefined the meter as the distance that light travels in vacuum during 1/299,792,458 s.Unit of length (meter), NIST ==Characteristics== Krypton is characterized by several sharp emission lines (spectral signatures) the strongest being green and yellow. Krypton, like the other noble gases, is used in lighting and photography. ",0.68,14,"""0.0245""",-1.0,7,B
+" If $125 \mathrm{~cm}^3$ of hydrogen gas effuses through a small hole in 135 seconds, how long will it take the same volume of oxygen gas to effuse under the same temperature and pressure?","Here \Delta P is the gas pressure difference across the barrier, A is the area of the hole, N_A is the Avogadro constant, R is the gas constant and T is the absolute temperature. In other words, the ratio of the rates of effusion of two gases at the same temperature and pressure is given by the inverse ratio of the square roots of the masses of the gas particles. : {\mbox{Rate of effusion of gas}_1 \over \mbox{Rate of effusion of gas}_2}=\sqrt{M_2 \over M_1} where M_1 and M_2 represent the molar masses of the gases. The effusion rate for a gas depends directly on the average velocity of its particles. The rate \Phi_N at which a gas of molar mass M effuses (typically expressed as the number of molecules passing through the hole per second) is thenPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W.H.Freeman 2006) p.756 : \Phi_N = \frac{\Delta PAN_A}{\sqrt{2\pi MRT}}. If a small area A on the container is punched to become a small hole, the effusive flow rate will be \begin{align} Q_\text{effusion} &= J_\text{impingement} \times A \\\ &= \frac{P A}{\sqrt{2 \pi m k_{B} T}} \\\ &= \frac{P A N_A}{\sqrt{2 \pi M R T}} \end{align} where M is the molar mass, N_A is the Avogadro constant, and R = N_A k_B is the gas constant. Assuming the pressure difference between the two sides of the barrier is much smaller than P_{\rm avg}, the average absolute pressure in the system (i.e. \Delta P\ll P_{\rm avg}), it is possible to express effusion flow as a volumetric flow rate as follows: :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi k_BT}{32m}} or :\Phi_V=\frac{\Delta P d^2}{P_{\rm avg}}\sqrt{\frac{\pi RT}{32M}} where \Phi_V is the volumetric flow rate of the gas, P_{\rm avg} is the average pressure on either side of the orifice, and d is the hole diameter. ==Effect of molecular weight== At constant pressure and temperature, the root-mean-square speed and therefore the effusion rate are inversely proportional to the square root of the molecular weight. Gases with a lower molecular weight effuse more rapidly than gases with a higher molecular weight, so that the number of lighter molecules passing through the hole per unit time is greater. ===Graham's law=== Scottish chemist Thomas Graham (1805–1869) found experimentally that the rate of effusion of a gas is inversely proportional to the square root of the mass of its particles. The Lockman Hole is an area of the sky in which minimal amounts of neutral hydrogen gas are observed. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus Such a hole is often described as a pinhole and the escape of the gas is due to the pressure difference between the container and the exterior. In this region, the typical column density of neutral hydrogen is NH = 0.6 x 1020 cm−2. In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules.K.J. Laidler and J.H. Meiser, Physical Chemistry, Benjamin/Cummings 1982, p.18. Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. The number of atomic or molecular collisions with a wall of a container per unit area per unit time (impingement rate) is given by: J_\text{impingement} = \frac{P}{\sqrt{2 \pi m k_{B} T}}. assuming mean free path is much greater than pinhole diameter and the gas can be treated as an ideal gas. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. Pressure vessels for gas storage may also be classified by volume. Conversely, when the diameter is larger than the mean free path of the gas, flow obeys the Sampson flow law. Forming gas is used as an atmosphere for processes that need the properties of hydrogen gas. Under these conditions, essentially all molecules which arrive at the hole continue and pass through the hole, since collisions between molecules in the region of the hole are negligible. Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. ", 258.14,0.0029,"""15.0""",537,52,D
+The vibrational wavenumber of $\mathrm{Br}_2$ is $323.2 \mathrm{~cm}^{-1}$. Evaluate the vibrational partition function explicitly (without approximation) and plot its value as a function of temperature. At what temperature is the value within 5 per cent of the value calculated from the approximate formula?,"The vibrational temperature is used commonly when finding the vibrational partition function. In terms of the vibrational wavenumbers we can write the partition function as Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{ u}_j}{k_\text{B} T}} } It is convenient to define a characteristic vibrational temperature \Theta_{i,\text{vib}} = \frac{h u_i}{k_\text{B}} where u is experimentally determined for each vibrational mode by taking a spectrum or by calculation. Using this approximation we can derive a closed form expression for the vibrational partition function. The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. ==Definition== For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_{j,n} is the energy of j-th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The vibrational partition functionDonald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 traditionally refers to the component of the canonical partition function resulting from the vibrational degrees of freedom of a system. In the Central England Temperature series, dating back to 1659, at the time it was the 2nd hottest July on record, the hottest since 1783. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics For an isolated molecule of n atoms, the number of vibrational modes (i.e. values of j) is 3n − 5 for linear molecules and 3n − 6 for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons. ==Approximations== ===Quantum harmonic oscillator=== The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } where E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j is total vibrational zero point energy of the system. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of . By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} == References == ==See also== * Partition function (mathematics) Category:Partition functions The specific heat capacity has a sharp peak as the temperature approaches the lambda point. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. Often the wavenumber, \tilde{ u} with units of cm−1 is given instead of the angular frequency of a vibrational mode and also often misnamed frequency. The 1808 United Kingdom heat wave was a period of exceptionally high temperatures during July 1808. Given a solution of the heat equation, the value of for a small positive value of may be approximated as times the average value of the function over a sphere of very small radius centered at . ===Character of the solutions=== right|frame|Solution of a 1D heat partial differential equation. A quantum harmonic oscillator has an energy spectrum characterized by: E_{j,n} = \hbar\omega_j\left(n_j + \frac{1}{2}\right) where j runs over vibrational modes and n_j is the vibrational quantum number in the j-th mode, \hbar is Planck's constant, h, divided by 2 \pi and \omega_j is the angular frequency of the j'th mode. ",+37,432,"""-5.0""",4500,-0.75,D
 "A thermodynamic study of $\mathrm{DyCl}_3$ (E.H.P. Cordfunke, et al., J. Chem. Thermodynamics 28, 1387 (1996)) determined its standard enthalpy of formation from the following information
 (1) $\mathrm{DyCl}_3(\mathrm{~s}) \rightarrow \mathrm{DyCl}_3(\mathrm{aq}$, in $4.0 \mathrm{M} \mathrm{HCl}) \quad \Delta_{\mathrm{r}} H^{\ominus}=-180.06 \mathrm{~kJ} \mathrm{~mol}^{-1}$
 (2) $\mathrm{Dy}(\mathrm{s})+3 \mathrm{HCl}(\mathrm{aq}, 4.0 \mathrm{~m}) \rightarrow \mathrm{DyCl}_3(\mathrm{aq}$, in $4.0 \mathrm{M} \mathrm{HCl}(\mathrm{aq}))+\frac{3}{2} \mathrm{H}_2(\mathrm{~g})$ $\Delta_{\mathrm{r}} H^{\ominus}=-699.43 \mathrm{~kJ} \mathrm{~mol}^{-1}$
 (3) $\frac{1}{2} \mathrm{H}_2(\mathrm{~g})+\frac{1}{2} \mathrm{Cl}_2(\mathrm{~g}) \rightarrow \mathrm{HCl}(\mathrm{aq}, 4.0 \mathrm{M}) \quad \Delta_{\mathrm{r}} H^{\ominus}=-158.31 \mathrm{~kJ} \mathrm{~mol}^{-1}$
-Determine $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{DyCl}_3, \mathrm{~s}\right)$ from these data.","Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by : M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T and : c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. ==Definitions== The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 : \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} : c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} where : t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} measures the temperature relative to the critical point. ==Derivation== For the magnetic analogue of the Maxwell relations for the response functions, the relation : \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 follows, and with thermodynamic stability requiring that c_H, c_M\mbox{ and }\chi_T \geq 0 , one has : c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 which, under the conditions H=0, t>0 and the definition of the critical exponents gives : (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} which gives the Rushbrooke inequality : \alpha' + 2\beta + \gamma' \geq 2. This page provides supplementary chemical data on Ytterbium(III) chloride == Structure and properties data == Structure and properties Standard reduction potential, E° 1.620 eV Crystallographic constants a = 6.73 b = 11.65 c = 6.38 β = 110.4° Effective nuclear charge 3.290 Bond strength 1194±7 kJ/mol Bond length 2.434 (Yb-Cl) Bond angle 111.5° (Cl-Yb-Cl) Magnetic susceptibility 4.4 μB == Thermodynamic properties == Phase behavior Std enthalpy change of fusionΔfusH ~~o~~ 58.1±11.6 kJ/mol Std entropy change of fusionΔfusS ~~o~~ 50.3±10.1 J/(mol•K) Std enthalpy change of atomizationΔatH ~~o~~ 1166.5±4.3 kJ/mol Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −959.5±3.0 kJ/mol Standard molar entropy S ~~o~~ solid 163.5 J/(mol•K) Heat capacity cp 101.4 J/(mol•K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid −212.8 kJ/mol Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas −651.1±5.0 kJ/mol == Spectral data == UV-Vis λmax 268 nm Extinction coefficient 303 M−1cm−1 IR Major absorption bands ν1 = 368.0 cm−1 ν2 = 178.4 cm−1 ν3 = 330.7 cm−1 ν4 = 117.8 cm−1 MS Ionization potentials from electron impact YbCl3+ = 10.9±0.1 eV YbCl2+ = 11.6±0.1 eV YbCl+ = 14.3±0.1 eV ==References== Category:Chemical data pages Category:Chemical data pages cleanup J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. * J.A. Dean (ed) in Lange's Handbook of Chemistry, McGraw-Hill, New York, USA, 14th edition, 1992. This page provides supplementary chemical data on gold(III) chloride == Thermodynamic properties == Phase behavior Triple point ? It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. This page provides supplementary chemical data on Lutetium(III) oxide == Thermodynamic properties == Phase behavior Triple point ? ",0.318,-2.99,122.0,449,-994.3,E
-"Calculate $\Delta_{\mathrm{r}} G^{\ominus}(375 \mathrm{~K})$ for the reaction $2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})$ from the values of $\Delta_{\mathrm{r}} G^{\ominus}(298 \mathrm{~K})$ : and $\Delta_{\mathrm{r}} H^{\ominus}(298 \mathrm{~K})$, and the GibbsHelmholtz equation.","For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. The molecular formula C17H14N2O2 (molar mass: 278.305 g/mol, exact mass: 278.1055 u) may refer to: * Bimakalim * Sudan Red G Category:Molecular formulas The molecular formula C16H10N2Na2O7S2 (molar mass: 452.369 g/mol) may refer to: * Orange G * Orange GGN * Sunset Yellow FCF Category:Molecular formulas The molecular formula C18H26O (molar mass: 258.40 g/mol, exact mass: 258.1984 u) may refer to: * Galaxolide (HHCB) * Xibornol The molecular formula C12H22O10 (molar mass: 326.29 g/mol, exact mass: 326.121297 u) may refer to: * Neohesperidose or 2-O-alpha-L-Rhamnopyranosyl-D- glucopyranose * Robinose * Rutinose or 6-O-alpha-L-Rhamnopyranosyl-D- glucupyranose A closely related technique is the use of an electroanalytical voltaic cell, which can be used to measure the Gibbs energy for certain reactions as a function of temperature, yielding K_\mathrm{eq}(T) and thereby \Delta_{\text {rxn}} H^\ominus . Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., ""Physical Chemistry"" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., ""Atkins' Physical Chemistry"" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). This method is based on Hess's law, which states that the enthalpy change is the same for a chemical reaction which occurs as a single reaction or in several steps. If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. This method is only approximate, however, because a reported bond energy is only an average value for different molecules with bonds between the same elements.Petrucci, Harwood and Herring, pages 422–423 ==References== Category:Enthalpy Category:Thermochemistry Category:Thermodynamics pl:Standardowe molowe ciepło tworzenia Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. One large class of reactions for which such measurements are common is the combustion of organic compounds by reaction with molecular oxygen (O2) to form carbon dioxide and water (H2O). We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions. ",+93.4,-501,0.000226,12,+2.9,B
-"The vapour pressure of benzene is $53.3 \mathrm{kPa}$ at $60.6^{\circ} \mathrm{C}$, but it fell to $51.5 \mathrm{kPa}$ when $19.0 \mathrm{~g}$ of an non-volatile organic compound was dissolved in $500 \mathrm{~g}$ of benzene. Calculate the molar mass of the compound.","The molecular formula C25H25NO4 (molar mass: 403.47 g/mol, exact mass: 403.1784 u) may refer to: * Benzhydrocodone * 7-Spiroindanyloxymorphone (SIOM) The molecular formula C25H25NO (molar mass: 355.47 g/mol, exact mass: 355.1936 u) may refer to: * JWH-007 * JWH-019 * JWH-047 * JWH-122 Category:Molecular formulas The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C25H27N (molar mass: 341.49 g/mol, exact mass: 341.2143 u) may refer to: * JWH-184 * JWH-196 The molecular formula C2H6ClO2PS (molar mass: 160.56 g/mol, exact mass: 159.9515 u) may refer to: * Dimethyl chlorothiophosphate * Dimethyl phosphorochloridothioate The molecular formula C24H23NO (molar mass: 341.44 g/mol, exact mass: 341.1780 u) may refer to: * JWH-018, also known as 1-pentyl-3-(1-naphthoyl)indole or AM-678 * JWH-148 Category:Molecular formulas The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 ",5.5,0.5,85.0,3.38,13,C
-"J.G. Dojahn, et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. The ground state of $\mathrm{F}_2^{-}$is ${ }^2 \sum_{\mathrm{u}}^{+}$with a fundamental vibrational wavenumber of $450.0 \mathrm{~cm}^{-1}$ and equilibrium internuclear distance of $190.0 \mathrm{pm}$. The first two excited states are at 1.609 and $1.702 \mathrm{eV}$ above the ground state. Compute the standard molar entropy of $\mathrm{F}_2^{-}$at $298 \mathrm{~K}$.","Constants of Diatomic Molecules, by K. P. Huber and Gerhard Herzberg (Van nostrand Reinhold company, New York, 1979, ), is a classic comprehensive multidisciplinary reference text contains a critical compilation of available data for all diatomic molecules and ions known at the time of publication - over 900 diatomic species in all - including electronic energies, vibrational and rotational constants, and observed transitions. Extensive footnotes discuss the reliability of these data and additional detailed informationon potential energy curves, spin- coupling constants, /\\-type doubling, perturbations between electronic states, hyperfine structure, rotational g factors, dipole moments, radiative lifetimes, oscillator strengths, dissociation energies and ionization potentials when available, and other aspects. {\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}\\!| cdf =\Gamma\\!\left(\frac{ u}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\\!\left(\frac{ u}{2}\right)\\!| mean =\frac{1}{ u-2}\\! for u >2\\!| median = \approx \dfrac{1}{ u\bigg(1-\dfrac{2}{9 u}\bigg)^3}| mode =\frac{1}{ u+2}\\!| variance =\frac{2}{( u-2)^2 ( u-4)}\\! for u >4\\!| skewness =\frac{4}{ u-6}\sqrt{2( u-4)}\\! for u >6\\!| kurtosis =\frac{12(5 u-22)}{( u-6)( u-8)}\\! for u >8\\!| entropy =\frac{ u}{2} \\!+\\!\ln\\!\left(\frac{ u}{2}\Gamma\\!\left(\frac{ u}{2}\right)\right) \\!-\\!\left(1\\!+\\!\frac{ u}{2}\right)\psi\\!\left(\frac{ u}{2}\right)| mgf =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-t}{2i}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2t}\right); does not exist as real valued function| char =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-it}{2}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2it}\right)| }} In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-valued random variable. The Rydberg–Klein–Rees method is a procedure used in the analysis of rotational-vibrational spectra of diatomic molecules to obtain a potential energy curve from the experimentally-known line positions. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10���2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10��7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. :From this source with some modifications and additions of later data: :*W.S. Fyfe, Geochemistry, Oxford University Press, (1974). Molecular Spectra and Molecular Structure IV. Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. In particular for integer valued degrees of freedom u we have: For u >1 even, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 5 \cdot 3} { 2 \sqrt{ u}( u -2)( u -4)\cdots 4 \cdot 2\,}\cdot For u >1 odd, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 4 \cdot 2} {\pi \sqrt{ u}( u -2)( u -4)\cdots 5 \cdot 3\,}\cdot\\! For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. Geometrically frustrated systems in particular often exhibit residual entropy. (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). A great deal of research has thus been undertaken into finding other systems that exhibit residual entropy. One of the interesting properties of geometrically frustrated magnetic materials such as spin ice is that the level of residual entropy can be controlled by the application of an external magnetic field. An alternative formula, valid for t^2 < u, is :\int_{-\infty}^t f(u)\,du = \tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}( u+1) \right)} {\sqrt{\pi u}\,\Gamma \left(\tfrac{ u}{2}\right)} \, {}_2F_1 \left( \tfrac{1}{2}, \tfrac{1}{2}( u+1); \tfrac{3}{2}; -\tfrac{t^2}{ u} \right), where 2F1 is a particular case of the hypergeometric function. This material is thus analogous to water ice, with the exception that the spins on the corners of the tetrahedra can point into or out of the tetrahedra, thereby producing the same 2-in, 2-out rule as in water ice, and therefore the same residual entropy. However, it turns out that for a large number of water molecules in this configuration, the hydrogen atoms have a large number of possible configurations that meet the 2-in 2-out rule (each oxygen atom must have two 'near' (or 'in') hydrogen atoms, and two far (or 'out') hydrogen atoms). The first definition yields a probability density function given by : f_1(x; u) = \frac{2^{- u/2}}{\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}, while the second definition yields the density function : f_2(x; u) = \frac{( u/2)^{ u/2}}{\Gamma( u/2)} x^{- u/2-1} e^{- u/(2 x)} . ",0.405,-30,199.4,6.6,−1.642876,C
-"The duration of a $90^{\circ}$ or $180^{\circ}$ pulse depends on the strength of the $\mathscr{B}_1$ field. If a $180^{\circ}$ pulse requires $12.5 \mu \mathrm{s}$, what is the strength of the $\mathscr{B}_1$ field? ","This minimum value depends on the definition used for the duration and on the shape of the pulse. The intensity functions—temporal I(t) and spectral S(\omega) —determine the time duration and spectrum bandwidth of the pulse. A pulsed field gradient is a short, timed pulse with spatial-dependent field intensity. The interval between the 50% points of the final amplitude is usually used to determine or define pulse duration, and this is understood to be the case unless otherwise specified. 500px|thumb|right|The duration-bandwidth product depends on the shape of the power spectrum of the pulse. PSR B1919+21 is a pulsar with a period of 1.3373 seconds and a pulse width of 0.04 seconds. For different pulse shapes, the minimum duration-bandwidth product is different. right|thumb|300px|Pulse duration using 50% peak amplitude. thumb|300px|DECT phone pulduration measurement (100 Hz / 10 mS) on channel 8 In signal processing and telecommunication, pulse duration is the interval between the time, during the first transition, that the amplitude of the pulse reaches a specified fraction (level) of its final amplitude, and the time the pulse amplitude drops, on the last transition, to the same level. Other fractions of the final amplitude, e.g., 90% or 1/e, may also be used, as may the root mean square (rms) value of the pulse amplitude. In radar, the pulse duration is the time the radar's transmitter is energized during each cycle. ==References== * * Category:Signal processing Category:Telecommunication theory The length of a pulse thereby is determined by its complex spectral components, which include not just their relative intensities, but also the relative positions (spectral phase) of these spectral components. For example, \mathrm{sech^2} pulses have a minimum duration-bandwidth product of 0.315 while gaussian pulses have a minimum value of 0.441. Usually the attribute 'ultrashort' applies to pulses with a duration of a few tens of femtoseconds, but in a larger sense any pulse which lasts less than a few picoseconds can be considered ultrashort. Intensity autocorrelation gives the pulse width when a particular pulse shape is assumed. The signal had a -second period (not in 1967, but in 1991) and 0.04-second pulsewidth. For a given spectrum, the minimum time-bandwidth product, and therefore the shortest pulse, is obtained by a transform-limited pulse, i.e., for a constant spectral phase \phi(\omega) . In the specialized literature, ""ultrashort"" refers to the femtosecond (fs) and picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses artificially generated. The terms in \gamma_x and \gamma_y describe the walk-off of the pulse; the coefficient \gamma_x ~ (\gamma_y ) is the ratio of the component of the group velocity x ~ (y) and the unit vector in the direction of propagation of the pulse (z-axis). In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. A pulse shaper can be used to make more complicated alterations on both the phase and the amplitude of ultrashort pulses. There is no standard definition of ultrashort pulse. A bandwidth-limited pulse (also known as Fourier- transform-limited pulse, or more commonly, transform-limited pulse) is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. ",5.9,-1.0,1.95,2,0.44,A
-"In 1976 it was mistakenly believed that the first of the 'superheavy' elements had been discovered in a sample of mica. Its atomic number  was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)","* Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913). The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. In this case, it is the poor shielding capacity of the 3d-electrons which affects the atomic radii and chemistries of the elements immediately following the first row of the transition metals, from gallium (Z = 31) to bromine (Z = 35). ==Calculated atomic radius== The following table shows atomic radii computed from theoretical models, as published by Enrico Clementi and others in 1967. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. These dimensions are much smaller than the diameter of the atom itself (nucleus + electron cloud), by a factor of about 26,634 (uranium atomic radius is about ())""Uranium"" IDC Technologies. to about 60,250 (hydrogen atomic radius is about ).26,634 derives from x / ; 60,250 derives from x / The branch of physics concerned with the study and understanding of the atomic nucleus, including its composition and the forces that bind it together, is called nuclear physics. ==Introduction== ===History=== The nucleus was discovered in 1911, as a result of Ernest Rutherford's efforts to test Thomson's ""plum pudding model"" of the atom. The elements immediately following the lanthanides have atomic radii which are smaller than would be expected and which are almost identical to the atomic radii of the elements immediately above them. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Accuracy of ±5 pm.The way the atomic radius varies with increasing atomic number can be explained by the arrangement of electrons in shells of fixed capacity. Similarly, the distance from shell-closure explains the unusual instability of isotopes which have far from stable numbers of these particles, such as the radioactive elements 43 (technetium) and 61 (promethium), each of which is preceded and followed by 17 or more stable elements. HE 1327-2326, discovered in 2005 by Anna Frebel and collaborators, was the star with the lowest known iron abundance until SMSS J031300.36−670839.3 was discovered. These trends of the atomic radii (and of various other chemical and physical properties of the elements) can be explained by the electron shell theory of the atom; they provided important evidence for the development and confirmation of quantum theory. ==Atomic radius== Note: All measurements given are in picometers (pm). The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. Under most definitions the radii of isolated neutral atoms range between 30 and 300 pm (trillionths of a meter), or between 0.3 and 3 ångströms. Bohemium was the name assigned to the element with atomic number 93, now known as neptunium, when its discovery was first incorrectly alleged. The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Although the model itself is now obsolete, the Bohr radius for the hydrogen atom is still regarded as an important physical constant. ==Empirically measured atomic radius== The following table shows empirically measured covalent radii for the elements, as published by J. C. Slater in 1964. Therefore, the radius of an atom is more than 10,000 times the radius of its nucleus (1–10 fm), and less than 1/1000 of the wavelength of visible light (400–700 nm). The largest known completely stable nucleus (i.e. stable to alpha, beta, and gamma decay) is lead-208 which contains a total of 208 nucleons (126 neutrons and 82 protons). Such predictions are especially useful for elements whose radii cannot be measured experimentally (e.g. those that have not been discovered, or that have too short of a half- life). ==References== Category:Atomic radius Category:Properties of chemical elements The value of the radius may depend on the atom's state and context. Data derived from other sources with different assumptions cannot be compared. * † to an accuracy of about 5 pm * (b) 12 coordinate * (c) gallium has an anomalous crystal structure * (d) 10 coordinate * (e) uranium, neptunium and plutonium have irregular structures *Triple bond mean-square deviation 3pm. ==References== Data is as quoted at http://www.webelements.com/ from these sources: ===Covalent radii (single bond)=== * * * * * ===Metallic radius=== Category:Properties of chemical elements Category:Chemical element data pages Category:Atomic radius ",313,311875200,0.42,2.24,0,C
-The ground level of $\mathrm{Cl}$ is ${ }^2 \mathrm{P}_{3 / 2}$ and a ${ }^2 \mathrm{P}_{1 / 2}$ level lies $881 \mathrm{~cm}^{-1}$ above it. Calculate the electronic contribution to the molar Gibbs energy of $\mathrm{Cl}$ atoms at  $500 \mathrm{~K}$.,"Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . These tables list values of molar ionization energies, measured in kJ⋅mol−1. :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: :\mathrm{d}\mathbf{G} =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The first molar ionization energy applies to the neutral atoms. If multiple phases of matter are present, the chemical potentials across a phase boundary are equal.Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium (i.e. that the temperature and pressure is constant throughout the system), we recover the Gibbs' phase rule. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. When pressure and temperature are variable, only I-1 of I components have independent values for chemical potential and Gibbs' phase rule follows. One particularly useful expression arises when considering binary solutions.The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, At constant P (isobaric) and T (isothermal) it becomes: :0= N_1 \mathrm{d}\mu_1 + N_2 \mathrm{d}\mu_2 or, normalizing by total number of moles in the system N_1 + N_2, substituting in the definition of activity coefficient \gamma and using the identity x_1 + x_2 = 1 : :0= x_1 \mathrm{d}\ln(\gamma_1) + x_2 \mathrm{d}\ln(\gamma_2) Chemical Thermodynamics of Materials, 2004 Svein Stølen, p. 79, This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for the vapor pressure of a fluid mixture from limited experimental data. == Ternary and multicomponent solutions and mixtures== Lawrence Stamper Darken has shown that the Gibbs-Duhem equation can be applied to the determination of chemical potentials of components from a multicomponent system from experimental data regarding the chemical potential \bar {G_2} of only one component (here component 2) at all compositions. The law can also be written as a function of the total number of atoms N in the sample: :C/N = 3k_{\rm B}, where kB is Boltzmann constant. ==Application limits== thumb|upright=2.2|The molar heat capacity plotted of most elements at 25°C plotted as a function of atomic number. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. In modern terms, Dulong and Petit found that the heat capacity of a mole of many solid elements is about 3R, where R is the universal gas constant. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained. ",3,96.4365076099,-6.42,7.00,0.0408,C
-Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately $0.92 \mathrm{~g} \mathrm{~cm}^{-3}$ and that of liquid water is $1.00 \mathrm{~g} \mathrm{~cm}^{-3}$.,"The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. The pressure melting point of ice is the temperature at which ice melts at a given pressure. thumb|Phase diagram of water Ice VI is a form of ice that exists at high pressure at the order of about 1 GPa (= 10 000 bar) and temperatures ranging from 130 up to 355 Kelvin (−143 °C up to 82 °C); see also the phase diagram of water. As the pressure increases with depth in a glacier from the weight of the ice above, the pressure melting point of ice decreases within bounds, as shown in the diagram. The level where ice can start melting is where the pressure melting point equals the actual temperature. The triple point of ice VI with ice VII and liquid water is at about 82 °C and 2.22 GPa and its triple point with ice V and liquid water is at 0.16 °C and 0.6324 GPa = 6324 bar.Water Phase Diagram www1.lsbu.ac.uk, version of 9 September 2019, retrieved 3 October 2019 Ice VI undergoes phase transitions into ices XV and XIX upon cooling depending on pressure as hydrochloric acid is doped. == See also == * Ice phases (overview) == References == == External links == * Physik des Eises (PDF in German, iktp.tu-dresden.de) * Ice phases (www.idc- online.com) Category:Water ice With increasing pressure above 10 MPa, the pressure melting point decreases to a minimum of −21.9 °C at 209.9 MPa. Thereafter, the pressure melting point rises rapidly with pressure, passing back through 0 °C at 632.4 MPa. ==Pressure melting point in glaciers== Glaciers are subject to geothermal heat flux from below and atmospheric warming or cooling from above. Water vapor (H2O) 200px Liquid state Water Solid state Ice Properties Molecular formula H2O Molar mass 18.01528(33) g/mol Melting point Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C) and boils at 373.1339 K (99.9839 °C) Boiling point specific gas constant 461.5 J/(kg·K) Heat of vaporization 2.27 MJ/kg Heat capacity 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous phase of water. A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). Humidity ranges from 0 grams per cubic metre in dry air to 30 grams per cubic metre (0.03 ounce per cubic foot) when the vapor is saturated at 30 °C. === Sublimation === Sublimation is the process by which water molecules directly leave the surface of ice without first becoming liquid water. The temperature range that is determined can then be averaged to gain the melting point of the sample being examined. Melting occurs on both the top and the bottom of the ice. Ice X, within physical chemistry, is a cubic crystalline form of ice formed in the same manner as ice VII, but at pressures as high as about 70 GPa. The formula is valid from about −50 to 102 °C; however there are a very limited number of measurements of the vapor pressure of water over supercooled liquid water. It would also be the level of the base of an ice shelf, or the ice-water interface of a subglacial lake. ==References== Category:Glaciology To estimate ice area, scientists calculate the percentage of sea ice in each pixel, multiply by the pixel area, and total the amounts. Sea ice extent is the area of sea with a specified amount of ice, usually 15%. thumb|A Fisher–Johns apparatus A melting-point apparatus is a scientific instrument used to determine the melting point of a substance. Sea ice rejects salt over time and becomes less salty resulting in a higher melting point. If there is a net increase of heat, then the ice will thin. In static equilibrium conditions, this would be the highest level where water can exist in a glacier. ",0.318, -31.95,6.3,272.8,7,D
-What is the temperature of a two-level system of energy separation equivalent to $400 \mathrm{~cm}^{-1}$ when the population of the upper state is one-third that of the lower state?,"At equilibrium, only a thermally isolating boundary can support a temperature difference. ==See also== * Closed system * Dynamical system * Mechanically isolated system * Open system * Thermodynamic system * Isolated system ==References== Category:Thermodynamic systems Equilibrium isotope fractionation is the partial separation of isotopes between two or more substances in chemical equilibrium. thumb|right|250 px|Vertical cross-section of a thermal low Thermal lows, or heat lows, are non-frontal low-pressure areas that occur over the continents in the subtropics during the warm season, as the result of intense heating when compared to their surrounding environments.Glossary of Meteorology (2009). thumb|The approximate temperature in the solar atmosphere plotted against height The solar transition region is a region of the Sun's atmosphere between the upper chromosphere and corona. The vibrational temperature is used commonly when finding the vibrational partition function. This is a collection of temperature conversion formulas and comparisons among eight different temperature scales, several of which have long been obsolete. The opposite of a thermally isolated system is a thermally open system, which allows the transfer of heat energy and entropy. Thermally open systems may vary, however, in the rate at which they equilibrate, depending on the nature of the boundary of the open system. Thermal lows which develop near sea level can build in height during the warm season, or summer, to the elevation of the 700 hPa pressure surface,David R. Rowson and Stephen J. Colucci (1992). At equilibrium, the temperatures on both sides of a thermally open boundary are equal. Thermal Low. Some numbers in this table have been rounded. === Graphical representation === :File:Comparison of temperature scales blank.svg|845x580px| circle 40 330 4 0 K / 0 °R (−273.15 °C) circle 504 330 4 0 °F (−17.78 °C) circle 537 270 4 150 °D circle 537 317 4 32 °F circle 537 325 4 7.5 °Rø circle 537 332 4 0 °C / 0 °Ré / 0 °N circle 718 245 4 212 °F circle 718 290 4 100 °C circle 718 298 4 80 °Ré circle 718 306 4 60 °Rø circle 718 317 4 33 °N circle 718 330 4 0 °D rect 0 0 845 580 :File:Comparison of temperature scales blank.svg desc none Rankine (°R) Kelvin (K) Fahrenheit (°F) Celsius (°C) Réaumur (°Ré) Rømer (°Rø) Newton (°N) Delisle (°D) Absolute zero Lowest recorded surface temperature on Earth Fahrenheit's ice/water/salt mixture Melting point of ice (at standard pressure) Average surface temperature on Earth (15 °C) Average human body temperature (37 °C) Highest recorded surface temperature on Earth Boiling point of water (at standard pressure) ==Conversion table between the different temperature units== ==See also== * Degree of frost * Conversion of units * Gas mark == Notes and references== Category:Scales of temperature Category:Conversion of units of measurement Temperatures on scales that either do not share a numeric zero or are nonlinearly related cannot correctly be mathematically equated (related using the symbol =), and thus temperatures on different scales are more correctly described as corresponding (related using the symbol ≘). == Celsius scale == == Kelvin scale == == Fahrenheit scale == == Rankine scale == == Delisle scale == == Sir Isaac Newton's degree of temperature == == Réaumur scale == == Rømer scale == ==Comparison values chart== Celsius Fahrenheit Kelvin Rankine Delisle Newton Réaumur Rømer 500.00 932.00 773.15 1391.67 −600.00 165.00 400.00 270.00 490.00 914.00 763.15 1373.67 −585.00 161.70 392.00 264.75 480.00 896.00 753.15 1355.67 −570.00 158.40 384.00 259.50 470.00 878.00 743.15 1337.67 −555.00 155.10 376.00 254.25 460.00 860.00 733.15 1319.67 −540.00 151.80 368.00 249.00 450.00 842.00 723.15 1301.67 −525.00 148.50 360.00 243.75 440.00 824.00 713.15 1283.67 −510.00 145.20 352.00 238.50 430.00 806.00 703.15 1265.67 −495.00 141.90 344.00 233.25 420.00 788.00 693.15 1247.67 −480.00 138.60 336.00 228.00 410.00 770.00 683.15 1229.67 −465.00 135.30 328.00 222.75 400.00 752.00 673.15 1211.67 −450.00 132.00 320.00 217.50 390.00 734.00 663.15 1193.67 −435.00 128.70 312.00 212.25 380.00 716.00 653.15 1175.67 −420.00 125.40 304.00 207.00 370.00 698.00 643.15 1157.67 −405.00 122.10 296.00 201.75 360.00 680.00 633.15 1139.67 −390.00 118.80 288.00 196.50 350.00 662.00 623.15 1121.67 −375.00 115.50 280.00 191.25 340.00 644.00 613.15 1103.67 −360.00 112.20 272.00 186.00 330.00 626.00 603.15 1085.67 −345.00 108.90 264.00 180.75 320.00 608.00 593.15 1067.67 −330.00 105.60 256.00 175.50 310.00 590.00 583.15 1049.67 −315.00 102.30 248.00 170.25 300.00 572.00 573.15 1031.67 −300.00 99.00 240.00 165.00 290.00 554.00 563.15 1013.67 −285.00 95.70 232.00 159.75 280.00 536.00 553.15 995.67 −270.00 92.40 224.00 154.50 270.00 518.00 543.15 977.67 −255.00 89.10 216.00 149.25 260.00 500.00 533.15 959.67 −240.00 85.80 208.00 144.00 250.00 482.00 523.15 941.67 −225.00 82.50 200.00 138.75 240.00 464.00 513.15 923.67 −210.00 79.20 192.00 133.50 230.00 446.00 503.15 905.67 −195.00 75.90 184.00 128.25 220.00 428.00 493.15 887.67 −180.00 72.60 176.00 123.00 210.00 410.00 483.15 869.67 −165.00 69.30 168.00 117.75 200.00 392.00 473.15 851.67 −150.00 66.00 160.00 112.50 190.00 374.00 463.15 833.67 −135.00 62.70 152.00 107.25 180.00 356.00 453.15 815.67 −120.00 59.40 144.00 102.00 170.00 338.00 443.15 797.67 −105.00 56.10 136.00 96.75 160.00 320.00 433.15 779.67 −90.00 52.80 128.00 91.50 150.00 302.00 423.15 761.67 −75.00 49.50 120.00 86.25 140.00 284.00 413.15 743.67 −60.00 46.20 112.00 81.00 130.00 266.00 403.15 725.67 −45.00 42.90 104.00 75.75 120.00 248.00 393.15 707.67 −30.00 39.60 96.00 70.50 110.00 230.00 383.15 689.67 −15.00 36.30 88.00 65.25 100.00 212.00 373.15 671.67 0.00 33.00 80.00 60.00 90.00 194.00 363.15 653.67 15.00 29.70 72.00 54.75 80.00 176.00 353.15 635.67 30.00 26.40 64.00 49.50 70.00 158.00 343.15 617.67 45.00 23.10 56.00 44.25 60.00 140.00 333.15 599.67 60.00 19.80 48.00 39.00 50.00 122.00 323.15 581.67 75.00 16.50 40.00 33.75 40.00 104.00 313.15 563.67 90.00 13.20 32.00 28.50 30.00 86.00 303.15 545.67 105.00 9.90 24.00 23.25 20.00 68.00 293.15 527.67 120.00 6.60 16.00 18.00 10.00 50.00 283.15 509.67 135.00 3.30 8.00 12.75 0.00 32.00 273.15 491.67 150.00 0.00 0.00 7.50 −10.00 14.00 263.15 473.67 165.00 −3.30 −8.00 2.25 -14.26 6.29 258.86 465.96 171.43 -4.71 -11.43 0.00 -17.78 0.00 255.37 459.67 176.67 -5.87 -14.22 -1.83 −20.00 −4.00 253.15 455.67 180.00 −6.60 −16.00 −3.00 −30.00 −22.00 243.15 437.67 195.00 −9.90 −24.00 −8.25 −40.00 −40.00 233.15 419.67 210.00 −13.20 −32.00 −13.50 −50.00 −58.00 223.15 401.67 225.00 −16.50 −40.00 −18.75 −60.00 −76.00 213.15 383.67 240.00 −19.80 −48.00 −24.00 −70.00 −94.00 203.15 365.67 255.00 −23.10 −56.00 −29.25 −80.00 −112.00 193.15 347.67 270.00 −26.40 −64.00 −34.50 −90.00 −130.00 183.15 329.67 285.00 −29.70 −72.00 −39.75 −100.00 −148.00 173.15 311.67 300.00 −33.00 −80.00 −45.00 −110.00 −166.00 163.15 293.67 315.00 −36.30 −88.00 −50.25 −120.00 −184.00 153.15 275.67 330.00 −39.60 −96.00 −55.50 −130.00 −202.00 143.15 257.67 345.00 −42.90 −104.00 −60.75 −140.00 −220.00 133.15 239.67 360.00 −46.20 −112.00 −66.00 −150.00 −238.00 123.15 221.67 375.00 −49.50 −120.00 −71.25 −160.00 −256.00 113.15 203.67 390.00 −52.80 −128.00 −76.50 −170.00 −274.00 103.15 185.67 405.00 −56.10 −136.00 −81.75 −180.00 −292.00 93.15 167.67 420.00 −59.40 −144.00 −87.00 −190.00 −310.00 83.15 149.67 435.00 −62.70 −152.00 −92.25 −200.00 −328.00 73.15 131.67 450.00 −66.00 −160.00 −97.50 −210.00 −346.00 63.15 113.67 465.00 −69.30 −168.00 −102.75 −220.00 −364.00 53.15 95.67 480.00 −72.60 −176.00 −108.00 −230.00 −382.00 43.15 77.67 495.00 −75.90 −184.00 −113.25 −240.00 −400.00 33.15 59.67 510.00 −79.20 −192.00 −118.50 −250.00 −418.00 23.15 41.67 525.00 −82.50 −200.00 −123.75 −260.00 −436.00 13.15 23.67 540.00 −85.80 −208.00 −129.00 −270.00 −454.00 3.15 5.67 555.00 −89.10 −216.00 −134.25 −273.15 −459.67 0.00 0.00 559.725 −90.1395 −218.52 −135.90375 Celsius Fahrenheit Kelvin Rankine Delisle Newton Réaumur Rømer ==Comparison of temperature scales== Comparison of temperature scales Comment Kelvin Celsius Fahrenheit Rankine Delisle Newton Réaumur Rømer Absolute zero 0.00 −273.15 −459.67 0.00 559.73 −90.14 −218.52 −135.90 Lowest recorded surface temperature on EarthThe Coldest Inhabited Places on Earth; researchers of the Vostok Station recorded the coldest known temperature on Earth on July 21st 1983: −89.2 °C (−128.6 °F). 184 −89.2 −128.6 331 284 −29 −71 −39 Fahrenheit's ice/salt mixture 255.37 −17.78 0.00 459.67 176.67 −5.87 −14.22 −1.83 Ice melts (at standard pressure) 273.15 0.00 32.00 491.67 150.00 0.00 0.00 7.50 Triple point of water 273.16 0.01 32.018 491.688 149.985 0.0033 0.008 7.50525 Average surface temperature on Earth 288 15 59 519 128 5 12 15 Average human body temperature* 310 37 98 558 95 12 29 27 Highest recorded surface temperature on Earth 331 58 136.4 596 63 19 46 38 Water boils (at standard pressure) 373.1339 99.9839 211.97102 671.64102 0.00 33.00 80.00 60.00 Titanium melts 1941 1668 3034 3494 −2352 550 1334 883 The surface of the Sun 5800 5500 9900 10400 −8100 1800 4400 2900 * Normal human body temperature is 36.8 °C ±0.7 °C, or 98.2 °F ±1.3 °F. The commonly given value 98.6 °F is simply the exact conversion of the nineteenth-century German standard of 37 °C. Horita J. and Wesolowski D.J. (1994) Liquid- vapor fractionation of oxygen and hydrogen isotopes of water from the freezing to the critical temperature. The internal energy of a thermally isolated system may therefore change due to the exchange of work energy. This condition holds at the top of the chromosphere, where the equilibrium temperature is a few tens of thousands of kelvins. Equilibrium fractionation is strongest at low temperatures, and (along with kinetic isotope effects) forms the basis of the most widely used isotopic paleothermometers (or climate proxies): D/H and 18O/16O records from ice cores, and 18O/16O records from calcium carbonate. Thermal lows occur near the Sonoran Desert, on the Mexican plateau, in California's Great Central Valley, in the Sahara, in the Kalahari, over north- west Argentina, in South America, over the Kimberley region of north-west Australia, over the Iberian peninsula, and over the Tibetan plateau. An example of equilibrium isotope fractionation is the concentration of heavy isotopes of oxygen in liquid water, relative to water vapor, :{H2{^{16}O}{(l)}} + {H2{^{18}O}{(g)}} <=> {H2{^{18}O}{(l)}} + {H2{^{16}O}{(g)}} At 20 °C, the equilibrium fractionation factor for this reaction is :\alpha = \frac\ce{(^{18}O/^{16}O)_{Liquid}}\ce{(^{18}O/^{16}O)_{Vapor}} = 1.0098 Equilibrium fractionation is a type of mass-dependent isotope fractionation, while mass-independent fractionation is usually assumed to be a non- equilibrium process. * Animated explanation of the temperature of the Transition Region (and Chromosphere) (University of South Wales). In thermodynamics, a thermally isolated system can exchange no mass or heat energy with its environment. ",5.5,524,6.6,0.32,2.00,B
-"At $300 \mathrm{~K}$ and $20 \mathrm{~atm}$, the compression factor of a gas is 0.86 . Calculate the volume occupied by $8.2 \mathrm{mmol}$ of the gas under these conditions.","For an ideal gas the compressibility factor is Z=1 per definition. The compression ratio is the ratio between the volume of the cylinder and combustion chamber in an internal combustion engine at their maximum and minimum values. The reduced specific volume is defined by, : u_R = \frac{ u_\text{actual}}{RT_\text{cr}/P_\text{cr}} where u_\text{actual} is the specific volume. page 140 Once two of the three reduced properties are found, the compressibility chart can be used. thumb|right|225px|Static compression ratio is determined using the cylinder volume when the piston is at the top and bottom of its travel. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. The compressibility factor should not be confused with the compressibility (also known as coefficient of compressibility or isothermal compressibility) of a material, which is the measure of the relative volume change of a fluid or solid in response to a pressure change. ==Definition and physical significance== thumb|A graphical representation of the behavior of gases and how that behavior relates to compressibility factor. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . For a gas that is a mixture of two or more pure gases (air or natural gas, for example), the gas composition must be known before compressibility can be calculated. The closer the gas is to its critical point or its boiling point, the more Z deviates from the ideal case. === Fugacity === The compressibility factor is linked to the fugacity by the relation: : f = P \exp\left(\int \frac{Z-1}{P} dP\right) ==Generalized compressibility factor graphs for pure gases== thumb|400px|Generalized compressibility factor diagram. For example, if the static compression ratio is 10:1, and the dynamic compression ratio is 7.5:1, a useful value for cylinder pressure would be 7.51.3 × atmospheric pressure, or 13.7 bar (relative to atmospheric pressure). In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. thumb|400px|Definition of formation volume factor Bo and gas/oil ratio Rs for oil When oil is produced to surface temperature and pressure it is usual for some natural gas to come out of solution. Experimental values for the compressibility factor confirm this. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. Racing engines burning methanol and ethanol fuel often have a compression ratio of 14:1 to 16:1. == Mathematical formula == In a piston engine, the static compression ratio (CR) is the ratio between the volume of the cylinder and combustion chamber when the piston is at the bottom of its stroke, and the volume of the combustion chamber when the piston is at the top of its stroke. * Real Gases includes a discussion of compressibility factors. If either the reduced pressure or temperature is unknown, the reduced specific volume must be found. The alveolar gas equation is the method for calculating partial pressure of alveolar oxygen (PAO2). In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature, P_r and T_r, are used to normalize the compressibility factor data. This is the volume of the space in the cylinder left at the end of the compression stroke. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known. ",-4.37 ,20.2,1.16,1.95 ,8.7,E
-"A very crude model of the buckminsterfullerene molecule $\left(\mathrm{C}_{60}\right)$ is to treat it as a collection of electrons in a cube with sides of length equal to the mean diameter of the molecule $(0.7 \mathrm{~nm})$. Suppose that only the $\pi$ electrons of the carbon atoms contribute, and predict the wavelength of the first excitation of $\mathrm{C}_{60}$. (The actual value is $730 \mathrm{~nm}$.)","The nucleus to nucleus diameter of a buckminsterfullerene molecule is about 0.71 nm. A related fullerene molecule, named buckminsterfullerene (C60 fullerene), consists of 60 carbon atoms. The buckminsterfullerene molecule has two bond lengths. The van der Waals diameter of a buckminsterfullerene molecule is about 1.1 nanometers (nm). C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. thumb|Model of the C60 fullerene (buckminsterfullerene).|alt= thumb|Model of the C20 fullerene.|alt= thumb|right|Model of a carbon nanotube. thumb|C60 fullerite (bulk solid C60).|alt= A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The empirical formula of buckminsterfullerene is and its structure is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. C60 fullerene has 60 π electrons but a closed shell configuration requires 72 electrons. Its average bond length is 1.4 Å. ====Other fullerenes==== Another fairly common fullerene has empirical formula , but fullerenes with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained. Buckminsterfullerene-2D-skeletal numbered.svg|(-Ih)[5,6]fullerene Carbon numbering. In 2019, ionized C60 molecules were detected with the Hubble Space Telescope in the space between those stars. ==Types== There are two major families of fullerenes, with fairly distinct properties and applications: the closed buckyballs and the open-ended cylindrical carbon nanotubes. In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. The number of fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure is unique. The vertices of the 26-fullerene graph can be labeled with sequences of 12 bits, in such a way that distance in the graph equals half of the Hamming distance between these bitvectors. The 26-fullerene graph is one of only five fullerenes with such an embedding.. right|200px|Fullerene C60 Fullerene chemistry is a field of organic chemistry devoted to the chemical properties of fullerenes. In a humorously speculative 1966 column for New Scientist, David Jones suggested the possibility of making giant hollow carbon molecules by distorting a plane hexagonal net with the addition of impurity atoms. ==See also== *Buckypaper *Carbocatalysis *Dodecahedrane *Fullerene ligand *Goldberg–Coxeter construction *Lonsdaleite *Triumphene *Truncated rhombic triacontahedron ==References== ==External links== * Nanocarbon: From Graphene to Buckyballs Interactive 3D models of cyclohexane, benzene, graphene, graphite, chiral & non-chiral nanotubes, and C60 Buckyballs - WeCanFigureThisOut.org. *Properties of fullerene *Richard Smalley's autobiography at Nobel.se *Sir Harry Kroto's webpage *Simple model of Fullerene *Introduction to fullerites *Bucky Balls, a short video explaining the structure of by the Vega Science Trust *Giant Fullerenes, a short video looking at Giant Fullerenes *Graphene, 15 September 2010, BBC Radio program Discovery Category:Emerging technologies The 26-fullerene graph has many perfect matchings. Fullerenes with fewer than 60 carbons do not obey isolated pentagon rule (IPR). C70fullerene-2D-skeletal numbered.svg|(-D5h(6))[5,6]fullerene Carbon numbering. Note that only one form of , buckminsterfullerene, has no pair of adjacent pentagons (the smallest such fullerene). The family is named after buckminsterfullerene (C60), the most famous member, which in turn is named after Buckminster Fuller. ",2.14,2.567,1.6,0.264,+2.9,C
-"Consider the half-cell reaction $\operatorname{AgCl}(s)+\mathrm{e}^{-} \rightarrow$ $\operatorname{Ag}(s)+\mathrm{Cl}^{-}(a q)$. If $\mu^{\circ}(\mathrm{AgCl}, s)=-109.71 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and if $E^{\circ}=+0.222 \mathrm{~V}$ for this half-cell, calculate the standard Gibbs energy of formation of $\mathrm{Cl}^{-}(a q)$.","The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: :\mathrm{d}\mathbf{G} =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Silver chloride is a chemical compound with the chemical formula AgCl. \left( \sqrt{\lambda}, \sqrt{x} \right) with Marcum Q-function Q_M(a,b) | mean =k+\lambda\,| median =| mode =| variance =2(k+2\lambda)\,| skewness =\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}| kurtosis =\frac{12(k+4\lambda)}{(k+2\lambda)^2}| entropy =| mgf =\frac{\exp\left(\frac{\lambda t}{1-2t }\right)}{(1-2 t)^{k/2}} \text{ for }2t<1| char =\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}} }} In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. Above 7.5 GPa, silver chloride transitions into a monoclinic KOH phase. The first few central moments are: :\mu_2=2(k+2\lambda)\, :\mu_3=8(k+3\lambda)\, :\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\, The nth cumulant is :\kappa_n=2^{n-1}(n-1)!(k+n\lambda).\, Hence :\mu'_n = 2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1} \frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\lambda )\mu'_{n-j}. === Cumulative distribution function === Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as :P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} The research presented at the Gibbs Conference is focused on understanding biological process through quantitative thermodynamic analysis. ""Standard potential of the silver-silver chloride electrode"". This reaction is used in photography and film and is the following: :Cl− \+ hν → Cl + e− (excitation of the chloride ion, which gives up its extra electron into the conduction band) :Ag+ \+ e− → Ag (liberation of a silver ion, which gains an electron to become a silver atom) The process is not reversible because the silver atom liberated is typically found at a crystal defect or an impurity site so that the electron's energy is lowered enough that it is ""trapped"". ==Uses== ===Silver chloride electrode=== Silver chloride is a constituent of the silver chloride electrode which is a common reference electrode in electrochemistry. Most complexes derived from AgCl are two-, three-, and, in rare cases, four-coordinate, adopting linear, trigonal planar, and tetrahedral coordination geometries, respectively. :3AgCl(s) + Na3AsO3(aq) -> Ag3AsO3(s) + 3NaCl(aq) :3AgCl(s) +Na3AsO4(aq) -> Ag3AsO4(s) + 3NaCl(aq) The above 2 reactions are particularly important in the qualitative analysis of AgCl in labs as AgCl is white, which changes to Ag3AsO3 (silver arsenite) which is yellow, or Ag3AsO4(Silver arsenate) which is reddish brown. ==Chemistry== thumb|right|Silver chloride decomposes over time with exposure to UV light In one of the most famous reactions in chemistry, the addition of colorless aqueous silver nitrate to an equally colorless solution of sodium chloride produces an opaque white precipitate of AgCl:More info on Chlorine test :Ag+ (aq) + Cl^- (aq) -> AgCl (s) This conversion is a common test for the presence of chloride in solution. The equation can also be expressed in terms of the thermal wavelength \Lambda: : \frac{S}{k_{\rm B}N} = \ln\left(\frac{V}{N\Lambda^3}\right)+\frac{5}{2} , For a derivation of the Sackur–Tetrode equation, see the Gibbs paradox. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The indices in the series are (1 + 2i) + (k − 1) = k + 2i as required. == Properties == === Moment generating function === The moment-generating function is given by :M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}. === Moments === The first few raw moments are: :\mu'_1=k+\lambda :\mu'_2=(k+\lambda)^2 + 2(k + 2\lambda) :\mu'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda) :\mu'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda). Silver chloride is also produced as a byproduct of the Miller process where silver metal is reacted with chlorine gas at elevated temperatures. ==History== Silver chloride was known since ancient times. The entropy predicted by the Sackur–Tetrode equation approaches negative infinity as the temperature approaches zero. ==Sackur–Tetrode constant== The Sackur–Tetrode constant, written S0/R, is equal to S/kBN evaluated at a temperature of T = 1 kelvin, at standard pressure (100 kPa or 101.325 kPa, to be specified), for one mole of an ideal gas composed of particles of mass equal to the atomic mass constant (). It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF – this is left as an exercise. For AgBr and AgI, the Ksp values are 5.2 x 10−13 and 8.3 x 10−17, respectively. ",2.50,24.4,0.14,-131.1,537,D
-"$\mathrm{N}_2 \mathrm{O}_3$ dissociates according to the equilibrium $\mathrm{N}_2 \mathrm{O}_3(\mathrm{~g}) \rightleftharpoons \mathrm{NO}_2(\mathrm{~g})+\mathrm{NO}(\mathrm{g})$. At $298 \mathrm{~K}$ and one bar pressure, the degree of dissociation defined as the ratio of moles of $\mathrm{NO}_2(g)$ or $\mathrm{NO}(g)$ to the moles of the reactant assuming no dissociation occurs is $3.5 \times 10^{-3}$. Calculate $\Delta G_R^{\circ}$ for this reaction.","The molecular formula C19H22N2O3 (molar mass: 326.39 g/mol, exact mass: 326.1630 u) may refer to: * Bumadizone * 25CN-NBOMe right|thumb|Nitrogen dioxide Nitryl is the nitrogen dioxide (NO2) moiety when it occurs in a larger compound as a univalent fragment. An alternative method is reaction of Nb2O5 with Nb powder at 1100 °C.Pradyot Patnaik (2002), Handbook of Inorganic Chemicals,McGraw-Hill Professional, == Properties == The room temperature form of NbO2 has a tetragonal, rutile-like structure with short Nb-Nb distances, indicating Nb-Nb bonding.Wells A.F. (1984) Structural Inorganic Chemistry 5th edition Oxford Science Publications The high temperature form also has a rutile-like structure with short Nb-Nb distances. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup An algorithm for computing a 4/3-approximation of the dissociation number in bipartite graphs was published in 2022. The problem of computing diss(G) (dissociation number problem) was firstly studied by Yannakakis. Niobium dioxide, is the chemical compound with the formula NbO2. The molecular formula C21H29NO3 (molar mass: 343.46 g/mol, exact mass: 343.2147 u) may refer to: * CAR-226,086 * CAR-301,060 * 25iP-NBOMe * 25P-NBOMe Category:Molecular formulas The number of vertices in a maximum cardinality dissociation set in G is called the dissociation number of G, denoted by diss(G). thumb|Examples for the definition of the dissociation number In the mathematical discipline of graph theory, a subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. NbO2 is insoluble in water and is a powerful reducing agent, reducing carbon dioxide to carbon and sulfur dioxide to sulfur. It is a bluish-black non-stoichiometric solid with a composition range of NbO1.94-NbO2.09. It can be prepared by reducing Nb2O5 with H2 at 800–1350 °C. Like nitrogen dioxide, the nitryl moiety contains a nitrogen atom with two bonds to the two oxygen atoms, and a third bond shared equally between the nitrogen and the two oxygen atoms. The dissociation number is a special case of the more general Maximum k-dependent Set Problem for k=1. In an industrial process for the production of niobium metal, NbO2 is produced as an intermediate, by the hydrogen reduction of Nb2O5.Patent EP1524252, Sintered bodies based on niobium suboxide, Schnitter C, Wötting G The NbO2 is subsequently reacted with magnesium vapor to produce niobium metal.Method for producing tantalum/niobium metal powders by the reduction of their oxides by gaseous magnesium, US patent 6171363 (2001), Shekhter L.N., Tripp T.B., Lanin L.L. (H. C. Starck, Inc.) ==References== Category:Niobium(IV) compounds Category:Non-stoichiometric compounds Category:Transition metal oxides Examples include nitryl fluoride (NO2F) and nitryl chloride (NO2Cl). Two high-pressure phases have been reported: one with a rutile-like structure (again, with short Nb-Nb distances); and a higher pressure with baddeleyite-related structure. The nitrogen-centred radical is then free to form a bond with another univalent fragment (X) to produce an N−X bond, where X can be F, Cl, OH, etc. In organic nomenclature, the nitryl moiety is known as the nitro group. For instance, nitryl benzene is normally called nitrobenzene (PhNO2). ==See also== * Dinitrogen tetroxide * Nitro compound * Nitrosyl (R−N=O) * Isocyanide (R−N≡C) * Nitryl fluoride * Nitrate ==References== Category:Inorganic nitrogen compounds Category:Oxides Category:Free radicals Category:Nitrogen–oxygen compounds The problem asks for the size of a largest subset S of the vertices of a graph G, so that the induced subgraph G[S] has maximum degree k. == Notes == == References == * * * Category:Graph invariants ",nan,4.4,0.24995,0.42,28,E
-"Approximately how many oxygen molecules arrive each second at the mitochondrion of an active person with a mass of $84 \mathrm{~kg}$ ? The following data are available: Oxygen consumption is about $40 . \mathrm{mL}$ of $\mathrm{O}_2$ per minute per kilogram of body weight, measured at $T=300 . \mathrm{K}$ and $P=1.00 \mathrm{~atm}$. In an adult there are about $1.6 \times 10^{10}$ cells per kg body mass. Each cell contains about 800 . mitochondria.","By convention, 1 MET is considered equivalent to the consumption of 3.5 ml O2·kg−1·min−1 (or 3.5 ml of oxygen per kilogram of body mass per minute) and is roughly equivalent to the expenditure of 1 kcal per kilogram of body weight per hour. Within aerobic respiration, the P/O ratio continues to be debated; however, current figures place it at 2.5 ATP per 1/2(O2) reduced to water, though some claim the ratio is 3. Mitochondria are commonly between 0.75 and 3 μm in cross section, but vary considerably in size and structure. Other definitions which roughly produce the same numbers have been devised, such as: : \text{1 MET}\ = 1 \, \frac{\text{kcal}}{\text{kg}\times\text{h}}\ = 4.184 \, \frac{\text{kJ}}{\text{kg}\times\text{h}} = 1.162 \, \frac{\text{W}}{\text{kg}} where * kcal = kilocalorie * kg = kilogram * h = hour * kJ = kilojoule * W = watt ===Based on watts produced and body surface area=== Still another definition is based on the body surface area, BSA, and energy itself, where the BSA is expressed in m2: : \text{1 MET}\ = 58.2 \, \frac{\text{J}}{\text{s}\times\text{BSA}}\ = 58.2 \, \frac{\text{W}}{\text{m}^2} = 18.4 \, \frac{\text{Btu}}{\text{h}\times\text{ft}^2} which is equal to the rate of energy produced per unit surface area of an average person seated at rest. The theoretical maximum value of is 21%, because the efficiency of glucose oxidation is only 42%, and half of the ATP so produced is wasted. == Criticism of explanations == Kozłowski and Konarzewski have argued that attempts to explain Kleiber's law via any sort of limiting factor is flawed, because metabolic rates vary by factors of 4-5 between rest and activity. Health and fitness studies often bracket cohort activity levels in MET⋅hours/week. ==Quantitative definitions== ===Based on oxygen utilization and body mass=== The original definition of metabolic equivalent of task is the oxygen used by a person in milliliters per minute per kilogram body mass divided by 3.5. The resulting P/O ratio would be the ratio of H/O and H/P; which is 10/3.67 or 2.73 for NADH-linked respiration, and 6/3.67 or 1.64 for UQH2-linked respiration, with actual values being somewhere between. == Notes == == References == *Garrett RH & Grisham CM (2010). Air is typically around 21% oxygen, and at sea level, the PO2 of air is typically around 159 mmHg. A MET also is defined as oxygen uptake in ml/kg/min with one MET equal to the oxygen cost of sitting quietly, equivalent to 3.5 ml/kg/min. A single mitochondrion is often found in unicellular organisms, while human liver cells have about 1000–2000 mitochondria per cell, making up 1/5 of the cell volume. Mitochondria stripped of their outer membrane are called mitoplasts. ===Outer membrane=== The outer mitochondrial membrane, which encloses the entire organelle, is 60 to 75 angstroms (Å) thick. The metabolic equivalent of task (MET) is the objective measure of the ratio of the rate at which a person expends energy, relative to the mass of that person, while performing some specific physical activity compared to a reference, currently set by convention at an absolute 3.5 mL of oxygen per kg per minute, which is the energy expended when sitting quietly by a reference individual, chosen to be roughly representative of the general population, and thereby suited to epidemiological surveys. This value was first experimentally derived from the resting oxygen consumption of a particular subject (a healthy 40-year-old, 70 kg man) and must therefore be treated as a convention. Such studies estimate that at the MAM, which may comprise up to 20% of the mitochondrial outer membrane, the ER and mitochondria are separated by a mere 10–25 nm and held together by protein tethering complexes. The number of mitochondria in a cell can vary widely by organism, tissue, and cell type. The MAM thus offers a perspective on mitochondria that diverges from the traditional view of this organelle as a static, isolated unit appropriated for its metabolic capacity by the cell.Csordás et al., Trends Cell Biol. 2018 Jul;28(7):523-540. . If the oxygen level is too low, mitochondria cannot metabolize nutrients for energy via aerobic metabolism. Mitochondria 10-0 The partial pressure of oxygen in mitochondria is generally assumed to be lower than the surroundings because the mitochondria consume oxygen. Taking this into account, it takes 8/3 +1 or 3.67 protons for vertebrate mitochondria to synthesize one ATP in the cytoplasm from ADP and Pi in the cytoplasm. Since the RMR of a person depends mainly on lean body mass (and not total weight) and other physiological factors such as health status and age, actual RMR (and thus 1-MET energy equivalents) may vary significantly from the kcal/(kg·h) rule of thumb. In respiratory physiology, the oxygen cascade describes the flow of oxygen from air to mitochondria, where it is consumed in aerobic respiration to release energy. RMR measurements by calorimetry in medical surveys have shown that the conventional 1-MET value overestimates the actual resting O2 consumption and energy expenditures by about 20% to 30% on the average; body composition (ratio of body fat to lean body mass) accounted for most of the variance. ==Standardized definition for research== The Compendium of Physical Activities was developed for use in epidemiologic studies to standardize the assignment of MET intensities in physical activity questionnaires. ",0.19, 7.42,144.0,0,1.27,E
+Determine $\Delta_{\mathrm{f}} H^{\ominus}\left(\mathrm{DyCl}_3, \mathrm{~s}\right)$ from these data.","Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by : M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T and : c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. ==Definitions== The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 : \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} : c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} where : t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} measures the temperature relative to the critical point. ==Derivation== For the magnetic analogue of the Maxwell relations for the response functions, the relation : \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 follows, and with thermodynamic stability requiring that c_H, c_M\mbox{ and }\chi_T \geq 0 , one has : c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 which, under the conditions H=0, t>0 and the definition of the critical exponents gives : (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} which gives the Rushbrooke inequality : \alpha' + 2\beta + \gamma' \geq 2. This page provides supplementary chemical data on Ytterbium(III) chloride == Structure and properties data == Structure and properties Standard reduction potential, E° 1.620 eV Crystallographic constants a = 6.73 b = 11.65 c = 6.38 β = 110.4° Effective nuclear charge 3.290 Bond strength 1194±7 kJ/mol Bond length 2.434 (Yb-Cl) Bond angle 111.5° (Cl-Yb-Cl) Magnetic susceptibility 4.4 μB == Thermodynamic properties == Phase behavior Std enthalpy change of fusionΔfusH ~~o~~ 58.1±11.6 kJ/mol Std entropy change of fusionΔfusS ~~o~~ 50.3±10.1 J/(mol•K) Std enthalpy change of atomizationΔatH ~~o~~ 1166.5±4.3 kJ/mol Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −959.5±3.0 kJ/mol Standard molar entropy S ~~o~~ solid 163.5 J/(mol•K) Heat capacity cp 101.4 J/(mol•K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid −212.8 kJ/mol Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas −651.1±5.0 kJ/mol == Spectral data == UV-Vis λmax 268 nm Extinction coefficient 303 M−1cm−1 IR Major absorption bands ν1 = 368.0 cm−1 ν2 = 178.4 cm−1 ν3 = 330.7 cm−1 ν4 = 117.8 cm−1 MS Ionization potentials from electron impact YbCl3+ = 10.9±0.1 eV YbCl2+ = 11.6±0.1 eV YbCl+ = 14.3±0.1 eV ==References== Category:Chemical data pages Category:Chemical data pages cleanup J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows: : T = \frac{1}{2} \left\\{ u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) \right\\} \left\\{ v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} \right\\} : V = \frac{w_{1}(q_{1}) + w_{2}(q_{2}) + \cdots + w_{s}(q_{s}) }{u_{1}(q_{1}) + u_{2}(q_{2}) + \cdots + u_{s}(q_{s}) } The solution of this system consists of a set of separably integrable equations : \frac{\sqrt{2}}{Y}\, dt = \frac{d\varphi_{1}}{\sqrt{E \chi_{1} - \omega_{1} + \gamma_{1}}} = \frac{d\varphi_{2}}{\sqrt{E \chi_{2} - \omega_{2} + \gamma_{2}}} = \cdots = \frac{d\varphi_{s}}{\sqrt{E \chi_{s} - \omega_{s} + \gamma_{s}}} where E = T + V is the conserved energy and the \gamma_{s} are constants. J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Multiplying both sides by 2 Y \dot{\varphi}_{r}, re-arranging, and exploiting the relation 2T = YF yields the equation : 2 Y \dot{\varphi}_{r} \frac{d}{dt} \left(Y \dot{\varphi}_{r}\right) = 2T\dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 Y \dot{\varphi}_{r} \frac{\partial V}{\partial \varphi_{r}} = 2 \dot{\varphi}_{r} \frac{\partial}{\partial \varphi_{r}} \left[ (E-V) Y \right], which may be written as : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2 E \dot{\varphi}_{r} \frac{\partial Y}{\partial \varphi_{r}} - 2 \dot{\varphi}_{r} \frac{\partial W}{\partial \varphi_{r}} = 2E \dot{\varphi}_{r} \frac{d\chi_{r} }{d\varphi_{r}} - 2 \dot{\varphi}_{r} \frac{d\omega_{r}}{d\varphi_{r}}, where E = T + V is the (conserved) total energy. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals : Y = \cosh^{2} \xi - \cos^{2} \eta and the function W equals : W = -\mu_{1} \left( \cosh \xi + \cos \eta \right) - \mu_{2} \left( \cosh \xi - \cos \eta \right) Using the general solution for a Liouville dynamical system below, one obtains : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\xi}^{2} = E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma : \frac{ma^{2}}{2} \left( \cosh^{2} \xi - \cos^{2} \eta \right)^{2} \dot{\eta}^{2} = -E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma Introducing a parameter u by the formula : du = \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}, gives the parametric solution : u = \int \frac{d\xi}{\sqrt{E \cosh^{2} \xi + \left( \frac{\mu_{1} + \mu_{2}}{a} \right) \cosh \xi - \gamma}} = \int \frac{d\eta}{\sqrt{-E \cos^{2} \eta + \left( \frac{\mu_{1} - \mu_{2}}{a} \right) \cos \eta + \gamma}}. Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u. ===Constant of motion=== The bicentric problem has a constant of motion, namely, : r_{1}^{2} r_{2}^{2} \left( \frac{d\theta_{1}}{dt} \right) \left( \frac{d\theta_{2}}{dt} \right) - 2c \left[ \mu_{1} \cos \theta_{1} + \mu_{2} \cos \theta_{2} \right], from which the problem can be solved using the method of the last multiplier. ==Derivation== ===New variables=== To eliminate the v functions, the variables are changed to an equivalent set : \varphi_{r} = \int dq_{r} \sqrt{v_{r}(q_{r})}, giving the relation : v_{1}(q_{1}) \dot{q}_{1}^{2} + v_{2}(q_{2}) \dot{q}_{2}^{2} + \cdots + v_{s}(q_{s}) \dot{q}_{s}^{2} = \dot{\varphi}_{1}^{2} + \dot{\varphi}_{2}^{2} + \cdots + \dot{\varphi}_{s}^{2} = F, which defines a new variable F. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. * J.A. Dean (ed) in Lange's Handbook of Chemistry, McGraw-Hill, New York, USA, 14th edition, 1992. This page provides supplementary chemical data on gold(III) chloride == Thermodynamic properties == Phase behavior Triple point ? It follows that : \frac{d}{dt} \left(Y^{2} \dot{\varphi}_{r}^{2} \right) = 2\frac{d}{dt} \left( E \chi_{r} - \omega_{r} \right), which may be integrated once to yield : \frac{1}{2} Y^{2} \dot{\varphi}_{r}^{2} = E \chi_{r} - \omega_{r} + \gamma_{r}, where the \gamma_{r} are constants of integration subject to the energy conservation : \sum_{r=1}^{s} \gamma_{r} = 0. This page provides supplementary chemical data on Lutetium(III) oxide == Thermodynamic properties == Phase behavior Triple point ? ",0.318,-2.99,"""122.0""",449,-994.3,E
+"Calculate $\Delta_{\mathrm{r}} G^{\ominus}(375 \mathrm{~K})$ for the reaction $2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_2(\mathrm{~g}) \rightarrow 2 \mathrm{CO}_2(\mathrm{~g})$ from the values of $\Delta_{\mathrm{r}} G^{\ominus}(298 \mathrm{~K})$ : and $\Delta_{\mathrm{r}} H^{\ominus}(298 \mathrm{~K})$, and the GibbsHelmholtz equation.","For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. ==The Physical Properties of Greenhouse Gases== Gas Alternate Name Formula 1998 Level Increase since 1750 Radiative forcing (Wm−2) Specific heat at STP (J kg−1) Carbon dioxide (CO2) 365ppm 87 ppm 1.46 819 Methane (CH4) 1,745ppb 1,045ppb 0.48 2191 Nitrous oxide (N2O) 314ppb 44ppb 0.15 880 Tetrafluoromethane Carbon tetrafluoride (CF4) 80ppt 40ppt 0.003 1330 Hexafluoroethane (C2F6) 3 ppt 3ppt 0.001 Sulfur hexafluoride (SF6) 4.2ppt 4.2ppt 0.002 HFC-23* Trifluoromethane (CHF3) 14ppt 14ppt 0.002 HFC-134a* 1,1,1,2-tetrafluoroethane (C2H2F4) 7.5ppt 7.5ppt 0.001 HFC-152a* 1,1-Difluoroethane (C2H4F2) 0.5ppt 0.5ppt 0.000 Water vapour (H2O(G)) 2000 Category:Greenhouse gases The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. The molecular formula C17H14N2O2 (molar mass: 278.305 g/mol, exact mass: 278.1055 u) may refer to: * Bimakalim * Sudan Red G Category:Molecular formulas The molecular formula C16H10N2Na2O7S2 (molar mass: 452.369 g/mol) may refer to: * Orange G * Orange GGN * Sunset Yellow FCF Category:Molecular formulas The molecular formula C18H26O (molar mass: 258.40 g/mol, exact mass: 258.1984 u) may refer to: * Galaxolide (HHCB) * Xibornol The molecular formula C12H22O10 (molar mass: 326.29 g/mol, exact mass: 326.121297 u) may refer to: * Neohesperidose or 2-O-alpha-L-Rhamnopyranosyl-D- glucopyranose * Robinose * Rutinose or 6-O-alpha-L-Rhamnopyranosyl-D- glucupyranose A closely related technique is the use of an electroanalytical voltaic cell, which can be used to measure the Gibbs energy for certain reactions as a function of temperature, yielding K_\mathrm{eq}(T) and thereby \Delta_{\text {rxn}} H^\ominus . Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., ""Physical Chemistry"" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., ""Atkins' Physical Chemistry"" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The enthalpy of formation of one mole of ethane gas refers to the reaction 2 C (graphite) + 3 H2 (g) → C2H6 (g). * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). This method is based on Hess's law, which states that the enthalpy change is the same for a chemical reaction which occurs as a single reaction or in several steps. If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. This method is only approximate, however, because a reported bond energy is only an average value for different molecules with bonds between the same elements.Petrucci, Harwood and Herring, pages 422–423 ==References== Category:Enthalpy Category:Thermochemistry Category:Thermodynamics pl:Standardowe molowe ciepło tworzenia Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. One large class of reactions for which such measurements are common is the combustion of organic compounds by reaction with molecular oxygen (O2) to form carbon dioxide and water (H2O). We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions. ",+93.4,-501,"""0.000226""",12,+2.9,B
+"The vapour pressure of benzene is $53.3 \mathrm{kPa}$ at $60.6^{\circ} \mathrm{C}$, but it fell to $51.5 \mathrm{kPa}$ when $19.0 \mathrm{~g}$ of an non-volatile organic compound was dissolved in $500 \mathrm{~g}$ of benzene. Calculate the molar mass of the compound.","The molecular formula C25H25NO4 (molar mass: 403.47 g/mol, exact mass: 403.1784 u) may refer to: * Benzhydrocodone * 7-Spiroindanyloxymorphone (SIOM) The molecular formula C25H25NO (molar mass: 355.47 g/mol, exact mass: 355.1936 u) may refer to: * JWH-007 * JWH-019 * JWH-047 * JWH-122 Category:Molecular formulas The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C25H27N (molar mass: 341.49 g/mol, exact mass: 341.2143 u) may refer to: * JWH-184 * JWH-196 The molecular formula C2H6ClO2PS (molar mass: 160.56 g/mol, exact mass: 159.9515 u) may refer to: * Dimethyl chlorothiophosphate * Dimethyl phosphorochloridothioate The molecular formula C24H23NO (molar mass: 341.44 g/mol, exact mass: 341.1780 u) may refer to: * JWH-018, also known as 1-pentyl-3-(1-naphthoyl)indole or AM-678 * JWH-148 Category:Molecular formulas The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 The molecular formula C26H40O2 (molar mass: 384.59 g/mol, exact mass: 384.3028 u) may refer to: * L-759,656 * L-759,633 ",5.5,0.5,"""85.0""",3.38,13,C
+"J.G. Dojahn, et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. The ground state of $\mathrm{F}_2^{-}$is ${ }^2 \sum_{\mathrm{u}}^{+}$with a fundamental vibrational wavenumber of $450.0 \mathrm{~cm}^{-1}$ and equilibrium internuclear distance of $190.0 \mathrm{pm}$. The first two excited states are at 1.609 and $1.702 \mathrm{eV}$ above the ground state. Compute the standard molar entropy of $\mathrm{F}_2^{-}$at $298 \mathrm{~K}$.","Constants of Diatomic Molecules, by K. P. Huber and Gerhard Herzberg (Van nostrand Reinhold company, New York, 1979, ), is a classic comprehensive multidisciplinary reference text contains a critical compilation of available data for all diatomic molecules and ions known at the time of publication - over 900 diatomic species in all - including electronic energies, vibrational and rotational constants, and observed transitions. Extensive footnotes discuss the reliability of these data and additional detailed informationon potential energy curves, spin- coupling constants, /\\-type doubling, perturbations between electronic states, hyperfine structure, rotational g factors, dipole moments, radiative lifetimes, oscillator strengths, dissociation energies and ionization potentials when available, and other aspects. {\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}\\!| cdf =\Gamma\\!\left(\frac{ u}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\\!\left(\frac{ u}{2}\right)\\!| mean =\frac{1}{ u-2}\\! for u >2\\!| median = \approx \dfrac{1}{ u\bigg(1-\dfrac{2}{9 u}\bigg)^3}| mode =\frac{1}{ u+2}\\!| variance =\frac{2}{( u-2)^2 ( u-4)}\\! for u >4\\!| skewness =\frac{4}{ u-6}\sqrt{2( u-4)}\\! for u >6\\!| kurtosis =\frac{12(5 u-22)}{( u-6)( u-8)}\\! for u >8\\!| entropy =\frac{ u}{2} \\!+\\!\ln\\!\left(\frac{ u}{2}\Gamma\\!\left(\frac{ u}{2}\right)\right) \\!-\\!\left(1\\!+\\!\frac{ u}{2}\right)\psi\\!\left(\frac{ u}{2}\right)| mgf =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-t}{2i}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2t}\right); does not exist as real valued function| char =\frac{2}{\Gamma(\frac{ u}{2})} \left(\frac{-it}{2}\right)^{\\!\\!\frac{ u}{4}} K_{\frac{ u}{2}}\\!\left(\sqrt{-2it}\right)| }} In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ) is a continuous probability distribution of a positive-valued random variable. The Rydberg–Klein–Rees method is a procedure used in the analysis of rotational-vibrational spectra of diatomic molecules to obtain a potential energy curve from the experimentally-known line positions. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. :From this source with some modifications and additions of later data: :*W.S. Fyfe, Geochemistry, Oxford University Press, (1974). Molecular Spectra and Molecular Structure IV. Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. In particular for integer valued degrees of freedom u we have: For u >1 even, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 5 \cdot 3} { 2 \sqrt{ u}( u -2)( u -4)\cdots 4 \cdot 2\,}\cdot For u >1 odd, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 4 \cdot 2} {\pi \sqrt{ u}( u -2)( u -4)\cdots 5 \cdot 3\,}\cdot\\! For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. Geometrically frustrated systems in particular often exhibit residual entropy. (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). A great deal of research has thus been undertaken into finding other systems that exhibit residual entropy. One of the interesting properties of geometrically frustrated magnetic materials such as spin ice is that the level of residual entropy can be controlled by the application of an external magnetic field. An alternative formula, valid for t^2 < u, is :\int_{-\infty}^t f(u)\,du = \tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}( u+1) \right)} {\sqrt{\pi u}\,\Gamma \left(\tfrac{ u}{2}\right)} \, {}_2F_1 \left( \tfrac{1}{2}, \tfrac{1}{2}( u+1); \tfrac{3}{2}; -\tfrac{t^2}{ u} \right), where 2F1 is a particular case of the hypergeometric function. This material is thus analogous to water ice, with the exception that the spins on the corners of the tetrahedra can point into or out of the tetrahedra, thereby producing the same 2-in, 2-out rule as in water ice, and therefore the same residual entropy. However, it turns out that for a large number of water molecules in this configuration, the hydrogen atoms have a large number of possible configurations that meet the 2-in 2-out rule (each oxygen atom must have two 'near' (or 'in') hydrogen atoms, and two far (or 'out') hydrogen atoms). The first definition yields a probability density function given by : f_1(x; u) = \frac{2^{- u/2}}{\Gamma( u/2)}\,x^{- u/2-1} e^{-1/(2 x)}, while the second definition yields the density function : f_2(x; u) = \frac{( u/2)^{ u/2}}{\Gamma( u/2)} x^{- u/2-1} e^{- u/(2 x)} . ",0.405,-30,"""199.4""",6.6,−1.642876,C
+"The duration of a $90^{\circ}$ or $180^{\circ}$ pulse depends on the strength of the $\mathscr{B}_1$ field. If a $180^{\circ}$ pulse requires $12.5 \mu \mathrm{s}$, what is the strength of the $\mathscr{B}_1$ field? ","This minimum value depends on the definition used for the duration and on the shape of the pulse. The intensity functions—temporal I(t) and spectral S(\omega) —determine the time duration and spectrum bandwidth of the pulse. A pulsed field gradient is a short, timed pulse with spatial-dependent field intensity. The interval between the 50% points of the final amplitude is usually used to determine or define pulse duration, and this is understood to be the case unless otherwise specified. 500px|thumb|right|The duration-bandwidth product depends on the shape of the power spectrum of the pulse. PSR B1919+21 is a pulsar with a period of 1.3373 seconds and a pulse width of 0.04 seconds. For different pulse shapes, the minimum duration-bandwidth product is different. right|thumb|300px|Pulse duration using 50% peak amplitude. thumb|300px|DECT phone pulduration measurement (100 Hz / 10 mS) on channel 8 In signal processing and telecommunication, pulse duration is the interval between the time, during the first transition, that the amplitude of the pulse reaches a specified fraction (level) of its final amplitude, and the time the pulse amplitude drops, on the last transition, to the same level. Other fractions of the final amplitude, e.g., 90% or 1/e, may also be used, as may the root mean square (rms) value of the pulse amplitude. In radar, the pulse duration is the time the radar's transmitter is energized during each cycle. ==References== * * Category:Signal processing Category:Telecommunication theory The length of a pulse thereby is determined by its complex spectral components, which include not just their relative intensities, but also the relative positions (spectral phase) of these spectral components. For example, \mathrm{sech^2} pulses have a minimum duration-bandwidth product of 0.315 while gaussian pulses have a minimum value of 0.441. Usually the attribute 'ultrashort' applies to pulses with a duration of a few tens of femtoseconds, but in a larger sense any pulse which lasts less than a few picoseconds can be considered ultrashort. Intensity autocorrelation gives the pulse width when a particular pulse shape is assumed. The signal had a -second period (not in 1967, but in 1991) and 0.04-second pulsewidth. For a given spectrum, the minimum time-bandwidth product, and therefore the shortest pulse, is obtained by a transform-limited pulse, i.e., for a constant spectral phase \phi(\omega) . In the specialized literature, ""ultrashort"" refers to the femtosecond (fs) and picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses artificially generated. The terms in \gamma_x and \gamma_y describe the walk-off of the pulse; the coefficient \gamma_x ~ (\gamma_y ) is the ratio of the component of the group velocity x ~ (y) and the unit vector in the direction of propagation of the pulse (z-axis). In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. A pulse shaper can be used to make more complicated alterations on both the phase and the amplitude of ultrashort pulses. There is no standard definition of ultrashort pulse. A bandwidth-limited pulse (also known as Fourier- transform-limited pulse, or more commonly, transform-limited pulse) is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. ",5.9,-1.0,"""1.95""",2,0.44,A
+"In 1976 it was mistakenly believed that the first of the 'superheavy' elements had been discovered in a sample of mica. Its atomic number  was believed to be 126. What is the most probable distance of the innermost electrons from the nucleus of an atom of this element? (In such elements, relativistic effects are very important, but ignore them here.)","* Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913). The atomic radius of a chemical element is the distance from the center of the nucleus to the outermost shell of an electron. In this case, it is the poor shielding capacity of the 3d-electrons which affects the atomic radii and chemistries of the elements immediately following the first row of the transition metals, from gallium (Z = 31) to bromine (Z = 35). ==Calculated atomic radius== The following table shows atomic radii computed from theoretical models, as published by Enrico Clementi and others in 1967. Almost all of the mass of an atom is located in the nucleus, with a very small contribution from the electron cloud. These dimensions are much smaller than the diameter of the atom itself (nucleus + electron cloud), by a factor of about 26,634 (uranium atomic radius is about ())""Uranium"" IDC Technologies. to about 60,250 (hydrogen atomic radius is about ).26,634 derives from x / ; 60,250 derives from x / The branch of physics concerned with the study and understanding of the atomic nucleus, including its composition and the forces that bind it together, is called nuclear physics. ==Introduction== ===History=== The nucleus was discovered in 1911, as a result of Ernest Rutherford's efforts to test Thomson's ""plum pudding model"" of the atom. The elements immediately following the lanthanides have atomic radii which are smaller than would be expected and which are almost identical to the atomic radii of the elements immediately above them. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. van der Waals radii Element radius (Å) Hydrogen 1.2 (1.09) Carbon 1.7 Nitrogen 1.55 Oxygen 1.52 Fluorine 1.47 Phosphorus 1.8 Sulfur 1.8 Chlorine 1.75 Copper 1.4 van der Waals radii taken from Bondi's compilation (1964). Accuracy of ±5 pm.The way the atomic radius varies with increasing atomic number can be explained by the arrangement of electrons in shells of fixed capacity. Similarly, the distance from shell-closure explains the unusual instability of isotopes which have far from stable numbers of these particles, such as the radioactive elements 43 (technetium) and 61 (promethium), each of which is preceded and followed by 17 or more stable elements. HE 1327-2326, discovered in 2005 by Anna Frebel and collaborators, was the star with the lowest known iron abundance until SMSS J031300.36−670839.3 was discovered. These trends of the atomic radii (and of various other chemical and physical properties of the elements) can be explained by the electron shell theory of the atom; they provided important evidence for the development and confirmation of quantum theory. ==Atomic radius== Note: All measurements given are in picometers (pm). The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. Under most definitions the radii of isolated neutral atoms range between 30 and 300 pm (trillionths of a meter), or between 0.3 and 3 ångströms. Bohemium was the name assigned to the element with atomic number 93, now known as neptunium, when its discovery was first incorrectly alleged. The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Although the model itself is now obsolete, the Bohr radius for the hydrogen atom is still regarded as an important physical constant. ==Empirically measured atomic radius== The following table shows empirically measured covalent radii for the elements, as published by J. C. Slater in 1964. Therefore, the radius of an atom is more than 10,000 times the radius of its nucleus (1–10 fm), and less than 1/1000 of the wavelength of visible light (400–700 nm). The largest known completely stable nucleus (i.e. stable to alpha, beta, and gamma decay) is lead-208 which contains a total of 208 nucleons (126 neutrons and 82 protons). Such predictions are especially useful for elements whose radii cannot be measured experimentally (e.g. those that have not been discovered, or that have too short of a half- life). ==References== Category:Atomic radius Category:Properties of chemical elements The value of the radius may depend on the atom's state and context. Data derived from other sources with different assumptions cannot be compared. * † to an accuracy of about 5 pm * (b) 12 coordinate * (c) gallium has an anomalous crystal structure * (d) 10 coordinate * (e) uranium, neptunium and plutonium have irregular structures *Triple bond mean-square deviation 3pm. ==References== Data is as quoted at http://www.webelements.com/ from these sources: ===Covalent radii (single bond)=== * * * * * ===Metallic radius=== Category:Properties of chemical elements Category:Chemical element data pages Category:Atomic radius ",313,311875200,"""0.42""",2.24,0,C
+The ground level of $\mathrm{Cl}$ is ${ }^2 \mathrm{P}_{3 / 2}$ and a ${ }^2 \mathrm{P}_{1 / 2}$ level lies $881 \mathrm{~cm}^{-1}$ above it. Calculate the electronic contribution to the molar Gibbs energy of $\mathrm{Cl}$ atoms at  $500 \mathrm{~K}$.,"Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . These tables list values of molar ionization energies, measured in kJ⋅mol−1. :ΔG˚ = -RTln(Keq) :ΔG˚ = (2.303)RT(pKeq) :ΔG˚ = -nFE˚1/2 Useful conversion factors: :-23.06 (kcal/mol)(e−)−1(V)−1 :1.37(pKeq) kcal/mol :1.37[-log(Keq)] kcal/mol == References == Category:Thermodynamic cycles The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: :\mathrm{d}\mathbf{G} =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system:A to Z of Thermodynamics Pierre Perrot :\sum_{i=1}^I N_i \mathrm{d}\mu_i = - S \mathrm{d}T + V \mathrm{d}p where N_i is the number of moles of component i, \mathrm{d}\mu_i the infinitesimal increase in chemical potential for this component, S the entropy, T the absolute temperature, V volume and p the pressure. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The first molar ionization energy applies to the neutral atoms. If multiple phases of matter are present, the chemical potentials across a phase boundary are equal.Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 Combining expressions for the Gibbs–Duhem equation in each phase and assuming systematic equilibrium (i.e. that the temperature and pressure is constant throughout the system), we recover the Gibbs' phase rule. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. When pressure and temperature are variable, only I-1 of I components have independent values for chemical potential and Gibbs' phase rule follows. One particularly useful expression arises when considering binary solutions.The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, At constant P (isobaric) and T (isothermal) it becomes: :0= N_1 \mathrm{d}\mu_1 + N_2 \mathrm{d}\mu_2 or, normalizing by total number of moles in the system N_1 + N_2, substituting in the definition of activity coefficient \gamma and using the identity x_1 + x_2 = 1 : :0= x_1 \mathrm{d}\ln(\gamma_1) + x_2 \mathrm{d}\ln(\gamma_2) Chemical Thermodynamics of Materials, 2004 Svein Stølen, p. 79, This equation is instrumental in the calculation of thermodynamically consistent and thus more accurate expressions for the vapor pressure of a fluid mixture from limited experimental data. == Ternary and multicomponent solutions and mixtures== Lawrence Stamper Darken has shown that the Gibbs-Duhem equation can be applied to the determination of chemical potentials of components from a multicomponent system from experimental data regarding the chemical potential \bar {G_2} of only one component (here component 2) at all compositions. The law can also be written as a function of the total number of atoms N in the sample: :C/N = 3k_{\rm B}, where kB is Boltzmann constant. ==Application limits== thumb|upright=2.2|The molar heat capacity plotted of most elements at 25°C plotted as a function of atomic number. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. In modern terms, Dulong and Petit found that the heat capacity of a mole of many solid elements is about 3R, where R is the universal gas constant. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained. ",3,96.4365076099,"""-6.42""",7.00,0.0408,C
+Calculate the melting point of ice under a pressure of 50 bar. Assume that the density of ice under these conditions is approximately $0.92 \mathrm{~g} \mathrm{~cm}^{-3}$ and that of liquid water is $1.00 \mathrm{~g} \mathrm{~cm}^{-3}$.,"The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. The pressure melting point of ice is the temperature at which ice melts at a given pressure. thumb|Phase diagram of water Ice VI is a form of ice that exists at high pressure at the order of about 1 GPa (= 10 000 bar) and temperatures ranging from 130 up to 355 Kelvin (−143 °C up to 82 °C); see also the phase diagram of water. As the pressure increases with depth in a glacier from the weight of the ice above, the pressure melting point of ice decreases within bounds, as shown in the diagram. The level where ice can start melting is where the pressure melting point equals the actual temperature. The triple point of ice VI with ice VII and liquid water is at about 82 °C and 2.22 GPa and its triple point with ice V and liquid water is at 0.16 °C and 0.6324 GPa = 6324 bar.Water Phase Diagram www1.lsbu.ac.uk, version of 9 September 2019, retrieved 3 October 2019 Ice VI undergoes phase transitions into ices XV and XIX upon cooling depending on pressure as hydrochloric acid is doped. == See also == * Ice phases (overview) == References == == External links == * Physik des Eises (PDF in German, iktp.tu-dresden.de) * Ice phases (www.idc- online.com) Category:Water ice With increasing pressure above 10 MPa, the pressure melting point decreases to a minimum of −21.9 °C at 209.9 MPa. Thereafter, the pressure melting point rises rapidly with pressure, passing back through 0 °C at 632.4 MPa. ==Pressure melting point in glaciers== Glaciers are subject to geothermal heat flux from below and atmospheric warming or cooling from above. Water vapor (H2O) 200px Liquid state Water Solid state Ice Properties Molecular formula H2O Molar mass 18.01528(33) g/mol Melting point Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C) and boils at 373.1339 K (99.9839 °C) Boiling point specific gas constant 461.5 J/(kg·K) Heat of vaporization 2.27 MJ/kg Heat capacity 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous phase of water. A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). Humidity ranges from 0 grams per cubic metre in dry air to 30 grams per cubic metre (0.03 ounce per cubic foot) when the vapor is saturated at 30 °C. === Sublimation === Sublimation is the process by which water molecules directly leave the surface of ice without first becoming liquid water. The temperature range that is determined can then be averaged to gain the melting point of the sample being examined. Melting occurs on both the top and the bottom of the ice. Ice X, within physical chemistry, is a cubic crystalline form of ice formed in the same manner as ice VII, but at pressures as high as about 70 GPa. The formula is valid from about −50 to 102 °C; however there are a very limited number of measurements of the vapor pressure of water over supercooled liquid water. It would also be the level of the base of an ice shelf, or the ice-water interface of a subglacial lake. ==References== Category:Glaciology To estimate ice area, scientists calculate the percentage of sea ice in each pixel, multiply by the pixel area, and total the amounts. Sea ice extent is the area of sea with a specified amount of ice, usually 15%. thumb|A Fisher–Johns apparatus A melting-point apparatus is a scientific instrument used to determine the melting point of a substance. Sea ice rejects salt over time and becomes less salty resulting in a higher melting point. If there is a net increase of heat, then the ice will thin. In static equilibrium conditions, this would be the highest level where water can exist in a glacier. ",0.318, -31.95,"""6.3""",272.8,7,D
+What is the temperature of a two-level system of energy separation equivalent to $400 \mathrm{~cm}^{-1}$ when the population of the upper state is one-third that of the lower state?,"At equilibrium, only a thermally isolating boundary can support a temperature difference. ==See also== * Closed system * Dynamical system * Mechanically isolated system * Open system * Thermodynamic system * Isolated system ==References== Category:Thermodynamic systems Equilibrium isotope fractionation is the partial separation of isotopes between two or more substances in chemical equilibrium. thumb|right|250 px|Vertical cross-section of a thermal low Thermal lows, or heat lows, are non-frontal low-pressure areas that occur over the continents in the subtropics during the warm season, as the result of intense heating when compared to their surrounding environments.Glossary of Meteorology (2009). thumb|The approximate temperature in the solar atmosphere plotted against height The solar transition region is a region of the Sun's atmosphere between the upper chromosphere and corona. The vibrational temperature is used commonly when finding the vibrational partition function. This is a collection of temperature conversion formulas and comparisons among eight different temperature scales, several of which have long been obsolete. The opposite of a thermally isolated system is a thermally open system, which allows the transfer of heat energy and entropy. Thermally open systems may vary, however, in the rate at which they equilibrate, depending on the nature of the boundary of the open system. Thermal lows which develop near sea level can build in height during the warm season, or summer, to the elevation of the 700 hPa pressure surface,David R. Rowson and Stephen J. Colucci (1992). At equilibrium, the temperatures on both sides of a thermally open boundary are equal. Thermal Low. Some numbers in this table have been rounded. === Graphical representation === :File:Comparison of temperature scales blank.svg|845x580px| circle 40 330 4 0 K / 0 °R (−273.15 °C) circle 504 330 4 0 °F (−17.78 °C) circle 537 270 4 150 °D circle 537 317 4 32 °F circle 537 325 4 7.5 °Rø circle 537 332 4 0 °C / 0 °Ré / 0 °N circle 718 245 4 212 °F circle 718 290 4 100 °C circle 718 298 4 80 °Ré circle 718 306 4 60 °Rø circle 718 317 4 33 °N circle 718 330 4 0 °D rect 0 0 845 580 :File:Comparison of temperature scales blank.svg desc none Rankine (°R) Kelvin (K) Fahrenheit (°F) Celsius (°C) Réaumur (°Ré) Rømer (°Rø) Newton (°N) Delisle (°D) Absolute zero Lowest recorded surface temperature on Earth Fahrenheit's ice/water/salt mixture Melting point of ice (at standard pressure) Average surface temperature on Earth (15 °C) Average human body temperature (37 °C) Highest recorded surface temperature on Earth Boiling point of water (at standard pressure) ==Conversion table between the different temperature units== ==See also== * Degree of frost * Conversion of units * Gas mark == Notes and references== Category:Scales of temperature Category:Conversion of units of measurement Temperatures on scales that either do not share a numeric zero or are nonlinearly related cannot correctly be mathematically equated (related using the symbol =), and thus temperatures on different scales are more correctly described as corresponding (related using the symbol ≘). == Celsius scale == == Kelvin scale == == Fahrenheit scale == == Rankine scale == == Delisle scale == == Sir Isaac Newton's degree of temperature == == Réaumur scale == == Rømer scale == ==Comparison values chart== Celsius Fahrenheit Kelvin Rankine Delisle Newton Réaumur Rømer 500.00 932.00 773.15 1391.67 −600.00 165.00 400.00 270.00 490.00 914.00 763.15 1373.67 −585.00 161.70 392.00 264.75 480.00 896.00 753.15 1355.67 −570.00 158.40 384.00 259.50 470.00 878.00 743.15 1337.67 −555.00 155.10 376.00 254.25 460.00 860.00 733.15 1319.67 −540.00 151.80 368.00 249.00 450.00 842.00 723.15 1301.67 −525.00 148.50 360.00 243.75 440.00 824.00 713.15 1283.67 −510.00 145.20 352.00 238.50 430.00 806.00 703.15 1265.67 −495.00 141.90 344.00 233.25 420.00 788.00 693.15 1247.67 −480.00 138.60 336.00 228.00 410.00 770.00 683.15 1229.67 −465.00 135.30 328.00 222.75 400.00 752.00 673.15 1211.67 −450.00 132.00 320.00 217.50 390.00 734.00 663.15 1193.67 −435.00 128.70 312.00 212.25 380.00 716.00 653.15 1175.67 −420.00 125.40 304.00 207.00 370.00 698.00 643.15 1157.67 −405.00 122.10 296.00 201.75 360.00 680.00 633.15 1139.67 −390.00 118.80 288.00 196.50 350.00 662.00 623.15 1121.67 −375.00 115.50 280.00 191.25 340.00 644.00 613.15 1103.67 −360.00 112.20 272.00 186.00 330.00 626.00 603.15 1085.67 −345.00 108.90 264.00 180.75 320.00 608.00 593.15 1067.67 −330.00 105.60 256.00 175.50 310.00 590.00 583.15 1049.67 −315.00 102.30 248.00 170.25 300.00 572.00 573.15 1031.67 −300.00 99.00 240.00 165.00 290.00 554.00 563.15 1013.67 −285.00 95.70 232.00 159.75 280.00 536.00 553.15 995.67 −270.00 92.40 224.00 154.50 270.00 518.00 543.15 977.67 −255.00 89.10 216.00 149.25 260.00 500.00 533.15 959.67 −240.00 85.80 208.00 144.00 250.00 482.00 523.15 941.67 −225.00 82.50 200.00 138.75 240.00 464.00 513.15 923.67 −210.00 79.20 192.00 133.50 230.00 446.00 503.15 905.67 −195.00 75.90 184.00 128.25 220.00 428.00 493.15 887.67 −180.00 72.60 176.00 123.00 210.00 410.00 483.15 869.67 −165.00 69.30 168.00 117.75 200.00 392.00 473.15 851.67 −150.00 66.00 160.00 112.50 190.00 374.00 463.15 833.67 −135.00 62.70 152.00 107.25 180.00 356.00 453.15 815.67 −120.00 59.40 144.00 102.00 170.00 338.00 443.15 797.67 −105.00 56.10 136.00 96.75 160.00 320.00 433.15 779.67 −90.00 52.80 128.00 91.50 150.00 302.00 423.15 761.67 −75.00 49.50 120.00 86.25 140.00 284.00 413.15 743.67 −60.00 46.20 112.00 81.00 130.00 266.00 403.15 725.67 −45.00 42.90 104.00 75.75 120.00 248.00 393.15 707.67 −30.00 39.60 96.00 70.50 110.00 230.00 383.15 689.67 −15.00 36.30 88.00 65.25 100.00 212.00 373.15 671.67 0.00 33.00 80.00 60.00 90.00 194.00 363.15 653.67 15.00 29.70 72.00 54.75 80.00 176.00 353.15 635.67 30.00 26.40 64.00 49.50 70.00 158.00 343.15 617.67 45.00 23.10 56.00 44.25 60.00 140.00 333.15 599.67 60.00 19.80 48.00 39.00 50.00 122.00 323.15 581.67 75.00 16.50 40.00 33.75 40.00 104.00 313.15 563.67 90.00 13.20 32.00 28.50 30.00 86.00 303.15 545.67 105.00 9.90 24.00 23.25 20.00 68.00 293.15 527.67 120.00 6.60 16.00 18.00 10.00 50.00 283.15 509.67 135.00 3.30 8.00 12.75 0.00 32.00 273.15 491.67 150.00 0.00 0.00 7.50 −10.00 14.00 263.15 473.67 165.00 −3.30 −8.00 2.25 -14.26 6.29 258.86 465.96 171.43 -4.71 -11.43 0.00 -17.78 0.00 255.37 459.67 176.67 -5.87 -14.22 -1.83 −20.00 −4.00 253.15 455.67 180.00 −6.60 −16.00 −3.00 −30.00 −22.00 243.15 437.67 195.00 −9.90 −24.00 −8.25 −40.00 −40.00 233.15 419.67 210.00 −13.20 −32.00 −13.50 −50.00 −58.00 223.15 401.67 225.00 −16.50 −40.00 −18.75 −60.00 −76.00 213.15 383.67 240.00 −19.80 −48.00 −24.00 −70.00 −94.00 203.15 365.67 255.00 −23.10 −56.00 −29.25 −80.00 −112.00 193.15 347.67 270.00 −26.40 −64.00 −34.50 −90.00 −130.00 183.15 329.67 285.00 −29.70 −72.00 −39.75 −100.00 −148.00 173.15 311.67 300.00 −33.00 −80.00 −45.00 −110.00 −166.00 163.15 293.67 315.00 −36.30 −88.00 −50.25 −120.00 −184.00 153.15 275.67 330.00 −39.60 −96.00 −55.50 −130.00 −202.00 143.15 257.67 345.00 −42.90 −104.00 −60.75 −140.00 −220.00 133.15 239.67 360.00 −46.20 −112.00 −66.00 −150.00 −238.00 123.15 221.67 375.00 −49.50 −120.00 −71.25 −160.00 −256.00 113.15 203.67 390.00 −52.80 −128.00 −76.50 −170.00 −274.00 103.15 185.67 405.00 −56.10 −136.00 −81.75 −180.00 −292.00 93.15 167.67 420.00 −59.40 −144.00 −87.00 −190.00 −310.00 83.15 149.67 435.00 −62.70 −152.00 −92.25 −200.00 −328.00 73.15 131.67 450.00 −66.00 −160.00 −97.50 −210.00 −346.00 63.15 113.67 465.00 −69.30 −168.00 −102.75 −220.00 −364.00 53.15 95.67 480.00 −72.60 −176.00 −108.00 −230.00 −382.00 43.15 77.67 495.00 −75.90 −184.00 −113.25 −240.00 −400.00 33.15 59.67 510.00 −79.20 −192.00 −118.50 −250.00 −418.00 23.15 41.67 525.00 −82.50 −200.00 −123.75 −260.00 −436.00 13.15 23.67 540.00 −85.80 −208.00 −129.00 −270.00 −454.00 3.15 5.67 555.00 −89.10 −216.00 −134.25 −273.15 −459.67 0.00 0.00 559.725 −90.1395 −218.52 −135.90375 Celsius Fahrenheit Kelvin Rankine Delisle Newton Réaumur Rømer ==Comparison of temperature scales== Comparison of temperature scales Comment Kelvin Celsius Fahrenheit Rankine Delisle Newton Réaumur Rømer Absolute zero 0.00 −273.15 −459.67 0.00 559.73 −90.14 −218.52 −135.90 Lowest recorded surface temperature on EarthThe Coldest Inhabited Places on Earth; researchers of the Vostok Station recorded the coldest known temperature on Earth on July 21st 1983: −89.2 °C (−128.6 °F). 184 −89.2 −128.6 331 284 −29 −71 −39 Fahrenheit's ice/salt mixture 255.37 −17.78 0.00 459.67 176.67 −5.87 −14.22 −1.83 Ice melts (at standard pressure) 273.15 0.00 32.00 491.67 150.00 0.00 0.00 7.50 Triple point of water 273.16 0.01 32.018 491.688 149.985 0.0033 0.008 7.50525 Average surface temperature on Earth 288 15 59 519 128 5 12 15 Average human body temperature* 310 37 98 558 95 12 29 27 Highest recorded surface temperature on Earth 331 58 136.4 596 63 19 46 38 Water boils (at standard pressure) 373.1339 99.9839 211.97102 671.64102 0.00 33.00 80.00 60.00 Titanium melts 1941 1668 3034 3494 −2352 550 1334 883 The surface of the Sun 5800 5500 9900 10400 −8100 1800 4400 2900 * Normal human body temperature is 36.8 °C ±0.7 °C, or 98.2 °F ±1.3 °F. The commonly given value 98.6 °F is simply the exact conversion of the nineteenth-century German standard of 37 °C. Horita J. and Wesolowski D.J. (1994) Liquid- vapor fractionation of oxygen and hydrogen isotopes of water from the freezing to the critical temperature. The internal energy of a thermally isolated system may therefore change due to the exchange of work energy. This condition holds at the top of the chromosphere, where the equilibrium temperature is a few tens of thousands of kelvins. Equilibrium fractionation is strongest at low temperatures, and (along with kinetic isotope effects) forms the basis of the most widely used isotopic paleothermometers (or climate proxies): D/H and 18O/16O records from ice cores, and 18O/16O records from calcium carbonate. Thermal lows occur near the Sonoran Desert, on the Mexican plateau, in California's Great Central Valley, in the Sahara, in the Kalahari, over north- west Argentina, in South America, over the Kimberley region of north-west Australia, over the Iberian peninsula, and over the Tibetan plateau. An example of equilibrium isotope fractionation is the concentration of heavy isotopes of oxygen in liquid water, relative to water vapor, :{H2{^{16}O}{(l)}} + {H2{^{18}O}{(g)}} <=> {H2{^{18}O}{(l)}} + {H2{^{16}O}{(g)}} At 20 °C, the equilibrium fractionation factor for this reaction is :\alpha = \frac\ce{(^{18}O/^{16}O)_{Liquid}}\ce{(^{18}O/^{16}O)_{Vapor}} = 1.0098 Equilibrium fractionation is a type of mass-dependent isotope fractionation, while mass-independent fractionation is usually assumed to be a non- equilibrium process. * Animated explanation of the temperature of the Transition Region (and Chromosphere) (University of South Wales). In thermodynamics, a thermally isolated system can exchange no mass or heat energy with its environment. ",5.5,524,"""6.6""",0.32,2.00,B
+"At $300 \mathrm{~K}$ and $20 \mathrm{~atm}$, the compression factor of a gas is 0.86 . Calculate the volume occupied by $8.2 \mathrm{mmol}$ of the gas under these conditions.","For an ideal gas the compressibility factor is Z=1 per definition. The compression ratio is the ratio between the volume of the cylinder and combustion chamber in an internal combustion engine at their maximum and minimum values. The reduced specific volume is defined by, : u_R = \frac{ u_\text{actual}}{RT_\text{cr}/P_\text{cr}} where u_\text{actual} is the specific volume. page 140 Once two of the three reduced properties are found, the compressibility chart can be used. thumb|right|225px|Static compression ratio is determined using the cylinder volume when the piston is at the top and bottom of its travel. The compressibility factor is defined in thermodynamics and engineering frequently as: :Z = \frac{p}{\rho R_\text{specific} T}, where p is the pressure, \rho is the density of the gas and R_\text{specific} = \frac{R}{M} is the specific gas constant, page 327 M being the molar mass, and the T is the absolute temperature (kelvin or Rankine scale). Alternatively, the compressibility factor for specific gases can be read from generalized compressibility charts that plot Z as a function of pressure at constant temperature. The compressibility factor should not be confused with the compressibility (also known as coefficient of compressibility or isothermal compressibility) of a material, which is the measure of the relative volume change of a fluid or solid in response to a pressure change. ==Definition and physical significance== thumb|A graphical representation of the behavior of gases and how that behavior relates to compressibility factor. For example, methyl chloride, a highly polar molecule and therefore with significant intermolecular forces, the experimental value for the compressibility factor is Z=0.9152 at a pressure of 10 atm and temperature of 100 °C. page 3-268 For air (small non-polar molecules) at approximately the same conditions, the compressibility factor is only Z=1.0025 (see table below for 10 bars, 400 K). ===Compressibility of air=== Normal air comprises in crude numbers 80 percent nitrogen and 20 percent oxygen . For a gas that is a mixture of two or more pure gases (air or natural gas, for example), the gas composition must be known before compressibility can be calculated. The closer the gas is to its critical point or its boiling point, the more Z deviates from the ideal case. === Fugacity === The compressibility factor is linked to the fugacity by the relation: : f = P \exp\left(\int \frac{Z-1}{P} dP\right) ==Generalized compressibility factor graphs for pure gases== thumb|400px|Generalized compressibility factor diagram. For example, if the static compression ratio is 10:1, and the dynamic compression ratio is 7.5:1, a useful value for cylinder pressure would be 7.51.3 × atmospheric pressure, or 13.7 bar (relative to atmospheric pressure). In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. thumb|400px|Definition of formation volume factor Bo and gas/oil ratio Rs for oil When oil is produced to surface temperature and pressure it is usual for some natural gas to come out of solution. Experimental values for the compressibility factor confirm this. As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, T_r, and reduced pressure, P_r, should have the same compressibility factor. Racing engines burning methanol and ethanol fuel often have a compression ratio of 14:1 to 16:1. == Mathematical formula == In a piston engine, the static compression ratio (CR) is the ratio between the volume of the cylinder and combustion chamber when the piston is at the bottom of its stroke, and the volume of the combustion chamber when the piston is at the top of its stroke. * Real Gases includes a discussion of compressibility factors. If either the reduced pressure or temperature is unknown, the reduced specific volume must be found. The alveolar gas equation is the method for calculating partial pressure of alveolar oxygen (PAO2). In order to obtain a generalized graph that can be used for many different gases, the reduced pressure and temperature, P_r and T_r, are used to normalize the compressibility factor data. This is the volume of the space in the cylinder left at the end of the compression stroke. The quantum gases hydrogen, helium, and neon do not conform to the corresponding-states behavior and the reduced pressure and temperature for those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the generalized graphs: : T_r = \frac{T}{T_c + 8} and P_r = \frac{P}{P_c + 8} where the temperatures are in kelvins and the pressures are in atmospheres. ==Reading a generalized compressibility chart== In order to read a compressibility chart, the reduced pressure and temperature must be known. ",-4.37 ,20.2,"""1.16""",1.95 ,8.7,E
+"A very crude model of the buckminsterfullerene molecule $\left(\mathrm{C}_{60}\right)$ is to treat it as a collection of electrons in a cube with sides of length equal to the mean diameter of the molecule $(0.7 \mathrm{~nm})$. Suppose that only the $\pi$ electrons of the carbon atoms contribute, and predict the wavelength of the first excitation of $\mathrm{C}_{60}$. (The actual value is $730 \mathrm{~nm}$.)","The nucleus to nucleus diameter of a buckminsterfullerene molecule is about 0.71 nm. A related fullerene molecule, named buckminsterfullerene (C60 fullerene), consists of 60 carbon atoms. The buckminsterfullerene molecule has two bond lengths. The van der Waals diameter of a buckminsterfullerene molecule is about 1.1 nanometers (nm). C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. thumb|Model of the C60 fullerene (buckminsterfullerene).|alt= thumb|Model of the C20 fullerene.|alt= thumb|right|Model of a carbon nanotube. thumb|C60 fullerite (bulk solid C60).|alt= A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The empirical formula of buckminsterfullerene is and its structure is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge. C60 fullerene has 60 π electrons but a closed shell configuration requires 72 electrons. Its average bond length is 1.4 Å. ====Other fullerenes==== Another fairly common fullerene has empirical formula , but fullerenes with 72, 76, 84 and even up to 100 carbon atoms are commonly obtained. Buckminsterfullerene-2D-skeletal numbered.svg|(-Ih)[5,6]fullerene Carbon numbering. In 2019, ionized C60 molecules were detected with the Hubble Space Telescope in the space between those stars. ==Types== There are two major families of fullerenes, with fairly distinct properties and applications: the closed buckyballs and the open-ended cylindrical carbon nanotubes. In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. The number of fullerenes with a given even number of vertices grows quickly in the number of vertices; 26 is the largest number of vertices for which the fullerene structure is unique. The vertices of the 26-fullerene graph can be labeled with sequences of 12 bits, in such a way that distance in the graph equals half of the Hamming distance between these bitvectors. The 26-fullerene graph is one of only five fullerenes with such an embedding.. right|200px|Fullerene C60 Fullerene chemistry is a field of organic chemistry devoted to the chemical properties of fullerenes. In a humorously speculative 1966 column for New Scientist, David Jones suggested the possibility of making giant hollow carbon molecules by distorting a plane hexagonal net with the addition of impurity atoms. ==See also== *Buckypaper *Carbocatalysis *Dodecahedrane *Fullerene ligand *Goldberg–Coxeter construction *Lonsdaleite *Triumphene *Truncated rhombic triacontahedron ==References== ==External links== * Nanocarbon: From Graphene to Buckyballs Interactive 3D models of cyclohexane, benzene, graphene, graphite, chiral & non-chiral nanotubes, and C60 Buckyballs - WeCanFigureThisOut.org. *Properties of fullerene *Richard Smalley's autobiography at Nobel.se *Sir Harry Kroto's webpage *Simple model of Fullerene *Introduction to fullerites *Bucky Balls, a short video explaining the structure of by the Vega Science Trust *Giant Fullerenes, a short video looking at Giant Fullerenes *Graphene, 15 September 2010, BBC Radio program Discovery Category:Emerging technologies The 26-fullerene graph has many perfect matchings. Fullerenes with fewer than 60 carbons do not obey isolated pentagon rule (IPR). C70fullerene-2D-skeletal numbered.svg|(-D5h(6))[5,6]fullerene Carbon numbering. Note that only one form of , buckminsterfullerene, has no pair of adjacent pentagons (the smallest such fullerene). The family is named after buckminsterfullerene (C60), the most famous member, which in turn is named after Buckminster Fuller. ",2.14,2.567,"""1.6""",0.264,+2.9,C
+"Consider the half-cell reaction $\operatorname{AgCl}(s)+\mathrm{e}^{-} \rightarrow$ $\operatorname{Ag}(s)+\mathrm{Cl}^{-}(a q)$. If $\mu^{\circ}(\mathrm{AgCl}, s)=-109.71 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and if $E^{\circ}=+0.222 \mathrm{~V}$ for this half-cell, calculate the standard Gibbs energy of formation of $\mathrm{Cl}^{-}(a q)$.","The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. Since the Gibbs free energy is the Legendre transformation of the internal energy, the derivatives can be replaced by their definitions, transforming the above equation into: :\mathrm{d}\mathbf{G} =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i The chemical potential is simply another name for the partial molar Gibbs free energy (or the partial Gibbs free energy, depending on whether N is in units of moles or particles). Silver chloride is a chemical compound with the chemical formula AgCl. \left( \sqrt{\lambda}, \sqrt{x} \right) with Marcum Q-function Q_M(a,b) | mean =k+\lambda\,| median =| mode =| variance =2(k+2\lambda)\,| skewness =\frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}| kurtosis =\frac{12(k+4\lambda)}{(k+2\lambda)^2}| entropy =| mgf =\frac{\exp\left(\frac{\lambda t}{1-2t }\right)}{(1-2 t)^{k/2}} \text{ for }2t<1| char =\frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}} }} In probability theory and statistics, the noncentral chi-squared distribution (or noncentral chi-square distribution, noncentral \chi^2 distribution) is a noncentral generalization of the chi-squared distribution. Above 7.5 GPa, silver chloride transitions into a monoclinic KOH phase. The first few central moments are: :\mu_2=2(k+2\lambda)\, :\mu_3=8(k+3\lambda)\, :\mu_4=12(k+2\lambda)^2+48(k+4\lambda)\, The nth cumulant is :\kappa_n=2^{n-1}(n-1)!(k+n\lambda).\, Hence :\mu'_n = 2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1} \frac{(n-1)!2^{j-1}}{(n-j)!}(k+j\lambda )\mu'_{n-j}. === Cumulative distribution function === Again using the relation between the central and noncentral chi-squared distributions, the cumulative distribution function (cdf) can be written as :P(x; k, \lambda ) = e^{-\lambda/2}\; \sum_{j=0}^\infty \frac{(\lambda/2)^j}{j!} The research presented at the Gibbs Conference is focused on understanding biological process through quantitative thermodynamic analysis. ""Standard potential of the silver-silver chloride electrode"". This reaction is used in photography and film and is the following: :Cl− \+ hν → Cl + e− (excitation of the chloride ion, which gives up its extra electron into the conduction band) :Ag+ \+ e− → Ag (liberation of a silver ion, which gains an electron to become a silver atom) The process is not reversible because the silver atom liberated is typically found at a crystal defect or an impurity site so that the electron's energy is lowered enough that it is ""trapped"". ==Uses== ===Silver chloride electrode=== Silver chloride is a constituent of the silver chloride electrode which is a common reference electrode in electrochemistry. Most complexes derived from AgCl are two-, three-, and, in rare cases, four-coordinate, adopting linear, trigonal planar, and tetrahedral coordination geometries, respectively. :3AgCl(s) + Na3AsO3(aq) -> Ag3AsO3(s) + 3NaCl(aq) :3AgCl(s) +Na3AsO4(aq) -> Ag3AsO4(s) + 3NaCl(aq) The above 2 reactions are particularly important in the qualitative analysis of AgCl in labs as AgCl is white, which changes to Ag3AsO3 (silver arsenite) which is yellow, or Ag3AsO4(Silver arsenate) which is reddish brown. ==Chemistry== thumb|right|Silver chloride decomposes over time with exposure to UV light In one of the most famous reactions in chemistry, the addition of colorless aqueous silver nitrate to an equally colorless solution of sodium chloride produces an opaque white precipitate of AgCl:More info on Chlorine test :Ag+ (aq) + Cl^- (aq) -> AgCl (s) This conversion is a common test for the presence of chloride in solution. The equation can also be expressed in terms of the thermal wavelength \Lambda: : \frac{S}{k_{\rm B}N} = \ln\left(\frac{V}{N\Lambda^3}\right)+\frac{5}{2} , For a derivation of the Sackur–Tetrode equation, see the Gibbs paradox. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. The indices in the series are (1 + 2i) + (k − 1) = k + 2i as required. == Properties == === Moment generating function === The moment-generating function is given by :M(t;k,\lambda)=\frac{\exp\left(\frac{ \lambda t}{1-2t }\right)}{(1-2 t)^{k/2}}. === Moments === The first few raw moments are: :\mu'_1=k+\lambda :\mu'_2=(k+\lambda)^2 + 2(k + 2\lambda) :\mu'_3=(k+\lambda)^3 + 6(k+\lambda)(k+2\lambda)+8(k+3\lambda) :\mu'_4=(k+\lambda)^4+12(k+\lambda)^2(k+2\lambda)+4(11k^2+44k\lambda+36\lambda^2)+48(k+4\lambda). Silver chloride is also produced as a byproduct of the Miller process where silver metal is reacted with chlorine gas at elevated temperatures. ==History== Silver chloride was known since ancient times. The entropy predicted by the Sackur–Tetrode equation approaches negative infinity as the temperature approaches zero. ==Sackur–Tetrode constant== The Sackur–Tetrode constant, written S0/R, is equal to S/kBN evaluated at a temperature of T = 1 kelvin, at standard pressure (100 kPa or 101.325 kPa, to be specified), for one mole of an ideal gas composed of particles of mass equal to the atomic mass constant (). It remains to plug in the MGF for the non-central chi square distributions into the product and compute the new MGF – this is left as an exercise. For AgBr and AgI, the Ksp values are 5.2 x 10−13 and 8.3 x 10−17, respectively. ",2.50,24.4,"""0.14""",-131.1,537,D
+"$\mathrm{N}_2 \mathrm{O}_3$ dissociates according to the equilibrium $\mathrm{N}_2 \mathrm{O}_3(\mathrm{~g}) \rightleftharpoons \mathrm{NO}_2(\mathrm{~g})+\mathrm{NO}(\mathrm{g})$. At $298 \mathrm{~K}$ and one bar pressure, the degree of dissociation defined as the ratio of moles of $\mathrm{NO}_2(g)$ or $\mathrm{NO}(g)$ to the moles of the reactant assuming no dissociation occurs is $3.5 \times 10^{-3}$. Calculate $\Delta G_R^{\circ}$ for this reaction.","The molecular formula C19H22N2O3 (molar mass: 326.39 g/mol, exact mass: 326.1630 u) may refer to: * Bumadizone * 25CN-NBOMe right|thumb|Nitrogen dioxide Nitryl is the nitrogen dioxide (NO2) moiety when it occurs in a larger compound as a univalent fragment. An alternative method is reaction of Nb2O5 with Nb powder at 1100 °C.Pradyot Patnaik (2002), Handbook of Inorganic Chemicals,McGraw-Hill Professional, == Properties == The room temperature form of NbO2 has a tetragonal, rutile-like structure with short Nb-Nb distances, indicating Nb-Nb bonding.Wells A.F. (1984) Structural Inorganic Chemistry 5th edition Oxford Science Publications The high temperature form also has a rutile-like structure with short Nb-Nb distances. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup An algorithm for computing a 4/3-approximation of the dissociation number in bipartite graphs was published in 2022. The problem of computing diss(G) (dissociation number problem) was firstly studied by Yannakakis. Niobium dioxide, is the chemical compound with the formula NbO2. The molecular formula C21H29NO3 (molar mass: 343.46 g/mol, exact mass: 343.2147 u) may refer to: * CAR-226,086 * CAR-301,060 * 25iP-NBOMe * 25P-NBOMe Category:Molecular formulas The number of vertices in a maximum cardinality dissociation set in G is called the dissociation number of G, denoted by diss(G). thumb|Examples for the definition of the dissociation number In the mathematical discipline of graph theory, a subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. NbO2 is insoluble in water and is a powerful reducing agent, reducing carbon dioxide to carbon and sulfur dioxide to sulfur. It is a bluish-black non-stoichiometric solid with a composition range of NbO1.94-NbO2.09. It can be prepared by reducing Nb2O5 with H2 at 800–1350 °C. Like nitrogen dioxide, the nitryl moiety contains a nitrogen atom with two bonds to the two oxygen atoms, and a third bond shared equally between the nitrogen and the two oxygen atoms. The dissociation number is a special case of the more general Maximum k-dependent Set Problem for k=1. In an industrial process for the production of niobium metal, NbO2 is produced as an intermediate, by the hydrogen reduction of Nb2O5.Patent EP1524252, Sintered bodies based on niobium suboxide, Schnitter C, Wötting G The NbO2 is subsequently reacted with magnesium vapor to produce niobium metal.Method for producing tantalum/niobium metal powders by the reduction of their oxides by gaseous magnesium, US patent 6171363 (2001), Shekhter L.N., Tripp T.B., Lanin L.L. (H. C. Starck, Inc.) ==References== Category:Niobium(IV) compounds Category:Non-stoichiometric compounds Category:Transition metal oxides Examples include nitryl fluoride (NO2F) and nitryl chloride (NO2Cl). Two high-pressure phases have been reported: one with a rutile-like structure (again, with short Nb-Nb distances); and a higher pressure with baddeleyite-related structure. The nitrogen-centred radical is then free to form a bond with another univalent fragment (X) to produce an N−X bond, where X can be F, Cl, OH, etc. In organic nomenclature, the nitryl moiety is known as the nitro group. For instance, nitryl benzene is normally called nitrobenzene (PhNO2). ==See also== * Dinitrogen tetroxide * Nitro compound * Nitrosyl (R−N=O) * Isocyanide (R−N≡C) * Nitryl fluoride * Nitrate ==References== Category:Inorganic nitrogen compounds Category:Oxides Category:Free radicals Category:Nitrogen–oxygen compounds The problem asks for the size of a largest subset S of the vertices of a graph G, so that the induced subgraph G[S] has maximum degree k. == Notes == == References == * * * Category:Graph invariants ",,4.4,"""0.24995""",0.42,28,E
+"Approximately how many oxygen molecules arrive each second at the mitochondrion of an active person with a mass of $84 \mathrm{~kg}$ ? The following data are available: Oxygen consumption is about $40 . \mathrm{mL}$ of $\mathrm{O}_2$ per minute per kilogram of body weight, measured at $T=300 . \mathrm{K}$ and $P=1.00 \mathrm{~atm}$. In an adult there are about $1.6 \times 10^{10}$ cells per kg body mass. Each cell contains about 800 . mitochondria.","By convention, 1 MET is considered equivalent to the consumption of 3.5 ml O2·kg−1·min−1 (or 3.5 ml of oxygen per kilogram of body mass per minute) and is roughly equivalent to the expenditure of 1 kcal per kilogram of body weight per hour. Within aerobic respiration, the P/O ratio continues to be debated; however, current figures place it at 2.5 ATP per 1/2(O2) reduced to water, though some claim the ratio is 3. Mitochondria are commonly between 0.75 and 3 μm in cross section, but vary considerably in size and structure. Other definitions which roughly produce the same numbers have been devised, such as: : \text{1 MET}\ = 1 \, \frac{\text{kcal}}{\text{kg}\times\text{h}}\ = 4.184 \, \frac{\text{kJ}}{\text{kg}\times\text{h}} = 1.162 \, \frac{\text{W}}{\text{kg}} where * kcal = kilocalorie * kg = kilogram * h = hour * kJ = kilojoule * W = watt ===Based on watts produced and body surface area=== Still another definition is based on the body surface area, BSA, and energy itself, where the BSA is expressed in m2: : \text{1 MET}\ = 58.2 \, \frac{\text{J}}{\text{s}\times\text{BSA}}\ = 58.2 \, \frac{\text{W}}{\text{m}^2} = 18.4 \, \frac{\text{Btu}}{\text{h}\times\text{ft}^2} which is equal to the rate of energy produced per unit surface area of an average person seated at rest. The theoretical maximum value of is 21%, because the efficiency of glucose oxidation is only 42%, and half of the ATP so produced is wasted. == Criticism of explanations == Kozłowski and Konarzewski have argued that attempts to explain Kleiber's law via any sort of limiting factor is flawed, because metabolic rates vary by factors of 4-5 between rest and activity. Health and fitness studies often bracket cohort activity levels in MET⋅hours/week. ==Quantitative definitions== ===Based on oxygen utilization and body mass=== The original definition of metabolic equivalent of task is the oxygen used by a person in milliliters per minute per kilogram body mass divided by 3.5. The resulting P/O ratio would be the ratio of H/O and H/P; which is 10/3.67 or 2.73 for NADH-linked respiration, and 6/3.67 or 1.64 for UQH2-linked respiration, with actual values being somewhere between. == Notes == == References == *Garrett RH & Grisham CM (2010). Air is typically around 21% oxygen, and at sea level, the PO2 of air is typically around 159 mmHg. A MET also is defined as oxygen uptake in ml/kg/min with one MET equal to the oxygen cost of sitting quietly, equivalent to 3.5 ml/kg/min. A single mitochondrion is often found in unicellular organisms, while human liver cells have about 1000–2000 mitochondria per cell, making up 1/5 of the cell volume. Mitochondria stripped of their outer membrane are called mitoplasts. ===Outer membrane=== The outer mitochondrial membrane, which encloses the entire organelle, is 60 to 75 angstroms (Å) thick. The metabolic equivalent of task (MET) is the objective measure of the ratio of the rate at which a person expends energy, relative to the mass of that person, while performing some specific physical activity compared to a reference, currently set by convention at an absolute 3.5 mL of oxygen per kg per minute, which is the energy expended when sitting quietly by a reference individual, chosen to be roughly representative of the general population, and thereby suited to epidemiological surveys. This value was first experimentally derived from the resting oxygen consumption of a particular subject (a healthy 40-year-old, 70 kg man) and must therefore be treated as a convention. Such studies estimate that at the MAM, which may comprise up to 20% of the mitochondrial outer membrane, the ER and mitochondria are separated by a mere 10–25 nm and held together by protein tethering complexes. The number of mitochondria in a cell can vary widely by organism, tissue, and cell type. The MAM thus offers a perspective on mitochondria that diverges from the traditional view of this organelle as a static, isolated unit appropriated for its metabolic capacity by the cell.Csordás et al., Trends Cell Biol. 2018 Jul;28(7):523-540. . If the oxygen level is too low, mitochondria cannot metabolize nutrients for energy via aerobic metabolism. Mitochondria 10-0 The partial pressure of oxygen in mitochondria is generally assumed to be lower than the surroundings because the mitochondria consume oxygen. Taking this into account, it takes 8/3 +1 or 3.67 protons for vertebrate mitochondria to synthesize one ATP in the cytoplasm from ADP and Pi in the cytoplasm. Since the RMR of a person depends mainly on lean body mass (and not total weight) and other physiological factors such as health status and age, actual RMR (and thus 1-MET energy equivalents) may vary significantly from the kcal/(kg·h) rule of thumb. In respiratory physiology, the oxygen cascade describes the flow of oxygen from air to mitochondria, where it is consumed in aerobic respiration to release energy. RMR measurements by calorimetry in medical surveys have shown that the conventional 1-MET value overestimates the actual resting O2 consumption and energy expenditures by about 20% to 30% on the average; body composition (ratio of body fat to lean body mass) accounted for most of the variance. ==Standardized definition for research== The Compendium of Physical Activities was developed for use in epidemiologic studies to standardize the assignment of MET intensities in physical activity questionnaires. ",0.19, 7.42,"""144.0""",0,1.27,E
 "In a FRET experiment designed to monitor conformational changes in T4 lysozyme, the fluorescence intensity fluctuates between 5000 and 10,000 counts per second.
-Assuming that 7500 counts represents a FRET efficiency of 0.5 , what is the change in FRET pair separation distance during the reaction? For the tetramethylrhodamine/texas red FRET pair employed $r_0=50 . Å$.","In practice, FRET systems are characterized by the Förster's radius (R0): the distance between the fluorophores at which FRET efficiency is 50%. For many FRET fluorophore pairs, R0 lies between 20 and 90 Å, depending on the acceptor used and the spatial arrangements of the fluorophores within the assay. Single-molecule FRET measurements are typically performed on fluorescence microscopes, either using surface-immobilized or freely-diffusing molecules. The high time resolution of confocal single-molecule FRET measurements allows observers to potentially detect dynamics on time scales as low as 10 μs. FRET studies calculate corresponding FRET efficiencies as a result of time-resolved observation of protein folding events. Once the single molecule intensities vs. time are available the FRET efficiency can be computed for each FRET pair as a function of time and thereby it is possible to follow kinetic events on the single molecule scale and to build FRET histograms showing the distribution of states in each molecule. These FRET efficiencies can then be used to infer distances between molecules as a function of time. :FRET= \frac {\tfrac {I_A} {\eta_A Q_A} }{\tfrac {I_A} {\eta_A Q_A} + \tfrac {I_D} {\eta_D Q_D}}. where FRET is the FRET efficiency of the two-dye system at a period of time, I_A and I_D are measured photon counts of the acceptor and donor channel respectively at the same period of time, \eta_A and \eta_D are the photon collection efficiencies of the two channels, and Q_A and Q_D are quantum yield of the two dyes. The FRET efficiency is the number of photons emitted from the acceptor dye over the sum of the emissions of the donor and the acceptor dye. The FRET signal is weaker than with fluorescence, but has the advantage that there is only signal during a reaction (aside from autofluorescence). ===Scanning FCS === In Scanning fluorescence correlation spectroscopy (sFCS) the measurement volume is moved across the sample in a defined way. Time-resolved fluorescence energy transfer (TR-FRET) is the practical combination of time-resolved fluorometry (TRF) with Förster resonance energy transfer (FRET) that offers a powerful tool for drug discovery researchers. Using two acceptor fluorophores rather than one, FRET can observe multiple sites for correlated movements and spatial changes in any complex molecule. Single FRET pairs are illuminated using intense light sources, typically lasers, in order to generate sufficient fluorescence signals to enable single-molecule detection. In order to obtain statistical confidence of the FRET values, tens to hundreds of photons are required, which put the best possible time resolution to the order of 1 microsecond. This issue, however, is not particularly relevant when the distance estimation of the two fluorophores does not need to be determined with exact and absolute precision. The average molecular brightness (\langle \epsilon\rangle) is related to the variance (\sigma^2) and the average intensity (\langle I\rangle ) as follows: : \ \langle \varepsilon\rangle =\frac{\sigma^2 - \langle I\rangle}{\langle I\rangle} = \sum_i f_i \varepsilon_i Here f_i and \epsilon_i are the fractional intensity and molecular brightness, respectively, of species i. ===FRET-FCS=== Another FCS based approach to studying molecular interactions uses fluorescence resonance energy transfer (FRET) instead of fluorescence, and is called FRET-FCS. FRET involves two fluorophores, a donor and an acceptor. Unlike ensemble FRET, single-molecule FRET allows real-time monitoring of target binding events. The FRET aspect of the technology is driven by several factors, including spectral overlap and the proximity of the fluorophores involved, wherein energy transfer occurs only when the distance between the donor and the acceptor is small enough. Normally, the fluorescent emission of both donor and acceptor fluorophores is detected by two independent detectors and the FRET signal is computed from the ratio of intensities in the two channels. Fluorescent dye \ D [10−10 m2 s−1] T [°C] Excitation wavelength [nm] Reference Rhodamine 6G 2.8, 3.0, 4.14 ± 0.05, 4.20 ± 0.06 25 514 Rhodamine 110 2.7 488 Tetramethyl rhodamine 2.6 543 Cy3 2.8 543 Cy5 2.5, 3.7 ± 0.15 25 633 carboxyfluorescein 3.2 488 Alexa 488 1.96, 4.35 22.5±0.5 488 Atto 655-maleimide 4.07 ± 0.1 25 663 Atto 655-carboxylicacid 4.26 ± 0.08 25 663 2′, 7′-difluorofluorescein (Oregon Green 488) 4.11 ± 0.06 25 498 ==Variations of FCS== FCS almost always refers to the single point, single channel, temporal autocorrelation measurement, although the term ""fluorescence correlation spectroscopy"" out of its historical scientific context implies no such restriction. However, because the donor species used in a TR-FRET assay has a fluorescent lifetime that is many orders of magnitude longer than background fluorescence or scattered light, emission signal resulting from energy transfer can be measured after any interfering signal has completely decayed. ",24,2.567,12.0,30,8.44,C
-"An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \times 10^3 \mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?","Performance of absorption refrigerator chillers is typically much lower, as they are not heat pumps relying on compression, but instead rely on chemical reactions driven by heat. == Equation == The equation is: :{\rm COP} = \frac{|Q|}{ W} where * Q \ is the useful heat supplied or removed by the considered system (machine). Heat pumps are measured by the efficiency with which they give off heat to the hot reservoir, COPheating; refrigerators and air conditioners by the efficiency with which they take up heat from the cold space, COPcooling: :\mathrm{COP}_{\mathrm{heating}} \equiv \frac{|Q_{\rm H}|}{W_{\rm in}} = \frac{Q_{\rm C} + W_{\rm in}}{W_{\rm in}} = \mathrm{COP}_{\mathrm{cooling}}+1\, :\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_{\rm C}}{W_{\rm in}}\, The reason the term ""coefficient of performance"" is used instead of ""efficiency"" is that, since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work, so the COP can be greater than 1 (100%). Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is: :\mathrm{COP}_{\mathrm{heating}} \le \frac{T_{\rm H}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{heating,Carnot} :\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_{\rm C}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{cooling,Carnot} The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator since :\mathrm{COP}_{\mathrm{heating}} = \mathrm{COP}_{\mathrm{cooling}} + 1 This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted by-product. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. Not stating whether an efficiency is HHV or LHV renders such numbers very misleading. ==Heat pumps and refrigerators== Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. At maximum theoretical efficiency, therefore : {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.Borgnakke, C., & Sonntag, R. (2013). For energy- conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the annual fuel use efficiency (AFUE).HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, US, 2004 ===Heat exchangers=== A counter flow heat exchanger is the most efficient type of heat exchanger in transferring heat energy from one circuit to the other. Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, : {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{- \ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}} {\rm COP}_{\rm heating} applies to heat pumps and {\rm COP}_{\rm cooling} applies to air conditioners and refrigerators. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is taken up from the cold reservoir (QC) is equal to the magnitude of the total heat energy given off to the hot reservoir (|QH|) :|Q_{\rm H}| = Q_{\rm C} + W_{\rm in} Their efficiency is measured by a coefficient of performance (COP). However, for a more complete picture of heat exchanger efficiency, exergetic considerations must be taken into account. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. However, for heating, the COP is the ratio of the magnitude of the heat given off to the hot reservoir (which is the heat taken up from the cold reservoir plus the input work) to the input work: : {\rm COP}_{\rm cooling}=\frac{|Q_{\rm C}|}{ W}=\frac{Q_{\rm C}}{ W} : {\rm COP}_{\rm heating}=\frac{| Q_{\rm H}|}{ W}=\frac{Q_{\rm C} + W}{ W} = {\rm COP}_{\rm cooling} + 1 where * Q_{\rm C} > 0 \ is the heat removed from the cold reservoir and added to the system; * Q_{\rm H} < 0 \ is the heat given off to the hot reservoir; it is lost by the system and therefore negative. (see heat). So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. Sometimes, the term efficiency is used for the ratio of the achieved COP to the Carnot COP, which can not exceed 100%. ==Energy efficiency== The 'thermal efficiency' is sometimes called the energy efficiency. Examples are: *friction of moving parts *inefficient combustion *heat loss from the combustion chamber *departure of the working fluid from the thermodynamic properties of an ideal gas *aerodynamic drag of air moving through the engine *energy used by auxiliary equipment like oil and water pumps. *inefficient compressors and turbines *imperfect valve timing These factors may be accounted when analyzing thermodynamic cycles, however discussion of how to do so is outside the scope of this article. ==Energy conversion== For a device that converts energy from another form into thermal energy (such as an electric heater, boiler, or furnace), the thermal efficiency is :\eta_{\rm th} \equiv \frac{|Q_{\rm out}|}{Q_{\rm in}} where the Q quantities are heat-equivalent values. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout < 0 into the surroundings: :Q_{in} = |W_{\rm out}| + |Q_{\rm out}| The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. An electric resistance heater has a thermal efficiency close to 100%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. ",-3.141592,4.85,2.9,0.5,0.6957,B
-"An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \times 10^3 \mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?","Performance of absorption refrigerator chillers is typically much lower, as they are not heat pumps relying on compression, but instead rely on chemical reactions driven by heat. == Equation == The equation is: :{\rm COP} = \frac{|Q|}{ W} where * Q \ is the useful heat supplied or removed by the considered system (machine). Heat pumps are measured by the efficiency with which they give off heat to the hot reservoir, COPheating; refrigerators and air conditioners by the efficiency with which they take up heat from the cold space, COPcooling: :\mathrm{COP}_{\mathrm{heating}} \equiv \frac{|Q_{\rm H}|}{W_{\rm in}} = \frac{Q_{\rm C} + W_{\rm in}}{W_{\rm in}} = \mathrm{COP}_{\mathrm{cooling}}+1\, :\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_{\rm C}}{W_{\rm in}}\, The reason the term ""coefficient of performance"" is used instead of ""efficiency"" is that, since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work, so the COP can be greater than 1 (100%). Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is: :\mathrm{COP}_{\mathrm{heating}} \le \frac{T_{\rm H}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{heating,Carnot} :\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_{\rm C}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{cooling,Carnot} The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator since :\mathrm{COP}_{\mathrm{heating}} = \mathrm{COP}_{\mathrm{cooling}} + 1 This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted by-product. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. For energy- conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the annual fuel use efficiency (AFUE).HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, US, 2004 ===Heat exchangers=== A counter flow heat exchanger is the most efficient type of heat exchanger in transferring heat energy from one circuit to the other. Not stating whether an efficiency is HHV or LHV renders such numbers very misleading. ==Heat pumps and refrigerators== Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. At maximum theoretical efficiency, therefore : {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.Borgnakke, C., & Sonntag, R. (2013). Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, : {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{- \ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}} {\rm COP}_{\rm heating} applies to heat pumps and {\rm COP}_{\rm cooling} applies to air conditioners and refrigerators. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is taken up from the cold reservoir (QC) is equal to the magnitude of the total heat energy given off to the hot reservoir (|QH|) :|Q_{\rm H}| = Q_{\rm C} + W_{\rm in} Their efficiency is measured by a coefficient of performance (COP). However, for a more complete picture of heat exchanger efficiency, exergetic considerations must be taken into account. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. Sometimes, the term efficiency is used for the ratio of the achieved COP to the Carnot COP, which can not exceed 100%. ==Energy efficiency== The 'thermal efficiency' is sometimes called the energy efficiency. Examples are: *friction of moving parts *inefficient combustion *heat loss from the combustion chamber *departure of the working fluid from the thermodynamic properties of an ideal gas *aerodynamic drag of air moving through the engine *energy used by auxiliary equipment like oil and water pumps. *inefficient compressors and turbines *imperfect valve timing These factors may be accounted when analyzing thermodynamic cycles, however discussion of how to do so is outside the scope of this article. ==Energy conversion== For a device that converts energy from another form into thermal energy (such as an electric heater, boiler, or furnace), the thermal efficiency is :\eta_{\rm th} \equiv \frac{|Q_{\rm out}|}{Q_{\rm in}} where the Q quantities are heat-equivalent values. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout < 0 into the surroundings: :Q_{in} = |W_{\rm out}| + |Q_{\rm out}| The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. An electric resistance heater has a thermal efficiency close to 100%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. Seasonal efficiency gives an indication on how efficiently a heat pump operates over an entire cooling or heating season. ==See also== * Seasonal energy efficiency ratio (SEER) * Seasonal thermal energy storage (STES) * Heating seasonal performance factor (HSPF) * Power usage effectiveness (PUE) * Thermal efficiency * Vapor-compression refrigeration * Air conditioner * HVAC ==Notes== == External links == *Discussion on changes to COP of a heat pump depending on input and output temperatures *See COP definition in Cap XII of the book Industrial Energy Management - Principles and Applications Category:Heat pumps Category:Heating, ventilation, and air conditioning Category:Dimensionless numbers of thermodynamics Category:Engineering ratios ",0.5117,4.85,7.136,9.13,234.4,B
-"You have collected a tissue specimen that you would like to preserve by freeze drying. To ensure the integrity of the specimen, the temperature should not exceed $-5.00{ }^{\circ} \mathrm{C}$. The vapor pressure of ice at $273.16 \mathrm{~K}$ is $624 \mathrm{~Pa}$. What is the maximum pressure at which the freeze drying can be carried out?","Usually, the freezing temperatures are between and . ===Primary drying=== During the primary drying phase, the pressure is lowered (to the range of a few millibars), and enough heat is supplied to the material for the ice to sublimate. It is important to note that, in this range of pressure, the heat is brought mainly by conduction or radiation; the convection effect is negligible, due to the low air density. ===Secondary drying=== thumb|A benchtop manifold freeze-drier The secondary drying phase aims to remove unfrozen water molecules, since the ice was removed in the primary drying phase. After the freeze-drying process is complete, the vacuum is usually broken with an inert gas, such as nitrogen, before the material is sealed. For increased efficiency, the condenser temperature should be 20 °C (36°F) less than the product during primary drying and have a defrosting mechanism to ensure that the maximum amount of water vapor in the air is condensed. ==== Shelf fluid ==== The amount of heat energy needed at times of the primary and secondary drying phase is regulated by an external heat exchanger. The freezing phase is the most critical in the whole freeze-drying process, as the freezing method can impact the speed of reconstitution, duration of freeze-drying cycle, product stability, and appropriate crystallization. In this phase, the temperature is raised higher than in the primary drying phase, and can even be above , to break any physico-chemical interactions that have formed between the water molecules and the frozen material. Therefore, freeze-drying is often reserved for materials that are heat- sensitive, such as proteins, enzymes, microorganisms, and blood plasma. At the end of the operation, the final residual water content in the product is extremely low, around 1–4%. == Applications of freeze drying == Freeze-drying causes less damage to the substance than other dehydration methods using higher temperatures. In bacteriology freeze-drying is used to conserve special strains. The frost point for a given parcel of air is always higher than the dew point, as breaking the stronger bonding between water molecules on the surface of ice compared to the surface of (supercooled) liquid water requires a higher temperature. == See also == * Bubble point * Carburetor heat * Hydrocarbon dew point * Psychrometrics * Thermodynamic diagrams ==References== ==External links== * Often Needed Answers about Temp, Humidity & Dew Point from the sci.geo.meteorology Category:Atmospheric thermodynamics Category:Temperature Category:Psychrometrics Category:Threshold temperatures Category:Gases Category:Meteorological data and networks Category:Humidity and hygrometry sv:Luftfuktighet#Daggpunkt In April 1966, the first human body was frozen—though it had been embalmed for two months—by being placed in liquid nitrogen and stored at just above freezing. Cryogenicists use the Kelvin or Rankine temperature scale, both of which measure from absolute zero, rather than more usual scales such as Celsius which measures from the freezing point of water at sea levelCelsius, Anders (1742) ""Observationer om twänne beständiga grader på en thermometer"" (Observations about two stable degrees on a thermometer), Kungliga Svenska Vetenskapsakademiens Handlingar (Proceedings of the Royal Swedish Academy of Sciences), 3 : 171–180 and Fig. 1.Don Rittner; Ronald A. Bailey (2005): Encyclopedia of Chemistry. Modern freeze drying began as early as 1890 by Richard Altmann who devised a method to freeze dry tissues (either plant or animal), but went virtually unnoticed until the 1930s. Because the final freeze dried product is porous, complete re-hydration can occur in the food. A significant turning point for freeze drying occurred during World War II when blood plasma and penicillin were needed to treat the wounded in the field. Freeze drying, also known as lyophilization or cryodesiccation, is a low temperature dehydration process that involves freezing the product and lowering pressure, removing the ice by sublimation. Cryonics uses temperatures below −130 °C, called cryopreservation, in an attempt to preserve enough brain information to permit the future revival of the cryopreserved person. thumb|Phase diagram of water Ice VI is a form of ice that exists at high pressure at the order of about 1 GPa (= 10 000 bar) and temperatures ranging from 130 up to 355 Kelvin (−143 °C up to 82 °C); see also the phase diagram of water. Freeze-drying is commonly used to preserve crustaceans, fish, amphibians, reptiles, insects, and smaller mammals. Freeze-drying is known to result in the highest quality of foods amongst all drying techniques because structural integrity is maintained along with preservation of flavors. The U.S. National Institute of Standards and Technology considers the field of cryogenics as that involving temperatures below -153 Celsius (120K; -243.4 Fahrenheit) Discovery of superconducting materials with critical temperatures significantly above the boiling point of nitrogen has provided new interest in reliable, low cost methods of producing high temperature cryogenic refrigeration. Hence, to avoid this issue, mass spectrometers are used to identify vapors released by silicone oil to immediately take corrective action and prevent contamination of the product. === Products === Mammalian cells generally do not survive freeze drying even though they still can be preserved. ==Equipment and types of freeze dryers== thumb|Unloading trays of freeze-dried material from a small cabinet-type freeze-dryer thumb|A residential freeze-dryer, along with the vacuum pump, and a cooling fan for the pump There are many types of freeze- dryers available, however, they usually contain a few essential components. ",2,14,425.0,1.7,-20,C
-The molar constant volume heat capacity for $\mathrm{I}_2(\mathrm{~g})$ is $28.6 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. What is the vibrational contribution to the heat capacity? You can assume that the contribution from the electronic degrees of freedom is negligible.,"On the other hand, the vibration modes only start to become active around 350 K (77 °C) Accordingly, the molar heat capacity cP,m is nearly constant at 29.1 J⋅K−1⋅mol−1 from 100 K to about 300 °C. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. The molar heat capacity is an ""intensive"" property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. So, for example, the atom-molar heat capacity of water is 1/3 of its molar heat capacity, namely 25.3 J⋅K−1⋅mol−1. Generally, the most notable constant parameter is the volumetric heat capacity (at least for solids) which is around the value of 3 megajoule per cubic meter per kelvin:Ashby, Shercliff, Cebon, Materials, Cambridge University Press, Chapter 12: Atoms in vibration: material and heat \rho c_p \simeq 3\,\text{MJ}/(\text{m}^3{\cdot}\text{K})\quad \text{(solid)} Note that the especially high molar values, as for paraffin, gasoline, water and ammonia, result from calculating specific heats in terms of moles of molecules. The molar heat capacity of a substance has the same dimension as the heat capacity of an object; namely, L2⋅M⋅T−2⋅Θ−1, or M(L/T)2/Θ. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where ""amol"" denotes an amount of the solid that contains the Avogadro number of atoms. As in the case f gases, some of the vibration modes will be ""frozen out"" at low temperatures, especially in solids with light and tightly bound atoms, causing the atom-molar heat capacity to be less than this theoretical limit. Here, it predicts higher heat capacities than are actually found, with the difference due to higher-energy vibrational modes not being populated at room temperatures in these substances. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . These extra degrees of freedom contribute to the molar heat capacity of the substance. The modern theory of the heat capacity of solids states that it is due to lattice vibrations in the solid. == History == Experimentally Pierre Louis Dulong and Alexis Thérèse Petit had found in 1819 that the heat capacity per weight (the mass-specific heat capacity) for 13 measured elements was close to a constant value, after it had been multiplied by a number representing the presumed relative atomic weight of the element. The molar heat capacity of the gas will then be determined only by the ""active"" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. If specific heat is expressed per mole of atoms for these substances, none of the constant-volume values exceed, to any large extent, the theoretical Dulong–Petit limit of 25 J⋅mol−1⋅K−1 = 3 R per mole of atoms (see the last column of this table). Since the molar heat capacity of a substance is the specific heat c times the molar mass of the substance M/N its numerical value is generally smaller than that of the specific heat. This agreement is because in the classical statistical theory of Ludwig Boltzmann, the heat capacity of solids approaches a maximum of 3R per mole of atoms because full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom, each corresponding to a quadratic kinetic energy term and a quadratic potential energy term. The vibrational temperature is used commonly when finding the vibrational partition function. The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. ",7.82,13.2,6.0,420,635.7,A
-The diffusion coefficient for $\mathrm{CO}_2$ at $273 \mathrm{~K}$ and $1 \mathrm{~atm}$ is $1.00 \times 10^{-5} \mathrm{~m}^2 \mathrm{~s}^{-1}$. Estimate the collisional cross section of $\mathrm{CO}_2$ given this diffusion coefficient.,"Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. For a reaction between A and B, the collision frequency calculated with the hard-sphere model with the unit number of collisions per m3 per second is: : Z = n_\text{A} n_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} = 10^6N_A^2\text{[A][B]} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} where: *σAB is the reaction cross section (unit m2), the area when two molecules collide with each other, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where rA the radius of A and rB the radius of B in unit m. * kB is the Boltzmann constant unit J⋅K−1. Jixin Chen proposed a finite-time solution to the diffusion flux in 2022 which significantly changes the estimated collision frequency of two particles in a solution. ==Rate equations== The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is : r(T) = kn_\text{A}n_\text{B}= Z \rho \exp \left( \frac{-E_\text{a}}{RT} \right) where: *k is the rate constant in units of (number of molecules)−1⋅s−1⋅m3. * nA is the number density of A in the gas in units of m−3. * nB is the number density of B in the gas in units of m−3. For the diffusive collision, at the infinite time limit when the molecular flux can be calculated from the Fick's laws of diffusion, in 1916 Smoluchowski derived a collision frequency between molecule A and B in a diluted solution: : Z_{AB} = 4 \pi R D_r C_A C_B where: * Z_{AB} is the collision frequency, unit #collisions/s in 1 m3 of solution. Thus the probability of collision should be calculated using the Brownian motion model, which can be approximated to a diffusive flux using various boundary conditions that yield different equations in the Smoluchowski model and the JChen Model. * [A] and [B] are the molar concentration of A and B respectively, unit mol/L. * k is the diffusive collision rate constant, unit L4/3 mol-4/3 s−1. ==See also== * Two-dimensional gas * Rate equation ==References== == External links == *Introduction to Collision Theory Category:Chemical kinetics * A is the area of the collision cross-section in unit m2. * \beta is the product of the unitless fractions of reactive surface area on A and B. * D_r is the relative diffusion constant between A and B, unit m2/s. * D_r is the relative diffusion constant between A and B, unit m2/s, and D_r = D_A + D_B. * C_A and C_B are the number concentrations of molecules A and B in the solution respectively, unit #molecule/m3. or : Z_{AB} = 1000 N_A * 4 \pi R D_r [A] [B] = k [A] [B] where: * Z_{AB} is in unit mole collisions/s in 1 L of solution. For a diluted solution in the gas or the liquid phase, the collision equation developed for neat gas is not suitable when diffusion takes control of the collision frequency, i.e., the direct collision between the two molecules no longer dominates. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. * [A] and [B] are the molar concentration of A and B respectively, unit mol/L. * k is the diffusive collision rate constant, unit L mol−1 s−1. This hypothesis yields a fractal reaction kinetic rate equation of diffusive collision in a diluted solution: : Z_{AB} = (1000 N_A)^{4/3} * 8 \pi^{-1} A \beta D_r ([A] + [B])^{1/3}[A] [B] = k ([A] + [B])^{1/3}[A] [B] where: * Z_{AB} is in unit mole collisions/s in 1 L of solution. * R is the radius of the collision cross-section, unit m. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by \left(\frac{R_1}{R_2}\right)^{{3}/{8}} Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material. * e = coefficient of restitution * Sy = dynamic yield strength (dynamic ""elastic limit"") * E′ = effective elastic modulus * ρ = density * v = velocity at impact * μ = Poisson's ratio e = 3.1 \left(\frac{S_\text{y}}{1}\right)^{5/8} \left(\frac{1}{E'}\right)^{1/2} \left(\frac{1}{v}\right)^{1/4} \left(\frac{1}{\rho}\right)^{1/8} E' = \frac{E}{1-\mu^2} This equation overestimates the actual COR. : Experimental rate constants compared to the ones predicted by collision theory for reactions in solutionE.A. Moelwyn- Hughes, The kinetics of reactions in solution, 2nd ed, page 71. Reaction Solvent A, 1011 s−1⋅M−1 Z, 1011 s−1⋅M−1 Steric factor C2H5Br + OH− ethanol 4.30 3.86 1.11 C2H5O− \+ CH3I ethanol 2.42 1.93 1.25 ClCH2CO2− \+ OH− water 4.55 2.86 1.59 C3H6Br2 \+ I− methanol 1.07 1.39 0.77 HOCH2CH2Cl + OH− water 25.5 2.78 9.17 4-CH3C6H4O− \+ CH3I ethanol 8.49 1.99 4.27 CH3(CH2)2Cl + I− acetone 0.085 1.57 0.054 C5H5N + CH3I C2H2Cl4 — — 2.0 10 ==Alternative collision models for diluted solutions== Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion. Theoretical models to calculate the collision frequency in solutions have been proposed by Marian Smoluchowski in a seminal 1916 publication at the infinite time limit, and Jixin Chen in 2022 at a finite-time approximation. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. A scheme of comparing the rate equations in pure gas and solution is shown in the right figure. thumb|A scheme comparing direct collision and diffusive collision, with corresponding rate equations. ",0.318,4.5,76.0, 7.0,234.4,A
-"Benzoic acid, $1.35 \mathrm{~g}$, is reacted with oxygen in a constant volume calorimeter to form $\mathrm{H}_2 \mathrm{O}(l)$ and $\mathrm{CO}_2(g)$ at $298 \mathrm{~K}$. The mass of the water in the inner bath is $1.55 \times$ $10^3 \mathrm{~g}$. The temperature of the calorimeter and its contents rises $2.76 \mathrm{~K}$ as a result of this reaction. Calculate the calorimeter constant.","The theoretical bases of indirect calorimetry: a review."" Another source gives :Energy (kcal/min) = (respiration in L/min times change in percentage oxygen) / 20 This corresponds to: :Metabolic rate (cal per minute) = 5 (VO2 in mL/min) ==References== ==Further reading== * Category:Calorimetry In SI units, the calorimeter constant is then calculated by dividing the change in enthalpy (ΔH) in joules by the change in temperature (ΔT) in kelvins or degrees Celsius: :C_\mathrm{cal} = \frac{\Delta{H}}{\Delta{T}} The calorimeter constant is usually presented in units of joules per degree Celsius (J/°C) or joules per kelvin (J/K). Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The formula can also be written for units of calories per day where VO2 is oxygen consumption expressed in millilitres per minute and VCO2 is the rate of carbon dioxide production in millilitres per minute. Regardless of the specific chemical process, with a known calorimeter constant and a known change in temperature the heat added to the system may be calculated by multiplying the calorimeter constant by that change in temperature. ==See also== *Thermodynamics ==References== Category:Calorimetry Category:Thermochemistry Metabolism. 1988 Mar;37(3):287-301. ==Scientific background== Indirect calorimetry measures O2 consumption and CO2 production. The Weir formula is a formula used in indirect calorimetry, relating metabolic rate to oxygen consumption and carbon dioxide production. Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). thumb|Original RC1 Calorimeter A reaction calorimeter is a calorimeter that measures the amount of energy released (exothermic) or absorbed (endothermic) by a chemical reaction. Journal of Thermal Analysis and Calorimetry, 147(17), 9301–9351. Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics ""Measuring RMR with Indirect Calorimetry (IC)."" Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. We know from the heat balance equation that: :Q = mf.Cpf.Tin \- Tout We also know that from the heat flow equation that :Q = U.A.LMTD We can therefore rearrange this such that :U = mf.Cpf.Tin \- Tout /A.LMTD This will allow us therefore to monitor U as a function of time. ==Continuous Reaction Calorimeter== thumb|Original Contiplant Calorimeter The Continuous Reaction Calorimeter is especially suitable to obtain thermodynamic information for a scale-up of continuous processes in tubular reactors. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). According to original source, it says: :Metabolic rate (kcal per day) = 1440 (3.9 VO2 \+ 1.1 VCO2) where VO2 is oxygen consumption in litres per minute and VCO2 is the rate of carbon dioxide production in litres per minute. A calorimeter constant (denoted Ccal) is a constant that quantifies the heat capacity of a calorimeter. The profile of the curve is determined by the c-value, which is calculated using the equation: ::: c = n K_a M where n is the stoichiometry of the binding, K_a is the association constant and M is the concentration of the molecule in the cell.Quick Start: Isothermal Titration Calorimetry (ITC) (2016). The molecular formula C20H40O2 (molar mass: 312.53 g/mol, exact mass: 312.3028 u) may refer to: * Arachidic acid, also called eicosanoic acid * Phytanic acid Category:Molecular formulas It may be calculated by applying a known amount of heat to the calorimeter and measuring the calorimeter's corresponding change in temperature. The calorimeter has a gas collector that adapts to the subject and through a unidirectional valve minute by minute collects and quantifies the volume and concentration of O2 inspired and CO2 expired by the subject. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. ",1.27,1.60,313.0,0.5,6.64,E
-The activation energy for a reaction is $50 . \mathrm{J} \mathrm{mol}^{-1}$. Determine the effect on the rate constant for this reaction with a change in temperature from $273 \mathrm{~K}$ to $298 \mathrm{~K}$.,"The rate constant as a function of thermodynamic temperature is then given by: k(T) = Ae^{- E_\mathrm{a}/RT} The reaction rate is given by: r = Ae^{ - E_\mathrm{a}/RT}[\mathrm{A}]^m[\mathrm{B}]^n, where Ea is the activation energy, and R is the gas constant, and m and n are experimentally determined partial orders in [A] and [B], respectively. For the above reaction, one can expect the change of the reaction rate constant (based either on mole fraction or on molar concentration) with pressure at constant temperature to be: : \left(\frac{\partial \ln k_x}{\partial P} \right)_T = -\frac{\Delta V^{\ddagger}} {RT} In practice, the matter can be complicated because the partial molar volumes and the activation volume can themselves be a function of pressure. For a given reaction, the ratio of its rate constant at a higher temperature to its rate constant at a lower temperature is known as its temperature coefficient, (Q). For a one-step process taking place at room temperature, the corresponding Gibbs free energy of activation (ΔG‡) is approximately 23 kcal/mol. ==Dependence on temperature== The Arrhenius equation is an elementary treatment that gives the quantitative basis of the relationship between the activation energy and the reaction rate at which a reaction proceeds. This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics The second molecule of H2 does not appear in the rate equation because it reacts in the third step, which is a rapid step after the rate-determining step, so that it does not affect the overall reaction rate. ==Temperature dependence== Each reaction rate coefficient k has a temperature dependency, which is usually given by the Arrhenius equation: : k = A e^{ - \frac{E_\mathrm{a}}{RT} }. It can be done with the help of computer simulation software. ==Rate constant calculations== Rate constant can be calculated for elementary reactions by molecular dynamics simulations. The temperature dependence of ΔG‡ is used to compute these parameters, the enthalpy of activation ΔH‡ and the entropy of activation ΔS‡, based on the defining formula ΔG‡ = ΔH‡ − TΔS‡. The reaction rate thus defined has the units of mol/L/s. In these equations k(T) is the reaction rate coefficient or rate constant, although it is not really a constant, because it includes all the parameters that affect reaction rate, except for time and concentration. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. As temperature increases, the kinetic energy of the reactants increases. A rise of ten degrees Celsius results in approximately twice the reaction rate. For a reaction between reactants A and B to form a product C, where :A and B are reactants :C is a product :a, b, and c are stoichiometric coefficients, the reaction rate is often found to have the form: r = k[\mathrm{A}]^m [\mathrm{B}]^{n} Here is the reaction rate constant that depends on temperature, and [A] and [B] are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution. A=\frac{k}{e^{-E_a/RT}} The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction. The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit time. As a rule of thumb, reaction rates for many reactions double for every ten degrees Celsius increase in temperature. For a unimolecular step the reaction rate is described by r = k_1[\mathrm{A}], where k_1 is a unimolecular rate constant. By using the mass balance for the system in which the reaction occurs, an expression for the rate of change in concentration can be derived. The rate ratio at a temperature increase of 10 degrees (marked by points) is equal to the Q10 coefficient. For a bimolecular step the reaction rate is described by r=k_2[\mathrm{A}][\mathrm{B}], where k_2 is a bimolecular rate constant. Substitution of this equation in the previous equation leads to a rate equation expressed in terms of the original reactants : v = k_2 K_1 [\ce{H2}] [\ce{NO}]^2 \, This agrees with the form of the observed rate equation if it is assumed that . ",0.15,11,6.1,4500,8.87,A
-"How long will it take to pass $200 . \mathrm{mL}$ of $\mathrm{H}_2$ at $273 \mathrm{~K}$ through a $10 . \mathrm{cm}$-long capillary tube of $0.25 \mathrm{~mm}$ if the gas input and output pressures are 1.05 and $1.00 \mathrm{~atm}$, respectively?","This pressure difference can be calculated from Laplace's pressure equation, :\Delta P=\frac{2 \gamma}{R}. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. This time the equation is solved for the height of the meniscus level which again can be used to give the capillary length. One can reorganize to show the capillary length as a function of surface tension and gravity. :\lambda_{\rm c}^2=\frac{hr}{2\cos\theta}, with h the height of the liquid, r the radius of the capillary tube, and \theta the contact angle. If the temperature is 20o then \lambda_c= 2.71mm The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. On the other hand, the capillary length would be {\lambda \scriptscriptstyle c} = 6.68mm for water-air on the moon. For molecular fluids, the interfacial tensions and density differences are typically of the order of 10-100 mN m−1 and 0.1-1 g mL−1 respectively resulting in a capillary length of \sim3 mm for water and air at room temperature on earth. thumb|292x292px|The capillary length will vary for different liquids and different conditions. As above, the Laplace and hydrostatic pressure are equated resulting in :R= \frac{\gamma}{\Delta \rho g e_0}=\frac{\lambda_{\rm c}^2}{e_0}. Using the same premises of capillary rise, one can find the capillary length as a function of the volume increase, and wetting perimeter of the capillary walls. ====Association with a sessile droplet==== Another way to find the capillary length is using different pressure points inside a sessile droplet, with each point having a radius of curvature, and equate them to the Laplace pressure equation. 200px|right|thumb|Diagram of a balloon catheter. The equation for \lambda_{\rm c} can also be found with an extra \sqrt{2} term, most often used when normalising the capillary height. == Origin == ===Theoretical=== One way to theoretically derive the capillary length, is to imagine a liquid droplet at the point where surface tension balances gravity. There exists a pressure difference either side of this curvature, and when this balances out the pressure due to gravity, one can rearrange to find the capillary length. For a soap bubble, the surface tension must be divided by the mean thickness, resulting in a capillary length of about 3 meters in air! This was a mathematical explanation of the work published by James Jurin in 1719, where he quantified a relationship between the maximum height taken by a liquid in a capillary tube and its diameter – Jurin's law. The capillary length can then be worked out the same way except that the thickness of the film, e_0 must be taken into account as the bubble has a hollow center, unlike the droplet which is a solid. However if these conditions are known, the capillary length can be considered a constant for any given liquid, and be used in numerous fluid mechanical problems to scale the derived equations such that they are valid for any fluid. When a capillary tube is inserted into a liquid, the liquid will rise or fall in the tube, due to an imbalance in pressure. thumb|upright=0.4|right|Diagram of a Durham Tube Durham tubes are used in microbiology to detect production of gas by microorganisms. Therefore the bond number can be written as :\mathrm{Bo}=\left(\frac{L}{\lambda_{\rm c}}\right)^2, with \lambda_{\rm c} the capillary length. This property is usually used by physicists to estimate the height a liquid will rise in a particular capillary tube, radius known, without the need for an experiment. The pressure due to gravity (hydrostatic pressure) P_{\rm h} of a column of liquid is given by :P_{\rm h}=\rho g h=2\rho g\lambda_{\rm c} , where \rho is the droplet density, g the gravitational acceleration, and h=2\lambda_{\rm c} is the height of the droplet. ",22,0.9984,635.7,-75,0.0245,A
-Calculate the Debye-Hückel screening length $1 / \kappa$ at $298 \mathrm{~K}$ in a $0.0075 \mathrm{~m}$ solution of $\mathrm{K}_3 \mathrm{PO}_4$.,"Today, \kappa^{-1} is called the Debye screening length. The extended Debye–Hückel equation provides accurate results for μ ≤ 0.1. In the context of solids, Thomas–Fermi screening length may be required instead of Debye length. == See also == * Bjerrum length * Debye–Falkenhagen effect * Plasma oscillation * Shielding effect * Screening effect == References == == Further reading == * * Category:Electricity Category:Electronics concepts Category:Colloidal chemistry Category:Plasma parameters Category:Electrochemistry Category:Length Category:Peter Debye In plasmas and electrolytes, the Debye length \lambda_{\rm D} (Debye radius or Debye–Hückel screening length), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. The corresponding Debye screening wave vector k_{\rm D}=1/\lambda_{\rm D} for particles of density n, charge q at a temperature T is given by k_{\rm D}^2=4\pi n q^2/(k_{\rm B}T) in Gaussian units. P^\text{ex} = -\frac{k_\text{B} T \kappa_\text{cgs}^3}{24\pi} = -\frac{k_\text{B} T \left(\frac{4\pi \sum_j c_j q_j}{\varepsilon_0 \varepsilon_r k_\text{B} T }\right)^{3/2}}{24\pi}. This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of Meyer-Vernet N (1993) Aspects of Debye shielding. The first is what could be called the square of the reduced inverse screening length, (\kappa a)^2. The Debye–Hückel length may be expressed in terms of the Bjerrum length \lambda_{\rm B} as \lambda_{\rm D} = \left(4 \pi \, \lambda_{\rm B} \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2}, where z_j = q_j/e is the integer charge number that relates the charge on the j-th ionic species to the elementary charge e. == In a plasma == For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. The equation is \ln(\gamma_i) = -\frac{z_i^2 q^2 \kappa}{8 \pi \varepsilon_r \varepsilon_0 k_\text{B} T} = -\frac{z_i^2 q^3 N^{1/2}_\text{A}}{4 \pi (\varepsilon_r \varepsilon_0 k_\text{B} T)^{3/2}} \sqrt{10^3\frac{I}{2}} = -A z_i^2 \sqrt{I}, where * z_i is the charge number of ion species i, * q is the elementary charge, * \kappa is the inverse of the Debye screening length \lambda_{\rm D} (defined below), * \varepsilon_r is the relative permittivity of the solvent, * \varepsilon_0 is the permittivity of free space, * k_\text{B} is the Boltzmann constant, * T is the temperature of the solution, * N_\mathrm{A} is the Avogadro constant, * I is the ionic strength of the solution (defined below), * A is a constant that depends on temperature. The Debye length of semiconductors is given: L_{\rm D} = \sqrt{\frac{\varepsilon k_{\rm B} T}{q^2 N_{\rm dop}}} where * ε is the dielectric constant, * kB is the Boltzmann constant, * T is the absolute temperature in kelvins, * q is the elementary charge, and * Ndop is the net density of dopants (either donors or acceptors). Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). It is possible that Debye–Hückel equation is not able to foresee this behavior because of the linearization of the Poisson–Boltzmann equation, or maybe not: studies about this have been started only during the last years of the 20th century because before it wasn't possible to investigate the 10−4 M region, so it is possible that during the next years new theories will be born. ==Extensions of the theory== A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry One such extended Debye–Hückel equation is given by: \- \log_{10}(\gamma) = \frac{A|z_+z_-|\sqrt{I}}{1 + Ba\sqrt{I}} where \gamma as its common logarithm is the activity coefficient, z is the integer charge of the ion (1 for H+, 2 for Mg2+ etc.), As the only characteristic length scale in the Debye–Hückel equation, \lambda_D sets the scale for variations in the potential and in the concentrations of charged species. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. thumb|left|σ Persei in optical light Sigma Persei (Sigma Per, σ Persei, σ Per) is an orange K-type giant with an apparent magnitude of +4.36. The excess osmotic pressure obtained from Debye–Hückel theory is in cgs units:http://homepages.rpi.edu/~keblip/THERMO/chapters/Chapter33.pdf, page 9. Alternatively, \kappa^{-1} = \frac{1}{\sqrt{8\pi \lambda_{\rm B} N_{\rm A} \times 10^{-24} I}} where \lambda_{\rm B} is the Bjerrum length of the medium in nm, and the factor 10^{-24} derives from transforming unit volume from cubic dm to cubic nm. The molecular formula C20H28FN3O3 (molar mass: 377.453 g/mol, exact mass: 377.2115 u) may refer to: * 5F-ADB * 5F-EMB-PINACA Category:Molecular formulas Factor out the scalar potential and assign the leftovers, which are constant, to \kappa^2. The term in parentheses divided by \varepsilon, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale \lambda_{\rm D} = \left(\frac{\varepsilon \, k_{\rm B} T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2} that commonly is referred to as the Debye–Hückel length. It has been solved by the Swedish mathematician Thomas Hakon Gronwall and his collaborators physicical chemists V. K. La Mer and Karl Sandved in a 1928 article from Physikalische Zeitschrift dealing with extensions to Debye–Huckel theory, which resorted to Taylor series expansion. ",1,0.6296296296,1.4,46.7,122,C
-"A system consisting of $82.5 \mathrm{~g}$ of liquid water at $300 . \mathrm{K}$ is heated using an immersion heater at a constant pressure of 1.00 bar. If a current of $1.75 \mathrm{~A}$ passes through the $25.0 \mathrm{ohm}$ resistor for 100 .s, what is the final temperature of the water?","thumb |upright=1.35 |Upper ocean heat content (OHC) has been increasing because oceans have absorbed a large portion of the excess heat trapped in the atmosphere by human-caused global warming. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The partnership between Argo and Jason measurements has yielded ongoing improvements to estimates of OHC and other global ocean properties. ==Causes for heat uptake== thumb|320px|Over 90% of the thermal energy that has accumulated on Earth from global heating since 1970 is stored in the ocean. thumbnail|300px|Global Heat Content in the top 2000 meters of the ocean since 1958 The more abundant equatorial solar irradiance which is absorbed by Earth's tropical surface waters drives the overall poleward propagation of ocean heat. From this, Ohm determined his law of proportionality and published his results. thumb|Internal resistance model In modern notation we would write, I = \frac {\mathcal E}{r+R}, where \mathcal E is the open-circuit emf of the thermocouple, r is the internal resistance of the thermocouple and R is the resistance of the test wire. thumb|170px|A typical glass-tube immersion style aquarium heater An aquarium heater is a device used in the fishkeeping hobby to warm the temperature of water in aquariums. To calculate the ocean heat content, measurements of ocean temperature at many different locations and depths are required. To calculate the ocean heat content, measurements of ocean temperature at many different locations and depths are required. Ocean heat content (OHC) is the energy absorbed and stored by oceans. thumb|400px|Diagram showing pressure difference induced by a temperature difference. The increase in OHC accounts for 30–40% of global sea-level rise from 1900 to 2020 because of thermal expansion. Additionally, a study from 2022 on anthropogenic warming in the ocean indicates that 62% of the warming from the years between 1850 and 2018 in the North Atlantic along 25°N is kept in the water below 700 m, where a major percentage of the ocean's surplus heat is stored. In 2022, the world’s oceans, as given by OHC, were again the hottest in the historical record and exceeded the previous 2021 record maximum. 50x50px Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License Ocean heat content and sea level rise are important indicators of climate change. Between 1971 and 2018, the rise in OHC accounted for over 90% of Earth’s excess thermal energy from global heating. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. A study in 2015 concluded that ocean heat content increases by the Pacific Ocean were compensated by an abrupt distribution of OHC into the Indian Ocean. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. The upper ocean (0–700 m) has warmed since 1971, while it is very likely that warming has occurred at intermediate depths (700–2000 m) and likely that deep ocean (below 2000 m) temperatures have increased. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). Measurements of temperature versus ocean depth generally show an upper mixed layer (0–200 m), a thermocline (200–1500 m), and a deep ocean layer (>1500 m). thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. ",14.44,322,-131.1,0.132,6.6,B
-"For an ensemble consisting of a mole of particles having two energy levels separated by $1000 . \mathrm{cm}^{-1}$, at what temperature will the internal energy equal $3.00 \mathrm{~kJ}$ ?","Approximate values of kT at 298 K Units kT = J kT = pN⋅nm kT = cal kT = meV kT=-174 dBm/Hz kT/hc ≈ cm−1 kT/e = 25.7 mV RT = kT ⋅ NA = kJ⋅mol−1 RT = 0.593 kcal⋅mol−1 h/kT = 0.16 ps kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, : :\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} :\, p(k)= e^{-k^2\over 2mT}. Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, quasi- particles, and photons. == Critical temperature == This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: :T_{\rm c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_{\rm B}} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_{\rm B}} where: : \,T_{\rm c} is the critical temperature, \,n the particle density, \,m the mass per boson, \hbar the reduced Planck constant, \,k_{\rm B} the Boltzmann constant and \,\zeta the Riemann zeta function; \,\zeta(3/2)\approx 2.6124. MJ/kg may refer to: * megajoules per kilogram * Specific kinetic energy * Heat of fusion * Heat of combustion kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion At temperature T, a particle will have a lesser probability to be in state |1\rangle by e^{-E/kT}. thumb|upright=1.5|Schematic Bose–Einstein condensation versus temperature of the energy diagram In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system by k. On the critical region it is possible to define a critical temperature and thermal wavelength: :\lambda_c^3=g_{3/2}(1)v=\zeta(3/2)v :T_c=\frac{2\pi \hbar^2 }{m k_B \lambda_c^2} recovering the value indicated on the previous section. The SI units for RT are joules per mole (J/mol). This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. == Derivation == === Ideal Bose gas === For an ideal Bose gas we have the equation of state: :\frac{1}{v}=\frac{1}{\lambda^3}g_{3/2}(f)+\frac{1}{V}\frac{f}{1-f} where v=V/N is the per particle volume, \lambda the thermal wavelength, f the fugacity and :g_\alpha (f)=\sum \limits_{n=1}^\infty \frac{f^n}{n^\alpha} It is noticeable that g_{3/2} is a monotonically growing function of f in f \in [0, 1], which are the only values for which the series converge. For a system in equilibrium in canonical ensemble, the probability of the system being in state with energy E is proportional to . As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. It satisfies : \frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{ \left(-\frac{\Delta E}{k T_{\rm ex}} \right) }, where * is the number of particles in an upper (e.g. excited) state; * is the statistical weight of those upper-state particles; * is the number of particles in a lower (e.g. ground) state; * is the statistical weight of those lower-state particles; * is the exponential function; * is the Boltzmann constant; * is the difference in energy between the upper and lower states. When the integral (also known as Bose–Einstein integral) is evaluated with factors of k_B and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section. ",1310,35,6.1,-1,11000,A
-A muscle fiber contracts by $3.5 \mathrm{~cm}$ and in doing so lifts a weight. Calculate the work performed by the fiber. Assume the muscle fiber obeys Hooke's law $F=-k x$ with a force constant $k$ of $750 . \mathrm{N} \mathrm{m}^{-1}$.,"The actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The work of the net force is calculated as the product of its magnitude and the particle displacement. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. Therefore, work need only be computed for the gravitational forces acting on the bodies. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . Notice that the work done by gravity depends only on the vertical movement of the object. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. From Newton's second law, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy corresponding to the linear velocity and angular velocity of that body, W = \Delta E_\text{k}. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. In [Erdemir, 2007] a list of possible cost functions with a brief rationale and the suggested model validation technique is available. ==== Clarification on the use of the maximum isometric force ==== Muscle contraction can be eccentric (velocity of contraction v<0), concentric (v >0) or isometric (v=0). If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). ",0.46,62.8318530718,2.0,4.85,0.75,A
+Assuming that 7500 counts represents a FRET efficiency of 0.5 , what is the change in FRET pair separation distance during the reaction? For the tetramethylrhodamine/texas red FRET pair employed $r_0=50 . Å$.","In practice, FRET systems are characterized by the Förster's radius (R0): the distance between the fluorophores at which FRET efficiency is 50%. For many FRET fluorophore pairs, R0 lies between 20 and 90 Å, depending on the acceptor used and the spatial arrangements of the fluorophores within the assay. Single-molecule FRET measurements are typically performed on fluorescence microscopes, either using surface-immobilized or freely-diffusing molecules. The high time resolution of confocal single-molecule FRET measurements allows observers to potentially detect dynamics on time scales as low as 10 μs. FRET studies calculate corresponding FRET efficiencies as a result of time-resolved observation of protein folding events. Once the single molecule intensities vs. time are available the FRET efficiency can be computed for each FRET pair as a function of time and thereby it is possible to follow kinetic events on the single molecule scale and to build FRET histograms showing the distribution of states in each molecule. These FRET efficiencies can then be used to infer distances between molecules as a function of time. :FRET= \frac {\tfrac {I_A} {\eta_A Q_A} }{\tfrac {I_A} {\eta_A Q_A} + \tfrac {I_D} {\eta_D Q_D}}. where FRET is the FRET efficiency of the two-dye system at a period of time, I_A and I_D are measured photon counts of the acceptor and donor channel respectively at the same period of time, \eta_A and \eta_D are the photon collection efficiencies of the two channels, and Q_A and Q_D are quantum yield of the two dyes. The FRET efficiency is the number of photons emitted from the acceptor dye over the sum of the emissions of the donor and the acceptor dye. The FRET signal is weaker than with fluorescence, but has the advantage that there is only signal during a reaction (aside from autofluorescence). ===Scanning FCS === In Scanning fluorescence correlation spectroscopy (sFCS) the measurement volume is moved across the sample in a defined way. Time-resolved fluorescence energy transfer (TR-FRET) is the practical combination of time-resolved fluorometry (TRF) with Förster resonance energy transfer (FRET) that offers a powerful tool for drug discovery researchers. Using two acceptor fluorophores rather than one, FRET can observe multiple sites for correlated movements and spatial changes in any complex molecule. Single FRET pairs are illuminated using intense light sources, typically lasers, in order to generate sufficient fluorescence signals to enable single-molecule detection. In order to obtain statistical confidence of the FRET values, tens to hundreds of photons are required, which put the best possible time resolution to the order of 1 microsecond. This issue, however, is not particularly relevant when the distance estimation of the two fluorophores does not need to be determined with exact and absolute precision. The average molecular brightness (\langle \epsilon\rangle) is related to the variance (\sigma^2) and the average intensity (\langle I\rangle ) as follows: : \ \langle \varepsilon\rangle =\frac{\sigma^2 - \langle I\rangle}{\langle I\rangle} = \sum_i f_i \varepsilon_i Here f_i and \epsilon_i are the fractional intensity and molecular brightness, respectively, of species i. ===FRET-FCS=== Another FCS based approach to studying molecular interactions uses fluorescence resonance energy transfer (FRET) instead of fluorescence, and is called FRET-FCS. FRET involves two fluorophores, a donor and an acceptor. Unlike ensemble FRET, single-molecule FRET allows real-time monitoring of target binding events. The FRET aspect of the technology is driven by several factors, including spectral overlap and the proximity of the fluorophores involved, wherein energy transfer occurs only when the distance between the donor and the acceptor is small enough. Normally, the fluorescent emission of both donor and acceptor fluorophores is detected by two independent detectors and the FRET signal is computed from the ratio of intensities in the two channels. Fluorescent dye \ D [10−10 m2 s−1] T [°C] Excitation wavelength [nm] Reference Rhodamine 6G 2.8, 3.0, 4.14 ± 0.05, 4.20 ± 0.06 25 514 Rhodamine 110 2.7 488 Tetramethyl rhodamine 2.6 543 Cy3 2.8 543 Cy5 2.5, 3.7 ± 0.15 25 633 carboxyfluorescein 3.2 488 Alexa 488 1.96, 4.35 22.5±0.5 488 Atto 655-maleimide 4.07 ± 0.1 25 663 Atto 655-carboxylicacid 4.26 ± 0.08 25 663 2′, 7′-difluorofluorescein (Oregon Green 488) 4.11 ± 0.06 25 498 ==Variations of FCS== FCS almost always refers to the single point, single channel, temporal autocorrelation measurement, although the term ""fluorescence correlation spectroscopy"" out of its historical scientific context implies no such restriction. However, because the donor species used in a TR-FRET assay has a fluorescent lifetime that is many orders of magnitude longer than background fluorescence or scattered light, emission signal resulting from energy transfer can be measured after any interfering signal has completely decayed. ",24,2.567,"""12.0""",30,8.44,C
+"An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \times 10^3 \mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?","Performance of absorption refrigerator chillers is typically much lower, as they are not heat pumps relying on compression, but instead rely on chemical reactions driven by heat. == Equation == The equation is: :{\rm COP} = \frac{|Q|}{ W} where * Q \ is the useful heat supplied or removed by the considered system (machine). Heat pumps are measured by the efficiency with which they give off heat to the hot reservoir, COPheating; refrigerators and air conditioners by the efficiency with which they take up heat from the cold space, COPcooling: :\mathrm{COP}_{\mathrm{heating}} \equiv \frac{|Q_{\rm H}|}{W_{\rm in}} = \frac{Q_{\rm C} + W_{\rm in}}{W_{\rm in}} = \mathrm{COP}_{\mathrm{cooling}}+1\, :\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_{\rm C}}{W_{\rm in}}\, The reason the term ""coefficient of performance"" is used instead of ""efficiency"" is that, since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work, so the COP can be greater than 1 (100%). Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is: :\mathrm{COP}_{\mathrm{heating}} \le \frac{T_{\rm H}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{heating,Carnot} :\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_{\rm C}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{cooling,Carnot} The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator since :\mathrm{COP}_{\mathrm{heating}} = \mathrm{COP}_{\mathrm{cooling}} + 1 This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted by-product. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. Not stating whether an efficiency is HHV or LHV renders such numbers very misleading. ==Heat pumps and refrigerators== Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. At maximum theoretical efficiency, therefore : {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.Borgnakke, C., & Sonntag, R. (2013). For energy- conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the annual fuel use efficiency (AFUE).HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, US, 2004 ===Heat exchangers=== A counter flow heat exchanger is the most efficient type of heat exchanger in transferring heat energy from one circuit to the other. Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, : {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{- \ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}} {\rm COP}_{\rm heating} applies to heat pumps and {\rm COP}_{\rm cooling} applies to air conditioners and refrigerators. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is taken up from the cold reservoir (QC) is equal to the magnitude of the total heat energy given off to the hot reservoir (|QH|) :|Q_{\rm H}| = Q_{\rm C} + W_{\rm in} Their efficiency is measured by a coefficient of performance (COP). However, for a more complete picture of heat exchanger efficiency, exergetic considerations must be taken into account. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. However, for heating, the COP is the ratio of the magnitude of the heat given off to the hot reservoir (which is the heat taken up from the cold reservoir plus the input work) to the input work: : {\rm COP}_{\rm cooling}=\frac{|Q_{\rm C}|}{ W}=\frac{Q_{\rm C}}{ W} : {\rm COP}_{\rm heating}=\frac{| Q_{\rm H}|}{ W}=\frac{Q_{\rm C} + W}{ W} = {\rm COP}_{\rm cooling} + 1 where * Q_{\rm C} > 0 \ is the heat removed from the cold reservoir and added to the system; * Q_{\rm H} < 0 \ is the heat given off to the hot reservoir; it is lost by the system and therefore negative. (see heat). So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. Sometimes, the term efficiency is used for the ratio of the achieved COP to the Carnot COP, which can not exceed 100%. ==Energy efficiency== The 'thermal efficiency' is sometimes called the energy efficiency. Examples are: *friction of moving parts *inefficient combustion *heat loss from the combustion chamber *departure of the working fluid from the thermodynamic properties of an ideal gas *aerodynamic drag of air moving through the engine *energy used by auxiliary equipment like oil and water pumps. *inefficient compressors and turbines *imperfect valve timing These factors may be accounted when analyzing thermodynamic cycles, however discussion of how to do so is outside the scope of this article. ==Energy conversion== For a device that converts energy from another form into thermal energy (such as an electric heater, boiler, or furnace), the thermal efficiency is :\eta_{\rm th} \equiv \frac{|Q_{\rm out}|}{Q_{\rm in}} where the Q quantities are heat-equivalent values. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout < 0 into the surroundings: :Q_{in} = |W_{\rm out}| + |Q_{\rm out}| The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. An electric resistance heater has a thermal efficiency close to 100%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. ",-3.141592,4.85,"""2.9""",0.5,0.6957,B
+"An air conditioner is a refrigerator with the inside of the house acting as the cold reservoir and the outside atmosphere acting as the hot reservoir. Assume that an air conditioner consumes $1.70 \times 10^3 \mathrm{~W}$ of electrical power, and that it can be idealized as a reversible Carnot refrigerator. If the coefficient of performance of this device is 3.30 , how much heat can be extracted from the house in a day?","Performance of absorption refrigerator chillers is typically much lower, as they are not heat pumps relying on compression, but instead rely on chemical reactions driven by heat. == Equation == The equation is: :{\rm COP} = \frac{|Q|}{ W} where * Q \ is the useful heat supplied or removed by the considered system (machine). Heat pumps are measured by the efficiency with which they give off heat to the hot reservoir, COPheating; refrigerators and air conditioners by the efficiency with which they take up heat from the cold space, COPcooling: :\mathrm{COP}_{\mathrm{heating}} \equiv \frac{|Q_{\rm H}|}{W_{\rm in}} = \frac{Q_{\rm C} + W_{\rm in}}{W_{\rm in}} = \mathrm{COP}_{\mathrm{cooling}}+1\, :\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_{\rm C}}{W_{\rm in}}\, The reason the term ""coefficient of performance"" is used instead of ""efficiency"" is that, since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work, so the COP can be greater than 1 (100%). Less work is required to move heat than for conversion into heat, and because of this, heat pumps, air conditioners and refrigeration systems can have a coefficient of performance greater than one. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is: :\mathrm{COP}_{\mathrm{heating}} \le \frac{T_{\rm H}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{heating,Carnot} :\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_{\rm C}}{T_{\rm H} - T_{\rm C}}=\mathrm{COP}_\mathrm{cooling,Carnot} The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator since :\mathrm{COP}_{\mathrm{heating}} = \mathrm{COP}_{\mathrm{cooling}} + 1 This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted by-product. For example, if an automobile engine burns gasoline at a temperature of T_{\rm H} = 816^\circ \text{C} = 1500^\circ \text{F} = 1089 \text{K} and the ambient temperature is T_{\rm C} = 21^\circ \text{C} = 70^\circ \text{F} = 294 \text{K}, then its maximum possible efficiency is: :\eta_{\rm th} \le \left (1 - \frac{294 K}{1089 K} \right ) 100\% = 73.0\% It can be seen that since T_{\rm C} is fixed by the environment, the only way for a designer to increase the Carnot efficiency of an engine is to increase T_{\rm H}, the temperature at which the heat is added to the engine. The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work (energy) required. For energy- conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the annual fuel use efficiency (AFUE).HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, US, 2004 ===Heat exchangers=== A counter flow heat exchanger is the most efficient type of heat exchanger in transferring heat energy from one circuit to the other. Not stating whether an efficiency is HHV or LHV renders such numbers very misleading. ==Heat pumps and refrigerators== Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. At maximum theoretical efficiency, therefore : {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} which is equal to the reciprocal of the thermal efficiency of an ideal heat engine, because a heat pump is a heat engine operating in reverse.Borgnakke, C., & Sonntag, R. (2013). Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency, : {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{- \ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}} {\rm COP}_{\rm heating} applies to heat pumps and {\rm COP}_{\rm cooling} applies to air conditioners and refrigerators. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is taken up from the cold reservoir (QC) is equal to the magnitude of the total heat energy given off to the hot reservoir (|QH|) :|Q_{\rm H}| = Q_{\rm C} + W_{\rm in} Their efficiency is measured by a coefficient of performance (COP). However, for a more complete picture of heat exchanger efficiency, exergetic considerations must be taken into account. Thermal efficiency is defined as :\eta_{\rm th} \equiv \frac{|W_{\rm out}|}{Q_{\rm in}} = \frac{ {Q_{\rm in}} - |Q_{\rm out}|} {Q_{\rm in}} = 1 - \frac{|Q_{\rm out}|}{Q_{\rm in}} The efficiency of even the best heat engines is low; usually below 50% and often far below. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. Sometimes, the term efficiency is used for the ratio of the achieved COP to the Carnot COP, which can not exceed 100%. ==Energy efficiency== The 'thermal efficiency' is sometimes called the energy efficiency. Examples are: *friction of moving parts *inefficient combustion *heat loss from the combustion chamber *departure of the working fluid from the thermodynamic properties of an ideal gas *aerodynamic drag of air moving through the engine *energy used by auxiliary equipment like oil and water pumps. *inefficient compressors and turbines *imperfect valve timing These factors may be accounted when analyzing thermodynamic cycles, however discussion of how to do so is outside the scope of this article. ==Energy conversion== For a device that converts energy from another form into thermal energy (such as an electric heater, boiler, or furnace), the thermal efficiency is :\eta_{\rm th} \equiv \frac{|Q_{\rm out}|}{Q_{\rm in}} where the Q quantities are heat-equivalent values. So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout < 0 into the surroundings: :Q_{in} = |W_{\rm out}| + |Q_{\rm out}| The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants, in which a steam turbine is powered by exhaust heat from a gas turbine, it can approach 60%. An electric resistance heater has a thermal efficiency close to 100%. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. Seasonal efficiency gives an indication on how efficiently a heat pump operates over an entire cooling or heating season. ==See also== * Seasonal energy efficiency ratio (SEER) * Seasonal thermal energy storage (STES) * Heating seasonal performance factor (HSPF) * Power usage effectiveness (PUE) * Thermal efficiency * Vapor-compression refrigeration * Air conditioner * HVAC ==Notes== == External links == *Discussion on changes to COP of a heat pump depending on input and output temperatures *See COP definition in Cap XII of the book Industrial Energy Management - Principles and Applications Category:Heat pumps Category:Heating, ventilation, and air conditioning Category:Dimensionless numbers of thermodynamics Category:Engineering ratios ",0.5117,4.85,"""7.136""",9.13,234.4,B
+"You have collected a tissue specimen that you would like to preserve by freeze drying. To ensure the integrity of the specimen, the temperature should not exceed $-5.00{ }^{\circ} \mathrm{C}$. The vapor pressure of ice at $273.16 \mathrm{~K}$ is $624 \mathrm{~Pa}$. What is the maximum pressure at which the freeze drying can be carried out?","Usually, the freezing temperatures are between and . ===Primary drying=== During the primary drying phase, the pressure is lowered (to the range of a few millibars), and enough heat is supplied to the material for the ice to sublimate. It is important to note that, in this range of pressure, the heat is brought mainly by conduction or radiation; the convection effect is negligible, due to the low air density. ===Secondary drying=== thumb|A benchtop manifold freeze-drier The secondary drying phase aims to remove unfrozen water molecules, since the ice was removed in the primary drying phase. After the freeze-drying process is complete, the vacuum is usually broken with an inert gas, such as nitrogen, before the material is sealed. For increased efficiency, the condenser temperature should be 20 °C (36°F) less than the product during primary drying and have a defrosting mechanism to ensure that the maximum amount of water vapor in the air is condensed. ==== Shelf fluid ==== The amount of heat energy needed at times of the primary and secondary drying phase is regulated by an external heat exchanger. The freezing phase is the most critical in the whole freeze-drying process, as the freezing method can impact the speed of reconstitution, duration of freeze-drying cycle, product stability, and appropriate crystallization. In this phase, the temperature is raised higher than in the primary drying phase, and can even be above , to break any physico-chemical interactions that have formed between the water molecules and the frozen material. Therefore, freeze-drying is often reserved for materials that are heat- sensitive, such as proteins, enzymes, microorganisms, and blood plasma. At the end of the operation, the final residual water content in the product is extremely low, around 1–4%. == Applications of freeze drying == Freeze-drying causes less damage to the substance than other dehydration methods using higher temperatures. In bacteriology freeze-drying is used to conserve special strains. The frost point for a given parcel of air is always higher than the dew point, as breaking the stronger bonding between water molecules on the surface of ice compared to the surface of (supercooled) liquid water requires a higher temperature. == See also == * Bubble point * Carburetor heat * Hydrocarbon dew point * Psychrometrics * Thermodynamic diagrams ==References== ==External links== * Often Needed Answers about Temp, Humidity & Dew Point from the sci.geo.meteorology Category:Atmospheric thermodynamics Category:Temperature Category:Psychrometrics Category:Threshold temperatures Category:Gases Category:Meteorological data and networks Category:Humidity and hygrometry sv:Luftfuktighet#Daggpunkt In April 1966, the first human body was frozen—though it had been embalmed for two months—by being placed in liquid nitrogen and stored at just above freezing. Cryogenicists use the Kelvin or Rankine temperature scale, both of which measure from absolute zero, rather than more usual scales such as Celsius which measures from the freezing point of water at sea levelCelsius, Anders (1742) ""Observationer om twänne beständiga grader på en thermometer"" (Observations about two stable degrees on a thermometer), Kungliga Svenska Vetenskapsakademiens Handlingar (Proceedings of the Royal Swedish Academy of Sciences), 3 : 171–180 and Fig. 1.Don Rittner; Ronald A. Bailey (2005): Encyclopedia of Chemistry. Modern freeze drying began as early as 1890 by Richard Altmann who devised a method to freeze dry tissues (either plant or animal), but went virtually unnoticed until the 1930s. Because the final freeze dried product is porous, complete re-hydration can occur in the food. A significant turning point for freeze drying occurred during World War II when blood plasma and penicillin were needed to treat the wounded in the field. Freeze drying, also known as lyophilization or cryodesiccation, is a low temperature dehydration process that involves freezing the product and lowering pressure, removing the ice by sublimation. Cryonics uses temperatures below −130 °C, called cryopreservation, in an attempt to preserve enough brain information to permit the future revival of the cryopreserved person. thumb|Phase diagram of water Ice VI is a form of ice that exists at high pressure at the order of about 1 GPa (= 10 000 bar) and temperatures ranging from 130 up to 355 Kelvin (−143 °C up to 82 °C); see also the phase diagram of water. Freeze-drying is commonly used to preserve crustaceans, fish, amphibians, reptiles, insects, and smaller mammals. Freeze-drying is known to result in the highest quality of foods amongst all drying techniques because structural integrity is maintained along with preservation of flavors. The U.S. National Institute of Standards and Technology considers the field of cryogenics as that involving temperatures below -153 Celsius (120K; -243.4 Fahrenheit) Discovery of superconducting materials with critical temperatures significantly above the boiling point of nitrogen has provided new interest in reliable, low cost methods of producing high temperature cryogenic refrigeration. Hence, to avoid this issue, mass spectrometers are used to identify vapors released by silicone oil to immediately take corrective action and prevent contamination of the product. === Products === Mammalian cells generally do not survive freeze drying even though they still can be preserved. ==Equipment and types of freeze dryers== thumb|Unloading trays of freeze-dried material from a small cabinet-type freeze-dryer thumb|A residential freeze-dryer, along with the vacuum pump, and a cooling fan for the pump There are many types of freeze- dryers available, however, they usually contain a few essential components. ",2,14,"""425.0""",1.7,-20,C
+The molar constant volume heat capacity for $\mathrm{I}_2(\mathrm{~g})$ is $28.6 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. What is the vibrational contribution to the heat capacity? You can assume that the contribution from the electronic degrees of freedom is negligible.,"On the other hand, the vibration modes only start to become active around 350 K (77 °C) Accordingly, the molar heat capacity cP,m is nearly constant at 29.1 J⋅K−1⋅mol−1 from 100 K to about 300 °C. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. The molar heat capacity is an ""intensive"" property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. So, for example, the atom-molar heat capacity of water is 1/3 of its molar heat capacity, namely 25.3 J⋅K−1⋅mol−1. Generally, the most notable constant parameter is the volumetric heat capacity (at least for solids) which is around the value of 3 megajoule per cubic meter per kelvin:Ashby, Shercliff, Cebon, Materials, Cambridge University Press, Chapter 12: Atoms in vibration: material and heat \rho c_p \simeq 3\,\text{MJ}/(\text{m}^3{\cdot}\text{K})\quad \text{(solid)} Note that the especially high molar values, as for paraffin, gasoline, water and ammonia, result from calculating specific heats in terms of moles of molecules. The molar heat capacity of a substance has the same dimension as the heat capacity of an object; namely, L2⋅M⋅T−2⋅Θ−1, or M(L/T)2/Θ. Said another way, the atom-molar heat capacity of a solid substance is expected to be 3R = 24.94 J⋅K−1⋅mol−1, where ""amol"" denotes an amount of the solid that contains the Avogadro number of atoms. As in the case f gases, some of the vibration modes will be ""frozen out"" at low temperatures, especially in solids with light and tightly bound atoms, causing the atom-molar heat capacity to be less than this theoretical limit. Here, it predicts higher heat capacities than are actually found, with the difference due to higher-energy vibrational modes not being populated at room temperatures in these substances. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . These extra degrees of freedom contribute to the molar heat capacity of the substance. The modern theory of the heat capacity of solids states that it is due to lattice vibrations in the solid. == History == Experimentally Pierre Louis Dulong and Alexis Thérèse Petit had found in 1819 that the heat capacity per weight (the mass-specific heat capacity) for 13 measured elements was close to a constant value, after it had been multiplied by a number representing the presumed relative atomic weight of the element. The molar heat capacity of the gas will then be determined only by the ""active"" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.Quantum Physics and the Physics of large systems, Part 1A Physics, University of Cambridge, C.G. Smith, 2008. right|thumb|upright=1.25|Constant- volume specific heat capacity of a diatomic gas (idealised). According to Mayer's relation, the molar heat capacity at constant pressure would be :cP,m = cV,m \+ R = fR + R = (f + 2)R Thus, each additional degree of freedom will contribute R to the molar heat capacity of the gas (both cV,m and cP,m). Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. If specific heat is expressed per mole of atoms for these substances, none of the constant-volume values exceed, to any large extent, the theoretical Dulong–Petit limit of 25 J⋅mol−1⋅K−1 = 3 R per mole of atoms (see the last column of this table). Since the molar heat capacity of a substance is the specific heat c times the molar mass of the substance M/N its numerical value is generally smaller than that of the specific heat. This agreement is because in the classical statistical theory of Ludwig Boltzmann, the heat capacity of solids approaches a maximum of 3R per mole of atoms because full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom, each corresponding to a quadratic kinetic energy term and a quadratic potential energy term. The vibrational temperature is used commonly when finding the vibrational partition function. The following table shows the experimental molar heat capacities at constant pressure cP,m of the above polyatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2: 25 °C 25 °C 500 °C 500 °C 5000 °C 5000 °C Gas cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* cP,m J⋅K−1⋅mol−1 f* f Notes N≡N=O 38.6 7.3 51.8 10.5 62.0 12.9 13 ""Nitrous oxide"" NIST Chemistry WebBook, SRD 69, online. ",7.82,13.2,"""6.0""",420,635.7,A
+The diffusion coefficient for $\mathrm{CO}_2$ at $273 \mathrm{~K}$ and $1 \mathrm{~atm}$ is $1.00 \times 10^{-5} \mathrm{~m}^2 \mathrm{~s}^{-1}$. Estimate the collisional cross section of $\mathrm{CO}_2$ given this diffusion coefficient.,"Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. For a reaction between A and B, the collision frequency calculated with the hard-sphere model with the unit number of collisions per m3 per second is: : Z = n_\text{A} n_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} = 10^6N_A^2\text{[A][B]} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}} where: *σAB is the reaction cross section (unit m2), the area when two molecules collide with each other, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where rA the radius of A and rB the radius of B in unit m. * kB is the Boltzmann constant unit J⋅K−1. Jixin Chen proposed a finite-time solution to the diffusion flux in 2022 which significantly changes the estimated collision frequency of two particles in a solution. ==Rate equations== The rate for a bimolecular gas-phase reaction, A + B → product, predicted by collision theory is : r(T) = kn_\text{A}n_\text{B}= Z \rho \exp \left( \frac{-E_\text{a}}{RT} \right) where: *k is the rate constant in units of (number of molecules)−1⋅s−1⋅m3. * nA is the number density of A in the gas in units of m−3. * nB is the number density of B in the gas in units of m−3. For the diffusive collision, at the infinite time limit when the molecular flux can be calculated from the Fick's laws of diffusion, in 1916 Smoluchowski derived a collision frequency between molecule A and B in a diluted solution: : Z_{AB} = 4 \pi R D_r C_A C_B where: * Z_{AB} is the collision frequency, unit #collisions/s in 1 m3 of solution. Thus the probability of collision should be calculated using the Brownian motion model, which can be approximated to a diffusive flux using various boundary conditions that yield different equations in the Smoluchowski model and the JChen Model. * [A] and [B] are the molar concentration of A and B respectively, unit mol/L. * k is the diffusive collision rate constant, unit L4/3 mol-4/3 s−1. ==See also== * Two-dimensional gas * Rate equation ==References== == External links == *Introduction to Collision Theory Category:Chemical kinetics * A is the area of the collision cross-section in unit m2. * \beta is the product of the unitless fractions of reactive surface area on A and B. * D_r is the relative diffusion constant between A and B, unit m2/s. * D_r is the relative diffusion constant between A and B, unit m2/s, and D_r = D_A + D_B. * C_A and C_B are the number concentrations of molecules A and B in the solution respectively, unit #molecule/m3. or : Z_{AB} = 1000 N_A * 4 \pi R D_r [A] [B] = k [A] [B] where: * Z_{AB} is in unit mole collisions/s in 1 L of solution. For a diluted solution in the gas or the liquid phase, the collision equation developed for neat gas is not suitable when diffusion takes control of the collision frequency, i.e., the direct collision between the two molecules no longer dominates. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. * [A] and [B] are the molar concentration of A and B respectively, unit mol/L. * k is the diffusive collision rate constant, unit L mol−1 s−1. This hypothesis yields a fractal reaction kinetic rate equation of diffusive collision in a diluted solution: : Z_{AB} = (1000 N_A)^{4/3} * 8 \pi^{-1} A \beta D_r ([A] + [B])^{1/3}[A] [B] = k ([A] + [B])^{1/3}[A] [B] where: * Z_{AB} is in unit mole collisions/s in 1 L of solution. * R is the radius of the collision cross-section, unit m. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by \left(\frac{R_1}{R_2}\right)^{{3}/{8}} Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material. * e = coefficient of restitution * Sy = dynamic yield strength (dynamic ""elastic limit"") * E′ = effective elastic modulus * ρ = density * v = velocity at impact * μ = Poisson's ratio e = 3.1 \left(\frac{S_\text{y}}{1}\right)^{5/8} \left(\frac{1}{E'}\right)^{1/2} \left(\frac{1}{v}\right)^{1/4} \left(\frac{1}{\rho}\right)^{1/8} E' = \frac{E}{1-\mu^2} This equation overestimates the actual COR. : Experimental rate constants compared to the ones predicted by collision theory for reactions in solutionE.A. Moelwyn- Hughes, The kinetics of reactions in solution, 2nd ed, page 71. Reaction Solvent A, 1011 s−1⋅M−1 Z, 1011 s−1⋅M−1 Steric factor C2H5Br + OH− ethanol 4.30 3.86 1.11 C2H5O− \+ CH3I ethanol 2.42 1.93 1.25 ClCH2CO2− \+ OH− water 4.55 2.86 1.59 C3H6Br2 \+ I− methanol 1.07 1.39 0.77 HOCH2CH2Cl + OH− water 25.5 2.78 9.17 4-CH3C6H4O− \+ CH3I ethanol 8.49 1.99 4.27 CH3(CH2)2Cl + I− acetone 0.085 1.57 0.054 C5H5N + CH3I C2H2Cl4 — — 2.0 10 ==Alternative collision models for diluted solutions== Collision in diluted gas or liquid solution is regulated by diffusion instead of direct collisions, which can be calculated from Fick's laws of diffusion. Theoretical models to calculate the collision frequency in solutions have been proposed by Marian Smoluchowski in a seminal 1916 publication at the infinite time limit, and Jixin Chen in 2022 at a finite-time approximation. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. A scheme of comparing the rate equations in pure gas and solution is shown in the right figure. thumb|A scheme comparing direct collision and diffusive collision, with corresponding rate equations. ",0.318,4.5,"""76.0""", 7.0,234.4,A
+"Benzoic acid, $1.35 \mathrm{~g}$, is reacted with oxygen in a constant volume calorimeter to form $\mathrm{H}_2 \mathrm{O}(l)$ and $\mathrm{CO}_2(g)$ at $298 \mathrm{~K}$. The mass of the water in the inner bath is $1.55 \times$ $10^3 \mathrm{~g}$. The temperature of the calorimeter and its contents rises $2.76 \mathrm{~K}$ as a result of this reaction. Calculate the calorimeter constant.","The theoretical bases of indirect calorimetry: a review."" Another source gives :Energy (kcal/min) = (respiration in L/min times change in percentage oxygen) / 20 This corresponds to: :Metabolic rate (cal per minute) = 5 (VO2 in mL/min) ==References== ==Further reading== * Category:Calorimetry In SI units, the calorimeter constant is then calculated by dividing the change in enthalpy (ΔH) in joules by the change in temperature (ΔT) in kelvins or degrees Celsius: :C_\mathrm{cal} = \frac{\Delta{H}}{\Delta{T}} The calorimeter constant is usually presented in units of joules per degree Celsius (J/°C) or joules per kelvin (J/K). Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The formula can also be written for units of calories per day where VO2 is oxygen consumption expressed in millilitres per minute and VCO2 is the rate of carbon dioxide production in millilitres per minute. Regardless of the specific chemical process, with a known calorimeter constant and a known change in temperature the heat added to the system may be calculated by multiplying the calorimeter constant by that change in temperature. ==See also== *Thermodynamics ==References== Category:Calorimetry Category:Thermochemistry Metabolism. 1988 Mar;37(3):287-301. ==Scientific background== Indirect calorimetry measures O2 consumption and CO2 production. The Weir formula is a formula used in indirect calorimetry, relating metabolic rate to oxygen consumption and carbon dioxide production. Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). thumb|Original RC1 Calorimeter A reaction calorimeter is a calorimeter that measures the amount of energy released (exothermic) or absorbed (endothermic) by a chemical reaction. Journal of Thermal Analysis and Calorimetry, 147(17), 9301–9351. Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics ""Measuring RMR with Indirect Calorimetry (IC)."" Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. We know from the heat balance equation that: :Q = mf.Cpf.Tin \- Tout We also know that from the heat flow equation that :Q = U.A.LMTD We can therefore rearrange this such that :U = mf.Cpf.Tin \- Tout /A.LMTD This will allow us therefore to monitor U as a function of time. ==Continuous Reaction Calorimeter== thumb|Original Contiplant Calorimeter The Continuous Reaction Calorimeter is especially suitable to obtain thermodynamic information for a scale-up of continuous processes in tubular reactors. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). According to original source, it says: :Metabolic rate (kcal per day) = 1440 (3.9 VO2 \+ 1.1 VCO2) where VO2 is oxygen consumption in litres per minute and VCO2 is the rate of carbon dioxide production in litres per minute. A calorimeter constant (denoted Ccal) is a constant that quantifies the heat capacity of a calorimeter. The profile of the curve is determined by the c-value, which is calculated using the equation: ::: c = n K_a M where n is the stoichiometry of the binding, K_a is the association constant and M is the concentration of the molecule in the cell.Quick Start: Isothermal Titration Calorimetry (ITC) (2016). The molecular formula C20H40O2 (molar mass: 312.53 g/mol, exact mass: 312.3028 u) may refer to: * Arachidic acid, also called eicosanoic acid * Phytanic acid Category:Molecular formulas It may be calculated by applying a known amount of heat to the calorimeter and measuring the calorimeter's corresponding change in temperature. The calorimeter has a gas collector that adapts to the subject and through a unidirectional valve minute by minute collects and quantifies the volume and concentration of O2 inspired and CO2 expired by the subject. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. ",1.27,1.60,"""313.0""",0.5,6.64,E
+The activation energy for a reaction is $50 . \mathrm{J} \mathrm{mol}^{-1}$. Determine the effect on the rate constant for this reaction with a change in temperature from $273 \mathrm{~K}$ to $298 \mathrm{~K}$.,"The rate constant as a function of thermodynamic temperature is then given by: k(T) = Ae^{- E_\mathrm{a}/RT} The reaction rate is given by: r = Ae^{ - E_\mathrm{a}/RT}[\mathrm{A}]^m[\mathrm{B}]^n, where Ea is the activation energy, and R is the gas constant, and m and n are experimentally determined partial orders in [A] and [B], respectively. For the above reaction, one can expect the change of the reaction rate constant (based either on mole fraction or on molar concentration) with pressure at constant temperature to be: : \left(\frac{\partial \ln k_x}{\partial P} \right)_T = -\frac{\Delta V^{\ddagger}} {RT} In practice, the matter can be complicated because the partial molar volumes and the activation volume can themselves be a function of pressure. For a given reaction, the ratio of its rate constant at a higher temperature to its rate constant at a lower temperature is known as its temperature coefficient, (Q). For a one-step process taking place at room temperature, the corresponding Gibbs free energy of activation (ΔG‡) is approximately 23 kcal/mol. ==Dependence on temperature== The Arrhenius equation is an elementary treatment that gives the quantitative basis of the relationship between the activation energy and the reaction rate at which a reaction proceeds. This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics The second molecule of H2 does not appear in the rate equation because it reacts in the third step, which is a rapid step after the rate-determining step, so that it does not affect the overall reaction rate. ==Temperature dependence== Each reaction rate coefficient k has a temperature dependency, which is usually given by the Arrhenius equation: : k = A e^{ - \frac{E_\mathrm{a}}{RT} }. It can be done with the help of computer simulation software. ==Rate constant calculations== Rate constant can be calculated for elementary reactions by molecular dynamics simulations. The temperature dependence of ΔG‡ is used to compute these parameters, the enthalpy of activation ΔH‡ and the entropy of activation ΔS‡, based on the defining formula ΔG‡ = ΔH‡ − TΔS‡. The reaction rate thus defined has the units of mol/L/s. In these equations k(T) is the reaction rate coefficient or rate constant, although it is not really a constant, because it includes all the parameters that affect reaction rate, except for time and concentration. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. As temperature increases, the kinetic energy of the reactants increases. A rise of ten degrees Celsius results in approximately twice the reaction rate. For a reaction between reactants A and B to form a product C, where :A and B are reactants :C is a product :a, b, and c are stoichiometric coefficients, the reaction rate is often found to have the form: r = k[\mathrm{A}]^m [\mathrm{B}]^{n} Here is the reaction rate constant that depends on temperature, and [A] and [B] are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution. A=\frac{k}{e^{-E_a/RT}} The units of the pre-exponential factor A are identical to those of the rate constant and will vary depending on the order of the reaction. The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit time. As a rule of thumb, reaction rates for many reactions double for every ten degrees Celsius increase in temperature. For a unimolecular step the reaction rate is described by r = k_1[\mathrm{A}], where k_1 is a unimolecular rate constant. By using the mass balance for the system in which the reaction occurs, an expression for the rate of change in concentration can be derived. The rate ratio at a temperature increase of 10 degrees (marked by points) is equal to the Q10 coefficient. For a bimolecular step the reaction rate is described by r=k_2[\mathrm{A}][\mathrm{B}], where k_2 is a bimolecular rate constant. Substitution of this equation in the previous equation leads to a rate equation expressed in terms of the original reactants : v = k_2 K_1 [\ce{H2}] [\ce{NO}]^2 \, This agrees with the form of the observed rate equation if it is assumed that . ",0.15,11,"""6.1""",4500,8.87,A
+"How long will it take to pass $200 . \mathrm{mL}$ of $\mathrm{H}_2$ at $273 \mathrm{~K}$ through a $10 . \mathrm{cm}$-long capillary tube of $0.25 \mathrm{~mm}$ if the gas input and output pressures are 1.05 and $1.00 \mathrm{~atm}$, respectively?","This pressure difference can be calculated from Laplace's pressure equation, :\Delta P=\frac{2 \gamma}{R}. 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. This time the equation is solved for the height of the meniscus level which again can be used to give the capillary length. One can reorganize to show the capillary length as a function of surface tension and gravity. :\lambda_{\rm c}^2=\frac{hr}{2\cos\theta}, with h the height of the liquid, r the radius of the capillary tube, and \theta the contact angle. If the temperature is 20o then \lambda_c= 2.71mm The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. On the other hand, the capillary length would be {\lambda \scriptscriptstyle c} = 6.68mm for water-air on the moon. For molecular fluids, the interfacial tensions and density differences are typically of the order of 10-100 mN m−1 and 0.1-1 g mL−1 respectively resulting in a capillary length of \sim3 mm for water and air at room temperature on earth. thumb|292x292px|The capillary length will vary for different liquids and different conditions. As above, the Laplace and hydrostatic pressure are equated resulting in :R= \frac{\gamma}{\Delta \rho g e_0}=\frac{\lambda_{\rm c}^2}{e_0}. Using the same premises of capillary rise, one can find the capillary length as a function of the volume increase, and wetting perimeter of the capillary walls. ====Association with a sessile droplet==== Another way to find the capillary length is using different pressure points inside a sessile droplet, with each point having a radius of curvature, and equate them to the Laplace pressure equation. 200px|right|thumb|Diagram of a balloon catheter. The equation for \lambda_{\rm c} can also be found with an extra \sqrt{2} term, most often used when normalising the capillary height. == Origin == ===Theoretical=== One way to theoretically derive the capillary length, is to imagine a liquid droplet at the point where surface tension balances gravity. There exists a pressure difference either side of this curvature, and when this balances out the pressure due to gravity, one can rearrange to find the capillary length. For a soap bubble, the surface tension must be divided by the mean thickness, resulting in a capillary length of about 3 meters in air! This was a mathematical explanation of the work published by James Jurin in 1719, where he quantified a relationship between the maximum height taken by a liquid in a capillary tube and its diameter – Jurin's law. The capillary length can then be worked out the same way except that the thickness of the film, e_0 must be taken into account as the bubble has a hollow center, unlike the droplet which is a solid. However if these conditions are known, the capillary length can be considered a constant for any given liquid, and be used in numerous fluid mechanical problems to scale the derived equations such that they are valid for any fluid. When a capillary tube is inserted into a liquid, the liquid will rise or fall in the tube, due to an imbalance in pressure. thumb|upright=0.4|right|Diagram of a Durham Tube Durham tubes are used in microbiology to detect production of gas by microorganisms. Therefore the bond number can be written as :\mathrm{Bo}=\left(\frac{L}{\lambda_{\rm c}}\right)^2, with \lambda_{\rm c} the capillary length. This property is usually used by physicists to estimate the height a liquid will rise in a particular capillary tube, radius known, without the need for an experiment. The pressure due to gravity (hydrostatic pressure) P_{\rm h} of a column of liquid is given by :P_{\rm h}=\rho g h=2\rho g\lambda_{\rm c} , where \rho is the droplet density, g the gravitational acceleration, and h=2\lambda_{\rm c} is the height of the droplet. ",22,0.9984,"""635.7""",-75,0.0245,A
+Calculate the Debye-Hückel screening length $1 / \kappa$ at $298 \mathrm{~K}$ in a $0.0075 \mathrm{~m}$ solution of $\mathrm{K}_3 \mathrm{PO}_4$.,"Today, \kappa^{-1} is called the Debye screening length. The extended Debye–Hückel equation provides accurate results for μ ≤ 0.1. In the context of solids, Thomas–Fermi screening length may be required instead of Debye length. == See also == * Bjerrum length * Debye–Falkenhagen effect * Plasma oscillation * Shielding effect * Screening effect == References == == Further reading == * * Category:Electricity Category:Electronics concepts Category:Colloidal chemistry Category:Plasma parameters Category:Electrochemistry Category:Length Category:Peter Debye In plasmas and electrolytes, the Debye length \lambda_{\rm D} (Debye radius or Debye–Hückel screening length), is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. The corresponding Debye screening wave vector k_{\rm D}=1/\lambda_{\rm D} for particles of density n, charge q at a temperature T is given by k_{\rm D}^2=4\pi n q^2/(k_{\rm B}T) in Gaussian units. P^\text{ex} = -\frac{k_\text{B} T \kappa_\text{cgs}^3}{24\pi} = -\frac{k_\text{B} T \left(\frac{4\pi \sum_j c_j q_j}{\varepsilon_0 \varepsilon_r k_\text{B} T }\right)^{3/2}}{24\pi}. This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of Meyer-Vernet N (1993) Aspects of Debye shielding. The first is what could be called the square of the reduced inverse screening length, (\kappa a)^2. The Debye–Hückel length may be expressed in terms of the Bjerrum length \lambda_{\rm B} as \lambda_{\rm D} = \left(4 \pi \, \lambda_{\rm B} \, \sum_{j = 1}^N n_j^0 \, z_j^2\right)^{-1/2}, where z_j = q_j/e is the integer charge number that relates the charge on the j-th ionic species to the elementary charge e. == In a plasma == For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. The equation is \ln(\gamma_i) = -\frac{z_i^2 q^2 \kappa}{8 \pi \varepsilon_r \varepsilon_0 k_\text{B} T} = -\frac{z_i^2 q^3 N^{1/2}_\text{A}}{4 \pi (\varepsilon_r \varepsilon_0 k_\text{B} T)^{3/2}} \sqrt{10^3\frac{I}{2}} = -A z_i^2 \sqrt{I}, where * z_i is the charge number of ion species i, * q is the elementary charge, * \kappa is the inverse of the Debye screening length \lambda_{\rm D} (defined below), * \varepsilon_r is the relative permittivity of the solvent, * \varepsilon_0 is the permittivity of free space, * k_\text{B} is the Boltzmann constant, * T is the temperature of the solution, * N_\mathrm{A} is the Avogadro constant, * I is the ionic strength of the solution (defined below), * A is a constant that depends on temperature. The Debye length of semiconductors is given: L_{\rm D} = \sqrt{\frac{\varepsilon k_{\rm B} T}{q^2 N_{\rm dop}}} where * ε is the dielectric constant, * kB is the Boltzmann constant, * T is the absolute temperature in kelvins, * q is the elementary charge, and * Ndop is the net density of dopants (either donors or acceptors). Debye length is an important parameter in plasma physics, electrolytes, and colloids (DLVO theory). It is possible that Debye–Hückel equation is not able to foresee this behavior because of the linearization of the Poisson–Boltzmann equation, or maybe not: studies about this have been started only during the last years of the 20th century because before it wasn't possible to investigate the 10−4 M region, so it is possible that during the next years new theories will be born. ==Extensions of the theory== A number of approaches have been proposed to extend the validity of the law to concentration ranges as commonly encountered in chemistry One such extended Debye–Hückel equation is given by: \- \log_{10}(\gamma) = \frac{A|z_+z_-|\sqrt{I}}{1 + Ba\sqrt{I}} where \gamma as its common logarithm is the activity coefficient, z is the integer charge of the ion (1 for H+, 2 for Mg2+ etc.), As the only characteristic length scale in the Debye–Hückel equation, \lambda_D sets the scale for variations in the potential and in the concentrations of charged species. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. thumb|left|σ Persei in optical light Sigma Persei (Sigma Per, σ Persei, σ Per) is an orange K-type giant with an apparent magnitude of +4.36. The excess osmotic pressure obtained from Debye–Hückel theory is in cgs units:http://homepages.rpi.edu/~keblip/THERMO/chapters/Chapter33.pdf, page 9. Alternatively, \kappa^{-1} = \frac{1}{\sqrt{8\pi \lambda_{\rm B} N_{\rm A} \times 10^{-24} I}} where \lambda_{\rm B} is the Bjerrum length of the medium in nm, and the factor 10^{-24} derives from transforming unit volume from cubic dm to cubic nm. The molecular formula C20H28FN3O3 (molar mass: 377.453 g/mol, exact mass: 377.2115 u) may refer to: * 5F-ADB * 5F-EMB-PINACA Category:Molecular formulas Factor out the scalar potential and assign the leftovers, which are constant, to \kappa^2. The term in parentheses divided by \varepsilon, has the units of an inverse length squared and by dimensional analysis leads to the definition of the characteristic length scale \lambda_{\rm D} = \left(\frac{\varepsilon \, k_{\rm B} T}{\sum_{j = 1}^N n_j^0 \, q_j^2}\right)^{1/2} that commonly is referred to as the Debye–Hückel length. It has been solved by the Swedish mathematician Thomas Hakon Gronwall and his collaborators physicical chemists V. K. La Mer and Karl Sandved in a 1928 article from Physikalische Zeitschrift dealing with extensions to Debye–Huckel theory, which resorted to Taylor series expansion. ",1,0.6296296296,"""1.4""",46.7,122,C
+"A system consisting of $82.5 \mathrm{~g}$ of liquid water at $300 . \mathrm{K}$ is heated using an immersion heater at a constant pressure of 1.00 bar. If a current of $1.75 \mathrm{~A}$ passes through the $25.0 \mathrm{ohm}$ resistor for 100 .s, what is the final temperature of the water?","thumb |upright=1.35 |Upper ocean heat content (OHC) has been increasing because oceans have absorbed a large portion of the excess heat trapped in the atmosphere by human-caused global warming. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The partnership between Argo and Jason measurements has yielded ongoing improvements to estimates of OHC and other global ocean properties. ==Causes for heat uptake== thumb|320px|Over 90% of the thermal energy that has accumulated on Earth from global heating since 1970 is stored in the ocean. thumbnail|300px|Global Heat Content in the top 2000 meters of the ocean since 1958 The more abundant equatorial solar irradiance which is absorbed by Earth's tropical surface waters drives the overall poleward propagation of ocean heat. From this, Ohm determined his law of proportionality and published his results. thumb|Internal resistance model In modern notation we would write, I = \frac {\mathcal E}{r+R}, where \mathcal E is the open-circuit emf of the thermocouple, r is the internal resistance of the thermocouple and R is the resistance of the test wire. thumb|170px|A typical glass-tube immersion style aquarium heater An aquarium heater is a device used in the fishkeeping hobby to warm the temperature of water in aquariums. To calculate the ocean heat content, measurements of ocean temperature at many different locations and depths are required. To calculate the ocean heat content, measurements of ocean temperature at many different locations and depths are required. Ocean heat content (OHC) is the energy absorbed and stored by oceans. thumb|400px|Diagram showing pressure difference induced by a temperature difference. The increase in OHC accounts for 30–40% of global sea-level rise from 1900 to 2020 because of thermal expansion. Additionally, a study from 2022 on anthropogenic warming in the ocean indicates that 62% of the warming from the years between 1850 and 2018 in the North Atlantic along 25°N is kept in the water below 700 m, where a major percentage of the ocean's surplus heat is stored. In 2022, the world’s oceans, as given by OHC, were again the hottest in the historical record and exceeded the previous 2021 record maximum. 50x50px Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License Ocean heat content and sea level rise are important indicators of climate change. Between 1971 and 2018, the rise in OHC accounted for over 90% of Earth’s excess thermal energy from global heating. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. A study in 2015 concluded that ocean heat content increases by the Pacific Ocean were compensated by an abrupt distribution of OHC into the Indian Ocean. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. The upper ocean (0–700 m) has warmed since 1971, while it is very likely that warming has occurred at intermediate depths (700–2000 m) and likely that deep ocean (below 2000 m) temperatures have increased. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). Measurements of temperature versus ocean depth generally show an upper mixed layer (0–200 m), a thermocline (200–1500 m), and a deep ocean layer (>1500 m). thumb|upright=1.8|Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. While the light is turned off, the value of q for the tungsten filament would be zero. == Solving the heat equation using Fourier series == right|thumb|300px|Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions. ",14.44,322,"""-131.1""",0.132,6.6,B
+"For an ensemble consisting of a mole of particles having two energy levels separated by $1000 . \mathrm{cm}^{-1}$, at what temperature will the internal energy equal $3.00 \mathrm{~kJ}$ ?","Approximate values of kT at 298 K Units kT = J kT = pN⋅nm kT = cal kT = meV kT=-174 dBm/Hz kT/hc ≈ cm−1 kT/e = 25.7 mV RT = kT ⋅ NA = kJ⋅mol−1 RT = 0.593 kcal⋅mol−1 h/kT = 0.16 ps kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, : :\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} :\, p(k)= e^{-k^2\over 2mT}. Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, quasi- particles, and photons. == Critical temperature == This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: :T_{\rm c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_{\rm B}} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_{\rm B}} where: : \,T_{\rm c} is the critical temperature, \,n the particle density, \,m the mass per boson, \hbar the reduced Planck constant, \,k_{\rm B} the Boltzmann constant and \,\zeta the Riemann zeta function; \,\zeta(3/2)\approx 2.6124. MJ/kg may refer to: * megajoules per kilogram * Specific kinetic energy * Heat of fusion * Heat of combustion kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion At temperature T, a particle will have a lesser probability to be in state |1\rangle by e^{-E/kT}. thumb|upright=1.5|Schematic Bose–Einstein condensation versus temperature of the energy diagram In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system by k. On the critical region it is possible to define a critical temperature and thermal wavelength: :\lambda_c^3=g_{3/2}(1)v=\zeta(3/2)v :T_c=\frac{2\pi \hbar^2 }{m k_B \lambda_c^2} recovering the value indicated on the previous section. The SI units for RT are joules per mole (J/mol). This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. == Derivation == === Ideal Bose gas === For an ideal Bose gas we have the equation of state: :\frac{1}{v}=\frac{1}{\lambda^3}g_{3/2}(f)+\frac{1}{V}\frac{f}{1-f} where v=V/N is the per particle volume, \lambda the thermal wavelength, f the fugacity and :g_\alpha (f)=\sum \limits_{n=1}^\infty \frac{f^n}{n^\alpha} It is noticeable that g_{3/2} is a monotonically growing function of f in f \in [0, 1], which are the only values for which the series converge. For a system in equilibrium in canonical ensemble, the probability of the system being in state with energy E is proportional to . As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. It satisfies : \frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{ \left(-\frac{\Delta E}{k T_{\rm ex}} \right) }, where * is the number of particles in an upper (e.g. excited) state; * is the statistical weight of those upper-state particles; * is the number of particles in a lower (e.g. ground) state; * is the statistical weight of those lower-state particles; * is the exponential function; * is the Boltzmann constant; * is the difference in energy between the upper and lower states. When the integral (also known as Bose–Einstein integral) is evaluated with factors of k_B and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section. ",1310,35,"""6.1""",-1,11000,A
+A muscle fiber contracts by $3.5 \mathrm{~cm}$ and in doing so lifts a weight. Calculate the work performed by the fiber. Assume the muscle fiber obeys Hooke's law $F=-k x$ with a force constant $k$ of $750 . \mathrm{N} \mathrm{m}^{-1}$.,"The actual force increased from 0 to F during the deformation; the work done can be computed by integration in distance. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The work of the net force is calculated as the product of its magnitude and the particle displacement. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. Therefore, work need only be computed for the gravitational forces acting on the bodies. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . Notice that the work done by gravity depends only on the vertical movement of the object. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. From Newton's second law, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy corresponding to the linear velocity and angular velocity of that body, W = \Delta E_\text{k}. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. The work–energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. In [Erdemir, 2007] a list of possible cost functions with a brief rationale and the suggested model validation technique is available. ==== Clarification on the use of the maximum isometric force ==== Muscle contraction can be eccentric (velocity of contraction v<0), concentric (v >0) or isometric (v=0). If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). ",0.46,62.8318530718,"""2.0""",4.85,0.75,A
 "A gas sample is known to be a mixture of ethane and butane. A bulb having a $230.0 \mathrm{~cm}^3$ capacity is filled with the gas to a pressure of $97.5 \times 10^3 \mathrm{~Pa}$ at $23.1^{\circ} \mathrm{C}$. If the mass of the gas in the bulb is $0.3554 \mathrm{~g}$, what is the mole percent of butane in the mixture?
-","Butane is one of a group of liquefied petroleum gases (LP gases). Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. For example, the density of liquid propane is 571.8±1 kg/m3 (for pressures up to 2MPa and temperature 27±0.2 °C), while the density of liquid butane is 625.5±0.7 kg/m3 (for pressures up to 2MPa and temperature -13±0.2 °C). alt=Density of liquid and vaporized butane|none|thumb|500x500px|Propane and butane density data == Isomers == Common name normal butane unbranched butane n-butane isobutane i-butane IUPAC name butane methylpropane Molecular diagram 150px 120px Skeletal diagram 120px 100px Rotation about the central C−C bond produces two different conformations (trans and gauche) for n-butane. == Reactions == When oxygen is plentiful, butane burns to form carbon dioxide and water vapor; when oxygen is limited, carbon (soot) or carbon monoxide may also be formed. The relative rates of the chlorination is partially explained by the differing bond dissociation energies, 425 and 411 kJ/mol for the two types of C-H bonds. == Uses == Normal butane can be used for gasoline blending, as a fuel gas, fragrance extraction solvent, either alone or in a mixture with propane, and as a feedstock for the manufacture of ethylene and butadiene, a key ingredient of synthetic rubber. When there is sufficient oxygen: : 2 C4H10 \+ 13 O2 → 8 CO2 \+ 10 H2O When oxygen is limited: : 2 C4H10 \+ 9 O2 → 8 CO + 10 H2O By weight, butane contains about or by liquid volume . The molecular formula C4H10 (molar mass: 58.12 g/mol, exact mass: 58.0783 u) may refer to: * Butane, or n-butane * Isobutane, also known as methylpropane or 2-methylpropane Butane is a highly flammable, colorless, easily liquefied gas that quickly vaporizes at room temperature and pressure. Butane is denser than air. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Butane did not have much practical use until the 1910s, when W. Snelling identified butane and propane as components in gasoline and found that, if they were cooled, they could be stored in a volume-reduced liquified state in pressurized containers. == Density == The density of butane is highly dependent on temperature and pressure in the reservoir. Butane is also used as lighter fuel for common lighters or butane torches and is sold bottled as a fuel for cooking, barbecues and camping stoves. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . For gasoline blending, n-butane is the main component used to manipulate the Reid vapor pressure (RVP). Butane is the most commonly abused volatile substance in the UK, and was the cause of 52% of solvent related deaths in 2000. Instead of a mole the constant can be expressed by considering the normal cubic meter. Butane () or n-butane is an alkane with the formula C4H10. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. File:Photo D2.jpg | Butane fuel canisters for use in camping stoves File:The Green Lighter 1 cropped.jpg | Butane lighter, showing liquid butane reservoir File:Aerosol.png | An aerosol spray can, which may be using butane as a propellant File:ButaneGasCylinder WhiteBack.jpg | Butane gas cylinder used for cooking == Effects and health issues == Inhalation of butane can cause euphoria, drowsiness, unconsciousness, asphyxia, cardiac arrhythmia, fluctuations in blood pressure and temporary memory loss, when abused directly from a highly pressurized container, and can result in death from asphyxiation and ventricular fibrillation. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. It provides a molar mass for air of 28.9625 g/mol, and provides a composition for standard dry air as a footnote. == References == Category:Gases When blended with propane and other hydrocarbons, the mixture may be referred to commercially as liquefied petroleum gas (LPG). ",32,0.6321205588,672.4,-214,6,A
-"One liter of fully oxygenated blood can carry 0.18 liters of $\mathrm{O}_2$ measured at $T=298 \mathrm{~K}$ and $P=1.00 \mathrm{~atm}$. Calculate the number of moles of $\mathrm{O}_2$ carried per liter of blood. Hemoglobin, the oxygen transport protein in blood has four oxygen binding sites. How many hemoglobin molecules are required to transport the $\mathrm{O}_2$ in $1.0 \mathrm{~L}$ of fully oxygenated blood?","Each hemoglobin molecule has the capacity to carry four oxygen molecules. The oxygen-carrying capacity of hemoglobin is determined by the type of hemoglobin present in the blood. Although binding of oxygen to hemoglobin continues to some extent for pressures about 50 mmHg, as oxygen partial pressures decrease in this steep area of the curve, the oxygen is unloaded to peripheral tissue readily as the hemoglobin's affinity diminishes. Venous blood with an oxygen concentration of 15 mL/100 mL would therefore lead to typical values of the a-vO2 diff at rest of around 5 mL/100 mL. To see the relative affinities of each successive oxygen as you remove/add oxygen from/to the hemoglobin from the curve compare the relative increase/decrease in p(O2) needed for the corresponding increase/decrease in s(O2). ==Factors that affect the standard dissociation curve== The strength with which oxygen binds to hemoglobin is affected by several factors. So, one will have a lesser hemoglobin saturation percentage for the same [O2] or a higher partial pressure of oxygen. The amount of oxygen bound to the hemoglobin at any time is related, in large part, to the partial pressure of oxygen to which the hemoglobin is exposed. The oxygen bound to the hemoglobin is released into the blood's plasma and absorbed into the tissues, and the carbon dioxide in the tissues is bound to the hemoglobin. The partial pressure of oxygen in the blood at which the hemoglobin is 50% saturated, typically about 26.6 mmHg (3.5 kPa) for a healthy person, is known as the P50. The binding affinity of hemoglobin to O2 is greatest under a relatively high pH. === Carbon dioxide === Carbon dioxide affects the curve in two ways. Arterial blood will generally contain an oxygen concentration of around 20 mL/100 mL. A hemoglobin molecule can bind up to four oxygen molecules in a reversible method. The T state has a lower affinity for oxygen than the R state, so with increased acidity, the hemoglobin binds less O2 for a given PO2 (and more H+). Specifically, the oxyhemoglobin dissociation curve relates oxygen saturation (SO2) and partial pressure of oxygen in the blood (PO2), and is determined by what is called ""hemoglobin affinity for oxygen""; that is, how readily hemoglobin acquires and releases oxygen molecules into the fluid that surrounds it. thumb|Structure of oxyhemoglobin ==Background== Hemoglobin (Hb) is the primary vehicle for transporting oxygen in the blood. The a-vO2 diff is usually measured in millilitres of oxygen per 100 millilitres of blood (mL/100 mL).Malpeli, Physical Education, Chapter 4: Acute Responses to Exercise, p. 106. ==Measurement== The arteriovenous oxygen difference is usually taken by comparing the difference in the oxygen concentration of oxygenated blood in the femoral, brachial, or radial artery and the oxygen concentration in the deoxygenated blood from the mixed supply found in the pulmonary artery (as an indicator of the typical mixed venous supply). As the blood circulates to other body tissue in which the partial pressure of oxygen is less, the hemoglobin releases the oxygen into the tissue because the hemoglobin cannot maintain its full bound capacity of oxygen in the presence of lower oxygen partial pressures. ==Sigmoid shape== thumb|Hemoglobin saturation curve The curve is usually best described by a sigmoid plot, using a formula of the kind: :S(t) = \frac{1}{1 + e^{-t}}. In the capillaries, where carbon dioxide is produced, oxygen bound to the hemoglobin is released into the blood's plasma and absorbed into the tissues. Hemoglobin's affinity for oxygen increases as successive molecules of oxygen bind. HbF then delivers that bound oxygen to tissues that have even lower partial pressures where it can be released. ==See also== * Automated analyzer * Bohr effect ==Notes== ==References== ==External links== * * The Interactive Oxyhemoglobin Dissociation Curve * Simulation of the parameters CO2, pH and temperature on the oxygen–hemoglobin dissociation curve (left or right shift) Category:Respiratory physiology Category:Chemical pathology Category:Hematology Category:Oxygen The 'plateau' portion of the oxyhemoglobin dissociation curve is the range that exists at the pulmonary capillaries (minimal reduction of oxygen transported until the p(O2) falls 50 mmHg). Solid oxygen forms at normal atmospheric pressure at a temperature below 54.36 K (−218.79 °C, −361.82 °F). The phosphate/oxygen ratio, or P/O ratio, refers to the amount of ATP produced from the movement of two electrons through a defined electron transport chain, terminated by reduction of an oxygen atom.Garrett & Grisham 2010, p.620. ",-4564.7,1.11,4.946,3.7,6.283185307,B
-Consider a collection of molecules where each molecule has two nondegenerate energy levels that are separated by $6000 . \mathrm{cm}^{-1}$. Measurement of the level populations demonstrates that there are exactly 8 times more molecules in the ground state than in the upper state. What is the temperature of the collection?,"Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, : :\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} :\, p(k)= e^{-k^2\over 2mT}. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. thumb|upright=1.5|Schematic Bose–Einstein condensation versus temperature of the energy diagram In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. Average yearly temperature is 22.4°C, ranging from an average minimum of 12.2°C to a maximum of 29.9°C. In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. At temperature T, a particle will have a lesser probability to be in state |1\rangle by e^{-E/kT}. Only 8% of atoms are in the ground state of the trap near absolute zero, rather than the 100% of a true condensate. Some of the warmest temperatures can be found in the thermosphere, due to its reception of strong ionizing radiation at the level of the Van Allen radiation belt. ==Temperature range== The variation in temperature that occurs from the highs of the day to the cool of nights is called diurnal temperature variation. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values. == In fiction == * In the 2016 film Spectral, the US military battles mysterious enemy creatures fashioned out of Bose–Einstein condensates. Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, quasi- particles, and photons. == Critical temperature == This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: :T_{\rm c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_{\rm B}} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_{\rm B}} where: : \,T_{\rm c} is the critical temperature, \,n the particle density, \,m the mass per boson, \hbar the reduced Planck constant, \,k_{\rm B} the Boltzmann constant and \,\zeta the Riemann zeta function; \,\zeta(3/2)\approx 2.6124. Atmospheric temperature is a measure of temperature at different levels of the Earth's atmosphere. Average maximum yearly temperature is 28.7°C and average minimum is 21.9°C. The average temperature range is 5.7°C only. Junction temperature, short for transistor junction temperature, is the highest operating temperature of the actual semiconductor in an electronic device. Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. The average temperature range is 11.4 degrees. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. ",1.07,1.44,0.3333333,"102,965.21",4152,E
-Calculate $\Delta S^{\circ}$ for the reaction $3 \mathrm{H}_2(g)+\mathrm{N}_2(g) \rightarrow$ $2 \mathrm{NH}_3(g)$ at $725 \mathrm{~K}$. Omit terms in the temperature-dependent heat capacities higher than $T^2 / \mathrm{K}^2$.,"Since \Delta_r G^\ominus = - RT \ln K_{eq}, the temperature dependence of both terms can be described by Van t'Hoff equations as a function of T. Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., ""Physical Chemistry"" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., ""Atkins' Physical Chemistry"" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The definite integral between temperatures and is then :\ln \frac{K_2}{K_1} = \frac{\Delta_r H^\ominus}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right). The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. Since in reality \Delta_r H^\ominus and the standard reaction entropy \Delta_r S^\ominus do vary with temperature for most processes, the integrated equation is only approximate. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. For a change from reactants to products at constant temperature and pressure the equation becomes \Delta G = \Delta H - T\Delta S. Differentiation of this expression with respect to the variable while assuming that both \Delta_r H^\ominus and \Delta_r S^\ominus are independent of yields the Van 't Hoff equation. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. However, the relation loses its validity when the aggregation number is also temperature-dependent, and the following relation should be used instead: :RT^2\left(\frac{\partial}{\partial T}\ln\mathrm{CMC}\right)_P = -\Delta_r H^\ominus_\mathrm{m}(N) + T\left(\frac{\partial}{\partial N}\left(G_{N+1} - G_N\right)\right)_{T,P}\left(\frac{\partial N}{\partial T}\right)_P, with and being the free energies of the surfactant in a micelle with aggregation number and respectively. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. In this equation is the equilibrium constant at absolute temperature , and is the equilibrium constant at absolute temperature . ===Development from thermodynamics=== Combining the well-known formula for the Gibbs free energy of reaction : \Delta_r G^\ominus = \Delta_r H^\ominus - T\Delta_r S^\ominus, where is the entropy of the system, with the Gibbs free energy isotherm equation: :\Delta_r G^\ominus = -RT \ln K_\mathrm{eq}, we obtain :\ln K_\mathrm{eq} = -\frac{\Delta_r H^\ominus}{RT} + \frac{\Delta_r S^\ominus}{R}. The Van 't Hoff plot can be used to find the enthalpy and entropy change for each mechanism and the favored mechanism under different temperatures. :\begin{align} \Delta_r H_1 &= - R \times \text{slope}_1, & \Delta_r S_1 &= R \times \text{intercept}_1; \\\\[5pt] \Delta_r H_2 &= - R \times \text{slope}_2, & \Delta_r S_2 &= R \times \text{intercept}_2. \end{align} In the example figure, the reaction undergoes mechanism 1 at high temperature and mechanism 2 at low temperature. ===Temperature dependence=== thumb|right|x275px|Temperature-dependent Van 't Hoff plot The Van 't Hoff plot is linear based on the tacit assumption that the enthalpy and entropy are constant with temperature changes. The slope of the line may be multiplied by the gas constant to obtain the standard enthalpy change of the reaction, and the intercept may be multiplied by to obtain the standard entropy change. ===Van 't Hoff isotherm=== The Van 't Hoff isotherm can be used to determine the temperature dependence of the Gibbs free energy of reaction for non- standard state reactions at a constant temperature: :\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_\mathrm{r}G = \Delta_\mathrm{r}G^\ominus + RT \ln Q_\mathrm{r}, where \Delta_\mathrm{r}G is the Gibbs free energy of reaction under non-standard states at temperature T, \Delta_r G^\ominus is the Gibbs free energy for the reaction at (T,P^0), \xi is the extent of reaction, and is the thermodynamic reaction quotient. We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. In practice, the equation is often integrated between two temperatures under the assumption that the standard reaction enthalpy \Delta_r H^\ominus is constant (and furthermore, this is also often assumed to be equal to its value at standard temperature). Thus, for an exothermic reaction, the Van 't Hoff plot should always have a positive slope. === Error propagation === At first glance, using the fact that it would appear that two measurements of would suffice to be able to obtain an accurate value of : :\Delta_r H^\ominus = R \frac{\ln K_1 - \ln K_2}{\frac{1}{T_2} - \frac{1}{T_1}}, where and are the equilibrium constant values obtained at temperatures and respectively. Knowing the slope and intercept from the Van 't Hoff plot, the enthalpy and entropy of a reaction can be easily obtained using :\begin{align} \Delta_r H &= - R \times \text{slope}, \\\ \Delta_r S &= R \times \text{intercept}. \end{align} The Van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. When a reaction is at equilibrium, and \Delta_\mathrm{r}G = 0. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. A first-order approximation is to assume that the two different reaction products have different heat capacities. The data should have a linear relationship, the equation for which can be found by fitting the data using the linear form of the Van 't Hoff equation :\ln K_\mathrm{eq} = -\frac{\Delta_r H^\ominus}{RT} + \frac{\Delta_r S^\ominus}{R}. ",9.30,6.9,2.0,-20,-191.2,E
-"The thermal conductivities of acetylene $\left(\mathrm{C}_2 \mathrm{H}_2\right)$ and $\mathrm{N}_2$ at $273 \mathrm{~K}$ and $1 \mathrm{~atm}$ are 0.01866 and $0.0240 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~m}^{-1} \mathrm{~s}^{-1}$, respectively. Based on these data, what is the ratio of the collisional cross section of acetylene relative to $\mathrm{N}_2$ ?","Under these assumptions, an elementary calculation yields for the thermal conductivity : k = \beta \rho \lambda c_v \sqrt{\frac{2k_\text{B} T}{\pi m}}, where \beta is a numerical constant of order 1, k_\text{B} is the Boltzmann constant, and \lambda is the mean free path, which measures the average distance a molecule travels between collisions. In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature (T). : \frac \kappa \sigma = LT Theoretically, the proportionality constant L, known as the Lorenz number, is equal to : L = \frac \kappa {\sigma T} = \frac{\pi^2} 3 \left(\frac{k_{\rm B}} e \right)^2 = 2.44\times 10^{-8}\;\mathrm{V}^2\mathrm{K}^{-2}, where kB is Boltzmann's constant and e is the elementary charge. In summary, for a plate of thermal conductivity k, area A and thickness L, *thermal conductance = kA/L, measured in W⋅K−1. **thermal resistance = L/(kA), measured in K⋅W−1. *heat transfer coefficient = k/L, measured in W⋅K−1⋅m−2. **thermal insulance = L/k, measured in K⋅m2⋅W−1. ""The Thermal Conductivity of Air at Reduced Pressures and Length Scales,"" Electronics Cooling, November 2002, http://www.electronics- cooling.com/2002/11/the-thermal-conductivity-of-air-at-reduced-pressures-and- length-scales/ Retrieved 05:20, 10 April 2016 (UTC). 273-293-298 300 600 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 hiAerosols2.95-loAerosols7.83×10−15 (78.03%N2,21%O2,+0.93%Ar,+0.04%CO2) (1 atm) The plate distance is one centimeter, the special conductivity values were calculated from the Lasance approximation formula in The Thermal conductivity of Air at Reduced Pressures and Length Scales and the primary values were taken from Weast at the normal pressure tables in the CRC handbook on page E2. From considerations of energy conservation, the heat flow between the two bodies in contact, bodies A and B, is found as: One may observe that the heat flow is directly related to the thermal conductivities of the bodies in contact, k_A and k_B, the contact area A, and the thermal contact resistance, 1/h_c, which, as previously noted, is the inverse of the thermal conductance coefficient, h_c. ==Importance== Most experimentally determined values of the thermal contact resistance fall between 0.000005 and 0.0005 m2 K/W (the corresponding range of thermal contact conductance is 200,000 to 2000 W/m2 K). For a plate of thermal conductivity k, area A and thickness L, the conductance is kA/L, measured in W⋅K−1.Bejan, p. 34 The relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity. ==Experimental values== thumb|upright=2.5|Experimental values of thermal conductivity The thermal conductivities of common substances span at least four orders of magnitude. This conductance, known as thermal boundary conductance, is due to the differences in electronic and vibrational properties between the contacting materials. Then : \kappa = L T \sigma = {\pi^2 k_{\rm B}^2 \over 3e^2}\times {n e^2 \over h} T = {\pi^2 k_{\rm B}^2 \over 3h} n T = g_0 n, where g_0 is the thermal conductance quantum defined above. ==References== ==See also== * Thermal properties of nanostructures Category:Mesoscopic physics Category:Nanotechnology Category:Quantum mechanics Category:Condensed matter physics Category:Physical quantities Category:Heat conduction Many pure metals have a peak thermal conductivity between 2 K and 10 K. In imperial units, thermal conductivity is measured in BTU/(h⋅ft⋅°F).1 Btu/(h⋅ft⋅°F) = 1.730735 W/(m⋅K) The dimension of thermal conductivity is M1L1T−3Θ−1, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ). The reciprocal of the heat transfer coefficient is thermal insulance. This table shows thermal conductivity in SI units of watts per metre-kelvin (W·m−1·K−1). The thermal contact conductance coefficient, h_c, is a property indicating the thermal conductivity, or ability to conduct heat, between two bodies in contact. Therefore, :\frac \kappa \sigma = \frac{c_V m^2 \, \langle {v} \rangle^2}{3e^2} = \frac{8}{\pi} \frac{k_{\rm B}^2T}{e^2}, which is the Wiedemann–Franz law with an erroneous proportionality constant \frac{8}{\pi}\approx 2.55; ===Free electron model=== After taking into account the quantum effects, as in the free electron model, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to \frac{\pi^2} 3\approx3.29, which agrees with experimental values. ==Temperature dependence== The value L0 = 2.44×10−8 V2K−2 results from the fact that at low temperatures (T\rightarrow 0 K) the heat and charge currents are carried by the same quasi-particles: electrons or holes. *It happens that the online record has the thermal conductivity at 30 Kelvins and \parallel to the c axis posted at 1.36 W⋅cm−1 K−1 and 78.0 Btu hr−1 ft−1 F−1 which is incorrect. NBS 6.00 \parallel to c axis, 3.90 \perp to c axis 5.00 \parallel to c axis, 3.41 \perp to c axis 4.47 \parallel to c axis, 3.12 \perp to c axis 4.19 \parallel to c axis, 3.04 \perp to c axis List 311 366 422 500 600 700 800 The noted authorities have reported some values in three digits as cited here in metric translation but they have not demonstrated three digit measurement.R.W.Powell, C.Y.Ho and P.E.Liley, Thermal Conductivity of Selected Materials, NSRDS-NBS 8, Issued 25 November 1966, pages 69, 99>Link Text Errata: The numbered references in the NSRDS-NBS-8 pdf are found near the end of the TPRC Data Book Volume 2 and not somewhere in Volume 3 like it says. In a gas, thermal conduction is mediated by discrete molecular collisions. Finally, thermal diffusivity \alpha combines thermal conductivity with density and specific heat: :\alpha = \frac{ k }{ \rho c_{p} }. In many materials, q is observed to be directly proportional to the temperature difference and inversely proportional to the separation distance L: : q = -k \cdot \frac{T_2 - T_1}{L}. In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the Wiedemann–Franz law. Following the Wiedemann–Franz law, thermal conductivity of metals is approximately proportional to the absolute temperature (in kelvins) times electrical conductivity. ",4.5,+5.41,1.33,0.375,2.25,C
-"Consider the gas phase thermal decomposition of 1.0 atm of $\left(\mathrm{CH}_3\right)_3 \mathrm{COOC}\left(\mathrm{CH}_3\right)_3(\mathrm{~g})$ to acetone $\left(\mathrm{CH}_3\right)_2 \mathrm{CO}(\mathrm{g})$ and ethane $\left(\mathrm{C}_2 \mathrm{H}_6\right)(\mathrm{g})$, which occurs with a rate constant of $0.0019 \mathrm{~s}^{-1}$. After initiation of the reaction, at what time would you expect the pressure to be $1.8 \mathrm{~atm}$ ?","However, they observed that the thermodynamics became favourable for crystalline solid acetone at the melting point (−96 °C). One litre of acetone can dissolve around 250 litres of acetylene at a pressure of .Mine Safety and Health Administration (MSHA) – Safety Hazard Information – Special Hazards of Acetylene . At temperatures greater than acetone's flash point of , air mixtures of between 2.5% and 12.8% acetone, by volume, may explode or cause a flash fire. In acetone vapor at ambient temperature, only 2.4% of the molecules are in the enol form. :300px ===Aldol condensation=== In the presence of suitable catalysts, two acetone molecules also combine to form the compound diacetone alcohol , which on dehydration gives mesityl oxide . The synthesis involves the condensation of acetone with phenol: :(CH3)2CO + 2 C6H5OH -> (CH3)2C(C6H4OH)2 + H2O Many millions of kilograms of acetone are consumed in the production of the solvents methyl isobutyl alcohol and methyl isobutyl ketone. Since thermal decomposition is a kinetic process, the observed temperature of its beginning in most instances will be a function of the experimental conditions and sensitivity of the experimental setup. The flame temperature of pure acetone is 1980 °C.Haynes, p. 15.49 ===Toxicity=== Acetone has been studied extensively and is believed to exhibit only slight toxicity in normal use. thumb|280px|Temperature-dependency of the heats of vaporization for water, methanol, benzene, and acetone In thermodynamics, the enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. CH3)2CO -> (CH3)2C(OH)CH2C(O)CH3 Condensation with acetylene gives 2-methylbut-3-yn-2-ol, precursor to synthetic terpenes and terpenoids. ===Laboratory=== ====Chemical research==== In the laboratory, acetone is used as a polar, aprotic solvent in a variety of organic reactions, such as SN2 reactions. Octyl acetate, or octyl ethanoate, is an organic compound with the formula CH3(CH2)7O2CCH3. thumb|upright=1.75|Processes in the thermal degradation of organic matter at atmospheric pressure. In 1960, Soviet chemists observed that the thermodynamics of this process is unfavourable for liquid acetone, so that it (unlike thioacetone and formol) is not expected to polymerise spontaneously, even with catalysts. The technique, called acetone vapor bath smoothing, involves placing the printed part in a sealed chamber containing a small amount of acetone, and heating to around 80 degrees Celsius for 10 minutes. The use of acetone solvent is critical for the Jones oxidation. Acetone can be cooled with dry ice to −78 °C without freezing; acetone/dry ice baths are commonly used to conduct reactions at low temperatures. Msha.gov. Retrieved on 2012-11-26.History – Acetylene dissolved in acetone . As the liquid and gas are in equilibrium at the boiling point (Tb), ΔvG = 0, which leads to: :\Delta_\text{v} S = S_\text{gas} - S_\text{liquid} = \frac{\Delta_\text{v} H}{T_\text{b}} As neither entropy nor enthalpy vary greatly with temperature, it is normal to use the tabulated standard values without any correction for the difference in temperature from 298 K. The reaction is usually endothermic as heat is required to break chemical bonds in the compound undergoing decomposition. The decomposition temperature of a substance is the temperature at which the substance chemically decomposes. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics *The compound with the highest known decomposition temperature is carbon monoxide at ≈3870 °C (≈7000 °F). ===Decomposition of nitrates, nitrites and ammonium compounds=== * Ammonium dichromate on heating yields nitrogen, water and chromium(III) oxide. During World War I, Chaim Weizmann developed the process for industrial production of acetone (Weizmann Process).Chaim Weizmann chemistryexplained.com ==Production== In 2010, the worldwide production capacity for acetone was estimated at 6.7 million tonnes per year. ",4.4,5040,131.0,269,2598960,D
-"Autoclaves that are used to sterilize surgical tools require a temperature of $120 .{ }^{\circ} \mathrm{C}$ to kill some bacteria. If water is used for this purpose, at what pressure must the autoclave operate?","The operator is required to manually perform steam pulsing at certain pressures as indicated by the gauge. == In medicine == thumb|Dental equipment in an autoclave to be sterilized for 2 hours at 150 to 180 degrees Celsius A medical autoclave is a device that uses steam to sterilize equipment and other objects. thumb|Cutaway illustration of a cylindrical-chamber autoclave An autoclave is a machine used to carry out industrial and scientific processes requiring elevated temperature and pressure in relation to ambient pressure and/or temperature. Many autoclaves are used to sterilize equipment and supplies by subjecting them to pressurized saturated steam at for around 30-60 minutes at a pressure of 15 psi (103 kPa or 1.02 atm) depending on the size of the load and the contents. There are physical, chemical, and biological indicators that can be used to ensure that an autoclave reaches the correct temperature for the correct amount of time. Autoclave tape works by changing color after exposure to temperatures commonly used in sterilization processes, typically 121°C in a steam autoclave. (A properly calibrated medical-grade autoclave uses thousands of gallons of water each day, independent of task, with correspondingly high electric power consumption.) ==In research== Autoclaves used in education, research, biomedical research, pharmaceutical research and industrial settings (often called ""research-grade"" autoclaves) are used to sterilize lab instruments, glassware, culture media, and liquid media. UCR's research-grade autoclaves performed the same tasks with equal effectiveness, but used 83% less energy and 97% less water. ==Quality assurance== In order to sterilize items effectively, it is important to use optimal parameters when running an autoclave cycle. Since exact temperature control is difficult, the temperature is monitored, and the sterilization time adjusted accordingly. ==Additional images== Image:Autoclave stove top.jpg|Stovetop autoclaves, also known as pressure cooker—the simplest of autoclaves File:Autoclave machine.jpg|The machine on the right is an autoclave used for processing substantial quantities of laboratory equipment prior to reuse, and infectious material prior to disposal. Autoclaves are found in many medical settings, laboratories, and other places that need to ensure the sterility of an object. Super heating conditions and steam generation are achieved by variable pressure control, which cycles between ambient and negative pressure within the sterilization vessel. Some computer-controlled autoclaves use an F0 (F-nought) value to control the sterilization cycle. The high heat and pressure that autoclaves generate help to ensure that the best possible physical properties are repeatable. Research-grade autoclaves—which are not approved for use in sterilizing instruments that will be directly used on humans—are primarily designed for efficiency, flexibility, and ease-of-use. Machines in this category largely operate under the same principles as conventional autoclaves in that they are able to neutralize potentially infectious agents by using pressurized steam and superheated water. Autoclaves are used before surgical procedures to perform sterilization and in the chemical industry to cure coatings and vulcanize rubber and for hydrothermal synthesis. With this process, waste enters and the product leaves the autoclave without the loss of temperature or pressure in the vessel. Some autoclaves, also referred to as waste converters, can operate in the atmospheric pressure range to achieve full sterilization of pathogenic waste. For steam sterilization to occur, the entire item must completely reach and maintain 121°C for 15–20 minutes with proper steam exposure to ensure sterilization. If the autoclave does not reach the right temperature, the spores will germinate when incubated and their metabolism will change the color of a pH-sensitive chemical. In most of the industrialized world medical-grade autoclaves are regulated medical devices. Other types of autoclaves are used to grow crystals under high temperatures and pressures. In dentistry, autoclaves provide sterilization of dental instruments. ",1.1,14.80,29.9,1.95 ,0.020,D
-Imagine gaseous $\mathrm{Ar}$ at $298 \mathrm{~K}$ confined to move in a two-dimensional plane of area $1.00 \mathrm{~cm}^2$. What is the value of the translational partition function?,"In statistical mechanics, the translational partition function, q_T is that part of the partition function resulting from the movement (translation) of the center of mass. The partition function is a function of the temperature T and the microstate energies E1, E2, E3, etc. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. The partition function has many physical meanings, as discussed in Meaning and significance. == Canonical partition function == === Definition === Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles are fixed. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies. ==Grand canonical partition function== We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. When considering a set of N non-interacting but identical atoms or molecules, when QT ≫ N , or equivalently when ρ Λ ≪ 1 where ρ is the density of particles, the total translational partition function can be written : Q_T(T,N) = \frac{ q_T(T)^N }{N!} A plane partition may be represented visually by the placement of a stack of \pi_{i,j} unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. In this case we must describe the partition function using an integral rather than a sum. The asymptotics for plane partitions were first calculated by E. M. Wright.E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189. In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Plane partitions are a generalization of partitions of an integer. By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} == References == ==See also== * Partition function (mathematics) Category:Partition functions The sum of a plane partition is : n=\sum_{i,j} \pi_{i,j} . In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. This partition function is closely related to the grand potential, \Phi_{\rm G}, by the relation : -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. The usefulness of the partition function stems from the fact that the macroscopic thermodynamic quantities of a system can be related to its microscopic details through the derivatives of its partition function. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. From this point of view, a plane partition can be defined as a finite subset \mathcal{P} of positive integer lattice points (i, j, k) in \mathbb{N}^3, such that if (r, s, t) lies in \mathcal{P} and if (i, j, k) satisfies 1\leq i\leq r, 1\leq j\leq s, and 1\leq k\leq t, then (i, j, k) also lies in \mathcal{P}. The partition function can be related to thermodynamic properties because it has a very important statistical meaning. Using this approximation we can derive a closed form expression for the vibrational partition function. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous. ====Classical discrete system==== For a canonical ensemble that is classical and discrete, the canonical partition function is defined as Z = \sum_i e^{-\beta E_i}, where * i is the index for the microstates of the system; * e is Euler's number; * \beta is the thermodynamic beta, defined as \tfrac{1}{k_\text{B} T} where k_\text{B} is Boltzmann's constant; * E_i is the total energy of the system in the respective microstate. ",-87.8,1.69,7.0,3.9,1.45,D
-"Determine the equilibrium constant for the dissociation of sodium at $298 \mathrm{~K}: \mathrm{Na}_2(g) \rightleftharpoons 2 \mathrm{Na}(g)$. For $\mathrm{Na}_2$, $B=0.155 \mathrm{~cm}^{-1}, \widetilde{\nu}=159 \mathrm{~cm}^{-1}$, the dissociation energy is $70.4 \mathrm{~kJ} / \mathrm{mol}$, and the ground-state electronic degeneracy for $\mathrm{Na}$ is 2 .","Bioanalytical Chemistry Textbook De Gruyter 2021 https://doi.org/10.1515/9783110589160-206 For a general reaction: : A_\mathit{x} B_\mathit{y} <=> \mathit{x} A{} + \mathit{y} B in which a complex \ce{A}_x \ce{B}_y breaks down into x A subunits and y B subunits, the dissociation constant is defined as : K_D = \frac{[\ce A]^x [\ce B]^y}{[\ce A_x \ce B_y]} where [A], [B], and [Ax By] are the equilibrium concentrations of A, B, and the complex Ax By, respectively. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). Nonadiabatic transition state theory (NA-TST) is a powerful tool to predict rates of chemical reactions from a computational standpoint. (The symbol K_a, used for the acid dissociation constant, can lead to confusion with the association constant and it may be necessary to see the reaction or the equilibrium expression to know which is meant.) In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. A molecule can have several acid dissociation constants. In the special case of salts, the dissociation constant can also be called an ionization constant. For example, for the ionic crystal NaCl, there arise two Madelung constants - one for Na and another for Cl. The molecular formula C17H14N2O2 (molar mass: 278.305 g/mol, exact mass: 278.1055 u) may refer to: * Bimakalim * Sudan Red G Category:Molecular formulas The dissociation constant is the inverse of the association constant. : Ab + Ag <=> AbAg : K_A = \frac{\left[ \ce{AbAg} \right]}{\left[ \ce{Ab} \right] \left[ \ce{Ag} \right]} = \frac{1}{K_D} This chemical equilibrium is also the ratio of the on-rate (kforward or ka) and off-rate (kback or kd) constants. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants. == Formula == A fast converging formula for the Madelung constant of NaCl is :12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right) == Generalization == It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. The binding constant, or affinity constant/association constant, is a special case of the equilibrium constant K, and is the inverse of the dissociation constant. The dissociation constant has molar units (M) and corresponds to the ligand concentration [L] at which half of the proteins are occupied at equilibrium, i.e., the concentration of ligand at which the concentration of protein with ligand bound [LP] equals the concentration of protein with no ligand bound [P]. [B^{-3}] \over [H B^{-2}]} & \mathrm{p}K_3 &= -\log K_3 \end{align} ==Dissociation constant of water== The dissociation constant of water is denoted Kw: :K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-] The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration. The properties of this ion are strongly related to the surface potential present on a corresponding solid. NA-TST can be reduced to the traditional TST in the limit of unit probability. ==References== Category:Chemical physics Examples of Madelung constants Ion in crystalline compound M (based on ) \overline{M} (based on ) Cl− and Cs+ in CsCl ±1.762675 ±2.035362 Cl− and Na+ in rocksalt NaCl ±1.747565 ±3.495129 S2− and Zn2+ in sphalerite ZnS ±3.276110 ±7.56585 F− in fluorite CaF2 1.762675 4.070723 Ca2+ in fluorite CaF2 -3.276110 −7.56585 The continuous reduction of with decreasing coordination number for the three cubic AB compounds (when accounting for the doubled charges in ZnS) explains the observed propensity of alkali halides to crystallize in the structure with highest compatible with their ionic radii. Sub-picomolar dissociation constants as a result of non-covalent binding interactions between two molecules are rare. For the binding of receptor and ligand molecules in solution, the molar Gibbs free energy ΔG, or the binding affinity is related to the dissociation constant Kd via :\Delta G = R T\ln{{K_{\rm d} \over c^{\ominus}}}, in which R is the ideal gas constant, T temperature and the standard reference concentration c ~~o~~ = 1 mol/L. == See also == * Binding coefficient Category:Equilibrium chemistry ",−1.642876,24,2.25,6.9,537,C
-"At $298.15 \mathrm{~K}, \Delta G_f^{\circ}(\mathrm{HCOOH}, g)=-351.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta G_f^{\circ}(\mathrm{HCOOH}, l)=-361.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the vapor pressure of formic acid at this temperature.","Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds"", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The second solution is switching to another vapor pressure equation with more than three parameters. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The formula is valid from about −50 to 102 °C; however there are a very limited number of measurements of the vapor pressure of water over supercooled liquid water. (The small differences in the results are only caused by the used limited precision of the coefficients). ==Extension of the Antoine equations== To overcome the limits of the Antoine equation some simple extension by additional terms are used: : \begin{align} P &= \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)} \\\ P &= \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right). \end{align} The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Regarding the measurement of upper-air humidity, this publication also reads (in Section 12.5.1): The saturation with respect to water cannot be measured much below –50 °C, so manufacturers should use one of the following expressions for calculating saturation vapour pressure relative to water at the lowest temperatures – Wexler (1976, 1977), reported by Flatau et al. (1992)., Hyland and Wexler (1983) or Sonntag (1994) – and not the Goff- Gratch equation recommended in earlier WMO publications. ==Experimental correlation== The original Goff–Gratch (1945) experimental correlation reads as follows: : \log\ e^*\ = -7.90298(T_\mathrm{st}/T-1)\ +\ 5.02808\ \log(T_\mathrm{st}/T) -\ 1.3816\times10^{-7}(10^{11.344(1-T/T_\mathrm{st})}-1) +\ 8.1328\times10^{-3}(10^{-3.49149(T_\mathrm{st}/T-1)}-1)\ +\ \log\ e^*_\mathrm{st} where: :log refers to the logarithm in base 10 :e* is the saturation water vapor pressure (hPa) :T is the absolute air temperature in kelvins :Tst is the steam-point (i.e. boiling point at 1 atm.) temperature (373.15 K) :e*st is e* at the steam-point pressure (1 atm = 1013.25 hPa) Similarly, the correlation for the saturation water vapor pressure over ice is: : \log\ e^*_i\ = -9.09718(T_0/T-1)\ -\ 3.56654\ \log(T_0/T) +\ 0.876793(1-T/T_0) +\ \log\ e^*_{i0} where: :log stands for the logarithm in base 10 :e*i is the saturation water vapor pressure over ice (hPa) :T is the air temperature (K) :T0 is the ice-point (triple point) temperature (273.16 K) :e*i0 is e* at the ice-point pressure (6.1173 hPa) ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Tetens equation *Lee–Kesler method ==References== * Goff, J. A., and Gratch, S. (1946) Low- pressure properties of water from −160 to 212 °F, in Transactions of the American Society of Heating and Ventilating Engineers, pp 95–122, presented at the 52nd annual meeting of the American Society of Heating and Ventilating Engineers, New York, 1946. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Water vapor (H2O) 200px Liquid state Water Solid state Ice Properties Molecular formula H2O Molar mass 18.01528(33) g/mol Melting point Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C) and boils at 373.1339 K (99.9839 °C) Boiling point specific gas constant 461.5 J/(kg·K) Heat of vaporization 2.27 MJ/kg Heat capacity 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous phase of water. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., ""Antoine Vapor Pressure Constants of Pure Compounds"" ** Yaws C. L., Yang H.-C., ""To Estimate Vapor Pressure Easily. The maximum partial pressure (saturation pressure) of water vapor in air varies with temperature of the air and water vapor mixture. The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: :T = \frac{B}{A-\log_{10}\, p} - C ==Validity range== Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. The Goff–Gratch equation is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the saturation water vapor pressure at a given temperature. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). ",1.51,2.00,6.07,358800,-167,A
-The collisional cross section of $\mathrm{N}_2$ is $0.43 \mathrm{~nm}^2$. What is the diffusion coefficient of $\mathrm{N}_2$ at a pressure of $1 \mathrm{~atm}$ and a temperature of $298 \mathrm{~K}$ ?,"The diffusion profile therefore can be depicted by the following equation. (dln\bar{c}/dy^{6/5})^{5/3}=0.66(D_1/t)^{1/2}(1/D_b\delta) To further determine D_b , two common methods were used. The first equation represents diffusion in the volume, while the second shows diffusion along the grain boundary, respectively. \frac{\partial c}{\partial t}=D\left({\partial^2 c\over\partial x^2}+{\partial^2 c\over\partial y^2}\right) where |x|>\delta/2 \frac{\partial c_b}{\partial t}=D_b\left({\partial^2 c_b\over\partial y^2}\right)+\frac{2D}{\delta}\left(\frac{\partial c}{\partial x}\right)_{x=\delta/2} where c(x, y, t) is the volume concentration of the diffusing atoms and c_b(y, t) is their concentration in the grain boundary. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). thumb|Thermal diffusion coefficients vs. temperature, for air at normal pressure The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. * Method 2: To compare the length of penetration of a given concentration at the boundary \ \Delta y with the length of lattice penetration from the surface far from the boundary. == References == == See also == * Kirkendall effect * Phase transformations in solids * Mass diffusivity Category:Diffusion The grain boundary diffusion coefficient is the diffusion coefficient of a diffusant along a grain boundary in a polycrystalline solid.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965. ""On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)"". The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux.S. Rehfeldt, J. Stichlmair: Measurement and calculation of multicomponent diffusion coefficients in liquids, Fluid Phase Equilibria, 2007, 256, 99–104 The molecular friction between two components is proportional to their difference in speed and their mole fractions. Suppose that the thickness of the slab is \delta, the length is y, and the depth is a unit length, the diffusion process can be described as the following formula. The effective diffusion coefficient of a in atomic diffusion of solid polycrystalline materials like metal alloys is often represented as a weighted average of the grain boundary diffusion coefficient and the lattice diffusion coefficient.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965. However, at temperatures below 700 °C, the values of D_b with polycrystal silver consistently lie above the values of D_b with a single crystal. == Measurement == The general way to measure grain boundary diffusion coefficients was suggested by Fisher. It dominates the effective diffusion rate at lower temperatures in metals and metal alloys. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. The Boltzmann–Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8 Tmelt (of an alloy), the grain boundary component can be neglected. ==Modeling== The effective diffusion coefficient can be modeled using Hart's equation when lattice diffusion is dominant (type A kinetics): : D_\text{eff} = f D_\text{gb} + (1-f) D_\ell where :D_\text{eff} = {}effective diffusion coefficient :D_\text{gb} = {}grain boundary diffusion coefficient :D_\ell = {}lattice diffusion coefficient :f = \frac{q \delta}{d} : q = {}value based on grain shape, 1 for parallel grains, 3 for square grains : d = {}average grain size :\delta = {}grain boundary width, often assumed to be 0.5 nm Grain boundary diffusion is significant in face-centered cubic metals below about 0.8 Tmelt (Absolute). The Mathematics of Diffusion. This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law. == Matano’s method == Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4–0.6, so as temperature is lowered, the grain boundary diffusion component increases. Assuming a diffusion coefficient D that is in general a function of concentration c, Fick's second law is :\frac{\partial c}{\partial t} = \frac{\partial}{\partial x} \underbrace{\left[ D(c)\frac{\partial c}{\partial x} \right]}_\text{flux}, where t is time, and x is distance. The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. Another example of double diffusion is the formation of false bottoms at the interface of sea ice and under-ice meltwater layers. ",5654.86677646,0.88,1.6,1.06,"102,965.21",D
+","Butane is one of a group of liquefied petroleum gases (LP gases). Forming gas is a mixture of hydrogen (mole fraction varies) and nitrogen. For example, the density of liquid propane is 571.8±1 kg/m3 (for pressures up to 2MPa and temperature 27±0.2 °C), while the density of liquid butane is 625.5±0.7 kg/m3 (for pressures up to 2MPa and temperature -13±0.2 °C). alt=Density of liquid and vaporized butane|none|thumb|500x500px|Propane and butane density data == Isomers == Common name normal butane unbranched butane n-butane isobutane i-butane IUPAC name butane methylpropane Molecular diagram 150px 120px Skeletal diagram 120px 100px Rotation about the central C−C bond produces two different conformations (trans and gauche) for n-butane. == Reactions == When oxygen is plentiful, butane burns to form carbon dioxide and water vapor; when oxygen is limited, carbon (soot) or carbon monoxide may also be formed. The relative rates of the chlorination is partially explained by the differing bond dissociation energies, 425 and 411 kJ/mol for the two types of C-H bonds. == Uses == Normal butane can be used for gasoline blending, as a fuel gas, fragrance extraction solvent, either alone or in a mixture with propane, and as a feedstock for the manufacture of ethylene and butadiene, a key ingredient of synthetic rubber. When there is sufficient oxygen: : 2 C4H10 \+ 13 O2 → 8 CO2 \+ 10 H2O When oxygen is limited: : 2 C4H10 \+ 9 O2 → 8 CO + 10 H2O By weight, butane contains about or by liquid volume . The molecular formula C4H10 (molar mass: 58.12 g/mol, exact mass: 58.0783 u) may refer to: * Butane, or n-butane * Isobutane, also known as methylpropane or 2-methylpropane Butane is a highly flammable, colorless, easily liquefied gas that quickly vaporizes at room temperature and pressure. Butane is denser than air. The gas constant is expressed in the same unit as are molar entropy and molar heat. ==Dimensions== From the ideal gas law PV = nRT we get: :R = \frac{PV}{nT} where P is pressure, V is volume, n is number of moles of a given substance, and T is temperature. Butane did not have much practical use until the 1910s, when W. Snelling identified butane and propane as components in gasoline and found that, if they were cooled, they could be stored in a volume-reduced liquified state in pressurized containers. == Density == The density of butane is highly dependent on temperature and pressure in the reservoir. Butane is also used as lighter fuel for common lighters or butane torches and is sold bottled as a fuel for cooking, barbecues and camping stoves. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . For gasoline blending, n-butane is the main component used to manipulate the Reid vapor pressure (RVP). Butane is the most commonly abused volatile substance in the UK, and was the cause of 52% of solvent related deaths in 2000. Instead of a mole the constant can be expressed by considering the normal cubic meter. Butane () or n-butane is an alkane with the formula C4H10. The gas constant occurs in the ideal gas law: PV = nRT = m R_{\rm specific} T where P is the absolute pressure, V is the volume of gas, n is the amount of substance, m is the mass, and T is the thermodynamic temperature. Also note that this was well before the 2019 SI redefinition, through which the constant was given an exact value. ==References== ==External links== * Ideal gas calculator – Ideal gas calculator provides the correct information for the moles of gas involved. File:Photo D2.jpg | Butane fuel canisters for use in camping stoves File:The Green Lighter 1 cropped.jpg | Butane lighter, showing liquid butane reservoir File:Aerosol.png | An aerosol spray can, which may be using butane as a propellant File:ButaneGasCylinder WhiteBack.jpg | Butane gas cylinder used for cooking == Effects and health issues == Inhalation of butane can cause euphoria, drowsiness, unconsciousness, asphyxia, cardiac arrhythmia, fluctuations in blood pressure and temporary memory loss, when abused directly from a highly pressurized container, and can result in death from asphyxiation and ventricular fibrillation. However, following the 2019 redefinition of the SI base units, R now has an exact value defined in terms of other exactly defined physical constants. ==Specific gas constant== Rspecific for dry air Unit 287.052874 J⋅kg−1⋅K−1 53.3523 ft⋅lbf⋅lb−1⋅°R−1 1,716.46 ft⋅lbf⋅slug−1⋅°R−1 The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture. It provides a molar mass for air of 28.9625 g/mol, and provides a composition for standard dry air as a footnote. == References == Category:Gases When blended with propane and other hydrocarbons, the mixture may be referred to commercially as liquefied petroleum gas (LPG). ",32,0.6321205588,"""672.4""",-214,6,A
+"One liter of fully oxygenated blood can carry 0.18 liters of $\mathrm{O}_2$ measured at $T=298 \mathrm{~K}$ and $P=1.00 \mathrm{~atm}$. Calculate the number of moles of $\mathrm{O}_2$ carried per liter of blood. Hemoglobin, the oxygen transport protein in blood has four oxygen binding sites. How many hemoglobin molecules are required to transport the $\mathrm{O}_2$ in $1.0 \mathrm{~L}$ of fully oxygenated blood?","Each hemoglobin molecule has the capacity to carry four oxygen molecules. The oxygen-carrying capacity of hemoglobin is determined by the type of hemoglobin present in the blood. Although binding of oxygen to hemoglobin continues to some extent for pressures about 50 mmHg, as oxygen partial pressures decrease in this steep area of the curve, the oxygen is unloaded to peripheral tissue readily as the hemoglobin's affinity diminishes. Venous blood with an oxygen concentration of 15 mL/100 mL would therefore lead to typical values of the a-vO2 diff at rest of around 5 mL/100 mL. To see the relative affinities of each successive oxygen as you remove/add oxygen from/to the hemoglobin from the curve compare the relative increase/decrease in p(O2) needed for the corresponding increase/decrease in s(O2). ==Factors that affect the standard dissociation curve== The strength with which oxygen binds to hemoglobin is affected by several factors. So, one will have a lesser hemoglobin saturation percentage for the same [O2] or a higher partial pressure of oxygen. The amount of oxygen bound to the hemoglobin at any time is related, in large part, to the partial pressure of oxygen to which the hemoglobin is exposed. The oxygen bound to the hemoglobin is released into the blood's plasma and absorbed into the tissues, and the carbon dioxide in the tissues is bound to the hemoglobin. The partial pressure of oxygen in the blood at which the hemoglobin is 50% saturated, typically about 26.6 mmHg (3.5 kPa) for a healthy person, is known as the P50. The binding affinity of hemoglobin to O2 is greatest under a relatively high pH. === Carbon dioxide === Carbon dioxide affects the curve in two ways. Arterial blood will generally contain an oxygen concentration of around 20 mL/100 mL. A hemoglobin molecule can bind up to four oxygen molecules in a reversible method. The T state has a lower affinity for oxygen than the R state, so with increased acidity, the hemoglobin binds less O2 for a given PO2 (and more H+). Specifically, the oxyhemoglobin dissociation curve relates oxygen saturation (SO2) and partial pressure of oxygen in the blood (PO2), and is determined by what is called ""hemoglobin affinity for oxygen""; that is, how readily hemoglobin acquires and releases oxygen molecules into the fluid that surrounds it. thumb|Structure of oxyhemoglobin ==Background== Hemoglobin (Hb) is the primary vehicle for transporting oxygen in the blood. The a-vO2 diff is usually measured in millilitres of oxygen per 100 millilitres of blood (mL/100 mL).Malpeli, Physical Education, Chapter 4: Acute Responses to Exercise, p. 106. ==Measurement== The arteriovenous oxygen difference is usually taken by comparing the difference in the oxygen concentration of oxygenated blood in the femoral, brachial, or radial artery and the oxygen concentration in the deoxygenated blood from the mixed supply found in the pulmonary artery (as an indicator of the typical mixed venous supply). As the blood circulates to other body tissue in which the partial pressure of oxygen is less, the hemoglobin releases the oxygen into the tissue because the hemoglobin cannot maintain its full bound capacity of oxygen in the presence of lower oxygen partial pressures. ==Sigmoid shape== thumb|Hemoglobin saturation curve The curve is usually best described by a sigmoid plot, using a formula of the kind: :S(t) = \frac{1}{1 + e^{-t}}. In the capillaries, where carbon dioxide is produced, oxygen bound to the hemoglobin is released into the blood's plasma and absorbed into the tissues. Hemoglobin's affinity for oxygen increases as successive molecules of oxygen bind. HbF then delivers that bound oxygen to tissues that have even lower partial pressures where it can be released. ==See also== * Automated analyzer * Bohr effect ==Notes== ==References== ==External links== * * The Interactive Oxyhemoglobin Dissociation Curve * Simulation of the parameters CO2, pH and temperature on the oxygen–hemoglobin dissociation curve (left or right shift) Category:Respiratory physiology Category:Chemical pathology Category:Hematology Category:Oxygen The 'plateau' portion of the oxyhemoglobin dissociation curve is the range that exists at the pulmonary capillaries (minimal reduction of oxygen transported until the p(O2) falls 50 mmHg). Solid oxygen forms at normal atmospheric pressure at a temperature below 54.36 K (−218.79 °C, −361.82 °F). The phosphate/oxygen ratio, or P/O ratio, refers to the amount of ATP produced from the movement of two electrons through a defined electron transport chain, terminated by reduction of an oxygen atom.Garrett & Grisham 2010, p.620. ",-4564.7,1.11,"""4.946""",3.7,6.283185307,B
+Consider a collection of molecules where each molecule has two nondegenerate energy levels that are separated by $6000 . \mathrm{cm}^{-1}$. Measurement of the level populations demonstrates that there are exactly 8 times more molecules in the ground state than in the upper state. What is the temperature of the collection?,"Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, : :\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} :\, p(k)= e^{-k^2\over 2mT}. The molecular-scale temperature is the defining property of the U.S. Standard Atmosphere, 1962. thumb|upright=1.5|Schematic Bose–Einstein condensation versus temperature of the energy diagram In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. Average yearly temperature is 22.4°C, ranging from an average minimum of 12.2°C to a maximum of 29.9°C. In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. At temperature T, a particle will have a lesser probability to be in state |1\rangle by e^{-E/kT}. Only 8% of atoms are in the ground state of the trap near absolute zero, rather than the 100% of a true condensate. Some of the warmest temperatures can be found in the thermosphere, due to its reception of strong ionizing radiation at the level of the Van Allen radiation belt. ==Temperature range== The variation in temperature that occurs from the highs of the day to the cool of nights is called diurnal temperature variation. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values. == In fiction == * In the 2016 film Spectral, the US military battles mysterious enemy creatures fashioned out of Bose–Einstein condensates. Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, quasi- particles, and photons. == Critical temperature == This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by: :T_{\rm c}=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_{\rm B}} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_{\rm B}} where: : \,T_{\rm c} is the critical temperature, \,n the particle density, \,m the mass per boson, \hbar the reduced Planck constant, \,k_{\rm B} the Boltzmann constant and \,\zeta the Riemann zeta function; \,\zeta(3/2)\approx 2.6124. Atmospheric temperature is a measure of temperature at different levels of the Earth's atmosphere. Average maximum yearly temperature is 28.7°C and average minimum is 21.9°C. The average temperature range is 5.7°C only. Junction temperature, short for transistor junction temperature, is the highest operating temperature of the actual semiconductor in an electronic device. Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. The average temperature range is 11.4 degrees. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. ",1.07,1.44,"""0.3333333""","102,965.21",4152,E
+Calculate $\Delta S^{\circ}$ for the reaction $3 \mathrm{H}_2(g)+\mathrm{N}_2(g) \rightarrow$ $2 \mathrm{NH}_3(g)$ at $725 \mathrm{~K}$. Omit terms in the temperature-dependent heat capacities higher than $T^2 / \mathrm{K}^2$.,"Since \Delta_r G^\ominus = - RT \ln K_{eq}, the temperature dependence of both terms can be described by Van t'Hoff equations as a function of T. Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., ""Physical Chemistry"" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., ""Atkins' Physical Chemistry"" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The definite integral between temperatures and is then :\ln \frac{K_2}{K_1} = \frac{\Delta_r H^\ominus}{R} \left( \frac{1}{T_1} - \frac{1}{T_2} \right). The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. Since in reality \Delta_r H^\ominus and the standard reaction entropy \Delta_r S^\ominus do vary with temperature for most processes, the integrated equation is only approximate. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. For a change from reactants to products at constant temperature and pressure the equation becomes \Delta G = \Delta H - T\Delta S. Differentiation of this expression with respect to the variable while assuming that both \Delta_r H^\ominus and \Delta_r S^\ominus are independent of yields the Van 't Hoff equation. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. However, the relation loses its validity when the aggregation number is also temperature-dependent, and the following relation should be used instead: :RT^2\left(\frac{\partial}{\partial T}\ln\mathrm{CMC}\right)_P = -\Delta_r H^\ominus_\mathrm{m}(N) + T\left(\frac{\partial}{\partial N}\left(G_{N+1} - G_N\right)\right)_{T,P}\left(\frac{\partial N}{\partial T}\right)_P, with and being the free energies of the surfactant in a micelle with aggregation number and respectively. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. In this equation is the equilibrium constant at absolute temperature , and is the equilibrium constant at absolute temperature . ===Development from thermodynamics=== Combining the well-known formula for the Gibbs free energy of reaction : \Delta_r G^\ominus = \Delta_r H^\ominus - T\Delta_r S^\ominus, where is the entropy of the system, with the Gibbs free energy isotherm equation: :\Delta_r G^\ominus = -RT \ln K_\mathrm{eq}, we obtain :\ln K_\mathrm{eq} = -\frac{\Delta_r H^\ominus}{RT} + \frac{\Delta_r S^\ominus}{R}. The Van 't Hoff plot can be used to find the enthalpy and entropy change for each mechanism and the favored mechanism under different temperatures. :\begin{align} \Delta_r H_1 &= - R \times \text{slope}_1, & \Delta_r S_1 &= R \times \text{intercept}_1; \\\\[5pt] \Delta_r H_2 &= - R \times \text{slope}_2, & \Delta_r S_2 &= R \times \text{intercept}_2. \end{align} In the example figure, the reaction undergoes mechanism 1 at high temperature and mechanism 2 at low temperature. ===Temperature dependence=== thumb|right|x275px|Temperature-dependent Van 't Hoff plot The Van 't Hoff plot is linear based on the tacit assumption that the enthalpy and entropy are constant with temperature changes. The slope of the line may be multiplied by the gas constant to obtain the standard enthalpy change of the reaction, and the intercept may be multiplied by to obtain the standard entropy change. ===Van 't Hoff isotherm=== The Van 't Hoff isotherm can be used to determine the temperature dependence of the Gibbs free energy of reaction for non- standard state reactions at a constant temperature: :\left(\frac {dG}{d\xi}\right)_{T,p} = \Delta_\mathrm{r}G = \Delta_\mathrm{r}G^\ominus + RT \ln Q_\mathrm{r}, where \Delta_\mathrm{r}G is the Gibbs free energy of reaction under non-standard states at temperature T, \Delta_r G^\ominus is the Gibbs free energy for the reaction at (T,P^0), \xi is the extent of reaction, and is the thermodynamic reaction quotient. We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. In practice, the equation is often integrated between two temperatures under the assumption that the standard reaction enthalpy \Delta_r H^\ominus is constant (and furthermore, this is also often assumed to be equal to its value at standard temperature). Thus, for an exothermic reaction, the Van 't Hoff plot should always have a positive slope. === Error propagation === At first glance, using the fact that it would appear that two measurements of would suffice to be able to obtain an accurate value of : :\Delta_r H^\ominus = R \frac{\ln K_1 - \ln K_2}{\frac{1}{T_2} - \frac{1}{T_1}}, where and are the equilibrium constant values obtained at temperatures and respectively. Knowing the slope and intercept from the Van 't Hoff plot, the enthalpy and entropy of a reaction can be easily obtained using :\begin{align} \Delta_r H &= - R \times \text{slope}, \\\ \Delta_r S &= R \times \text{intercept}. \end{align} The Van 't Hoff plot can be used to quickly determine the enthalpy of a chemical reaction both qualitatively and quantitatively. When a reaction is at equilibrium, and \Delta_\mathrm{r}G = 0. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. A first-order approximation is to assume that the two different reaction products have different heat capacities. The data should have a linear relationship, the equation for which can be found by fitting the data using the linear form of the Van 't Hoff equation :\ln K_\mathrm{eq} = -\frac{\Delta_r H^\ominus}{RT} + \frac{\Delta_r S^\ominus}{R}. ",9.30,6.9,"""2.0""",-20,-191.2,E
+"The thermal conductivities of acetylene $\left(\mathrm{C}_2 \mathrm{H}_2\right)$ and $\mathrm{N}_2$ at $273 \mathrm{~K}$ and $1 \mathrm{~atm}$ are 0.01866 and $0.0240 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~m}^{-1} \mathrm{~s}^{-1}$, respectively. Based on these data, what is the ratio of the collisional cross section of acetylene relative to $\mathrm{N}_2$ ?","Under these assumptions, an elementary calculation yields for the thermal conductivity : k = \beta \rho \lambda c_v \sqrt{\frac{2k_\text{B} T}{\pi m}}, where \beta is a numerical constant of order 1, k_\text{B} is the Boltzmann constant, and \lambda is the mean free path, which measures the average distance a molecule travels between collisions. In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution of the thermal conductivity (κ) to the electrical conductivity (σ) of a metal is proportional to the temperature (T). : \frac \kappa \sigma = LT Theoretically, the proportionality constant L, known as the Lorenz number, is equal to : L = \frac \kappa {\sigma T} = \frac{\pi^2} 3 \left(\frac{k_{\rm B}} e \right)^2 = 2.44\times 10^{-8}\;\mathrm{V}^2\mathrm{K}^{-2}, where kB is Boltzmann's constant and e is the elementary charge. In summary, for a plate of thermal conductivity k, area A and thickness L, *thermal conductance = kA/L, measured in W⋅K−1. **thermal resistance = L/(kA), measured in K⋅W−1. *heat transfer coefficient = k/L, measured in W⋅K−1⋅m−2. **thermal insulance = L/k, measured in K⋅m2⋅W−1. ""The Thermal Conductivity of Air at Reduced Pressures and Length Scales,"" Electronics Cooling, November 2002, http://www.electronics- cooling.com/2002/11/the-thermal-conductivity-of-air-at-reduced-pressures-and- length-scales/ Retrieved 05:20, 10 April 2016 (UTC). 273-293-298 300 600 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 233.2 266.5 299.9 hiAerosols2.95-loAerosols7.83×10−15 (78.03%N2,21%O2,+0.93%Ar,+0.04%CO2) (1 atm) The plate distance is one centimeter, the special conductivity values were calculated from the Lasance approximation formula in The Thermal conductivity of Air at Reduced Pressures and Length Scales and the primary values were taken from Weast at the normal pressure tables in the CRC handbook on page E2. From considerations of energy conservation, the heat flow between the two bodies in contact, bodies A and B, is found as: One may observe that the heat flow is directly related to the thermal conductivities of the bodies in contact, k_A and k_B, the contact area A, and the thermal contact resistance, 1/h_c, which, as previously noted, is the inverse of the thermal conductance coefficient, h_c. ==Importance== Most experimentally determined values of the thermal contact resistance fall between 0.000005 and 0.0005 m2 K/W (the corresponding range of thermal contact conductance is 200,000 to 2000 W/m2 K). For a plate of thermal conductivity k, area A and thickness L, the conductance is kA/L, measured in W⋅K−1.Bejan, p. 34 The relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity. ==Experimental values== thumb|upright=2.5|Experimental values of thermal conductivity The thermal conductivities of common substances span at least four orders of magnitude. This conductance, known as thermal boundary conductance, is due to the differences in electronic and vibrational properties between the contacting materials. Then : \kappa = L T \sigma = {\pi^2 k_{\rm B}^2 \over 3e^2}\times {n e^2 \over h} T = {\pi^2 k_{\rm B}^2 \over 3h} n T = g_0 n, where g_0 is the thermal conductance quantum defined above. ==References== ==See also== * Thermal properties of nanostructures Category:Mesoscopic physics Category:Nanotechnology Category:Quantum mechanics Category:Condensed matter physics Category:Physical quantities Category:Heat conduction Many pure metals have a peak thermal conductivity between 2 K and 10 K. In imperial units, thermal conductivity is measured in BTU/(h⋅ft⋅°F).1 Btu/(h⋅ft⋅°F) = 1.730735 W/(m⋅K) The dimension of thermal conductivity is M1L1T−3Θ−1, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ). The reciprocal of the heat transfer coefficient is thermal insulance. This table shows thermal conductivity in SI units of watts per metre-kelvin (W·m−1·K−1). The thermal contact conductance coefficient, h_c, is a property indicating the thermal conductivity, or ability to conduct heat, between two bodies in contact. Therefore, :\frac \kappa \sigma = \frac{c_V m^2 \, \langle {v} \rangle^2}{3e^2} = \frac{8}{\pi} \frac{k_{\rm B}^2T}{e^2}, which is the Wiedemann–Franz law with an erroneous proportionality constant \frac{8}{\pi}\approx 2.55; ===Free electron model=== After taking into account the quantum effects, as in the free electron model, the heat capacity, mean free path and average speed of electrons are modified and the proportionality constant is then corrected to \frac{\pi^2} 3\approx3.29, which agrees with experimental values. ==Temperature dependence== The value L0 = 2.44×10−8 V2K−2 results from the fact that at low temperatures (T\rightarrow 0 K) the heat and charge currents are carried by the same quasi-particles: electrons or holes. *It happens that the online record has the thermal conductivity at 30 Kelvins and \parallel to the c axis posted at 1.36 W⋅cm−1 K−1 and 78.0 Btu hr−1 ft−1 F−1 which is incorrect. NBS 6.00 \parallel to c axis, 3.90 \perp to c axis 5.00 \parallel to c axis, 3.41 \perp to c axis 4.47 \parallel to c axis, 3.12 \perp to c axis 4.19 \parallel to c axis, 3.04 \perp to c axis List 311 366 422 500 600 700 800 The noted authorities have reported some values in three digits as cited here in metric translation but they have not demonstrated three digit measurement.R.W.Powell, C.Y.Ho and P.E.Liley, Thermal Conductivity of Selected Materials, NSRDS-NBS 8, Issued 25 November 1966, pages 69, 99>Link Text Errata: The numbered references in the NSRDS-NBS-8 pdf are found near the end of the TPRC Data Book Volume 2 and not somewhere in Volume 3 like it says. In a gas, thermal conduction is mediated by discrete molecular collisions. Finally, thermal diffusivity \alpha combines thermal conductivity with density and specific heat: :\alpha = \frac{ k }{ \rho c_{p} }. In many materials, q is observed to be directly proportional to the temperature difference and inversely proportional to the separation distance L: : q = -k \cdot \frac{T_2 - T_1}{L}. In the metallic phase, the electronic contribution to thermal conductivity was much smaller than what would be expected from the Wiedemann–Franz law. Following the Wiedemann–Franz law, thermal conductivity of metals is approximately proportional to the absolute temperature (in kelvins) times electrical conductivity. ",4.5,+5.41,"""1.33""",0.375,2.25,C
+"Consider the gas phase thermal decomposition of 1.0 atm of $\left(\mathrm{CH}_3\right)_3 \mathrm{COOC}\left(\mathrm{CH}_3\right)_3(\mathrm{~g})$ to acetone $\left(\mathrm{CH}_3\right)_2 \mathrm{CO}(\mathrm{g})$ and ethane $\left(\mathrm{C}_2 \mathrm{H}_6\right)(\mathrm{g})$, which occurs with a rate constant of $0.0019 \mathrm{~s}^{-1}$. After initiation of the reaction, at what time would you expect the pressure to be $1.8 \mathrm{~atm}$ ?","However, they observed that the thermodynamics became favourable for crystalline solid acetone at the melting point (−96 °C). One litre of acetone can dissolve around 250 litres of acetylene at a pressure of .Mine Safety and Health Administration (MSHA) – Safety Hazard Information – Special Hazards of Acetylene . At temperatures greater than acetone's flash point of , air mixtures of between 2.5% and 12.8% acetone, by volume, may explode or cause a flash fire. In acetone vapor at ambient temperature, only 2.4% of the molecules are in the enol form. :300px ===Aldol condensation=== In the presence of suitable catalysts, two acetone molecules also combine to form the compound diacetone alcohol , which on dehydration gives mesityl oxide . The synthesis involves the condensation of acetone with phenol: :(CH3)2CO + 2 C6H5OH -> (CH3)2C(C6H4OH)2 + H2O Many millions of kilograms of acetone are consumed in the production of the solvents methyl isobutyl alcohol and methyl isobutyl ketone. Since thermal decomposition is a kinetic process, the observed temperature of its beginning in most instances will be a function of the experimental conditions and sensitivity of the experimental setup. The flame temperature of pure acetone is 1980 °C.Haynes, p. 15.49 ===Toxicity=== Acetone has been studied extensively and is believed to exhibit only slight toxicity in normal use. thumb|280px|Temperature-dependency of the heats of vaporization for water, methanol, benzene, and acetone In thermodynamics, the enthalpy of vaporization (symbol ), also known as the (latent) heat of vaporization or heat of evaporation, is the amount of energy (enthalpy) that must be added to a liquid substance to transform a quantity of that substance into a gas. CH3)2CO -> (CH3)2C(OH)CH2C(O)CH3 Condensation with acetylene gives 2-methylbut-3-yn-2-ol, precursor to synthetic terpenes and terpenoids. ===Laboratory=== ====Chemical research==== In the laboratory, acetone is used as a polar, aprotic solvent in a variety of organic reactions, such as SN2 reactions. Octyl acetate, or octyl ethanoate, is an organic compound with the formula CH3(CH2)7O2CCH3. thumb|upright=1.75|Processes in the thermal degradation of organic matter at atmospheric pressure. In 1960, Soviet chemists observed that the thermodynamics of this process is unfavourable for liquid acetone, so that it (unlike thioacetone and formol) is not expected to polymerise spontaneously, even with catalysts. The technique, called acetone vapor bath smoothing, involves placing the printed part in a sealed chamber containing a small amount of acetone, and heating to around 80 degrees Celsius for 10 minutes. The use of acetone solvent is critical for the Jones oxidation. Acetone can be cooled with dry ice to −78 °C without freezing; acetone/dry ice baths are commonly used to conduct reactions at low temperatures. Msha.gov. Retrieved on 2012-11-26.History – Acetylene dissolved in acetone . As the liquid and gas are in equilibrium at the boiling point (Tb), ΔvG = 0, which leads to: :\Delta_\text{v} S = S_\text{gas} - S_\text{liquid} = \frac{\Delta_\text{v} H}{T_\text{b}} As neither entropy nor enthalpy vary greatly with temperature, it is normal to use the tabulated standard values without any correction for the difference in temperature from 298 K. The reaction is usually endothermic as heat is required to break chemical bonds in the compound undergoing decomposition. The decomposition temperature of a substance is the temperature at which the substance chemically decomposes. Thermokinetics deals with the study of thermal decomposition kinetics. ==See also== *Thermogravimetry *Differential thermal analysis *Differential scanning calorimetry ==References== * * Category:Chemical kinetics *The compound with the highest known decomposition temperature is carbon monoxide at ≈3870 °C (≈7000 °F). ===Decomposition of nitrates, nitrites and ammonium compounds=== * Ammonium dichromate on heating yields nitrogen, water and chromium(III) oxide. During World War I, Chaim Weizmann developed the process for industrial production of acetone (Weizmann Process).Chaim Weizmann chemistryexplained.com ==Production== In 2010, the worldwide production capacity for acetone was estimated at 6.7 million tonnes per year. ",4.4,5040,"""131.0""",269,2598960,D
+"Autoclaves that are used to sterilize surgical tools require a temperature of $120 .{ }^{\circ} \mathrm{C}$ to kill some bacteria. If water is used for this purpose, at what pressure must the autoclave operate?","The operator is required to manually perform steam pulsing at certain pressures as indicated by the gauge. == In medicine == thumb|Dental equipment in an autoclave to be sterilized for 2 hours at 150 to 180 degrees Celsius A medical autoclave is a device that uses steam to sterilize equipment and other objects. thumb|Cutaway illustration of a cylindrical-chamber autoclave An autoclave is a machine used to carry out industrial and scientific processes requiring elevated temperature and pressure in relation to ambient pressure and/or temperature. Many autoclaves are used to sterilize equipment and supplies by subjecting them to pressurized saturated steam at for around 30-60 minutes at a pressure of 15 psi (103 kPa or 1.02 atm) depending on the size of the load and the contents. There are physical, chemical, and biological indicators that can be used to ensure that an autoclave reaches the correct temperature for the correct amount of time. Autoclave tape works by changing color after exposure to temperatures commonly used in sterilization processes, typically 121°C in a steam autoclave. (A properly calibrated medical-grade autoclave uses thousands of gallons of water each day, independent of task, with correspondingly high electric power consumption.) ==In research== Autoclaves used in education, research, biomedical research, pharmaceutical research and industrial settings (often called ""research-grade"" autoclaves) are used to sterilize lab instruments, glassware, culture media, and liquid media. UCR's research-grade autoclaves performed the same tasks with equal effectiveness, but used 83% less energy and 97% less water. ==Quality assurance== In order to sterilize items effectively, it is important to use optimal parameters when running an autoclave cycle. Since exact temperature control is difficult, the temperature is monitored, and the sterilization time adjusted accordingly. ==Additional images== Image:Autoclave stove top.jpg|Stovetop autoclaves, also known as pressure cooker—the simplest of autoclaves File:Autoclave machine.jpg|The machine on the right is an autoclave used for processing substantial quantities of laboratory equipment prior to reuse, and infectious material prior to disposal. Autoclaves are found in many medical settings, laboratories, and other places that need to ensure the sterility of an object. Super heating conditions and steam generation are achieved by variable pressure control, which cycles between ambient and negative pressure within the sterilization vessel. Some computer-controlled autoclaves use an F0 (F-nought) value to control the sterilization cycle. The high heat and pressure that autoclaves generate help to ensure that the best possible physical properties are repeatable. Research-grade autoclaves—which are not approved for use in sterilizing instruments that will be directly used on humans—are primarily designed for efficiency, flexibility, and ease-of-use. Machines in this category largely operate under the same principles as conventional autoclaves in that they are able to neutralize potentially infectious agents by using pressurized steam and superheated water. Autoclaves are used before surgical procedures to perform sterilization and in the chemical industry to cure coatings and vulcanize rubber and for hydrothermal synthesis. With this process, waste enters and the product leaves the autoclave without the loss of temperature or pressure in the vessel. Some autoclaves, also referred to as waste converters, can operate in the atmospheric pressure range to achieve full sterilization of pathogenic waste. For steam sterilization to occur, the entire item must completely reach and maintain 121°C for 15–20 minutes with proper steam exposure to ensure sterilization. If the autoclave does not reach the right temperature, the spores will germinate when incubated and their metabolism will change the color of a pH-sensitive chemical. In most of the industrialized world medical-grade autoclaves are regulated medical devices. Other types of autoclaves are used to grow crystals under high temperatures and pressures. In dentistry, autoclaves provide sterilization of dental instruments. ",1.1,14.80,"""29.9""",1.95 ,0.020,D
+Imagine gaseous $\mathrm{Ar}$ at $298 \mathrm{~K}$ confined to move in a two-dimensional plane of area $1.00 \mathrm{~cm}^2$. What is the value of the translational partition function?,"In statistical mechanics, the translational partition function, q_T is that part of the partition function resulting from the movement (translation) of the center of mass. The partition function is a function of the temperature T and the microstate energies E1, E2, E3, etc. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. The partition function has many physical meanings, as discussed in Meaning and significance. == Canonical partition function == === Definition === Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles are fixed. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies. ==Grand canonical partition function== We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. When considering a set of N non-interacting but identical atoms or molecules, when QT ≫ N , or equivalently when ρ Λ ≪ 1 where ρ is the density of particles, the total translational partition function can be written : Q_T(T,N) = \frac{ q_T(T)^N }{N!} A plane partition may be represented visually by the placement of a stack of \pi_{i,j} unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. In this case we must describe the partition function using an integral rather than a sum. The asymptotics for plane partitions were first calculated by E. M. Wright.E. M. Wright, Asymptotic partition formulae I. Plane partitions, The Quarterly Journal of Mathematics 1 (1931) 177–189. In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Plane partitions are a generalization of partitions of an integer. By taking the zero point energy as the reference point to which other energies are measured, the expression for the partition function becomes Q_\text{vib}(T) = \prod_{i=1}^f \frac{1}{1-e^{-\Theta_{\text{vib},i}/T}} == References == ==See also== * Partition function (mathematics) Category:Partition functions The sum of a plane partition is : n=\sum_{i,j} \pi_{i,j} . In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. This partition function is closely related to the grand potential, \Phi_{\rm G}, by the relation : -k_B T \ln \mathcal{Z} = \Phi_{\rm G} = \langle E \rangle - TS - \mu \langle N\rangle. The usefulness of the partition function stems from the fact that the macroscopic thermodynamic quantities of a system can be related to its microscopic details through the derivatives of its partition function. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. From this point of view, a plane partition can be defined as a finite subset \mathcal{P} of positive integer lattice points (i, j, k) in \mathbb{N}^3, such that if (r, s, t) lies in \mathcal{P} and if (i, j, k) satisfies 1\leq i\leq r, 1\leq j\leq s, and 1\leq k\leq t, then (i, j, k) also lies in \mathcal{P}. The partition function can be related to thermodynamic properties because it has a very important statistical meaning. Using this approximation we can derive a closed form expression for the vibrational partition function. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous. ====Classical discrete system==== For a canonical ensemble that is classical and discrete, the canonical partition function is defined as Z = \sum_i e^{-\beta E_i}, where * i is the index for the microstates of the system; * e is Euler's number; * \beta is the thermodynamic beta, defined as \tfrac{1}{k_\text{B} T} where k_\text{B} is Boltzmann's constant; * E_i is the total energy of the system in the respective microstate. ",-87.8,1.69,"""7.0""",3.9,1.45,D
+"Determine the equilibrium constant for the dissociation of sodium at $298 \mathrm{~K}: \mathrm{Na}_2(g) \rightleftharpoons 2 \mathrm{Na}(g)$. For $\mathrm{Na}_2$, $B=0.155 \mathrm{~cm}^{-1}, \widetilde{\nu}=159 \mathrm{~cm}^{-1}$, the dissociation energy is $70.4 \mathrm{~kJ} / \mathrm{mol}$, and the ground-state electronic degeneracy for $\mathrm{Na}$ is 2 .","Bioanalytical Chemistry Textbook De Gruyter 2021 https://doi.org/10.1515/9783110589160-206 For a general reaction: : A_\mathit{x} B_\mathit{y} <=> \mathit{x} A{} + \mathit{y} B in which a complex \ce{A}_x \ce{B}_y breaks down into x A subunits and y B subunits, the dissociation constant is defined as : K_D = \frac{[\ce A]^x [\ce B]^y}{[\ce A_x \ce B_y]} where [A], [B], and [Ax By] are the equilibrium concentrations of A, B, and the complex Ax By, respectively. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). Nonadiabatic transition state theory (NA-TST) is a powerful tool to predict rates of chemical reactions from a computational standpoint. (The symbol K_a, used for the acid dissociation constant, can lead to confusion with the association constant and it may be necessary to see the reaction or the equilibrium expression to know which is meant.) In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. A molecule can have several acid dissociation constants. In the special case of salts, the dissociation constant can also be called an ionization constant. For example, for the ionic crystal NaCl, there arise two Madelung constants - one for Na and another for Cl. The molecular formula C17H14N2O2 (molar mass: 278.305 g/mol, exact mass: 278.1055 u) may refer to: * Bimakalim * Sudan Red G Category:Molecular formulas The dissociation constant is the inverse of the association constant. : Ab + Ag <=> AbAg : K_A = \frac{\left[ \ce{AbAg} \right]}{\left[ \ce{Ab} \right] \left[ \ce{Ag} \right]} = \frac{1}{K_D} This chemical equilibrium is also the ratio of the on-rate (kforward or ka) and off-rate (kback or kd) constants. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants. == Formula == A fast converging formula for the Madelung constant of NaCl is :12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right) == Generalization == It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. The binding constant, or affinity constant/association constant, is a special case of the equilibrium constant K, and is the inverse of the dissociation constant. The dissociation constant has molar units (M) and corresponds to the ligand concentration [L] at which half of the proteins are occupied at equilibrium, i.e., the concentration of ligand at which the concentration of protein with ligand bound [LP] equals the concentration of protein with no ligand bound [P]. [B^{-3}] \over [H B^{-2}]} & \mathrm{p}K_3 &= -\log K_3 \end{align} ==Dissociation constant of water== The dissociation constant of water is denoted Kw: :K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-] The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration. The properties of this ion are strongly related to the surface potential present on a corresponding solid. NA-TST can be reduced to the traditional TST in the limit of unit probability. ==References== Category:Chemical physics Examples of Madelung constants Ion in crystalline compound M (based on ) \overline{M} (based on ) Cl− and Cs+ in CsCl ±1.762675 ±2.035362 Cl− and Na+ in rocksalt NaCl ±1.747565 ±3.495129 S2− and Zn2+ in sphalerite ZnS ±3.276110 ±7.56585 F− in fluorite CaF2 1.762675 4.070723 Ca2+ in fluorite CaF2 -3.276110 −7.56585 The continuous reduction of with decreasing coordination number for the three cubic AB compounds (when accounting for the doubled charges in ZnS) explains the observed propensity of alkali halides to crystallize in the structure with highest compatible with their ionic radii. Sub-picomolar dissociation constants as a result of non-covalent binding interactions between two molecules are rare. For the binding of receptor and ligand molecules in solution, the molar Gibbs free energy ΔG, or the binding affinity is related to the dissociation constant Kd via :\Delta G = R T\ln{{K_{\rm d} \over c^{\ominus}}}, in which R is the ideal gas constant, T temperature and the standard reference concentration c ~~o~~ = 1 mol/L. == See also == * Binding coefficient Category:Equilibrium chemistry ",−1.642876,24,"""2.25""",6.9,537,C
+"At $298.15 \mathrm{~K}, \Delta G_f^{\circ}(\mathrm{HCOOH}, g)=-351.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta G_f^{\circ}(\mathrm{HCOOH}, l)=-361.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the vapor pressure of formic acid at this temperature.","Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds"", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The second solution is switching to another vapor pressure equation with more than three parameters. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The formula is valid from about −50 to 102 °C; however there are a very limited number of measurements of the vapor pressure of water over supercooled liquid water. (The small differences in the results are only caused by the used limited precision of the coefficients). ==Extension of the Antoine equations== To overcome the limits of the Antoine equation some simple extension by additional terms are used: : \begin{align} P &= \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)} \\\ P &= \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right). \end{align} The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Regarding the measurement of upper-air humidity, this publication also reads (in Section 12.5.1): The saturation with respect to water cannot be measured much below –50 °C, so manufacturers should use one of the following expressions for calculating saturation vapour pressure relative to water at the lowest temperatures – Wexler (1976, 1977), reported by Flatau et al. (1992)., Hyland and Wexler (1983) or Sonntag (1994) – and not the Goff- Gratch equation recommended in earlier WMO publications. ==Experimental correlation== The original Goff–Gratch (1945) experimental correlation reads as follows: : \log\ e^*\ = -7.90298(T_\mathrm{st}/T-1)\ +\ 5.02808\ \log(T_\mathrm{st}/T) -\ 1.3816\times10^{-7}(10^{11.344(1-T/T_\mathrm{st})}-1) +\ 8.1328\times10^{-3}(10^{-3.49149(T_\mathrm{st}/T-1)}-1)\ +\ \log\ e^*_\mathrm{st} where: :log refers to the logarithm in base 10 :e* is the saturation water vapor pressure (hPa) :T is the absolute air temperature in kelvins :Tst is the steam-point (i.e. boiling point at 1 atm.) temperature (373.15 K) :e*st is e* at the steam-point pressure (1 atm = 1013.25 hPa) Similarly, the correlation for the saturation water vapor pressure over ice is: : \log\ e^*_i\ = -9.09718(T_0/T-1)\ -\ 3.56654\ \log(T_0/T) +\ 0.876793(1-T/T_0) +\ \log\ e^*_{i0} where: :log stands for the logarithm in base 10 :e*i is the saturation water vapor pressure over ice (hPa) :T is the air temperature (K) :T0 is the ice-point (triple point) temperature (273.16 K) :e*i0 is e* at the ice-point pressure (6.1173 hPa) ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Tetens equation *Lee–Kesler method ==References== * Goff, J. A., and Gratch, S. (1946) Low- pressure properties of water from −160 to 212 °F, in Transactions of the American Society of Heating and Ventilating Engineers, pp 95–122, presented at the 52nd annual meeting of the American Society of Heating and Ventilating Engineers, New York, 1946. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Water vapor (H2O) 200px Liquid state Water Solid state Ice Properties Molecular formula H2O Molar mass 18.01528(33) g/mol Melting point Vienna Standard Mean Ocean Water (VSMOW), used for calibration, melts at 273.1500089(10) K (0.000089(10) °C) and boils at 373.1339 K (99.9839 °C) Boiling point specific gas constant 461.5 J/(kg·K) Heat of vaporization 2.27 MJ/kg Heat capacity 1.864 kJ/(kg·K) Water vapor, water vapour or aqueous vapor is the gaseous phase of water. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., ""Antoine Vapor Pressure Constants of Pure Compounds"" ** Yaws C. L., Yang H.-C., ""To Estimate Vapor Pressure Easily. The maximum partial pressure (saturation pressure) of water vapor in air varies with temperature of the air and water vapor mixture. The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: :T = \frac{B}{A-\log_{10}\, p} - C ==Validity range== Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. The Goff–Gratch equation is one (arguably the first reliable in history) amongst many experimental correlation proposed to estimate the saturation water vapor pressure at a given temperature. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. A variety of empirical formulas exist for this quantity; the most used reference formula is the Goff-Gratch equation for the SVP over liquid water below zero degrees Celsius: :\begin{align} \log_{10} \left( p \right) = & -7.90298 \left( \frac{373.16}{T}-1 \right) + 5.02808 \log_{10} \frac{373.16}{T} \\\ & \- 1.3816 \times 10^{-7} \left( 10^{11.344 \left( 1-\frac{T}{373.16} \right)} -1 \right) \\\ & \+ 8.1328 \times 10^{-3} \left( 10^{-3.49149 \left( \frac{373.16}{T}-1 \right)} -1 \right) \\\ & \+ \log_{10} \left( 1013.246 \right) \end{align} where , temperature of the moist air, is given in units of kelvin, and is given in units of millibars (hectopascals). ",1.51,2.00,"""6.07""",358800,-167,A
+The collisional cross section of $\mathrm{N}_2$ is $0.43 \mathrm{~nm}^2$. What is the diffusion coefficient of $\mathrm{N}_2$ at a pressure of $1 \mathrm{~atm}$ and a temperature of $298 \mathrm{~K}$ ?,"The diffusion profile therefore can be depicted by the following equation. (dln\bar{c}/dy^{6/5})^{5/3}=0.66(D_1/t)^{1/2}(1/D_b\delta) To further determine D_b , two common methods were used. The first equation represents diffusion in the volume, while the second shows diffusion along the grain boundary, respectively. \frac{\partial c}{\partial t}=D\left({\partial^2 c\over\partial x^2}+{\partial^2 c\over\partial y^2}\right) where |x|>\delta/2 \frac{\partial c_b}{\partial t}=D_b\left({\partial^2 c_b\over\partial y^2}\right)+\frac{2D}{\delta}\left(\frac{\partial c}{\partial x}\right)_{x=\delta/2} where c(x, y, t) is the volume concentration of the diffusing atoms and c_b(y, t) is their concentration in the grain boundary. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). thumb|Thermal diffusion coefficients vs. temperature, for air at normal pressure The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. * Method 2: To compare the length of penetration of a given concentration at the boundary \ \Delta y with the length of lattice penetration from the surface far from the boundary. == References == == See also == * Kirkendall effect * Phase transformations in solids * Mass diffusivity Category:Diffusion The grain boundary diffusion coefficient is the diffusion coefficient of a diffusant along a grain boundary in a polycrystalline solid.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965. ""On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)"". The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux.S. Rehfeldt, J. Stichlmair: Measurement and calculation of multicomponent diffusion coefficients in liquids, Fluid Phase Equilibria, 2007, 256, 99–104 The molecular friction between two components is proportional to their difference in speed and their mole fractions. Suppose that the thickness of the slab is \delta, the length is y, and the depth is a unit length, the diffusion process can be described as the following formula. The effective diffusion coefficient of a in atomic diffusion of solid polycrystalline materials like metal alloys is often represented as a weighted average of the grain boundary diffusion coefficient and the lattice diffusion coefficient.P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965. However, at temperatures below 700 °C, the values of D_b with polycrystal silver consistently lie above the values of D_b with a single crystal. == Measurement == The general way to measure grain boundary diffusion coefficients was suggested by Fisher. It dominates the effective diffusion rate at lower temperatures in metals and metal alloys. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. The Boltzmann–Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8 Tmelt (of an alloy), the grain boundary component can be neglected. ==Modeling== The effective diffusion coefficient can be modeled using Hart's equation when lattice diffusion is dominant (type A kinetics): : D_\text{eff} = f D_\text{gb} + (1-f) D_\ell where :D_\text{eff} = {}effective diffusion coefficient :D_\text{gb} = {}grain boundary diffusion coefficient :D_\ell = {}lattice diffusion coefficient :f = \frac{q \delta}{d} : q = {}value based on grain shape, 1 for parallel grains, 3 for square grains : d = {}average grain size :\delta = {}grain boundary width, often assumed to be 0.5 nm Grain boundary diffusion is significant in face-centered cubic metals below about 0.8 Tmelt (Absolute). The Mathematics of Diffusion. This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law. == Matano’s method == Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4–0.6, so as temperature is lowered, the grain boundary diffusion component increases. Assuming a diffusion coefficient D that is in general a function of concentration c, Fick's second law is :\frac{\partial c}{\partial t} = \frac{\partial}{\partial x} \underbrace{\left[ D(c)\frac{\partial c}{\partial x} \right]}_\text{flux}, where t is time, and x is distance. The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. Another example of double diffusion is the formation of false bottoms at the interface of sea ice and under-ice meltwater layers. ",5654.86677646,0.88,"""1.6""",1.06,"102,965.21",D
 "A vessel contains $1.15 \mathrm{~g}$ liq $\mathrm{H}_2 \mathrm{O}$ in equilibrium with water vapor at $30 .{ }^{\circ} \mathrm{C}$. At this temperature, the vapor pressure of $\mathrm{H}_2 \mathrm{O}$ is 31.82 torr. What volume increase is necessary for all the water to evaporate?
-","The rate of evaporation in an open system is related to the vapor pressure found in a closed system. The boiling temperature of water for atmospheric pressures can be approximated by the Antoine equation: :\log_{10}\left(\frac{P}{1\text{ Torr}}\right) = 8.07131 - \frac{1730.63\ {}^\circ\text{C}}{233.426\ {}^\circ\text{C} + T_b} or transformed into this temperature-explicit form: :T_b = \frac{1730.63\ {}^\circ\text{C}}{8.07131 - \log_{10}\left(\frac{P}{1\text{ Torr}}\right)} - 233.426\ {}^\circ\text{C} where the temperature T_b is the boiling point in degrees Celsius and the pressure P is in torr. ==Dühring's rule== Dühring's rule states that a linear relationship exists between the temperatures at which two solutions exert the same vapor pressure. ==Examples== The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units). Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations On the computation of saturation vapour pressure. The evaporation will continue until an equilibrium is reached when the evaporation of the liquid is equal to its condensation. (Alternate title: ""Water Vapor Myths: A Brief Tutorial"".) ==See also== * Absolute humidity * Antoine equation * Lee–Kesler method * Osmotic coefficient * Raoult's law: vapor pressure lowering in solution * Reid vapor pressure * Relative humidity * Relative volatility * Saturation vapor density * Triple point * True vapor pressure * Vapor–liquid equilibrium * Vapor pressures of the elements (data page) * Vapour pressure of water ==References== ==External links== *Fluid Characteristics Chart, Engineer's Edge *Vapor Pressure, Hyperphysics *Vapor Pressure, The MSDS HyperGlossary *Online vapor pressure calculation tool (Requires Registration) *Prediction of Vapor Pressures of Pure Liquid Organic Compounds Category:Engineering thermodynamics Category:Gases Category:Meteorological concepts Category:Pressure Category:Thermodynamic properties Air is given a vapour density of one. The equilibrium vapor pressure is an indication of a liquid's thermodynamic tendency to evaporate. Many of the molecules return to the liquid, with returning molecules becoming more frequent as the density and pressure of the vapor increases. At the normal boiling point of a liquid, the vapor pressure is equal to the standard atmospheric pressure defined as 1 atmosphere, 760Torr, 101.325kPa, or 14.69595psi. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidity affects rate of evaporation of water. Evaporation also tends to proceed more quickly with higher flow rates between the gaseous and liquid phase and in liquids with higher vapor pressure. Actually, as stated by Dalton's law (known since 1802), the partial pressure of water vapor or any substance does not depend on air at all, and the relevant temperature is that of the liquid. In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. ;Concentration of the substance evaporating in the air: If the air already has a high concentration of the substance evaporating, then the given substance will evaporate more slowly. The pressure exhibited by vapor present above a liquid surface is known as vapor pressure. According to Monteith and Unsworth, ""Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C."" Even at lower temperatures, individual molecules of a liquid can evaporate if they have more than the minimum amount of kinetic energy required for vaporization. == Factors influencing the rate of evaporation == Note: Air is used here as a common example of the surrounding gas; however, other gases may hold that role. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and cause the liquid to form vapor bubbles. When only a small proportion of the molecules meet these criteria, the rate of evaporation is low. In an enclosed environment, a liquid will evaporate until the surrounding air is saturated. For a system consisting of vapor and liquid of a pure substance, this equilibrium state is directly related to the vapor pressure of the substance, as given by the Clausius–Clapeyron relation: : \ln \left( \frac{ P_2 }{ P_1 } \right) = - \frac{ \Delta H_{\rm vap } }{ R } \left( \frac{ 1 }{ T_2 } - \frac{ 1 }{ T_1 } \right) where P1, P2 are the vapor pressures at temperatures T1, T2 respectively, ΔHvap is the enthalpy of vaporization, and R is the universal gas constant. ",15.425,2,3.03,-383,37.9,E
-A cell is roughly spherical with a radius of $20.0 \times 10^{-6} \mathrm{~m}$. Calculate the work required to expand the cell surface against the surface tension of the surroundings if the radius increases by a factor of three. Assume the cell is surrounded by pure water and that $T=298.15 \mathrm{~K}$.,"In that case, in order to increase the surface area of a mass of liquid by an amount, , a quantity of work, , is needed (where is the surface energy density of the liquid). The work associated with the first step (unstrained) is W_1 = 2 \gamma_0 A_0, where \gamma_0 and A_0 are the excess free energy and area of each of new surfaces. For the second step, work (w_2), equals the work needed to elastically deform the total bulk volume and the four (two original and two newly formed) surfaces. A suggestion is surface stress define as association with the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface instead of up definition. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. Surface stress was first defined by Josiah Willard Gibbs (1839-1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface. Incorporating this value into the surface energy equation allows for the surface energy to be estimated. The driving force for a change in the surface concentration associated with a contraction of the surface is proportional to the difference between surface stress and surface free energy. In order to move a cube from the bulk of a material to the surface, energy is required. Based on the contact angle results and knowing the surface tension of the liquids, the surface energy can be calculated. A common approach to achieving this is known as the workcell. However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy. A workcell is an arrangement of resources in a manufacturing environment to improve the quality, speed and cost of the process. The following equation can be used as a reasonable estimate for surface energy: :\gamma \approx \frac{-\Delta_\text{sub} H\left(z_\sigma - z_\beta\right)}{a_0 N_\text{A} z_\beta} == Interfacial energy == The presence of an interface influences generally all thermodynamic parameters of a system. The surface energy of the facets can thus be found to within a scaling constant by measuring the relative sizes of the facets. ===Calculation=== ====Deformed solid==== In the deformation of solids, surface energy can be treated as the ""energy required to create one unit of surface area"", and is a function of the difference between the total energies of the system before and after the deformation: :\gamma = \frac{1}{A} \left(E_1 - E_0\right). This energy cost is incorporated into the surface energy of the material, which is quantified by: thumb|center|480x240px|Cube model. The surface wants to expand creating a compressive stress. Surface area can be determined by squaring the cube root of the volume of the molecule: :a_0 = V_\text{molecule}^\frac{2}{3} = \left(\frac{\bar{M}}{\rho N_\text{A}}\right)^\frac{2}{3} Here, corresponds to the molar mass of the molecule, corresponds to the density, and is the Avogadro constant. The cube model can be used to model pure, uniform materials or an individual molecular component to estimate their surface energy. :\gamma = \frac{\left(z_\sigma - z_\beta\right) \frac{1}{2}W_\text{AA}}{a_0} where and are coordination numbers corresponding to the surface and the bulk regions of the material, and are equal to 5 and 6, respectively; is the surface area of an individual molecule, and is the pairwise intermolecular energy. Experimental setup for measuring relative surface energy and its function can be seen in the video. ===Estimation from the heat of sublimation=== To estimate the surface energy of a pure, uniform material, an individual region of the material can be modeled as a cube. There are several different models for calculating the surface energy based on the contact angle readings. ",14,4943,49.0,2.89,2.10,D
-"A vessel is filled completely with liquid water and sealed at $13.56^{\circ} \mathrm{C}$ and a pressure of 1.00 bar. What is the pressure if the temperature of the system is raised to $82.0^{\circ} \mathrm{C}$ ? Under these conditions, $\beta_{\text {water }}=2.04 \times 10^{-4} \mathrm{~K}^{-1}$, $\beta_{\text {vessel }}=1.42 \times 10^{-4} \mathrm{~K}^{-1}$, and $\kappa_{\text {water }}=4.59 \times 10^{-5} \mathrm{bar}^{-1}$.","Details of the calculation: \left( \frac{\partial P}{\partial T} \right)_{V} = -\left( \frac{\partial V}{\partial T} \right)_{p}\left( \frac{\partial P}{\partial V} \right)_{T} = - (V\alpha) \left(\frac{-1}{\kappa_T}\right) = \alpha\kappa_T \left( \frac{\partial P}{\partial T} \right)_{V} = \frac{\frac{1}{V}\left( \frac{\partial V}{\partial T} \right)_{p}}{\frac{-1}{V} \left( \frac{\partial V}{\partial P} \right)_{T}} = \frac{\alpha}{\beta} ==The utility of the thermal pressure== thumb|upright=1.6|Figure 1: Thermal pressure as a function of temperature normalized to A of the few compounds commonly used in the study of Geophysics. Specific heat capacity at constant pressure also increases with temperature, from 4.187 kJ/kg at 25 °C to 8.138 kJ/kg at 350 °C. thumb|400px|Diagram showing pressure difference induced by a temperature difference. Thus, the thermal pressure of a solid due to moderate temperature change above the Debye temperature can be approximated by assuming a constant value of \alpha and \kappa_T. Angel, Ross J., Miozzi Francesca, and Alvaro Matteo (2019). American Academy of Arts & Sciences. . ==Thermal pressure at high temperature== As mentioned above, \alpha\kappa_T is one of the most common formulations for the thermal pressure coefficient. Some formulations for the thermal pressure coefficient include: \left( \frac{\partial P}{\partial T} \right)_{v} = \alpha\kappa_T = \frac{\gamma}{V}C_V = \frac{\alpha}{\beta_T} Where \alpha is the volume thermal expansion, \kappa_T the isothermal bulk modulus, \gamma the Grüneisen parameter, \beta_T the compressibility and C_Vthe constant-volume heat capacity. Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations This pressure is given by the saturated vapour pressure, and can be looked up in steam tables, or calculated. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. As a guide, the saturated vapour pressure at 121 °C is 200 kPa, 150 °C is 470 kPa, and 200 °C is 1,550 kPa. Int J Thermophys 43, 169 (2022). https://doi.org/10.1007/s10765-022-03089-8 authors demonstrated that,at ambient pressure, the pressure predicted of Au and MgO from a constant value of \alpha\kappa_T deviates from the experimental data, and the higher temperature, the more deviation. In thermodynamics, thermal pressure (also known as the thermal pressure coefficient) is a measure of the relative pressure change of a fluid or a solid as a response to a temperature change at constant volume. According to Monteith and Unsworth, ""Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C."" Above about 300 °C, water starts to behave as a near-critical liquid, and physical properties such as density start to change more significantly with pressure. The combined effect of a change in pressure and temperature is described by the strain tensor \varepsilon_{ij}:\varepsilon_{ij}= \alpha_{ij} dT- \beta_{ij} dP Where \alpha_{ij} is the volume thermal expansion tensor and \beta_{ij} is the compressibility tensor. Commonly the thermal pressure coefficient may be expressed as functions of temperature and volume. ""A New Conception of Thermal Pressure and a Theory of Solutions"". Thus, the study of the thermal pressure coefficient provides a useful basis for understanding the nature of liquid and solid. The thermal pressure coefficient is used to calculate results that are applied widely in industry, and they would further accelerate the development of thermodynamic theory. The Tetens equation is an equation to calculate the saturation vapour pressure of water over liquid and ice. For example, to heat water from 25 °C to steam at 250 °C at 1 atm requires 2869 kJ/kg. There are two main types of calculation of the thermal pressure coefficient: one is the Virial theorem and its derivatives; the other is the Van der Waals type and its derivatives. ",93.4,15.1,1.5,5040,7,A
+","The rate of evaporation in an open system is related to the vapor pressure found in a closed system. The boiling temperature of water for atmospheric pressures can be approximated by the Antoine equation: :\log_{10}\left(\frac{P}{1\text{ Torr}}\right) = 8.07131 - \frac{1730.63\ {}^\circ\text{C}}{233.426\ {}^\circ\text{C} + T_b} or transformed into this temperature-explicit form: :T_b = \frac{1730.63\ {}^\circ\text{C}}{8.07131 - \log_{10}\left(\frac{P}{1\text{ Torr}}\right)} - 233.426\ {}^\circ\text{C} where the temperature T_b is the boiling point in degrees Celsius and the pressure P is in torr. ==Dühring's rule== Dühring's rule states that a linear relationship exists between the temperatures at which two solutions exert the same vapor pressure. ==Examples== The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units). Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations On the computation of saturation vapour pressure. The evaporation will continue until an equilibrium is reached when the evaporation of the liquid is equal to its condensation. (Alternate title: ""Water Vapor Myths: A Brief Tutorial"".) ==See also== * Absolute humidity * Antoine equation * Lee–Kesler method * Osmotic coefficient * Raoult's law: vapor pressure lowering in solution * Reid vapor pressure * Relative humidity * Relative volatility * Saturation vapor density * Triple point * True vapor pressure * Vapor–liquid equilibrium * Vapor pressures of the elements (data page) * Vapour pressure of water ==References== ==External links== *Fluid Characteristics Chart, Engineer's Edge *Vapor Pressure, Hyperphysics *Vapor Pressure, The MSDS HyperGlossary *Online vapor pressure calculation tool (Requires Registration) *Prediction of Vapor Pressures of Pure Liquid Organic Compounds Category:Engineering thermodynamics Category:Gases Category:Meteorological concepts Category:Pressure Category:Thermodynamic properties Air is given a vapour density of one. The equilibrium vapor pressure is an indication of a liquid's thermodynamic tendency to evaporate. Many of the molecules return to the liquid, with returning molecules becoming more frequent as the density and pressure of the vapor increases. At the normal boiling point of a liquid, the vapor pressure is equal to the standard atmospheric pressure defined as 1 atmosphere, 760Torr, 101.325kPa, or 14.69595psi. High concentration of the evaporating substance in the surrounding gas significantly slows down evaporation, such as when humidity affects rate of evaporation of water. Evaporation also tends to proceed more quickly with higher flow rates between the gaseous and liquid phase and in liquids with higher vapor pressure. Actually, as stated by Dalton's law (known since 1802), the partial pressure of water vapor or any substance does not depend on air at all, and the relevant temperature is that of the liquid. In thermodynamics, the entropy of vaporization is the increase in entropy upon vaporization of a liquid. ;Concentration of the substance evaporating in the air: If the air already has a high concentration of the substance evaporating, then the given substance will evaporate more slowly. The pressure exhibited by vapor present above a liquid surface is known as vapor pressure. According to Monteith and Unsworth, ""Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C."" Even at lower temperatures, individual molecules of a liquid can evaporate if they have more than the minimum amount of kinetic energy required for vaporization. == Factors influencing the rate of evaporation == Note: Air is used here as a common example of the surrounding gas; however, other gases may hold that role. With any incremental increase in that temperature, the vapor pressure becomes sufficient to overcome atmospheric pressure and cause the liquid to form vapor bubbles. When only a small proportion of the molecules meet these criteria, the rate of evaporation is low. In an enclosed environment, a liquid will evaporate until the surrounding air is saturated. For a system consisting of vapor and liquid of a pure substance, this equilibrium state is directly related to the vapor pressure of the substance, as given by the Clausius–Clapeyron relation: : \ln \left( \frac{ P_2 }{ P_1 } \right) = - \frac{ \Delta H_{\rm vap } }{ R } \left( \frac{ 1 }{ T_2 } - \frac{ 1 }{ T_1 } \right) where P1, P2 are the vapor pressures at temperatures T1, T2 respectively, ΔHvap is the enthalpy of vaporization, and R is the universal gas constant. ",15.425,2,"""3.03""",-383,37.9,E
+A cell is roughly spherical with a radius of $20.0 \times 10^{-6} \mathrm{~m}$. Calculate the work required to expand the cell surface against the surface tension of the surroundings if the radius increases by a factor of three. Assume the cell is surrounded by pure water and that $T=298.15 \mathrm{~K}$.,"In that case, in order to increase the surface area of a mass of liquid by an amount, , a quantity of work, , is needed (where is the surface energy density of the liquid). The work associated with the first step (unstrained) is W_1 = 2 \gamma_0 A_0, where \gamma_0 and A_0 are the excess free energy and area of each of new surfaces. For the second step, work (w_2), equals the work needed to elastically deform the total bulk volume and the four (two original and two newly formed) surfaces. A suggestion is surface stress define as association with the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface instead of up definition. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. Surface stress was first defined by Josiah Willard Gibbs (1839-1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface. Incorporating this value into the surface energy equation allows for the surface energy to be estimated. The driving force for a change in the surface concentration associated with a contraction of the surface is proportional to the difference between surface stress and surface free energy. In order to move a cube from the bulk of a material to the surface, energy is required. Based on the contact angle results and knowing the surface tension of the liquids, the surface energy can be calculated. A common approach to achieving this is known as the workcell. However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy. A workcell is an arrangement of resources in a manufacturing environment to improve the quality, speed and cost of the process. The following equation can be used as a reasonable estimate for surface energy: :\gamma \approx \frac{-\Delta_\text{sub} H\left(z_\sigma - z_\beta\right)}{a_0 N_\text{A} z_\beta} == Interfacial energy == The presence of an interface influences generally all thermodynamic parameters of a system. The surface energy of the facets can thus be found to within a scaling constant by measuring the relative sizes of the facets. ===Calculation=== ====Deformed solid==== In the deformation of solids, surface energy can be treated as the ""energy required to create one unit of surface area"", and is a function of the difference between the total energies of the system before and after the deformation: :\gamma = \frac{1}{A} \left(E_1 - E_0\right). This energy cost is incorporated into the surface energy of the material, which is quantified by: thumb|center|480x240px|Cube model. The surface wants to expand creating a compressive stress. Surface area can be determined by squaring the cube root of the volume of the molecule: :a_0 = V_\text{molecule}^\frac{2}{3} = \left(\frac{\bar{M}}{\rho N_\text{A}}\right)^\frac{2}{3} Here, corresponds to the molar mass of the molecule, corresponds to the density, and is the Avogadro constant. The cube model can be used to model pure, uniform materials or an individual molecular component to estimate their surface energy. :\gamma = \frac{\left(z_\sigma - z_\beta\right) \frac{1}{2}W_\text{AA}}{a_0} where and are coordination numbers corresponding to the surface and the bulk regions of the material, and are equal to 5 and 6, respectively; is the surface area of an individual molecule, and is the pairwise intermolecular energy. Experimental setup for measuring relative surface energy and its function can be seen in the video. ===Estimation from the heat of sublimation=== To estimate the surface energy of a pure, uniform material, an individual region of the material can be modeled as a cube. There are several different models for calculating the surface energy based on the contact angle readings. ",14,4943,"""49.0""",2.89,2.10,D
+"A vessel is filled completely with liquid water and sealed at $13.56^{\circ} \mathrm{C}$ and a pressure of 1.00 bar. What is the pressure if the temperature of the system is raised to $82.0^{\circ} \mathrm{C}$ ? Under these conditions, $\beta_{\text {water }}=2.04 \times 10^{-4} \mathrm{~K}^{-1}$, $\beta_{\text {vessel }}=1.42 \times 10^{-4} \mathrm{~K}^{-1}$, and $\kappa_{\text {water }}=4.59 \times 10^{-5} \mathrm{bar}^{-1}$.","Details of the calculation: \left( \frac{\partial P}{\partial T} \right)_{V} = -\left( \frac{\partial V}{\partial T} \right)_{p}\left( \frac{\partial P}{\partial V} \right)_{T} = - (V\alpha) \left(\frac{-1}{\kappa_T}\right) = \alpha\kappa_T \left( \frac{\partial P}{\partial T} \right)_{V} = \frac{\frac{1}{V}\left( \frac{\partial V}{\partial T} \right)_{p}}{\frac{-1}{V} \left( \frac{\partial V}{\partial P} \right)_{T}} = \frac{\alpha}{\beta} ==The utility of the thermal pressure== thumb|upright=1.6|Figure 1: Thermal pressure as a function of temperature normalized to A of the few compounds commonly used in the study of Geophysics. Specific heat capacity at constant pressure also increases with temperature, from 4.187 kJ/kg at 25 °C to 8.138 kJ/kg at 350 °C. thumb|400px|Diagram showing pressure difference induced by a temperature difference. Thus, the thermal pressure of a solid due to moderate temperature change above the Debye temperature can be approximated by assuming a constant value of \alpha and \kappa_T. Angel, Ross J., Miozzi Francesca, and Alvaro Matteo (2019). American Academy of Arts & Sciences. . ==Thermal pressure at high temperature== As mentioned above, \alpha\kappa_T is one of the most common formulations for the thermal pressure coefficient. Some formulations for the thermal pressure coefficient include: \left( \frac{\partial P}{\partial T} \right)_{v} = \alpha\kappa_T = \frac{\gamma}{V}C_V = \frac{\alpha}{\beta_T} Where \alpha is the volume thermal expansion, \kappa_T the isothermal bulk modulus, \gamma the Grüneisen parameter, \beta_T the compressibility and C_Vthe constant-volume heat capacity. Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations This pressure is given by the saturated vapour pressure, and can be looked up in steam tables, or calculated. The psychrometric constant \gamma relates the partial pressure of water in air to the air temperature. As a guide, the saturated vapour pressure at 121 °C is 200 kPa, 150 °C is 470 kPa, and 200 °C is 1,550 kPa. Int J Thermophys 43, 169 (2022). https://doi.org/10.1007/s10765-022-03089-8 authors demonstrated that,at ambient pressure, the pressure predicted of Au and MgO from a constant value of \alpha\kappa_T deviates from the experimental data, and the higher temperature, the more deviation. In thermodynamics, thermal pressure (also known as the thermal pressure coefficient) is a measure of the relative pressure change of a fluid or a solid as a response to a temperature change at constant volume. According to Monteith and Unsworth, ""Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C."" Above about 300 °C, water starts to behave as a near-critical liquid, and physical properties such as density start to change more significantly with pressure. The combined effect of a change in pressure and temperature is described by the strain tensor \varepsilon_{ij}:\varepsilon_{ij}= \alpha_{ij} dT- \beta_{ij} dP Where \alpha_{ij} is the volume thermal expansion tensor and \beta_{ij} is the compressibility tensor. Commonly the thermal pressure coefficient may be expressed as functions of temperature and volume. ""A New Conception of Thermal Pressure and a Theory of Solutions"". Thus, the study of the thermal pressure coefficient provides a useful basis for understanding the nature of liquid and solid. The thermal pressure coefficient is used to calculate results that are applied widely in industry, and they would further accelerate the development of thermodynamic theory. The Tetens equation is an equation to calculate the saturation vapour pressure of water over liquid and ice. For example, to heat water from 25 °C to steam at 250 °C at 1 atm requires 2869 kJ/kg. There are two main types of calculation of the thermal pressure coefficient: one is the Virial theorem and its derivatives; the other is the Van der Waals type and its derivatives. ",93.4,15.1,"""1.5""",5040,7,A
 "A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height $h$ ) can be obtained by considering the potential energy due to gravity:
 $$
 P(h)=e^{-m g h / k T}
 $$
-In this expression $m$ is the per-particle mass of the gas, $g$ is the acceleration due to gravity, $k$ is a constant equal to $1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}$, and $T$ is temperature. Determine $\langle h\rangle$ for methane $\left(\mathrm{CH}_4\right)$ using this distribution function.","Atmospheric methane is the methane present in Earth's atmosphere. When methane reaches the surface and the atmosphere, it is known as atmospheric methane. Methane's heat of combustion is 55.5 MJ/kg.Energy Content of some Combustibles (in MJ/kg) . Methane has a boiling point of −161.5 °C at a pressure of one atmosphere. Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). [A] Net production of O3 CH4 \+ ·OH → CH3· + H2O CH3· + O2 \+ M → CH3O2· + M CH3O2· + NO → NO2 \+ CH3O· CH3O· + O2 → HO2· + HCHO HO2· + NO → NO2 \+ ·OH (2x) NO2 \+ hv → O(3P) + NO (2x) O(3P) + O2 \+ M → O3 \+ M [NET: CH4 \+ 4O2 → HCHO + 2O3 \+ H2O] [B] No net change of O3 CH4 \+ ·OH → CH3· + H2O CH3· + O2 \+ M → CH3O2· + M CH3O2· + HO2· + M → CH3O2H + O2 \+ M CH3O2H + hv → CH3O· + ·OH CH3O· + O2 → HO2· + HCHO [NET: CH4 \+ O2 → HCHO + H2O] ==See also== * Climate change * Global warming * Permafrost * Methane * Methane emissions ==Notes== ==References== ==External links== * * * * * * * * * * * Category:Methane Category:Atmosphere Category:Greenhouse gases A Chapman function describes the integration of atmospheric absorption along a slant path on a spherical earth, relative to the vertical case. The reaction of methane with hydroxyl in the troposphere or stratosphere creates the methyl radical ·CH3 and water vapor. Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. Methane is an important greenhouse gas with a global warming potential of 40 compared to (potential of 1) over a 100-year period, and above 80 over a 20-year period. The Chapman function is named after Sydney Chapman, who introduced the function in 1931. == Definition == In an isothermal model of the atmosphere, the density \varrho(h) varies exponentially with altitude h according to the Barometric formula: :\varrho(h) = \varrho_0 \exp\left(- \frac h H \right), where \varrho_0 denotes the density at sea level (h=0) and H the so-called scale height. The IPCC reports that the global warming potential (GWP) for methane is about 84 in terms of its impact over a 20-year timeframe See Table 8.7.—that means it traps 84 times more heat per mass unit than carbon dioxide (CO2) and 105 times the effect when accounting for aerosol interactions. The globally averaged concentration of methane in Earth's atmosphere increased by about 150% from 722 ± 25 ppb in 1750 to 1803.1 ± 0.6 ppb in 2011. Etminan et al. published their new calculations for methane's radiative forcing (RF) in a 2016 Geophysical Research Letters journal article which incorporated the shortwave bands of CH4 in measuring forcing, not used in previous, simpler IPCC methods. Image:Isothermal-barotropic atmosphere model.png ===The U.S. Standard Atmosphere=== The U.S. Standard Atmosphere model starts with many of the same assumptions as the isothermal-barotropic model, including ideal gas behavior, and constant molecular weight, but it differs by defining a more realistic temperature function, consisting of eight data points connected by straight lines; i.e. regions of constant temperature gradient. Methane is a strong GHG with a global warming potential 84 times greater than CO2 in a 20-year time frame. This global destruction of atmospheric methane mainly occurs in the troposphere. The annual average for methane (CH4) was 1866 ppb in 2019 and scientists reported with ""very high confidence"" that concentrations of CH4 were higher than at any time in at least 800,000 years. As air rises in the tropics, methane is carried upwards through the tropospherethe lowest portion of Earth's atmosphere which is to from the Earth's surface, into the lower stratospherethe ozone layerand then the upper portion of the stratosphere. (The total air mass below a certain altitude is calculated by integrating over the density function.) This geopotential altitude h is then used instead of geometric altitude z in the hydrostatic equations. ==Common models== * COSPAR International Reference Atmosphere * International Standard Atmosphere * Jacchia Reference Atmosphere, an older model still commonly used in spacecraft dynamics * Jet standard atmosphere * NRLMSISE-00 is a recent model from NRL often used in the atmospheric sciences * US Standard Atmosphere ==See also== * Standard temperature and pressure * Upper- atmospheric models ==References== ==External links== *Public Domain Aeronautical Software – Derivation of hydrostatic equations used in the 1976 US Standard Atmosphere *FORTRAN code to calculate the US Standard Atmosphere *NASA GSFC Atmospheric Models overview *Various models at NASA GSFC ModelWeb *Earth Global Reference Atmospheric Model (Earth-GRAM 2010) Category:Atmospheric sciences ",1.6,-1.0,62.8318530718,226,11,A
-"A camper stranded in snowy weather loses heat by wind convection. The camper is packing emergency rations consisting of $58 \%$ sucrose, $31 \%$ fat, and $11 \%$ protein by weight. Using the data provided in Problem P4.32 and assuming the fat content of the rations can be treated with palmitic acid data and the protein content similarly by the protein data in Problem P4.32, how much emergency rations must the camper consume in order to compensate for a reduction in body temperature of $3.5 \mathrm{~K}$ ? Assume the heat capacity of the body equals that of water. Assume the camper weighs $67 \mathrm{~kg}$.","Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The ration was two-thirds of a pound (302 g) of bread and two-thirds of a pound of meat. fourpence (4d) was deducted daily from the soldiers' pay. Vegetables and boiled starchy foods should be cooked without added salt ==== Daily Messing Rate ==== The Daily Messing Rate (DMR) is used to provide the following daily calorific intake; Daily Messing Rate Type Calorific Intake Basic DMR 3000 Kcal Exercise (Field) DMR. 4000 Kcal Overseas Exercise (Field) DMR. 4000 Kcal Operational DMR. 4000 Kcal Nijmegen Marches. 4000 Kcal Norway DMR. 5000 Kcal The current Daily Messing Rate is; * £2.73 in the United Kingdom * £3.60 outside the United Kingdom ==== Catering for diversity ==== In accordance with current UK legislation and Government guidelines it is incumbent on the Armed Forces to cater for all personnel irrespective of gender, race, religious belief, medical requirements and committed lifestyle choices. ==United States== During the American Revolution, the Continental Congress regulated garrison rations, stipulating in the Militia Law of 1775 that they should consist of: :One pound of beef, or 3/4 of a pound of pork or one pound of fish, per day. Rations in camp. The theoretical bases of indirect calorimetry: a review."" The daily ration scale in September 1941 was as follows; ==== Food ==== Meat Bacon and Ham Butter and margarine Cheese Cooking fats Sugar Tea Preserves Army rations Home Service Scale (Men) 12 oz (340 g) 1.14 oz (32 g) 1.89 oz (53 g) 0.57 oz (16 g) 0.28 oz (7 g) 4.28 oz (121 g) 0.57 oz (16 g) 1.14 oz (32 g) Army rations Home Service Scale (Women) 6 oz (170 g) 1.28 oz (36 g) 1.5 oz (42 g) (margarine only) 0.57 oz (16 g) - 2 oz (56 g) 0.28 oz (7 g) 1 oz (28 g) === Modern === ==== UK MOD Nutrition Policy Statement ==== Joint Service Publication (JSP) 456 Part 2 Volume 1 of December 2014, the Ministry of Defence policy on nutrition is as follows; The UK Ministry of Defence (MOD) undertakes to provide military personnel with a basic knowledge of nutrition, with the aim of optimising physical and mental function, long-term health, and morale. After a volume is met, Resting Energy Expenditure is calculated by the Weir formula and results are displayed in software attached to the system. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. Data on per capita food supplies are expressed in terms of quantity and by applying appropriate food composition factors for all primary and processed products also in terms of dietary energy value, protein and fat content. Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics ""Measuring RMR with Indirect Calorimetry (IC)."" By World War I, the American garrison ration had improved dramatically, including 137 grams of protein, 129 grams of fat, and 539 grams of carbohydrate every day, with a total of roughly 4,000 calories. The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. In the West Indies troops were issued with salt beef on five days with fresh meat being issued for two days a week. === Crimean War === Following initial disasters in the supply system, reforms were made and British troops were issued the following; 24 oz (680 g) of bread, 16 oz (453 g) meat, 2 oz (56 g) Rice, 2 oz (56 g) Sugar, 3 oz (85 g) Coffee, 1 Gill (0.118l) spirits and ½ oz (14 g) salt. === First World War === During the First World War British troops were issued the following daily ration; 1¼ pound (567 g) of meat, 1 pound (453 g) preserved meat, 1¼ (567 g) pound of bread, (or 1 pound (453 g) of biscuit and 4 oz (113 g) of bacon), 4 oz (113 g) Jam, 3 oz (85 g) sugar, ⅝ oz (17 g) tea, 8 oz (226 g) vegetables and 2 oz (56 g) of butter (weekly) ==== Horse Rations ==== As horses were a principal form of transport for the British Army, horses also had a scale of rations issued. A garrison ration (or mess ration for food rations of this type) is a type of military ration. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). The per capita supply of each such food item available for human consumption is then obtained by dividing the respective quantity by the related data on the population actually partaking in it. Further advances in nutrition led to the replacement of the garrison ration in 1933 with the New Army ration, which ultimately developed into the rations system described at United States military ration. *Canopy (dilution): The dilution technique is considered the gold standard technology for Resting Energy Expenditure measurement in clinical nutrition. From 1815 to 1854 the daily ration for a British soldier in the United Kingdom was 1 pound of bread (453 g) and ¾ of a pound of meat (340 g). Food Item Ration I Ration II Ration III Ration IV Rye bread 700g (1.54 lb) 700g (1.54 lb) 700g (1.54 lb) 600g (1.32 lb) Fresh meat with bones 136g (4.8 oz) 107g (3.7 oz) 90g (3.17 oz) 56g (2 oz) Soy bean flour 7g (0.24 oz) 7g (0.24 oz) 7g (0.24 oz) 7g (0.24 oz) Headless fish 30g (1 oz) 30g (1 oz) 30g (1 oz) 30g (1 oz) Fresh vegetables and fruits 250g (8.8 oz) 250g (8.8 oz) 250g (8.8 oz) 250g (8.8 oz) Potatoes 320g (11.29 oz) 320g (11.29 oz) 320g (11.29 oz) 320g (11.29 oz) Legumes 80g (2.8 oz) 80g (2.8 oz) 80g (2.8 oz) 80g (2.8 oz) Pudding powder 20g (0.70 oz) 20g (0.70 oz) 20g (0.70 oz) 20g (0.70 oz) Sweetened condensed skim milk 25g (0.88 oz) 25g (0.88 oz) 25g (0.88 oz) 25g (0.88 oz) Salt 15g (0.5 oz) 15g (0.5 oz) 15g (0.5 oz) 15g (0.5 oz) Other seasonings 3g (0.1 oz) 3g (0.1 oz) 3g (0.1 oz) 3g (0.1 oz) Spices 1g (0.03 oz) 1g (0.03 oz) 1g (0.03 oz) 1g (0.03 oz) Fats and bread spreads 60g (2.11 oz) 50g (1.76 oz) 40g (1.41 oz) 35g (1.23 oz) Coffee 9g (0.32 oz) 9g (0.32 oz) 9g (0.32 oz) 9g (0.32 oz) Sugar 40g (1.4 oz) 35g (1.23 oz) 30g (1.05 oz) 30g (1.05 oz) Supplementary allowances 2g (0.07 oz) 2g (0.07 oz) 2g (0.07 oz) 2g (0.07 oz) Total Maximum Ration in grams 1698 1654 1622 1483 Total Maximum Ration in Pounds 3.74 3.64 3.57 3.26 == United Kingdom == In 1689 the first Royal warrant was published concerning the messing provisions for troops. ",0.318,+11,49.0, -6.8,-1.0,C
-"At 303 . K, the vapor pressure of benzene is 120 . Torr and that of hexane is 189 Torr. Calculate the vapor pressure of a solution for which $x_{\text {benzene }}=0.28$ assuming ideal behavior.","Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The second solution is switching to another vapor pressure equation with more than three parameters. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds"", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The vapor phase is also assumed to behave like an ideal gas, so :v_v = \frac{k T}{P}, where k is the Boltzmann constant. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). (The small differences in the results are only caused by the used limited precision of the coefficients). ==Extension of the Antoine equations== To overcome the limits of the Antoine equation some simple extension by additional terms are used: : \begin{align} P &= \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)} \\\ P &= \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right). \end{align} The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The equation was presented in 1888 by the French engineer (1825–1897). ==Equation== The Antoine equation is :\log_{10} p = A-\frac{B}{C+T}. where p is the vapor pressure, is temperature (in °C or in K according to the value of C) and , and are component-specific constants. Let v_l and v_v be the volume occupied by one molecule in the liquid phase and vapor phase respectively. The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: :T = \frac{B}{A-\log_{10}\, p} - C ==Validity range== Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. Equilibrium vapor pressure depends on droplet size. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., ""Antoine Vapor Pressure Constants of Pure Compounds"" ** Yaws C. L., Yang H.-C., ""To Estimate Vapor Pressure Easily. This may be written in the following form, known as the Ostwald–Freundlich equation: \ln \frac{p}{p_{\rm sat}} = \frac{2 \gamma V_\text{m}}{rRT}, where p is the actual vapour pressure, p_{\rm sat} is the saturated vapour pressure when the surface is flat, \gamma is the liquid/vapor surface tension, V_\text{m} is the molar volume of the liquid, R is the universal gas constant, r is the radius of the droplet, and T is temperature. Image:VaporPressureFitAugust.png | Deviations of an August equation fit (2 parameters) Image:VaporPressureFitAntoine.png | Deviations of an Antoine equation fit (3 parameters) Image:VaporPressureFitDIPPR101.png | Deviations of a DIPPR 105 equation fit (4 parameters) ==Example parameters== Parameterisation for T in °C and P in mmHg A B C T min. (°C) T max. (°C) Water 8.07131 1730.63 233.426 1 100 Water 8.14019 1810.94 244.485 99 374 Ethanol 8.20417 1642.89 230.300 −57 80 Ethanol 7.68117 1332.04 199.200 77 243 ===Example calculation=== The normal boiling point of ethanol is TB = 78.32 °C. :\begin{align} P &= 10^{\left(8.20417 - \frac{1642.89}{78.32 + 230.300}\right)} = 760.0\ \text{mmHg} \\\ P &= 10^{\left(7.68117 - \frac{1332.04}{78.32 + 199.200}\right)} = 761.0\ \text{mmHg} \end{align} (760mmHg = 101.325kPa = 1.000atm = normal pressure) This example shows a severe problem caused by using two different sets of coefficients. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Thus, the change in the Gibbs free energy for one molecule is :\Delta g = - k T \int\limits_{P_{sat}}^{P} \frac {dP}{P}, where P_{sat} is the saturated vapor pressure of x over a flat surface and P is the actual vapor pressure over the liquid. ", 0.4,0.69,170.0,24,6,C
-"Determine the molar standard Gibbs energy for ${ }^{35} \mathrm{Cl}^{35} \mathrm{Cl}$ where $\widetilde{\nu}=560 . \mathrm{cm}^{-1}, B=0.244 \mathrm{~cm}^{-1}$, and the ground electronic state is nondegenerate.","The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. These tables list values of molar ionization energies, measured in kJ⋅mol−1. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m��K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. Unlike standard enthalpies of formation, the value of is absolute. LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. The first molar ionization energy applies to the neutral atoms. That is, an element in its standard state has a definite, nonzero value of at room temperature. Under identical conditions, it is greater for a heavier gas. ==See also== *Entropy *Heat *Gibbs free energy *Helmholtz free energy *Standard state *Third law of thermodynamics ==References== ==External links== *Standard Thermodynamic Properties of Chemical Substances Table Category:Chemical properties Category:Thermodynamic entropy The notation (P=0) denotes low pressure limiting values. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. All data from rutherfordium onwards is predicted. == All Ionization Energies == Number Symbol Name 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 1 H hydrogen 1312.0 2 He helium 2372.3 5250.5 3 Li lithium 520.2 7298.1 11,815.0 4 Be beryllium 899.5 1757.1 14,848.7 21,006.6 5 B boron 800.6 2427.1 3659.7 25,025.8 32,826.7 6 C carbon 1086.5 2352.6 4620.5 6222.7 37,831 47,277.0 7 N nitrogen 1402.3 2856 4578.1 7475.0 9444.9 53,266.6 64,360 8 O oxygen 1313.9 3388.3 5300.5 7469.2 10,989.5 13,326.5 71,330 84,078.0 9 F fluorine 1681.0 3374.2 6050.4 8407.7 11,022.7 15,164.1 17,868 92,038.1 106,434.3 10 Ne neon 2080.7 3952.3 6122 9371 12,177 15,238.90 19,999.0 23,069.5 115,379.5 131,432 11 Na sodium 495.8 4562 6910.3 9543 13,354 16,613 20,117 25,496 28,932 141,362 12 Mg magnesium 737.7 1450.7 7732.7 10,542.5 13,630 18,020 21,711 25,661 31,653 35,458 13 Al aluminium 577.5 1816.7 2744.8 11,577 14,842 18,379 23,326 27,465 31,853 38,473 14 Si silicon 786.5 1577.1 3231.6 4355.5 16,091 19,805 23,780 29,287 33,878 38,726 15 P phosphorus 1011.8 1907 2914.1 4963.6 6273.9 21,267 25,431 29,872 35,905 40,950 16 S sulfur 999.6 2252 3357 4556 7004.3 8495.8 27,107 31,719 36,621 43,177 17 Cl chlorine 1251.2 2298 3822 5158.6 6542 9362 11,018 33,604 38,600 43,961 18 Ar argon 1520.6 2665.8 3931 5771 7238 8781 11,995 13,842 40,760 46,186 19 K potassium 418.8 3052 4420 5877 7975 9590 11,343 14,944 16,963.7 48,610 20 Ca calcium 589.8 1145.4 4912.4 6491 8153 10,496 12,270 14,206 18,191 20,385 21 Sc scandium 633.1 1235.0 2388.6 7090.6 8843 10,679 13,310 15,250 17,370 21,726 22 Ti titanium 658.8 1309.8 2652.5 4174.6 9581 11,533 13,590 16,440 18,530 20,833 23 V vanadium 650.9 1414 2830 4507 6298.7 12,363 14,530 16,730 19,860 22,240 24 Cr chromium 652.9 1590.6 2987 4743 6702 8744.9 15,455 17,820 20,190 23,580 25 Mn manganese 717.3 1509.0 3248 4940 6990 9220 11,500 18,770 21,400 23,960 26 Fe iron 762.5 1561.9 2957 5290 7240 9560 12,060 14,580 22,540 25,290 27 Co cobalt 760.4 1648 3232 4950 7670 9840 12,440 15,230 17,959 26,570 28 Ni nickel 737.1 1753.0 3395 5300 7339 10,400 12,800 15,600 18,600 21,670 29 Cu copper 745.5 1957.9 3555 5536 7700 9900 13,400 16,000 19,200 22,400 30 Zn zinc 906.4 1733.3 3833 5731 7970 10,400 12,900 16,800 19,600 23,000 31 Ga gallium 578.8 1979.3 2963 6180 32 Ge germanium 762 1537.5 3302.1 4411 9020 33 As arsenic 947.0 1798 2735 4837 6043 12,310 34 Se selenium 941.0 2045 2973.7 4144 6590 7880 14,990 35 Br bromine 1139.9 2103 3470 4560 5760 8550 9940 18,600 36 Kr krypton 1350.8 2350.4 3565 5070 6240 7570 10,710 12,138 22,274 25,880 37 Rb rubidium 403.0 2633 3860 5080 6850 8140 9570 13,120 14,500 26,740 38 Sr strontium 549.5 1064.2 4138 5500 6910 8760 10,230 11,800 15,600 17,100 39 Y yttrium 600 1180 1980 5847 7430 8970 11,190 12,450 14,110 18,400 40 Zr zirconium 640.1 1270 2218 3313 7752 9500 41 Nb niobium 652.1 1380 2416 3700 4877 9847 12,100 42 Mo molybdenum 684.3 1560 2618 4480 5257 6640.8 12,125 13,860 15,835 17,980 43 Tc technetium 702 1470 2850 44 Ru ruthenium 710.2 1620 2747 45 Rh rhodium 719.7 1740 2997 46 Pd palladium 804.4 1870 3177 47 Ag silver 731.0 2070 3361 48 Cd cadmium 867.8 1631.4 3616 49 In indium 558.3 1820.7 2704 5210 50 Sn tin 708.6 1411.8 2943.0 3930.3 7456 51 Sb antimony 834 1594.9 2440 4260 5400 10,400 52 Te tellurium 869.3 1790 2698 3610 5668 6820 13,200 53 I iodine 1008.4 1845.9 3180 54 Xe xenon 1170.4 2046.4 3099.4 55 Cs caesium 375.7 2234.3 3400 56 Ba barium 502.9 965.2 3600 57 La lanthanum 538.1 1067 1850.3 4819 5940 58 Ce cerium 534.4 1050 1949 3547 6325 7490 59 Pr praseodymium 527 1020 2086 3761 5551 60 Nd neodymium 533.1 1040 2130 3900 61 Pm promethium 540 1050 2150 3970 62 Sm samarium 544.5 1070 2260 3990 63 Eu europium 547.1 1085 2404 4120 64 Gd gadolinium 593.4 1170 1990 4250 65 Tb terbium 565.8 1110 2114 3839 66 Dy dysprosium 573.0 1130 2200 3990 67 Ho holmium 581.0 1140 2204 4100 68 Er erbium 589.3 1150 2194 4120 69 Tm thulium 596.7 1160 2285 4120 70 Yb ytterbium 603.4 1174.8 2417 4203 71 Lu lutetium 523.5 1340 2022.3 4370 6445 72 Hf hafnium 658.5 1440 2250 3216 73 Ta tantalum 761 1500 74 W tungsten 770 1700 75 Re rhenium 760 1260 2510 3640 76 Os osmium 840 1600 77 Ir iridium 880 1600 78 Pt platinum 870 1791 79 Au gold 890.1 1980 80 Hg mercury 1007.1 1810 3300 81 Tl thallium 589.4 1971 2878 82 Pb lead 715.6 1450.5 3081.5 4083 6640 83 Bi bismuth 703 1610 2466 4370 5400 8520 84 Po polonium 812.1 85 At astatine 899.003 86 Rn radon 1037 87 Fr francium 393 88 Ra radium 509.3 979.0 89 Ac actinium 499 1170 1900 4700 90 Th thorium 587 1110 1978 2780 91 Pa protactinium 568 1128 1814 2991 92 U uranium 597.6 1420 1900 3145 93 Np neptunium 604.5 1128 1997 3242 94 Pu plutonium 584.7 1128 2084 3338 95 Am americium 578 1158 2132 3493 96 Cm curium 581 1196 2026 3550 97 Bk berkelium 601 1186 2152 3434 98 Cf californium 608 1206 2267 3599 99 Es einsteinium 619 1216 2334 3734 100 Fm fermium 629 1225 2363 3792 101 Md mendelevium 636 1235 2470 3840 102 No nobelium 639 1254 2643 3956 103 Lr lawrencium 479 1428 2228 4910 104 Rf rutherfordium 580 1390 2300 3080 105 Db dubnium 665 1547 2378 3299 4305 106 Sg seaborgium 757 1733 2484 3416 4562 5716 107 Bh bohrium 740 1690 2570 3600 4730 5990 7230 108 Hs hassium 730 1760 2830 3640 4940 6180 7540 8860 109 Mt meitnerium 800 1820 2900 3900 4900 110 Ds darmstadtium 960 1890 3030 4000 5100 111 Rg roentgenium 1020 2070 3080 4100 5300 112 Cn copernicium 1155 2170 3160 4200 5500 113 Nh nihonium 707.2 2309 3226 4382 5638 114 Fl flerovium 832.2 1600 3370 4400 5850 115 Mc moscovium 538.3 1760 2650 4680 5720 116 Lv livermorium 663.9 1330 2850 3810 6080 117 Ts tennessine 736.9 1435.4 2161.9 4012.9 5076.4 118 Og oganesson 860.1 1560 119 Uue ununennium 463.1 1700 120 Ubn unbinilium 563.3 121 Ubu unbiunium 429.4 1110 1710 4270 122 Ubb unbibium 545 1090 1848 2520 == 11th-20th ionisation energies == number symbol name 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th 11 Na sodium 159,076 12 Mg magnesium 169,988 189,368 13 Al aluminium 42,647 201,266 222,316 14 Si silicon 45,962 50,502 235,196 257,923 15 P phosphorus 46,261 54,110 59,024 271,791 296,195 16 S sulfur 48,710 54,460 62,930 68,216 311,048 337,138 17 Cl chlorine 51,068 57,119 63,363 72,341 78,095 352,994 380,760 18 Ar argon 52,002 59,653 66,199 72,918 82,473 88,576 397,605 427,066 19 K potassium 54,490 60,730 68,950 75,900 83,080 93,400 99,710 444,880 476,063 20 Ca calcium 57,110 63,410 70,110 78,890 86,310 94,000 104,900 111,711 494,850 527,762 21 Sc scandium 24,102 66,320 73,010 80,160 89,490 97,400 105,600 117,000 124,270 547,530 22 Ti titanium 25,575 28,125 76,015 83,280 90,880 100,700 109,100 117,800 129,900 137,530 23 V vanadium 24,670 29,730 32,446 86,450 94,170 102,300 112,700 121,600 130,700 143,400 24 Cr chromium 26,130 28,750 34,230 37,066 97,510 105,800 114,300 125,300 134,700 144,300 25 Mn manganese 27,590 30,330 33,150 38,880 41,987 109,480 118,100 127,100 138,600 148,500 26 Fe iron 28,000 31,920 34,830 37,840 44,100 47,206 122,200 131,000 140,500 152,600 27 Co cobalt 29,400 32,400 36,600 39,700 42,800 49,396 52,737 134,810 145,170 154,700 28 Ni nickel 30,970 34,000 37,100 41,500 44,800 48,100 55,101 58,570 148,700 159,000 29 Cu copper 25,600 35,600 38,700 42,000 46,700 50,200 53,700 61,100 64,702 163,700 30 Zn zinc 26,400 29,990 40,490 43,800 47,300 52,300 55,900 59,700 67,300 71,200 36 Kr krypton 29,700 33,800 37,700 43,100 47,500 52,200 57,100 61,800 75,800 80,400 38 Sr strontium 31,270 39 Y yttrium 19,900 36,090 42 Mo molybdenum 20,190 22,219 26,930 29,196 52,490 55,000 61,400 67,700 74,000 80,400 == 21st-30th ionisation energies == number symbol name 21st 22nd 23rd 24th 25th 26th 27th 28th 29th 30th 21 Sc scandium 582,163 22 Ti titanium 602,930 639,294 23 V vanadium 151,440 661,050 699,144 24 Cr chromium 157,700 166,090 721,870 761,733 25 Mn manganese 158,600 172,500 181,380 785,450 827,067 26 Fe iron 163,000 173,600 188,100 195,200 851,800 895,161 27 Co cobalt 167,400 178,100 189,300 204,500 214,100 920,870 966,023 28 Ni nickel 169,400 182,700 194,000 205,600 221,400 231,490 992,718 1,039,668 29 Cu copper 174,100 184,900 198,800 210,500 222,700 239,100 249,660 1,067,358 1,116,105 30 Zn zinc 179,100 36 Kr krypton 85,300 90,400 96,300 101,400 111,100 116,290 282,500 296,200 311,400 326,200 42 Mo molybdenum 87,000 93,400 98,420 104,400 121,900 127,700 133,800 139,800 148,100 154,500 == References == * Ionization energies of the elements (data page) * (for predictions) * * (for predictions) Category:Properties of chemical elements ",-57.2,226,12.0,7.27,-0.55,A
-"For the reaction $\mathrm{C}($ graphite $)+\mathrm{H}_2 \mathrm{O}(g) \rightleftharpoons$ $\mathrm{CO}(g)+\mathrm{H}_2(g), \Delta H_R^{\circ}=131.28 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298.15 \mathrm{~K}$. Use the values of $C_{P, m}^{\circ}$ at $298.15 \mathrm{~K}$ in the data tables to calculate $\Delta H_R^{\circ}$ at $125.0^{\circ} \mathrm{C}$.","== Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup Graphane is a two-dimensional polymer of carbon and hydrogen with the formula unit (CH)n where n is large. P-doped graphane is proposed to be a high-temperature BCS theory superconductor with a Tc above 90 K. ==Variants== Partial hydrogenation leads to hydrogenated graphene rather than (fully hydrogenated) graphane. The structure was found, using a cluster expansion method, to be the most stable of all the possible hydrogenation ratios of graphene. * Values from CRC refer to ""100 kPa (1 bar or 0.987 standard atmospheres)"". :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Hydrogenation decreases the dependence of the lattice constant on temperature, which indicates a possible application in precision instruments. ==References== ==External links== * Sep 14, 2010 Hydrogen vacancies induce stable ferromagnetism in graphane * May 25, 2010 Graphane yields new potential * May 02 2010 Doped Graphane Should Superconduct at 90K Category:Two-dimensional nanomaterials Category:Polymers Category:Superconductors Category:Hydrocarbons Density functional theory calculations suggested that hydrogenated and fluorinated forms of other group IV (Si, Ge and Sn) nanosheets present properties similar to graphane. ==Potential applications== p-Doped graphane is postulated to be a high-temperature BCS theory superconductor with a Tc above 90 K. Graphane has been proposed for hydrogen storage. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Partial hydrogenation results in hydrogenated graphene, which was reported by Elias et al in 2009 by a TEM study to be ""direct evidence for a new graphene-based derivative"". * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Above 750 K Tc values may be in error by 10 K or more. Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages In the last case mechanical exfoliation of hydrogenated top layers can be used. ==Structure== The first theoretical description of graphane was reported in 2003. CR2: Values refer to 300 K and a pressure of ""100 kPa (1 bar)"", or to the saturation vapor pressure if that is less than 100 kPa. ",30,8.8,1855.0,0.000216,132.9,E
-Calculate the mean ionic activity of a $0.0350 \mathrm{~m} \mathrm{Na}_3 \mathrm{PO}_4$ solution for which the mean activity coefficient is 0.685 .,"It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient. Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. : Aqua ions in seawater (salinity = 35) Ion Concentration (mol kg−1) 0.469 0.0102 0.0528 0.0103 Many other aqua ions are present in seawater in concentrations ranging from ppm to ppt. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. Journal of Solution Chemistry, 26, 791–815. ==External links== * For easy calculation of activity coefficients in (non- micellar) solutions, check out the IUPAC open project Aq-solutions (freeware). D&H; say that, due to the ""mutual electrostatic forces between the ions"", it is necessary to modify the Guldberg–Waage equation by replacing K with \gamma K, where \gamma is an overall activity coefficient, not a ""special"" activity coefficient (a separate activity coefficient associated with each species)—which is what is used in modern chemistry . Typical values are 3Å for ions such as H+, Cl−, CN−, and HCOO−. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. The electrostatic energy of the ion at site then is the product of its charge with the potential acting at its site :E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. In these solutions the activity coefficient may actually increase with ionic strength. center|The Debye–Hückel plot with different values for ion charge Z and ion diameter a The Debye–Hückel equation cannot be used in the solutions of surfactants where the presence of micelles influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%). ==See also== * Strong electrolyte * Weak electrolyte * Ionic atmosphere * Debye–Hückel theory * Poisson–Boltzmann equation ==Notes== ==References== * Alt URL * * * * Malatesta, F., and Zamboni, R. (1997). Ionic potential is also a measure of the polarising power of a cation. Ions fall into four groups. This factor takes into account the interaction energy of ions in solution. == Debye–Hückel limiting law == In order to calculate the activity a_C of an ion C in a solution, one must know the concentration and the activity coefficient: a_C = \gamma \frac\mathrm{[C]}\mathrm{[C^\ominus]}, where * \gamma is the activity coefficient of C, * \mathrm{[C^\ominus]} is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used, * \mathrm{[C]} is a measure of the concentration of C. Dividing \mathrm{[C]} with \mathrm{[C^\ominus]} gives a dimensionless quantity. thumb|250px|Distribution of ions in a solution The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. A metal ion in aqueous solution or aqua ion is a cation, dissolved in water, of chemical formula [M(H2O)n]z+. Zn2+ −0.751 Ga3+ −0.53 Ge2+ +0.1 Rb −2.98 Sr2+ −2.899 Y3+ −2.37 ... Hg2+ −0.854 Tl3+ +0.73 Pb2+ −0.126 Bi3+ +0.16 Po4+ +0.76 Fr −2.9 Ra2+ −2.8 Lr3+ −1.96 La3+ −2.52 Ce3+ −2.32 Pr3+ −2.34 Nd3+ −2.32 Pm3+ −2.30 Sm3+ −2.28 Eu3+ −1.98 Gd3+ −2.27 Tb3+ −2.27 Dy3+ −2.32 Ho3+ −2.37 Er3+ −2.33 Tm3+ −2.30 Yb3+ −2.23 Ac3+ −2.18 Th4+ −1.83 Pa4+ −1.46 U4+ −1.51 Np4+ −1.33 Pu4+ −1.80 Am3+ −2.06 Cm3+ −2.07 Bk3+ −2.03 Cf3+ −2.01 Es3+ −1.99 Fm3+ −1.97 Md3+ −1.65 No3+ −1.20 : Standard electrode potentials /V for 1st. row transition metal ions Couple Ti V Cr Mn Fe Co Ni Cu M2+ / M −1.63 −1.18 −0.91 −1.18 −0.473 −0.28 −0.228 +0.345 M3+ / M −1.37 −0.87 −0.74 −0.28 −0.06 +0.41 : Miscellaneous standard electrode potentials /V Ag+ / Ag Pd2+ / Pd Pt2+ / Pt Zr4+ / Zr Hf4+ / Hf Au3+ / Au Ce4+ / Ce +0.799 +0.915 +1.18 −1.53 −1.70 +1.50 −1.32 As the standard electrode potential is more negative the aqua ion is more difficult to reduce. For ions in solution Shannon's ""effective ionic radius"" is the measure most often used.. Ions are 6-coordinate unless indicated differently in parentheses (e.g. 146 (4) for 4-coordinate N3−). Ions are 6-coordinate unless indicated differently in parentheses (e.g. 146 (4) for 4-coordinate N3−). Li+ -118.8 Na+ -87.4 Mg2+ -267.8 Al3+ -464.4 K+ -51.9 Ca2+ -209.2 ... ",0.0547,399,34.0,0.2115,7.136,A
-" Consider the transition between two forms of solid tin, $\mathrm{Sn}(s$, gray $) \rightarrow \mathrm{Sn}(s$, white $)$. The two phases are in equilibrium at 1 bar and $18^{\circ} \mathrm{C}$. The densities for gray and white tin are 5750 and $7280 \mathrm{~kg} \mathrm{~m}^{-3}$, respectively, and the molar entropies for gray and white tin are 44.14 and $51.18 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, respectively. Calculate the temperature at which the two phases are in equilibrium at 350. bar.","J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Transition temperature is the temperature at which a material changes from one crystal state (allotrope) to another. Tin(II) chloride, also known as stannous chloride, is a white crystalline solid with the formula . K (? °C), ? K (? °C), ? * Handbook of Chemistry and Physics, 71st edition, CRC Press, Ann Arbor, Michigan, 1990. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? It was first discovered in 1822. == Preparation == To obtain tin(II) acetate, tin(II) oxide is dissolved in glacial acetic acid and refluxed to obtain yellow Sn(CH3COO)2·2CH3COOH when cooled. More formally, it is the temperature at which two crystalline forms of a substance can co-exist in equilibrium. White tin may also refer specifically to β-tin, the metallic allotrope of the pure element, as opposed to the nonmetallic allotrope α-tin (also known as gray tin), which occurs at temperatures below , a transformation known as tin pest). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? At 95.6 °C the two forms can co-exist. Tin(II) acetate is the acetate salt of tin(II), with the chemical formula of Sn(CH3COO)2. Another example is tin, which transitions from a cubic crystal below 13.2 °C to a tetragonal crystal above that temperature. White tin is refined metallic tin. Tin(II) chloride should not be confused with the other chloride of tin; tin(IV) chloride or stannic chloride (SnCl4). ==Chemical structure== SnCl2 has a lone pair of electrons, such that the molecule in the gas phase is bent. The main part of the molecule stacks into double layers in the crystal lattice, with the ""second"" water sandwiched between the layers. thumb|460px|left|Structures of tin(II) chloride and related compounds ==Chemical properties== Tin(II) chloride can dissolve in less than its own mass of water without apparent decomposition, but as the solution is diluted, hydrolysis occurs to form an insoluble basic salt: :SnCl2 (aq) + H2O (l) Sn(OH)Cl (s) + HCl (aq) Therefore, if clear solutions of tin(II) chloride are to be used, it must be dissolved in hydrochloric acid (typically of the same or greater molarity as the stannous chloride) to maintain the equilibrium towards the left-hand side (using Le Chatelier's principle). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? Solutions of tin(II) chloride can also serve simply as a source of Sn2+ ions, which can form other tin(II) compounds via precipitation reactions. Complex tin (II) acetates. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? In the case of ferroelectric or ferromagnetic crystals, a transition temperature may be known as the Curie temperature. == See also == * Crystal system Category:Crystallography Category:Threshold temperatures ",479,2.24,-242.6,7.82,-3.5,E
-"The densities of pure water and ethanol are 997 and $789 \mathrm{~kg} \mathrm{~m}^{-3}$, respectively. For $x_{\text {ethanol }}=0.35$, the partial molar volumes of ethanol and water are 55.2 and $17.8 \times 10^{-3} \mathrm{~L} \mathrm{~mol}^{-1}$, respectively. Calculate the change in volume relative to the pure components when $2.50 \mathrm{~L}$ of a solution with $x_{\text {ethanol }}=0.35$ is prepared.","It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. The volume of alcohol in the solution can then be estimated. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. Mixing two solutions of alcohol of different strengths usually causes a change in volume. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Therefore, one can use the following equation to convert between ABV and ABW: \text{ABV} = \text{ABW} \times \frac{\text{density of beverage}}{\text{density of alcohol}} At relatively low ABV, the alcohol percentage by weight is about 4/5 of the ABV (e.g., 3.2% ABW is about 4% ABV). The following formulas can be used to calculate the volumes of solute (Vsolute) and solvent (Vsolvent) to be used: *Vsolute = Vtotal / F *Vsolvent = Vtotal \- Vsolute , where: *Vtotal is the desired total volume *F is the desired dilution factor number (the number in the position of F if expressed as ""1:F dilution factor"" or ""xF dilution"") However, some solutions and mixtures take up slightly less volume than their components. The International Organization of Legal Metrology has tables of density of water–ethanol mixtures at different concentrations and temperatures. At 0% and 100% ABV is equal to ABW, but at values in between ABV is always higher, up to ~13% higher around 60% ABV. ==See also== *Apparent molar property *Excess molar quantity *Standard drink *Unit of alcohol *Volume fraction == Notes == == References == ==Bibliography== * * * * ==External links== * * Category:Alcohol measurement The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Mixing pure water with a solution less than 24% by mass causes a slight increase in total volume, whereas the mixing of two solutions above 24% causes a decrease in volume. Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. The amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object. However, because of the miscibility of alcohol and water, the conversion factor is not constant but rather depends upon the concentration of alcohol. It is defined as the number of millilitres (mL) of pure ethanol present in of solution at . The density of sugar in water is greater than the density of alcohol in water. The volume was rounded up to 750 mL and then was used as the base size for French wine containers, with all subdivisions and multiples figured from it. In some countries, e.g. France, alcohol by volume is often referred to as degrees Gay-Lussac (after the French chemist Joseph Louis Gay-Lussac), although there is a slight difference since the Gay- Lussac convention uses the International Standard Atmosphere value for temperature, . ==Volume change== thumb|upright=1.48|Change in volume with increasing ABV. In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. Originally there were different standard gallons depending on the type of alcohol. ",0.8561,-0.10,0.88,-36.5,8.7,B
-"For $\mathrm{N}_2$ at $298 \mathrm{~K}$, what fraction of molecules has a speed between 200. and $300 . \mathrm{m} / \mathrm{s}$ ?","To help compare different orders of magnitude, the following list describes various speed levels between approximately 2.2 m/s and 3.0 m/s (the speed of light). These atoms effuse out of a hole in the oven with average speeds on the order of hundreds of m/s and large velocity distributions (due to their high temperature). The inch per second is a unit of speed or velocity. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:chem.libretexts.org: Collision Frequency : Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, which has units of [volume][time]−1. Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. Current fastest macroscopic man-made system. 450,000 1,600,000 1,000,000 0.0015 Typical speed of a particle of the solar wind, relative to the Sun. 552,000 1,990,000 1,230,000 0.0018 Speed of the Milky Way, relative to the cosmic microwave background. 617,700 2,224,000 1,382,000 0.0021 Escape velocity from the surface of the Sun. 106 1,000,000 3,600,000 2,200,000 0.0030 Typical speed of a Moreton wave across the surface of the Sun. 1,610,000 5,800,000 3,600,000 0.0054 Speed of hypervelocity star PSR B2224+65, which currently seems to be leaving the Milky Way. 5,000,000 18,000,000 11,000,000 0.017 Estimated minimum speed of star S2 at its closest approach to Sagittarius A*. 107 14,000,000 50,000,000 31,000,000 0.047 Typical speed of a fast neutron. 30,000,000 100,000,000 70,000,000 0.1 Typical speed of an electron in a cathode ray tube. 108 100,000,000 360,000,000 220,000,000 0.3 The escape velocity of a neutron star. 100,000,000 360,000,000 220,000,000 0.3 Typical speed of the return stroke of lightning (cf. stepped leader above). 124,000,000 447,000,000 277,000,000 0.4 Speed of light in a diamond (Refractive index 2.417). 200,000,000 720,000,000 440,000,000 0.7 Speed of a signal in an optical fiber. 299,792,456 1,079,252,840 670,615,282 1 − 9 Speed of the 7 TeV protons in the Large Hadron Collider at full power. 299,792,457.996 1,079,252,848.786 670,616,629.38 1 − 1 Maximal speed of an electron in LEP (104.5 GeV). 299,792,458 − 1.5×10−15 1,079,252,848.8 − 5.4×10−15 670,616,629.4 1 − 4.9×10−24 Speed of the Oh-My-God particle ultra-high-energy cosmic ray. 299,792,458 1,079,252,848.8 670,616,629.4 1 Speed of light or other electromagnetic radiation in a vacuum or massless particles. >299,792,458 >1,079,252,848.8 >670,616,629.4 >1 Expansion rate of the universe between objects farther apart than the Hubble radius ==See also== *Typical projectile speeds - also showing the corresponding kinetic energy per unit mass *Neutron temperature ==References== Category:Units of velocity Category:Physical quantities Speed The newton-second (also newton second; symbol: N⋅s or N s) is the unit of impulse in the International System of Units (SI). In the constant deceleration approach we get: ::v\left(z\right)=\sqrt{v_{i}^{2}-2az} ::B\left(z\right)=\frac{\hbar k}{\mu'}v+\frac{\hbar \delta}{\mu'}=\frac{\hbar kv_{i}}{\mu'}\sqrt{1-\frac{2a}{v_{i}^{2}}z}+\frac{\hbar \delta}{\mu'} where v_{i} is the maximum velocity class that will be slowed; all the atoms in the velocity distribution that have velocities v will be slowed, and those with velocities v>v_{i} will not be slowed at all. Mass (kg) Speed (m/s) Momentum (N⋅s) Explanation 0.42 2.4 1 A football (FIFA specified weight for outdoor size 5) kicked to a speed of . 0.42 38 16 The momentum of the famous football kick of the Brazilian player Roberto Carlos in the match against France in 1997. Kinetic diameter is a measure applied to atoms and molecules that expresses the likelihood that a molecule in a gas will collide with another molecule. Here, the Maxwell–Boltzmann distribution of energies must be considered, which leads to the modified expression,Freude, p. 4 :d^2 = {1 \over \sqrt 2 \pi l n} ==List of diameters== The following table lists the kinetic diameters of some common molecules; Molecule Molecule Molecular mass Kinetic diameter (pm) ref Name Formula Molecular mass Kinetic diameter (pm) ref Hydrogen H2 2 289 Helium He 4 260 Matteucci et al., p. 6 Methane CH4 16 380 Ammonia NH3 17 260 Breck Water H2O 18 265 Neon Ne 20 275 Acetylene C2H2 26 330 Nitrogen N2 28 364 Carbon monoxide CO 28 376 Ethylene C2H4 28 390 Nitric oxide NO 30 317 Oxygen O2 32 346 Hydrogen sulfide H2S 34 360 Hydrogen chloride HCl 36 320 Argon Ar 40 340 Propylene C3H6 42 450 Carbon dioxide CO2 44 330 Nitrous oxide N2O 44 330 Propane C3H8 44 430 Sulfur dioxide SO2 64 360 Chlorine Cl2 70 320 Benzene C6H6 78 585 Li & Talu, p. 373 Hydrogen bromide HBr 81 350 Krypton Kr 84 360 Xenon Xe 131 396 Sulfur hexafluoride SF6 146 550 Carbon tetrachloride CCl4 154 590 Bromine Br2 160 350 ==Dissimilar particles== Collisions between two dissimilar particles occur when a beam of fast particles is fired into a gas consisting of another type of particle, or two dissimilar molecules randomly collide in a gas mixture. For particles of different size, more elaborate expressions can be derived for estimating u. ==References== Category:Chemical kinetics thumb|A Zeeman slower before its incorporation into a larger cold atom experiment. The average acceleration (due to many photon absorption events over time) of an atom with mass, M, a cycling transition with frequency, \omega=ck+\delta, and linewidth, \gamma, that is in the presence of a laser beam that has wavenumber, k, and intensity I=s_{0}I_{s} (where I_s=\hbar c \gamma k^{3}/12\pi is the saturation intensity of the laser) is :: \vec{a}=\frac{\hbar\vec{k}\gamma}{2M}\frac{s_{0}}{1+s_{0}+\left(2\delta'/\gamma\right)^2} In the rest frame of the atoms with velocity, v, in the atomic beam, the frequency of the laser beam is shifted by k_{L}v. The SSERVI - Impact Dust Accelerator Facility at the University of Colorado, 47th Lunar and Planetary Science Conference (2016), accessed May 30, 2017 140,000 540,000 313,170 0.00047 Approaching velocity of Messier 98 to our galaxy. 192,000 690,000 430,000 0.00064 Predicted top speed of the Parker Solar Probe at its closest perihelion in 2024. 200,000 700,000 450,000 0.00070 Orbital speed of the Solar System in the Milky Way galaxy. 308,571 1,080,000 694,288 0.001 Approaching velocity of Andromeda Galaxy to our galaxy. 440,000 1,600,000 980,000 0.0015 Typical speed of the stepped leader of lightning (cf. return stroke below). 445,000 1,600,000 995,000 0.0015 Max velocity of the remaining shell (mass about 0.1 mg) of an inertial confinement fusion capsule driven by the National Ignition Facility for the 'Bigfoot' capsule campaign. Thus it aims at a final velocity of about 10 m/s (depending on the atom used), starting with a beam of atoms with a velocity of a few hundred meters per second. For a fast moving particle (that is, one moving much faster than the particles it is moving through) the kinetic diameter is given by,Ismail et al., p. 14 :d^2 = {1 \over \pi l n} :where, :d is the kinetic diameter, :r is the kinetic radius, r = d/2, :l is the mean free path, and :n is the number density of particles However, a more usual situation is that the colliding particle being considered is indistinguishable from the population of particles in general. It is dimensionally equivalent to the momentum unit kilogram-metre per second (kg⋅m/s). One newton-second corresponds to a one-newton force applied for one second. :\vec F \cdot t = \Delta m \vec v It can be used to identify the resultant velocity of a mass if a force accelerates the mass for a specific time interval. ==Definition== Momentum is given by the formula: :\mathbf{p} = m \mathbf{v}, * \mathbf{p} is the momentum in newton-seconds (N⋅s) or ""kilogram-metres per second"" (kg⋅m/s) * m is the mass in kilograms (kg) * \mathbf{v} is the velocity in metres per second (m/s) ==Examples== This table gives the magnitudes of some momenta for various masses and speeds. The kinetic diameter is not the same as atomic diameter defined in terms of the size of the atom's electron shell, which is generally a lot smaller, depending on the exact definition used. Abbreviations include in/s, in/sec, ips, and less frequently in s−1. ==Conversions== 1 inch per second is equivalent to: : = 0.0254 metres per second (exactly) : = or 0.083 feet per second (exactly) : = or 0.05681 miles per hour (exactly) : = 0.09144 km·h−1 (exactly) 1 metre per second ≈ 39.370079 inches per second (approximately) 1 foot per second = 12 inches per second (exactly) 1 mile per hour = 17.6 inches per second (exactly) 1 kilometre per hour ≈ 10.936133 inches per second (approximately) ==Uses== In magnetic tape sound recording, magnetic tape speed is often quoted in inches per second (abbreviated ""ips""). ",+4.1,4500,41.4,0,0.132,E
+In this expression $m$ is the per-particle mass of the gas, $g$ is the acceleration due to gravity, $k$ is a constant equal to $1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}$, and $T$ is temperature. Determine $\langle h\rangle$ for methane $\left(\mathrm{CH}_4\right)$ using this distribution function.","Atmospheric methane is the methane present in Earth's atmosphere. When methane reaches the surface and the atmosphere, it is known as atmospheric methane. Methane's heat of combustion is 55.5 MJ/kg.Energy Content of some Combustibles (in MJ/kg) . Methane has a boiling point of −161.5 °C at a pressure of one atmosphere. Using the values T=273 K and M=29 g/mol as characteristic of the Earth's atmosphere, H = RT/Mg = (8.315*273)/(29*9.8) = 7.99, or about 8 km, which coincidentally is approximate height of Mt. Everest. Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). [A] Net production of O3 CH4 \+ ·OH → CH3· + H2O CH3· + O2 \+ M → CH3O2· + M CH3O2· + NO → NO2 \+ CH3O· CH3O· + O2 → HO2· + HCHO HO2· + NO → NO2 \+ ·OH (2x) NO2 \+ hv → O(3P) + NO (2x) O(3P) + O2 \+ M → O3 \+ M [NET: CH4 \+ 4O2 → HCHO + 2O3 \+ H2O] [B] No net change of O3 CH4 \+ ·OH → CH3· + H2O CH3�� + O2 \+ M → CH3O2· + M CH3O2· + HO2· + M → CH3O2H + O2 \+ M CH3O2H + hv → CH3O· + ·OH CH3O· + O2 → HO2· + HCHO [NET: CH4 \+ O2 → HCHO + H2O] ==See also== * Climate change * Global warming * Permafrost * Methane * Methane emissions ==Notes== ==References== ==External links== * * * * * * * * * * * Category:Methane Category:Atmosphere Category:Greenhouse gases A Chapman function describes the integration of atmospheric absorption along a slant path on a spherical earth, relative to the vertical case. The reaction of methane with hydroxyl in the troposphere or stratosphere creates the methyl radical ·CH3 and water vapor. Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. Methane is an important greenhouse gas with a global warming potential of 40 compared to (potential of 1) over a 100-year period, and above 80 over a 20-year period. The Chapman function is named after Sydney Chapman, who introduced the function in 1931. == Definition == In an isothermal model of the atmosphere, the density \varrho(h) varies exponentially with altitude h according to the Barometric formula: :\varrho(h) = \varrho_0 \exp\left(- \frac h H \right), where \varrho_0 denotes the density at sea level (h=0) and H the so-called scale height. The IPCC reports that the global warming potential (GWP) for methane is about 84 in terms of its impact over a 20-year timeframe See Table 8.7.—that means it traps 84 times more heat per mass unit than carbon dioxide (CO2) and 105 times the effect when accounting for aerosol interactions. The globally averaged concentration of methane in Earth's atmosphere increased by about 150% from 722 ± 25 ppb in 1750 to 1803.1 ± 0.6 ppb in 2011. Etminan et al. published their new calculations for methane's radiative forcing (RF) in a 2016 Geophysical Research Letters journal article which incorporated the shortwave bands of CH4 in measuring forcing, not used in previous, simpler IPCC methods. Image:Isothermal-barotropic atmosphere model.png ===The U.S. Standard Atmosphere=== The U.S. Standard Atmosphere model starts with many of the same assumptions as the isothermal-barotropic model, including ideal gas behavior, and constant molecular weight, but it differs by defining a more realistic temperature function, consisting of eight data points connected by straight lines; i.e. regions of constant temperature gradient. Methane is a strong GHG with a global warming potential 84 times greater than CO2 in a 20-year time frame. This global destruction of atmospheric methane mainly occurs in the troposphere. The annual average for methane (CH4) was 1866 ppb in 2019 and scientists reported with ""very high confidence"" that concentrations of CH4 were higher than at any time in at least 800,000 years. As air rises in the tropics, methane is carried upwards through the tropospherethe lowest portion of Earth's atmosphere which is to from the Earth's surface, into the lower stratospherethe ozone layerand then the upper portion of the stratosphere. (The total air mass below a certain altitude is calculated by integrating over the density function.) This geopotential altitude h is then used instead of geometric altitude z in the hydrostatic equations. ==Common models== * COSPAR International Reference Atmosphere * International Standard Atmosphere * Jacchia Reference Atmosphere, an older model still commonly used in spacecraft dynamics * Jet standard atmosphere * NRLMSISE-00 is a recent model from NRL often used in the atmospheric sciences * US Standard Atmosphere ==See also== * Standard temperature and pressure * Upper- atmospheric models ==References== ==External links== *Public Domain Aeronautical Software – Derivation of hydrostatic equations used in the 1976 US Standard Atmosphere *FORTRAN code to calculate the US Standard Atmosphere *NASA GSFC Atmospheric Models overview *Various models at NASA GSFC ModelWeb *Earth Global Reference Atmospheric Model (Earth-GRAM 2010) Category:Atmospheric sciences ",1.6,-1.0,"""62.8318530718""",226,11,A
+"A camper stranded in snowy weather loses heat by wind convection. The camper is packing emergency rations consisting of $58 \%$ sucrose, $31 \%$ fat, and $11 \%$ protein by weight. Using the data provided in Problem P4.32 and assuming the fat content of the rations can be treated with palmitic acid data and the protein content similarly by the protein data in Problem P4.32, how much emergency rations must the camper consume in order to compensate for a reduction in body temperature of $3.5 \mathrm{~K}$ ? Assume the heat capacity of the body equals that of water. Assume the camper weighs $67 \mathrm{~kg}$.","Another formula used is: citing :M=VO_2\left(\frac{RQ-0.7}{0.3}e_c+\frac{1-RQ}{0.3}e_f\right) where RQ is the respiratory quotient (ratio of volume CO2 produced to volume of O2 consumed), e_c is , the heat released per litre of oxygen by the oxidation of carbohydrate, and e_f is , the value for fat. The ration was two-thirds of a pound (302 g) of bread and two-thirds of a pound of meat. fourpence (4d) was deducted daily from the soldiers' pay. Vegetables and boiled starchy foods should be cooked without added salt ==== Daily Messing Rate ==== The Daily Messing Rate (DMR) is used to provide the following daily calorific intake; Daily Messing Rate Type Calorific Intake Basic DMR 3000 Kcal Exercise (Field) DMR. 4000 Kcal Overseas Exercise (Field) DMR. 4000 Kcal Operational DMR. 4000 Kcal Nijmegen Marches. 4000 Kcal Norway DMR. 5000 Kcal The current Daily Messing Rate is; * £2.73 in the United Kingdom * £3.60 outside the United Kingdom ==== Catering for diversity ==== In accordance with current UK legislation and Government guidelines it is incumbent on the Armed Forces to cater for all personnel irrespective of gender, race, religious belief, medical requirements and committed lifestyle choices. ==United States== During the American Revolution, the Continental Congress regulated garrison rations, stipulating in the Militia Law of 1775 that they should consist of: :One pound of beef, or 3/4 of a pound of pork or one pound of fish, per day. Rations in camp. The theoretical bases of indirect calorimetry: a review."" The daily ration scale in September 1941 was as follows; ==== Food ==== Meat Bacon and Ham Butter and margarine Cheese Cooking fats Sugar Tea Preserves Army rations Home Service Scale (Men) 12 oz (340 g) 1.14 oz (32 g) 1.89 oz (53 g) 0.57 oz (16 g) 0.28 oz (7 g) 4.28 oz (121 g) 0.57 oz (16 g) 1.14 oz (32 g) Army rations Home Service Scale (Women) 6 oz (170 g) 1.28 oz (36 g) 1.5 oz (42 g) (margarine only) 0.57 oz (16 g) - 2 oz (56 g) 0.28 oz (7 g) 1 oz (28 g) === Modern === ==== UK MOD Nutrition Policy Statement ==== Joint Service Publication (JSP) 456 Part 2 Volume 1 of December 2014, the Ministry of Defence policy on nutrition is as follows; The UK Ministry of Defence (MOD) undertakes to provide military personnel with a basic knowledge of nutrition, with the aim of optimising physical and mental function, long-term health, and morale. After a volume is met, Resting Energy Expenditure is calculated by the Weir formula and results are displayed in software attached to the system. This gives the same result as the Weir formula at RQ = 1 (burning only carbohydrates), and almost the same value at RQ = 0.7 (burning only fat). ⊅Σ′ ==History== Antoine Lavoisier noted in 1780 that heat production, in some cases, can be predicted from oxygen consumption, using multiple regression. Data on per capita food supplies are expressed in terms of quantity and by applying appropriate food composition factors for all primary and processed products also in terms of dietary energy value, protein and fat content. Ambient and diluted fractions of O2 and CO2 are measured for a known ventilation rate, and O2 consumption and CO2 production are determined and converted into Resting Energy Expenditure.Academy of Nutrition and Dietetics ""Measuring RMR with Indirect Calorimetry (IC)."" By World War I, the American garrison ration had improved dramatically, including 137 grams of protein, 129 grams of fat, and 539 grams of carbohydrate every day, with a total of roughly 4,000 calories. The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. In the West Indies troops were issued with salt beef on five days with fresh meat being issued for two days a week. === Crimean War === Following initial disasters in the supply system, reforms were made and British troops were issued the following; 24 oz (680 g) of bread, 16 oz (453 g) meat, 2 oz (56 g) Rice, 2 oz (56 g) Sugar, 3 oz (85 g) Coffee, 1 Gill (0.118l) spirits and ½ oz (14 g) salt. === First World War === During the First World War British troops were issued the following daily ration; 1¼ pound (567 g) of meat, 1 pound (453 g) preserved meat, 1¼ (567 g) pound of bread, (or 1 pound (453 g) of biscuit and 4 oz (113 g) of bacon), 4 oz (113 g) Jam, 3 oz (85 g) sugar, ⅝ oz (17 g) tea, 8 oz (226 g) vegetables and 2 oz (56 g) of butter (weekly) ==== Horse Rations ==== As horses were a principal form of transport for the British Army, horses also had a scale of rations issued. A garrison ration (or mess ration for food rations of this type) is a type of military ration. thumb|Indirect calorimetry metabolic cart measuring oxygen uptake (O2) and carbon dioxide production (CO2) of a spontaneously breathing subject (dilution method with canopy hood). Indirect calorimetry calculates heat that living organisms produce by measuring either their production of carbon dioxide and nitrogen waste (frequently ammonia in aquatic organisms, or urea in terrestrial ones), or from their consumption of oxygen. Indirect calorimetry estimates the type and rate of substrate utilization and energy metabolism in vivo starting from gas exchange measurements (oxygen consumption and carbon dioxide production during rest and steady-state exercise). The per capita supply of each such food item available for human consumption is then obtained by dividing the respective quantity by the related data on the population actually partaking in it. Further advances in nutrition led to the replacement of the garrison ration in 1933 with the New Army ration, which ultimately developed into the rations system described at United States military ration. *Canopy (dilution): The dilution technique is considered the gold standard technology for Resting Energy Expenditure measurement in clinical nutrition. From 1815 to 1854 the daily ration for a British soldier in the United Kingdom was 1 pound of bread (453 g) and ¾ of a pound of meat (340 g). Food Item Ration I Ration II Ration III Ration IV Rye bread 700g (1.54 lb) 700g (1.54 lb) 700g (1.54 lb) 600g (1.32 lb) Fresh meat with bones 136g (4.8 oz) 107g (3.7 oz) 90g (3.17 oz) 56g (2 oz) Soy bean flour 7g (0.24 oz) 7g (0.24 oz) 7g (0.24 oz) 7g (0.24 oz) Headless fish 30g (1 oz) 30g (1 oz) 30g (1 oz) 30g (1 oz) Fresh vegetables and fruits 250g (8.8 oz) 250g (8.8 oz) 250g (8.8 oz) 250g (8.8 oz) Potatoes 320g (11.29 oz) 320g (11.29 oz) 320g (11.29 oz) 320g (11.29 oz) Legumes 80g (2.8 oz) 80g (2.8 oz) 80g (2.8 oz) 80g (2.8 oz) Pudding powder 20g (0.70 oz) 20g (0.70 oz) 20g (0.70 oz) 20g (0.70 oz) Sweetened condensed skim milk 25g (0.88 oz) 25g (0.88 oz) 25g (0.88 oz) 25g (0.88 oz) Salt 15g (0.5 oz) 15g (0.5 oz) 15g (0.5 oz) 15g (0.5 oz) Other seasonings 3g (0.1 oz) 3g (0.1 oz) 3g (0.1 oz) 3g (0.1 oz) Spices 1g (0.03 oz) 1g (0.03 oz) 1g (0.03 oz) 1g (0.03 oz) Fats and bread spreads 60g (2.11 oz) 50g (1.76 oz) 40g (1.41 oz) 35g (1.23 oz) Coffee 9g (0.32 oz) 9g (0.32 oz) 9g (0.32 oz) 9g (0.32 oz) Sugar 40g (1.4 oz) 35g (1.23 oz) 30g (1.05 oz) 30g (1.05 oz) Supplementary allowances 2g (0.07 oz) 2g (0.07 oz) 2g (0.07 oz) 2g (0.07 oz) Total Maximum Ration in grams 1698 1654 1622 1483 Total Maximum Ration in Pounds 3.74 3.64 3.57 3.26 == United Kingdom == In 1689 the first Royal warrant was published concerning the messing provisions for troops. ",0.318,+11,"""49.0""", -6.8,-1.0,C
+"At 303 . K, the vapor pressure of benzene is 120 . Torr and that of hexane is 189 Torr. Calculate the vapor pressure of a solution for which $x_{\text {benzene }}=0.28$ assuming ideal behavior.","Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The second solution is switching to another vapor pressure equation with more than three parameters. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds"", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The vapor phase is also assumed to behave like an ideal gas, so :v_v = \frac{k T}{P}, where k is the Boltzmann constant. The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). (The small differences in the results are only caused by the used limited precision of the coefficients). ==Extension of the Antoine equations== To overcome the limits of the Antoine equation some simple extension by additional terms are used: : \begin{align} P &= \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)} \\\ P &= \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right). \end{align} The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The equation was presented in 1888 by the French engineer (1825–1897). ==Equation== The Antoine equation is :\log_{10} p = A-\frac{B}{C+T}. where p is the vapor pressure, is temperature (in °C or in K according to the value of C) and , and are component-specific constants. Let v_l and v_v be the volume occupied by one molecule in the liquid phase and vapor phase respectively. The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: :T = \frac{B}{A-\log_{10}\, p} - C ==Validity range== Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. Equilibrium vapor pressure depends on droplet size. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., ""Antoine Vapor Pressure Constants of Pure Compounds"" ** Yaws C. L., Yang H.-C., ""To Estimate Vapor Pressure Easily. This may be written in the following form, known as the Ostwald–Freundlich equation: \ln \frac{p}{p_{\rm sat}} = \frac{2 \gamma V_\text{m}}{rRT}, where p is the actual vapour pressure, p_{\rm sat} is the saturated vapour pressure when the surface is flat, \gamma is the liquid/vapor surface tension, V_\text{m} is the molar volume of the liquid, R is the universal gas constant, r is the radius of the droplet, and T is temperature. Image:VaporPressureFitAugust.png | Deviations of an August equation fit (2 parameters) Image:VaporPressureFitAntoine.png | Deviations of an Antoine equation fit (3 parameters) Image:VaporPressureFitDIPPR101.png | Deviations of a DIPPR 105 equation fit (4 parameters) ==Example parameters== Parameterisation for T in °C and P in mmHg A B C T min. (°C) T max. (°C) Water 8.07131 1730.63 233.426 1 100 Water 8.14019 1810.94 244.485 99 374 Ethanol 8.20417 1642.89 230.300 −57 80 Ethanol 7.68117 1332.04 199.200 77 243 ===Example calculation=== The normal boiling point of ethanol is TB = 78.32 °C. :\begin{align} P &= 10^{\left(8.20417 - \frac{1642.89}{78.32 + 230.300}\right)} = 760.0\ \text{mmHg} \\\ P &= 10^{\left(7.68117 - \frac{1332.04}{78.32 + 199.200}\right)} = 761.0\ \text{mmHg} \end{align} (760mmHg = 101.325kPa = 1.000atm = normal pressure) This example shows a severe problem caused by using two different sets of coefficients. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Thus, the change in the Gibbs free energy for one molecule is :\Delta g = - k T \int\limits_{P_{sat}}^{P} \frac {dP}{P}, where P_{sat} is the saturated vapor pressure of x over a flat surface and P is the actual vapor pressure over the liquid. ", 0.4,0.69,"""170.0""",24,6,C
+"Determine the molar standard Gibbs energy for ${ }^{35} \mathrm{Cl}^{35} \mathrm{Cl}$ where $\widetilde{\nu}=560 . \mathrm{cm}^{-1}, B=0.244 \mathrm{~cm}^{-1}$, and the ground electronic state is nondegenerate.","The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The standard Gibbs free energy of formation (Gf°) of a compound is the change of Gibbs free energy that accompanies the formation of 1 mole of a substance in its standard state from its constituent elements in their standard states (the most stable form of the element at 1 bar of pressure and the specified temperature, usually 298.15 K or 25 °C). The NIST Chemistry WebBook (see link below) is an online resource that contains standard enthalpy of formation for various compounds along with the standard absolute entropy for these compounds from which the standard Gibbs free energy of formation can be calculated. These tables list values of molar ionization energies, measured in kJ⋅mol−1. Note that all values are in kJ/mol. Far more extensive tables can be found in the CRC Handbook of Chemistry and Physics and the NIST JANAF tables.M. W. Chase, NIST – JANAF Thermochemical Tables, 4th Edition, Journal of Physical and Chemical Reference Data, Monograph 9, 1998. In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The table below lists the standard Gibbs function of formation for several elements and chemical compounds and is taken from Lange's Handbook of Chemistry. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. Unlike standard enthalpies of formation, the value of is absolute. LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. The first molar ionization energy applies to the neutral atoms. That is, an element in its standard state has a definite, nonzero value of at room temperature. Under identical conditions, it is greater for a heavier gas. ==See also== *Entropy *Heat *Gibbs free energy *Helmholtz free energy *Standard state *Third law of thermodynamics ==References== ==External links== *Standard Thermodynamic Properties of Chemical Substances Table Category:Chemical properties Category:Thermodynamic entropy The notation (P=0) denotes low pressure limiting values. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. All data from rutherfordium onwards is predicted. == All Ionization Energies == Number Symbol Name 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 1 H hydrogen 1312.0 2 He helium 2372.3 5250.5 3 Li lithium 520.2 7298.1 11,815.0 4 Be beryllium 899.5 1757.1 14,848.7 21,006.6 5 B boron 800.6 2427.1 3659.7 25,025.8 32,826.7 6 C carbon 1086.5 2352.6 4620.5 6222.7 37,831 47,277.0 7 N nitrogen 1402.3 2856 4578.1 7475.0 9444.9 53,266.6 64,360 8 O oxygen 1313.9 3388.3 5300.5 7469.2 10,989.5 13,326.5 71,330 84,078.0 9 F fluorine 1681.0 3374.2 6050.4 8407.7 11,022.7 15,164.1 17,868 92,038.1 106,434.3 10 Ne neon 2080.7 3952.3 6122 9371 12,177 15,238.90 19,999.0 23,069.5 115,379.5 131,432 11 Na sodium 495.8 4562 6910.3 9543 13,354 16,613 20,117 25,496 28,932 141,362 12 Mg magnesium 737.7 1450.7 7732.7 10,542.5 13,630 18,020 21,711 25,661 31,653 35,458 13 Al aluminium 577.5 1816.7 2744.8 11,577 14,842 18,379 23,326 27,465 31,853 38,473 14 Si silicon 786.5 1577.1 3231.6 4355.5 16,091 19,805 23,780 29,287 33,878 38,726 15 P phosphorus 1011.8 1907 2914.1 4963.6 6273.9 21,267 25,431 29,872 35,905 40,950 16 S sulfur 999.6 2252 3357 4556 7004.3 8495.8 27,107 31,719 36,621 43,177 17 Cl chlorine 1251.2 2298 3822 5158.6 6542 9362 11,018 33,604 38,600 43,961 18 Ar argon 1520.6 2665.8 3931 5771 7238 8781 11,995 13,842 40,760 46,186 19 K potassium 418.8 3052 4420 5877 7975 9590 11,343 14,944 16,963.7 48,610 20 Ca calcium 589.8 1145.4 4912.4 6491 8153 10,496 12,270 14,206 18,191 20,385 21 Sc scandium 633.1 1235.0 2388.6 7090.6 8843 10,679 13,310 15,250 17,370 21,726 22 Ti titanium 658.8 1309.8 2652.5 4174.6 9581 11,533 13,590 16,440 18,530 20,833 23 V vanadium 650.9 1414 2830 4507 6298.7 12,363 14,530 16,730 19,860 22,240 24 Cr chromium 652.9 1590.6 2987 4743 6702 8744.9 15,455 17,820 20,190 23,580 25 Mn manganese 717.3 1509.0 3248 4940 6990 9220 11,500 18,770 21,400 23,960 26 Fe iron 762.5 1561.9 2957 5290 7240 9560 12,060 14,580 22,540 25,290 27 Co cobalt 760.4 1648 3232 4950 7670 9840 12,440 15,230 17,959 26,570 28 Ni nickel 737.1 1753.0 3395 5300 7339 10,400 12,800 15,600 18,600 21,670 29 Cu copper 745.5 1957.9 3555 5536 7700 9900 13,400 16,000 19,200 22,400 30 Zn zinc 906.4 1733.3 3833 5731 7970 10,400 12,900 16,800 19,600 23,000 31 Ga gallium 578.8 1979.3 2963 6180 32 Ge germanium 762 1537.5 3302.1 4411 9020 33 As arsenic 947.0 1798 2735 4837 6043 12,310 34 Se selenium 941.0 2045 2973.7 4144 6590 7880 14,990 35 Br bromine 1139.9 2103 3470 4560 5760 8550 9940 18,600 36 Kr krypton 1350.8 2350.4 3565 5070 6240 7570 10,710 12,138 22,274 25,880 37 Rb rubidium 403.0 2633 3860 5080 6850 8140 9570 13,120 14,500 26,740 38 Sr strontium 549.5 1064.2 4138 5500 6910 8760 10,230 11,800 15,600 17,100 39 Y yttrium 600 1180 1980 5847 7430 8970 11,190 12,450 14,110 18,400 40 Zr zirconium 640.1 1270 2218 3313 7752 9500 41 Nb niobium 652.1 1380 2416 3700 4877 9847 12,100 42 Mo molybdenum 684.3 1560 2618 4480 5257 6640.8 12,125 13,860 15,835 17,980 43 Tc technetium 702 1470 2850 44 Ru ruthenium 710.2 1620 2747 45 Rh rhodium 719.7 1740 2997 46 Pd palladium 804.4 1870 3177 47 Ag silver 731.0 2070 3361 48 Cd cadmium 867.8 1631.4 3616 49 In indium 558.3 1820.7 2704 5210 50 Sn tin 708.6 1411.8 2943.0 3930.3 7456 51 Sb antimony 834 1594.9 2440 4260 5400 10,400 52 Te tellurium 869.3 1790 2698 3610 5668 6820 13,200 53 I iodine 1008.4 1845.9 3180 54 Xe xenon 1170.4 2046.4 3099.4 55 Cs caesium 375.7 2234.3 3400 56 Ba barium 502.9 965.2 3600 57 La lanthanum 538.1 1067 1850.3 4819 5940 58 Ce cerium 534.4 1050 1949 3547 6325 7490 59 Pr praseodymium 527 1020 2086 3761 5551 60 Nd neodymium 533.1 1040 2130 3900 61 Pm promethium 540 1050 2150 3970 62 Sm samarium 544.5 1070 2260 3990 63 Eu europium 547.1 1085 2404 4120 64 Gd gadolinium 593.4 1170 1990 4250 65 Tb terbium 565.8 1110 2114 3839 66 Dy dysprosium 573.0 1130 2200 3990 67 Ho holmium 581.0 1140 2204 4100 68 Er erbium 589.3 1150 2194 4120 69 Tm thulium 596.7 1160 2285 4120 70 Yb ytterbium 603.4 1174.8 2417 4203 71 Lu lutetium 523.5 1340 2022.3 4370 6445 72 Hf hafnium 658.5 1440 2250 3216 73 Ta tantalum 761 1500 74 W tungsten 770 1700 75 Re rhenium 760 1260 2510 3640 76 Os osmium 840 1600 77 Ir iridium 880 1600 78 Pt platinum 870 1791 79 Au gold 890.1 1980 80 Hg mercury 1007.1 1810 3300 81 Tl thallium 589.4 1971 2878 82 Pb lead 715.6 1450.5 3081.5 4083 6640 83 Bi bismuth 703 1610 2466 4370 5400 8520 84 Po polonium 812.1 85 At astatine 899.003 86 Rn radon 1037 87 Fr francium 393 88 Ra radium 509.3 979.0 89 Ac actinium 499 1170 1900 4700 90 Th thorium 587 1110 1978 2780 91 Pa protactinium 568 1128 1814 2991 92 U uranium 597.6 1420 1900 3145 93 Np neptunium 604.5 1128 1997 3242 94 Pu plutonium 584.7 1128 2084 3338 95 Am americium 578 1158 2132 3493 96 Cm curium 581 1196 2026 3550 97 Bk berkelium 601 1186 2152 3434 98 Cf californium 608 1206 2267 3599 99 Es einsteinium 619 1216 2334 3734 100 Fm fermium 629 1225 2363 3792 101 Md mendelevium 636 1235 2470 3840 102 No nobelium 639 1254 2643 3956 103 Lr lawrencium 479 1428 2228 4910 104 Rf rutherfordium 580 1390 2300 3080 105 Db dubnium 665 1547 2378 3299 4305 106 Sg seaborgium 757 1733 2484 3416 4562 5716 107 Bh bohrium 740 1690 2570 3600 4730 5990 7230 108 Hs hassium 730 1760 2830 3640 4940 6180 7540 8860 109 Mt meitnerium 800 1820 2900 3900 4900 110 Ds darmstadtium 960 1890 3030 4000 5100 111 Rg roentgenium 1020 2070 3080 4100 5300 112 Cn copernicium 1155 2170 3160 4200 5500 113 Nh nihonium 707.2 2309 3226 4382 5638 114 Fl flerovium 832.2 1600 3370 4400 5850 115 Mc moscovium 538.3 1760 2650 4680 5720 116 Lv livermorium 663.9 1330 2850 3810 6080 117 Ts tennessine 736.9 1435.4 2161.9 4012.9 5076.4 118 Og oganesson 860.1 1560 119 Uue ununennium 463.1 1700 120 Ubn unbinilium 563.3 121 Ubu unbiunium 429.4 1110 1710 4270 122 Ubb unbibium 545 1090 1848 2520 == 11th-20th ionisation energies == number symbol name 11th 12th 13th 14th 15th 16th 17th 18th 19th 20th 11 Na sodium 159,076 12 Mg magnesium 169,988 189,368 13 Al aluminium 42,647 201,266 222,316 14 Si silicon 45,962 50,502 235,196 257,923 15 P phosphorus 46,261 54,110 59,024 271,791 296,195 16 S sulfur 48,710 54,460 62,930 68,216 311,048 337,138 17 Cl chlorine 51,068 57,119 63,363 72,341 78,095 352,994 380,760 18 Ar argon 52,002 59,653 66,199 72,918 82,473 88,576 397,605 427,066 19 K potassium 54,490 60,730 68,950 75,900 83,080 93,400 99,710 444,880 476,063 20 Ca calcium 57,110 63,410 70,110 78,890 86,310 94,000 104,900 111,711 494,850 527,762 21 Sc scandium 24,102 66,320 73,010 80,160 89,490 97,400 105,600 117,000 124,270 547,530 22 Ti titanium 25,575 28,125 76,015 83,280 90,880 100,700 109,100 117,800 129,900 137,530 23 V vanadium 24,670 29,730 32,446 86,450 94,170 102,300 112,700 121,600 130,700 143,400 24 Cr chromium 26,130 28,750 34,230 37,066 97,510 105,800 114,300 125,300 134,700 144,300 25 Mn manganese 27,590 30,330 33,150 38,880 41,987 109,480 118,100 127,100 138,600 148,500 26 Fe iron 28,000 31,920 34,830 37,840 44,100 47,206 122,200 131,000 140,500 152,600 27 Co cobalt 29,400 32,400 36,600 39,700 42,800 49,396 52,737 134,810 145,170 154,700 28 Ni nickel 30,970 34,000 37,100 41,500 44,800 48,100 55,101 58,570 148,700 159,000 29 Cu copper 25,600 35,600 38,700 42,000 46,700 50,200 53,700 61,100 64,702 163,700 30 Zn zinc 26,400 29,990 40,490 43,800 47,300 52,300 55,900 59,700 67,300 71,200 36 Kr krypton 29,700 33,800 37,700 43,100 47,500 52,200 57,100 61,800 75,800 80,400 38 Sr strontium 31,270 39 Y yttrium 19,900 36,090 42 Mo molybdenum 20,190 22,219 26,930 29,196 52,490 55,000 61,400 67,700 74,000 80,400 == 21st-30th ionisation energies == number symbol name 21st 22nd 23rd 24th 25th 26th 27th 28th 29th 30th 21 Sc scandium 582,163 22 Ti titanium 602,930 639,294 23 V vanadium 151,440 661,050 699,144 24 Cr chromium 157,700 166,090 721,870 761,733 25 Mn manganese 158,600 172,500 181,380 785,450 827,067 26 Fe iron 163,000 173,600 188,100 195,200 851,800 895,161 27 Co cobalt 167,400 178,100 189,300 204,500 214,100 920,870 966,023 28 Ni nickel 169,400 182,700 194,000 205,600 221,400 231,490 992,718 1,039,668 29 Cu copper 174,100 184,900 198,800 210,500 222,700 239,100 249,660 1,067,358 1,116,105 30 Zn zinc 179,100 36 Kr krypton 85,300 90,400 96,300 101,400 111,100 116,290 282,500 296,200 311,400 326,200 42 Mo molybdenum 87,000 93,400 98,420 104,400 121,900 127,700 133,800 139,800 148,100 154,500 == References == * Ionization energies of the elements (data page) * (for predictions) * * (for predictions) Category:Properties of chemical elements ",-57.2,226,"""12.0""",7.27,-0.55,A
+"For the reaction $\mathrm{C}($ graphite $)+\mathrm{H}_2 \mathrm{O}(g) \rightleftharpoons$ $\mathrm{CO}(g)+\mathrm{H}_2(g), \Delta H_R^{\circ}=131.28 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298.15 \mathrm{~K}$. Use the values of $C_{P, m}^{\circ}$ at $298.15 \mathrm{~K}$ in the data tables to calculate $\Delta H_R^{\circ}$ at $125.0^{\circ} \mathrm{C}$.","== Heat of fusion == kJ/mol 1 H hydrogen (H2) use (H2) 0.1 CRC (H2) 0.12 LNG (H2) 0.117 WEL (per mol H atoms) 0.0585 2 He helium use 0.0138 LNG 0.0138 WEL 0.02 3 Li lithium use 3.00 CRC 3.00 LNG 3.00 WEL 3.0 4 Be beryllium use 7.895 CRC 7.895 LNG 7.895 WEL 7.95 5 B boron use 50.2 CRC 50.2 LNG 50.2 WEL 50 6 C carbon (graphite) use (graphite) 117 CRC (graphite) 117 LNG (graphite) 117 7 N nitrogen (N2) use (N2) 0.720 CRC (N2) 0.71 LNG (N2) 0.720 WEL (per mol N atoms) 0.36 8 O oxygen (O2) use (O2) 0.444 CRC (O2) 0.44 LNG (O2) 0.444 WEL (per mol O atoms) 0.222 9 F fluorine (F2) use (F2) 0.510 CRC (F2) 0.51 LNG (F2) 0.510 WEL (per mol F atoms) 0.26 10 Ne neon use 0.335 CRC 0.328 LNG 0.335 WEL 0.34 11 Na sodium use 2.60 CRC 2.60 LNG 2.60 WEL 2.60 12 Mg magnesium use 8.48 CRC 8.48 LNG 8.48 WEL 8.7 13 Al aluminium use 10.71 CRC 10.789 LNG 10.71 WEL 10.7 14 Si silicon use 50.21 CRC 50.21 LNG 50.21 WEL 50.2 15 P phosphorus use 0.66 CRC (white) 0.66 LNG 0.66 WEL 0.64 16 S sulfur use (mono) 1.727 CRC (mono) 1.72 LNG (mono) 1.727 WEL 1.73 17 Cl chlorine (Cl2) use (Cl2) 6.406 CRC (Cl2) 6.40 LNG (Cl2) 6.406 WEL (per mol Cl atoms) 3.2 18 Ar argon use 1.18 CRC 1.18 LNG 1.12 WEL 1.18 19 K potassium use 2.321 CRC 2.33 LNG 2.321 WEL 2.33 20 Ca calcium use 8.54 CRC 8.54 LNG 8.54 WEL 8.54 21 Sc scandium use 14.1 CRC 14.1 LNG 14.1 WEL 16 22 Ti titanium use 14.15 CRC 14.15 LNG 14.15 WEL 18.7 23 V vanadium use 21.5 CRC 21.5 LNG 21.5 WEL 22.8 24 Cr chromium use 21.0 CRC 21.0 LNG 21.0 WEL 20.5 25 Mn manganese use 12.91 CRC 12.91 LNG 12.9 WEL 13.2 26 Fe iron use 13.81 CRC 13.81 LNG 13.81 WEL 13.8 27 Co cobalt use 16.06 CRC 16.06 LNG 16.2 WEL 16.2 28 Ni nickel use 17.48 CRC 17.04 LNG 17.48 WEL 17.2 29 Cu copper use 13.26 CRC 12.93 LNG 13.26 WEL 13.1 30 Zn zinc use 7.32 CRC 7.068 LNG 7.32 WEL 7.35 31 Ga gallium use 5.59 CRC 5.576 LNG 5.59 WEL 5.59 32 Ge germanium use 36.94 CRC 36.94 LNG 36.94 WEL 31.8 33 As arsenic use (gray) 24.44 CRC (gray) 24.44 LNG 24.44 WEL 27.7 34 Se selenium use (gray) 6.69 CRC (gray) 6.69 LNG 6.69 WEL 5.4 35 Br bromine (Br2) use (Br2) 10.57 CRC (Br2) 10.57 LNG (Br2) 10.57 WEL (per mol Br atoms) 5.8 36 Kr krypton use 1.64 CRC 1.64 LNG 1.37 WEL 1.64 37 Rb rubidium use 2.19 CRC 2.19 LNG 2.19 WEL 2.19 38 Sr strontium use 7.43 CRC 7.43 LNG 7.43 WEL 8 39 Y yttrium use 11.42 CRC 11.4 LNG 11.42 WEL 11.4 40 Zr zirconium use 14 13.96 ± 0.36 CRC 21.00 LNG 21.00 WEL 21 41 Nb niobium use 30 CRC 30 LNG 30 WEL 26.8 42 Mo molybdenum use 37.48 CRC 37.48 LNG 37.48 WEL 36 43 Tc technetium use 33.29 CRC 33.29 LNG 33.29 WEL 23 44 Ru ruthenium use 38.59 CRC 38.59 LNG 38.59 WEL 25.7 45 Rh rhodium use 26.59 CRC 26.59 LNG 26.59 WEL 21.7 46 Pd palladium use 16.74 CRC 16.74 LNG 16.74 WEL 16.7 47 Ag silver use 11.28 CRC 11.28 LNG 11.95 WEL 11.3 48 Cd cadmium use 6.21 CRC 6.21 LNG 6.19 WEL 6.3 49 In indium use 3.281 CRC 3.281 LNG 3.28 WEL 3.26 50 Sn tin use (white) 7.03 CRC (white) 7.173 LNG (white) 7.03 WEL 7.0 51 Sb antimony use 19.79 CRC 19.79 LNG 19.87 WEL 19.7 52 Te tellurium use 17.49 CRC 17.49 LNG 17.49 WEL 17.5 53 I iodine (I2) use (I2) 15.52 CRC (I2) 15.52 LNG (I2) 150.66 [sic] WEL (per mol I atoms) 7.76 54 Xe xenon use 2.27 CRC 2.27 LNG 1.81 WEL 2.30 55 Cs caesium use 2.09 CRC 2.09 LNG 2.09 WEL 2.09 56 Ba barium use 7.12 CRC 7.12 LNG 7.12 WEL 8.0 57 La lanthanum use 6.20 CRC 6.20 LNG 6.20 WEL 6.2 58 Ce cerium use 5.46 CRC 5.46 LNG 5.46 WEL 5.5 59 Pr praseodymium use 6.89 CRC 6.89 LNG 6.89 WEL 6.9 60 Nd neodymium use 7.14 CRC 7.14 LNG 7.14 WEL 7.1 61 Pm promethium use 7.13 LNG 7.13 WEL about 7.7 62 Sm samarium use 8.62 CRC 8.62 LNG 8.62 WEL 8.6 63 Eu europium use 9.21 CRC 9.21 LNG 9.21 WEL 9.2 64 Gd gadolinium use 10.05 CRC 10.0 LNG 10.05 WEL 10.0 65 Tb terbium use 10.15 CRC 10.15 LNG 10.15 WEL 10.8 66 Dy dysprosium use 11.06 CRC 11.06 LNG 11.06 WEL 11.1 67 Ho holmium use 17.0 CRC 17.0 LNG 16.8 WEL 17.0 68 Er erbium use 19.90 CRC 19.9 LNG 19.90 WEL 19.9 69 Tm thulium use 16.84 CRC 16.84 LNG 16.84 WEL 16.8 70 Yb ytterbium use 7.66 CRC 7.66 LNG 7.66 WEL 7.7 71 Lu lutetium use ca. 22 CRC 22 LNG (22) WEL about 22 72 Hf hafnium use 27.2 CRC 27.2 LNG 27.2 WEL 25.5 73 Ta tantalum use 36.57 CRC 36.57 LNG 36.57 WEL 36 74 W tungsten use 52.31 CRC 52.31 LNG 52.31 WEL 35 75 Re rhenium use 60.43 CRC 60.43 LNG 60.43 WEL 33 76 Os osmium use 57.85 CRC 57.85 LNG 57.85 WEL 31 77 Ir iridium use 41.12 CRC 41.12 LNG 41.12 WEL 26 78 Pt platinum use 22.17 CRC 22.17 LNG 22.17 WEL 20 79 Au gold use 12.55 CRC 12.72 LNG 12.55 WEL 12.5 80 Hg mercury use 2.29 CRC 2.29 LNG 2.29 WEL 2.29 81 Tl thallium use 4.14 CRC 4.14 LNG 4.14 WEL 4.2 82 Pb lead use 4.77 CRC 4.782 LNG 4.77 WEL 4.77 83 Bi bismuth use 11.30 CRC 11.145 LNG 11.30 WEL 10.9 84 Po polonium use ca. 13 WEL about 13 85 At astatine use WEL (per mol At atoms) about 6 86 Rn radon use 3.247 LNG 3.247 WEL 3 87 Fr francium use ca. 2 WEL about 2 88 Ra radium use 8.5 LNG 8.5 WEL about 8 89 Ac actinium use 14 WEL 14 90 Th thorium use 13.81 CRC 13.81 LNG 13.81 WEL 16 91 Pa protactinium use 12.34 CRC 12.34 LNG 12.34 WEL 15 92 U uranium use 9.14 CRC 9.14 LNG 9.14 WEL 14 93 Np neptunium use 3.20 CRC 3.20 LNG 3.20 WEL 10 94 Pu plutonium use 2.82 CRC 2.82 LNG 2.82 95 Am americium use 14.39 CRC 14.39 LNG 14.39 == Notes == * Values refer to the enthalpy change between the liquid phase and the most stable solid phase at the melting point (normal, 101.325 kPa). == References == === CRC === As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. ==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup Graphane is a two-dimensional polymer of carbon and hydrogen with the formula unit (CH)n where n is large. P-doped graphane is proposed to be a high-temperature BCS theory superconductor with a Tc above 90 K. ==Variants== Partial hydrogenation leads to hydrogenated graphene rather than (fully hydrogenated) graphane. The structure was found, using a cluster expansion method, to be the most stable of all the possible hydrogenation ratios of graphene. * Values from CRC refer to ""100 kPa (1 bar or 0.987 standard atmospheres)"". :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). LNG: Values refer to 300 K. * Ref. WEL: Values refer to 25 °C. ==References== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Hydrogenation decreases the dependence of the lattice constant on temperature, which indicates a possible application in precision instruments. ==References== ==External links== * Sep 14, 2010 Hydrogen vacancies induce stable ferromagnetism in graphane * May 25, 2010 Graphane yields new potential * May 02 2010 Doped Graphane Should Superconduct at 90K Category:Two-dimensional nanomaterials Category:Polymers Category:Superconductors Category:Hydrocarbons Density functional theory calculations suggested that hydrogenated and fluorinated forms of other group IV (Si, Ge and Sn) nanosheets present properties similar to graphane. ==Potential applications== p-Doped graphane is postulated to be a high-temperature BCS theory superconductor with a Tc above 90 K. Graphane has been proposed for hydrogen storage. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). Partial hydrogenation results in hydrogenated graphene, which was reported by Elias et al in 2009 by a TEM study to be ""direct evidence for a new graphene-based derivative"". * A.M. James and M.P. Lord in Macmillan's Chemical and Physical Data, Macmillan, London, UK, 1992. Boca Raton, Florida, 2003; Section 6, Fluid Properties; Enthalpy of Fusion === LNG === As quoted from various sources in: * J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds === WEL === As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Above 750 K Tc values may be in error by 10 K or more. Table 259, Critical Temperatures, Pressures, and Densities of Gases == See also == Category:Phase transitions Category:Units of pressure Category:Temperature Category:Properties of chemical elements Category:Chemical element data pages In the last case mechanical exfoliation of hydrogenated top layers can be used. ==Structure== The first theoretical description of graphane was reported in 2003. CR2: Values refer to 300 K and a pressure of ""100 kPa (1 bar)"", or to the saturation vapor pressure if that is less than 100 kPa. ",30,8.8,"""1855.0""",0.000216,132.9,E
+Calculate the mean ionic activity of a $0.0350 \mathrm{~m} \mathrm{Na}_3 \mathrm{PO}_4$ solution for which the mean activity coefficient is 0.685 .,"It is important to note that because the ions in the solution act together, the activity coefficient obtained from this equation is actually a mean activity coefficient. Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. : Aqua ions in seawater (salinity = 35) Ion Concentration (mol kg−1) 0.469 0.0102 0.0528 0.0103 Many other aqua ions are present in seawater in concentrations ranging from ppm to ppt. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. Journal of Solution Chemistry, 26, 791–815. ==External links== * For easy calculation of activity coefficients in (non- micellar) solutions, check out the IUPAC open project Aq-solutions (freeware). D&H; say that, due to the ""mutual electrostatic forces between the ions"", it is necessary to modify the Guldberg–Waage equation by replacing K with \gamma K, where \gamma is an overall activity coefficient, not a ""special"" activity coefficient (a separate activity coefficient associated with each species)—which is what is used in modern chemistry . Typical values are 3Å for ions such as H+, Cl−, CN−, and HCOO−. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. The electrostatic energy of the ion at site then is the product of its charge with the potential acting at its site :E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. In these solutions the activity coefficient may actually increase with ionic strength. center|The Debye–Hückel plot with different values for ion charge Z and ion diameter a The Debye–Hückel equation cannot be used in the solutions of surfactants where the presence of micelles influences on the electrochemical properties of the system (even rough judgement overestimates γ for ~50%). ==See also== * Strong electrolyte * Weak electrolyte * Ionic atmosphere * Debye–Hückel theory * Poisson–Boltzmann equation ==Notes== ==References== * Alt URL * * * * Malatesta, F., and Zamboni, R. (1997). Ionic potential is also a measure of the polarising power of a cation. Ions fall into four groups. This factor takes into account the interaction energy of ions in solution. == Debye–Hückel limiting law == In order to calculate the activity a_C of an ion C in a solution, one must know the concentration and the activity coefficient: a_C = \gamma \frac\mathrm{[C]}\mathrm{[C^\ominus]}, where * \gamma is the activity coefficient of C, * \mathrm{[C^\ominus]} is the concentration of the chosen standard state, e.g. 1 mol/kg if molality is used, * \mathrm{[C]} is a measure of the concentration of C. Dividing \mathrm{[C]} with \mathrm{[C^\ominus]} gives a dimensionless quantity. thumb|250px|Distribution of ions in a solution The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. A metal ion in aqueous solution or aqua ion is a cation, dissolved in water, of chemical formula [M(H2O)n]z+. Zn2+ −0.751 Ga3+ −0.53 Ge2+ +0.1 Rb −2.98 Sr2+ −2.899 Y3+ −2.37 ... Hg2+ −0.854 Tl3+ +0.73 Pb2+ −0.126 Bi3+ +0.16 Po4+ +0.76 Fr −2.9 Ra2+ −2.8 Lr3+ −1.96 La3+ −2.52 Ce3+ −2.32 Pr3+ −2.34 Nd3+ −2.32 Pm3+ −2.30 Sm3+ −2.28 Eu3+ −1.98 Gd3+ −2.27 Tb3+ −2.27 Dy3+ −2.32 Ho3+ −2.37 Er3+ −2.33 Tm3+ −2.30 Yb3+ −2.23 Ac3+ −2.18 Th4+ −1.83 Pa4+ −1.46 U4+ −1.51 Np4+ −1.33 Pu4+ −1.80 Am3+ −2.06 Cm3+ −2.07 Bk3+ −2.03 Cf3+ −2.01 Es3+ −1.99 Fm3+ −1.97 Md3+ −1.65 No3+ −1.20 : Standard electrode potentials /V for 1st. row transition metal ions Couple Ti V Cr Mn Fe Co Ni Cu M2+ / M −1.63 −1.18 −0.91 −1.18 −0.473 −0.28 −0.228 +0.345 M3+ / M −1.37 −0.87 −0.74 −0.28 −0.06 +0.41 : Miscellaneous standard electrode potentials /V Ag+ / Ag Pd2+ / Pd Pt2+ / Pt Zr4+ / Zr Hf4+ / Hf Au3+ / Au Ce4+ / Ce +0.799 +0.915 +1.18 −1.53 −1.70 +1.50 −1.32 As the standard electrode potential is more negative the aqua ion is more difficult to reduce. For ions in solution Shannon's ""effective ionic radius"" is the measure most often used.. Ions are 6-coordinate unless indicated differently in parentheses (e.g. 146 (4) for 4-coordinate N3−). Ions are 6-coordinate unless indicated differently in parentheses (e.g. 146 (4) for 4-coordinate N3−). Li+ -118.8 Na+ -87.4 Mg2+ -267.8 Al3+ -464.4 K+ -51.9 Ca2+ -209.2 ... ",0.0547,399,"""34.0""",0.2115,7.136,A
+" Consider the transition between two forms of solid tin, $\mathrm{Sn}(s$, gray $) \rightarrow \mathrm{Sn}(s$, white $)$. The two phases are in equilibrium at 1 bar and $18^{\circ} \mathrm{C}$. The densities for gray and white tin are 5750 and $7280 \mathrm{~kg} \mathrm{~m}^{-3}$, respectively, and the molar entropies for gray and white tin are 44.14 and $51.18 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$, respectively. Calculate the temperature at which the two phases are in equilibrium at 350. bar.","J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Transition temperature is the temperature at which a material changes from one crystal state (allotrope) to another. Tin(II) chloride, also known as stannous chloride, is a white crystalline solid with the formula . K (? °C), ? K (? °C), ? * Handbook of Chemistry and Physics, 71st edition, CRC Press, Ann Arbor, Michigan, 1990. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? It was first discovered in 1822. == Preparation == To obtain tin(II) acetate, tin(II) oxide is dissolved in glacial acetic acid and refluxed to obtain yellow Sn(CH3COO)2·2CH3COOH when cooled. More formally, it is the temperature at which two crystalline forms of a substance can co-exist in equilibrium. White tin may also refer specifically to β-tin, the metallic allotrope of the pure element, as opposed to the nonmetallic allotrope α-tin (also known as gray tin), which occurs at temperatures below , a transformation known as tin pest). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? At 95.6 °C the two forms can co-exist. Tin(II) acetate is the acetate salt of tin(II), with the chemical formula of Sn(CH3COO)2. Another example is tin, which transitions from a cubic crystal below 13.2 °C to a tetragonal crystal above that temperature. White tin is refined metallic tin. Tin(II) chloride should not be confused with the other chloride of tin; tin(IV) chloride or stannic chloride (SnCl4). ==Chemical structure== SnCl2 has a lone pair of electrons, such that the molecule in the gas phase is bent. The main part of the molecule stacks into double layers in the crystal lattice, with the ""second"" water sandwiched between the layers. thumb|460px|left|Structures of tin(II) chloride and related compounds ==Chemical properties== Tin(II) chloride can dissolve in less than its own mass of water without apparent decomposition, but as the solution is diluted, hydrolysis occurs to form an insoluble basic salt: :SnCl2 (aq) + H2O (l) Sn(OH)Cl (s) + HCl (aq) Therefore, if clear solutions of tin(II) chloride are to be used, it must be dissolved in hydrochloric acid (typically of the same or greater molarity as the stannous chloride) to maintain the equilibrium towards the left-hand side (using Le Chatelier's principle). J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? Solutions of tin(II) chloride can also serve simply as a source of Sn2+ ions, which can form other tin(II) compounds via precipitation reactions. Complex tin (II) acetates. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? In the case of ferroelectric or ferromagnetic crystals, a transition temperature may be known as the Curie temperature. == See also == * Crystal system Category:Crystallography Category:Threshold temperatures ",479,2.24,"""-242.6""",7.82,-3.5,E
+"The densities of pure water and ethanol are 997 and $789 \mathrm{~kg} \mathrm{~m}^{-3}$, respectively. For $x_{\text {ethanol }}=0.35$, the partial molar volumes of ethanol and water are 55.2 and $17.8 \times 10^{-3} \mathrm{~L} \mathrm{~mol}^{-1}$, respectively. Calculate the change in volume relative to the pure components when $2.50 \mathrm{~L}$ of a solution with $x_{\text {ethanol }}=0.35$ is prepared.","It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. The volume of alcohol in the solution can then be estimated. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. Mixing two solutions of alcohol of different strengths usually causes a change in volume. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Therefore, one can use the following equation to convert between ABV and ABW: \text{ABV} = \text{ABW} \times \frac{\text{density of beverage}}{\text{density of alcohol}} At relatively low ABV, the alcohol percentage by weight is about 4/5 of the ABV (e.g., 3.2% ABW is about 4% ABV). The following formulas can be used to calculate the volumes of solute (Vsolute) and solvent (Vsolvent) to be used: *Vsolute = Vtotal / F *Vsolvent = Vtotal \- Vsolute , where: *Vtotal is the desired total volume *F is the desired dilution factor number (the number in the position of F if expressed as ""1:F dilution factor"" or ""xF dilution"") However, some solutions and mixtures take up slightly less volume than their components. The International Organization of Legal Metrology has tables of density of water–ethanol mixtures at different concentrations and temperatures. At 0% and 100% ABV is equal to ABW, but at values in between ABV is always higher, up to ~13% higher around 60% ABV. ==See also== *Apparent molar property *Excess molar quantity *Standard drink *Unit of alcohol *Volume fraction == Notes == == References == ==Bibliography== * * * * ==External links== * * Category:Alcohol measurement The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Mixing pure water with a solution less than 24% by mass causes a slight increase in total volume, whereas the mixing of two solutions above 24% causes a decrease in volume. Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. The amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object. However, because of the miscibility of alcohol and water, the conversion factor is not constant but rather depends upon the concentration of alcohol. It is defined as the number of millilitres (mL) of pure ethanol present in of solution at . The density of sugar in water is greater than the density of alcohol in water. The volume was rounded up to 750 mL and then was used as the base size for French wine containers, with all subdivisions and multiples figured from it. In some countries, e.g. France, alcohol by volume is often referred to as degrees Gay-Lussac (after the French chemist Joseph Louis Gay-Lussac), although there is a slight difference since the Gay- Lussac convention uses the International Standard Atmosphere value for temperature, . ==Volume change== thumb|upright=1.48|Change in volume with increasing ABV. In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. Originally there were different standard gallons depending on the type of alcohol. ",0.8561,-0.10,"""0.88""",-36.5,8.7,B
+"For $\mathrm{N}_2$ at $298 \mathrm{~K}$, what fraction of molecules has a speed between 200. and $300 . \mathrm{m} / \mathrm{s}$ ?","To help compare different orders of magnitude, the following list describes various speed levels between approximately 2.2 m/s and 3.0 m/s (the speed of light). These atoms effuse out of a hole in the oven with average speeds on the order of hundreds of m/s and large velocity distributions (due to their high temperature). The inch per second is a unit of speed or velocity. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:chem.libretexts.org: Collision Frequency : Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, which has units of [volume][time]−1. Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. Current fastest macroscopic man-made system. 450,000 1,600,000 1,000,000 0.0015 Typical speed of a particle of the solar wind, relative to the Sun. 552,000 1,990,000 1,230,000 0.0018 Speed of the Milky Way, relative to the cosmic microwave background. 617,700 2,224,000 1,382,000 0.0021 Escape velocity from the surface of the Sun. 106 1,000,000 3,600,000 2,200,000 0.0030 Typical speed of a Moreton wave across the surface of the Sun. 1,610,000 5,800,000 3,600,000 0.0054 Speed of hypervelocity star PSR B2224+65, which currently seems to be leaving the Milky Way. 5,000,000 18,000,000 11,000,000 0.017 Estimated minimum speed of star S2 at its closest approach to Sagittarius A*. 107 14,000,000 50,000,000 31,000,000 0.047 Typical speed of a fast neutron. 30,000,000 100,000,000 70,000,000 0.1 Typical speed of an electron in a cathode ray tube. 108 100,000,000 360,000,000 220,000,000 0.3 The escape velocity of a neutron star. 100,000,000 360,000,000 220,000,000 0.3 Typical speed of the return stroke of lightning (cf. stepped leader above). 124,000,000 447,000,000 277,000,000 0.4 Speed of light in a diamond (Refractive index 2.417). 200,000,000 720,000,000 440,000,000 0.7 Speed of a signal in an optical fiber. 299,792,456 1,079,252,840 670,615,282 1 − 9 Speed of the 7 TeV protons in the Large Hadron Collider at full power. 299,792,457.996 1,079,252,848.786 670,616,629.38 1 − 1 Maximal speed of an electron in LEP (104.5 GeV). 299,792,458 − 1.5×10−15 1,079,252,848.8 − 5.4×10−15 670,616,629.4 1 − 4.9×10−24 Speed of the Oh-My-God particle ultra-high-energy cosmic ray. 299,792,458 1,079,252,848.8 670,616,629.4 1 Speed of light or other electromagnetic radiation in a vacuum or massless particles. >299,792,458 >1,079,252,848.8 >670,616,629.4 >1 Expansion rate of the universe between objects farther apart than the Hubble radius ==See also== *Typical projectile speeds - also showing the corresponding kinetic energy per unit mass *Neutron temperature ==References== Category:Units of velocity Category:Physical quantities Speed The newton-second (also newton second; symbol: N⋅s or N s) is the unit of impulse in the International System of Units (SI). In the constant deceleration approach we get: ::v\left(z\right)=\sqrt{v_{i}^{2}-2az} ::B\left(z\right)=\frac{\hbar k}{\mu'}v+\frac{\hbar \delta}{\mu'}=\frac{\hbar kv_{i}}{\mu'}\sqrt{1-\frac{2a}{v_{i}^{2}}z}+\frac{\hbar \delta}{\mu'} where v_{i} is the maximum velocity class that will be slowed; all the atoms in the velocity distribution that have velocities v will be slowed, and those with velocities v>v_{i} will not be slowed at all. Mass (kg) Speed (m/s) Momentum (N⋅s) Explanation 0.42 2.4 1 A football (FIFA specified weight for outdoor size 5) kicked to a speed of . 0.42 38 16 The momentum of the famous football kick of the Brazilian player Roberto Carlos in the match against France in 1997. Kinetic diameter is a measure applied to atoms and molecules that expresses the likelihood that a molecule in a gas will collide with another molecule. Here, the Maxwell–Boltzmann distribution of energies must be considered, which leads to the modified expression,Freude, p. 4 :d^2 = {1 \over \sqrt 2 \pi l n} ==List of diameters== The following table lists the kinetic diameters of some common molecules; Molecule Molecule Molecular mass Kinetic diameter (pm) ref Name Formula Molecular mass Kinetic diameter (pm) ref Hydrogen H2 2 289 Helium He 4 260 Matteucci et al., p. 6 Methane CH4 16 380 Ammonia NH3 17 260 Breck Water H2O 18 265 Neon Ne 20 275 Acetylene C2H2 26 330 Nitrogen N2 28 364 Carbon monoxide CO 28 376 Ethylene C2H4 28 390 Nitric oxide NO 30 317 Oxygen O2 32 346 Hydrogen sulfide H2S 34 360 Hydrogen chloride HCl 36 320 Argon Ar 40 340 Propylene C3H6 42 450 Carbon dioxide CO2 44 330 Nitrous oxide N2O 44 330 Propane C3H8 44 430 Sulfur dioxide SO2 64 360 Chlorine Cl2 70 320 Benzene C6H6 78 585 Li & Talu, p. 373 Hydrogen bromide HBr 81 350 Krypton Kr 84 360 Xenon Xe 131 396 Sulfur hexafluoride SF6 146 550 Carbon tetrachloride CCl4 154 590 Bromine Br2 160 350 ==Dissimilar particles== Collisions between two dissimilar particles occur when a beam of fast particles is fired into a gas consisting of another type of particle, or two dissimilar molecules randomly collide in a gas mixture. For particles of different size, more elaborate expressions can be derived for estimating u. ==References== Category:Chemical kinetics thumb|A Zeeman slower before its incorporation into a larger cold atom experiment. The average acceleration (due to many photon absorption events over time) of an atom with mass, M, a cycling transition with frequency, \omega=ck+\delta, and linewidth, \gamma, that is in the presence of a laser beam that has wavenumber, k, and intensity I=s_{0}I_{s} (where I_s=\hbar c \gamma k^{3}/12\pi is the saturation intensity of the laser) is :: \vec{a}=\frac{\hbar\vec{k}\gamma}{2M}\frac{s_{0}}{1+s_{0}+\left(2\delta'/\gamma\right)^2} In the rest frame of the atoms with velocity, v, in the atomic beam, the frequency of the laser beam is shifted by k_{L}v. The SSERVI - Impact Dust Accelerator Facility at the University of Colorado, 47th Lunar and Planetary Science Conference (2016), accessed May 30, 2017 140,000 540,000 313,170 0.00047 Approaching velocity of Messier 98 to our galaxy. 192,000 690,000 430,000 0.00064 Predicted top speed of the Parker Solar Probe at its closest perihelion in 2024. 200,000 700,000 450,000 0.00070 Orbital speed of the Solar System in the Milky Way galaxy. 308,571 1,080,000 694,288 0.001 Approaching velocity of Andromeda Galaxy to our galaxy. 440,000 1,600,000 980,000 0.0015 Typical speed of the stepped leader of lightning (cf. return stroke below). 445,000 1,600,000 995,000 0.0015 Max velocity of the remaining shell (mass about 0.1 mg) of an inertial confinement fusion capsule driven by the National Ignition Facility for the 'Bigfoot' capsule campaign. Thus it aims at a final velocity of about 10 m/s (depending on the atom used), starting with a beam of atoms with a velocity of a few hundred meters per second. For a fast moving particle (that is, one moving much faster than the particles it is moving through) the kinetic diameter is given by,Ismail et al., p. 14 :d^2 = {1 \over \pi l n} :where, :d is the kinetic diameter, :r is the kinetic radius, r = d/2, :l is the mean free path, and :n is the number density of particles However, a more usual situation is that the colliding particle being considered is indistinguishable from the population of particles in general. It is dimensionally equivalent to the momentum unit kilogram-metre per second (kg⋅m/s). One newton-second corresponds to a one-newton force applied for one second. :\vec F \cdot t = \Delta m \vec v It can be used to identify the resultant velocity of a mass if a force accelerates the mass for a specific time interval. ==Definition== Momentum is given by the formula: :\mathbf{p} = m \mathbf{v}, * \mathbf{p} is the momentum in newton-seconds (N⋅s) or ""kilogram-metres per second"" (kg⋅m/s) * m is the mass in kilograms (kg) * \mathbf{v} is the velocity in metres per second (m/s) ==Examples== This table gives the magnitudes of some momenta for various masses and speeds. The kinetic diameter is not the same as atomic diameter defined in terms of the size of the atom's electron shell, which is generally a lot smaller, depending on the exact definition used. Abbreviations include in/s, in/sec, ips, and less frequently in s−1. ==Conversions== 1 inch per second is equivalent to: : = 0.0254 metres per second (exactly) : = or 0.083 feet per second (exactly) : = or 0.05681 miles per hour (exactly) : = 0.09144 km·h−1 (exactly) 1 metre per second ≈ 39.370079 inches per second (approximately) 1 foot per second = 12 inches per second (exactly) 1 mile per hour = 17.6 inches per second (exactly) 1 kilometre per hour ≈ 10.936133 inches per second (approximately) ==Uses== In magnetic tape sound recording, magnetic tape speed is often quoted in inches per second (abbreviated ""ips""). ",+4.1,4500,"""41.4""",0,0.132,E
 "Calculate the pressure exerted by Ar for a molar volume of $1.31 \mathrm{~L} \mathrm{~mol}^{-1}$ at $426 \mathrm{~K}$ using the van der Waals equation of state. The van der Waals parameters $a$ and $b$ for Ar are 1.355 bar dm${ }^6 \mathrm{~mol}^{-2}$ and $0.0320 \mathrm{dm}^3 \mathrm{~mol}^{-1}$, respectively. Is the attractive or repulsive portion of the potential dominant under these conditions?
-","The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). In Van der Waals' original derivation, given below, b' is four times the proper volume of the particle. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). For example, the pairwise attractive van der Waals interaction energy between H atoms in different H2 molecules equals 0.06 kJ/mol (0.6 meV) and the pairwise attractive interaction energy between O atoms in different O2 molecules equals 0.44 kJ/mol (4.6 meV). Accordingly, van der Waals forces can range from weak to strong interactions, and support integral structural loads when multitudes of such interactions are present. The main characteristics of van der Waals forces are: * They are weaker than normal covalent and ionic bonds. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The molar van der Waals volume should not be confused with the molar volume of the substance. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. * Van der Waals forces are additive and cannot be saturated. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. ",26.9,144,4.68,14,840,A
-"For water, $\Delta H_{\text {vaporization }}$ is $40.656 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and the normal boiling point is $373.12 \mathrm{~K}$. Calculate the boiling point for water on the top of Mt. Everest (elevation $8848 \mathrm{~m}$ ), where the barometric pressure is 253 Torr.","For comparison, on top of Mount Everest, at elevation, the pressure is about and the boiling point of water is . At higher elevations, where the atmospheric pressure is much lower, the boiling point is also lower. For every increase in elevation, water's boiling point is lowered by approximately 0.5 °C. Because of this, water boils at under standard pressure at sea level, but at at altitude. At in elevation, water boils at just . Charles Darwin commented on this phenomenon in The Voyage of the Beagle:Journal and remarks, Chapter XV, March 21, 1835 by Charles Darwin. ==Boiling point of pure water at elevated altitudes== Based on standard sea- level atmospheric pressure (courtesy, NOAA): Altitude, ft (m) Boiling point of water, °F (°C) 0 (0 m) 212°F (100°C) 500 (150 m) 211.1°F (99.5°C) 1,000 (305 m) 210.2°F (99°C) 2,000 (610 m) 208.4°F (98°C) 5,000 (1,524 m) 203°F (95°C) 6,000 (1,829 m) 201.1°F (94°C) 8,000 (2,438 m) 197.4°F (91.9°C) 10,000 (3,048 m) 193.6°F (89.8°C) 12,000 (3,658 m) 189.8°F (87.6°C) 14,000 (4,267 m) 185.9°F (85.5°C) 15,000 (4,572 m) 184.1°F (84.5°C) Source: NASA. ==References== ==External links== *Is it true that you can't make a decent cup of tea up a mountain? physics.org, accessed 2012-11-02 Category:Cooking techniques The normal boiling point (also called the atmospheric boiling point or the atmospheric pressure boiling point) of a liquid is the special case in which the vapor pressure of the liquid equals the defined atmospheric pressure at sea level, one atmosphere.General Chemistry Glossary Purdue University website page At that temperature, the vapor pressure of the liquid becomes sufficient to overcome atmospheric pressure and allow bubbles of vapor to form inside the bulk of the liquid. If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. At elevated altitudes, any cooking that involves boiling or steaming generally requires compensation for lower temperatures because the boiling point of water is lower at higher altitudes due to the decreased atmospheric pressure. Mount Everest is the world's highest mountain, with a peak at 8,849 metres (29,031.7 ft) above sea level. The boiling point corresponds to the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure. There are two conventions regarding the standard boiling point of water: The normal boiling point is at a pressure of 1 atm (i.e., 101.325 kPa). By comparison, reasonable base elevations for Everest range from on the south side to on the Tibetan Plateau, yielding a height above base in the range of .Mount Everest (1:50,000 scale map), prepared under the direction of Bradford Washburn for the Boston Museum of Science, the Swiss Foundation for Alpine Research, and the National Geographic Society, 1991, . The air pressure at the summit is generally about one-third what it is at sea level. The primary peak of Mount Everest is elevation above sea level. == Overview == The peak is a dome-shaped peak of snow and ice, and is connected to the summit of Mount Everest by the Cornice Traverse and Hillary Step, approximately from the higher peak. It also has the lowest normal boiling point (−24.2 °C), which is where the vapor pressure curve of methyl chloride (the blue line) intersects the horizontal pressure line of one atmosphere (atm) of absolute vapor pressure. In the expedition, the summit's altitude was measured as 8848.13 metres. * 1975 British Mount Everest Southwest Face expedition - On September 24, a British expedition led by Chris Bonington achieved the first ascent of the Southwest Face. The atmospheric pressure at the top of Everest is about a third of sea level pressure or , resulting in the availability of only about a third as much oxygen to breathe. The standard boiling point has been defined by IUPAC since 1982 as the temperature at which boiling occurs under a pressure of one bar. The boiling point of a liquid varies depending upon the surrounding environmental pressure. Boiling points may be published with respect to the NIST, USA standard pressure of 101.325 kPa (or 1 atm), or the IUPAC standard pressure of 100.000 kPa. Towards the end of the season, due to a stalled high-pressure system, conditions on Everest were better than usual, being warmer, drier, and less windy, facilitating a higher-than-usual summitting success rate of 70%. ", 7.0,0,273.0,9,344,E
-"An ideal solution is formed by mixing liquids $\mathrm{A}$ and $B$ at $298 \mathrm{~K}$. The vapor pressure of pure A is 151 Torr and that of pure B is 84.3 Torr. If the mole fraction of $\mathrm{A}$ in the vapor is 0.610 , what is the mole fraction of $\mathrm{A}$ in the solution?","The boiling temperature of water for atmospheric pressures can be approximated by the Antoine equation: :\log_{10}\left(\frac{P}{1\text{ Torr}}\right) = 8.07131 - \frac{1730.63\ {}^\circ\text{C}}{233.426\ {}^\circ\text{C} + T_b} or transformed into this temperature-explicit form: :T_b = \frac{1730.63\ {}^\circ\text{C}}{8.07131 - \log_{10}\left(\frac{P}{1\text{ Torr}}\right)} - 233.426\ {}^\circ\text{C} where the temperature T_b is the boiling point in degrees Celsius and the pressure P is in torr. ==Dühring's rule== Dühring's rule states that a linear relationship exists between the temperatures at which two solutions exert the same vapor pressure. ==Examples== The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units). The mole fraction is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and volume fraction (percentage by volume, vol%) are others. ==Properties== Mole fraction is used very frequently in the construction of phase diagrams. When Raoult's law and Dalton's law hold for the mixture, the K factor is defined as the ratio of the vapor pressure to the total pressure of the system: :K_i = \frac{P'_i}{P} Given either of x_i or y_i and either the temperature or pressure of a two-component system, calculations can be performed to determine the unknown information. ==References== ==See also== * Phase diagram * Azeotrope * Dew point Category:Temperature Category:Phase transitions Category:Gases The mole fraction is also called the amount fraction. It states that the activity (pressure or fugacity) of a single-phase mixture is equal to the mole-fraction-weighted sum of the components' vapor pressures: : P_{\rm tot} =\sum_i P y_i = \sum_i P_i^{\rm sat} x_i \, where P_{\rm tot} is the mixture's vapor pressure, x_i is the mole fraction of component i in the liquid phase and y_i is the mole fraction of component i in the vapor phase respectively. At the normal boiling point of a liquid, the vapor pressure is equal to the standard atmospheric pressure defined as 1 atmosphere, 760Torr, 101.325kPa, or 14.69595psi. If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. This is important for volatile inhalational anesthetics, most of which are liquids at body temperature, but with a relatively high vapor pressure. ==Estimating vapor pressures with Antoine equation== The Antoine equationWhat is the Antoine Equation? Whereas mole fraction is a ratio of moles to moles, molar concentration is a quotient of moles to volume. In chemistry, the mole fraction or molar fraction (xi or ) is defined as unit of the amount of a constituent (expressed in moles), ni, divided by the total amount of all constituents in a mixture (also expressed in moles), ntot. The vapor pressure of a liquid at its boiling point equals the pressure of its surrounding environment. ==Liquid mixtures: Raoult's law== Raoult's law gives an approximation to the vapor pressure of mixtures of liquids. The basic form of the equation is: :\log P = A-\frac{B}{C+T} and it can be transformed into this temperature-explicit form: :T = \frac{B}{A-\log P} - C where: * P is the absolute vapor pressure of a substance * T is the temperature of the substance * A, B and C are substance-specific coefficients (i.e., constants or parameters) * \log is typically either \log_{10} or \log_e A simpler form of the equation with only two coefficients is sometimes used: :\log P = A- \frac{B}{T} which can be transformed to: :T = \frac{B}{A-\log P} Sublimations and vaporizations of the same substance have separate sets of Antoine coefficients, as do components in mixtures. When the volatilities of both key components are equal, \alpha = 1 and separation of the two by distillation would be impossible under the given conditions because the compositions of the liquid and the vapor phase are the same (azeotrope). This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., ""Antoine Vapor Pressure Constants of Pure Compounds"" ** Yaws C. L., Yang H.-C., ""To Estimate Vapor Pressure Easily. The second solution is switching to another vapor pressure equation with more than three parameters. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds"", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. Experimental measurement of vapor pressure is a simple procedure for common pressures between 1 and 200 kPa. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. Actually, as stated by Dalton's law (known since 1802), the partial pressure of water vapor or any substance does not depend on air at all, and the relevant temperature is that of the liquid. thumb|300px|Boiling waterThe boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. ",0.28209479,0.466,7.0,6.3,1.51,B
-"The mean solar flux at Earth's surface is $\sim 2.00 \mathrm{~J}$ $\mathrm{cm}^{-2} \mathrm{~min}^{-1}$. In a nonfocusing solar collector, the temperature reaches a value of $79.5^{\circ} \mathrm{C}$. A heat engine is operated using the collector as the hot reservoir and a cold reservoir at $298 \mathrm{~K}$. Calculate the area of the collector needed to produce 1000. W. Assume that the engine operates at the maximum Carnot efficiency.","Solar energy – Solar thermal collectors – Test methods International Organization for Standardization, Geneva, Switzerland states that the efficiency of solar thermal collectors should be measured in terms of gross area and this might favour flat plates in respect to evacuated tube collectors in direct comparisons. thumb|An array of evacuated flat plate collectors next to compact solar concentrators thumb|A comparison of the energy output (kW.h/day) of a flat plate collector (blue lines; Thermodynamics S42-P; absorber 2.8 m2) and an evacuated tube collector (green lines; SunMaxx 20EVT; absorber 3.1 m2. In non-concentrating collectors, the aperture area (i.e., the area that receives the solar radiation) is roughly the same as the absorber area (i.e., the area absorbing the radiation). A solar thermal collector collects heat by absorbing sunlight. Transpired solar collectors are usually wall- mounted to capture the lower sun angle in the winter heating months as well as sun reflection off the snow and achieve their optimum performance and return on investment when operating at flow rates of between 4 and 8 CFM per square foot (72 to 144 m3/h.m2) of collector area. The extensive monitoring by Natural Resources Canada and NREL has shown that transpired solar collector systems reduce between 10-50% of the conventional heating load and that RETScreen is an accurate predictor of system performance. Its value is about 86%, which is the Chambadal-Novikov efficiency, an approximation related to the Carnot limit, based on the temperature of the photons emitted by the Sun's surface. == Effect of band gap energy == Solar cells operate as quantum energy conversion devices, and are therefore subject to the thermodynamic efficiency limit. This value depends on the size of the storage unit (hot water tank or storage battery), the size of the harvesting surface (sun collection surface or surface area of photovoltaic modules), and on the amount of energy required. The collector absorbs the incoming solar radiation, converting it into thermal energy. The solar energy flux (irradiance) incident on the Earth's surface has a variable and relatively low surface density, usually not exceeding 1100 W/m² without concentration systems. For a solar cell powered by the Sun's unconcentrated black-body radiation, the theoretical maximum efficiency is 43% whereas for a solar cell powered by the Sun's full concentrated radiation, the efficiency limit is up to 85%. Solar collector may refer to: * Solar thermal collector, a solar collector that collects heat by absorbing sunlight * Solar Collector (sculpture), a 2008 interactive light art installation in Cambridge, Ontario, Canada ==See also== *Concentrating solar power *Renewable heat *Solar air heating *Solar water heating *Solar panel They offer the highest energy conversion efficiency of any non-concentrating solar thermal collector, but require sophisticated technology for manufacturing. The exterior surface of a transpired solar collector consists of thousands of tiny micro-perforations that allow the boundary layer of heat to be captured and uniformly drawn into an air cavity behind the exterior panels. The internal combustion engine efficiency is determined by its two temperature reservoirs, the temperatures of ambient air and its upper limit operating temperature. Engine efficiency of thermal engines is the relationship between the total energy contained in the fuel, and the amount of energy used to perform useful work. In locations with average available solar energy, flat plate collectors are sized approximately 1.2 to 2.4 square decimeter per liter of one day's hot water use. === Applications === The main use of this technology is in residential buildings where the demand for hot water has a large impact on energy bills. In 1954 the solar constant was evaluated as 2.00 cal/min/cm2 ± 2%. The term ""solar collector"" commonly refers to a device for solar hot water heating, but may refer to large power generating installations such as solar parabolic troughs and solar towers or non water heating devices such as solar cooker, solar air heaters. With the increasing drive to install renewable energy systems on buildings, transpired solar collectors are now used across the entire building stock because of high energy production (up to 750 peak thermal Watts/square metre), high solar conversion (up to 90%) and lower capital costs when compared against solar photovoltaic and solar water heating. thumb|Solar air heating is a solar thermal technology in which the energy from the sun, solar insolation, is captured by an absorbing medium and used to heat air. Thermodynamic efficiency limit is the absolute maximum theoretically possible conversion efficiency of sunlight to electricity. Solar thermal collectors are either non-concentrating or concentrating. The solar coverage rate is the percentage of an amount of energy that is provided by the sun. ",4.16,19.4,-2.0,0.72,344,B
-A hiker caught in a thunderstorm loses heat when her clothing becomes wet. She is packing emergency rations that if completely metabolized will release $35 \mathrm{~kJ}$ of heat per gram of rations consumed. How much rations must the hiker consume to avoid a reduction in body temperature of $2.5 \mathrm{~K}$ as a result of heat loss? Assume the heat capacity of the body equals that of water and that the hiker weighs $51 \mathrm{~kg}$.,"The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion International Journal of Heat and Mass Transfer is a peer-reviewed scientific journal in the field of heat transfer and mass transfer, published by Elsevier. Heat and Mass Transfer is a peer-reviewed scientific journal published by Springer. Heatwork is the combined effect of temperature and time. Dietary energy supply is correlated with the rate of obesity. ==Regions== Daily dietary energy supply per capita: Region 1964-1966 1974-1976 1984-1986 1997-1999 World Sub-Saharan Africa ==Countries== Country 1979-1981 1989-1991 2001-2003 Canada China Eswatini United Kingdom United States ==See also== *Diet and obesity thumb|300x300px|Average dietary energy supply by region ==References== ==External links== * Category:Obesity Category:Diets *Temperature equivalents table & description of Orton pyrometric cones. As of 1995 the title Wärme- und Stoffübertragung was changed to Heat and Mass Transfer. == Indexing == Among others, the journal is indexed in Google Scholar, INIS Atomindex, Journal Citation Reports/Science Edition, OCLC, PASCAL, Science Citation Index, Science Citation Index Expanded (SciSearch) and Scopus. == External links == *Heat and Mass Transfer Category:Energy and fuel journals Category:Engineering journals Category:English-language journals Category:Monthly journals Category:Springer Science+Business Media academic journals Heatwork is taught in material science courses, but is not a precise measurement or a valid scientific concept. == External links == *Temperature equivalents table & description of Bullers Rings. *Temperature Equivalents, °F & °C for Bullers Ring. It gives an overestimate of the total amount of food consumed as it reflects both food consumed and food wasted. It serves the circulation of new developments in the field of basic research of heat and mass transfer phenomena, as well as related material properties and their measurements. *Temperature equivalents table & description of Nimra Cerglass pyrometric cones. Within tolerances, firing can be undertaken at lower temperatures for a longer period to achieve comparable results. *Temperature equivalents table of Seger pyrometric cones. When the amount of heatwork of two firings is the same, the pieces may look identical, but there may be differences not visible, such as mechanical strength and microstructure. It is important to several industries: *Ceramics *Glass and metal annealing *Metal heat treating Pyrometric devices can be used to gauge heat work as they deform or contract due to heatwork to produce temperature equivalents. Category:Glass physics Category:Pottery Category:Metallurgy Category:Ceramic engineering The editor-in-chief is T. S. Zhao (Hong Kong University of Science and Technology). ==Abstracting and indexing== The journal is abstracted and indexed in: According to the Journal Citation Reports, the journal has a 2020 impact factor of 5.584. ==References== ==External links== * Category:Physics journals Category:English-language journals Category:Engineering journals Category:Elsevier academic journals Category:Academic journals established in 1960 Category:Monthly journals Thereby applications to engineering problems are promoted. The journal publishes original research reports. It varies markedly between different regions and countries of the world. ",7.00,7200,-88.0,15,4.3,D
-Calculate the degree of dissociation of $\mathrm{N}_2 \mathrm{O}_4$ in the reaction $\mathrm{N}_2 \mathrm{O}_4(g) \rightleftharpoons 2 \mathrm{NO}_2(g)$ at 300 . $\mathrm{K}$ and a total pressure of 1.50 bar. Do you expect the degree of dissociation to increase or decrease as the temperature is increased to 550 . K? Assume that $\Delta H_R^{\circ}$ is independent of temperature.,"Figure 8 shows the corresponding enthalpy drop for the reaction = 0 case. none|310px|thumb|Figure 8. Stage enthalpy diagram for degree of reaction = 1⁄2 in a turbine and pump. left|thumb|Figure 6. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Hence from Tds = dh - \frac{dp}{\rho}, 400px|alt=enthalpy diagram|thumb|Figure 1. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. An algorithm for computing a 4/3-approximation of the dissociation number in bipartite graphs was published in 2022. The dissociation constant is the inverse of the association constant. From the relation for degree of reaction, || α2 > β3. right|380px|thumb|Figure 7. The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity. In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. thumb|Examples for the definition of the dissociation number In the mathematical discipline of graph theory, a subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. [B^{-3}] \over [H B^{-2}]} & \mathrm{p}K_3 &= -\log K_3 \end{align} ==Dissociation constant of water== The dissociation constant of water is denoted Kw: :K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-] The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration. The degree of reaction now depends only on ϕ and \tan{\beta_m} which again depend on geometrical parameters β3 and β2 i.e. the vane angles of stator outlet and rotor outlet. (The symbol K_a, used for the acid dissociation constant, can lead to confusion with the association constant and it may be necessary to see the reaction or the equilibrium expression to know which is meant.) The value of Kw varies with temperature, as shown in the table below. From the relation for degree of reaction,|| α2 < β3 which is also shown in corresponding Figure 7. === Reaction = zero === This is special case used for impulse turbine which suggest that entire pressure drop in the turbine is obtained in the stator. The problem of computing diss(G) (dissociation number problem) was firstly studied by Yannakakis. : Water temperature Kw pKw 0 °C 0.112 14.95 25 °C 1.023 13.99 50 °C 5.495 13.26 75 °C 19.95 12.70 100 °C 56.23 12.25 == See also == * Acid * Equilibrium constant * Ki Database * Competitive inhibition * pH * Scatchard plot * Ligand binding * Avidity ==References== Category:Equilibrium chemistry Category:Enzyme kinetics A molecule can have several acid dissociation constants. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. The number of vertices in a maximum cardinality dissociation set in G is called the dissociation number of G, denoted by diss(G). ",1.5,1.56,0.44,0.241,0.5061,D
-Calculate $\Delta G$ for the isothermal expansion of $2.25 \mathrm{~mol}$ of an ideal gas at $325 \mathrm{~K}$ from an initial pressure of 12.0 bar to a final pressure of 2.5 bar.,"For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work given by W = -\int_{2V_0}^{V_0} P\,\mathrm{d}V = - \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V = nRT\ln 2 = T \Delta S_\text{gas}. This type of expansion is named after James Prescott Joule who used this expansion, in 1845, in his study for the mechanical equivalent of heat, but this expansion was known long before Joule e.g. by John Leslie, in the beginning of the 19th century, and studied by Joseph-Louis Gay-Lussac in 1807 with similar results as obtained by Joule.D.S.L. Cardwell, From Watt to Clausius, Heinemann, London (1957)M.J. Klein, Principles of the theory of heat, D. Reidel Pub.Cy., Dordrecht (1986) The Joule expansion should not be confused with the Joule–Thomson expansion or throttling process which refers to the steady flow of a gas from a region of higher pressure to one of lower pressure via a valve or porous plug. ==Description== The process begins with gas under some pressure, P_{\mathrm{i}}, at temperature T_{\mathrm{i}}, confined to one half of a thermally isolated container (see the top part of the drawing at the beginning of this article). We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. Heating the gas up to the initial temperature increases the entropy by the amount \Delta S = n \int_{T}^{T_i} C_\mathrm{V} \frac{\mathrm{d}T'}{T'} = nR \ln 2. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. ===Ideal gases=== If the gas is ideal, both the initial (T_{\mathrm{i}}, P_{\mathrm{i}}, V_{\mathrm{i}}) and final (T_{\mathrm{f}}, P_{\mathrm{f}}, V_{\mathrm{f}}) conditions follow the Ideal Gas Law, so that initially P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}} and then, after the tap is opened, P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}. For the above defined path we have that and thus , and hence the increase in entropy for the Joule expansion is \Delta S=\int_i^f\mathrm{d}S=\int_{V_0}^{2V_0} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2. For a monatomic ideal gas , with the molar heat capacity at constant volume. thumb|240px|The Joule expansion, in which a volume is expanded to a volume in a thermally isolated chamber. thumb|180px|A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas. The Joule expansion, treated as a thought experiment involving ideal gases, is a useful exercise in classical thermodynamics. Here n is the number of moles of gas and R is the molar ideal gas constant. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. For other gases this ""Joule inversion temperature"" appears to be extremely high. ==Entropy production== Entropy is a function of state, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. thumb|400px|Diagram showing pressure difference induced by a temperature difference. After thermal equilibrium is reached, we then let the gas undergo another free expansion by and wait until thermal equilibrium is reached. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The Joule expansion (also called free expansion) is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container (via a small partition), with the other side of the container being evacuated. The fact that the temperature does not change makes it easy to compute the change in entropy of the universe for this process. ===Real gases=== Unlike ideal gases, the temperature of a real gas will change during a Joule expansion. ",226,0.333333,0.4908,-9.54,260,D
-Determine the total collisional frequency for $\mathrm{CO}_2$ at $1 \mathrm{~atm}$ and $298 \mathrm{~K}$.,"Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:chem.libretexts.org: Collision Frequency : Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, which has units of [volume][time]−1. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text{Coefficient of restitution } (e) = \frac{\left | \text{Relative velocity after collision} \right |}{\left | \text{Relative velocity before collision}\right |} The mathematics were developed by Sir Isaac Newton in 1687. The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. Setting the above estimate for \Delta m_e v_\perp equal to mv, we find the lower cut-off to the impact parameter to be about :b_0 = \frac{Ze^2}{4\pi\epsilon_0} \, \frac{1}{m_e v^2} We can also use \pi b_0^2 as an estimate of the cross section for large-angle collisions. If a given object collides with two different objects, each collision would have its own COR. Escande DF, Elskens Y, Doveil F (2015) Uniform derivation of Coulomb collisional transport thanks to Debye shielding. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material. Collision-induced emission and absorption by simultaneous collisions of three or more particles generally do involve pairwise-additive dipole components, as well as important irreducible dipole contributions and their spectra. == Historical sketch == Collision-induced absorption was first reported in compressed oxygen gas in 1949 by Harry Welsch and associates at frequencies of the fundamental band of the O2 molecule. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. Collision-induced spectra appear at the frequencies of the rotovibrational and electronic transition bands of the unperturbed molecules, and also at sums and differences of such transition frequencies: simultaneous transitions in two (or more) interacting molecules are well known to generate optical transitions of molecular complexes. == Virial expansions of spectral intensities == Intensities of spectra of individual atoms or molecules typically vary linearly with the numerical gas density. It is possible that e = \infty for a perfect explosion of a rigid system. === Paired objects === The COR is a property of a pair of objects in a collision, not a single object. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. (Note that an unperturbed O2 molecule, like all other diatomic homonuclear molecules, is infrared inactive on account of the inversion symmetry and does thus not possess a ""dipole allowed"" rotovibrational spectrum at any frequency). == Collision-induced spectra == Molecular fly-by collisions take little time, something like 10−13 s. The upper cut-off to the impact parameter should thus be approximately equal to the Debye length: : \lambda_D = \sqrt{\frac{\epsilon_0 kT_e}{n_e e^2}} == Coulomb logarithm == The integral of 1/b thus yields the logarithm of the ratio of the upper and lower cut-offs. Within the same approximations, a more elegant derivation of the collisional transport coefficients was provided, by using the Balescu–Lenard equation (see Sec. 8.4 of Balescu, R. 1997 Statistical Dynamics: Matter Out of Equilibrium. These are the collision-induced spectra of two-body (and quite possibly three-body,...) collisional complexes. The collision-induced spectra have sometimes been separated from the continua of individual atoms and molecules, based on the characteristic density dependences. Also, some recent articles have described superelastic collisions in which it is argued that the COR can take a value greater than one in a special case of oblique collisions. ",8.44,-6.9,-0.38,4.85,5.828427125,A
-"The volatile liquids $A$ and $\mathrm{B}$, for which $P_A^*=165$ Torr and $P_B^*=85.1$ Torr are confined to a piston and cylinder assembly. Initially, only the liquid phase is present. As the pressure is reduced, the first vapor is observed at a total pressure of 110 . Torr. Calculate $x_{\mathrm{A}}$","If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. thumb|300px|A representative pressure–volume diagram for a refrigeration cycle Vapour-compression refrigeration or vapor-compression refrigeration system (VCRS), in which the refrigerant undergoes phase changes, is one of the many refrigeration cycles and is the most widely used method for air conditioning of buildings and automobiles. Between point 3 and point 4, the vapor travels through the remainder of the condenser and is condensed into a high temperature, high pressure subcooled liquid. When the volatilities of both key components are equal, \alpha = 1 and separation of the two by distillation would be impossible under the given conditions because the compositions of the liquid and the vapor phase are the same (azeotrope). thumb|right|300px|Otto cycle pressure–volume diagram An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. The presence of other volatile components in a mixture affects the vapor pressures and thus boiling points and dew points of all the components in the mixture. Relative volatilities are not used in separation or absorption processes that involve components reacting with each other (for example, the absorption of gaseous carbon dioxide in aqueous solutions of sodium hydroxide). ==Definition== For a liquid mixture of two components (called a binary mixture) at a given temperature and pressure, the relative volatility is defined as :\alpha=\frac {(y_i/x_i)}{(y_j/x_j)} = K_i/K_j where: \alpha = the relative volatility of the more volatile component i to the less volatile component j y_i = the vapor–liquid equilibrium mole fraction of component i in the vapor phase x_i = the vapor–liquid equilibrium mole fraction of component i in the liquid phase y_j = the vapor–liquid equilibrium concentration of component j in the vapor phase x_j = the vapor–liquid equilibrium concentration of component j in the liquid phase (y/x) = Henry's law constant (also called the K value or vapor-liquid distribution ratio) of a component When their liquid concentrations are equal, more volatile components have higher vapor pressures than less volatile components. As the piston is capable of moving along the cylinder, the volume of the air changes with its position in the cylinder. thumb|250px|Diagram of cylinder and piston valve. If the pressure in a system remains constant (isobaric), a vapor at saturation temperature will begin to condense into its liquid phase as thermal energy (heat) is removed. That pressure reduction results in the adiabatic flash evaporation of a part of the liquid refrigerant. thumb|300px|Boiling waterThe boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. Relative volatility is a measure comparing the vapor pressures of the components in a liquid mixture of chemicals. Similarly, a liquid at saturation pressure and temperature will tend to flash into its vapor phase as system pressure is decreased. Similarly, a liquid at saturation temperature and pressure will boil into its vapor phase as additional thermal energy is applied. For a given pressure, different liquids will boil at different temperatures. The system, in this case, is defined to be the fluid (gas) within the cylinder. Furthermore, at any given temperature, the composition of the vapor is different from the composition of the liquid in most such cases. The condensed liquid refrigerant, in the thermodynamic state known as a saturated liquid, is next routed through an expansion valve where it undergoes an abrupt reduction in pressure. The cold refrigerant liquid and vapor mixture is then routed through the coil or tubes in the evaporator. thumb|right|400px|An illustration of fluid simulation using VOF method. The heat of vaporization is the energy required to transform a given quantity (a mol, kg, pound, etc.) of a substance from a liquid into a gas at a given pressure (often atmospheric pressure). ",0.312,2.57,36.0,3.333333333,-8,A
-The osmotic pressure of an unknown substance is measured at $298 \mathrm{~K}$. Determine the molecular weight if the concentration of this substance is $31.2 \mathrm{~kg} \mathrm{~m}^{-3}$ and the osmotic pressure is $5.30 \times 10^4 \mathrm{~Pa}$. The density of the solution is $997 \mathrm{~kg} \mathrm{~m}^{-3}$.,"Both sodium and chloride ions affect the osmotic pressure of the solution. The osmotic coefficient based on molality m is defined by: \phi = \frac{\mu_A^* - \mu_A}{RTM_A \sum_i m_i} and on a mole fraction basis by: \phi = -\frac{\mu_A^* - \mu_A}{RT \ln x_A} where \mu_A^* is the chemical potential of the pure solvent and \mu_A is the chemical potential of the solvent in a solution, MA is its molar mass, xA its mole fraction, R the gas constant and T the temperature in Kelvin. The transfer of solvent molecules will continue until equilibrium is attained. ==Theory and measurement== Jacobus van 't Hoff found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation: :\Pi = icRT where \Pi is osmotic pressure, i is the dimensionless van 't Hoff index, c is the molar concentration of solute, R is the ideal gas constant, and T is the absolute temperature (usually in kelvins). This value allows the measurement of the osmotic pressure of a solution and the determination of how the solvent will diffuse across a semipermeable membrane (osmosis) separating two solutions of different osmotic concentration. == Unit == The unit of osmotic concentration is the osmole. Osmotic concentration, formerly known as osmolarity, is the measure of solute concentration, defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). This is a non-SI unit of measurement that defines the number of moles of solute that contribute to the osmotic pressure of a solution. The Pfeffer cell was developed for the measurement of osmotic pressure. == Applications == thumb|upright=1.15|Osmotic pressure on red blood cells Osmotic pressure measurement may be used for the determination of molecular weights. The proportionality to concentration means that osmotic pressure is a colligative property. In order to find \Pi, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water. :\mu_v(x_v,p+\Pi) = \mu_v^0(p). Potential osmotic pressure is the maximum osmotic pressure that could develop in a solution if it were separated from its pure solvent by a semipermeable membrane. For aqueous solutions, the osmotic coefficients can be calculated theoretically by Pitzer equationsI. The osmotic pressure of ocean water is approximately 27 atm. Reverse osmosis desalinates fresh water from ocean salt water. == Derivation of the van 't Hoff formula == Consider the system at the point when it has reached equilibrium. For example, a 3 Osm solution might consist of: 3 moles glucose, or 1.5 moles NaCl, or 1 mole glucose + 1 mole NaCl, or 2 moles glucose + 0.5 mole NaCl, or any other such combination. ==Definition== The osmolarity of a solution, given in osmoles per liter (osmol/L) is calculated from the following expression: \mathrm{osmolarity} = \sum_i \varphi_i \, n_i C_i where * is the osmotic coefficient, which accounts for the degree of non-ideality of the solution. Temperature(°F) Specific weight (lbf/ft3) 32 62.42 40 62.43 50 62.41 60 62.37 70 62.30 80 62.22 90 62.11 100 62.00 110 61.86 120 61.71 130 61.55 140 61.38 150 61.20 160 61.00 170 60.80 180 60.58 190 60.36 200 60.12 212 59.83 ==Specific weight of air== Specific weight of air at standard sea-level atmospheric pressure (Metric units) Temperature(°C) Specific weight (N/m3) −40 14.86 −20 13.86 0 12.68 10 12.24 20 11.82 30 11.43 40 11.06 60 10.4 80 9.81 100 9.28 200 7.33 Specific weight of air at standard sea-level atmospheric pressure (English units) Temperature(°F) Specific Weight (lbf/ft3) −40 −20 0.0903 0 0.08637 10 0.08453 20 0.08277 30 0.08108 40 0.07945 50 0.0779 60 0.0764 70 0.07495 80 0.07357 90 0.07223 100 0.07094 120 0.06849 140 0.0662 160 0.06407 180 0.06206 200 0.06018 250 0.05594 ==References== ==External links== * Submerged weight calculator * Specific weight calculator * http://www.engineeringtoolbox.com/density-specific-weight-gravity-d_290.html * http://www.themeter.net/pesi-spec_e.htm Category:Soil mechanics Category:Fluid mechanics Category:Physical chemistry Category:Physical quantities Category:Density For example, the intracellular fluid and extracellular can be hyperosmotic, but isotonic – if the total concentration of solutes in one compartment is different from that of the other, but one of the ions can cross the membrane (in other words, a penetrating solute), drawing water with it, thus causing no net change in solution volume. ==Plasma osmolarity vs. osmolality== Plasma osmolarity can be calculated from plasma osmolality by the following equation: where: * is the density of the solution in g/ml, which is 1.025 g/ml for blood plasma. * is the (anhydrous) solute concentration in g/ml – not to be confused with the density of dried plasma According to IUPAC, osmolality is the quotient of the negative natural logarithm of the rational activity of water and the molar mass of water, whereas osmolarity is the product of the osmolality and the mass density of water (also known as osmotic concentration). Here, the difference in pressure of the two compartments \Pi \equiv p' - p is defined as the osmotic pressure exerted by the solutes. Also, the molar volume V_m may be written as volume per mole, V_m = V/n_v. In soil mechanics, specific weight may refer to: ===Civil and mechanical engineering=== Specific weight can be used in civil engineering and mechanical engineering to determine the weight of a structure designed to carry certain loads while remaining intact and remaining within limits regarding deformation. ==Specific weight of water== Specific weight of water at standard sea-level atmospheric pressure (Metric units) Temperature(°C) Specific weight (kN/m3) 0 9.805 5 9.807 10 9.804 15 9.798 20 9.789 25 9.777 30 9.765 40 9.731 50 9.690 60 9.642 70 9.589 80 9.530 90 9.467 100 9.399 Specific weight of water at standard sea-level atmospheric pressure (English units) Finnemore, J. E. (2002). The compartment containing the pure solvent has a chemical potential of \mu^0(p), where p is the pressure. The molecular formula C27H34O3 (molar mass: 406.56 g/mol, exact mass: 406.2508 u) may refer to: * Nandrolone phenylpropionate (NPP), or nandrolone phenpropionate * Testosterone phenylacetate The specific weight, also known as the unit weight, is the weight per unit volume of a material. Osmotic pressure is the basis of filtering (""reverse osmosis""), a process commonly used in water purification. ",-233,0.2553,8.8,1.45,1.19,D
- One mole of Ar initially at 310 . K undergoes an adiabatic expansion against a pressure $P_{\text {external }}=0$ from a volume of $8.5 \mathrm{~L}$ to a volume of $82.0 \mathrm{~L}$. Calculate the final temperature using the ideal gas,"We assume the expansion occurs without exchange of heat (adiabatic expansion). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). With our present knowledge of the thermodynamic properties of air Refprop, software package developed by National Institute of Standards and Technology (NIST) we can calculate that the temperature of the air should drop by about 3 degrees Celsius when the volume is doubled under adiabatic conditions. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. In the constant volume adiabatic flame temperature case, the volume of the system is held constant and hence there is no work occurring: : {}_RW_P = \int\limits_R^P {pdV} = 0 There is also no heat transfer because the process is defined to be adiabatic: {}_RQ_P = 0 . The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_{0}, usually . During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to: : dh = v \, dp. For a monatomic ideal gas , with the molar heat capacity at constant volume. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. For an imperfect or non-ideal gas, Chandrasekhar defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure: : \begin{align} \Gamma_1 &= \left.\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}, \\\\[2pt] \frac{\Gamma_2 - 1}{\Gamma_2} &= \left.\frac{\partial \ln T}{\partial \ln P}\right|_{S}, \\\\[2pt] \Gamma_3 - 1 &= \left.\frac{\partial \ln T}{\partial \ln \rho}\right|_{S}. \end{align} All of these are equal to \gamma in the case of an ideal gas. == See also == * Relations between heat capacities * Heat capacity * Specific heat capacity * Speed of sound * Thermodynamic equations * Thermodynamics * Volumetric heat capacity == References == == Notes == Category:Thermodynamic properties Category:Physical quantities Category:Ratios Category:Thought experiments in physics There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as: T = T_i 2^{-R/C_V} = T_i2^{-2/3} for the monoatomic ideal gas. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. ",292,2.3,9.8,0,310,E
-"A refrigerator is operated by a $0.25-\mathrm{hp}(1 \mathrm{hp}=$ 746 watts) motor. If the interior is to be maintained at $4.50^{\circ} \mathrm{C}$ and the room temperature on a hot day is $38^{\circ} \mathrm{C}$, what is the maximum heat leak (in watts) that can be tolerated? Assume that the coefficient of performance is $50 . \%$ of the maximum theoretical value. What happens if the leak is greater than your calculated maximum value?","The basic SI units equation for deriving cooling capacity is of the form: :\dot{Q}=\dot{m}C_p\Delta T Where :\dot{Q} is the cooling capacity [kW] :\dot{m} is the mass rate [kg/s] :C_p is the specific heat capacity [kJ/kg K] :\Delta T is the temperature change [K] ==References== Category:Heating, ventilation, and air conditioning As the target temperature of the refrigerator approaches ambient temperature, without exceeding it, the refrigeration capacity increases thus increasing the refrigerator's COP. This is a table of specific heat capacities by magnitude. Cooling capacity is the measure of a cooling system's ability to remove heat. Modern cogeneration plants have power loss ratios of about 1/5 to 1/9 when delivering heat in the range of 80 °C-120 °C.Danny Harvey: Clean building - contribution from cogeneration, trigeneration and district energy, Cogeneration and On-Site Power Production, september–october 2006, pp. 107-115 (Fig. 1) That means in exchange of one kWh of electrical energy ca. 5 up to 9 kWh of useful heat are obtained. The power loss factor β describes the loss of electrical power in CHP systems with a variable power-to-heat ratio when an increasing heat flow is extracted from the main thermodynamic electricity generating process in order to provide useful heat. right|thumb|150px|Thermal mass refrigerator A thermal mass refrigerator is a refrigerator that is foreseen with thermal mass as well as insulation to decrease the energy use of the refrigerator. A refrigerator death is death by suffocation in a refrigerator or other air- tight appliance. A particularly popular thermal mass refrigerator was conceived by Michael Reynolds and detailed in the book ""Earthship Volume 3"". On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . This refrigerator was a DIY refrigerator designed around a (Sun-Frost) DC refrigeration unit run on P.V. panels.Earthship Volume 3 by Michael Reynolds ==Design details== The thermal mass used in Michael Reynolds' design is a combination of a liquid (i.e. water or beer) together with concrete mass. The partial steam flow, which goes into the heating condenser at high temperature can no longer work in the low-pressure section and is responsible for the loss of power. It is equivalent to the heat supplied to the evaporator/boiler part of the refrigeration cycle and may be called the ""rate of refrigeration"" or ""refrigeration capacity"". The Refrigerator Safety Act in 1956 was a U.S. law that required a change in the way refrigerator doors stay shut. The Refrigerator Safety Act was a factor in the decline, in combination with other factors such as ""reduced exposure and increased parental supervision"". ==Entrapment hazards== Hazardous items for refrigerator deaths are ""places with a poor air supply, a heavy lid or a self-latching door"". The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size- dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Parents or caregivers can lessen the risks of refrigerator deaths. Another unit common in non-metric regions or sectors is the ton of refrigeration, which describes the amount of water at freezing temperature that can be frozen in 24 hours, equivalent to or . In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. Usually, the power loss factor refers to extraction steam turbines in thermal power stations, which conduct a part of the steam in a heating condenser for the production of useful heat, instead of the low pressure part of the steam turbine where is could perform mechanical work. thumb|Power loss within an extraction steam turbine: CHP plant section (left) and T-s-diagram (right) \beta = \frac{\Delta P_\text{el}}{\dot Q_\text{utile}} The picture on the right shows in the left part the principle of steam extraction. Based on the equivalence of power loss and gain of heat, the power loss method assigns CO2 emissions and primary energy from the fuel to the useful heat and the electrical energy. == References == Category:Cogeneration Category:Energy conversion The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio. ==Derivation== If an infinitesimally small amount of heat \delta Q is supplied to a system in a reversible way then, according to the second law of thermodynamics, the entropy change of the system is given by: :dS = \frac{\delta Q}{T}\, Since :\delta Q = C dT\, where C is the heat capacity, it follows that: :T dS = C dT\, The heat capacity depends on how the external variables of the system are changed when the heat is supplied. ",-32,15.425,1.8763, 6.07,773,E
-"In order to get in shape for mountain climbing, an avid hiker with a mass of $60 . \mathrm{kg}$ ascends the stairs in the world's tallest structure, the $828 \mathrm{~m}$ tall Burj Khalifa in Dubai, United Arab Emirates. Assume that she eats energy bars on the way up and that her body is $25 \%$ efficient in converting the energy content of the bars into the work of climbing. How many energy bars does she have to eat if a single bar produces $1.08 \times 10^3 \mathrm{~kJ}$ of energy upon metabolizing?","Manufacturing of energy bars may supply nutrients in sufficient quantity to be used as meal replacements. ==Nutrition== A typical energy bar weighs between 30 and 50 g and is likely to supply about 200–300 Cal (840–1,300 kJ), 3–9 g of fat, 7–15 g of protein, and 20–40 g of carbohydrates — the three sources of energy in food. Energy bars are supplemental bars containing cereals, micronutrients, and flavor ingredients intended to supply quick food energy. Energy bars may be used as an energy source during athletic events such as marathons, triathlons and other activities which require a high energy expenditure for long periods of time. Because most energy bars contain added protein, carbohydrates, dietary fiber, and other nutrients, they may be marketed as functional foods. YouBar is an online nutrition bar company that makes customized energy bars. The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. thumb|Energy bars vary in size, ingredients, and nutritional benefits. CalorieMate (カロリーメイト karorīmeito) is a brand of nutritional energy bar and energy gel foods produced by Otsuka Pharmaceutical Co., in Japan. Physical Activity MET Light Intensity Activities < 3 sleeping 0.9 watching television 1.0 writing, desk work, typing 1.8 walking, 1.7 mph (2.7 km/h), level ground, strolling, very slow 2.3 walking, 2.5 mph (4 km/h) 2.9 Moderate Intensity Activities 3 to 6 bicycling, stationary, 50 watts, very light effort 3.0 walking 3.0 mph (4.8 km/h) 3.3 calisthenics, home exercise, light or moderate effort, general 3.5 walking 3.4 mph (5.5 km/h) 3.6 bicycling, <10 mph (16 km/h), leisure, to work or for pleasure 4.0 bicycling, stationary, 100 watts, light effort 5.5 Vigorous Intensity Activities > 6 jogging, general 7.0 calisthenics (e.g. pushups, situps, pullups, jumping jacks), heavy, vigorous effort 8.0 running jogging, in place 8.0 rope jumping 10.0 ==Fuel Used== The body uses different amounts of energy substrates (carbohydrates or fats) depending on the intensity of the exercise and the heart rate of the exerciser. One MET, which is equal to 3.5 mL/kg per minute, is considered to be the average resting energy expenditure of a typical human being. High intensity activity also yields a higher total caloric expenditure. For those who are malnourished, energy bars, such as Plumpy'nut, are an effective tool for treating malnutrition. == See also == * Candy bar * Protein bar * Energy gel * Sports drink * High energy biscuits * Flapjack (oat bar) * D ration ==References== Category:Dietary supplements Category:Energy food products Category:Snack foods Dietary energy supply is correlated with the rate of obesity. ==Regions== Daily dietary energy supply per capita: Region 1964-1966 1974-1976 1984-1986 1997-1999 World Sub-Saharan Africa ==Countries== Country 1979-1981 1989-1991 2001-2003 Canada China Eswatini United Kingdom United States ==See also== *Diet and obesity thumb|300x300px|Average dietary energy supply by region ==References== ==External links== * Category:Obesity Category:Diets An intensity of exercise equivalent to 6 METs means that the energy expenditure of the exercise is six times the resting energy expenditure.Vehrs, P., Ph.D. (2011). Intensity of exercise can be expressed as multiples of resting energy expenditure. On the other hand, high intensity activity utilizes a larger percentage of carbohydrates in the calories expended because its quick production of energy makes it the preferred energy substrate for high intensity exercise. Intensity (%MHR) Heart Rate (bpm) % Carbohydrate % Fat 65-70 130-140 15 85 70-75 140-150 35 65 75-80 150-160 65 35 80-85 160-170 80 20 85-90 170-180 90 10 90-95 180-190 95 5 100 190-200 100 - These estimates are valid only when glycogen reserves are able to cover the energy needs. Fats sources are often cocoa butter and dark chocolate. == Usage == Energy bars are used in a variety of contexts. right|300px Exercise intensity refers to how much energy is expended when exercising. Protein is a third energy substrate, but it contributes minimally and is therefore discounted in the percent contribution graphs reflecting different intensities of exercise. This table outlines the estimated distribution of energy consumption at different intensity levels for a healthy 20-year-old with a Max Heart Rate (MHR) of 200. thumb|A Lärabar bar Lärabar is a brand of energy bars produced by General Mills. ",1.8, -2.5,1.6,0.082,5300,A
-The half-life of ${ }^{238} \mathrm{U}$ is $4.5 \times 10^9$ years. How many disintegrations occur in $1 \mathrm{~min}$ for a $10 \mathrm{mg}$ sample of this element?,"The short half-life of 87.7 years of 238Pu means that a large amount of it decayed during its time inside his body, especially when compared to the 24,100 year half-life of 239Pu. The half-life of 242Pu is about 15 times that of 239Pu; so it is one-fifteenth as radioactive, and not one of the larger contributors to nuclear waste radioactivity. 242Pu's gamma ray emissions are also weaker than those of the other isotopes. Plutonium-242 decays via spontaneous fission in about 5.5 × 10−4% of casesChart of all nuclei which includes half life and mode of decay ==References== Category:Actinides Category:Nuclear materials Category:Isotopes of plutonium Plutonium-244 (244Pu) is an isotope of plutonium that has a half-life of 80 million years. Plutonium-238 (238Pu or Pu-238) is a fissile, radioactive isotope of plutonium that has a half-life of 87.7 years. Modern calculations of his lifetime absorbed dose give an incredible 64 Sv (6400 rem) total. ===Weapons=== The first application of 238Pu was its use in nuclear weapon components made at Mound Laboratories for Lawrence Radiation Laboratory (now Lawrence Livermore National Laboratory). Presence of 244Pu fission tracks can be established by using the initial ratio of 244Pu to 238U (Pu/U)0 at a time T0 = years, when Xe formation first began in meteorites, and by considering how the ratio of Pu/U fission tracks varies over time. Plutonium-242 (242Pu or Pu-242) is one of the isotopes of plutonium, the second longest-lived, with a half-life of 375,000 years. This is longer than any of the other isotopes of plutonium and longer than any other actinide isotope except for the three naturally abundant ones: uranium-235 (704 million years), uranium-238 (4.468 billion years), and thorium-232 (14.05 billion years). The amount of 244Pu in the pre-Solar nebula (4.57×109 years ago) was estimated as 0.8% the amount of 238U..As the age of the Earth is about 57 half-lives of 244Pu, the amount of plutonium-244 left should be very small; Hoffman et al. estimated its content in the rare-earth mineral bastnasite as = 1.0×10−18 g/g, which corresponded to the content in the Earth crust as low as 3×10−25 g/g (i.e. the total mass of plutonium-244 in Earth's crust is about 9 g). This gives a density for 238Pu of (1.66053906660×10−24g/dalton×238.0495599 daltons/atom×16 atoms/unit cell)/(319.96 Å3/unit cell × 10−24cc/Å3) or 19.8 g/cc. It decays by electron capture to stable cadmium-111 with a half-life of 2.8 days. However, 238Pu is far more dangerous than 239Pu due to its short half-life and being a strong alpha-emitter. Historically, most plutonium-238 has been produced by Savannah River in their weapons reactor, by irradiating with neutrons neptunium-237 (half life ). \+ → Neptunium-237 is a by-product of the production of plutonium-239 weapons-grade material, and when the site was shut down in 1988, 238Pu was mixed with about 16% 239Pu. ===Human radiation experiments=== Plutonium was first synthesized in 1940 and isolated in 1941 by chemists at the University of California, Berkeley. They also reported an even longer half-life for alpha decay of bismuth-209 to the first excited state of thallium-205 (at 204 keV), was estimated to be 1.66 years. In February 2013, a small amount of 238Pu was successfully produced by Oak Ridge's High Flux Isotope Reactor, and on December 22, 2015, they reported the production of of 238Pu. Although 209Bi holds the half-life record for alpha decay, bismuth does not have the longest half-life of any radionuclide to be found experimentally--this distinction belongs to tellurium-128 (128Te) with a half-life estimated at 7.7 × 1024 years by double β-decay (double beta decay). Bismuth-209 (209Bi) is the isotope of bismuth with the longest known half-life of any radioisotope that undergoes α-decay (alpha decay). Significant amounts of pure 238Pu could also be produced in a thorium fuel cycle. The density of plutonium-238 at room temperature is about 19.8 g/cc.Calculated from the atomic weight and the atomic volume. However, 242Pu's low cross section means that relatively little of it will be transmuted during one cycle in a thermal reactor. ==Decay== Plutonium-242 mainly decays into uranium-238 via alpha decay, before continuing along the uranium series. The material will generate about 0.57 watts per gram of 238Pu. ",1.43,1.2,313.0,1.5,-1.46,A
-"Calculate the ionic strength in a solution that is 0.0750 $m$ in $\mathrm{K}_2 \mathrm{SO}_4, 0.0085 \mathrm{~m}$ in $\mathrm{Na}_3 \mathrm{PO}_4$, and $0.0150 \mathrm{~m}$ in $\mathrm{MgCl}_2$.","For the electrolyte MgSO4, however, each ion is doubly- charged, leading to an ionic strength that is four times higher than an equivalent concentration of sodium chloride: :I = \frac{1}{2}[c(+2)^2+c(-2)^2] = \frac{1}{2}[4c + 4c] = 4c Generally multivalent ions contribute strongly to the ionic strength. ===Calculation example=== As a more complex example, the ionic strength of a mixed solution 0.050 M in Na2SO4 and 0.020 M in KCl is: : \begin{align} I & = \tfrac 1 2 \times \left[\begin{array}{l} \\{(\text{concentration of }\ce{Na2SO4}\text{ in M}) \times (\text{number of }\ce{Na+}) \times (\text{charge of }\ce{Na+})^2\\}\ + \\\ \\{(\text{concentration of }\ce{Na2SO4}\text{ in M}) \times (\text{number of }\ce{SO4^2-}) \times (\text{charge of }\ce{SO4^2-})^2\\} \ + \\\ \\{(\text{concentration of }\ce{KCl}\text{ in M}) \times (\text{number of }\ce{K+}) \times (\text{charge of }\ce{K+})^2\\}\ + \\\ \\{(\text{concentration of }\ce{KCl}\text{ in M}) \times (\text{number of }\ce{Cl-}) \times (\text{charge of }\ce{Cl-})^2\\} \end{array}\right] \\\ & = \tfrac 1 2 \times [\\{0.050 M \times 2 \times (+1)^2\\} + \\{0.050 M \times 1 \times (-2)^2\\} + \\{0.020 M \times 1 \times (+1)^2\\} + \\{0.020 M \times 1 \times (-1)^2\\}] \\\ & = 0.17 M \end{align} ==Non-ideal solutions== Because in non-ideal solutions volumes are no longer strictly additive it is often preferable to work with molality b (mol/kg of H2O) rather than molarity c (mol/L). Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic strength can be molar (mol/L solution) or molal (mol/kg solvent) and to avoid confusion the units should be stated explicitly. For a 1:1 electrolyte such as sodium chloride, where each ion is singly-charged, the ionic strength is equal to the concentration. One of the main characteristics of a solution with dissolved ions is the ionic strength. The concept of ionic strength was first introduced by Lewis and Randall in 1921 while describing the activity coefficients of strong electrolytes. ==Quantifying ionic strength== The molar ionic strength, I, of a solution is a function of the concentration of all ions present in that solution. In that case, molal ionic strength is defined as: : I = \frac{1}{2}\sum_{{i}=1}^{n} b_{i}z_{i}^{2} in which :i = ion identification number :z = charge of ion :b = molality (mol solute per Kg solvent)Standard definition of molality ==Importance== The ionic strength plays a central role in the Debye–Hückel theory that describes the strong deviations from ideality typically encountered in ionic solutions. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. That is, the Debye length, which is the inverse of the Debye parameter (κ), is inversely proportional to the square root of the ionic strength. In condensed matter physics and inorganic chemistry, the cation-anion radius ratio can be used to predict the crystal structure of an ionic compound based on the relative size of its atoms. Comparison between observed and calculated ion separations (in pm) MX Observed Soft-sphere model LiCl 257.0 257.2 LiBr 275.1 274.4 NaCl 282.0 281.9 NaBr 298.7 298.2 In the soft-sphere model, k has a value between 1 and 2. Magnesium orthosilicate is a chemical compound with the formula Mg2SiO4.Magnesium orthosilicate at Chemister It is the orthosilicate salt of magnesium. _Effective_ ionic radii in pm of elements as a function of ionic charge and spin (ls = low spin, hs = high spin). Curiously, no theoretical justification for the equation containing k has been given. == Non-spherical ions == The concept of ionic radii is based on the assumption of a spherical ion shape. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as ""effective"" ionic radii. One approach to improving the calculated accuracy is to model ions as ""soft spheres"" that overlap in the crystal. Natural waters such as mineral water and seawater have often a non-negligible ionic strength due to the presence of dissolved salts which significantly affects their properties. ==See also== * Activity (chemistry) * Activity coefficient * Bromley equation * Davies equation * Debye–Hückel equation * Debye–Hückel theory * Double layer (interfacial) * Double layer (electrode) * Double layer forces * Electrical double layer * Gouy-Chapman model * Flocculation * Peptization (the inverse of flocculation) * DLVO theory (from Derjaguin, Landau, Verwey and Overbeek) * Interface and colloid science * Osmotic coefficient * Pitzer equations * Poisson–Boltzmann equation * Specific ion Interaction Theory * Salting in * Salting out ==External links== * Ionic strength * Ionic strength introduction at the EPA web site == References == Category:Analytical chemistry Category:Colloidal chemistry Category:Electrochemical equations Category:Electrochemical concepts Category:Equilibrium chemistry Category:Physical chemistry Category:Physical quantities Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. All compounds crystallize in the NaCl structure. thumb|300 px|Relative radii of atoms and ions. ",1.88,0.7854,9.73,0.6749,0.321,E
-"The interior of a refrigerator is typically held at $36^{\circ} \mathrm{F}$ and the interior of a freezer is typically held at $0.00^{\circ} \mathrm{F}$. If the room temperature is $65^{\circ} \mathrm{F}$, by what factor is it more expensive to extract the same amount of heat from the freezer than from the refrigerator? Assume that the theoretical limit for the performance of a reversible refrigerator is valid in this case.","Direct Cool Vs Frost Free Refrigerators – Know the Differences Direct cool is less expensive in production and in operation, as it consumes less energy when compared to frost free refrigerators ==References== Category:Refrigerants 2\. While having the same total pressure throughout the system, the refrigerator maintains a low partial pressure of the refrigerant (therefore high evaporation rate) in the part of the system that draws heat out of the low-temperature interior of the refrigerator, but maintains the refrigerant at high partial pressure (therefore low evaporation rate) in the part of the system that expels heat to the ambient-temperature air outside the refrigerator. A single-pressure absorption refrigerator takes advantage of the fact that a liquid's evaporation rate depends upon the partial pressure of the vapor above the liquid and goes up with lower partial pressure. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. Heat flows from the hotter interior of the refrigerator to the colder liquid, promoting further evaporation. 3\. A refrigerator designed to reach cryogenic temperatures (below ) is often called a cryocooler. The refrigerator was less efficient than existing appliances, although having no moving parts made it more reliable; the introduction of non-toxic Freon — later found to be responsible for serious depletion of the Earth's ozone layer — to replace toxic refrigerant gases made it even less attractive commercially. Progress in the cryocooler field in recent decades is in large part due to development of new materials having high heat capacity below 10 K.T. Kuriyama, R. Hakamada, H. Nakagome, Y. Tokai, M. Sahashi, R. Li, O. Yoshida, K. Matsumoto, and T. Hashimoto, Advances in Cryogenic Engineering 35B, 1261 (1990) == Stirling refrigerators == ===Components=== 300px|thumb| Fig.1 Schematic diagram of a Stirling cooler. right|thumb|150px|Thermal mass refrigerator A thermal mass refrigerator is a refrigerator that is foreseen with thermal mass as well as insulation to decrease the energy use of the refrigerator. In the 1960s, absorption refrigeration saw a renaissance due to the substantial demand for refrigerators for caravans (travel trailers). Unlike more common vapor-compression refrigeration systems, an absorption refrigerator has no moving parts. ==History== In the early years of the 20th century, the vapor absorption cycle using water- ammonia systems was popular and widely used, but after the development of the vapor compression cycle it lost much of its importance because of its low coefficient of performance (about one fifth of that of the vapor compression cycle). An absorption refrigerator changes the gas back into a liquid using a method that needs only heat, and has no moving parts other than the fluids. 300px|right|Absorption cooling process The absorption cooling cycle can be described in three phases: #Evaporation: A liquid refrigerant evaporates in a low partial pressure environment, thus extracting heat from its surroundings (e.g. the refrigerator's compartment). Direct-cool refrigerators produce the cooling effect by a natural convection process from cooled surfaces in the insulated compartment that is being cooled. Compression refrigerators typically use an HCFC or HFC, while absorption refrigerators typically use ammonia or water and need at least a second fluid able to absorb the coolant, the absorbent, respectively water (for ammonia) or brine (for water). The water evaporated from the salt solution is re- condensed, and rerouted back to the evaporative cooler. ===Single pressure absorption refrigeration=== thumb|right|300px|Domestic absorption refrigerator. 1\. The refrigerator is a small unit placed over a campfire, that can later be used to cool of water to just above freezing for 24 hours in a environment. The main difference between the two systems is the way the refrigerant is changed from a gas back into a liquid so that the cycle can repeat. thumb|200px|Einstein's and Szilárd's patent application thumb|200px|Annotated patent drawing The Einstein–Szilard or Einstein refrigerator is an absorption refrigerator which has no moving parts, operates at constant pressure, and requires only a heat source to operate. This refrigerator was a DIY refrigerator designed around a (Sun-Frost) DC refrigeration unit run on P.V. panels.Earthship Volume 3 by Michael Reynolds ==Design details== The thermal mass used in Michael Reynolds' design is a combination of a liquid (i.e. water or beer) together with concrete mass. Direct cool is one of the two major types of techniques used in domestic refrigerators, the other being the ""frost-free"" type. The ""Tang-Dresselhaus Theory"" (Shuang Tang and Mildred Dresselhaus) has pointed out that anisotropic transport behaviors of quantum confined BiSb alloys nanostructures can optimize the pertinent thermoelectric cooling performance below 77 K for applications in satellites and space stations. ==See also== * Cryogenic processor * Adiabatic demagnetization refrigerator * Dilution refrigerator * Hampson-Linde cycle * Pulse tube refrigerator * Stirling engine (Stirling cryocooler) * Entropy production ==References== Category:Cooling technology Category:Cryogenics Category:Industrial gases ",-242.6,2.4,1.3,0.0547,13.2,B
-"Calculate the rotational partition function for $\mathrm{SO}_2$ at $298 \mathrm{~K}$ where $B_A=2.03 \mathrm{~cm}^{-1}, B_B=0.344 \mathrm{~cm}^{-1}$, and $B_C=0.293 \mathrm{~cm}^{-1}$","For each value of J, we have rotational degeneracy, g_j = (2J+1), so the rotational partition function is therefore \zeta^\text{rot} = \sum_{J=0}^\infty g_j e^{-E_J/k_\text{B} T} = \sum_{J=0}^\infty (2J+1) e^{-J(J+1) B / k_\text{B} T}. In terms of these constants, the rotational partition function can be written in the high temperature limit as G. Herzberg, ibid, Equation (V,29) \zeta^\text{rot} \approx \frac{\sqrt{\pi} }{\sigma} \sqrt{ \frac{ (k_\text{B} T)^3 }{ A B C }} with \sigma again known as the rotational symmetry number G. Herzberg, ibid; see Table 140 for values for common molecular point groups which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: \zeta^\text{rot} \approx \frac{5.34 \times 10^6}{\sigma} \sqrt{ \frac{T^3}{A B C}} where the prefactor comes from \sqrt{\frac{(\pi k_\text{B}) ^3}{h^3}} = 5.34 \times 10^6 when A, B, and C are expressed in units of MHz. The expressions for \zeta^\text{rot} works for asymmetric, symmetric and spherical top rotors. ==References== ==See also== * Translational partition function * Vibrational partition function * Partition function (mathematics) Category:Equations of physics Category:Partition functions In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. Atkins and J. de Paula ""Physical Chemistry"", 9th edition (W.H. Freeman 2010), p.597 :\theta_{\mathrm{R}} = \frac{hc \overline{B}}{k_{\mathrm{B}}} = \frac{\hbar^2}{2k_{\mathrm{B}}I}, where \overline{B} = B/hc is the rotational constant, is a molecular moment of inertia, is the Planck constant, is the speed of light, is the reduced Planck constant and is the Boltzmann constant. For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E_J^\text{rot} = \frac{\mathbf{J}^2}{2I} = \frac{J(J+1)\hbar^2}{2I} = J(J+1)B. J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0,1,2, \ldots, B = \frac{\hbar^2}{2I} is the rotational constant, and I is the moment of inertia. thumb|350px|Dini's Surface with constants a = 1, b = 0.5 and 0 ≤ u ≤ 4 and 0 It is named after Ulisse Dini and described by the following parametric equations: : \begin{align} x&=a \cos u \sin v \\\ y&=a \sin u \sin v \\\ z&=a \left(\cos v +\ln \tan \frac{v}{2} \right) + bu \end{align} thumb|350px|right|Dini's surface with 0 ≤ u ≤ 4 and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945, Equation (V,21) \zeta^\text{rot} = \frac{ k_\text{B} T}{B} + \frac{1}{3} + \frac{1}{15} \left( \frac{B}{ k_\text{B} T} \right) + \frac{4}{315} \left( \frac{B}{k_\text{B} T} \right)^2 + \frac{1}{315} \left( \frac{B}{k_\text{B} T} \right)^3 + \cdots . In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor \sigma = 2 with \sigma known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without. ==Nonlinear molecules== A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A, B, and C , which can often be determined by rotational spectroscopy. This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable. \zeta^\text{rot} \approx \int_0^{\infty} (2J+1)e^{-J(J+1) B /k_\text{B} T} dJ = \frac{ k_\text{B} T}{B} . In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k_\text{B} T . == Quantum symmetry effects == For a diatomic molecule with a center of symmetry, such as \rm H_2, N_2, CO_2, or \mathrm{ H_2 C_2} (i.e. D_{\infty h} point group), rotation of a molecule by \pi radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. The mean thermal rotational energy per molecule can now be computed by taking the derivative of \zeta^\text{rot} with respect to temperature T. The characteristic rotational temperature ( or ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. ASM Handbook; Alloy Phase Diagrams; v. 3; ASM International, USA; 1992, pp. 491–492. For molecules, under the assumption that total energy levels E_j can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom)Donald A. McQuarrie, ibid E_j = \sum_i E_j^i = E_j^\text{trans} + E_j^\text{ns} + E_j^\text{rot} + E_j^\text{vib} + E_j^\text{e} and the number of degenerate states are given as products of the single contributions g_j = \prod_i g_j^i = g_j^\text{trans} g_j^\text{ns} g_j^\text{rot} g_j^\text{vib} g_j^\text{e}, where ""trans"", ""ns"", ""rot"", ""vib"" and ""e"" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions \zeta = \sum_j g_j e^{-E_j/k_\text{B} T} can be written as a product itself \zeta = \prod_i \zeta^i = \zeta^\text{trans} \zeta^\text{ns} \zeta^\text{rot} \zeta^\text{vib}\zeta^\text{e}. == Linear molecules == Rotational energies are quantized. This alloy presents a eutectic temperature of 382 K (109 °C; 228.2 °F). On 19 February 1772, the agreement of partition was signed in Vienna. thumb|right|300px|Picture of Europe for July 1772, satirical British plate The Partition Sejm () was a Sejm lasting from 1773 to 1775 in the Polish–Lithuanian Commonwealth, convened by its three neighbours (the Russian Empire, Prussia and Austria) in order to legalize their First Partition of Poland. *BiIn2 (from 52.5 to 53.5 wt% of In), having a hexagonal structure with 2 atoms per unit cell. The Sejm on 30 September 1773 accepted the partition treaty. Atkins and J. de Paula ""Physical Chemistry"", 10th edition, Table 12D.1, p.987 ==References== ==See also== *Rotational spectroscopy *Vibrational temperature *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics ",0.312, 252.8,273.0,3.52,5840,E
-"For a two-level system where $v=1.50 \times 10^{13} \mathrm{~s}^{-1}$, determine the temperature at which the internal energy is equal to $0.25 \mathrm{Nhv}$, or $1 / 2$ the limiting value of $0.50 \mathrm{Nhv}$.","The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy. The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T : U = C_V n T, where C_V is the isochoric (at constant volume) molar heat capacity of the gas. Knowing temperature and pressure to be the derivatives T = \frac{\partial U}{\partial S}, P = -\frac{\partial U}{\partial V}, the ideal gas law PV = nRT immediately follows as below: : T = \frac{\partial U}{\partial S} = \frac{U}{C_V n} : P = -\frac{\partial U}{\partial V} = U \frac{R}{C_V V} : \frac{P}{T} = \frac{\frac{U R}{C_V V}}{\frac{U}{C_V n}} = \frac{n R}{V} : PV = nRT ==Internal energy of a closed thermodynamic system== The above summation of all components of change in internal energy assumes that a positive energy denotes heat added to the system or the negative of work done by the system on its surroundings. The differential internal energy may be written as :\mathrm{d} U = \frac{\partial U}{\partial S} \mathrm{d} S + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - P \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i, which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure P to be the negative of the similar derivative with respect to volume V, : T = \frac{\partial U}{\partial S}, : P = -\frac{\partial U}{\partial V}, and where the coefficients \mu_{i} are the chemical potentials for the components of type i in the system. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The expression relating changes in internal energy to changes in temperature and volume is : \mathrm{d}U =C_{V} \, \mathrm{d}T +\left[T\left(\frac{\partial P}{\partial T}\right)_{V} - P\right] \mathrm{d}V. Appendix D. ISBN 978-1305079113 Na sodium 107 K potassium 89 Rb rubidium 81 Cs caesium 76 Mg magnesium 148 Ca calcium 178 Sr strontium 164 Ba barium 180 Fe iron 416 Ni nickel 430 Cu copper 338 Zn zinc 131 Ag silver 285 W tungsten 849 Au gold 366 C graphite 717 C diamond 715 Si silicon 456 Sn tin 302 Pb lead 195 I2 iodine 62.4 C10H8 naphthalene 72.9 CO2 carbon dioxide 25 ==See also== * Heat * Sublimation (chemistry) * Phase transition * Clausius-Clapeyron equation == References == Category:Enthalpy For a closed system, with transfers only as heat and work, the change in the internal energy is : \mathrm{d} U = \delta Q - \delta W, expressing the first law of thermodynamics. Although the heat capacity has a peak, it does not tend towards infinity (contrary to what the graph may suggest), but has finite limiting values when approaching the transition from above and below. The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree: : U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots), where \alpha is a factor describing the growth of the system. For those phase transitions specific heat does tend to infinity. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U(n,T). If a real gas can be described by the van der Waals equation of state p = \frac{nRT}{V-nb} - a \frac{n^2}{V^2} it follows from the thermodynamic equation of state that \pi_T = a \frac{n^2}{V^2} Since the parameter a is always positive, so is its internal pressure: internal energy of a van der Waals gas always increases when it expands isothermally. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. On the energetics of maximum-entropy temperature profiles, Q. J. R. Meteorol. The equation of state is the ideal gas law :P V = n R T. Solve for pressure: :P = \frac{n R T}{V}. The change in internal energy becomes : \mathrm{d}U = T \, \mathrm{d}S - P \, \mathrm{d}V. ===Changes due to temperature and volume=== The expression relating changes in internal energy to changes in temperature and volume is This is useful if the equation of state is known. Generalized Thermodynamics, M.I.T. Press, Cambridge MA. Equilibrium Thermodynamics, second edition, McGraw-Hill, London, . In thermodynamics, the enthalpy of sublimation, or heat of sublimation, is the heat required to sublimate (change from solid to gas) one mole of a substance at a given combination of temperature and pressure, usually standard temperature and pressure (STP). ",2.2,655,0.000216,46.7,0.000226,B
-Calculate $K_P$ at $600 . \mathrm{K}$ for the reaction $\mathrm{N}_2 \mathrm{O}_4(l) \rightleftharpoons 2 \mathrm{NO}_2(g)$ assuming that $\Delta H_R^{\circ}$ is constant over the interval 298-725 K.,"The molecular formula C18H12O4 (molar mass: 292.28 g/mol, exact mass: 292.0736 u) may refer to: * Karanjin * Polyporic acid Category:Molecular formulas The molecular formula C6H6N4O4 (molar mass: 198.14 g/mol, exact mass: 198.0389 u) may refer to: * 2,4-Dinitrophenylhydrazine * Nitrofurazone The molecular formula C6H7KO6 (molar mass: 214.21 g/mol, exact mass: 213.9880 u) may refer to: * Potassium ascorbate * Potassium erythorbate The molecular formula C12H14O4 (molar mass: 222.23 g/mol, exact mass: 222.0892 u) may refer to: * Apiole * Blattellaquinone * Diethyl phthalate * Dillapiole * Monobutyl phthalate The molecular formula C27H33NO4 (molar mass: 435.56 g/mol, exact mass: 435.2410 u) may refer to: * Paxilline, a potassium channel blocker * BU-48 The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas ",0.33333333,1.7,2.25,0.00017,4.76,E
-"Count Rumford observed that using cannon boring machinery a single horse could heat $11.6 \mathrm{~kg}$ of ice water $(T=273 \mathrm{~K})$ to $T=355 \mathrm{~K}$ in 2.5 hours. Assuming the same rate of work, how high could a horse raise a $225 \mathrm{~kg}$ weight in 2.5 minutes? Assume the heat capacity of water is $4.18 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$.","Based on differences in the definition of what constitutes the ""power of a horse"", a horsepower-hour differs slightly from the German ""Pferdestärkenstunde"" (PSh): :1.014 PSh = 1 hp⋅h = 1,980,000 lbf⋅ft = 0.7457 kW⋅h. :1  The unit represents an amount of work a horse is supposed capable of delivering during an hour (1 horsepower integrated over a time interval of an hour). Pound per hour is a mass flow unit. For example, if Railroad A borrows a 2,500 horsepower locomotive from Railroad B and operates it for twelve hours, Railroad A owes a debt of (2,500 hp × 12 h) = 30,000 hp⋅h. In the US utility industry, steam and water flows throughout turbine cycles are typically expressed in PPH, while in Europe these mass flows are usually expressed in metric tonnes per hour: :1 lb/h = 0.4535927 kg/h = 126.00 mg/s Minimum fuel intake on a jumbo jet can be as low as 150 lb/h when idling; however, this is not enough to sustain flight. PSh = 0.73549875 kW⋅h = 2647.7955 kJ (exactly by definition) The horsepower-hour is still used in the railroad industry when sharing motive power (locomotives). The steam to oil ratio is a measure of the water and energy consumption related to oil production in cyclic steam stimulation and steam assisted gravity drainage oil production. Humber Fifteen 15 horsepower cars were medium to large cars, classified as medium weight, with a less powerful than usual engine which attracted less annual taxation and provided more stately progress. Their equivalent prewar car with an engine of 3.3 Litres had twin overhead camshafts. ===Bodies=== The 15.9 was available as a saloon or a 5-seater tourer. Ice Water (foaled 1963 in Ontario) was a Canadian Thoroughbred racehorse. ==Background== Ice Water was a bay mare owned and bred by George Gardiner. This means two to eight barrels of water converted into steam is used to produce one barrel of oil. == References == * Glossary at Schlumberger. Railroad A may repay the debt by loaning Railroad B a 3,000 horsepower locomotive for ten hours. ==References== Category:Imperial units Category:Units of energy The GWR 2021 Class was a class of 140 steam locomotives. Ice Water raced and won at age four and five, notably winning her second and third consecutive runnings of the Belle Mahone Stakes. ==Breeding record== She was retired to broodmare duty for the 1969 season at her owner's breeding farm where she had limited success. ==References== * Ice Water's pedigree and partial racing stats * Article on Gardiner Farms and Ice Water at the Jockey Club of Canada Category:1963 racehorse births Category:Thoroughbred family 13-c Category:Racehorses bred in Ontario Category:Racehorses trained in Canada A horsepower-hour (symbol: hp⋅h) is an outdated unit of energy, not used in the International System of Units. Ice Water's sire was Nearctic who also sired the most influential sire of the 20th Century, Northern Dancer. The car's steering was delightful but its brakes and suspension were only satisfactory. ==Fifteen 40== The 15-40-hp, a lightly revised 15.9, was displayed at the Olympia Motor Show in October 1924. ===Engine=== The Times noted some trouble had been taken to dampen engine vibration. The experiment was unpopular with engine crews, and the bodywork removed in 1911. ==See also== * GWR 0-6-0PT – list of classes of GWR 0-6-0 pannier tank, including table of preserved locomotives ==References== ==Sources== * Ian Allan ABC of British Railways Locomotives, 1948 edition, part 1, pp 16,51 * * 2021 Category:0-6-0ST locomotives Category:Railway locomotives introduced in 1897 Category:Standard gauge steam locomotives of Great Britain Category:Scrapped locomotives Against females, Ice Water won the Wonder Where Stakes and the Belle Mahone Stakes. They were superseded by the short-lived GWR 1600 Class, nominally a Hawksworth design, but in reality a straightforward update of the then 75-year-old design, with new boiler, bigger cab and bunker. ==Coachwork== When autotrains were introduced on the GWR, a trial was made of enclosing the engine in coachwork to resemble the coaches. The typical values are three to eight and two to five respectively. The newspaper noted that a few drops of oil two or three times a week ensures tappets run for a long time without shake otherwise they soon become noisy. ",30,0.3085,2.0,-0.0301,-2,A
-The vibrational frequency of $I_2$ is $208 \mathrm{~cm}^{-1}$. At what temperature will the population in the first excited state be half that of the ground state?,"The vibrational temperature is used commonly when finding the vibrational partition function. 2MASS J03480772−6022270 (abbreviated to 2MASS J0348−6022) is a brown dwarf of spectral class T7, located in the constellation Reticulum approximately 27.2 light-years from the Sun. The high estimated age of 2MASS J0348−6022 is due to its late T-type spectral class, which is generally expected to describe the later evolutionary stages of brown dwarfs as they cool. == Rotation == === Photometric variability and periodicity === 2MASS J0348−6022 is the fastest-rotating brown dwarf confirmed , with a photometric periodicity of hours. The near-infrared spectrum of 2MASS J0348−6022 also displays a pair of narrow absorption lines at 1.243 and 1.252 μm, which are attributed to the presence of neutral potassium (K I) in the brown dwarf's atmosphere. Photometric variability in 2MASS J0348−6022 was first reported in 2008 by Fraser Clarke and collaborators using the New Technology Telescope's (NTT) near-infrared spectrograph. 2MASS J09373487+2931409, or 2MASSI J0937347+293142 (abbreviated to 2MASS 0937+2931) is a brown dwarf of spectral class T6, located in the constellation Leo about 19.96 light-years from Earth. ==Discovery== 2MASS 0937+2931 was discovered in 2002 by Adam J. Burgasser et al. from Two Micron All-Sky Survey (2MASS), conducted from 1997 to 2001. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. 2MASS J12195156+3128497 (abbreviated to 2MASS J1219+3128) is a rapidly- rotating brown dwarf of spectral class L8, located in the constellation Coma Berenices about 66 light-years from Earth. Absorption bands of iron(I) hydride (FeH) have also been found in 2MASS J0348−6022's spectrum between 1.72–1.78 μm. The mass, radius, and age of 2MASS J0348−6022 are estimated by interpolation of brown dwarf evolutionary models based on effective temperature and surface gravity. 2MASS J02431371−2453298 (abbreviated to 2MASS 0243−2453) is a brown dwarf of spectral class T6, located in the constellation Fornax about 34.84 light-years from Earth. ==Discovery== 2MASS 0243−2453 was discovered in 2002 by Adam J. Burgasser et al. from Two Micron All-Sky Survey (2MASS), conducted from 1997 to 2001. The high spin rate and oblateness of 2MASS J0348−6022 places it at about 45% of its rotational stability limit, assuming a smoothly varying fluid interior. A less precise parallax of this object, measured under U.S. Naval Observatory Infrared Astrometry Program, was published in 2004 by Vrba et al. ==Properties== 2MASS 0937+2931 has an unusual spectrum, indicating a metal- poor atmosphere and/or a high surface gravity (high pressure at the surface). Its effective temperature is estimated at about 800 Kelvin. The inclination of 2MASS J0348−6022's spin axis to Earth is , derived from its v sin i value. A previous estimate by Burgasser and collaborators from the spectrophotometric relation of spectral type and near-infrared absolute magnitude resulted in a value of , based on 2MASS JHK-band photometry. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics In a 2021 study, Megan Tannock and collaborators compared the near-infrared spectrum of 2MASS J0348−6022 to various published photospheric models and derived multiple best-fit solutions for its effective temperature and surface gravity. The trigonometric parallax of 2MASS J0348−6022 has been measured to be milliarcseconds, from 16 observations by the New Technology Telescope (NTT) collected over 6.4 years. Given the distance estimate from trigonometric parallax, the corresponding tangential velocity is , consistent with the kinematics of the stars of the Galactic disk. == Spectral class == 2MASS J0348−6022 is classified as a late T-type brown dwarf with the spectral class T7, distinguished by the presence of strong methane (CH4) and water (H2O) absorption bands in its near-infrared spectrum between wavelengths 1.2 and 2.35 μm. This can be explained by the presence of CH4 in its atmosphere, which is opaque to wavelengths around 3.3 μm. === Physical effects === The spectral lines in 2MASS J0348−6022's spectrum are Doppler-broadened due to the brown dwarf's rapid rotation, consistent with its short photometric periodicity. The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. ",432,3,7.0,21, -6.04697,A
+","The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. We get : p = \frac{NkT}{V-Nb'} - \frac{N^2 a'}{V^2} \Rightarrow \left(p + \frac{N^2 a'}{V^2} \right)(V-Nb') = NkT \Rightarrow \left(p + \frac{n^2 a}{V^2} \right)(V-nb) = nRT. ==Maxwell equal area rule== Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The first form of the Van der Waals equation of state given above can be recast in the following reduced form: :\left(p_R + \frac{3}{v_R^2}\right)(3v_R - 1) = (8 T_R) This equation is invariant for all fluids; that is, the same reduced form equation of state applies, no matter what a and b may be for the particular fluid. Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). In Van der Waals' original derivation, given below, b' is four times the proper volume of the particle. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. The polarizability is related the van der Waals volume by the relation V_{\rm w} = {1\over{4\pi\varepsilon_0}}\alpha , so the van der Waals volume of helium V = = 0.2073 Å by this method, corresponding to r = 0.37 Å. Van der Waals forces are responsible for certain cases of pressure broadening (van der Waals broadening) of spectral lines and the formation of van der Waals molecules. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). For example, the pairwise attractive van der Waals interaction energy between H atoms in different H2 molecules equals 0.06 kJ/mol (0.6 meV) and the pairwise attractive interaction energy between O atoms in different O2 molecules equals 0.44 kJ/mol (4.6 meV). Accordingly, van der Waals forces can range from weak to strong interactions, and support integral structural loads when multitudes of such interactions are present. The main characteristics of van der Waals forces are: * They are weaker than normal covalent and ionic bonds. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The molar van der Waals volume should not be confused with the molar volume of the substance. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. * Van der Waals forces are additive and cannot be saturated. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for the finite size of the molecules. ",26.9,144,"""4.68""",14,840,A
+"For water, $\Delta H_{\text {vaporization }}$ is $40.656 \mathrm{~kJ} \mathrm{~mol}^{-1}$, and the normal boiling point is $373.12 \mathrm{~K}$. Calculate the boiling point for water on the top of Mt. Everest (elevation $8848 \mathrm{~m}$ ), where the barometric pressure is 253 Torr.","For comparison, on top of Mount Everest, at elevation, the pressure is about and the boiling point of water is . At higher elevations, where the atmospheric pressure is much lower, the boiling point is also lower. For every increase in elevation, water's boiling point is lowered by approximately 0.5 °C. Because of this, water boils at under standard pressure at sea level, but at at altitude. At in elevation, water boils at just . Charles Darwin commented on this phenomenon in The Voyage of the Beagle:Journal and remarks, Chapter XV, March 21, 1835 by Charles Darwin. ==Boiling point of pure water at elevated altitudes== Based on standard sea- level atmospheric pressure (courtesy, NOAA): Altitude, ft (m) Boiling point of water, °F (°C) 0 (0 m) 212°F (100°C) 500 (150 m) 211.1°F (99.5°C) 1,000 (305 m) 210.2°F (99°C) 2,000 (610 m) 208.4°F (98°C) 5,000 (1,524 m) 203°F (95°C) 6,000 (1,829 m) 201.1°F (94°C) 8,000 (2,438 m) 197.4°F (91.9°C) 10,000 (3,048 m) 193.6°F (89.8°C) 12,000 (3,658 m) 189.8°F (87.6°C) 14,000 (4,267 m) 185.9°F (85.5°C) 15,000 (4,572 m) 184.1°F (84.5°C) Source: NASA. ==References== ==External links== *Is it true that you can't make a decent cup of tea up a mountain? physics.org, accessed 2012-11-02 Category:Cooking techniques The normal boiling point (also called the atmospheric boiling point or the atmospheric pressure boiling point) of a liquid is the special case in which the vapor pressure of the liquid equals the defined atmospheric pressure at sea level, one atmosphere.General Chemistry Glossary Purdue University website page At that temperature, the vapor pressure of the liquid becomes sufficient to overcome atmospheric pressure and allow bubbles of vapor to form inside the bulk of the liquid. If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. At elevated altitudes, any cooking that involves boiling or steaming generally requires compensation for lower temperatures because the boiling point of water is lower at higher altitudes due to the decreased atmospheric pressure. Mount Everest is the world's highest mountain, with a peak at 8,849 metres (29,031.7 ft) above sea level. The boiling point corresponds to the temperature at which the vapor pressure of the liquid equals the surrounding environmental pressure. There are two conventions regarding the standard boiling point of water: The normal boiling point is at a pressure of 1 atm (i.e., 101.325 kPa). By comparison, reasonable base elevations for Everest range from on the south side to on the Tibetan Plateau, yielding a height above base in the range of .Mount Everest (1:50,000 scale map), prepared under the direction of Bradford Washburn for the Boston Museum of Science, the Swiss Foundation for Alpine Research, and the National Geographic Society, 1991, . The air pressure at the summit is generally about one-third what it is at sea level. The primary peak of Mount Everest is elevation above sea level. == Overview == The peak is a dome-shaped peak of snow and ice, and is connected to the summit of Mount Everest by the Cornice Traverse and Hillary Step, approximately from the higher peak. It also has the lowest normal boiling point (−24.2 °C), which is where the vapor pressure curve of methyl chloride (the blue line) intersects the horizontal pressure line of one atmosphere (atm) of absolute vapor pressure. In the expedition, the summit's altitude was measured as 8848.13 metres. * 1975 British Mount Everest Southwest Face expedition - On September 24, a British expedition led by Chris Bonington achieved the first ascent of the Southwest Face. The atmospheric pressure at the top of Everest is about a third of sea level pressure or , resulting in the availability of only about a third as much oxygen to breathe. The standard boiling point has been defined by IUPAC since 1982 as the temperature at which boiling occurs under a pressure of one bar. The boiling point of a liquid varies depending upon the surrounding environmental pressure. Boiling points may be published with respect to the NIST, USA standard pressure of 101.325 kPa (or 1 atm), or the IUPAC standard pressure of 100.000 kPa. Towards the end of the season, due to a stalled high-pressure system, conditions on Everest were better than usual, being warmer, drier, and less windy, facilitating a higher-than-usual summitting success rate of 70%. ", 7.0,0,"""273.0""",9,344,E
+"An ideal solution is formed by mixing liquids $\mathrm{A}$ and $B$ at $298 \mathrm{~K}$. The vapor pressure of pure A is 151 Torr and that of pure B is 84.3 Torr. If the mole fraction of $\mathrm{A}$ in the vapor is 0.610 , what is the mole fraction of $\mathrm{A}$ in the solution?","The boiling temperature of water for atmospheric pressures can be approximated by the Antoine equation: :\log_{10}\left(\frac{P}{1\text{ Torr}}\right) = 8.07131 - \frac{1730.63\ {}^\circ\text{C}}{233.426\ {}^\circ\text{C} + T_b} or transformed into this temperature-explicit form: :T_b = \frac{1730.63\ {}^\circ\text{C}}{8.07131 - \log_{10}\left(\frac{P}{1\text{ Torr}}\right)} - 233.426\ {}^\circ\text{C} where the temperature T_b is the boiling point in degrees Celsius and the pressure P is in torr. ==Dühring's rule== Dühring's rule states that a linear relationship exists between the temperatures at which two solutions exert the same vapor pressure. ==Examples== The following table is a list of a variety of substances ordered by increasing vapor pressure (in absolute units). The mole fraction is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and volume fraction (percentage by volume, vol%) are others. ==Properties== Mole fraction is used very frequently in the construction of phase diagrams. When Raoult's law and Dalton's law hold for the mixture, the K factor is defined as the ratio of the vapor pressure to the total pressure of the system: :K_i = \frac{P'_i}{P} Given either of x_i or y_i and either the temperature or pressure of a two-component system, calculations can be performed to determine the unknown information. ==References== ==See also== * Phase diagram * Azeotrope * Dew point Category:Temperature Category:Phase transitions Category:Gases The mole fraction is also called the amount fraction. It states that the activity (pressure or fugacity) of a single-phase mixture is equal to the mole-fraction-weighted sum of the components' vapor pressures: : P_{\rm tot} =\sum_i P y_i = \sum_i P_i^{\rm sat} x_i \, where P_{\rm tot} is the mixture's vapor pressure, x_i is the mole fraction of component i in the liquid phase and y_i is the mole fraction of component i in the vapor phase respectively. At the normal boiling point of a liquid, the vapor pressure is equal to the standard atmospheric pressure defined as 1 atmosphere, 760Torr, 101.325kPa, or 14.69595psi. If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. This is important for volatile inhalational anesthetics, most of which are liquids at body temperature, but with a relatively high vapor pressure. ==Estimating vapor pressures with Antoine equation== The Antoine equationWhat is the Antoine Equation? Whereas mole fraction is a ratio of moles to moles, molar concentration is a quotient of moles to volume. In chemistry, the mole fraction or molar fraction (xi or ) is defined as unit of the amount of a constituent (expressed in moles), ni, divided by the total amount of all constituents in a mixture (also expressed in moles), ntot. The vapor pressure of a liquid at its boiling point equals the pressure of its surrounding environment. ==Liquid mixtures: Raoult's law== Raoult's law gives an approximation to the vapor pressure of mixtures of liquids. The basic form of the equation is: :\log P = A-\frac{B}{C+T} and it can be transformed into this temperature-explicit form: :T = \frac{B}{A-\log P} - C where: * P is the absolute vapor pressure of a substance * T is the temperature of the substance * A, B and C are substance-specific coefficients (i.e., constants or parameters) * \log is typically either \log_{10} or \log_e A simpler form of the equation with only two coefficients is sometimes used: :\log P = A- \frac{B}{T} which can be transformed to: :T = \frac{B}{A-\log P} Sublimations and vaporizations of the same substance have separate sets of Antoine coefficients, as do components in mixtures. When the volatilities of both key components are equal, \alpha = 1 and separation of the two by distillation would be impossible under the given conditions because the compositions of the liquid and the vapor phase are the same (azeotrope). This doesn't affect the equation form. ==Sources for Antoine equation parameters== * NIST Chemistry WebBook * Dortmund Data Bank * Directory of reference books and data banks containing Antoine constants * Several reference books and publications, e. g. ** Lange's Handbook of Chemistry, McGraw-Hill Professional ** Wichterle I., Linek J., ""Antoine Vapor Pressure Constants of Pure Compounds"" ** Yaws C. L., Yang H.-C., ""To Estimate Vapor Pressure Easily. The second solution is switching to another vapor pressure equation with more than three parameters. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds"", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 ==See also== *Vapour pressure of water *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation *Raoult's law *Thermodynamic activity ==References== ==External links== * Gallica, scanned original paper * NIST Chemistry Web Book * Calculation of vapor pressures with the Antoine equation Category:Equations Category:Thermodynamic equations The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. The Antoine equation is a class of semi-empirical correlations describing the relation between vapor pressure and temperature for pure substances. Experimental measurement of vapor pressure is a simple procedure for common pressures between 1 and 200 kPa. Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. Actually, as stated by Dalton's law (known since 1802), the partial pressure of water vapor or any substance does not depend on air at all, and the relevant temperature is that of the liquid. thumb|300px|Boiling waterThe boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. ",0.28209479,0.466,"""7.0""",6.3,1.51,B
+"The mean solar flux at Earth's surface is $\sim 2.00 \mathrm{~J}$ $\mathrm{cm}^{-2} \mathrm{~min}^{-1}$. In a nonfocusing solar collector, the temperature reaches a value of $79.5^{\circ} \mathrm{C}$. A heat engine is operated using the collector as the hot reservoir and a cold reservoir at $298 \mathrm{~K}$. Calculate the area of the collector needed to produce 1000. W. Assume that the engine operates at the maximum Carnot efficiency.","Solar energy – Solar thermal collectors – Test methods International Organization for Standardization, Geneva, Switzerland states that the efficiency of solar thermal collectors should be measured in terms of gross area and this might favour flat plates in respect to evacuated tube collectors in direct comparisons. thumb|An array of evacuated flat plate collectors next to compact solar concentrators thumb|A comparison of the energy output (kW.h/day) of a flat plate collector (blue lines; Thermodynamics S42-P; absorber 2.8 m2) and an evacuated tube collector (green lines; SunMaxx 20EVT; absorber 3.1 m2. In non-concentrating collectors, the aperture area (i.e., the area that receives the solar radiation) is roughly the same as the absorber area (i.e., the area absorbing the radiation). A solar thermal collector collects heat by absorbing sunlight. Transpired solar collectors are usually wall- mounted to capture the lower sun angle in the winter heating months as well as sun reflection off the snow and achieve their optimum performance and return on investment when operating at flow rates of between 4 and 8 CFM per square foot (72 to 144 m3/h.m2) of collector area. The extensive monitoring by Natural Resources Canada and NREL has shown that transpired solar collector systems reduce between 10-50% of the conventional heating load and that RETScreen is an accurate predictor of system performance. Its value is about 86%, which is the Chambadal-Novikov efficiency, an approximation related to the Carnot limit, based on the temperature of the photons emitted by the Sun's surface. == Effect of band gap energy == Solar cells operate as quantum energy conversion devices, and are therefore subject to the thermodynamic efficiency limit. This value depends on the size of the storage unit (hot water tank or storage battery), the size of the harvesting surface (sun collection surface or surface area of photovoltaic modules), and on the amount of energy required. The collector absorbs the incoming solar radiation, converting it into thermal energy. The solar energy flux (irradiance) incident on the Earth's surface has a variable and relatively low surface density, usually not exceeding 1100 W/m² without concentration systems. For a solar cell powered by the Sun's unconcentrated black-body radiation, the theoretical maximum efficiency is 43% whereas for a solar cell powered by the Sun's full concentrated radiation, the efficiency limit is up to 85%. Solar collector may refer to: * Solar thermal collector, a solar collector that collects heat by absorbing sunlight * Solar Collector (sculpture), a 2008 interactive light art installation in Cambridge, Ontario, Canada ==See also== *Concentrating solar power *Renewable heat *Solar air heating *Solar water heating *Solar panel They offer the highest energy conversion efficiency of any non-concentrating solar thermal collector, but require sophisticated technology for manufacturing. The exterior surface of a transpired solar collector consists of thousands of tiny micro-perforations that allow the boundary layer of heat to be captured and uniformly drawn into an air cavity behind the exterior panels. The internal combustion engine efficiency is determined by its two temperature reservoirs, the temperatures of ambient air and its upper limit operating temperature. Engine efficiency of thermal engines is the relationship between the total energy contained in the fuel, and the amount of energy used to perform useful work. In locations with average available solar energy, flat plate collectors are sized approximately 1.2 to 2.4 square decimeter per liter of one day's hot water use. === Applications === The main use of this technology is in residential buildings where the demand for hot water has a large impact on energy bills. In 1954 the solar constant was evaluated as 2.00 cal/min/cm2 ± 2%. The term ""solar collector"" commonly refers to a device for solar hot water heating, but may refer to large power generating installations such as solar parabolic troughs and solar towers or non water heating devices such as solar cooker, solar air heaters. With the increasing drive to install renewable energy systems on buildings, transpired solar collectors are now used across the entire building stock because of high energy production (up to 750 peak thermal Watts/square metre), high solar conversion (up to 90%) and lower capital costs when compared against solar photovoltaic and solar water heating. thumb|Solar air heating is a solar thermal technology in which the energy from the sun, solar insolation, is captured by an absorbing medium and used to heat air. Thermodynamic efficiency limit is the absolute maximum theoretically possible conversion efficiency of sunlight to electricity. Solar thermal collectors are either non-concentrating or concentrating. The solar coverage rate is the percentage of an amount of energy that is provided by the sun. ",4.16,19.4,"""-2.0""",0.72,344,B
+A hiker caught in a thunderstorm loses heat when her clothing becomes wet. She is packing emergency rations that if completely metabolized will release $35 \mathrm{~kJ}$ of heat per gram of rations consumed. How much rations must the hiker consume to avoid a reduction in body temperature of $2.5 \mathrm{~K}$ as a result of heat loss? Assume the heat capacity of the body equals that of water and that the hiker weighs $51 \mathrm{~kg}$.,"The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion International Journal of Heat and Mass Transfer is a peer-reviewed scientific journal in the field of heat transfer and mass transfer, published by Elsevier. Heat and Mass Transfer is a peer-reviewed scientific journal published by Springer. Heatwork is the combined effect of temperature and time. Dietary energy supply is correlated with the rate of obesity. ==Regions== Daily dietary energy supply per capita: Region 1964-1966 1974-1976 1984-1986 1997-1999 World Sub-Saharan Africa ==Countries== Country 1979-1981 1989-1991 2001-2003 Canada China Eswatini United Kingdom United States ==See also== *Diet and obesity thumb|300x300px|Average dietary energy supply by region ==References== ==External links== * Category:Obesity Category:Diets *Temperature equivalents table & description of Orton pyrometric cones. As of 1995 the title Wärme- und Stoffübertragung was changed to Heat and Mass Transfer. == Indexing == Among others, the journal is indexed in Google Scholar, INIS Atomindex, Journal Citation Reports/Science Edition, OCLC, PASCAL, Science Citation Index, Science Citation Index Expanded (SciSearch) and Scopus. == External links == *Heat and Mass Transfer Category:Energy and fuel journals Category:Engineering journals Category:English-language journals Category:Monthly journals Category:Springer Science+Business Media academic journals Heatwork is taught in material science courses, but is not a precise measurement or a valid scientific concept. == External links == *Temperature equivalents table & description of Bullers Rings. *Temperature Equivalents, °F & °C for Bullers Ring. It gives an overestimate of the total amount of food consumed as it reflects both food consumed and food wasted. It serves the circulation of new developments in the field of basic research of heat and mass transfer phenomena, as well as related material properties and their measurements. *Temperature equivalents table & description of Nimra Cerglass pyrometric cones. Within tolerances, firing can be undertaken at lower temperatures for a longer period to achieve comparable results. *Temperature equivalents table of Seger pyrometric cones. When the amount of heatwork of two firings is the same, the pieces may look identical, but there may be differences not visible, such as mechanical strength and microstructure. It is important to several industries: *Ceramics *Glass and metal annealing *Metal heat treating Pyrometric devices can be used to gauge heat work as they deform or contract due to heatwork to produce temperature equivalents. Category:Glass physics Category:Pottery Category:Metallurgy Category:Ceramic engineering The editor-in-chief is T. S. Zhao (Hong Kong University of Science and Technology). ==Abstracting and indexing== The journal is abstracted and indexed in: According to the Journal Citation Reports, the journal has a 2020 impact factor of 5.584. ==References== ==External links== * Category:Physics journals Category:English-language journals Category:Engineering journals Category:Elsevier academic journals Category:Academic journals established in 1960 Category:Monthly journals Thereby applications to engineering problems are promoted. The journal publishes original research reports. It varies markedly between different regions and countries of the world. ",7.00,7200,"""-88.0""",15,4.3,D
+Calculate the degree of dissociation of $\mathrm{N}_2 \mathrm{O}_4$ in the reaction $\mathrm{N}_2 \mathrm{O}_4(g) \rightleftharpoons 2 \mathrm{NO}_2(g)$ at 300 . $\mathrm{K}$ and a total pressure of 1.50 bar. Do you expect the degree of dissociation to increase or decrease as the temperature is increased to 550 . K? Assume that $\Delta H_R^{\circ}$ is independent of temperature.,"Figure 8 shows the corresponding enthalpy drop for the reaction = 0 case. none|310px|thumb|Figure 8. Stage enthalpy diagram for degree of reaction = 1⁄2 in a turbine and pump. left|thumb|Figure 6. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. Hence from Tds = dh - \frac{dp}{\rho}, 400px|alt=enthalpy diagram|thumb|Figure 1. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. An algorithm for computing a 4/3-approximation of the dissociation number in bipartite graphs was published in 2022. The dissociation constant is the inverse of the association constant. From the relation for degree of reaction, || α2 > β3. right|380px|thumb|Figure 7. The dissociation constant for a particular ligand-protein interaction can change significantly with solution conditions (e.g., temperature, pH and salt concentration). Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity. In chemistry, biochemistry, and pharmacology, a dissociation constant (K_D) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. thumb|Examples for the definition of the dissociation number In the mathematical discipline of graph theory, a subset of vertices in a graph G is called dissociation if it induces a subgraph with maximum degree 1. [B^{-3}] \over [H B^{-2}]} & \mathrm{p}K_3 &= -\log K_3 \end{align} ==Dissociation constant of water== The dissociation constant of water is denoted Kw: :K_\mathrm{w} = [\ce{H}^+] [\ce{OH}^-] The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration. The degree of reaction now depends only on ϕ and \tan{\beta_m} which again depend on geometrical parameters β3 and β2 i.e. the vane angles of stator outlet and rotor outlet. (The symbol K_a, used for the acid dissociation constant, can lead to confusion with the association constant and it may be necessary to see the reaction or the equilibrium expression to know which is meant.) The value of Kw varies with temperature, as shown in the table below. From the relation for degree of reaction,|| α2 < β3 which is also shown in corresponding Figure 7. === Reaction = zero === This is special case used for impulse turbine which suggest that entire pressure drop in the turbine is obtained in the stator. The problem of computing diss(G) (dissociation number problem) was firstly studied by Yannakakis. : Water temperature Kw pKw 0 °C 0.112 14.95 25 °C 1.023 13.99 50 °C 5.495 13.26 75 °C 19.95 12.70 100 °C 56.23 12.25 == See also == * Acid * Equilibrium constant * Ki Database * Competitive inhibition * pH * Scatchard plot * Ligand binding * Avidity ==References== Category:Equilibrium chemistry Category:Enzyme kinetics A molecule can have several acid dissociation constants. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. The number of vertices in a maximum cardinality dissociation set in G is called the dissociation number of G, denoted by diss(G). ",1.5,1.56,"""0.44""",0.241,0.5061,D
+Calculate $\Delta G$ for the isothermal expansion of $2.25 \mathrm{~mol}$ of an ideal gas at $325 \mathrm{~K}$ from an initial pressure of 12.0 bar to a final pressure of 2.5 bar.,"For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work given by W = -\int_{2V_0}^{V_0} P\,\mathrm{d}V = - \int_{2V_0}^{V_0} \frac{nRT}{V} \mathrm{d}V = nRT\ln 2 = T \Delta S_\text{gas}. This type of expansion is named after James Prescott Joule who used this expansion, in 1845, in his study for the mechanical equivalent of heat, but this expansion was known long before Joule e.g. by John Leslie, in the beginning of the 19th century, and studied by Joseph-Louis Gay-Lussac in 1807 with similar results as obtained by Joule.D.S.L. Cardwell, From Watt to Clausius, Heinemann, London (1957)M.J. Klein, Principles of the theory of heat, D. Reidel Pub.Cy., Dordrecht (1986) The Joule expansion should not be confused with the Joule–Thomson expansion or throttling process which refers to the steady flow of a gas from a region of higher pressure to one of lower pressure via a valve or porous plug. ==Description== The process begins with gas under some pressure, P_{\mathrm{i}}, at temperature T_{\mathrm{i}}, confined to one half of a thermally isolated container (see the top part of the drawing at the beginning of this article). We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. Heating the gas up to the initial temperature increases the entropy by the amount \Delta S = n \int_{T}^{T_i} C_\mathrm{V} \frac{\mathrm{d}T'}{T'} = nR \ln 2. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. Thus, at low temperatures the Joule expansion process provides information on intermolecular forces. ===Ideal gases=== If the gas is ideal, both the initial (T_{\mathrm{i}}, P_{\mathrm{i}}, V_{\mathrm{i}}) and final (T_{\mathrm{f}}, P_{\mathrm{f}}, V_{\mathrm{f}}) conditions follow the Ideal Gas Law, so that initially P_{\mathrm{i}} V_{\mathrm{i}} = n R T_{\mathrm{i}} and then, after the tap is opened, P_{\mathrm{f}} V_{\mathrm{f}} = n R T_{\mathrm{f}}. For the above defined path we have that and thus , and hence the increase in entropy for the Joule expansion is \Delta S=\int_i^f\mathrm{d}S=\int_{V_0}^{2V_0} \frac{P\,\mathrm{d}V}{T}=\int_{V_0}^{2V_0} \frac{n R\,\mathrm{d}V}{V}=n R\ln 2. For a monatomic ideal gas , with the molar heat capacity at constant volume. thumb|240px|The Joule expansion, in which a volume is expanded to a volume in a thermally isolated chamber. thumb|180px|A free expansion of a gas can be achieved by moving the piston out faster than the fastest molecules in the gas. The Joule expansion, treated as a thought experiment involving ideal gases, is a useful exercise in classical thermodynamics. Here n is the number of moles of gas and R is the molar ideal gas constant. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. For other gases this ""Joule inversion temperature"" appears to be extremely high. ==Entropy production== Entropy is a function of state, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. Pressure gain combustion is a combustion process whereby gas expansion by heat release is constrained, causing a rise in stagnation pressure and allowing work extraction by expansion to the initial pressure. thumb|400px|Diagram showing pressure difference induced by a temperature difference. After thermal equilibrium is reached, we then let the gas undergo another free expansion by and wait until thermal equilibrium is reached. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The Joule expansion (also called free expansion) is an irreversible process in thermodynamics in which a volume of gas is kept in one side of a thermally isolated container (via a small partition), with the other side of the container being evacuated. The fact that the temperature does not change makes it easy to compute the change in entropy of the universe for this process. ===Real gases=== Unlike ideal gases, the temperature of a real gas will change during a Joule expansion. ",226,0.333333,"""0.4908""",-9.54,260,D
+Determine the total collisional frequency for $\mathrm{CO}_2$ at $1 \mathrm{~atm}$ and $298 \mathrm{~K}$.,"Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. In an ideal gas, assuming that the species behave like hard spheres, the collision frequency between entities of species A and species B is:chem.libretexts.org: Collision Frequency : Z = N_\text{A} N_\text{B} \sigma_\text{AB} \sqrt\frac{8 k_\text{B} T}{\pi \mu_\text{AB}}, which has units of [volume][time]−1. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text{Coefficient of restitution } (e) = \frac{\left | \text{Relative velocity after collision} \right |}{\left | \text{Relative velocity before collision}\right |} The mathematics were developed by Sir Isaac Newton in 1687. The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. Setting the above estimate for \Delta m_e v_\perp equal to mv, we find the lower cut-off to the impact parameter to be about :b_0 = \frac{Ze^2}{4\pi\epsilon_0} \, \frac{1}{m_e v^2} We can also use \pi b_0^2 as an estimate of the cross section for large-angle collisions. If a given object collides with two different objects, each collision would have its own COR. Escande DF, Elskens Y, Doveil F (2015) Uniform derivation of Coulomb collisional transport thanks to Debye shielding. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material. Collision-induced emission and absorption by simultaneous collisions of three or more particles generally do involve pairwise-additive dipole components, as well as important irreducible dipole contributions and their spectra. == Historical sketch == Collision-induced absorption was first reported in compressed oxygen gas in 1949 by Harry Welsch and associates at frequencies of the fundamental band of the O2 molecule. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. Collision-induced spectra appear at the frequencies of the rotovibrational and electronic transition bands of the unperturbed molecules, and also at sums and differences of such transition frequencies: simultaneous transitions in two (or more) interacting molecules are well known to generate optical transitions of molecular complexes. == Virial expansions of spectral intensities == Intensities of spectra of individual atoms or molecules typically vary linearly with the numerical gas density. It is possible that e = \infty for a perfect explosion of a rigid system. === Paired objects === The COR is a property of a pair of objects in a collision, not a single object. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. (Note that an unperturbed O2 molecule, like all other diatomic homonuclear molecules, is infrared inactive on account of the inversion symmetry and does thus not possess a ""dipole allowed"" rotovibrational spectrum at any frequency). == Collision-induced spectra == Molecular fly-by collisions take little time, something like 10−13 s. The upper cut-off to the impact parameter should thus be approximately equal to the Debye length: : \lambda_D = \sqrt{\frac{\epsilon_0 kT_e}{n_e e^2}} == Coulomb logarithm == The integral of 1/b thus yields the logarithm of the ratio of the upper and lower cut-offs. Within the same approximations, a more elegant derivation of the collisional transport coefficients was provided, by using the Balescu–Lenard equation (see Sec. 8.4 of Balescu, R. 1997 Statistical Dynamics: Matter Out of Equilibrium. These are the collision-induced spectra of two-body (and quite possibly three-body,...) collisional complexes. The collision-induced spectra have sometimes been separated from the continua of individual atoms and molecules, based on the characteristic density dependences. Also, some recent articles have described superelastic collisions in which it is argued that the COR can take a value greater than one in a special case of oblique collisions. ",8.44,-6.9,"""-0.38""",4.85,5.828427125,A
+"The volatile liquids $A$ and $\mathrm{B}$, for which $P_A^*=165$ Torr and $P_B^*=85.1$ Torr are confined to a piston and cylinder assembly. Initially, only the liquid phase is present. As the pressure is reduced, the first vapor is observed at a total pressure of 110 . Torr. Calculate $x_{\mathrm{A}}$","If the heat of vaporization and the vapor pressure of a liquid at a certain temperature are known, the boiling point can be calculated by using the Clausius–Clapeyron equation, thus: :T_\text{B} = \left(\frac{1}{T_0} - \frac{R\,\ln \frac{P}{P_0}}{\Delta H_\text{vap}}\right)^{-1} where: :T_\text{B} is the boiling point at the pressure of interest, :R is the ideal gas constant, :P is the vapor pressure of the liquid, :P_0 is some pressure where the corresponding T_0 is known (usually data available at 1 atm or 100 kPa), :\Delta H_\text{vap} is the heat of vaporization of the liquid, :T_0 is the boiling temperature, :\ln is the natural logarithm. thumb|300px|A representative pressure–volume diagram for a refrigeration cycle Vapour-compression refrigeration or vapor-compression refrigeration system (VCRS), in which the refrigerant undergoes phase changes, is one of the many refrigeration cycles and is the most widely used method for air conditioning of buildings and automobiles. Between point 3 and point 4, the vapor travels through the remainder of the condenser and is condensed into a high temperature, high pressure subcooled liquid. When the volatilities of both key components are equal, \alpha = 1 and separation of the two by distillation would be impossible under the given conditions because the compositions of the liquid and the vapor phase are the same (azeotrope). thumb|right|300px|Otto cycle pressure–volume diagram An Otto cycle is an idealized thermodynamic cycle that describes the functioning of a typical spark ignition piston engine. The presence of other volatile components in a mixture affects the vapor pressures and thus boiling points and dew points of all the components in the mixture. Relative volatilities are not used in separation or absorption processes that involve components reacting with each other (for example, the absorption of gaseous carbon dioxide in aqueous solutions of sodium hydroxide). ==Definition== For a liquid mixture of two components (called a binary mixture) at a given temperature and pressure, the relative volatility is defined as :\alpha=\frac {(y_i/x_i)}{(y_j/x_j)} = K_i/K_j where: \alpha = the relative volatility of the more volatile component i to the less volatile component j y_i = the vapor–liquid equilibrium mole fraction of component i in the vapor phase x_i = the vapor–liquid equilibrium mole fraction of component i in the liquid phase y_j = the vapor–liquid equilibrium concentration of component j in the vapor phase x_j = the vapor–liquid equilibrium concentration of component j in the liquid phase (y/x) = Henry's law constant (also called the K value or vapor-liquid distribution ratio) of a component When their liquid concentrations are equal, more volatile components have higher vapor pressures than less volatile components. As the piston is capable of moving along the cylinder, the volume of the air changes with its position in the cylinder. thumb|250px|Diagram of cylinder and piston valve. If the pressure in a system remains constant (isobaric), a vapor at saturation temperature will begin to condense into its liquid phase as thermal energy (heat) is removed. That pressure reduction results in the adiabatic flash evaporation of a part of the liquid refrigerant. thumb|300px|Boiling waterThe boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor. Relative volatility is a measure comparing the vapor pressures of the components in a liquid mixture of chemicals. Similarly, a liquid at saturation pressure and temperature will tend to flash into its vapor phase as system pressure is decreased. Similarly, a liquid at saturation temperature and pressure will boil into its vapor phase as additional thermal energy is applied. For a given pressure, different liquids will boil at different temperatures. The system, in this case, is defined to be the fluid (gas) within the cylinder. Furthermore, at any given temperature, the composition of the vapor is different from the composition of the liquid in most such cases. The condensed liquid refrigerant, in the thermodynamic state known as a saturated liquid, is next routed through an expansion valve where it undergoes an abrupt reduction in pressure. The cold refrigerant liquid and vapor mixture is then routed through the coil or tubes in the evaporator. thumb|right|400px|An illustration of fluid simulation using VOF method. The heat of vaporization is the energy required to transform a given quantity (a mol, kg, pound, etc.) of a substance from a liquid into a gas at a given pressure (often atmospheric pressure). ",0.312,2.57,"""36.0""",3.333333333,-8,A
+The osmotic pressure of an unknown substance is measured at $298 \mathrm{~K}$. Determine the molecular weight if the concentration of this substance is $31.2 \mathrm{~kg} \mathrm{~m}^{-3}$ and the osmotic pressure is $5.30 \times 10^4 \mathrm{~Pa}$. The density of the solution is $997 \mathrm{~kg} \mathrm{~m}^{-3}$.,"Both sodium and chloride ions affect the osmotic pressure of the solution. The osmotic coefficient based on molality m is defined by: \phi = \frac{\mu_A^* - \mu_A}{RTM_A \sum_i m_i} and on a mole fraction basis by: \phi = -\frac{\mu_A^* - \mu_A}{RT \ln x_A} where \mu_A^* is the chemical potential of the pure solvent and \mu_A is the chemical potential of the solvent in a solution, MA is its molar mass, xA its mole fraction, R the gas constant and T the temperature in Kelvin. The transfer of solvent molecules will continue until equilibrium is attained. ==Theory and measurement== Jacobus van 't Hoff found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation: :\Pi = icRT where \Pi is osmotic pressure, i is the dimensionless van 't Hoff index, c is the molar concentration of solute, R is the ideal gas constant, and T is the absolute temperature (usually in kelvins). This value allows the measurement of the osmotic pressure of a solution and the determination of how the solvent will diffuse across a semipermeable membrane (osmosis) separating two solutions of different osmotic concentration. == Unit == The unit of osmotic concentration is the osmole. Osmotic concentration, formerly known as osmolarity, is the measure of solute concentration, defined as the number of osmoles (Osm) of solute per litre (L) of solution (osmol/L or Osm/L). This is a non-SI unit of measurement that defines the number of moles of solute that contribute to the osmotic pressure of a solution. The Pfeffer cell was developed for the measurement of osmotic pressure. == Applications == thumb|upright=1.15|Osmotic pressure on red blood cells Osmotic pressure measurement may be used for the determination of molecular weights. The proportionality to concentration means that osmotic pressure is a colligative property. In order to find \Pi, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water. :\mu_v(x_v,p+\Pi) = \mu_v^0(p). Potential osmotic pressure is the maximum osmotic pressure that could develop in a solution if it were separated from its pure solvent by a semipermeable membrane. For aqueous solutions, the osmotic coefficients can be calculated theoretically by Pitzer equationsI. The osmotic pressure of ocean water is approximately 27 atm. Reverse osmosis desalinates fresh water from ocean salt water. == Derivation of the van 't Hoff formula == Consider the system at the point when it has reached equilibrium. For example, a 3 Osm solution might consist of: 3 moles glucose, or 1.5 moles NaCl, or 1 mole glucose + 1 mole NaCl, or 2 moles glucose + 0.5 mole NaCl, or any other such combination. ==Definition== The osmolarity of a solution, given in osmoles per liter (osmol/L) is calculated from the following expression: \mathrm{osmolarity} = \sum_i \varphi_i \, n_i C_i where * is the osmotic coefficient, which accounts for the degree of non-ideality of the solution. Temperature(°F) Specific weight (lbf/ft3) 32 62.42 40 62.43 50 62.41 60 62.37 70 62.30 80 62.22 90 62.11 100 62.00 110 61.86 120 61.71 130 61.55 140 61.38 150 61.20 160 61.00 170 60.80 180 60.58 190 60.36 200 60.12 212 59.83 ==Specific weight of air== Specific weight of air at standard sea-level atmospheric pressure (Metric units) Temperature(°C) Specific weight (N/m3) −40 14.86 −20 13.86 0 12.68 10 12.24 20 11.82 30 11.43 40 11.06 60 10.4 80 9.81 100 9.28 200 7.33 Specific weight of air at standard sea-level atmospheric pressure (English units) Temperature(°F) Specific Weight (lbf/ft3) −40 −20 0.0903 0 0.08637 10 0.08453 20 0.08277 30 0.08108 40 0.07945 50 0.0779 60 0.0764 70 0.07495 80 0.07357 90 0.07223 100 0.07094 120 0.06849 140 0.0662 160 0.06407 180 0.06206 200 0.06018 250 0.05594 ==References== ==External links== * Submerged weight calculator * Specific weight calculator * http://www.engineeringtoolbox.com/density-specific-weight-gravity-d_290.html * http://www.themeter.net/pesi-spec_e.htm Category:Soil mechanics Category:Fluid mechanics Category:Physical chemistry Category:Physical quantities Category:Density For example, the intracellular fluid and extracellular can be hyperosmotic, but isotonic – if the total concentration of solutes in one compartment is different from that of the other, but one of the ions can cross the membrane (in other words, a penetrating solute), drawing water with it, thus causing no net change in solution volume. ==Plasma osmolarity vs. osmolality== Plasma osmolarity can be calculated from plasma osmolality by the following equation: where: * is the density of the solution in g/ml, which is 1.025 g/ml for blood plasma. * is the (anhydrous) solute concentration in g/ml – not to be confused with the density of dried plasma According to IUPAC, osmolality is the quotient of the negative natural logarithm of the rational activity of water and the molar mass of water, whereas osmolarity is the product of the osmolality and the mass density of water (also known as osmotic concentration). Here, the difference in pressure of the two compartments \Pi \equiv p' - p is defined as the osmotic pressure exerted by the solutes. Also, the molar volume V_m may be written as volume per mole, V_m = V/n_v. In soil mechanics, specific weight may refer to: ===Civil and mechanical engineering=== Specific weight can be used in civil engineering and mechanical engineering to determine the weight of a structure designed to carry certain loads while remaining intact and remaining within limits regarding deformation. ==Specific weight of water== Specific weight of water at standard sea-level atmospheric pressure (Metric units) Temperature(°C) Specific weight (kN/m3) 0 9.805 5 9.807 10 9.804 15 9.798 20 9.789 25 9.777 30 9.765 40 9.731 50 9.690 60 9.642 70 9.589 80 9.530 90 9.467 100 9.399 Specific weight of water at standard sea-level atmospheric pressure (English units) Finnemore, J. E. (2002). The compartment containing the pure solvent has a chemical potential of \mu^0(p), where p is the pressure. The molecular formula C27H34O3 (molar mass: 406.56 g/mol, exact mass: 406.2508 u) may refer to: * Nandrolone phenylpropionate (NPP), or nandrolone phenpropionate * Testosterone phenylacetate The specific weight, also known as the unit weight, is the weight per unit volume of a material. Osmotic pressure is the basis of filtering (""reverse osmosis""), a process commonly used in water purification. ",-233,0.2553,"""8.8""",1.45,1.19,D
+ One mole of Ar initially at 310 . K undergoes an adiabatic expansion against a pressure $P_{\text {external }}=0$ from a volume of $8.5 \mathrm{~L}$ to a volume of $82.0 \mathrm{~L}$. Calculate the final temperature using the ideal gas,"We assume the expansion occurs without exchange of heat (adiabatic expansion). The constant volume adiabatic flame temperature is the temperature that results from a complete combustion process that occurs without any work, heat transfer or changes in kinetic or potential energy. The temperature may depend on pressure, because at lower pressure there will be more dissociation of the combustion products, implying a lower adiabatic temperature. ≈6332 Aluminium Oxygen 3732 6750 Lithium Oxygen 2438 4420 Phosphorus (white) Oxygen 2969 5376 Zirconium Oxygen 4005 7241 ==Thermodynamics== thumb|right|300px|First law of thermodynamics for a closed reacting system From the first law of thermodynamics for a closed reacting system we have :{}_RQ_P - {}_RW_P = U_P - U_R where, {}_RQ_P and {}_RW_P are the heat and work transferred from the system to the surroundings during the process, respectively, and U_R and U_P are the internal energy of the reactants and products, respectively. This is because some of the energy released during combustion goes, as work, into changing the volume of the control system. thumb|right|300px|Constant volume flame temperature of a number of fuels, with air If we make the assumption that combustion goes to completion (i.e. forming only and ), we can calculate the adiabatic flame temperature by hand either at stoichiometric conditions or lean of stoichiometry (excess air). For an ideal gas, the change in entropyTipler, P., and Mosca, G. Physics for Scientists and Engineers (with modern physics), 6th edition, 2008. pages 602 and 647. is the same as for isothermal expansion where all heat is converted to work: \Delta S = \int_\text{i}^\text{f} dS = \int_{V_\text{i}}^{V_\text{f}} \frac{P\,dV}{T} = \int_{V_\text{i}}^{V_\text{f}} \frac{n R\,dV}{V} = n R \ln \frac{V_\text{f}}{V_\text{i}} = N k_\text{B} \ln \frac{V_\text{f}}{V_\text{i}}. In the case of the constant pressure adiabatic flame temperature, the pressure of the system is held constant, which results in the following equation for the work: : {}_RW_P = \int\limits_R^P {pdV} = p\left( {V_P - V_R } \right) Again there is no heat transfer occurring because the process is defined to be adiabatic: {}_RQ_P = 0 . The above value of 1.4 is highly consistent with the measured adiabatic indices for dry air within a temperature range of 0–200 °C, exhibiting a deviation of only 0.2% (see tabulation above). With our present knowledge of the thermodynamic properties of air Refprop, software package developed by National Institute of Standards and Technology (NIST) we can calculate that the temperature of the air should drop by about 3 degrees Celsius when the volume is doubled under adiabatic conditions. There are a number of programs available that can calculate the adiabatic flame temperature taking into account dissociation through equilibrium constants (Stanjan, NASA CEA, AFTP). We see that the adiabatic flame temperature of the constant pressure process is lower than that of the constant volume process. In the constant volume adiabatic flame temperature case, the volume of the system is held constant and hence there is no work occurring: : {}_RW_P = \int\limits_R^P {pdV} = 0 There is also no heat transfer because the process is defined to be adiabatic: {}_RQ_P = 0 . The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_{0}, usually . During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion. For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to: : dh = v \, dp. For a monatomic ideal gas , with the molar heat capacity at constant volume. As a result, these substances will burn at a constant pressure, which allows the gas to expand during the process. == Common flame temperatures == Assuming initial atmospheric conditions (1bar and 20 °C), the following tableSee under ""Tables"" in the external references below. lists the flame temperature for various fuels under constant pressure conditions. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. In atmospheric science, equivalent temperature is the temperature of air in a parcel from which all the water vapor has been extracted by an adiabatic process. For an imperfect or non-ideal gas, Chandrasekhar defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure: : \begin{align} \Gamma_1 &= \left.\frac{\partial \ln P}{\partial \ln \rho}\right|_{S}, \\\\[2pt] \frac{\Gamma_2 - 1}{\Gamma_2} &= \left.\frac{\partial \ln T}{\partial \ln P}\right|_{S}, \\\\[2pt] \Gamma_3 - 1 &= \left.\frac{\partial \ln T}{\partial \ln \rho}\right|_{S}. \end{align} All of these are equal to \gamma in the case of an ideal gas. == See also == * Relations between heat capacities * Heat capacity * Specific heat capacity * Speed of sound * Thermodynamic equations * Thermodynamics * Volumetric heat capacity == References == == Notes == Category:Thermodynamic properties Category:Physical quantities Category:Ratios Category:Thought experiments in physics There are two types adiabatic flame temperature: constant volume and constant pressure, depending on how the process is completed. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as: T = T_i 2^{-R/C_V} = T_i2^{-2/3} for the monoatomic ideal gas. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. ",292,2.3,"""9.8""",0,310,E
+"A refrigerator is operated by a $0.25-\mathrm{hp}(1 \mathrm{hp}=$ 746 watts) motor. If the interior is to be maintained at $4.50^{\circ} \mathrm{C}$ and the room temperature on a hot day is $38^{\circ} \mathrm{C}$, what is the maximum heat leak (in watts) that can be tolerated? Assume that the coefficient of performance is $50 . \%$ of the maximum theoretical value. What happens if the leak is greater than your calculated maximum value?","The basic SI units equation for deriving cooling capacity is of the form: :\dot{Q}=\dot{m}C_p\Delta T Where :\dot{Q} is the cooling capacity [kW] :\dot{m} is the mass rate [kg/s] :C_p is the specific heat capacity [kJ/kg K] :\Delta T is the temperature change [K] ==References== Category:Heating, ventilation, and air conditioning As the target temperature of the refrigerator approaches ambient temperature, without exceeding it, the refrigeration capacity increases thus increasing the refrigerator's COP. This is a table of specific heat capacities by magnitude. Cooling capacity is the measure of a cooling system's ability to remove heat. Modern cogeneration plants have power loss ratios of about 1/5 to 1/9 when delivering heat in the range of 80 °C-120 °C.Danny Harvey: Clean building - contribution from cogeneration, trigeneration and district energy, Cogeneration and On-Site Power Production, september–october 2006, pp. 107-115 (Fig. 1) That means in exchange of one kWh of electrical energy ca. 5 up to 9 kWh of useful heat are obtained. The power loss factor β describes the loss of electrical power in CHP systems with a variable power-to-heat ratio when an increasing heat flow is extracted from the main thermodynamic electricity generating process in order to provide useful heat. right|thumb|150px|Thermal mass refrigerator A thermal mass refrigerator is a refrigerator that is foreseen with thermal mass as well as insulation to decrease the energy use of the refrigerator. A refrigerator death is death by suffocation in a refrigerator or other air- tight appliance. A particularly popular thermal mass refrigerator was conceived by Michael Reynolds and detailed in the book ""Earthship Volume 3"". On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . This refrigerator was a DIY refrigerator designed around a (Sun-Frost) DC refrigeration unit run on P.V. panels.Earthship Volume 3 by Michael Reynolds ==Design details== The thermal mass used in Michael Reynolds' design is a combination of a liquid (i.e. water or beer) together with concrete mass. The partial steam flow, which goes into the heating condenser at high temperature can no longer work in the low-pressure section and is responsible for the loss of power. It is equivalent to the heat supplied to the evaporator/boiler part of the refrigeration cycle and may be called the ""rate of refrigeration"" or ""refrigeration capacity"". The Refrigerator Safety Act in 1956 was a U.S. law that required a change in the way refrigerator doors stay shut. The Refrigerator Safety Act was a factor in the decline, in combination with other factors such as ""reduced exposure and increased parental supervision"". ==Entrapment hazards== Hazardous items for refrigerator deaths are ""places with a poor air supply, a heavy lid or a self-latching door"". The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size- dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Parents or caregivers can lessen the risks of refrigerator deaths. Another unit common in non-metric regions or sectors is the ton of refrigeration, which describes the amount of water at freezing temperature that can be frozen in 24 hours, equivalent to or . In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. Usually, the power loss factor refers to extraction steam turbines in thermal power stations, which conduct a part of the steam in a heating condenser for the production of useful heat, instead of the low pressure part of the steam turbine where is could perform mechanical work. thumb|Power loss within an extraction steam turbine: CHP plant section (left) and T-s-diagram (right) \beta = \frac{\Delta P_\text{el}}{\dot Q_\text{utile}} The picture on the right shows in the left part the principle of steam extraction. Based on the equivalence of power loss and gain of heat, the power loss method assigns CO2 emissions and primary energy from the fuel to the useful heat and the electrical energy. == References == Category:Cogeneration Category:Energy conversion The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio. ==Derivation== If an infinitesimally small amount of heat \delta Q is supplied to a system in a reversible way then, according to the second law of thermodynamics, the entropy change of the system is given by: :dS = \frac{\delta Q}{T}\, Since :\delta Q = C dT\, where C is the heat capacity, it follows that: :T dS = C dT\, The heat capacity depends on how the external variables of the system are changed when the heat is supplied. ",-32,15.425,"""1.8763""", 6.07,773,E
+"In order to get in shape for mountain climbing, an avid hiker with a mass of $60 . \mathrm{kg}$ ascends the stairs in the world's tallest structure, the $828 \mathrm{~m}$ tall Burj Khalifa in Dubai, United Arab Emirates. Assume that she eats energy bars on the way up and that her body is $25 \%$ efficient in converting the energy content of the bars into the work of climbing. How many energy bars does she have to eat if a single bar produces $1.08 \times 10^3 \mathrm{~kJ}$ of energy upon metabolizing?","Manufacturing of energy bars may supply nutrients in sufficient quantity to be used as meal replacements. ==Nutrition== A typical energy bar weighs between 30 and 50 g and is likely to supply about 200–300 Cal (840–1,300 kJ), 3–9 g of fat, 7–15 g of protein, and 20–40 g of carbohydrates — the three sources of energy in food. Energy bars are supplemental bars containing cereals, micronutrients, and flavor ingredients intended to supply quick food energy. Energy bars may be used as an energy source during athletic events such as marathons, triathlons and other activities which require a high energy expenditure for long periods of time. Because most energy bars contain added protein, carbohydrates, dietary fiber, and other nutrients, they may be marketed as functional foods. YouBar is an online nutrition bar company that makes customized energy bars. The dietary energy supply is the food available for human consumption, usually expressed in kilocalories or kilojoules per person per day. thumb|Energy bars vary in size, ingredients, and nutritional benefits. CalorieMate (カロリーメイト karorīmeito) is a brand of nutritional energy bar and energy gel foods produced by Otsuka Pharmaceutical Co., in Japan. Physical Activity MET Light Intensity Activities < 3 sleeping 0.9 watching television 1.0 writing, desk work, typing 1.8 walking, 1.7 mph (2.7 km/h), level ground, strolling, very slow 2.3 walking, 2.5 mph (4 km/h) 2.9 Moderate Intensity Activities 3 to 6 bicycling, stationary, 50 watts, very light effort 3.0 walking 3.0 mph (4.8 km/h) 3.3 calisthenics, home exercise, light or moderate effort, general 3.5 walking 3.4 mph (5.5 km/h) 3.6 bicycling, <10 mph (16 km/h), leisure, to work or for pleasure 4.0 bicycling, stationary, 100 watts, light effort 5.5 Vigorous Intensity Activities > 6 jogging, general 7.0 calisthenics (e.g. pushups, situps, pullups, jumping jacks), heavy, vigorous effort 8.0 running jogging, in place 8.0 rope jumping 10.0 ==Fuel Used== The body uses different amounts of energy substrates (carbohydrates or fats) depending on the intensity of the exercise and the heart rate of the exerciser. One MET, which is equal to 3.5 mL/kg per minute, is considered to be the average resting energy expenditure of a typical human being. High intensity activity also yields a higher total caloric expenditure. For those who are malnourished, energy bars, such as Plumpy'nut, are an effective tool for treating malnutrition. == See also == * Candy bar * Protein bar * Energy gel * Sports drink * High energy biscuits * Flapjack (oat bar) * D ration ==References== Category:Dietary supplements Category:Energy food products Category:Snack foods Dietary energy supply is correlated with the rate of obesity. ==Regions== Daily dietary energy supply per capita: Region 1964-1966 1974-1976 1984-1986 1997-1999 World Sub-Saharan Africa ==Countries== Country 1979-1981 1989-1991 2001-2003 Canada China Eswatini United Kingdom United States ==See also== *Diet and obesity thumb|300x300px|Average dietary energy supply by region ==References== ==External links== * Category:Obesity Category:Diets An intensity of exercise equivalent to 6 METs means that the energy expenditure of the exercise is six times the resting energy expenditure.Vehrs, P., Ph.D. (2011). Intensity of exercise can be expressed as multiples of resting energy expenditure. On the other hand, high intensity activity utilizes a larger percentage of carbohydrates in the calories expended because its quick production of energy makes it the preferred energy substrate for high intensity exercise. Intensity (%MHR) Heart Rate (bpm) % Carbohydrate % Fat 65-70 130-140 15 85 70-75 140-150 35 65 75-80 150-160 65 35 80-85 160-170 80 20 85-90 170-180 90 10 90-95 180-190 95 5 100 190-200 100 - These estimates are valid only when glycogen reserves are able to cover the energy needs. Fats sources are often cocoa butter and dark chocolate. == Usage == Energy bars are used in a variety of contexts. right|300px Exercise intensity refers to how much energy is expended when exercising. Protein is a third energy substrate, but it contributes minimally and is therefore discounted in the percent contribution graphs reflecting different intensities of exercise. This table outlines the estimated distribution of energy consumption at different intensity levels for a healthy 20-year-old with a Max Heart Rate (MHR) of 200. thumb|A Lärabar bar Lärabar is a brand of energy bars produced by General Mills. ",1.8, -2.5,"""1.6""",0.082,5300,A
+The half-life of ${ }^{238} \mathrm{U}$ is $4.5 \times 10^9$ years. How many disintegrations occur in $1 \mathrm{~min}$ for a $10 \mathrm{mg}$ sample of this element?,"The short half-life of 87.7 years of 238Pu means that a large amount of it decayed during its time inside his body, especially when compared to the 24,100 year half-life of 239Pu. The half-life of 242Pu is about 15 times that of 239Pu; so it is one-fifteenth as radioactive, and not one of the larger contributors to nuclear waste radioactivity. 242Pu's gamma ray emissions are also weaker than those of the other isotopes. Plutonium-242 decays via spontaneous fission in about 5.5 × 10−4% of casesChart of all nuclei which includes half life and mode of decay ==References== Category:Actinides Category:Nuclear materials Category:Isotopes of plutonium Plutonium-244 (244Pu) is an isotope of plutonium that has a half-life of 80 million years. Plutonium-238 (238Pu or Pu-238) is a fissile, radioactive isotope of plutonium that has a half-life of 87.7 years. Modern calculations of his lifetime absorbed dose give an incredible 64 Sv (6400 rem) total. ===Weapons=== The first application of 238Pu was its use in nuclear weapon components made at Mound Laboratories for Lawrence Radiation Laboratory (now Lawrence Livermore National Laboratory). Presence of 244Pu fission tracks can be established by using the initial ratio of 244Pu to 238U (Pu/U)0 at a time T0 = years, when Xe formation first began in meteorites, and by considering how the ratio of Pu/U fission tracks varies over time. Plutonium-242 (242Pu or Pu-242) is one of the isotopes of plutonium, the second longest-lived, with a half-life of 375,000 years. This is longer than any of the other isotopes of plutonium and longer than any other actinide isotope except for the three naturally abundant ones: uranium-235 (704 million years), uranium-238 (4.468 billion years), and thorium-232 (14.05 billion years). The amount of 244Pu in the pre-Solar nebula (4.57×109 years ago) was estimated as 0.8% the amount of 238U..As the age of the Earth is about 57 half-lives of 244Pu, the amount of plutonium-244 left should be very small; Hoffman et al. estimated its content in the rare-earth mineral bastnasite as = 1.0×10−18 g/g, which corresponded to the content in the Earth crust as low as 3×10−25 g/g (i.e. the total mass of plutonium-244 in Earth's crust is about 9 g). This gives a density for 238Pu of (1.66053906660×10−24g/dalton×238.0495599 daltons/atom×16 atoms/unit cell)/(319.96 Å3/unit cell × 10−24cc/Å3) or 19.8 g/cc. It decays by electron capture to stable cadmium-111 with a half-life of 2.8 days. However, 238Pu is far more dangerous than 239Pu due to its short half-life and being a strong alpha-emitter. Historically, most plutonium-238 has been produced by Savannah River in their weapons reactor, by irradiating with neutrons neptunium-237 (half life ). \+ → Neptunium-237 is a by-product of the production of plutonium-239 weapons-grade material, and when the site was shut down in 1988, 238Pu was mixed with about 16% 239Pu. ===Human radiation experiments=== Plutonium was first synthesized in 1940 and isolated in 1941 by chemists at the University of California, Berkeley. They also reported an even longer half-life for alpha decay of bismuth-209 to the first excited state of thallium-205 (at 204 keV), was estimated to be 1.66 years. In February 2013, a small amount of 238Pu was successfully produced by Oak Ridge's High Flux Isotope Reactor, and on December 22, 2015, they reported the production of of 238Pu. Although 209Bi holds the half-life record for alpha decay, bismuth does not have the longest half-life of any radionuclide to be found experimentally--this distinction belongs to tellurium-128 (128Te) with a half-life estimated at 7.7 × 1024 years by double β-decay (double beta decay). Bismuth-209 (209Bi) is the isotope of bismuth with the longest known half-life of any radioisotope that undergoes α-decay (alpha decay). Significant amounts of pure 238Pu could also be produced in a thorium fuel cycle. The density of plutonium-238 at room temperature is about 19.8 g/cc.Calculated from the atomic weight and the atomic volume. However, 242Pu's low cross section means that relatively little of it will be transmuted during one cycle in a thermal reactor. ==Decay== Plutonium-242 mainly decays into uranium-238 via alpha decay, before continuing along the uranium series. The material will generate about 0.57 watts per gram of 238Pu. ",1.43,1.2,"""313.0""",1.5,-1.46,A
+"Calculate the ionic strength in a solution that is 0.0750 $m$ in $\mathrm{K}_2 \mathrm{SO}_4, 0.0085 \mathrm{~m}$ in $\mathrm{Na}_3 \mathrm{PO}_4$, and $0.0150 \mathrm{~m}$ in $\mathrm{MgCl}_2$.","For the electrolyte MgSO4, however, each ion is doubly- charged, leading to an ionic strength that is four times higher than an equivalent concentration of sodium chloride: :I = \frac{1}{2}[c(+2)^2+c(-2)^2] = \frac{1}{2}[4c + 4c] = 4c Generally multivalent ions contribute strongly to the ionic strength. ===Calculation example=== As a more complex example, the ionic strength of a mixed solution 0.050 M in Na2SO4 and 0.020 M in KCl is: : \begin{align} I & = \tfrac 1 2 \times \left[\begin{array}{l} \\{(\text{concentration of }\ce{Na2SO4}\text{ in M}) \times (\text{number of }\ce{Na+}) \times (\text{charge of }\ce{Na+})^2\\}\ + \\\ \\{(\text{concentration of }\ce{Na2SO4}\text{ in M}) \times (\text{number of }\ce{SO4^2-}) \times (\text{charge of }\ce{SO4^2-})^2\\} \ + \\\ \\{(\text{concentration of }\ce{KCl}\text{ in M}) \times (\text{number of }\ce{K+}) \times (\text{charge of }\ce{K+})^2\\}\ + \\\ \\{(\text{concentration of }\ce{KCl}\text{ in M}) \times (\text{number of }\ce{Cl-}) \times (\text{charge of }\ce{Cl-})^2\\} \end{array}\right] \\\ & = \tfrac 1 2 \times [\\{0.050 M \times 2 \times (+1)^2\\} + \\{0.050 M \times 1 \times (-2)^2\\} + \\{0.020 M \times 1 \times (+1)^2\\} + \\{0.020 M \times 1 \times (-1)^2\\}] \\\ & = 0.17 M \end{align} ==Non-ideal solutions== Because in non-ideal solutions volumes are no longer strictly additive it is often preferable to work with molality b (mol/kg of H2O) rather than molarity c (mol/L). Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic strength can be molar (mol/L solution) or molal (mol/kg solvent) and to avoid confusion the units should be stated explicitly. For a 1:1 electrolyte such as sodium chloride, where each ion is singly-charged, the ionic strength is equal to the concentration. One of the main characteristics of a solution with dissolved ions is the ionic strength. The concept of ionic strength was first introduced by Lewis and Randall in 1921 while describing the activity coefficients of strong electrolytes. ==Quantifying ionic strength== The molar ionic strength, I, of a solution is a function of the concentration of all ions present in that solution. In that case, molal ionic strength is defined as: : I = \frac{1}{2}\sum_{{i}=1}^{n} b_{i}z_{i}^{2} in which :i = ion identification number :z = charge of ion :b = molality (mol solute per Kg solvent)Standard definition of molality ==Importance== The ionic strength plays a central role in the Debye–Hückel theory that describes the strong deviations from ideality typically encountered in ionic solutions. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. Ionic radius, rion, is the radius of a monatomic ion in an ionic crystal structure. That is, the Debye length, which is the inverse of the Debye parameter (κ), is inversely proportional to the square root of the ionic strength. In condensed matter physics and inorganic chemistry, the cation-anion radius ratio can be used to predict the crystal structure of an ionic compound based on the relative size of its atoms. Comparison between observed and calculated ion separations (in pm) MX Observed Soft-sphere model LiCl 257.0 257.2 LiBr 275.1 274.4 NaCl 282.0 281.9 NaBr 298.7 298.2 In the soft-sphere model, k has a value between 1 and 2. Magnesium orthosilicate is a chemical compound with the formula Mg2SiO4.Magnesium orthosilicate at Chemister It is the orthosilicate salt of magnesium. _Effective_ ionic radii in pm of elements as a function of ionic charge and spin (ls = low spin, hs = high spin). Curiously, no theoretical justification for the equation containing k has been given. == Non-spherical ions == The concept of ionic radii is based on the assumption of a spherical ion shape. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as ""effective"" ionic radii. One approach to improving the calculated accuracy is to model ions as ""soft spheres"" that overlap in the crystal. Natural waters such as mineral water and seawater have often a non-negligible ionic strength due to the presence of dissolved salts which significantly affects their properties. ==See also== * Activity (chemistry) * Activity coefficient * Bromley equation * Davies equation * Debye–Hückel equation * Debye–Hückel theory * Double layer (interfacial) * Double layer (electrode) * Double layer forces * Electrical double layer * Gouy-Chapman model * Flocculation * Peptization (the inverse of flocculation) * DLVO theory (from Derjaguin, Landau, Verwey and Overbeek) * Interface and colloid science * Osmotic coefficient * Pitzer equations * Poisson–Boltzmann equation * Specific ion Interaction Theory * Salting in * Salting out ==External links== * Ionic strength * Ionic strength introduction at the EPA web site == References == Category:Analytical chemistry Category:Colloidal chemistry Category:Electrochemical equations Category:Electrochemical concepts Category:Equilibrium chemistry Category:Physical chemistry Category:Physical quantities Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. Unit cell parameters (in pm, equal to two M–X bond lengths) for sodium and silver halides. All compounds crystallize in the NaCl structure. thumb|300 px|Relative radii of atoms and ions. ",1.88,0.7854,"""9.73""",0.6749,0.321,E
+"The interior of a refrigerator is typically held at $36^{\circ} \mathrm{F}$ and the interior of a freezer is typically held at $0.00^{\circ} \mathrm{F}$. If the room temperature is $65^{\circ} \mathrm{F}$, by what factor is it more expensive to extract the same amount of heat from the freezer than from the refrigerator? Assume that the theoretical limit for the performance of a reversible refrigerator is valid in this case.","Direct Cool Vs Frost Free Refrigerators – Know the Differences Direct cool is less expensive in production and in operation, as it consumes less energy when compared to frost free refrigerators ==References== Category:Refrigerants 2\. While having the same total pressure throughout the system, the refrigerator maintains a low partial pressure of the refrigerant (therefore high evaporation rate) in the part of the system that draws heat out of the low-temperature interior of the refrigerator, but maintains the refrigerant at high partial pressure (therefore low evaporation rate) in the part of the system that expels heat to the ambient-temperature air outside the refrigerator. A single-pressure absorption refrigerator takes advantage of the fact that a liquid's evaporation rate depends upon the partial pressure of the vapor above the liquid and goes up with lower partial pressure. An absorption refrigerator is a refrigerator that uses a heat source (e.g., solar energy, a fossil-fueled flame, waste heat from factories, or district heating systems) to provide the energy needed to drive the cooling process. Heat flows from the hotter interior of the refrigerator to the colder liquid, promoting further evaporation. 3\. A refrigerator designed to reach cryogenic temperatures (below ) is often called a cryocooler. The refrigerator was less efficient than existing appliances, although having no moving parts made it more reliable; the introduction of non-toxic Freon — later found to be responsible for serious depletion of the Earth's ozone layer — to replace toxic refrigerant gases made it even less attractive commercially. Progress in the cryocooler field in recent decades is in large part due to development of new materials having high heat capacity below 10 K.T. Kuriyama, R. Hakamada, H. Nakagome, Y. Tokai, M. Sahashi, R. Li, O. Yoshida, K. Matsumoto, and T. Hashimoto, Advances in Cryogenic Engineering 35B, 1261 (1990) == Stirling refrigerators == ===Components=== 300px|thumb| Fig.1 Schematic diagram of a Stirling cooler. right|thumb|150px|Thermal mass refrigerator A thermal mass refrigerator is a refrigerator that is foreseen with thermal mass as well as insulation to decrease the energy use of the refrigerator. In the 1960s, absorption refrigeration saw a renaissance due to the substantial demand for refrigerators for caravans (travel trailers). Unlike more common vapor-compression refrigeration systems, an absorption refrigerator has no moving parts. ==History== In the early years of the 20th century, the vapor absorption cycle using water- ammonia systems was popular and widely used, but after the development of the vapor compression cycle it lost much of its importance because of its low coefficient of performance (about one fifth of that of the vapor compression cycle). An absorption refrigerator changes the gas back into a liquid using a method that needs only heat, and has no moving parts other than the fluids. 300px|right|Absorption cooling process The absorption cooling cycle can be described in three phases: #Evaporation: A liquid refrigerant evaporates in a low partial pressure environment, thus extracting heat from its surroundings (e.g. the refrigerator's compartment). Direct-cool refrigerators produce the cooling effect by a natural convection process from cooled surfaces in the insulated compartment that is being cooled. Compression refrigerators typically use an HCFC or HFC, while absorption refrigerators typically use ammonia or water and need at least a second fluid able to absorb the coolant, the absorbent, respectively water (for ammonia) or brine (for water). The water evaporated from the salt solution is re- condensed, and rerouted back to the evaporative cooler. ===Single pressure absorption refrigeration=== thumb|right|300px|Domestic absorption refrigerator. 1\. The refrigerator is a small unit placed over a campfire, that can later be used to cool of water to just above freezing for 24 hours in a environment. The main difference between the two systems is the way the refrigerant is changed from a gas back into a liquid so that the cycle can repeat. thumb|200px|Einstein's and Szilárd's patent application thumb|200px|Annotated patent drawing The Einstein–Szilard or Einstein refrigerator is an absorption refrigerator which has no moving parts, operates at constant pressure, and requires only a heat source to operate. This refrigerator was a DIY refrigerator designed around a (Sun-Frost) DC refrigeration unit run on P.V. panels.Earthship Volume 3 by Michael Reynolds ==Design details== The thermal mass used in Michael Reynolds' design is a combination of a liquid (i.e. water or beer) together with concrete mass. Direct cool is one of the two major types of techniques used in domestic refrigerators, the other being the ""frost-free"" type. The ""Tang-Dresselhaus Theory"" (Shuang Tang and Mildred Dresselhaus) has pointed out that anisotropic transport behaviors of quantum confined BiSb alloys nanostructures can optimize the pertinent thermoelectric cooling performance below 77 K for applications in satellites and space stations. ==See also== * Cryogenic processor * Adiabatic demagnetization refrigerator * Dilution refrigerator * Hampson-Linde cycle * Pulse tube refrigerator * Stirling engine (Stirling cryocooler) * Entropy production ==References== Category:Cooling technology Category:Cryogenics Category:Industrial gases ",-242.6,2.4,"""1.3""",0.0547,13.2,B
+"Calculate the rotational partition function for $\mathrm{SO}_2$ at $298 \mathrm{~K}$ where $B_A=2.03 \mathrm{~cm}^{-1}, B_B=0.344 \mathrm{~cm}^{-1}$, and $B_C=0.293 \mathrm{~cm}^{-1}$","For each value of J, we have rotational degeneracy, g_j = (2J+1), so the rotational partition function is therefore \zeta^\text{rot} = \sum_{J=0}^\infty g_j e^{-E_J/k_\text{B} T} = \sum_{J=0}^\infty (2J+1) e^{-J(J+1) B / k_\text{B} T}. In terms of these constants, the rotational partition function can be written in the high temperature limit as G. Herzberg, ibid, Equation (V,29) \zeta^\text{rot} \approx \frac{\sqrt{\pi} }{\sigma} \sqrt{ \frac{ (k_\text{B} T)^3 }{ A B C }} with \sigma again known as the rotational symmetry number G. Herzberg, ibid; see Table 140 for values for common molecular point groups which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms. Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook: \zeta^\text{rot} \approx \frac{5.34 \times 10^6}{\sigma} \sqrt{ \frac{T^3}{A B C}} where the prefactor comes from \sqrt{\frac{(\pi k_\text{B}) ^3}{h^3}} = 5.34 \times 10^6 when A, B, and C are expressed in units of MHz. The expressions for \zeta^\text{rot} works for asymmetric, symmetric and spherical top rotors. ==References== ==See also== * Translational partition function * Vibrational partition function * Partition function (mathematics) Category:Equations of physics Category:Partition functions In chemistry, the rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. == Definition == The total canonical partition function Z of a system of N identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions \zeta:Donald A. McQuarrie, Statistical Mechanics, Harper & Row, 1973 Z = \frac{ \zeta^N }{ N! } with: \zeta = \sum_j g_j e^{ -E_j / k_\text{B} T} , where g_j is the degeneracy of the j-th quantum level of an individual particle, k_\text{B} is the Boltzmann constant, and T is the absolute temperature of system. Atkins and J. de Paula ""Physical Chemistry"", 9th edition (W.H. Freeman 2010), p.597 :\theta_{\mathrm{R}} = \frac{hc \overline{B}}{k_{\mathrm{B}}} = \frac{\hbar^2}{2k_{\mathrm{B}}I}, where \overline{B} = B/hc is the rotational constant, is a molecular moment of inertia, is the Planck constant, is the speed of light, is the reduced Planck constant and is the Boltzmann constant. For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are E_J^\text{rot} = \frac{\mathbf{J}^2}{2I} = \frac{J(J+1)\hbar^2}{2I} = J(J+1)B. J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e., J = 0,1,2, \ldots, B = \frac{\hbar^2}{2I} is the rotational constant, and I is the moment of inertia. thumb|350px|Dini's Surface with constants a = 1, b = 0.5 and 0 ≤ u ≤ 4 and 0 It is named after Ulisse Dini and described by the following parametric equations: : \begin{align} x&=a \cos u \sin v \\\ y&=a \sin u \sin v \\\ z&=a \left(\cos v +\ln \tan \frac{v}{2} \right) + bu \end{align} thumb|350px|right|Dini's surface with 0 ≤ u ≤ 4 and 0.01 ≤ v ≤ 1 and constants a = 1.0 and b = 0.2. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945, Equation (V,21) \zeta^\text{rot} = \frac{ k_\text{B} T}{B} + \frac{1}{3} + \frac{1}{15} \left( \frac{B}{ k_\text{B} T} \right) + \frac{4}{315} \left( \frac{B}{k_\text{B} T} \right)^2 + \frac{1}{315} \left( \frac{B}{k_\text{B} T} \right)^3 + \cdots . In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor \sigma = 2 with \sigma known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without. ==Nonlinear molecules== A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written A, B, and C , which can often be determined by rotational spectroscopy. This suggests we can approximate the sum by replacing the sum over J by an integral of J treated as a continuous variable. \zeta^\text{rot} \approx \int_0^{\infty} (2J+1)e^{-J(J+1) B /k_\text{B} T} dJ = \frac{ k_\text{B} T}{B} . In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is k_\text{B} T . == Quantum symmetry effects == For a diatomic molecule with a center of symmetry, such as \rm H_2, N_2, CO_2, or \mathrm{ H_2 C_2} (i.e. D_{\infty h} point group), rotation of a molecule by \pi radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. The mean thermal rotational energy per molecule can now be computed by taking the derivative of \zeta^\text{rot} with respect to temperature T. The characteristic rotational temperature ( or ) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. ASM Handbook; Alloy Phase Diagrams; v. 3; ASM International, USA; 1992, pp. 491–492. For molecules, under the assumption that total energy levels E_j can be partitioned into its contributions from different degrees of freedom (weakly coupled degrees of freedom)Donald A. McQuarrie, ibid E_j = \sum_i E_j^i = E_j^\text{trans} + E_j^\text{ns} + E_j^\text{rot} + E_j^\text{vib} + E_j^\text{e} and the number of degenerate states are given as products of the single contributions g_j = \prod_i g_j^i = g_j^\text{trans} g_j^\text{ns} g_j^\text{rot} g_j^\text{vib} g_j^\text{e}, where ""trans"", ""ns"", ""rot"", ""vib"" and ""e"" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions \zeta = \sum_j g_j e^{-E_j/k_\text{B} T} can be written as a product itself \zeta = \prod_i \zeta^i = \zeta^\text{trans} \zeta^\text{ns} \zeta^\text{rot} \zeta^\text{vib}\zeta^\text{e}. == Linear molecules == Rotational energies are quantized. This alloy presents a eutectic temperature of 382 K (109 °C; 228.2 °F). On 19 February 1772, the agreement of partition was signed in Vienna. thumb|right|300px|Picture of Europe for July 1772, satirical British plate The Partition Sejm () was a Sejm lasting from 1773 to 1775 in the Polish–Lithuanian Commonwealth, convened by its three neighbours (the Russian Empire, Prussia and Austria) in order to legalize their First Partition of Poland. *BiIn2 (from 52.5 to 53.5 wt% of In), having a hexagonal structure with 2 atoms per unit cell. The Sejm on 30 September 1773 accepted the partition treaty. Atkins and J. de Paula ""Physical Chemistry"", 10th edition, Table 12D.1, p.987 ==References== ==See also== *Rotational spectroscopy *Vibrational temperature *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics ",0.312, 252.8,"""273.0""",3.52,5840,E
+"For a two-level system where $v=1.50 \times 10^{13} \mathrm{~s}^{-1}$, determine the temperature at which the internal energy is equal to $0.25 \mathrm{Nhv}$, or $1 / 2$ the limiting value of $0.50 \mathrm{Nhv}$.","The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy. The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T : U = C_V n T, where C_V is the isochoric (at constant volume) molar heat capacity of the gas. Knowing temperature and pressure to be the derivatives T = \frac{\partial U}{\partial S}, P = -\frac{\partial U}{\partial V}, the ideal gas law PV = nRT immediately follows as below: : T = \frac{\partial U}{\partial S} = \frac{U}{C_V n} : P = -\frac{\partial U}{\partial V} = U \frac{R}{C_V V} : \frac{P}{T} = \frac{\frac{U R}{C_V V}}{\frac{U}{C_V n}} = \frac{n R}{V} : PV = nRT ==Internal energy of a closed thermodynamic system== The above summation of all components of change in internal energy assumes that a positive energy denotes heat added to the system or the negative of work done by the system on its surroundings. The differential internal energy may be written as :\mathrm{d} U = \frac{\partial U}{\partial S} \mathrm{d} S + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - P \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i, which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure P to be the negative of the similar derivative with respect to volume V, : T = \frac{\partial U}{\partial S}, : P = -\frac{\partial U}{\partial V}, and where the coefficients \mu_{i} are the chemical potentials for the components of type i in the system. The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The expression relating changes in internal energy to changes in temperature and volume is : \mathrm{d}U =C_{V} \, \mathrm{d}T +\left[T\left(\frac{\partial P}{\partial T}\right)_{V} - P\right] \mathrm{d}V. Appendix D. ISBN 978-1305079113 Na sodium 107 K potassium 89 Rb rubidium 81 Cs caesium 76 Mg magnesium 148 Ca calcium 178 Sr strontium 164 Ba barium 180 Fe iron 416 Ni nickel 430 Cu copper 338 Zn zinc 131 Ag silver 285 W tungsten 849 Au gold 366 C graphite 717 C diamond 715 Si silicon 456 Sn tin 302 Pb lead 195 I2 iodine 62.4 C10H8 naphthalene 72.9 CO2 carbon dioxide 25 ==See also== * Heat * Sublimation (chemistry) * Phase transition * Clausius-Clapeyron equation == References == Category:Enthalpy For a closed system, with transfers only as heat and work, the change in the internal energy is : \mathrm{d} U = \delta Q - \delta W, expressing the first law of thermodynamics. Although the heat capacity has a peak, it does not tend towards infinity (contrary to what the graph may suggest), but has finite limiting values when approaching the transition from above and below. The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree: : U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots), where \alpha is a factor describing the growth of the system. For those phase transitions specific heat does tend to infinity. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U(n,T). If a real gas can be described by the van der Waals equation of state p = \frac{nRT}{V-nb} - a \frac{n^2}{V^2} it follows from the thermodynamic equation of state that \pi_T = a \frac{n^2}{V^2} Since the parameter a is always positive, so is its internal pressure: internal energy of a van der Waals gas always increases when it expands isothermally. A Bordwell thermodynamic cycle use experimentally determined and reasonable estimates of Gibbs free energy (ΔG˚) values to determine unknown and experimentally inaccessible values. == Overview == Analogous to Hess's Law which deal with the summation of enthalpy (ΔH) values, Bordwell thermodynamic cycles deal with the summation of Gibbs free energy (ΔG) values. On the energetics of maximum-entropy temperature profiles, Q. J. R. Meteorol. The equation of state is the ideal gas law :P V = n R T. Solve for pressure: :P = \frac{n R T}{V}. The change in internal energy becomes : \mathrm{d}U = T \, \mathrm{d}S - P \, \mathrm{d}V. ===Changes due to temperature and volume=== The expression relating changes in internal energy to changes in temperature and volume is This is useful if the equation of state is known. Generalized Thermodynamics, M.I.T. Press, Cambridge MA. Equilibrium Thermodynamics, second edition, McGraw-Hill, London, . In thermodynamics, the enthalpy of sublimation, or heat of sublimation, is the heat required to sublimate (change from solid to gas) one mole of a substance at a given combination of temperature and pressure, usually standard temperature and pressure (STP). ",2.2,655,"""0.000216""",46.7,0.000226,B
+Calculate $K_P$ at $600 . \mathrm{K}$ for the reaction $\mathrm{N}_2 \mathrm{O}_4(l) \rightleftharpoons 2 \mathrm{NO}_2(g)$ assuming that $\Delta H_R^{\circ}$ is constant over the interval 298-725 K.,"The molecular formula C18H12O4 (molar mass: 292.28 g/mol, exact mass: 292.0736 u) may refer to: * Karanjin * Polyporic acid Category:Molecular formulas The molecular formula C6H6N4O4 (molar mass: 198.14 g/mol, exact mass: 198.0389 u) may refer to: * 2,4-Dinitrophenylhydrazine * Nitrofurazone The molecular formula C6H7KO6 (molar mass: 214.21 g/mol, exact mass: 213.9880 u) may refer to: * Potassium ascorbate * Potassium erythorbate The molecular formula C12H14O4 (molar mass: 222.23 g/mol, exact mass: 222.0892 u) may refer to: * Apiole * Blattellaquinone * Diethyl phthalate * Dillapiole * Monobutyl phthalate The molecular formula C27H33NO4 (molar mass: 435.56 g/mol, exact mass: 435.2410 u) may refer to: * Paxilline, a potassium channel blocker * BU-48 The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas The molecular formula C22H16N4O (molar mass: 352.39 g/mol, exact mass: 352.1324 u) may refer to: * Sudan III Category:Molecular formulas ",0.33333333,1.7,"""2.25""",0.00017,4.76,E
+"Count Rumford observed that using cannon boring machinery a single horse could heat $11.6 \mathrm{~kg}$ of ice water $(T=273 \mathrm{~K})$ to $T=355 \mathrm{~K}$ in 2.5 hours. Assuming the same rate of work, how high could a horse raise a $225 \mathrm{~kg}$ weight in 2.5 minutes? Assume the heat capacity of water is $4.18 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~g}^{-1}$.","Based on differences in the definition of what constitutes the ""power of a horse"", a horsepower-hour differs slightly from the German ""Pferdestärkenstunde"" (PSh): :1.014 PSh = 1 hp⋅h = 1,980,000 lbf⋅ft = 0.7457 kW⋅h. :1  The unit represents an amount of work a horse is supposed capable of delivering during an hour (1 horsepower integrated over a time interval of an hour). Pound per hour is a mass flow unit. For example, if Railroad A borrows a 2,500 horsepower locomotive from Railroad B and operates it for twelve hours, Railroad A owes a debt of (2,500 hp × 12 h) = 30,000 hp⋅h. In the US utility industry, steam and water flows throughout turbine cycles are typically expressed in PPH, while in Europe these mass flows are usually expressed in metric tonnes per hour: :1 lb/h = 0.4535927 kg/h = 126.00 mg/s Minimum fuel intake on a jumbo jet can be as low as 150 lb/h when idling; however, this is not enough to sustain flight. PSh = 0.73549875 kW⋅h = 2647.7955 kJ (exactly by definition) The horsepower-hour is still used in the railroad industry when sharing motive power (locomotives). The steam to oil ratio is a measure of the water and energy consumption related to oil production in cyclic steam stimulation and steam assisted gravity drainage oil production. Humber Fifteen 15 horsepower cars were medium to large cars, classified as medium weight, with a less powerful than usual engine which attracted less annual taxation and provided more stately progress. Their equivalent prewar car with an engine of 3.3 Litres had twin overhead camshafts. ===Bodies=== The 15.9 was available as a saloon or a 5-seater tourer. Ice Water (foaled 1963 in Ontario) was a Canadian Thoroughbred racehorse. ==Background== Ice Water was a bay mare owned and bred by George Gardiner. This means two to eight barrels of water converted into steam is used to produce one barrel of oil. == References == * Glossary at Schlumberger. Railroad A may repay the debt by loaning Railroad B a 3,000 horsepower locomotive for ten hours. ==References== Category:Imperial units Category:Units of energy The GWR 2021 Class was a class of 140 steam locomotives. Ice Water raced and won at age four and five, notably winning her second and third consecutive runnings of the Belle Mahone Stakes. ==Breeding record== She was retired to broodmare duty for the 1969 season at her owner's breeding farm where she had limited success. ==References== * Ice Water's pedigree and partial racing stats * Article on Gardiner Farms and Ice Water at the Jockey Club of Canada Category:1963 racehorse births Category:Thoroughbred family 13-c Category:Racehorses bred in Ontario Category:Racehorses trained in Canada A horsepower-hour (symbol: hp⋅h) is an outdated unit of energy, not used in the International System of Units. Ice Water's sire was Nearctic who also sired the most influential sire of the 20th Century, Northern Dancer. The car's steering was delightful but its brakes and suspension were only satisfactory. ==Fifteen 40== The 15-40-hp, a lightly revised 15.9, was displayed at the Olympia Motor Show in October 1924. ===Engine=== The Times noted some trouble had been taken to dampen engine vibration. The experiment was unpopular with engine crews, and the bodywork removed in 1911. ==See also== * GWR 0-6-0PT – list of classes of GWR 0-6-0 pannier tank, including table of preserved locomotives ==References== ==Sources== * Ian Allan ABC of British Railways Locomotives, 1948 edition, part 1, pp 16,51 * * 2021 Category:0-6-0ST locomotives Category:Railway locomotives introduced in 1897 Category:Standard gauge steam locomotives of Great Britain Category:Scrapped locomotives Against females, Ice Water won the Wonder Where Stakes and the Belle Mahone Stakes. They were superseded by the short-lived GWR 1600 Class, nominally a Hawksworth design, but in reality a straightforward update of the then 75-year-old design, with new boiler, bigger cab and bunker. ==Coachwork== When autotrains were introduced on the GWR, a trial was made of enclosing the engine in coachwork to resemble the coaches. The typical values are three to eight and two to five respectively. The newspaper noted that a few drops of oil two or three times a week ensures tappets run for a long time without shake otherwise they soon become noisy. ",30,0.3085,"""2.0""",-0.0301,-2,A
+The vibrational frequency of $I_2$ is $208 \mathrm{~cm}^{-1}$. At what temperature will the population in the first excited state be half that of the ground state?,"The vibrational temperature is used commonly when finding the vibrational partition function. 2MASS J03480772−6022270 (abbreviated to 2MASS J0348−6022) is a brown dwarf of spectral class T7, located in the constellation Reticulum approximately 27.2 light-years from the Sun. The high estimated age of 2MASS J0348−6022 is due to its late T-type spectral class, which is generally expected to describe the later evolutionary stages of brown dwarfs as they cool. == Rotation == === Photometric variability and periodicity === 2MASS J0348−6022 is the fastest-rotating brown dwarf confirmed , with a photometric periodicity of hours. The near-infrared spectrum of 2MASS J0348−6022 also displays a pair of narrow absorption lines at 1.243 and 1.252 μm, which are attributed to the presence of neutral potassium (K I) in the brown dwarf's atmosphere. Photometric variability in 2MASS J0348−6022 was first reported in 2008 by Fraser Clarke and collaborators using the New Technology Telescope's (NTT) near-infrared spectrograph. 2MASS J09373487+2931409, or 2MASSI J0937347+293142 (abbreviated to 2MASS 0937+2931) is a brown dwarf of spectral class T6, located in the constellation Leo about 19.96 light-years from Earth. ==Discovery== 2MASS 0937+2931 was discovered in 2002 by Adam J. Burgasser et al. from Two Micron All-Sky Survey (2MASS), conducted from 1997 to 2001. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. 2MASS J12195156+3128497 (abbreviated to 2MASS J1219+3128) is a rapidly- rotating brown dwarf of spectral class L8, located in the constellation Coma Berenices about 66 light-years from Earth. Absorption bands of iron(I) hydride (FeH) have also been found in 2MASS J0348−6022's spectrum between 1.72–1.78 μm. The mass, radius, and age of 2MASS J0348−6022 are estimated by interpolation of brown dwarf evolutionary models based on effective temperature and surface gravity. 2MASS J02431371−2453298 (abbreviated to 2MASS 0243−2453) is a brown dwarf of spectral class T6, located in the constellation Fornax about 34.84 light-years from Earth. ==Discovery== 2MASS 0243−2453 was discovered in 2002 by Adam J. Burgasser et al. from Two Micron All-Sky Survey (2MASS), conducted from 1997 to 2001. The high spin rate and oblateness of 2MASS J0348−6022 places it at about 45% of its rotational stability limit, assuming a smoothly varying fluid interior. A less precise parallax of this object, measured under U.S. Naval Observatory Infrared Astrometry Program, was published in 2004 by Vrba et al. ==Properties== 2MASS 0937+2931 has an unusual spectrum, indicating a metal- poor atmosphere and/or a high surface gravity (high pressure at the surface). Its effective temperature is estimated at about 800 Kelvin. The inclination of 2MASS J0348−6022's spin axis to Earth is , derived from its v sin i value. A previous estimate by Burgasser and collaborators from the spectrophotometric relation of spectral type and near-infrared absolute magnitude resulted in a value of , based on 2MASS JHK-band photometry. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics In a 2021 study, Megan Tannock and collaborators compared the near-infrared spectrum of 2MASS J0348−6022 to various published photospheric models and derived multiple best-fit solutions for its effective temperature and surface gravity. The trigonometric parallax of 2MASS J0348−6022 has been measured to be milliarcseconds, from 16 observations by the New Technology Telescope (NTT) collected over 6.4 years. Given the distance estimate from trigonometric parallax, the corresponding tangential velocity is , consistent with the kinematics of the stars of the Galactic disk. == Spectral class == 2MASS J0348−6022 is classified as a late T-type brown dwarf with the spectral class T7, distinguished by the presence of strong methane (CH4) and water (H2O) absorption bands in its near-infrared spectrum between wavelengths 1.2 and 2.35 μm. This can be explained by the presence of CH4 in its atmosphere, which is opaque to wavelengths around 3.3 μm. === Physical effects === The spectral lines in 2MASS J0348−6022's spectrum are Doppler-broadened due to the brown dwarf's rapid rotation, consistent with its short photometric periodicity. The vibrational temperature is commonly used in thermodynamics, to simplify certain equations. ",432,3,"""7.0""",21, -6.04697,A
 "One mole of $\mathrm{H}_2 \mathrm{O}(l)$ is compressed from a state described by $P=1.00$ bar and $T=350$. K to a state described by $P=590$. bar and $T=750$. K. In addition, $\beta=2.07 \times 10^{-4} \mathrm{~K}^{-1}$ and the density can be assumed to be constant at the value $997 \mathrm{~kg} \mathrm{~m}^{-3}$. Calculate $\Delta S$ for this transformation, assuming that $\kappa=0$.
-","J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The limit of this sum as N \rightarrow \infty becomes an integral: :S^\circ = \sum_{k=1}^N \Delta S_k = \sum_{k=1}^N \frac{dQ_k}{T} \rightarrow \int _0 ^{T_2} \frac{dS}{dT} dT = \int _0 ^{T_2} \frac {C_{p_k}}{T} dT In this example, T_2 =298.15 K and C_{p_k} is the molar heat capacity at a constant pressure of the substance in the reversible process . Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. From these sources: :*a - K.K. Kelley, Bur. Mines Bull. 383, (1935). :*b - :*c - Brewer, The thermodynamic and physical properties of the elements, Report for the Manhattan Project, (1946). :*d - :*e - :*f - :*g - ; :*h - :*i - . :*j - :*k - Int. National Critical Tables, vol. 3, p. 306, (1928). :*l - :*m - :*n - :*o - :*p - :*q - :*r - :*s - H.A. Jones, I. Langmuir, Gen. Electric Rev., vol. 30, p. 354, (1927). == See also == Category:Properties of chemical elements Category:Chemical element data pages The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. The following images show the density of the t-distribution for increasing values of u. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). ",524,7.27,0.195,-233,57.2,E
-"A mass of $34.05 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}(s)$ at $273 \mathrm{~K}$ is dropped into $185 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}(l)$ at $310 . \mathrm{K}$ in an insulated container at 1 bar of pressure. Calculate the temperature of the system once equilibrium has been reached. Assume that $C_{P, m}$ for $\mathrm{H}_2 \mathrm{O}(l)$ is constant at its values for $298 \mathrm{~K}$ throughout the temperature range of interest.","The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. The analysis here shows the simple unattributed formula and the Antoine equation are reasonably accurate at 100 °C, but quite poor for lower temperatures above freezing. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The equilibrium constants may be derived by best-fitting of the experimental data with a chemical model of the equilibrium system. == Experimental methods == There are four main experimental methods. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. The equilibrium constant value can be determined if any one of these concentrations can be measured. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Equilibrium constants are determined in order to quantify chemical equilibria. A large number of general-purpose computer programs for equilibrium constant calculation have been published. The former is an extremely simple Antoine equation, while the latter is a polynomial. ==Graphical pressure dependency on temperature== ==See also== *Dew point *Gas laws *Lee–Kesler method *Molar mass ==References== ==Further reading== * * * * ==External links== * * Category:Thermodynamic properties Category:Atmospheric thermodynamics The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_{0}, usually . It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out P = RT \left(\frac{1}{V_m} + \frac{B_{2}(T)}{V_m^2} + \cdots \right) This is the virial equation of state and describes a real gas. :H2O <=> H+ + OH-: K_\mathrm{W}^' = \frac{[H^+][OH^-]}{[H_2O]} With dilute solutions the concentration of water is assumed constant, so the equilibrium expression is written in the form of the ionic product of water. As expected, Buck's equation for > 0 °C is significantly more accurate than Tetens, and its superiority increases markedly above 50 °C, though it is more complicated to use. For this assumption to be valid, equilibrium constants must be determined in a medium of relatively high ionic strength. One or more equilibrium constants may be parameters of the refinement. ",12,292,24.4,-0.75,0.5,B
-"Calculate $\Delta H_f^{\circ}$ for $N O(g)$ at $975 \mathrm{~K}$, assuming that the heat capacities of reactants and products are constant over the temperature interval at their values at $298.15 \mathrm{~K}$.","Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., ""Physical Chemistry"" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., ""Atkins' Physical Chemistry"" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. \, Then :1-e^{-\hbar\omega_\alpha/k_{\rm B}T} \approx \hbar\omega_\alpha/k_{\rm B}T \, and we have :F=N\varepsilon_0+Nk_{\rm B}T\sum_{\alpha}\log\left(\frac{\hbar\omega_{\alpha}}{k_{\rm B}T}\right). 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity. ",12,91.7,-31.95,0.5,35.2,B
-A two-level system is characterized by an energy separation of $1.30 \times 10^{-18} \mathrm{~J}$. At what temperature will the population of the ground state be 5 times greater than that of the excited state?,"Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. An excited state is any state with energy greater than the ground state. The temperature of a group of particles is indicative of the level of excitation (with the notable exception of systems that exhibit negative temperature). The excitation temperature can even be negative for a system with inverted levels (such as a maser). It satisfies : \frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{ \left(-\frac{\Delta E}{k T_{\rm ex}} \right) }, where * is the number of particles in an upper (e.g. excited) state; * is the statistical weight of those upper-state particles; * is the number of particles in a lower (e.g. ground) state; * is the statistical weight of those lower-state particles; * is the exponential function; * is the Boltzmann constant; * is the difference in energy between the upper and lower states. Excitation refers to an increase in energy level above a chosen starting point, usually the ground state, but sometimes an already excited state. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature. == Absence of nodes in one dimension == In one dimension, the ground state of the Schrödinger equation can be proven to have no nodes. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. thumb|250px|Schematic picture of energy levels and examples of different states. In observations of the 21 cm line of hydrogen, the apparent value of the excitation temperature is often called the ""spin temperature"". ==References== Category:Temperature 300 px|thumb|A Jablonski diagram showing the excitation of molecule A to its singlet excited state (1A*) followed by intersystem crossing to the triplet state (3A) that relaxes to the ground state by phosphorescence. This phenomenon has been studied in the case of a two-dimensional gas in some detail, analyzing the time taken to relax to equilibrium. == Calculation of excited states == Excited states are often calculated using coupled cluster, Møller–Plesset perturbation theory, multi-configurational self-consistent field, configuration interaction, and time-dependent density functional theory. ==Excited-state absorption== The excitation of a system (an atom or molecule) from one excited state to a higher-energy excited state with the absorption of a photon is called excited-state absorption (ESA). Fortunately, it is often possible to define a quasi-Fermi level and quasi-temperature for a given location, that accurately describe the occupation of states in terms of a thermal distribution. The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. The ground state (blue) is x2–y2 orbitals; the excited orbitals are in green; the arrows illustrate inelastic x-ray spectroscopy. Relaxation of the excited state to its lowest vibrational level is called vibrational relaxation. The vibrational ground states of each electronic state are indicated with thick lines, the higher vibrational states with thinner lines. The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. Excited-state absorption measurements are done using pump–probe techniques such as flash photolysis. ",588313,-214,3.23,5.85,157.875,D
-At what temperature are there Avogadro's number of translational states available for $\mathrm{O}_2$ confined to a volume of 1000. $\mathrm{cm}^3$ ?,"The Avogadro number is the approximate number of nucleons (protons and neutrons) in one gram of ordinary matter. The Avogadro constant also relates the molar volume of a substance to the average volume nominally occupied by one of its particles, when both are expressed in the same units of volume. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The name Avogadro's number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 16 grams of oxygen. (The Avogadro number is closely related to the Loschmidt constant, and the two concepts are sometimes confused.) In older literature, the Avogadro number is denoted or , which is the number of particles that are contained in one mole, exactly . The Avogadro constant, commonly denoted or , is a ratio that relates the number of constituent particles (usually molecules, atoms, or ions) in a sample with the amount of substance in that sample. These definitions meant that the value of the Avogadro number depended on the experimentally determined value of the mass (in grams) of one atom of those elements, and therefore it was known only to a limited number of decimal digits. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained. Under the new definition, the mass of one mole of any substance (including hydrogen, carbon-12, and oxygen-16) is times the average mass of one of its constituent particles – a physical quantity whose precise value has to be determined experimentally for each substance. == History == === Origin of the concept === right|thumb|Jean Perrin in 1926 The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas. The value of the Avogadro constant was chosen so that the mass of one mole of a chemical compound, expressed in grams, is approximately the number of nucleons in one constituent particle of the substance. Thus, the Avogadro constant is the proportionality factor that relates the molar mass of a substance to the average mass of one molecule. This value, the number density of particles in an ideal gas, is now called the Loschmidt constant in his honor, and is related to the Avogadro constant, , by : n_0 = \frac{p_0N_{\rm A}}{R\,T_0}, where is the pressure, is the gas constant, and is the absolute temperature. The numeric value of the Avogadro constant expressed in reciprocal moles, a dimensionless number, is called the Avogadro number. As a consequence of this definition, in the SI system the Avogadro constant had the dimension reciprocal of amount of substance rather than of a pure number, and had the approximate value . thumb|The transformation of one phase from another by the growth of nuclei forming randomly in the parent phase The Avrami equation describes how solids transform from one phase to another at constant temperature. The goal of this definition was to make the mass of a mole of a substance, in grams, be numerically equal to the mass of one molecule relative to the mass of the hydrogen atom; which, because of the law of definite proportions, was the natural unit of atomic mass, and was assumed to be 1/16 of the atomic mass of oxygen. === First measurements === right|thumb|Josef Loschmidt The value of Avogadro's number (not yet known by that name) was first obtained indirectly by Josef Loschmidt in 1865, by estimating the number of particles in a given volume of gas. In general, for uniform nucleation and growth, n = D + 1, where D is the dimensionality of space in which crystallization occurs. == Interpretation of Avrami constants == Originally, n was held to have an integer value between 1 and 4, which reflected the nature of the transformation in question. Perrin himself determined the Avogadro number by several different experimental methods. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. Given the previous equations, this can be reduced to the more familiar form of the Avrami (JMAK) equation, which gives the fraction of transformed material after a hold time at a given temperature: : Y = 1 - \exp[-K\cdot t^n], where K = \pi\dot{N}\dot{G}^3/3, and n = 4. American Journal of Physics, 78 (4), 412-417 (https://doi.org/10.1119/1.3276053) and bound states in the continuum (red). ",0.000226,1.2,0.068,4.85,2,C
-The half-cell potential for the reaction $\mathrm{O}_2(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \rightarrow 2 \mathrm{H}_2 \mathrm{O}(l)$ is $+1.03 \mathrm{~V}$ at $298.15 \mathrm{~K}$ when $a_{\mathrm{O}_2}=1.00$. Determine $a_{\mathrm{H}^{+}}$,"The values below are standard apparent reduction potentials for electro- biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is Faraday's constant. At pH = 7, \+ → P680 ~ +1.0 Half-reaction independent of pH as no is involved in the reaction ==See also== * Nernst equation * Electron bifurcation * Pourbaix diagram * Reduction potential ** Dependency of reduction potential on pH * Standard electrode potential * Standard reduction potential * Standard reduction potential (data page) * Standard state ==References== ==Bibliography== ;Electrochemistry * ;Bio-electrochemistry * * * * * * ;Microbiology * * Category:Biochemistry Standard reduction potentials for half-reactions important in biochemistry Category:Electrochemical potentials Category:Thermodynamics databases Category:Biochemistry databases For standard conditions in electrochemistry (T = 25 °C, P = 1 atm and all concentrations being fixed at 1 mol/L, or 1 M) the standard reduction potential of hydrogen E^{\ominus}_\text{red H+} is fixed at zero by convention as it serves of reference. Definitions must be clearly expressed and carefully controlled, especially if the sources of data are different and arise from different fields (e.g., picking and directly mixing data from classical electrochemistry textbooks (E^{\ominus}_\text{red} versus SHE, pH = 0) and microbiology textbooks (E^{\ominus'}_\text{red} at pH = 7) without paying attention to the conventions on which they are based). ==Example in biochemistry== For example, in a two electrons couple like : the reduction potential becomes ~ 30 mV (or more exactly, 59.16 mV/2 = 29.6 mV) more positive for every power of ten increase in the ratio of the oxidised to the reduced form. ==Some important apparent potentials used in biochemistry== Half-reaction Δ°' (V) E' Physiological conditions References and notes −0.58 Many carboxylic acid: aldehyde redox reactions have a potential near this value 2 + 2 → −0.41 Non-zero value for the hydrogen potential because at pH = 7, [H+] = 10−7 M and not 1 M as in the standard hydrogen electrode (SHE), and that: → NADPH −0.320 −0.370 The ratio of :NADPH is maintained at around 1:50. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives: : : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 0 - \left(0.05916 \ \text{×} \ 7\right) = -0.414 \ V In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons () into hydrogen gas is no longer zero as with the standard hydrogen electrode (SHE) at 1 M (pH = 0) in classical electrochemistry, but that E_\text{red} = -0.414\mathrm V versus the standard hydrogen electrode (SHE). Te (aq) + 2 + 2 (s) + 4 1.02 2 . At pH = 7, when [] = 10−7 M, the reduction potential E_\text{red} of differs from zero because it depends on pH. Is it simply: * E_h = E_\text{red} calculated at pH 7 (with or without corrections for the activity coefficients), * E^{\ominus '}_\text{red}, a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it, * E^{\ominus '}_\text{red apparent at pH 7}, an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio \frac{h} {z} = \frac{\text{(number of involved protons)}} {\text{(number of exchanged electrons)}}. The same also applies for the reduction potential of oxygen: : For , E^{\ominus}_\text{red} = 1.229 V, so, applying the Nernst equation for pH = 7 gives: : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 1.229 - \left(0.05916 \ \text{×} \ 7\right) = 0.815 \ V For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. This is observed for the reduction of O2 into H2O, or OH−, and for reduction of H+ into H2. ==Formal standard reduction potential combined with the pH dependency== To obtain the reduction potential as a function of the measured concentrations of the redox- active species in solution, it is necessary to express the activities as a function of the concentrations. This equation predicts lower E_h at higher pH values. Fumarate + 2 + 2 → Succinate +0.03 +0.30 Formation of hydrogen peroxide from oxygen +0.82 In classical electrochemistry, E° for = +1.23 V with respect to the standard hydrogen electrode (SHE). The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . For the LJTS potential with r_\mathrm{end} = 2.5\,\sigma , the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon . At chemical equilibrium, the reaction quotient of the product activity (aRed) by the reagent activity (aOx) is equal to the equilibrium constant () of the half-reaction and in the absence of driving force () the potential () also becomes nul. This equation is the equation of a straight line for E_h as a function of pH with a slope of -0.05916\,\left(\frac{h}{z}\right) volt (pH has no units). The figure 8 shows the comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential and the 'Lennard-Jones truncated & shifted' potential. The activity coefficients \gamma_{red} and \gamma_{ox} are included in the formal potential E^{\ominus '}_\text{red}, and because they depend on experimental conditions such as temperature, ionic strength, and pH, E^{\ominus '}_\text{red} cannot be referred as an immuable standard potential but needs to be systematically determined for each specific set of experimental conditions. When the formal potential is measured under standard conditions (i.e. the activity of each dissolved species is 1 mol/L, T = 298.15 K = 25 °C = 77 °F, = 1 bar) it becomes de facto a standard potential. The properties of this ion are strongly related to the surface potential present on a corresponding solid. Without being able to correctly answering these questions, mixing data from different sources without appropriate conversion can lead to errors and confusion. ==Determination of the formal standard reduction potential when 1== The formal standard reduction potential E^{\ominus '}_\text{red} can be defined as the measured reduction potential E_\text{red} of the half-reaction at unity concentration ratio of the oxidized and reduced species (i.e., when 1) under given conditions. ",0.16,7200,22.2,4.16,3.0,D
-"The partial molar volumes of water and ethanol in a solution with $x_{\mathrm{H}_2 \mathrm{O}}=0.45$ at $25^{\circ} \mathrm{C}$ are 17.0 and $57.5 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$, respectively. Calculate the volume change upon mixing sufficient ethanol with $3.75 \mathrm{~mol}$ of water to give this concentration. The densities of water and ethanol are 0.997 and $0.7893 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively, at this temperature.","It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. Mixing two solutions of alcohol of different strengths usually causes a change in volume. The volume of alcohol in the solution can then be estimated. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Mixing pure water with a solution less than 24% by mass causes a slight increase in total volume, whereas the mixing of two solutions above 24% causes a decrease in volume. The International Organization of Legal Metrology has tables of density of water–ethanol mixtures at different concentrations and temperatures. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. Therefore, one can use the following equation to convert between ABV and ABW: \text{ABV} = \text{ABW} \times \frac{\text{density of beverage}}{\text{density of alcohol}} At relatively low ABV, the alcohol percentage by weight is about 4/5 of the ABV (e.g., 3.2% ABW is about 4% ABV). In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. At 0% and 100% ABV is equal to ABW, but at values in between ABV is always higher, up to ~13% higher around 60% ABV. ==See also== *Apparent molar property *Excess molar quantity *Standard drink *Unit of alcohol *Volume fraction == Notes == == References == ==Bibliography== * * * * ==External links== * * Category:Alcohol measurement The following formulas can be used to calculate the volumes of solute (Vsolute) and solvent (Vsolvent) to be used: *Vsolute = Vtotal / F *Vsolvent = Vtotal \- Vsolute , where: *Vtotal is the desired total volume *F is the desired dilution factor number (the number in the position of F if expressed as ""1:F dilution factor"" or ""xF dilution"") However, some solutions and mixtures take up slightly less volume than their components. The density of sugar in water is greater than the density of alcohol in water. The phenomenon of volume changes due to mixing dissimilar solutions is called ""partial molar volume"". thumb|The Mollier enthalpy–entropy diagram for water and steam. However, because of the miscibility of alcohol and water, the conversion factor is not constant but rather depends upon the concentration of alcohol. The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). In some countries, e.g. France, alcohol by volume is often referred to as degrees Gay-Lussac (after the French chemist Joseph Louis Gay-Lussac), although there is a slight difference since the Gay- Lussac convention uses the International Standard Atmosphere value for temperature, . ==Volume change== thumb|upright=1.48|Change in volume with increasing ABV. Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations and as long as the molar attenuation coefficients of the two components, and are known at both wavelengths. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal. ",1.8,-8,3.0,22,0.396,B
-"If the coefficient of static friction between the block and plane in the previous example is $\mu_s=0.4$, at what angle $\theta$ will the block start sliding if it is initially at rest?","The component of the force of gravity in the direction of the incline is given by: F_g = mg\sin{\theta} The normal force (perpendicular to the surface) is given by: N = mg\cos{\theta} Therefore, since the force of friction opposes the motion of the block, F_k =\mu_k \cdot mg\cos{\theta} To find the coefficient of kinetic friction on an inclined plane, one must find the moment where the force parallel to the plane is equal to the force perpendicular; this occurs when the block is moving at a constant velocity at some angle \theta \sum F = ma = 0 F_k = F_g or \mu_k mg\cos{\theta} = mg\sin{\theta} Here it is found that: \mu_k = \frac{mg\sin{\theta}}{mg\cos{\theta}} = \tan{\theta} where \theta is the angle at which the block begins moving at a constant velocity == References == Category:Classical mechanics The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: F_{k} = \mu_\mathrm{k} F_{n}. However, the magnitude of the friction force itself depends on the normal force, and hence on the mass of the block. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: F_\text{max} = \mu_\mathrm{s} F_\text{n}. Thus, a force is required to move the back of the contact, and frictional heat is released at the front. thumb|Angle of friction, θ, when block just starts to slide. ===Angle of friction=== For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. Sliding commences only after this frictional force reaches the value F_f = \mu N. For surfaces at rest relative to each other, \mu = \mu_\mathrm{s}, where \mu_\mathrm{s} is the coefficient of static friction. The friction increases as the applied force increases until the block moves. Coefficients of friction range from near zero to greater than one. Prior to sliding, this friction force is F_f = -P_x, where P_x is the horizontal component of the external force. After the block moves, it experiences kinetic friction, which is less than the maximum static friction. It is defined as: \tan{\theta} = \mu_\mathrm{s} and thus: \theta = \arctan{\mu_\mathrm{s}} where \theta is the angle from horizontal and μs is the static coefficient of friction between the objects. If an object is on a level surface and subjected to an external force P tending to cause it to slide, then the normal force between the object and the surface is just N = mg + P_y, where mg is the block's weight and P_y is the downward component of the external force. Sliding friction is almost always less than that of static friction; this is why it is easier to move an object once it starts moving rather than to get the object to begin moving from a rest position. This is called the angle of friction or friction angle. When there is no sliding occurring, the friction force can have any value from zero up to F_\text{max}. For surfaces in relative motion \mu = \mu_\mathrm{k}, where \mu_\mathrm{k} is the coefficient of kinetic friction. In fact, the friction force always satisfies F_f\le \mu N, with equality reached only at a critical ramp angle (given by \tan^{-1}\mu) that is steep enough to initiate sliding. This formula can also be used to calculate μs from empirical measurements of the friction angle. ===Friction at the atomic level=== Determining the forces required to move atoms past each other is a challenge in designing nanomachines. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. The coefficient of friction is an empirical measurementit has to be measured experimentally, and cannot be found through calculations. The coefficient of static friction, typically denoted as μs, is usually higher than the coefficient of kinetic friction. ",-8,0.4772,0.02,22,24,D
-"Halley's comet, which passed around the sun early in 1986, moves in a highly elliptical orbit with an eccentricity of 0.967 and a period of 76 years. Calculate its minimum  distances from the Sun.","The following is a list of comets with a very high eccentricity (generally 0.99 or higher) and a period of over 1,000 years that do not quite have a high enough velocity to escape the Solar System. On 23 March 2147 the comet will pass about from Earth with an uncertainty region of about ±2 million km. C/2001 OG108 (LONEOS) Closest Earth Approach on 2147-Mar-23 11:20 UT Date & time of closest approach Earth distance (AU) Sun distance (AU) Velocity wrt Earth (km/s) Velocity wrt Sun (km/s) Uncertainty region (3-sigma) Reference 2147-03-23 11:20 ± 13:38 40.3 35.3 ± 2 million km Horizons The comet has a rotational period of 2.38 ± 0.02 days (57.12 hr). The comet came to perihelion (closest approach to the Sun) on 15 March 2002. 170P/Christensen is a periodic comet in the Solar System. Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. C/ (LONEOS) is a Halley-type comet with an orbital period of 48.51 years. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. 1308 Halleria, provisional designation , is a carbonaceous Charis asteroid from the outer regions of the asteroid belt, approximately 43 kilometers in diameter. Using data from Fernandez (2004–2005) JPL lists the comet with an albedo of 0.05 and a diameter of 13.6 ± 1.0 km. 164P/Christensen is a periodic comet in the Solar System. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 164P/Christensen – Seiichi Yoshida @ aerith.net * Elements and Ephemeris for 164P/Christensen – Minor Planet Center Category:Periodic comets 0164 164P 20041221 It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 The actual orbit of these comets significantly differs from the provided coordinates. Of the short-period comets with known diameters and perihelion inside the orbit of Earth, C/ is the second largest after Comet Swift–Tuttle. In 2003, the comet was estimated to have a mean absolute V magnitude (H) of 13.05 ± 0.10, with an albedo of 0.03, giving an effective radius of 8.9 ± 0.7 km. A Solar System barycentric orbit computed at an epoch when the object is located beyond all the planets is a more accurate measurement of its long-term orbit. ==List of near-parabolic comets== Comet designation Comet name /discoverer Semimajor axis (AU) Eccentricity Inclination (°) Perihelion distance (AU) Absolute magnitude (M1/M2) Perihelion date Period (3) (years) Ref C/1680 V1 Great Comet of 1680 444.4285 0.999986 60.6784 0.006222 1680/12/18 9370 C/1769 P1 Messier 163.4554 0.999249 40.7338 0.122755 1769/10/08 2090 C/1785 E1 Méchain 120.6893 0.99646 92.639 0.42724 1785/04/08 1325 C/1807 R1 Great comet of 1807 143.2012 0.995488 63.1762 0.646124 1807/09/19 1710 C/1811 F1 Great Comet of 1811 212.3922 0.995125 106.9342 1.035412 1811/09/12 3100 C/1822 N1 Pons 310.8303 0.996316 127.3429 1.145099 1822/10/24 5480 C/1823 Y1 Great Comet of 1823 170 0.9987 103.68 0.2252 1823/12/09 2300 C/1825 K1 Gambart 246.605 0.996395 123.3414 0.889011 1825/05/31 3870 C/1825 N1 Pons 271.5793 0.995431 146.4353 1.240846 1825/12/11 4480 C/1826 P1 Pons 340.063 0.997492 25.9496 0.852878 1826/10/09 6270 C/1840 B1 Galle 180.8076 0.99325 120.7807 1.220451 1840/03/13 2430 C/1844 N1 Mauvais 3520.1687 0.999757 131.4092 0.855401 1844/10/17 208900 C/1844 Y1 Great Comet of 1844 358.9355 0.999302 45.5651 0.250537 1844/12/14 6800 C/1846 B1 de Vico 194.9063 0.992403 47.4257 1.480703 1846/01/22 2720 C/1847 C1 Hind 473.2556 0.99991 48.6636 0.042593 1847/03/30 10300 C/1847 N1 Mauvais 1251.6357 0.998589 96.5817 1.766058 1847/08/09 44280 C/1849 G1 Schweizer 568.5696 0.998427 66.9587 0.89436 1849/06/08 13560 C/1850 J1 Petersen 771.8979 0.998599 68.1848 1.081429 1850/07/24 21450 C/1854 R1 Klinkerfues 118.2650 0.993246 40.9201 0.798762 1854/10/28 1290 C/1854 Y1 Winnecke-Dien 156.4219 0.991309 14.152 1.359463 1854/12/16 1960 C/1857 Q1 Klinkerfues 182.3447 0.996913 123.9614 0.562898 1857/10/01 2460 C/1858 L1 Donati 156.132 0.996295 116.9512 0.578469 1858/09/30 1950 C/1863 G1 Klinkerfues 1269.962 0.999159 112.6209 1.068038 1863/04/05 45260 C/1863 G2 Respighi 682.7155 0.999079 85.4961 0.628781 1863/04/21 17840 C/1863 V1 Tempel 630.1632 0.998879 78.0817 0.706413 1863/11/09 15820 C/1864 N1 Tempel 249.1888 0.996351 178.1269 0.90929 1864/08/16 3930 C/1864 O1 Donati-Toussaint 1450.486 0.999358 109.7124 0.931212 1864/10/11 55240 C/1871 G1 Winnecke 299.3138 0.997814 87.6034 0.6543 1871/06/11 5180 C/1871 V1 Tempel 161.2851 0.995714 98.2992 0.691268 1871/12/20 2050 C/1873 Q1 Borrelly 225.7138 0.996482 95.9662 0.794061 1873/09/11 3390 C/1873 Q2 Henry 1425.6037 0.99973 121.4625 0.384913 1873/10/02 53830 C/1874 H1 Coggia 572.6966 0.99882 66.3439 0.675782 1874/07/07 13710 C/1874 O1 Borrelly 840.5894 0.998831 41.8266 0.982649 1874/08/27 24370 C/1877 G1 Winnecke 730.7538 0.9987 121.1548 0.94998 1877/04/18 19750 C/1877 G2 Swift 485.8223 0.997923 77.1916 1.009053 1877/04/27 10700 C/1881 K1 Great Comet of 1881 178 0.99589 63.4253 0.734547 1881/06/16 2390 C/1881 W1 Swift 195.8064 0.990169 144.8016 1.924973 1881/11/20 2740 C/1882 F1 Wells 10127.1667 0.999994 73.7977 0.060763 1882/06/11 1019140 C/1887 J1 Barnard 356.7476 0.996093 17.5479 1.393813 1887/06/17 7640 C/1888 D1 Sawerthal 169.3582 0.995874 42.2482 0.698772 1888/03/17 2200 C/1888 P1 Brooks 9806.8043 0.999908 74.1904 0.902226 1888/07/31 971160 C/1888 U1 Barnard 179.494 0.991488 56.3425 1.527853 1888/09/13 2400 C/1889 G1 Barnard 12393.3846 0.999818 163.8517 2.255596 1889/06/11 1379700 C/1889 O1 Davidson 435.2118 0.997611 65.9916 1.039721 1889/07/19 9080 C/1890 O2 Denning 1550.0898 0.999187 98.9373 1.260223 1890/09/25 61030 C/1890 V1 Zona 495.8077 0.995872 154.307 2.046694 1890/08/07 11040 C/1892 E1 Swift 809.1663 0.998731 38.7002 1.026832 1892/04/07 23020 C/1893 N1 Rordame-Quénisset 1249.1648 0.99946 159.9804 0.674549 1893/07/07 44150 C/1893 U1 Brooks 231.2706 0.996489 129.8233 0.811991 1893/09/19 3520 C/1898 V1 Chase 4634.0588 0.999507 22.5046 2.2846 1898/09/20 315460 C/1902 R1 Perrine 12533.5625 0.999968 156.3548 0.4011 1902/11/24 1403170 C/1903 A1 Giacobini 1244.1242 0.99967 30.9416 0.4106 1903/03/16 43880 C/1906 B1 Brooks 1926.3913 0.999327 126.4425 1.296 7.0 1905/12/22 84552.21 C/1907 L2 Daniel 424.6874 0.998794 8.9577 0.512173 1907/09/04 8750 C/1909 L1 Borrelly-Daniel 160.9095 0.994762 52.0803 0.842844 1909/06/05 2040 C/1910 A1 Great January comet of 1910 25795 0.999995 138.7812 0.128975 1910/01/17 4142890 C/1910 P1 Metcalf 9596.1232 0.999797 121.0556 1.948013 1910/09/16 940030 C/1911 N1 Kiess 184 0.9963 148.42 0.68383 1911/06/30 2500 C/1911 O1 Brooks 163.1454 0.997005 16.4153589 0.489429 1911/10/28 2090 C/1911 S2 Quénisset 429.6907 0.998167 108.1 0.787623 1911/11/12 8910 C/1913 J1 Schaumasse 309.1747 0.995288 152.3673 1.456831 1913/05/15 5436 C/1913 R1 Metcalf 555.7869 0.99756 143.3547 1.35612 1913/09/14 13100 C/1914 S1 Campbell? 534.2999 0.998666 77.836 0.712756 1914/08/05 12350 C/1916 G1 Wolf 2834.363 0.999405 25.6592 1.686446 1917/06/17 150900 C/1920 X1 Skjellerup 193.9334 0.994081 22.0303 1.147892 1920/12/11 2700 C/1922 B1 Reid 125.0064 0.986968 32.4456 1.629083 1921/10/28 1400 C/1922 W1 Skjellerup 147.4206 0.993735 23.3659 0.92359 1923/01/04 1790 C/1924 F1 Reid 1252.2789 0.998598 72.3273 1.755695 1924/03/13 44320 C/1925 F2 Reid 334.418 0.995116 26.9797 1.633299 1925/07/29 6120 C/1926 B1 Blathwayt 176.6645 0.992384 128.2986 1.345477 1926/01/02 2350 C/1927 E1 Stearns 2023.0104 0.998179 87.6525 3.683902 1927/03/22 90990 C/1927 X1 Skjellerup-Maristany 1100.9813 0.99984 85.1126 0.176157 1927/12/18 36530 C/1929 Y1 Wilk 691.5998 0.999028 124.5103 0.672235 1930/01/22 18190 C/1930 D1 Peltier-Schwassmann- Wachmann 581.6554 0.998131 99.883 1.087114 1930/11/15 14030 C/1931 P1 Ryves Comet 111.1632 0.999326 169.2881 0.074924 1931/08/25 1180 C/1936 K1 Peltier 133.7226 0.991775 78.5447 1.099868 1936/06/08 1550 C/1937 N1 Finsler 57516 0.999985 146.4156 0.862744 1937/08/15 13793870 C/1939 B1 Kozik-Peltier 146.3133 0.995103 63.5238 0.716496 1939/02/06 1770 C/1939 H1 Jurlof-Achmarof- Hassel 346.8588 0.998477 138.1212 0.528266 1939/04/10 6460 C/1939 V1 Friend 336.6129 0.997192 92.952 0.945209 1939/11/05 6180 C/1941 B2 de Kock- Paraskevopoulos 879.7695 0.999102 168.2039 0.790033 1941/01/27 26100 C/1942 X1 Whipple-Fedtke-Tevzadze 173.4555 0.992196 19.7127 1.353647 1943/02/06 2280 C/1944 H1 Väisälä 370.9009 0.9935 17.2882 2.410856 1945/01/04 7140 C/1947 F1 Rondanina-Bester 217.5667 0.997427 39.3015 0.559799 1947/05/20 3210 C/1947 X1-A Southern Comet of 1947 243.4336 0.999548 138.5419 0.110032 1947/12/02 3800 C/1947 X1-B Southern Comet of 1947 296.558 0.999629 138.5332 0.110023 1947/12/02 5110 C/1948 L1 Honda-Bernasconi 1661.024 0.999875 23.1489 0.207628 1948/05/15 67700 C/1948 N1 Wirtanen 3884.5787 0.999352 130.2675 2.517207 1949/05/01 242110 C/1948 V1 Eclipse Comet of 1948 2083.4 0.999935 23.117 0.135421 1948/10/27 95100 C/1948 W1 Bester 509.1675 0.997499 87.6054 1.273428 1948/10/22 11490 C/1949 N1 Bappu-Bok-Newkirk 1517.8296 0.998644 105.7686 2.058177 1949/10/26 59130 C/1951 P1 Wilson-Harrington 2836.8851 0.999739 152.5337 0.740427 1952/01/12 151100 C/1952 M1 Peltier 4605.0766 0.999739 45.5521 1.201925 1952/06/15 312500 C/1952 Q1 Harrington 407.0848 0.99591 59.1154 1.664977 1953/01/05 8210 C/1953 G1 Mrkos-Honda 391.7716 0.997391 93.8573 1.022132 1953/05/26 7750 C/1955 N1 Bakharev-Macfarlane-Krienke 244.7197 0.994167 50.0329 1.42745 1955/07/11 3830 C/1957 P1 Mrkos 558.9496 0.999365 93.9411 0.354933 1957/08/01 13210 C/1958 D1 Burnham 23205.0702 0.999943 15.7879 1.322689 1958/04/16 3534880 C/1958 R1 Burnham-Slaughter 12150.7313 0.999866 61.2576 1.628198 1959/03/11 1339380 C/1959 X1 Mrkos 4974.0634921 0.999748 19.6339 1.253464 1959/11/13 350810 C/1960 Y1 Candy 105.1101 0.9899 150.9552 1.061612 1961/02/08 1080 C/1961 O1 Wilson 1057.8684 0.999962 24.2116 0.040199 1961/07/17 34410 C/1961 R1 Humason 204.5261 0.989569 153.278 2.133412 1962/12/10 2920 C/1963 F1 Alcock 792.3201 0.99806 86.2194 1.537101 1963/05/05 22300 C/1964 L1 Tomita-Gerber-Honda 123.03 0.995933 161.8323 0.500363 1964/06/30 1360 C/1964 P1 Everhart 361.1342 0.996513 67.9689 1.259275 1964/08/23 6860 C/1965 S1-B Ikeya-Seki 103.7067 0.999925 141.861 0.007778 1965/10/21 1060 C/1966 P1 Kilston 3821.7115 0.999376 40.2648 2.384748 1966/10/28 236260 C/1966 P2 Barbon 1111.0435 0.998183 28.7058 2.018766 1966/04/17 37033 C/1967 Y1 Ikeya-Seki 2000.6851 0.999152 129.3153 1.696581 1968/02/25 89490 C/1968 H1 Tago-Honda-Yamamoto 174 0.9961 102.1698 0.680378 9.8 1968/05/16 2300 C/1968 Y1 Thomas 705.6891 0.995301 45.2291 3.316033 1969/01/12 18750 C/1969 O1-A Kohoutek 1964.6686 0.999125 86.3128 1.719085 1970/03/21 87080 C/1969 T1 Tago-Sato-Kosaka 6400 0.999926 75.81773 0.4726395 6.5 1969/12/21 508060 C/1969 Y1 Bennett 141.21513 0.996193 90.0394 0.537606 1970/03/20 1680 C/1972 E1 Bradfield 494.778 0.998126 123.693 0.927214 1972/03/27 11010 C/1972 F1 Gehrels 1071.8224 0.996943 175.616 3.276561 1971/01/06 35090 C/1972 X1 Araya 54008.3111 0.99991 113.0902 4.860748 1972/12/18 12551360 C/1973 D1 Kohoutek 1082.2388 0.998723 121.5982 1.382019 1973/06/07 35600 C/1974 C1 Bradfield 1660.6964 0.999697 61.2842 0.503191 1974/03/18 67680 C/1974 F1 Lovas 7566.4724 0.999602 50.6485 3.011456 1975/08/22 658170 C/1975 T1 Mori-Sato-Fujikawa 632 0.997461 97.6077 1.603934 5.5 1975/12/25 15880 C/1975 V1-A Comet West 6780.2069 0.999971 43.0664 0.196626 1967/02/25 558300 C/1976 D1 Bradfield 136.9866 0.993811 46.834 0.84781 1976/02/24 1600 C/1976 J1 Harlan 5143.859 0.999695 38.8063 1.568877 1976/11/03 368920 C/1977 R1 Kohler 2170 0.999543 48.71188 0.9905761 7.3 1977/11/10 101000 C/1977 V1 Tsuchinshan 9817.545 0.999633 168.5495 3.603039 1977/06/24 972760 C/1978 T1 Seargent 220 0.99832 67.828 0.36988 1978/09/14 3300 C/1980 V1 Meier 285 0.99468 100.9864 1.51956 7.2 1980/12/09 4820 C/1980 Y1 Bradfield 944.8109 0.999725 138.585 0.259823 1980/12/29 29040 C/1980 Y2 Panther 1640 0.998991 82.64774 1.657269 6.1 1981/01/27 66500 C/1981 H1 Bus 2510.8713 0.999021 160.664 2.458143 1981/07/30 125816 C/1981 M1 Gonzalez 3857.1917 0.999395 107.1467 2.333601 1981/03/25 239560 C/1982 M1 Austin 1072 0.999396 84.4951 0.6478114 8.8 1982/08/24 35100 C/1983 J1 Sugano-Saigusa- Fujikawa 4779.898 0.999901 96.623 0.471 12.3 1983/05/01 330473.13 C/1983 N1 IRAS 4168.9638 0.99942 138.8364 2.417999 1983/05/02 269180 C/1984 N1 Austin 1891.4545 0.999846 164.1533 0.291284 1984/08/12 82260 C/1984 U1 Shoemaker 1145.723 0.995209 179.2123 5.489159 1984/09/03 38780 C/1984 V1 Levy-Rudenko 1160 0.99921 65.7146 0.917949 9.4 1984/12/14 39600 C/1984 W2 Hartley 9501.7435 0.999579 89.3273 4.000234 1985/09/28 926200 C/1985 R1 Hartley-Good 5800 0.999881 79.9294 0.694577 8.4 1985/12/09 450000 C/1986 N1 Churyumov- Solodovnikov 5669.7575 0.999534 114.9293 2.642107 1986/05/06 426920 C/1986 V1 Sorrells 18913.7912 0.999909 160.5801 1.721155 1987/03/09 2601160 C/1987 B1 Nishikawa-Takamizawa-Tago 207 0.9958 172.22989 0.869589 7.4 1987/03/17 2980 C/1987 P1 Bradfield 165.2 0.99474 34.08809 0.868956 6 1987/11/07 2123 C/1987 U3 McNaught 406 0.99792 97.5751 0.84393 6.9 1987/12/02 8200 C/1988 A1 Liller 244.9295 0.996565 73.3224 0.841333 1988/03/31 3830 C/1988 F1 Levy 537.6264 0.997816 62.8074 1.174176 1987/11/29 12470 C/1988 J1 Shoemaker-Holt 541.2286 0.99783 62.8066 1.174466 1988/02/14 12590 C/1989 A1 Yanaka 1410 0.99866 52.4092 1.89458 5.1 1988/10/31 53000 C/1989 A5 Shoemaker 547.6156 0.99518 96.5548 2.639507 1989/02/26 12810 C/1989 T1 Helin-Roman-Alu 112.097 0.990657 46.0369 1.047322 1989/12/15 1190 C/1990 N1 Tsuchiya-Kiuchi 233.2246 0.995316 143.7839 1.092424 1990/09/28 3560 C/1991 A2 Masaru Arai 151.0756 0.990507 70.9783 1.434161 1990/12/10 1860 C/1991 B1 Shoemaker-Levy 348.8963 0.993508 77.2881 2.265035 1991/12/31 6520 C/1991 Q1 McNaught-Russell 589.3992 0.994581 90.5062 3.19395 1992/05/03 14310 C/1991 R1 McNaught-Russell 11160.2875 0.999374 104.5086 6.98634 1990/11/12 1179000 C/1991 T2 Shoemaker-Levy 6000 0.999860 113.49709 0.8362597 7.7 1992/07/24 4650000 C/1992 F1 Tanaka-Machholz 312.7164 0.995966 79.2924 1.261498 1992/04/22 5530 C/1992 J1 Spacewatch 77102.7179 0.999961 124.3187 3.007006 1993/09/05 21409400 C/1992 U1 Shoemaker 3928.1053 0.999411 65.9859 2.313654 1993/03/25 246190 C/1993 Y1 McNaught- Russell 134.8 0.99356 51.5866 0.8676358 12.2 1994/03/31 1564 C/1994 E2 Shoemaker-Levy 431.4296 0.997314 131.2547 1.15882 1994/05/27 8960 C/1994 G1-A Takamizawa-Levy 1549.8632 0.999123 132.8728 1.35923 1994/05/22 61020 C/1994 J2 Takamizawa 545.4374 0.996429 135.9611 1.947757 1994/06/29 12740 C/1994 T1 Machholz 3820.7081 0.999517 101.7379 1.845402 1994/10/02 236170 C/1995 O1 Comet Hale–Bopp 185.86 0.9950817 89.430154 0.9141335 2.3 1997/04/01 2534 C/1995 Q1 Bradfield 220.6208 0.998022 147.3942 0.436388 1995/08/31 3280 C/1996 B1 Szczepanski 156.9 0.99076 51.9189 1.448788 7.1 1996/02/06 1965 C/1996 B2 Comet Hyakutake 2270 0.9998987 124.92266 0.2302293 7.3 1996/05/01 108000 C/1996 Q1 Tabur 800 0.9989 73.359 0.83984 11.0 1996/11/03 22000 C/1996 R1 Hergenrother-Spahr 132 0.9856 145.8144 1.89920 5.8 1996/08/28 1510 C/1996 R3 Lagerkvist 404.015 0.987 39.2 5.24 10.5 1995/07/24 8120.91 Spacewatch 3081 0.998884 72.71704 3.436463 4.9 1999/11/27 171000 C/1997 G2 Montani 529 0.99417 69.83548 3.084966 5.3 1998/04/16 12160 C/1997 J1 Mueller 255.5 0.990991 122.96833 2.302132 8.6 1997/05/03 4085 C/1997 L1 Zhu-Balam 2420 0.99797 72.9914 4.89956 6.5 1996/11/22 119000 C/1997 T1 Utsunomiya 920 0.998523 127.99262 1.3591096 8.0 1997/12/10 27910 C/1998 H1 Stonehouse 710 0.9979 104.693 1.48729 10.0 1998/04/14 19000 C/1998 K2 LINEAR 3210 0.999276 64.45667 2.323479 8.6 1998/09/01 182000 C/1998 K3 LINEAR 1700 0.9979 160.2056 3.5463 10.0 1998/03/07 70000 C/1998 M1 LINEAR 431 0.99277 20.38455 3.11812 5.4 1998/10/28 8950 C/1998 M2 LINEAR 1215 0.997758 60.18232 2.725333 8.5 1998/08/13 42400 C/1998 M4 LINEAR 1100 0.998 154.572 2.6001 9.5 1997/12/10 30000 C/1998 M5 LINEAR 438.3 0.996025 82.22889 1.7422899 8.0 1999/01/24 9176 C/1998 M6 Montani 5400 0.9989 91.540 5.9787 7.5 1998/10/06 400000 C/1998 P1 Williams 1700 0.999325 145.72831 1.146108 8.0 1998/10/17 70000 C/1998 Q1 LINEAR 358 0.99559 32.3058 1.57788 14.0 1998/06/29 6770 C/1998 T1 LINEAR 1657 0.999114 170.15995 1.467728 9.5 1999/06/25 67400 C/1998 U5 LINEAR 102.88 0.987981 131.76474 1.2364530 10.9 1998/12/21 1043.5 C/1999 A1 Tilbrook 177 0.99587 89.481 0.730741 12.0 1999/01/29 2350 C/1999 F1 Catalina (CSS) 6700 0.999136 92.03554 5.787022 4.6 2002/02/13 548000 C/1999 F2 Dalcanton 2640 0.99821 56.42742 4.71807 7.6 1998/08/23 135000 C/1999 H1 Lee 2775 0.9997449 149.35290 0.70810722 9.4 1999/07/11 146200 C/1999 J3 LINEAR 1600 0.99939 101.6561 0.976809 11.3 1999/09/20 64000 C/1999 K2 Ferris 155 0.9658 82.191 5.2903 7.0 1999/04/10 1920 C/1999 K3 LINEAR 235 0.9918 92.274 1.92878 12.0 1999/02/27 3600 C/1999 K6 LINEAR 346.8 0.993532 46.34384 2.246976 11.3 1999/07/24 6459 C/1999 K7 LINEAR 700 0.9966 135.159 2.3227 13.0 1999/02/24 18000 C/1999 L2 LINEAR 390 0.9951 43.942 1.90476 13.0 1999/08/04 7800 C/1999 N2 Lynn 298 0.99745 111.6559 0.7612844 10.3 1999/07/23 5150 C/1999 T1 McNaught-Hartley 8100 0.999856 79.97521 1.1716989 8.6 2000/12/13 740000 C/2000 B2 LINEAR 6000 0.9994 93.647 3.7762 10.3 1999/11/10 500000 LINEAR 1916 0.998353 49.21252 3.155967 7.4 2001/06/19 83900 C/2000 K2 LINEAR 522.0 0.995332 25.63358 2.437066 9.3 2000/10/11 11930 C/2000 Y2 Skiff 490 0.99435 12.0875 2.76871 11.4 2001/03/21 10850 C/2001 A1 LINEAR 266 0.99095 59.941 2.4064 12.7 2000/09/17 4330 C/2001 A2-A LINEAR 2500 0.99969 36.487 0.779054 13 2001/05/24 130000 C/2001 A2-B LINEAR 1119 0.999304 36.47582 0.7790172 7 2001/05/24 37400 C/2001 C1 LINEAR 38000 0.99987 68.96470 5.10432 6.5 2002/03/28 7000000 LINEAR-NEAT 1193.5 0.9976606 163.212126 2.7920832 7.4 2003/07/09 41230 C/2001 K3 Skiff 2870 0.99893 52.0265 3.06012 9.4 2001/04/22 153000 C/2001 K5 LINEAR 11410 0.999546 72.590342 5.184246 4.4 2002/10/11 1220000 C/2001 O2 NEAT 2200 0.9978 90.9262 4.8194 6.6 1999/10/17 103000 C/2001 Q1 NEAT 171.2 0.96593 66.9504 5.83397 7.7 2001/09/20 2241 C/2001 U6 LINEAR 1149 0.99617 107.25550 4.40642 6.5 2002/08/08 39000 C/2001 W1 LINEAR 2100 0.9989 118.645 2.39924 13.7 2001/12/24 100000 C/2001 X1 LINEAR 570 0.99700 115.6268 1.69793 11.3 2002/01/08 13500 C/2002 B2 LINEAR 1400 0.9972 152.8726 3.8422 10.1 2002/04/06 50000 C/2002 C2 LINEAR 9000 0.99964 104.88143 3.25375 9.9 2002/04/10 860000 C/2002 F1 Utsunomiya 950 0.999539 80.8770 0.4382989 10.5 2002/04/22 29300 C/2002 H2 LINEAR 276 0.99407 110.5011 1.63484 10.5 2002/03/23 4570 C/2002 J4 NEAT 29000 0.999874 46.52550 3.633722 8.4 2003/10/03 4900000 C/2002 K1 NEAT 9000 0.9997 89.723 3.23024 11.4 2002/06/16 900000 C/2002 K2 LINEAR 763 0.99314 130.8957 5.23506 8.2 2002/06/05 21100 C/2002 L9 NEAT 4460 0.99842 68.44211 7.03301 4.7 2004/04/05 297000 C/2002 O6 SWAN 350 0.99858 58.6240 0.494648 13.0 2002/09/09 6500 C/2002 P1 NEAT 414 0.98422 34.6061 6.5302 8.2 2001/11/23 8420 C/2002 Q3-A LINEAR 465.333 0.997194 96.87858 1.30583 16.4 2002/08/19 10038.16 C/2002 V1 NEAT 1011 0.9999018 81.70600 0.0992581 10.4 2003/02/18 32100 C/2002 V2 LINEAR 5010 0.99864 166.77622 6.81203 8.4 2003/03/13 355000 LINEAR 202.14 0.966377 70.51612 6.796713 7.1 2006/02/06 2874.1 C/2002 X1 LINEAR 1376 0.998192 164.08943 2.4867001 9.8 2003/07/12 51020 C/2002 X5 Kudo-Fujikawa 1210 0.999843 94.15226 0.189935 10.6 2003/01/29 42000 C/2002 Y1 Juels-Holvorcem 250.6 0.997152 103.78154 0.7138096 9.8 2003/04/13 3967 C/2003 G2 LINEAR 440 0.9965 96.167 1.55337 16.0 2003/04/29 9000 C/2003 H1 LINEAR 2653 0.999156 138.667242 2.2396301 8.7 2004/02/22 136700 C/2003 H3 NEAT 13200 0.999780 42.81171 2.901441 9.6 2003/04/24 1510000 C/2003 J1 NEAT 577 0.99112 98.3135 5.12542 8.8 2003/10/10 13900 C/2003 L2 LINEAR 154.40 0.981446 82.05107 2.864801 9.9 2004/01/19 1918.7 C/2003 T2 LINEAR 6400 0.99972 87.5315 1.786352 9.8 2003/11/14 520000 C/2003 T3 Tabur 5730 0.999742 50.44443 1.4810758 5.8 2004/04/29 434000 C/2003 V1 LINEAR 603 0.99704 28.67513 1.78314 9.9 2003/03/11 14800 C/2004 F2 LINEAR 151.6 0.99056 104.9600 1.43044 13.2 2003/12/26 1870 C/2004 F4 Bradfield 238 0.999294 63.16456 0.168266 11.3 2004/04/17 3680 C/2004 G1 LINEAR 328.47 0.996 114.486 1.201 14.4 2004/06/04 5953.27 C/2004 K1 Catalina (CSS) 1819 0.998131 153.747521 3.399147 7.9 2005/07/05 77600 C/2004 L1 LINEAR 858 0.997615 159.36082 2.04741344 12.6 2005/03/30 25100 C/2004 L2 LINEAR 790 0.995215 62.51864 3.778629 8.3 2005/11/15 22190 C/2004 P1 NEAT 8100 0.99925 28.8163 6.01377 10.1 2003/08/08 720000 C/2004 Q1 Tucker 186.78 0.989042 56.08768 2.0467255 9.8 2004/12/06 2552.8 C/2004 Q2 Comet Machholz 2403 0.9994986 38.588963 1.2050414 9.9 2005/01/24 117800 LINEAR 700 0.997227 21.61823 1.942359 13.9 2005/03/03 18540 C/2004 T3 Siding Spring 5600 0.99842 71.9642 8.8644 6.6 2003/04/15 420000 C/2004 U1 LINEAR 3610 0.999264 130.62532 2.659321 9.0 2004/12/08 217000 C/2004 X2 LINEAR 1450 0.99738 72.118 3.79308 10.2 2004/08/24 55000 LINEAR 13000 0.99987 52.47641 1.781202 17.3 2005/03/03 1500000 C/2005 G1 LINEAR 18800 0.99974 108.41395 4.960798 7.7 2006/02/27 2600000 C/2005 L3 McNaught 13390 0.999582 139.449248 5.593622 6.4 2008/01/16 1550000 C/2005 N1 Juels-Holvorcem 729 0.998457 51.18017 1.125447 11.3 2005/08/22 19700 C/2005 R4 LINEAR 2067 0.99749 164.01260 5.188473 7.7 2006/03/08 94000 C/2005 S4 McNaught 5690 0.998972 107.95897 5.850109 7.9 2007/07/18 430000 C/2005 X1 Beshore 690 0.9958 91.944 2.8623 11.1 2005/07/05 18000 C/2005 YW LINEAR 190.4 0.989534 40.54361 1.9930109 7.4 2006/12/07 2628 C/2006 A1 Pojmański 2370 0.999765 92.73611 0.5553959 10.5 2006/02/22 115000 C/2006 A2 Catalina (CSS) 3800 0.99862 148.3226 5.3160 9.8 2005/05/20 240000 C/2006 B1 McNaught 1340 0.99776 134.28193 2.997591 10.3 2005/11/19 49100 Catalina (CSS) 216.32 0.991900 144.26278 1.7521694 12.3 2006/07/03 3182 C/2006 K4 NEAT 1818 0.998246 111.33346 3.188618 8.8 2007/11/29 77500 C/2006 L1 Garradd 551 0.997345 143.24257 1.462070 8.6 2006/10/18 12930 C/2006 M1 LINEAR 153.67 0.976859 54.87693 3.556199 9.6 2007/02/13 1905.0 C/2006 O2 Garradd 420 0.99634 43.0287 1.55479 12.7 2006/10/05 8700 C/2006 Q1 McNaught 6890 0.9995986 59.050380 2.7637144 7.0 2008/07/03 571000 C/2006 U6 Spacewatch 1931 0.998706 84.87894 2.4983978 8.8 2008/06/05 84900 C/2006 V1 Catalina (CSS) 257.6 0.989618 31.11947 2.674906 9.0 2007/11/26 4136 C/2006 W3 Christensen 17990 0.9998262 127.074692 3.1262325 6.7 2009/07/06 2410000 Lemmon 583.8 0.998987 152.70463 0.5912444 17.4 2007/04/28 14110 LINEAR 252.0 0.992839 30.62941 1.804374 7.4 2007/07/21 4000 C/2007 B2 Skiff 737.2 0.995965 27.49527 2.9749171 8.1 2008/08/20 20020 C/2007 D1 LINEAR 171366.7 0.99995 41.50701 8.793 8.9 2007/06/18 C/2007 D3 LINEAR 652 0.99201 45.92022 5.20897 9.2 2007/05/27 16650 C/2007 E2 Lovejoy 1330 0.99918 95.8830 1.092939 10.9 2007/03/27 49000 C/2007 K1 Lemmon 436 0.97880 108.4325 9.23905 8.6 2007/05/07 9100 C/2007 K6 McNaught 224 0.9847 105.064 3.4330 10.6 2007/07/01 3350 C/2007 M1 McNaught 1564 0.99522 139.72142 7.47465 5.6 2008/08/11 61900 C/2007 M2 Catalina (CSS) 5360 0.999339 80.94565 3.541050 9.0 2008/12/08 392000 C/2007 M3 LINEAR 171.45 0.979768 161.76086 3.468759 9.9 2007/09/04 2245 C/2007 N3 Lulin 72000 0.9999833 178.373611 1.21225837 9.7 2009/01/10 19500000 C/2007 T1 McNaught 4040 0.999760 117.64244 0.9685028 11.1 2007/12/12 256000 Spacewatch 12200 0.999603 86.99476 4.842732 7.1 2010/04/26 1350000 C/2007 Y2 McNaught 1210 0.99652 98.50321 4.20896 9.2 2008/04/08 42100 C/2008 C1 Chen-Gao 101627.54 0.9999876 61.7845 1.262343 11.7 2008/04/16 32398532.38 C/2008 E3 Garradd 3740 0.99852 105.07653 5.53103 5.1 2008/08/02 229000 C/2008 G1 Gibbs 365 0.98908 72.856 3.9898 10.5 2009/01/11 6980 C/2008 J1 Boattini 166.07 0.989617 61.78002 1.7242934 8.8 2008/07/13 2140.1 C/2008 L3 Hill 330 0.9939 100.201 2.0113 10.6 2008/04/22 5900 C/2008 N1 Holmes 973.1 0.997140 115.52100 2.7835117 9.9 2009/09/25 30360 C/2008 Q1 Maticic 593.6 0.995015 118.62662 2.959143 9.8 2008/12/30 14460 C/2008 Q3 Garradd 8900 0.999799 140.70663 1.7982291 6.1 2009/06/23 840000 C/2009 F1 Larson 106 0.9827 171.3755 1.8307 15.1 2009/06/25 1090 C/2009 F2 McNaught 346.1 0.98303 59.36694 5.87503 4.9 2009/11/14 6440 C/2009 F6 Yi-SWAN 512.2 0.997512 85.76481 1.274159 9.7 2009/05/07 11590 C/2009 K2 Catalina (CSS) 1460 0.997776 66.82192 3.246173 11.8 2010/02/07 55800 C/2009 O2 Catalina (CSS) 278.3 0.997501 107.96052 0.6955493 12.3 2010/03/24 4643 C/2009 T1 McNaught 3680 0.99831 89.89396 6.22041 8.5 2009/10/08 223000 C/2009 T3 LINEAR 4300 0.999470 148.74183 2.281140 13.5 2010/01/12 282000 C/2009 U3 Hill 167.88 0.991575 51.26077 1.414424 12.6 2010/03/20 2175 C/2009 U5 Grauer 10600 0.99943 25.4726 6.09424 9.1 2010/06/22 1090000 C/2009 W2 Boattini 16000 0.99956 164.49053 6.90713 6.9 2010/05/01 1900000 C/2009 Y1 Catalina (CSS) 375.4 0.993285 107.31660 2.5204945 6.5 2011/01/28 7273 C/2010 A4 Siding Spring 292.4 0.990638 96.73015 2.737999 7.4 2010/10/08 5001 C/2010 B1 Cardinal 2932 0.998997 101.97777 2.9414900 10.0 2011/02/07 158700 C/2010 D3 WISE 11600 0.99963 76.39488 4.24754 10.0 2010/09/03 1250000 C/2010 E1 Garradd 110.4 0.9759 71.698 2.66219 11.8 2009/11/07 1160 WISE-Garradd 299.3 0.990500 107.62532 2.842764 8.5 2010/11/07 5177 C/2010 G1 Boattini 480 0.9975 78.3870 1.20455 13.1 2010/04/02 10000 C/2010 G3 WISE 2630 0.99814 108.26760 4.90765 8.9 2010/04/11 135000 C/2010 H1 Garradd 8400 0.99967 36.5317 2.74555 12.4 2010/06/18 800000 C/2010 J2 McNaught 6300 0.999460 125.85156 3.386994 10.4 2010/06/03 500000 C/2010 L3 Catalina (CSS) 12800 0.99923 102.63105 9.88290 4.7 2010/11/10 1400000 C/2011 A3 Gibbs 1167 0.997992 26.07435 2.344839 9.7 2011/12/16 39900 C/2011 C1 McNaught 344.1 0.997433 16.82561 0.8833784 12.7 2011/04/18 6380 C/2011 C3 Gibbs 320 0.99527 49.3760 1.51689 14.1 2011/04/07 5700 C/2011 F1 LINEAR 2780 0.999345 56.61904 1.818266 8.3 2013/01/07 146000 C/2011 N2 McNaught 10000 0.9997 33.675 2.5634 6.3 2011/10/18 C/2011 O1 LINEAR 1210 0.996785 76.49889 3.890653 7.2 2012/08/18 42100 C/2011 Q1 PANSTARRS 3300 0.9979 94.8620 6.78009 7.5 2011/06/29 190000 C/2012 A2 LINEAR 978.2 0.996384 125.868509 3.5374738 8.4 2012/11/05 30590 C/2012 C1 McNaught 1274 0.99620 96.27770 4.837975 5.4 2013/02/04 45500 MOSS 139913.5 0.999991 27.74418 1.296092 11.1 2012/09/28 52335655.79 C/2012 E1 Hill 3760 0.99801 122.54208 7.50290 5.7 2011/07/04 231000 C/2012 E3 PANSTARRS 221 0.9827 105.658 3.8274 9.9 2011/05/12 3280 C/2012 F6 Lemmon 487.1 0.9984987 82.60885 0.7312382 5.5 2013/03/24 10750 C/2012 K5 LINEAR 774.4 0.9985256 92.848032 1.14181083 10.5 2012/11/28 21550 C/2012 K6 McNaught 4130 0.999188 135.21497 3.353033 8.8 2013/05/21 265000 C/2012 L1 LINEAR 767.3 0.997051 87.21917 2.262410 11.9 2012/12/25 21250 C/2012 L2 LINEAR 563.4 0.997322 70.98049 1.5085342 9.5 2013/05/09 13370 C/2012 L3 LINEAR 331 0.99079 134.19664 3.04503 9.0 2012/06/12 6020 Palomar 6350 0.99897 25.37958 6.53605 8.8 2015/08/16 505000 C/2012 OP Siding Spring 1054 0.99658 114.82872 3.60707 11.2 2012/12/04 34200 C/2012 S4 PANSTARRS 252223.8 0.999983 126.54131 4.34873 9.2 2013/06/28 126673944.62 C/2012 T4 McNaught 110 0.983 24.092 1.953 12.7 2012/10/10 1200 C/2012 U1 PANSTARRS 12200 0.99957 56.33902 5.26390 8.3 2014/07/04 1350000 C/2012 V1 PANSTARRS 3800 0.99945 157.8399 2.0890 11.5 2013/07/21 230000 C/2012 V2 LINEAR 616.7 0.997641 67.18470 1.4547602 8.4 2013/08/16 15320 C/2012 X1 LINEAR 156.71 0.989803 44.36218 1.597956 5.7 2014/02/21 1962 C/2013 E2 Iwamoto 233.07 0.993936 21.85771 1.413322 10.6 2013/03/09 3558 C/2013 F2 Catalina (CSS) 8400 0.99926 61.74927 6.21785 7.1 2013/04/19 770000 C/2013 F3 McNaught 759 0.99703 85.4445 2.252612 12.3 2013/05/25 20900 C/2013 G5 Catalina (CSS) 2700 0.99965 40.617 0.92894 14.5 2013/09/01 140000 C/2013 G6 Lemmon 387.1 0.994708 124.08435 2.048499 6.8 2013/07/25 7620 C/2013 G7 McNaught 2190 0.99786 105.11012 4.677404 6.2 2014/03/18 102200 C/2013 G8 PANSTARRS 3340 0.99846 27.61506 5.14118 8.4 2013/11/14 193000 C/2013 H1 La Sagra 181.5 0.98542 27.0895 2.64696 6.2 2013/05/19 2445 C/2013 J3 McNaught 1950 0.99795 118.2255 3.98869 5.8 2013/02/22 86000 C/2013 J5 Boattini 10000 0.999 136.011 4.9049 10.0 2012/11/29 C/2013 O3 McNaught 819 0.99612 102.83974 3.18010 11.1 2013/09/09 23400 C/2013 P2 PANSTARRS 2590 0.998904 125.53216 2.834925 11.8 2014/02/17 132000 C/2013 R1 Lovejoy 515.4 0.9984250 64.04094 0.81182562 11.6 2013/12/22 11702 Spacewatch 450.8 0.98707 31.40046 5.83064 6.7 2014/08/17 9570 C/2013 U2 Holvorcem 891 0.99426 43.09366 5.116745 5.3 2014/10/25 26590 C/2013 V5 Oukaimeden 488.1 0.9987183 154.88544 0.6255811 10.8 2014/09/28 10784 C/2013 Y2 PANSTARRS 219.9 0.991275 29.41474 1.919086 9.7 2014/06/13 3262 C/2014 A5 PANSTARRS 152.3 0.96848 31.9046 4.79991 11.6 2014/08/14 1879 C/2014 C3 NEOWISE 108.4 0.98283 151.7843 1.86203 12.0 2014/01/16 1129 C/2014 E2 Jacques 688 0.999035 156.392752 0.6639172 10.4 2014/07/02 18060 C/2014 F1 Hill 3600 0.9990 108.2529 3.49638 10.4 2013/10/04 210000 C/2014 F2 Tenagra 148.20 0.97089 119.06119 4.314460 5.6 2015/01/02 1804 C/2014 G1 PANSTARRS 1000 0.9943 165.6403 5.4685 6.0 2013/11/06 30000 C/2014 H1 Christensen 141 0.9849 99.936 2.1389 14.8 2014/04/15 1700 C/2014 M2 Christensen 980 0.99293 32.4062 6.9085 7.9 2014/07/18 30500 C/2014 M3 Catalina (CSS) 138 0.9824 164.90964 2.43428 12.9 2014/06/21 1630 C/2014 N2 PANSTARRS 4700 0.99954 133.0132 2.184401 12.0 2014/10/08 330000 C/2014 N3 NEOWISE 5800 0.999331 61.63825 3.882231 4.7 2015/03/13 442000 PANSTARRS 20602.48 0.99969 81.3473 6.2444 7.6 2016/12/10 2957246.18 C/2014 Q1 PANSTARRS 1129 0.999721 43.10685 0.314570 9.8 2015/07/06 38000 C/2014 Q2 Lovejoy 579.4 0.9977728 80.301302 1.2903578 7.9 2015/01/30 13946 C/2014 Q6 PANSTARRS 6883 0.999386 49.7968 4.222 6.5 2015/01/06 PANSTARRS 260 0.9913 124.818 2.2233 13.3 2014/07/09 4100 C/2014 R1 Borisov 179.4 0.992501 9.93289 1.345431 9.8 2014/11/19 2403 C/2014 R3 PANSTARRS 14434.53 0.9995 90.84 7.2756 6.3 2016/08/08 1734251.91 C/2014 R4 Gibbs 3200 0.99943 42.4116 1.81797 8.7 2014/10/21 180000 C/2014 S2 PANSTARRS 169.71 0.987622 64.67037 2.100644 5.0 2015/12/09 2210.9 C/2014 U3 Kowalski 1100 0.9976 152.9921 2.5588 12.4 2014/09/03 40000 C/2014 W2 PANSTARRS 1610 0.998341 81.998347 2.6702156 7.9 2016/03/10 64570 C/2014 W8 PANSTARRS 174.518 0.9711 42.111 5.044 10.5 2015/09/08 2305.52 PANSTARRS 902 0.99666 149.7827 3.01028 6.8 2015/04/05 27100 C/2015 C2 SWAN 471 0.99849 94.5013 0.711372 14.9 2015/03/04 10200 PANSTARRS 1484 0.999295 6.25965 1.046217 7.9 2017/05/09 57200 C/2015 F3 SWAN 232 0.99640 73.3865 0.83444 14.2 2015/03/09 3530 C/2015 F4 Jacques 116.37 0.985873 48.70495 1.6439255 11.4 2015/08/10 1255.3 C/2015 J2 PANSTARRS 246.9 0.98250 17.28183 4.32039 10.1 2015/09/08 3880 C/2015 K1 MASTER 180.6 0.98584 29.3817 2.55749 9.1 2014/10/13 2426 C/2015 K2 PANSTARRS 260 0.9944 29.110 1.45527 20.7 2015/06/08 4200 C/2015 M1 PANSTARRS 390 0.9946 57.310 2.0916 15.9 2015/05/15 8000 C/2015 M3 PANSTARRS 133.0 0.97328 65.95107 3.55241 11.5 2015/08/26 1533 C/2015 O1 PANSTARRS 651202.3 0.999994 127.211 3.7296 7.2 2018/02/19 C/2015 R3 PANSTARRS 3400 0.9985 83.6135 4.9033 5.0 2014/02/11 190000 LINEAR 1500 0.99906 11.3925 1.41314 10.8 2016/08/27 58000 C/2015 V3 PANSTARRS 822 0.99485 86.2318 4.23569 6.3 2015/11/24 23600 C/2015 WZ PANSTARRS 193.16 0.992873 134.13494 1.3766377 10.5 2016/04/15 2685 C/2015 Y1 LINEAR 292.5 0.99141 71.2196 2.514080 6.7 2016/05/15 5000 C/2016 A5 PANSTARRS 1200 0.9976 40.319 2.9469 12.8 2015/06/28 43000 C/2016 A6 PANSTARRS 217.53 0.9889 120.92 2.4124 7.8 2015/11/05 3208.44 C/2016 B1 NEOWISE 453 0.99293 50.4644 3.20625 5.9 2016/12/04 9700 C/2016 E2 Kowalski 138.88 0.992 135.95 1.074 19.5 2016/02/06 1636.74 C/2016 J2 Denneau 700 0.998 130.343 1.5184 15.3 2016/04/11 C/2016 KA Catalina (CSS) 6000 0.9990 104.6293 5.4009 8.8 2016/02/01 400000 C/2016 M1 PANSTARRS 1760 0.99875 90.99839 2.21103 8.1 2018/08/10 74000 C/2016 N4 MASTER 5315.30 0.99940 72.5573 3.19912 11.1 2017/09/16 387525 C/2016 N6 PANSTARRS 1600 0.9984 105.8345 2.6699 5.0 2018/07/18 67000 C/2016 P4 PANSTARRS 330 0.9819 29.89 5.888 10.7 2016/10/16 5900 C/2016 Q2 PANSTARRS 5467.19 0.9987 109.409 7.087 8.3 2021/05/10 404254.11 C/2016 R2 PANSTARRS 780 0.9967 58.2134 2.6020 5.1 2018/05/09 22000 C/2016 T1 Matheny 126.10 0.9818 126.095 2.3000 12.1 2017/02/01 1415.98 C/2016 T2 Matheny 101.74 0.98125 81.311 1.9078 13.8 2016/12/29 1026.30 C/2016 T3 PANSTARRS 144 0.9816 22.6727 2.6496 8.1 2017/09/06 1730 PANSTARRS 194.0 0.99531 24.0354 0.910285 18.7 2017/03/07 2700 PANSTARRS 111625.443 0.99992 32.431 9.2164 11.2 2018/02/17 37295204.74 C/2017 D2 Barros 1369.820 0.9982 31.26579 2.48587 11.1 2017/07/14 51000 C/2017 D5 PANSTARRS 112.2883 0.9806 131.03858 2.1672 14.6 2017/01/08 1200 C/2017 E4 Lovejoy 477.669 0.9989 88.1867 0.49357 15.6 2017/04/23 10000 C/2017 E5 Lemmon 388.6996 0.9954 122.6377 1.7829 12.0 2016/06/10 7600 C/2017 G3 PANSTARRS 287.3256 0.99098 159.051 2.59048 14.2 2017/04/15 4900 C/2017 K6 Jacques 1054.174 0.99810 57.2511 2.00279 10.7 2018/01/03 34000 C/2017 M3 PANSTARRS 173.8170 0.9732 77.5073 4.6561 6.2 2017/04/28 2292 C/2017 O1 ASASSN 439.1911 0.99658 39.849 1.4987 10.4 2017/10/14 9200 C/2017 P2 PANSTARRS 1210 0.997967 50.08486 2.461777 9.2 2017/12/06 42100 C/2017 T2 PANSTARRS 5007 0.99968 57.231 1.6151 10.2 2020/05/05 354300 C/2017 T3 ATLAS 1344 0.99939 88.10362 0.82522 11.1 2018/07/19 49280 C/2017 U2 Fuls 8555.39 0.99921 95.4291 6.700 8.8 2017/08/28 C/2017 Y1 PANSTARRS 3791 0.99902 55.2287 3.719 9.3 2017/08/31 234400 C/2017 Y2 PANSTARRS 2502.89 0.99841 124.67 3.957 8.0 2020/08/19 C/2018 A3 ATLAS 487.788 0.99328 139.56 3.277 9.2 2019/01/12 10773 C/2018 E2 Barros 1769.556 0.99778 97.7428 3.92 6.4 2017/12/23 74439 Lemmon 640.788 0.99757 84.694 1.55663 18.2 2018/05/23 16221.08 C/2018 F1 Grauer 322.415 0.9907 46.0706 2.993 13.7 2018/12/14 5789.36 Lemmon 833.97 0.99565 136.66655 3.627 12.2 2019/09/10 24084.39 C/2018 L2 ATLAS 246.853 0.9931 67.4235 1.712 8.1 2018/12/02 3879 C/2018 N1 NEOWISE 693.83 0.9981 159.44 1.307 15.0 2018/08/01 C/2018 R3 Lemmon 1970.10 0.99934 69.7154 1.29 11.3 2019/06/07 87446.17 C/2018 R4 Fuls 311.353 0.99451 11.68371 1.7093 11.8 2018/03/03 5494 C/2018 V4 Africano 214.059 0.98506 69.0028 3.19901 15.7 2019/03/01 3131.89 C/2018 X2 Fitzsimmons 155.971 0.9864 23.06 2.125 6.4 2019/07/08 1947.93 C/2018 Y1 Iwamoto 109.736 0.988 160.4 1.287 12.3 2019/02/07 1149.57 C/2019 B1 Africano 151.97 0.9895 123.36 1.597 14.6 2019/03/19 1873.55 C/2019 D1 Flewelling 137.6571 0.989 34.098 1.5775 11.8 2019/05/11 1615.12 C/2019 H1 NEOWISE 230.7855 0.99201 104.579 1.8448 13.5 2019/04/27 3506.07 C/2019 J2 Palomar 610.07 0.99717 105.138 1.7269 11.6 2019/07/19 ATLAS 355.31 0.99424 148.2972 2.045 15.1 2019/05/31 C/2019 K4 Ye 2370.96 0.9990 105.31 2.2594 12.8 2019/06/16 115449.93 C/2019 K5 Young 150.99 0.9865 15.315 2.035 12.3 2019/06/22 1855.41 C/2019 K8 ATLAS 1440.4914 0.998 93.222 3.195 11.3 2019/07/21 C/2019 N1 ATLAS 13156.57 0.99987 82.424 1.7047 9.0 2020/12/01 C/2019 T3 ATLAS 11484.78 0.99948 121.86 5.9468 6.6 2021/03/02 C/2019 T4 ATLAS 1007.58 0.9958 53.62 4.245 5.6 2022/06/09 31983.74 C/2019 U6 Lemmon 435.36 0.9979 61.0049 0.914 13.3 2020/06/18 9084.03 C/2019 V1 Borisov 3033.17 0.99898 61.8636 3.0968 14.5 2020/07/16 C/2019 Y1 ATLAS 231.099 0.9964 73.347 0.8378 12.4 2020/03/15 3513.22 C/2019 Y4 ATLAS 331.14 0.9992 45.380 0.253 7.9 2020/05/31 6025.89 C/2019 Y4-B ATLAS 665.948 0.99962 45.454 0.2525 15.8 2020/05/31 17185 C/2020 A2 Iwamoto 1070.02 0.9991 120.75 0.978 15.0 2020/01/08 35002.27 C/2020 A3 ATLAS 6807.33 0.9991 146.7 5.767 7.7 2019/06/29 C/2020 B3 Rankin 1919.5 0.99826 20.703 3.3446 14.5 2019/10/19 84101.96 C/2020 F3 NEOWISE 377.32 0.9992 128.937 0.295 12.3 2020/07/03 7329.46 C/2020 F6 PANSTARRS 405.3 0.99134 174.58 3.511 13.2 2020/04/11 8159.58 C/2020 F8 SWAN 6642.61 0.99994 110.80 0.430 11.6 2020/05/27 C/2020 H2 Pruyne 183.596 0.9955 125.04 0.834 19.8 2020/04/27 2487.72 C/2020 H4 Leonard 140.477 0.9933 84.320 0.9383 16.5 2020/08/29 1665.00 C/2020 H5 Robinson 2497.44 0.9963 70.204 9.3500 4.5 2020/12/05 C/2020 H7 Lemmon 1476.6 0.997 135.92 4.42 11.1 2020/06/02 56742.20 C/2020 H8 PANSTARRS 594.908 0.99214 99.65 4.6744 10.4 2020/06/04 14510 C/2020 H11 PANSTARRS Lemmon 10470 0.99927 151.41 7.631 7.4 2020/09/15 1070000 C/2020 J1 SONEAR 9376.42 0.9996 142.305 3.356 7.2 2021/04/18 C/2020 K1 PANSTARRS 3141.03 0.99902 89.646 3.078 5.6 2023/05/09 C/2020 K2 PANSTARRS 8380.85 0.99894 91.0288 8.8762 6.1 2020/08/05 C/2020 K3 Leonard 210.450 0.9924 128.72 1.593 14.8 2020/05/30 3053.03 C/2020 K6 Rankin 2876.55 0.998 103.619 5.8844 8.1 2021/09/11 C/2020 K7 PANSTARRS 108.4 0.9411 32.059 6.3847 7.9 2019/10/30 1128.70 C/2020 M5 ATLAS 4936.2 0.9994 93.223 3.005 6.9 2021/08/19 346814.64 C/2020 N2 ATLAS 108.74 0.9835 161.034 1.746 15.6 2020/08/23 1134.00 C/2020 P3 ATLAS 4910.32 0.9986 61.89 6.812 6.7 2021/04/20 C/2020 R2 ATLAS 398.332 0.9882 53.22 4.693 7.1 2022/02/24 7950.15 C/2020 R6 Rankin 451.023 0.9931 82.83 3.129 7.4 2019/09/10 C/2020 R7 ATLAS 6397.95 0.99953 114.893 2.957 10.7 2022/09/16 C/2020 S3 Erasmus 187.987 0.99788 19.861 0.3985 13.0 2020/12/12 2577.50 C/2020 S4 PANSTARRS 4962.260 0.99932 20.5750 3.3673 7.4 2023/02/09 C/2020 S8 PANSTARRS 271.2715 0.99129 108.517 2.3639 8.1 2021/04/10 4468.01 C/2020 T2 Palomar 323.34 0.99364 27.873 2.055 8.8 2021/07/11 5814.24 C/2020 T5 Lemmon 930.79 0.99797 66.604 1.889 16.2 2020/10/09 28398.06 C/2020 U5 Lemmon 79736.84 0.99995 97.280 3.756 9.7 2022/04/27 C/2020 Y2 ATLAS 1217.888 0.99743 101.281 3.132 6.4 2022/06/17 42502.95 C/2020 Y3 ATLAS 151.18 0.98678 83.097 1.999 14.6 2020/12/03 1858.91 C/2021 A2 NEOWISE 257.26 0.9945 106.978 1.413 14.7 2021/01/22 4126.29 C/2021 A6 PANSTARRS 11846.50 0.99933 75.605 7.929 7.1 2021/05/05 C/2021 A7 NEOWISE 5448.01 0.99964 78.149 1.968 13.5 2021/07/15 C/2021 B2 PANSTARRS 336.015 0.99252 38.094 2.513 4.8 2021/07/15 6159.5 C/2021 C1 Rankin 8030.77 0.99957 143.04 3.481 8.8 2020/12/07 C/2021 C4 ATLAS 4645.31 0.99903 132.84 4.504 6.9 2021/01/17 C/2021 C5 ATLAS 14042.0 0.99977 50.787 3.241 12.0 2023/02/10 C/2021 G2 ATLAS 4011.4 0.99876 48.478 4.976 5.7 2024/09/10 C/2021 N3 PANSTARRS 158.23 0.9640 26.74 5.701 7.1 2020/08/17 1990.4 C/2021 P2 PANSTARRS 2211.11 0.9977 150.02 5.072 5.4 2023/01/21 C/2021 P4 ATLAS 305.44 0.9965 56.31 1.080 8.7 2022/07/30 5338.32 C/2021 Q6 PANSTARRS 10.932 0.9992 161.85 8.716 6.9 2024/03/21 C/2021 R2 PANSTARRS 2265.45 0.9968 134.46 7.312 7.7 2021/12/25 C/2021 R7 PANSTARRS 989.93 0.9943 158.85 5.640 7.5 2021/04/14 C/2021 S3 PANSTARRS 3317.47 0.9996 58.55 1.318 6.8 2024/02/14 C/2021 S4 Tsuchinshan (*CTC) 161.91 0.9583 17.478 6.694 7.0 2023/12/31 2035.2 C/2021 T1 Lemmon 1463.83 0.9979 140.35 3.058 5.7 2021/10/14 56007.2 C/2021 T4 Lemmon 43170 0.9999 160.757 1.482 6.9 2023/07/31 8970000 C/2021 U5 Catalina 216.86 0.9891 39.05 2.363 6.9 2022/01/26 3193.50 C/2021 V1 Rankin 679.40 0.9956 71.441 3.014 15.9 2022/04/30 17709.00 C/2022 A1 Sarneczky 411.49 0.9970 116.51 1.253 19.2 2022/01/31 8347 C/2022 A3 Lemmon - ATLAS 1006.28 0.9963 88.360 3.703 5.3 2023/09/28 31921.61 C/2022 B4 382.80 0.9964 20.043 1.380 21.7 2022/01/29 C/2022 D2 Kowalski 356.11 0.9956 22.655 1.555 14.1 2022/03/27 6720 C/2022 H1 Kowalski 1158.59 0.9934 49.870 7.693 6.3 2024/01/18 C/2022 L1 Catalina 528.47 0.9970 123.468 1.591 13.3 2022/09/28 12150 C/2022 L4 PANSTARRS 254.18 0.9881 141.224 3.015 16.6 2021/12/08 C/2022 P3 ZTF 243.88 0.9894 59.519 2.561 14.5 2022/07/27 3808 C/2022 R2 ATLAS 422.92 0.9985 52.895 0.633 16.6 2022/10/25 8697 C/2022 T1 Lemmon 4655 0.9993 22.543 3.444 5.1 2024/02/17 318000 C/2022 U1 Leonard 5401 0.9992 128.126 4.203 6.7 2025/03/25 C/2022 U4 Bok 3727 0.9992 52.038 2.898 9.7 2023/08/03 227000 C/2022 W2 ATLAS 332.03 0.9906 63.533 3.123 14.1 2023/03/08 6050 C/2022 W3 Leonard 280.85 0.9950 103.560 1.398 14.0 2023/06/22 4707 C/2023 A1 Leonard 232.82 0.9921 94.744 1.835 7.8 2023/03/18 3552 C/2023 A2 SWAN 987.3 0.9990 94.708 0.948 12.5 2023/01/20 31000 C/2023 B2 ATLAS 589.595 0.9970 40.771 1.743 8.7 2023/03/10 14316 C/2023 C2 ATLAS 4219 0.9994 48.319 2.368 7.1 2024/11/16 27400 C/2023 F1 PanSTARRS 221.9 0.9923 131.744 1.708 7.8 2023/06/08 3304 C/2023 H1 PanSTARRS 1611 0.9972 21.777 4.44 13.2 2024/11/28 C/2023 H2 Lemmon 246.6 0.9963 113.75 0.894 10.0 2023/10/29 3872 Comet designation Comet name /discoverer Semimajor axis (AU) Eccentricity Inclination (°) Perihelion distance (AU) Absolute magnitude (H/M1/M2) Perihelion date Period (3) (years) Ref == See also == * List of comets by type * List of Halley-type comets * List of hyperbolic comets * List of long-period comets * List of numbered comets * List of periodic comets near parabolic Its orbit has an eccentricity of 0.01 and an inclination of 6° with respect to the ecliptic. Often, these comets, due to their extreme semimajor axes and eccentricity, will have small orbital interactions with planets and minor planets, most often ending up with the comets fluctuating significantly in their orbital path. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. Observations taken in January and February 2002 showed that the ""asteroid"" had developed a small amount of cometary activity as it approached perihelion. Damocloids have been studied as possible extinct cometary candidates due to the similarity of their orbital parameters with those of Halley-family comets. ==See also== * List of Halley-type comets == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris Category:Halley- type comets Category:Near-Earth comets Category:Damocloids 20010728 This comet probably represents the transition between typical Halley-family/long-period comets and extinct comets. The body's observation arc begins with its official discovery observation in March 1931. == Physical characteristics == Halleria is an assumed carbonaceous C-type asteroid, which agrees with the overall spectral type for members of the Charis family. === Rotation period === Between 2005 and 2011, three rotational lightcurves of Halleria were obtained from photometric observations by Donald Pray, René Roy, and Pierre Antonini (). ",0.0526315789,210,2.3613,14.34457,8.8,E
-"Next, we treat projectile motion in two dimensions, first without considering air resistance. Let the muzzle velocity of the projectile be $v_0$ and the angle of elevation be $\theta$ (Figure 2-7). Calculate the projectile's range.","The Range is maximum when angle \theta = 45°, i.e. \sin 2\theta=1. ==See also== * Atlatl * Ballistics * Gunpowder * Bullet * Impact depth * Kinetic bombardment * Shell (projectile) * Projectile point * Projectile use by animals * Arrow * Dart * Missile * Sling ammunition * Spear * Torpedo * Range of a projectile * Space debris * Trajectory of a projectile ==Notes== ==References== * ==External links== * Open Source Physics computer model * Projectile Motion Applet * Another projectile Motion Applet Category:Ammunition Category:Ballistics Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. There are various calculations for projectiles at a specific angle \theta: 1\. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Mathematically, it is given as t=U \sin\theta/g where g = acceleration due to gravity (app 9.81 m/s²), U = initial velocity (m/s) and \theta = angle made by the projectile with the horizontal axis. 2\. Range (R): The Range of a projectile is the horizontal distance covered (on the x-axis) by the projectile. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The second solution is the useful one for determining the range of the projectile. A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. In physics, a projectile launched with specific initial conditions will have a range. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. A range table was a list of angles of elevation a particular artillery gun barrel needed to be set to, to strike a target at a particular distance with a projectile of a particular weight using a propellant cartridge of a particular weight. The laser rangefinder and computer-based FCS make guns highly accurate. ==Superelevation== When firing a missile such as a MANPADS at an aircraft target, superelevation is an additional angle of elevation above the angle sighted on which corrects for the effect of gravity on the missile. ==See also== *Altitude (astronomy) *Pitching moment ==References== * Gunnery Instructions, U.S. Navy (1913), Register No. 4090 * Gunnery And Explosives For Artillery Officers (1911) * Fire Control Fundamentals, NAVPERS 91900 (1953), Part C: The Projectile in Flight - Exterior Ballistics * FM 6-40, Tactics, Techniques, and Procedures for Field Artillery Manual Cannon Gunnery (23 April 1996), Chapter 3 - Ballistics; Marine Corps Warfighting Publication No. 3-1.6.19 * FM 23-91, Mortar Gunnery (1 March 2000), Chapter 2 Fundamentals of Mortar Gunnery * Fundamentals of Naval Weapons Systems: Chapter 19 (Weapons and Systems Engineering Department United States Naval Academy) * Naval Ordnance and Gunnery (Vol.1 - Naval Ordnance) NAVPERS 10797-A (1957) * Naval Ordnance and Gunnery (Vol.2 - Fire Control) NAVPERS 10798-A (1957) * Naval Ordnance and Gunnery Category:Ballistics Category:Angle Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. thumb|A range of ""diabolo"" pellets with various nose profiles A pellet is a non-spherical projectile designed to be shot from an air gun, and an airgun that shoots such pellets is commonly known as a pellet gun. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. ",-3.5,15.425,72.0,5040,556,C
-Calculate the time needed for a spacecraft to make a Hohmann transfer from Earth to Mars,"* Ls 263 (Sol 505): Earth is closest to Mars (Sep 10, 1956). This is close to the modern value of 1/154 (many sources will cite somewhat different values, such as 1/193, because even a difference of only a couple of kilometers in the values of Mars' polar and equatorial radii gives a considerably different result). Solar time is a calculation of the passage of time based on the position of the Sun in the sky. Mars Year 1 is the first year of Martian timekeeping standard developed by Clancy et al. originally for the purposes of working with the cyclical temporal variations of meteorological phenomena of Mars, but later used for general timekeeping on Mars. They occur every 26, 79 and 100 years, and every 1,000 years or so there is an extra 53rd-year transit. ==Conjunctions== Transits of Earth from Mars usually occur in pairs, with one following the other after 79 years; rarely, there are three in the series. Start and End dates of Mars Years were determined for 1607-2141 by Piqueux et al. Earth and Mars dates can be converted in the Mars Climate Database, however, the Mars Years are only rational to apply to events that take place on Mars. The Observatory, 3 (1880), 471 * * SOLEX ==External links== * Transits of Earth on Mars – Fifteen millennium catalog: 5 000 BC – 10 000 AD * JPL HORIZONS System * Near miss of the Earth-moon system (2005-11-07) Earth from Mars Category:Earth Category:Mars However, Mars Year sols may be confused with rover mission times that are also expressed in sols. This short story was first published in the January 1971 issue of Playboy magazine.'Transit Of Earth' by Arthur C. Clarke read by himself, 16 October 2017. ==Dates of transits== Transits of Earth from Mars (grouped by series) November 10, 1595 May 5, 1621 May 8, 1700 November 9, 1800 November 12, 1879 May 8, 1905 May 11, 1984 November 10, 2084 November 15, 2163 May 10, 2189 May 13, 2268 November 13, 2368 May 10, 2394 November 17, 2447 May 13, 2473 May 16, 2552 November 15, 2652 May 13, 2678 ==Grazing and simultaneous transits== Sometimes Earth only grazes the Sun during a transit. A specific time within a day, always using UTC, is specified via a decimal fraction. ==References== ==External links== * Category:Types of year Category:Time in astronomy Year thumb|Transfer orbit from Earth to Mars. The last series ending was in 1211. ==View from Mars== No one has ever seen a transit of Earth from Mars, but the next transit will take place on November 10, 2084. Scientists generally use two sub-units of the Mars Year: * the Solar Longitude (Ls) system: 360 degrees per Mars Year that represent the position of Mars in its orbit around the Sun, or * the Sol system: 668 sols per Mars Year. In astronomy, a Julian year (symbol: a or aj) is a unit of measurement of time defined as exactly 365.25 days of SI seconds each.P. Kenneth Seidelmann, ed., The equivalent on Mars is termed Mars local true solar time (LTST). When the Sun has covered exactly 15 degrees (1/24 of a circle, both angles being measured in a plane perpendicular to Earth's axis), local apparent time is 13:00 exactly; after 15 more degrees it will be 14:00 exactly. * August 25, 2005: at 15:19:32 UTC, MRO was 100 million kilometers from Mars. Mean solar time is the hour angle of the mean Sun plus 12 hours. * January 29, 2006: at 06:59:24 UTC, MRO was 10 million kilometers from Mars. Date Duration in mean solar time February 11 24 hours March 26 24 hours − 18.1 seconds May 14 24 hours June 19 24 hours + 13.1 seconds July 25/26 24 hours September 16 24 hours − 21.3 seconds November 2/3 24 hours December 22 24 hours + 29.9 seconds These lengths will change slightly in a few years and significantly in thousands of years. ==Mean solar time== thumb|right|250px|The equation of time—above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow. For example, in the next 1000 years, seven days will be dropped from the Gregorian calendar but not from 1000 Julian years, so J3000.0 will be . == Julian calendar distinguished == The Julian year, being a uniform measure of duration, should not be confused with the variable length historical years in the Julian calendar. Also, better measurements have been made by using artificial satellites that have been put into orbit around Mars, including Mariner 9, Viking 1, Viking 2, and Soviet orbiters, and the more recent orbiters that have been sent from the Earth to Mars. ==In science fiction== A science fiction short story published in 1971 by Arthur C. Clarke, called ""Transit of Earth"", depicts a doomed astronaut on Mars observing the transit in 1984. ",0.2553,30,7.136,0.000216,2.24,E
-Calculate the maximum height change in the ocean tides caused by the Moon.,"File:High tide sun moon same side beginning.png|Spring tide: Sun and Moon on the same side (0°) File:Low tide sun moon 90 degrees.png|Neap tide: Sun and Moon at 90° File:High tide sun moon opposite side.png|Spring tide: Sun and Moon at opposite sides (180°) File:Low tide sun moon 270 degrees.png|Neap tide: Sun and Moon at 270° File:High tide sun moon same side end.png|Spring tide: Sun and Moon at the same side (cycle restarts) === Lunar distance === The changing distance separating the Moon and Earth also affects tide heights. The two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again lower high. To calculate the actual water depth, add the charted depth to the published tide height. The alternation in high tide heights becomes smaller, until they are the same (at the lunar equinox, the Moon is above the Equator), then redevelop but with the other polarity, waxing to a maximum difference and then waning again. === Current === The tides' influence on current or flow is much more difficult to analyze, and data is much more difficult to collect. The ocean bathymetry greatly influences the tide's exact time and height at a particular coastal point. Assuming (as a crude approximation) that the deceleration rate has been constant, this would imply that 70 million years ago, day length was on the order of 1% shorter with about 4 more days per year. === Bathymetry === The shape of the shoreline and the ocean floor changes the way that tides propagate, so there is no simple, general rule that predicts the time of high water from the Moon's position in the sky. At each place of interest, the tide heights are therefore measured for a period of time sufficiently long (usually more than a year in the case of a new port not previously studied) to enable the response at each significant tide-generating frequency to be distinguished by analysis, and to extract the tidal constants for a sufficient number of the strongest known components of the astronomical tidal forces to enable practical tide prediction. The daily inequality is not consistent and is generally small when the Moon is over the Equator. === Reference levels === The following reference tide levels can be defined, from the highest level to the lowest: * Highest astronomical tide (HAT) – The highest tide which can be predicted to occur. Furthermore, the coriolis and molecular mixing terms are omitted in the equations above since they are relatively small at the temporal and spatial scale of tides in shallow waters. The graph of Cook Strait's tides shows separately the high water and low water height and time, through November 2007; these are not measured values but instead are calculated from tidal parameters derived from years-old measurements. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. In (The Reckoning of Time) of 725 Bede linked semidurnal tides and the phenomenon of varying tidal heights to the Moon and its phases. Using a simple harmonic fitting algorithm with a moving time window of 25 hours, the water level amplitude of different tidal constituents can be found. A tidal height is a scalar quantity and varies smoothly over a wide region. Bede then observes that the height of tides varies over the month. Tides are the rise and fall of sea levels caused by gravitational forces exerted by the Moon and Sun and by Earth's rotation. He goes on to emphasise that in two lunar months (59 days) the Moon circles the Earth 57 times and there are 114 tides. They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. The difference between the height of a tide at perigean spring tide and the spring tide when the moon is at apogee depends on location but can be large as a foot higher. === Other constituents === These include solar gravitational effects, the obliquity (tilt) of the Earth's Equator and rotational axis, the inclination of the plane of the lunar orbit and the elliptical shape of the Earth's orbit of the Sun. Later the daily tides were explained more precisely by the interaction of the Moon's and the Sun's gravity. The two high waters on a given day are typically not the same height (the daily inequality); these are the higher high water and the lower high water in tide tables. 300px|thumb Tidal range is the difference in height between high tide and low tide. ",5,1.41,0.54,1855,0.241,C
-A particle of mass $m$ starts at rest on top of a smooth fixed hemisphere of radius $a$. Determine the angle at which the particle leaves the hemisphere.,"thumb|upright=1.5|Spherical pendulum: angles and velocities. thumb|150px|right|Equatorial Inertial wave pulse caused patterns of fluid flow inside a steadily-rotating spherical chamber. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. Trajectory of the mass on the sphere can be obtained from the expression for the total energy :E=\underbrace{\left[\frac{1}{2}ml^2\dot\theta^2 + \frac{1}{2}ml^2\sin^2\theta \dot \phi^2\right]}_{T}+\underbrace{ \bigg[-mgl\cos\theta\bigg]}_{V} by noting that the horizontal component of angular momentum L_z = ml^2\sin^2\\!\theta \,\dot\phi is a constant of motion, independent of time. Therefore, angle AOV measures 180° − θ. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. ==Lagrangian mechanics== Routinely, in order to write down the kinetic T=\tfrac{1}{2}mv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. thumb|226px|An epispiral with equation r(θ)=2sec(2θ) The epispiral is a plane curve with polar equation :\ r=a \sec{n\theta}. The angle \theta lies between two circles of latitude, where :E>\frac{1}{2}\frac{L_z^2}{ml^2\sin^2\theta}-mgl\cos\theta. ==See also== *Foucault pendulum *Conical pendulum *Newton's three laws of motion *Pendulum *Pendulum (mathematics) *Routhian mechanics ==References== ==Further reading== * * * * * * * * Category:Pendulums That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. Arrows on this cross section show the direction and strength of flow in the equatorial plane as the sphere continues to rotate clockwise on its axis which shown at left . thumb|250px|right|The Western Hemisphere The Western Hemisphere is the half of the planet Earth that lies west of the Prime Meridian (which crosses Greenwich, London, England) and east of the 180th meridian. Angle BOA is a central angle; call it θ. Its portion lying east of the 180th meridian is the only part of the country lying in the Western Hemisphere. Therefore, : 2 \psi + 180^\circ - \theta = 180^\circ. Subtract : (180^\circ - \theta) from both sides, : 2 \psi = \theta, where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB. ====Inscribed angles with the center of the circle in their interior==== thumb|Case: Center interior to angle Given a circle whose center is point O, choose three points V, C, and D on the circle. Angle DOC is a central angle, but so are angles DOE and EOC, and : \angle DOC = \angle DOE + \angle EOC. The angle θ does not change as its vertex is moved around on the circle. The last equation shows that angular momentum around the vertical axis, |\mathbf L_z| = l\sin\theta \times ml\sin\theta\,\dot\phi is conserved. Angle DOC is a central angle, but so are angles EOD and EOC, and : \angle DOC = \angle EOC - \angle EOD. In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Similarly, the Euler–Lagrange equation involving the azimuth \phi, : \frac{d}{dt}\frac{\partial}{\partial\dot\phi}L-\frac{\partial}{\partial\phi}L=0 gives : \frac{d}{dt} \left( ml^2\sin^2\theta \cdot \dot{\phi} \right) =0 . This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Combining these results with equation (4) yields : \theta_0 = 2 \psi_2 - 2 \psi_1 therefore, by equation (3), : \theta_0 = 2 \psi_0. thumb|400px|Animated gif of proof of the inscribed angle theorem. ",6.283185307,12,14.44,48.189685,35.2,D
-"Consider the first stage of a Saturn $V$ rocket used for the Apollo moon program. The initial mass is $2.8 \times 10^6 \mathrm{~kg}$, and the mass of the first-stage fuel is $2.1 \times 10^6$ kg. Assume a mean thrust of $37 \times 10^6 \mathrm{~N}$. The exhaust velocity is $2600 \mathrm{~m} / \mathrm{s}$. Calculate the final speed of the first stage at burnout. ","The S-I was the first stage of the Saturn I rocket used by NASA for the Apollo program. == Design == The S-I stage was powered by eight H-1 rocket engines burning RP-1 fuel with liquid oxygen (LOX) as oxidizer. Studied with the Saturn A-1 in 1959, the Saturn A-2 was deemed more powerful than the Saturn I rocket, consisting of a first stage, which actually flew on the Saturn IB, a second stage which contains four S-3 engines that flew on the Jupiter IRBM and a Centaur high-energy liquid-fueled third stage. == References == * Koelle, Heinz Hermann, Handbook of Astronautical Engineering, McGraw-Hill, New York, 1961. {{Infobox rocket |image = |imsize = |caption = |function = Launch vehicle for Project Horizon and Apollo |manufacturer = |country-origin = United States |height = (w/o payload) |diameter = |mass = gross (to LEO) |stages = |capacities = |family = Saturn |status = Study, not developed |sites = Kennedy Space Center |payloads = |stagedata = }} The Saturn C-2 was the second rocket in the Saturn C series studied from 1959 to 1962. Studied in 1959, the Saturn B-1, was a four-stage concept rocket similar to the Jupiter-C, and consisted of a Saturn IB first stage, a cluster of four Titan I first stages used for a second stage, a S-IV third stage and a Centaur high-energy liquid-fueled fourth stage. *Free return trajectory simulation, Robert A. Braeunig, August 2008 *Encyclopedia Astronautica Saturn C-2 C2 Category:Cancelled space launch vehicles It formed the second stage of the Saturn I and was powered by a cluster of six RL-10A-3 engines. The Army's original design used the S-III stage with two J-2 engines as the second stage; after the Saturn program was transferred to NASA, the second stage was replaced with an S-II second stage using four J-2 engines. The S-IV was the second stage of the Saturn I rocket used by NASA for early flights in the Apollo program. The Saturn C-8 was the largest member of the Saturn series of rockets to be designed. This saved up to 20% of structural weight. ==References== * * Category:Apollo program Category:Rocket stages The S-IV stage was a large LOX/LH2-fueled rocket stage used for the early test flights of the Saturn I rocket. The initial launch of the Saturn I consisted of an active S-I, an inactive S-IV and inactive S-V stage. Further development of the C-2 vehicle was cancelled on 23 June 1961. ==Launch vehicle design== The original Saturn C-2 design (1959-1960) was a four-stage launch vehicle, using an S-I first stage using eight Rocketdyne H-1 engines, later flown on the Saturn I. The design was for a four-stage launch vehicle that could launch 21,500 kg (47,300 lb) to low Earth orbit and send 6,800 kg (14,900 lb) to the Moon via Trans-Lunar Injection. The S-III stage would have been added atop the S-II, to convert the C-2 into the five-stage Saturn C-3. Later, a fifth J-2 engine was added to the S-II stage to be used on the Saturn C-5, which eventually was developed as the Saturn V launch vehicle. During a discussion on the Saturn program, several major problems were brought up: * The adequacy of the Saturn C-1 launch vehicle for the orbital qualification of the complete Apollo spacecraft was in question. Excellent account of the evolution, design, and development of the Saturn launch vehicles. Excellent account of the evolution, design, and development of the Saturn launch vehicles. Excellent account of the evolution, design, and development of the Saturn launch vehicles. While this S-V/Centaur stage would never fly on any Saturn rockets, it would be used on Atlas and Titan launch vehicles. The Saturn C-8 configuration was never taken further than the design process, as it was too large and costly. ==References== *Bilstein, Roger E, Stages to Saturn, US Government Printing Office, 1980. . ",3.8,2.16,35.0,3930,49,B
-How large should we take $n$ in order to guarantee that the Trapezoidal and Midpoint Rule approximations for $\int_1^2(1 / x) d x$ are accurate to within 0.0001 ?,"The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. Therefore the total error is bounded by \text{error} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right] = \frac{f(\xi)h^3N}{12}=\frac{f(\xi)(b-a)^3}{12N^2}. === Periodic and peak functions === The trapezoidal rule converges rapidly for periodic functions. As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule. thumb|Illustration of ""chained trapezoidal rule"" used on an irregularly-spaced partition of [a,b]. == History == A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic. == Numerical implementation == === Non-uniform grid === When the grid spacing is non-uniform, one can use the formula \int_{a}^{b} f(x)\, dx \approx \sum_{k=1}^N \frac{f(x_{k-1}) + f(x_k)}{2} \Delta x_k , wherein \Delta x_k = x_{k} - x_{k-1} . === Uniform grid === For a domain discretized into N equally spaced panels, considerable simplification may occur. It follows that \int_{a}^{b} f(x) \, dx \approx (b-a) \cdot \tfrac{1}{2}(f(a)+f(b)). thumb|right|An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The error in approximating an integral by Simpson's rule for n = 2 is -\frac{1}{90} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{2880} f^{(4)}(\xi), where \xi (the Greek letter xi) is some number between a and b. The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result: \text{E} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right] There exists a number ξ between a and b, such that \text{E} = -\frac{(b-a)^3}{12N^2} f(\xi) It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. Interestingly, in this case the trapezoidal rule often has sharper bounds than Simpson's rule for the same number of function evaluations. == Applicability and alternatives == The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points. Namely, composite Simpson's 1/3 rule requires 1.8 times more points to achieve the same accuracy as trapezoidal rule. It follows from the above formulas for the errors of the midpoint and trapezoidal rule that the leading error term vanishes if we take the weighted average \frac{2M + T}{3}. The trapezoidal rule states that the integral on the right- hand side can be approximated as \int_{t_n}^{t_{n+1}} f(t,y(t)) \,\mathrm{d}t \approx \tfrac12 h \Big( f(t_n,y(t_n)) + f(t_{n+1},y(t_{n+1})) \Big). Let \Delta x_k = \Delta x = \frac{b-a}{N} the approximation to the integral becomes \begin{align} \int_{a}^{b} f(x)\, dx &\approx \frac{\Delta x}{2} \sum_{k=1}^{N} \left( f(x_{k-1}) + f(x_{k}) \right) \\\\[1ex] &= \frac{\Delta x}{2} \Biggl( f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + \dotsb + 2f(x_{N-1}) + f(x_N) \Biggr) \\\\[1ex] &= \Delta x \left( \sum_{k=1}^{N-1} f(x_k) + \frac{f(x_N) + f(x_0) }{2} \right). \end{align} ==Error analysis== right|thumb|An animation showing how the trapezoidal rule approximation improves with more strips for an interval with a=2 and b=8. Several techniques can be used to analyze the error, including: #Fourier series #Residue calculus #Euler–Maclaurin summation formula #Polynomial interpolation It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions. === Proof === First suppose that h=\frac{b-a}{N} and a_k=a+(k-1)h. Note since it starts and ends at zero, this approximation yields zero area. alt=Two-piece approximation|thumb|Two-piece alt=Four-piece approximation|thumb|Four-piece alt=Eight-piece approximation|thumb|Eight-piece After trapezoid rule estimates are obtained, Richardson extrapolation is applied. Number of pieces Trapezoid estimates First iteration Second iteration Third iteration (4 MA − LA)/3* (16 MA − LA)/15 (64 MA − LA)/63 1 0 (4×16 − 0)/3 = 21.333... (16×34.667 − 21.333)/15 = 35.556... (64×42.489 − 35.556)/63 = 42.599... 2 16 (4×30 − 16)/3 = 34.666... (16×42 − 34.667)/15 = 42.489... 4 30 (4×39 − 30)/3 = 42 8 39 *MA stands for more accurate, LA stands for less accurate == Example == As an example, the Gaussian function is integrated from 0 to 1, i.e. the error function erf(1) ≈ 0.842700792949715. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. The approximation becomes more accurate as the resolution of the partition increases (that is, for larger N, all \Delta x_k decrease). Using another approximation (for example, the trapezoidal rule with twice as many points), it is possible to take a suitable weighted average and eliminate another error term. Simpson's 1/3 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{1}{3} h\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\\\ &= \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right], \end{align} where h = (b - a)/2 is the step size. (This derivation is essentially a less rigorous version of the quadratic interpolation derivation, where one saves significant calculation effort by guessing the correct functional form.) === Composite Simpson's 1/3 rule === If the interval of integration [a, b] is in some sense ""small"", then Simpson's rule with n = 2 subintervals will provide an adequate approximation to the exact integral. The result is then obtained by taking the mean of the two formulas. === Simpson's rules in the case of narrow peaks === In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule. ",6,41,-1.0,2.3,2.3,B
-Find the length of the cardioid $r=1+\sin \theta$.,"The foot of the perpendicular from point O on the tangent is point (r\cos \varphi, r\sin \varphi) with the still unknown distance r to the origin O. Inserting the point into the equation of the tangent yields (r\cos\varphi - 2a)\cos\varphi + r\sin^2\varphi = 2a \quad \rightarrow \quad r = 2a(1 + \cos \varphi) which is the polar equation of a cardioid. For the cardioid r(\varphi) = 2a (1 - \cos\varphi) = 4a \sin^2\left(\tfrac{\varphi}{2}\right) one gets \rho(\varphi) = \cdots = \frac{\left[16a^2\sin^2\frac{\varphi}{2}\right]^\frac{3}{2}} {24a^2 \sin^2\frac{\varphi}{2}} = \frac{8}{3}a\sin\frac{\varphi}{2} \ . }} == Properties == thumb|Chords of a cardioid === Chords through the cusp === ; C1: Chords through the cusp of the cardioid have the same length 4a. Hence the cardioid has the polar representation r(\varphi) = 1 - \cos\varphi and its inverse curve r(\varphi) = \frac{1}{1 - \cos\varphi}, which is a parabola (s. parabola in polar coordinates) with the equation x = \tfrac{1}{2}\left(y^2 - 1\right) in Cartesian coordinates. Their intersection point is x(t) = 2(1 + \cos t)\cos t,\quad y(t) = 2(1 + \cos t)\sin t, which is a point of the cardioid with polar equation r = 2(1 + \cos t). thumb|Cardioid as caustic: light source Z, light ray \vec s, reflected ray \vec r thumb|Cardioid as caustic of a circle with light source (right) on the perimeter === Cardioid as caustic of a circle === The considerations made in the previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid. In order to keep the calculations simple, the proof is given for the cardioid with polar representation r = 2(1 \mathbin{\color{red}+} \cos\varphi) (§ Cardioids in different positions). ===== Equation of the tangent of the cardioid with polar representation r = 2(1 + \cos\varphi) ===== From the parametric representation \begin{align} x(\varphi) &= 2(1 + \cos\varphi) \cos \varphi, \\\ y(\varphi) &= 2(1 + \cos\varphi) \sin \varphi \end{align} one gets the normal vector \vec n = \left(\dot y , -\dot x\right)^\mathsf{T}. For the cardioids with the equations r=2a(1-\cos\varphi) \; and r = 2b(1 + \cos\varphi)\ respectively one gets: \frac{dy_a}{dx} = \frac{\cos(\varphi) - \cos(2\varphi)}{\sin(2\varphi) - \sin(\varphi)} and \frac{dy_b}{dx} = -\frac{\cos(\varphi) + \cos(2\varphi)}{\sin(2\varphi) + \sin(\varphi)}\ . These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by -\tfrac{4}{3} a. The catacaustic of a circle with respect to a point on the circumference is a cardioid. thumb|upright=1.25|r=\frac{\sin \theta}{\theta}, -20<\theta<20 thumb|upright=1.25|cochleoid (solid) and its polar inverse (dashed) In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation :r=\frac{a \sin \theta}{\theta}, the Cartesian equation :(x^2+y^2)\arctan\frac{y}{x}=ay, or the parametric equations :x=\frac{a\sin t\cos t}{t}, \quad y=\frac{a\sin^2 t}{t}. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area) \; d\varphi = \cdots = 8a\int_0^\pi\sqrt{\tfrac{1}{2}(1 - \cos\varphi)}\; d\varphi = 8a\int_0^\pi\sin\left(\tfrac{\varphi}{2}\right) d\varphi = 16a. }} {r(\varphi)^2 + 2 \dot r(\varphi)^2 - r(\varphi) \ddot r(\varphi)} \ . thumb|A cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. For a cardioid one gets: : The evolute of a cardioid is another cardioid, one third as large, and facing the opposite direction (s. picture). === Proof === For the cardioid with parametric representation x(\varphi) = 2a (1 - \cos\varphi)\cos\varphi = 4a \sin^2\tfrac{\varphi}{2}\cos\varphi\, , y(\varphi) = 2a (1 - \cos\varphi)\sin\varphi = 4a \sin^2\tfrac{\varphi}{2}\sin\varphi the unit normal is \vec n(\varphi) = (-\sin\tfrac{3}{2}\varphi, \cos\tfrac{3}{2}\varphi) and the radius of curvature \rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . From here one gets the parametric representation above: \begin{array}{cclcccc} x(\varphi) &=& a\;(-\cos(2\varphi) + 2\cos\varphi - 1) &=& 2a(1 - \cos\varphi)\cdot\cos\varphi & & \\\ y(\varphi) &=& a\;(-\sin(2\varphi) + 2\sin\varphi) &=& 2a(1 - \cos\varphi)\cdot\sin\varphi &.& \end{array} (The trigonometric identities e^{i\varphi} = \cos\varphi + i\sin\varphi, \ (\cos\varphi)^2 + (\sin\varphi)^2 = 1, \cos(2\varphi) = (\cos\varphi)^2 - (\sin\varphi)^2, and \sin (2\varphi) = 2\sin\varphi\cos\varphi were used.) == Metric properties == For the cardioid as defined above the following formulas hold: * area A = 6\pi a^2, * arc length L = 16 a and * radius of curvature \rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . The reflected ray is part of the line with equation (see previous section) \cos\left(\tfrac{3}{2}\varphi\right) x + \sin \left(\tfrac{3}{2}\varphi\right) y = 4 \left(\cos\tfrac{1}{2}\varphi\right)^3 \, , which is tangent of the cardioid with polar equation r = 2(1 + \cos\varphi) from the previous section.}} For cardioids the following is true: : The orthogonal trajectories of the pencil of cardioids with equations r=2a(1-\cos\varphi)\ , \; a>0 \ , \ are the cardioids with equations r=2b(1+\cos\varphi)\ , \; b>0 \ . Hence any secant line of the circle, defined above, is a tangent of the cardioid, too: : The cardioid is the envelope of the chords of a circle. (Trigonometric formulae were used: \sin\tfrac{3}{2}\varphi = \sin\tfrac{\varphi}{2}\cos\varphi + \cos\tfrac{\varphi}{2}\sin\varphi\ ,\ \cos\tfrac{3}{2}\varphi = \cdots, \ \sin\varphi = 2\sin\tfrac{\varphi}{2}\cos\tfrac{\varphi}{2}, \ \cos\varphi= \cdots \ . ) == Orthogonal trajectories == 300px|thumb|Orthogonal cardioids An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. Remark: If point O is not on the perimeter of the circle k, one gets a limaçon of Pascal. == The evolute of a cardioid == thumb| The evolute of a curve is the locus of centers of curvature. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. The proofs of these statement use in both cases the polar representation of the cardioid. # The envelope of these chords is a cardioid. thumb|Cremona's generation of a cardioid ==== Proof ==== The following consideration uses trigonometric formulae for \cos\alpha + \cos\beta, \sin\alpha + \sin\beta, 1 + \cos 2\alpha , \cos 2\alpha, and \sin 2\alpha. Hence a cardioid is a special pedal curve of a circle. ==== Proof ==== In a Cartesian coordinate system circle k may have midpoint (2a,0) and radius 2a. ",2.89,15,0.11,8,4.979,D
-"Estimate the volume of the solid that lies above the square $R=[0,2] \times[0,2]$ and below the elliptic paraboloid $z=16-x^2-2 y^2$. Divide $R$ into four equal squares and choose the sample point to be the upper right corner of each square $R_{i j}$. ","The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. Hence the area of this development is thumb|cloister vault :B = \int_{0}^{\pi r} r\sin\left(\frac{\xi}{r}\right) \mathrm{d}\xi = 2r^2 and the total surface area is: :A=8\cdot B=16r^2. === Alternate proof of the volume formula === Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). This leads to :V = \int_{-r}^{r} (2x)^2 \mathrm{d}z = 4\cdot \int_{-r}^{r} x^2 \mathrm{d}z = 4\cdot \int_{-r}^{r} (r^2-z^2) \mathrm{d}z=\frac{16}{3} r^3. By the Pythagorean theorem, the radius of the cylinder is thumb|upright=1.2|Finding the measurements of the ring that is the horizontal cross-section. \sqrt{R^2 - \left(\frac{h}{2}\right)^2},\qquad\qquad(1) and the radius of the horizontal cross-section of the sphere at height y above the ""equator"" is \sqrt{R^2 - y^2}.\qquad\qquad(2) The cross-section of the band with the plane at height y is the region inside the larger circle of radius given by (2) and outside the smaller circle of radius given by (1). The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. thumb|right|150px|Animated depiction of a bicylinder == Bicylinder == 300px|thumb|The generation of a bicylinder 180px|thumb|Calculating the volume of a bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac{16}{3} r^3 and the ;surface area :A=16 r^2. The key for the determination of volume and surface area is the observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times. In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. The volume of the band is : \int_{-h/2}^{h/2} (\text{area of cross-section at height }y) \, dy, and that does not depend on R. File:Sphere volume derivation using bicylinder.jpg|Zu Chongzhi's method (similar to Cavalieri's principle) for calculating a sphere's volume includes calculating the volume of a bicylinder. After integrating these two functions with the disk method we would subtract them to yield the desired volume. The cross-section's area is therefore the area of the larger circle minus the area of the smaller circle: \begin{align} & {}\quad \pi(\text{larger radius})^2 - \pi(\text{smaller radius})^2 \\\ & = \pi\left(\sqrt{R^2 - y^2}\right)^2 - \pi\left(\sqrt{R^2 - \left(\frac{h}{2}\right)^2\,{}}\,\right)^2 = \pi\left(\left(\frac{h}{2}\right)^2 - y^2\right). \end{align} The radius R does not appear in the last quantity. thumb|Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. The volume of the 8 pyramids is: \textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 , and then we can calculate that the bicylinder volume is \textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3 == Tricylinder == 450px|thumb|Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. In this case the volume of the band is the volume of the whole sphere, which matches the formula given above. thumb|300px|Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a ""napkin ring"" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole. ",0.44,0.54,2.3613,10,34,E
-"Find the average value of the function $f(x)=1+x^2$ on the interval $[-1,2]$.","In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable. A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: . It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean of the data), in which case it may be known as mean square deviation. In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: : \bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx. The function provides the arithmetic mean. For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. The type of calculations used in adjusting general average gave rise to the use of ""average"" to mean ""arithmetic mean"". In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). thumb|right|A graph of the function f(x) = e^{-x^2} and the area between it and the x-axis, (i.e. the entire real line) which is equal to \sqrt{\pi}. (If there are an even number of numbers, the mean of the middle two is taken.) That is, \int_{-\infty}^{\infty} e^{-x^2} \, dx = 2\int_{0}^{\infty} e^{-x^2}\,dx. For this reason, it is recommended to avoid using the word ""average"" when discussing measures of central tendency. ==General properties== If all numbers in a list are the same number, then their average is also equal to this number. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of these two. A typical estimate for the sample variance from a set of sample values x_i uses a divisor of the number of values minus one, n-1, rather than n as in a simple quadratic mean, and this is still called the ""mean square"" (e.g. in analysis of variance): :s^2=\textstyle\frac{1}{n-1}\sum(x_i-\bar{x})^2 The second moment of a random variable, E(X^{2}) is also called the mean square. There is also a harmonic average of functions and a quadratic average (or root mean square) of functions. ==See also== *Mean Category:Means Category:Calculus ==References== By analogy, a defining property of the average value \bar{f} of a function over the interval [a,b] is that : \int_a^b\bar{f}\,dx = \int_a^bf(x)\,dx In other words, \bar{f} is the constant value which when integrated over [a,b] equals the result of integrating f(x) over [a,b]. The square root of a mean square is known as the root mean square (RMS or rms), and can be used as an estimate of the standard deviation of a random variable. ==References== Category:Means In mathematics, the mean value problem was posed by Stephen Smale in 1981. Most types of average, however, satisfy permutation- insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 6) is the same as that of (3, 2, 6, 4, 1). ==Pythagorean means== The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means. ==Statistical location== The mode, the median, and the mid- range are often used in addition to the mean as estimates of central tendency in descriptive statistics. Depending on the context, an average might be another statistic such as the median, or mode. ",0.3359,5.51,0.7812,210,2,E
-Find the area of the region enclosed by the parabolas $y=x^2$ and $y=2 x-x^2$,"thumb|right|200px|Two-dimensional plot (red curve) of the algebraic equation y = x^2 - x - 2. Adding the two equations together to get: : 8x = 16 which simplifies to : x = 2. The area under the curve decreases monotonically with increasing p. == Generalization == A natural generalization for the superparabola is to relax the constraint on the power of x. thumb|right|384px|In green, confocal parabolae opening upwards, 2y = \frac {x^2}{\sigma^2}-\sigma^2 In red, confocal parabolae opening downwards, 2y =-\frac{x^2}{\tau^2}+\tau^2 Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. Using the second equation: : 2x - y = 1 Subtracting 2x from each side of the equation: : \begin{align}2x - 2x - y & = 1 - 2x \\\ \- y & = 1 - 2x \end{align} and multiplying by −1: : y = 2x - 1. In mathematics, the definite integral :\int_a^b f(x)\, dx is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total. Using the method of exhaustion, it follows that the total area of the parabolic segment is given by :\text{Area}\;=\;T \,+\, \frac14T \,+\, \frac1{4^2}T \,+\, \frac1{4^3}T \,+\, \cdots. Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord. thumb|400x300px|Superparabola functions A superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points with :\frac{y}{b} = \lbrack1-\left(\frac{x}{a}\right)^2\rbrack^p, where , , and are positive integers. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. Divide both sides by 2: \frac{2x}{2} = \frac{8}{2} 5\. This simplifies to: 2x = 8 4\. When the center of gravity of the triangle is known, the equilibrium of the lever yields the area of the parabola in terms of the area of the triangle which has the same base and equal height. The superparabola can vary in shape from a rectangular function , to a semi- ellipse (, to a parabola , to a pulse function . == Mathematical properties == thumb|400x300px| Without loss of generality we can consider the canonical form of the superparabola :f(x;p)=\left(1-x^2 \right)^p When , the function describes a continuous differentiable curve on the plane. Area function may refer to: *Inverse hyperbolic function *Antiderivative Here, however, we have the analytic solution for the area under the curve. The foci of all these parabolae are located at the origin. An interesting property is that any superparabola raised to a power n is just another superparabola; thus :\int_{-1}^{1}f^n (x) = \psi(n p) The centroid of the area under the curve is given by :C = \frac{\mathbf {i}}{A} \int_{-1}^{1} x\int_{0}^{f(x)} dydx + \frac{\mathbf {j}}{A}\ \int_{-1}^{1} \int_{0}^{f(x)}y dy dx :=\frac{\mathbf{j}}{2A}\int_{-1}^{1} f^2 (x) dx =\mathbf{j}\frac{\psi (2p)}{2\psi(p)} where the x-component is zero by virtue of symmetry. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. The indefinite and definite integrals are given by :\int f(x)dx=x \cdot_{2}F_{1} (-p, 1/q; 1+ 1/q ; x^2) :\text{Area}=\int _{-1}^{1}f(x)dx=\Psi (p,q) where \Psi is a universal function valid for all q and p>-1. Because of this a quadratic equation must contain the term ax^2, which is known as the quadratic term. The curve can be described parametrically on the complex plane as :z=\sin(u)+i\cos^{2p}(u);\quad-\tfrac{\pi}{2}\leq u\leq\tfrac{\pi}{2} Derivatives of the superparabola are given by :f'(x;p)=-2px(1-x^2)^{p-1} :\frac{\partial f}{\partial p} = (1-x^2)^p\ln(1-x^2) = f(x)\ln\lbrack f(x; 1)\rbrack The area under the curve is given by :\text{Area} = \int_{-1}^{1}\int_{0}^{f(x)}dydx = \int_{-1}^{1} (1-x^2)^p dx = \psi(p) where is a global function valid for all , :\psi( p)=\frac {\sqrt{\pi}\, \Gamma(p+1)}{\Gamma(p+\frac{3}{2})} The area under a portion of the curve requires the indefinite integral : \int (1-x^2)^p dx = x\,{_2}F{_1} (1/2, -p; 3/2; x^2) where _2F_1 is the Gaussian hypergeometric function. ",2.8,117,0.333333333333333,5300,-3.5,C
-The region $\mathscr{R}$ enclosed by the curves $y=x$ and $y=x^2$ is rotated about the $x$-axis. Find the volume of the resulting solid.,"For example, the next figure shows the rotation along the -axis of the red ""leaf"" enclosed between the square- root and quadratic curves: thumb|Rotation about x-axis The volume of this solid is: :\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,. The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by :V = \pi \int_a^b \left| f(y)^2 - g(y)^2\right|\,dy\, . Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem). In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. The areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given :A_x = \int_\alpha^\beta 2 \pi r\sin{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , :A_y = \int_\alpha^\beta 2 \pi r\cos{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , : ==See also== * Gabriel's Horn * Guldinus theorem * Pseudosphere * Surface of revolution * Ungula ==Notes== == References == * * () * Category:Integral calculus Category:Solids The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by :V = 2\pi \int_a^b x |f(x) - g(x)|\, dx\, . This works only if the axis of rotation is horizontal (example: or some other constant). ===Function of === If the function to be revolved is a function of , the following integral will obtain the volume of the solid of revolution: :\pi\int_c^d R(y)^2\,dy where is the distance between the function and the axis of rotation. The surface created by this revolution and which bounds the solid is the surface of revolution. This works only if the axis of rotation is vertical (example: or some other constant). ===Washer method=== To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness , or a cylindrical shell of width ; and then find the limiting sum of these volumes as approaches 0, a value which may be found by evaluating a suitable integral. The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. If is the value of a horizontal axis, then the volume equals :\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,. The element is created by rotating a line segment (of length ) around some axis (located units away), so that a cylindrical volume of units is enclosed. ==Finding the volume== Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given by :A_x = \int_a^b 2 \pi y \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, , :A_y = \int_a^b 2 \pi x \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, . == Polar form == For a polar curve r=f(\theta) where \alpha\leq \theta\leq \beta, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are :V_x = \int_\alpha^\beta \left(\pi r^2\sin^2{\theta} \cos{\theta}\, \frac{dr}{d\theta}-\pi r^3\sin^3{\theta}\right)d\theta\,, :V_y = \int_\alpha^\beta \left(\pi r^2\sin{\theta} \cos^2{\theta}\, \frac{dr}{d\theta}+\pi r^3\cos^3{\theta}\right)d\theta \, . One simply must solve each equation for before one inserts them into the integration formula. ==See also== *Solid of revolution *Shell integration ==References== * * *Frank Ayres, Elliott Mendelson. In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution) that lies on the same plane. Volume solid is the term which indicates the solid proportion of the paint on a volume basis. For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. For example, to rotate the region between and along the axis , one would integrate as follows: :\pi\int_0^3\left(\left(4-\left(-2x+x^2\right)\right)^2 - (4-x)^2\right)\,dx\,. After integrating these two functions with the disk method we would subtract them to yield the desired volume. This method may be derived with the same triple integral, this time with a different order of integration: :V = \iiint_D dV = \int_a^b \int_{g(r)}^{f(r)} \int_0^{2\pi} r\,d\theta\,dz\,dr = 2\pi \int_a^b\int_{g(r)}^{f(r)} r\,dz\,dr = 2\pi\int_a^b r(f(r) - g(r))\,dr. ==Parametric form== When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by :V_x = \int_a^b \pi y^2 \, \frac{dx}{dt} \, dt \, , :V_y = \int_a^b \pi x^2 \, \frac{dy}{dt} \, dt \, . ",+7.3,0.000216,0.41887902047,3.54,2.9,C
-Use Simpson's Rule with $n=10$ to approximate $\int_1^2(1 / x) d x$.,"The error in approximating an integral by Simpson's rule for n = 2 is -\frac{1}{90} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{2880} f^{(4)}(\xi), where \xi (the Greek letter xi) is some number between a and b. Simpson's 1/3 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{1}{3} h\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\\\ &= \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right], \end{align} where h = (b - a)/2 is the step size. (This derivation is essentially a less rigorous version of the quadratic interpolation derivation, where one saves significant calculation effort by guessing the correct functional form.) === Composite Simpson's 1/3 rule === If the interval of integration [a, b] is in some sense ""small"", then Simpson's rule with n = 2 subintervals will provide an adequate approximation to the exact integral. This is called the trapezoidal rule \int_a^b f(x)\, dx \approx (b-a) \left(\frac{f(a) + f(b)}{2}\right). right|thumb|300px|Illustration of Simpson's rule. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]. Integration by Simpson's 1/3 rule can be represented as a weighted average with 2/3 of the value coming from integration by the trapezoidal rule with step h and 1/3 of the value coming from integration by the rectangle rule with step 2h. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error. This leads to the adaptive Simpson's method. == Simpson's 3/8 rule == Simpson's 3/8 rule, also called Simpson's second rule, is another method for numerical integration proposed by Thomas Simpson. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using Simpson's rule. If the 1/3 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 1/3 rule. The error committed by the composite Simpson's rule is -\frac{1}{180} h^4(b - a)f^{(4)}(\xi), where \xi is some number between a and b, and h = (b - a)/n is the ""step length"". thumb|right|Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red). thumb|right|An animation showing how Simpson's rule approximates the function with a parabola and the reduction in error with decreased step size thumb|right|An animation showing how Simpson's rule approximation improves with more strips. In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The formula above is obtained by combining the composite Simpson's 1/3 rule with the one consisting of using Simpson's 3/8 rule in the extreme subintervals and Simpson's 1/3 rule in the remaining subintervals. The 1/3 rule can be used for the remaining subintervals without changing the order of the error term (conversely, the 3/8 rule can be used with a composite 1/3 rule for odd-numbered subintervals). == Alternative extended Simpson's rule == This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding \int_a^b f(x)\, dx \approx \frac{1}{48} h\left[17f(x_0) + 59f(x_1) + 43f(x_2) + 49f(x_3) + 48 \sum_{i= 4 }^{n - 4} f(x_i) + 49f(x_{n - 3}) + 43f(x_{n - 2}) + 59f(x_{n - 1}) + 17f(x_n)\right]. If the 3/8 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 3/8 rule. Simpson's 3/8 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{3}{8} h\left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right]\\\ &= \frac{b - a}{8} \left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right], \end{align} where h = (b - a)/3 is the step size. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. See: * Simpson's rule, a method of numerical integration * Simpson's rules (ship stability) * Simpson–Kramer method In case of odd number N of subintervals, the above formula are used up to the second to last interval, and the last interval is handled separately by adding the following to the result: \alpha f_N + \beta f_{N - 1} - \eta f_{N - 2}, where \begin{align} \alpha &= \frac{2h_{N - 1}^2 + 3h_{N - 1} h_{N - 2}}{6(h_{N - 2} + h_{N - 1})},\\\\[1ex] \beta &= \frac{h_{N - 1}^2 + 3h_{N - 1} h_{N - 2}}{6h_{N - 2}},\\\\[1ex] \eta &= \frac{h_{N - 1}^3}{6 h_{N - 2}(h_{N - 2} + h_{N - 1})}. \end{align} Example implementation in Python from collections.abc import Sequence def simpson_nonuniform(x: Sequence[float], f: Sequence[float]) -> float: """""" Simpson rule for irregularly spaced data. :param x: Sampling points for the function values :param f: Function values at the sampling points :return: approximation for the integral See ``scipy.integrate.simpson`` and the underlying ``_basic_simpson`` for a more performant implementation utilizing numpy's broadcast. The result is then obtained by taking the mean of the two formulas. === Simpson's rules in the case of narrow peaks === In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule. ",0.33333333,-8,635013559600.0,0.693150,3,D
-"Use the Midpoint Rule with $m=n=2$ to estimate the value of the integral $\iint_R\left(x-3 y^2\right) d A$, where $R=\{(x, y) \mid 0 \leqslant x \leqslant 2,1 \leqslant y \leqslant 2\}$.","For example, doing the previous calculation with order reversed gives the same result: : \begin{align} \int_{11}^{14} \int_{7}^{10} \, \left(x^2 + 4y\right) \, dy\, dx & = \int_{11}^{14} \Big[x^2 y + 2y^2 \Big]_{y=7}^{y=10} \, dx \\\ &= \int_{11}^{14} \, (3x^2 + 102) \, dx \\\ &= \Big[x^3 + 102x \Big]_{x=11}^{x=14} \\\ &= 1719. \end{align} === Double integral over a normal domain === thumb|160px|right|Example: double integral over the normal region D Consider the region (please see the graphic in the example): :D = \\{ (x,y) \in \mathbf{R}^2 \ : \ x \ge 0, y \le 1, y \ge x^2 \\} Calculate :\iint_D (x+y) \, dx \, dy. Let and :D = \left\\{ (x,y) \in \R^2 \ : \ 2 \le x \le 4 \ ; \ 3 > \le y \le 6 \right\\} in which case :\int_3^6 \int_2^4 \ 2 \ dx\, dy > =2\int_3^6 \int_2^4 \ 1 \ dx\, dy= 2\cdot\operatorname{area}(D) = 2 \cdot (2 > \cdot 3) = 12, since by definition we have: :\int_3^6 \int_2^4 \ 1 \ dx\, > dy=\operatorname{area}(D). ===Use of symmetry=== When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is odd with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. I = \left.\int_0^{3a}\rho^4 > d\rho = \frac{\rho^5}{5}\right\vert_0^{3a} = \frac{243}{5}a^5, II = > \int_0^\pi \sin^3\theta \, d\theta = -\int_0^\pi \sin^2\theta \, d(\cos > \theta) = \int_0^\pi (\cos^2\theta-1) \, d(\cos \theta) = > \left.\frac{\cos^3\theta}{3}\right|^\pi_0 - \left.\cos\theta\right|^\pi_0 = > \frac{4}{3}, III = \int_0^{2\pi} d \varphi = 2\pi. Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. The result of this integral, which is a function depending only on , is then integrated with respect to . :\begin{align} \int_{11}^{14} \left(x^2 + 4y\right) \, dx & = \left [\frac13 x^3 + 4yx \right]_{x=11}^{x=14} \\\ &= \frac13(14)^3 + 4y(14) - \frac13(11)^3 - 4y(11) \\\ &= 471 + 12y \end{align} We then integrate the result with respect to . :\begin{align} \int_7^{10} (471 + 12y) \ dy & = \Big[471y + 6y^2\Big]_{y=7}^{y=10} \\\ &= 471(10)+ 6(10)^2 - 471(7) - 6(7)^2 \\\ &= 1719 \end{align} In cases where the double integral of the absolute value of the function is finite, the order of integration is interchangeable, that is, integrating with respect to x first and integrating with respect to y first produce the same result. It is now possible to apply the formula: :\iint_D (x+y) \, dx \, dy = \int_0^1 dx \int_{x^2}^1 (x+y) \, dy = \int_0^1 dx \ \left[xy + \frac{y^2}{2} \right]^1_{x^2} (at first the second integral is calculated considering x as a constant). Then > we get :\begin{align} \int_0^{2\pi} d\varphi \int_0^{3a} \rho^3 d\rho > \int_{-\sqrt{9a^2 - \rho^2}}^{\sqrt{9 a^2 - \rho^2}}\, dz &= 2 \pi > \int_0^{3a} 2 \rho^3 \sqrt{9 a^2 - \rho^2} \, d\rho \\\ &= -2 \pi \int_{9 > a^2}^0 (9 a^2 - t) \sqrt{t}\, dt && t = 9 a^2 - \rho^2 \\\ &= 2 \pi > \int_0^{9 a^2} \left ( 9 a^2 \sqrt{t} - t \sqrt{t} \right ) \, dt \\\ &= 2 > \pi \left( \int_0^{9 a^2} 9 a^2 \sqrt{t} \, dt - \int_0^{9 a^2} t \sqrt{t} > \, dt\right) \\\ &= 2 \pi \left[9 a^2 \frac23 t^{ \frac32 } - \frac{2}{5} > t^{ \frac{5}{2}} \right]_0^{9 a^2} \\\ &= 2 \cdot 27 \pi a^5 \left ( 6 - > \frac{18}{5} \right ) \\\ &= \frac{648 \pi}{5} a^5. \end{align} Thanks to > the passage to cylindrical coordinates it was possible to reduce the triple > integral to an easier one-variable integral. In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). See also the differential volume entry in nabla in cylindrical and spherical coordinates. ==Examples== === Double integral over a rectangle === Let us assume that we wish to integrate a multivariable function over a region : :A = \left \\{ (x,y) \in \mathbf{R}^2 \ : \ 11 \le x \le 14 \ ; \ 7 \le y \le 10 \right \\} \mbox{ and } f(x,y) = x^2 + 4y\, From this we formulate the iterated integral :\int_7^{10} \int_{11}^{14} (x^2 + 4y) \, dx\, dy The inner integral is performed first, integrating with respect to and taking as a constant, as it is not the variable of integration. right|thumb|Illustration of the midpoint method assuming that y_n equals the exact value y(t_n). The domain is the ball with center at the origin and radius , :D > = \left \\{ x^2 + y^2 + z^2 \le 9a^2 \right \\} and is the function to > integrate. Then, by Fubini's theorem: :\iint_D f(x,y)\, dx\, dy = \int_a^b dx \int_{ \alpha (x)}^{ \beta (x)} f(x,y)\, dy. ====-axis==== If is normal with respect to the -axis and is a continuous function; then and (both of which are defined on the interval ) are the two functions that determine . Collecting all parts, > \iiint_T \rho^4 \sin^3 \theta \, d\rho\, d\theta\, d\varphi = I\cdot II\cdot > III = \frac{243}{5}a^5\cdot \frac{4}{3}\cdot 2\pi = \frac{648}{5}\pi a^5. > Alternatively, this problem can be solved by using the passage to > cylindrical coordinates. Once the intervals are known, you have :\int_0^\pi \int_2^3 \rho^2 > \cos \varphi \, d \rho \, d \varphi = \int_0^\pi \cos \varphi \ d \varphi > \left[ \frac{\rho^3}{3} \right]_2^3 = \Big[ \sin \varphi \Big]_0^\pi \ > \left(9 - \frac{8}{3} \right) = 0. ====Cylindrical coordinates==== thumb|right|190px|Cylindrical coordinates. The explicit midpoint method is given by the formula the implicit midpoint method by for n=0, 1, 2, \dots Here, h is the step size -- a small positive number, t_n=t_0 + n h, and y_n is the computed approximate value of y(t_n). If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. Using the linearity property, the > integral can be decomposed into three pieces: :\iint_T \left(2\sin x - 3y^3 > + 5\right) \, dx \, dy = \iint_T 2 \sin x \, dx \, dy - \iint_T 3y^3 \, dx > \, dy + \iint_T 5 \, dx \, dy The function is an odd function in the > variable and the disc is symmetric with respect to the -axis, so the value > of the first integral is 0. thumb|right|Integral as area between two curves. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. * Sphere: The volume of a sphere with radius can be calculated by integrating the constant function 1 over the sphere, using spherical coordinates. ::\begin{align} \text{Volume} &= \iiint_D f(x,y,z) \, dx\, dy\, dz \\\ &= \iiint_D 1 \, dV \\\ &= \iiint_S \rho^2 \sin \varphi \, d\rho\, d\theta\, d\varphi \\\ &= \int_0^{2\pi} \, d \theta \int_0^{ \pi } \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\\ &= 2 \pi \int_0^\pi \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\\ &= 2 \pi \int_0^\pi \sin \varphi \frac{R^3}{3 }\, d \varphi \\\ &= \frac23 \pi R^3 \Big[-\cos \varphi\Big]_0^\pi = \frac43 \pi R^3. \end{align} * Tetrahedron (triangular pyramid or 3-simplex): The volume of a tetrahedron with its apex at the origin and edges of length along the -, - and -axes can be calculated by integrating the constant function 1 over the tetrahedron. ::\begin{align} \text{Volume} &= \int_0^\ell dx \int_0^{\ell-x}\, dy \int_0^{\ell-x-y }\, dz \\\ &= \int_0^\ell dx \int_0^{\ell-x } (\ell - x - y)\, dy \\\ &= \int_0^\ell \left( l^2 - 2 \ell x + x^2 - \frac{(\ell-x)^2 }{2}\right)\, dx \\\ &= \ell^3 - \ell \ell^2 + \frac{\ell^3}{3 } - \left[\frac{\ell^2 x}{2} - \frac{ \ell x^2}{2} + \frac{x^3}{6 }\right]_0^ \ell \\\ &= \frac{\ell^3}{3} - \frac{\ell^3}{6} = \frac{ \ell^3}{6}\end{align} :This is in agreement with the formula for the volume of a pyramid ::\mathrm{Volume} = \frac13 \times \text{base area} \times \text{height} = \frac13 \times \frac{\ell^2}{2} \times \ell = \frac{ \ell^3}{6}. thumb|right|140px|Example of an improper domain. ==Multiple improper integral== In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral. ==Multiple integrals and iterated integrals== Fubini's theorem states that if :\iint_{A\times B} \left|f(x,y)\right|\,d(x,y)<\infty, that is, if the integral is absolutely convergent, then the multiple integral will give the same result as either of the two iterated integrals: :\iint_{A\times B} f(x,y)\,d(x,y)=\int_A\left(\int_B f(x,y)\,dy\right)\,dx=\int_B\left(\int_A f(x,y)\,dx\right)\,dy. In the following example, the electric field produced by a distribution of charges given by the volume charge density is obtained by a triple integral of a vector function: :\vec E = \frac {1}{4 \pi \varepsilon_0} \iiint \frac {\vec r - \vec r'}{\left \| \vec r - \vec r' \right \|^3} \rho (\vec r')\, d^3 r'. ",-11.875,0.9731,-2.5,0.6749,1260,A
-"The base radius and height of a right circular cone are measured as $10 \mathrm{~cm}$ and $25 \mathrm{~cm}$, respectively, with a possible error in measurement of as much as $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated volume of the cone.","For 0 \le K' \le 1 : y = R\left({2 \left({x \over L}\right) - K'\left({x \over L}\right)^2 \over 2 - K'}\right) can vary anywhere between and , but the most common values used for nose cone shapes are: Parabola Type Value Cone Half Three Quarter Full For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same and , then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated. :\rho > {R^2 + L^2 \over 2R} and \alpha = \arccos \left({\sqrt{L^2 + R^2} \over 2\rho}\right)-\arctan \left({R \over L}\right) Then the radius at any point as varies from to is: :y = \sqrt{\rho^2 - (\rho\cos(\alpha) - x)^2} - \rho\sin(\alpha) If the chosen is less than the tangent ogive and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. The radius of the circle that forms the ogive is called the ogive radius, , and it is related to the length and base radius of the nose cone as expressed by the formula: :\rho = {R^2 + L^2\over 2R} The radius at any point , as varies from to is: :y = \sqrt{\rho^2 - (L - x)^2}+R - \rho The nose cone length, , must be less than or equal to . The tangency point where the sphere meets the cone can be found from: : x_t = \frac{L^2}{R} \sqrt{ \frac{r_n^2}{R^2 + L^2} } : y_t = \frac{x_t R}{L} where is the radius of the spherical nose cap. The failure is governed by crack growth in concrete, which forms a typical cone shape having the anchor's axis as revolution axis. ==Mechanical models== ===ACI 349-85=== Under tension loading, the concrete cone failure surface has 45° inclination. For , Haack nose cones bulge to a maximum diameter greater than the base diameter. The sides of a conic profile are straight lines, so the diameter equation is simply: : y = {xR \over L} Cones are sometimes defined by their half angle, : : \phi = \arctan \left({R \over L}\right) and y = x \tan(\phi)\; ==== Spherically blunted conic ==== In practical applications such as re-entry vehicles, a conical nose is often blunted by capping it with a segment of a sphere. A conical measure is a type of laboratory glassware which consists of a conical cup with a notch on the top to allow for the easy pouring of liquids, and graduated markings on the side to allow easy and accurate measurement of volumes of liquid. Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. alt=Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.|thumb|300x300px|General parameters used for constructing nose cone profiles. For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium. == Nose cone shapes and equations == === General dimensions === In all of the following nose cone shape equations, is the overall length of the nose cone and is the radius of the base of the nose cone. is the radius at any point , as varies from , at the tip of the nose cone, to . The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = f_{ct} {A_{N}} [N] Where: f_{ct} \- tensile strength of concrete A_{N} \- Cone's projected area === Concrete capacity design (CCD) approach for fastening to concrete=== Under tension loading, the concrete capacity of a single anchor is calculated assuming an inclination between the failure surface and surface of the concrete member of about 35°. The length/diameter relation is also often called the caliber of a nose cone. The center of the spherical nose cap, , can be found from: : x_o = x_t + \sqrt{ r_n^2 - y_t^2} And the apex point, can be found from: : x_a = x_o - r_n === Bi-conic === A bi-conic nose cone shape is simply a cone with length stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with length , where the base of the upper cone is equal in radius to the top radius of the smaller frustum with base radius . The use of the conical measure usually dictates its construction material. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. Conical measures are the most commonly used item of glassware used in the preparation of extemporaneous medicaments. While the equations describe the 'perfect' shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons. === Conic === A very common nose-cone shape is a simple cone. :L=L_1+L_2 :For 0 \le x \le L_1 : y = {xR_1 \over L_1} :For L_1 \le x \le L : y = R_1 + {(x - L_1)(R_2-R_1)\over L_2} Half angles: :\phi_1 = \arctan \left({R_1 \over L_1}\right) and y = x \tan(\phi_1)\; :\phi_2 = \arctan \left({R_2 - R_1 \over L_2}\right) and y = R_1 + (x - L_1) \tan(\phi_2)\; === Tangent ogive === Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. They are not as precise as graduated cylinders for measuring liquids, but make up for this in terms of easy pouring and ability to mix solutions within the measure itself. ==History== During his experiments, Abū al-Rayhān al-Bīrūnī (973-1048) invented the conical measure,Marshall Clagett (1961). ",0,5.0,62.8318530718,0.333333333333333,1110,C
-A force of $40 \mathrm{~N}$ is required to hold a spring that has been stretched from its natural length of $10 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $15 \mathrm{~cm}$ to $18 \mathrm{~cm}$ ?,"Let be the amount by which the free end of the spring was displaced from its ""relaxed"" position (when it is not being stretched). An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. The force an ideal spring would exert is exactly proportional to its extension or compression. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. The work of the net force is calculated as the product of its magnitude and the particle displacement. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The force is applied through the ends of the spring. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The work can be split into two terms \delta W = \delta W_\mathrm{s} + \delta W_\mathrm{b} where is the work done by surface forces while is the work done by body forces. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Explaining the Power of Springing Bodies, London, 1678. as: (""as the extension, so the force"" or ""the extension is proportional to the force""). Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. The manufacture normally specifies the spring rate. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. ",5.5,-0.0301,399.0,1.56,0.7854,D
+","J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). The limit of this sum as N \rightarrow \infty becomes an integral: :S^\circ = \sum_{k=1}^N \Delta S_k = \sum_{k=1}^N \frac{dQ_k}{T} \rightarrow \int _0 ^{T_2} \frac{dS}{dT} dT = \int _0 ^{T_2} \frac {C_{p_k}}{T} dT In this example, T_2 =298.15 K and C_{p_k} is the molar heat capacity at a constant pressure of the substance in the reversible process . Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. Thus the Gibbs free energy of a system can be calculated by collecting moles together carefully at a specified T, P and at a constant molar ratio composition (so that the chemical potential doesn't change as the moles are added together), i.e. : \mathbf{G} = \sum_{i=1}^I \mu_i N_i . In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. From these sources: :*a - K.K. Kelley, Bur. Mines Bull. 383, (1935). :*b - :*c - Brewer, The thermodynamic and physical properties of the elements, Report for the Manhattan Project, (1946). :*d - :*e - :*f - :*g - ; :*h - :*i - . :*j - :*k - Int. National Critical Tables, vol. 3, p. 306, (1928). :*l - :*m - :*n - :*o - :*p - :*q - :*r - :*s - H.A. Jones, I. Langmuir, Gen. Electric Rev., vol. 30, p. 354, (1927). == See also == Category:Properties of chemical elements Category:Chemical element data pages The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. The total differential of this expression is : \mathrm{d}\mathbf{G} = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i Combining the two expressions for the total differential of the Gibbs free energy gives : \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i =V \mathrm{d}p-S \mathrm{d}T+\sum_{i=1}^I \mu_i \mathrm{d}N_i which simplifies to the Gibbs–Duhem relation: : \sum_{i=1}^I N_i \mathrm{d}\mu_i = -S \mathrm{d}T + V \mathrm{d}p === Alternative derivation === Another way of deriving the Gibbs-Duhem equation can be found be taking the extensivity of energy into account. The following images show the density of the t-distribution for increasing values of u. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. The Gibbs−Duhem equation cannot be used for small thermodynamic systems due to the influence of surface effects and other microscopic phenomena. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). ",524,7.27,"""0.195""",-233,57.2,E
+"A mass of $34.05 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}(s)$ at $273 \mathrm{~K}$ is dropped into $185 \mathrm{~g}$ of $\mathrm{H}_2 \mathrm{O}(l)$ at $310 . \mathrm{K}$ in an insulated container at 1 bar of pressure. Calculate the temperature of the system once equilibrium has been reached. Assume that $C_{P, m}$ for $\mathrm{H}_2 \mathrm{O}(l)$ is constant at its values for $298 \mathrm{~K}$ throughout the temperature range of interest.","The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. The potential temperature is denoted \theta and, for a gas well-approximated as ideal, is given by : \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, where T is the current absolute temperature (in K) of the parcel, R is the gas constant of air, and c_p is the specific heat capacity at a constant pressure. The analysis here shows the simple unattributed formula and the Antoine equation are reasonably accurate at 100 °C, but quite poor for lower temperatures above freezing. For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state, pv = RT can be substituted into the 1st law yielding, after some rearrangement: : \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, where the dh = c_{p}dT was used and both terms were divided by the product pv Integrating yields: : \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, and solving for T_{0}, the temperature a parcel would acquire if moved adiabatically to the pressure level p_{0}, you get: : T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. == Potential virtual temperature == The potential virtual temperature \theta_{v}, defined by : \theta_v = \theta \left( 1 + 0.61 r - r_L \right), is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P0. ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. The equilibrium constants may be derived by best-fitting of the experimental data with a chemical model of the equilibrium system. == Experimental methods == There are four main experimental methods. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. The Boyle temperature is formally defined as the temperature for which the second virial coefficient, B_{2}(T), becomes zero. The equilibrium constant value can be determined if any one of these concentrations can be measured. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). Equilibrium constants are determined in order to quantify chemical equilibria. A large number of general-purpose computer programs for equilibrium constant calculation have been published. The former is an extremely simple Antoine equation, while the latter is a polynomial. ==Graphical pressure dependency on temperature== ==See also== *Dew point *Gas laws *Lee–Kesler method *Molar mass ==References== ==Further reading== * * * * ==External links== * * Category:Thermodynamic properties Category:Atmospheric thermodynamics The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_{0}, usually . It is at this temperature that the attractive forces and the repulsive forces acting on the gas particles balance out P = RT \left(\frac{1}{V_m} + \frac{B_{2}(T)}{V_m^2} + \cdots \right) This is the virial equation of state and describes a real gas. :H2O <=> H+ + OH-: K_\mathrm{W}^' = \frac{[H^+][OH^-]}{[H_2O]} With dilute solutions the concentration of water is assumed constant, so the equilibrium expression is written in the form of the ionic product of water. As expected, Buck's equation for > 0 °C is significantly more accurate than Tetens, and its superiority increases markedly above 50 °C, though it is more complicated to use. For this assumption to be valid, equilibrium constants must be determined in a medium of relatively high ionic strength. One or more equilibrium constants may be parameters of the refinement. ",12,292,"""24.4""",-0.75,0.5,B
+"Calculate $\Delta H_f^{\circ}$ for $N O(g)$ at $975 \mathrm{~K}$, assuming that the heat capacities of reactants and products are constant over the temperature interval at their values at $298.15 \mathrm{~K}$.","Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. Integration of this equation permits the evaluation of the heat of reaction at one temperature from measurements at another temperature.Laidler K.J. and Meiser J.H., ""Physical Chemistry"" (Benjamin/Cummings 1982), p.62Atkins P. and de Paula J., ""Atkins' Physical Chemistry"" (8th edn, W.H. Freeman 2006), p.56 :\Delta H^\circ \\! \left( T \right) = \Delta H^\circ \\! \left( T^\circ \right) + \int_{T^\circ}^{T} \Delta C_P^\circ \, \mathrm{d} T Pressure variation effects and corrections due to mixing are generally minimal unless a reaction involves non-ideal gases and/or solutes, or is carried out at extremely high pressures. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. In modern terms the mass m of the sample divided by molar mass M gives the number of moles n. :m/M = n Therefore, using uppercase C for the full heat capacity (in joule per kelvin), we have: :C(M/m) = C/n = K = 3R or :C/n = 3R. We have : \Delta U = \sum U_{products} - \sum U_{reactants} This also signifies that the amount of heat absorbed at constant volume could be identified with the change in the thermodynamic quantity internal energy. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Heat capacity ratio for various gases Gas Temp. [°C] H2 −181 1.597 −76 1.453 20 1.410 100 1.404 400 1.387 1000 1.358 2000 1.318 He 20 1.660 Ar −180 1.760 20 1.670 O2 −181 1.450 −76 1.415 20 1.400 100 1.399 200 1.397 400 1.394 N2 −181 1.470 Cl2 20 1.340 Ne 19 1.640 Xe 19 1.660 Kr 19 1.680 Hg 360 1.670 H2O 20 1.330 100 1.324 200 1.310 CO2 0 1.310 20 1.300 100 1.281 400 1.235 1000 1.195 CO 20 1.400 NO 20 1.400 N2O 20 1.310 CH4 −115 1.410 −74 1.350 20 1.320 NH3 15 1.310 SO2 15 1.290 C2H6 15 1.220 C3H8 16 1.130 Dry air -15 1.404 0 1.403 20 1.400 100 1.401 200 1.398 400 1.393 1000 1.365 In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volume (). Therefore, the heat capacity ratio in this example is 1.4. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. \, Then :1-e^{-\hbar\omega_\alpha/k_{\rm B}T} \approx \hbar\omega_\alpha/k_{\rm B}T \, and we have :F=N\varepsilon_0+Nk_{\rm B}T\sum_{\alpha}\log\left(\frac{\hbar\omega_{\alpha}}{k_{\rm B}T}\right). 'At, Astatine', system no. 8a, Springer-Verlag, Berlin, , pp. 116–117 ===LNG=== As quoted from various sources in: * J.A. Dean (ed.), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6, Thermodynamic Properties; Table 6.4, Heats of Fusion, Vaporization, and Sublimation and Specific Heat at Various Temperatures of the Elements and Inorganic Compounds ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T. H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity. ",12,91.7,"""-31.95""",0.5,35.2,B
+A two-level system is characterized by an energy separation of $1.30 \times 10^{-18} \mathrm{~J}$. At what temperature will the population of the ground state be 5 times greater than that of the excited state?,"Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. An excited state is any state with energy greater than the ground state. The temperature of a group of particles is indicative of the level of excitation (with the notable exception of systems that exhibit negative temperature). The excitation temperature can even be negative for a system with inverted levels (such as a maser). It satisfies : \frac{n_{\rm u}}{n_{\rm l}} = \frac{g_{\rm u}}{g_{\rm l}} \exp{ \left(-\frac{\Delta E}{k T_{\rm ex}} \right) }, where * is the number of particles in an upper (e.g. excited) state; * is the statistical weight of those upper-state particles; * is the number of particles in a lower (e.g. ground) state; * is the statistical weight of those lower-state particles; * is the exponential function; * is the Boltzmann constant; * is the difference in energy between the upper and lower states. Excitation refers to an increase in energy level above a chosen starting point, usually the ground state, but sometimes an already excited state. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature. == Absence of nodes in one dimension == In one dimension, the ground state of the Schrödinger equation can be proven to have no nodes. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. thumb|250px|Schematic picture of energy levels and examples of different states. In observations of the 21 cm line of hydrogen, the apparent value of the excitation temperature is often called the ""spin temperature"". ==References== Category:Temperature 300 px|thumb|A Jablonski diagram showing the excitation of molecule A to its singlet excited state (1A*) followed by intersystem crossing to the triplet state (3A) that relaxes to the ground state by phosphorescence. This phenomenon has been studied in the case of a two-dimensional gas in some detail, analyzing the time taken to relax to equilibrium. == Calculation of excited states == Excited states are often calculated using coupled cluster, Møller–Plesset perturbation theory, multi-configurational self-consistent field, configuration interaction, and time-dependent density functional theory. ==Excited-state absorption== The excitation of a system (an atom or molecule) from one excited state to a higher-energy excited state with the absorption of a photon is called excited-state absorption (ESA). Fortunately, it is often possible to define a quasi-Fermi level and quasi-temperature for a given location, that accurately describe the occupation of states in terms of a thermal distribution. The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. The ground state (blue) is x2–y2 orbitals; the excited orbitals are in green; the arrows illustrate inelastic x-ray spectroscopy. Relaxation of the excited state to its lowest vibrational level is called vibrational relaxation. The vibrational ground states of each electronic state are indicated with thick lines, the higher vibrational states with thinner lines. The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. Excited-state absorption measurements are done using pump–probe techniques such as flash photolysis. ",588313,-214,"""3.23""",5.85,157.875,D
+At what temperature are there Avogadro's number of translational states available for $\mathrm{O}_2$ confined to a volume of 1000. $\mathrm{cm}^3$ ?,"The Avogadro number is the approximate number of nucleons (protons and neutrons) in one gram of ordinary matter. The Avogadro constant also relates the molar volume of a substance to the average volume nominally occupied by one of its particles, when both are expressed in the same units of volume. thumb|upright=2.2|Molar heat capacity of most elements at 25 °C is in the range between 2.8 R and 3.4 R: Plot as a function of atomic number with a y range from 22.5 to 30 J/mol K. The name Avogadro's number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 16 grams of oxygen. (The Avogadro number is closely related to the Loschmidt constant, and the two concepts are sometimes confused.) In older literature, the Avogadro number is denoted or , which is the number of particles that are contained in one mole, exactly . The Avogadro constant, commonly denoted or , is a ratio that relates the number of constituent particles (usually molecules, atoms, or ions) in a sample with the amount of substance in that sample. These definitions meant that the value of the Avogadro number depended on the experimentally determined value of the mass (in grams) of one atom of those elements, and therefore it was known only to a limited number of decimal digits. The value of 3R is about 25 joules per kelvin, and Dulong and Petit essentially found that this was the heat capacity of certain solid elements per mole of atoms they contained. Under the new definition, the mass of one mole of any substance (including hydrogen, carbon-12, and oxygen-16) is times the average mass of one of its constituent particles – a physical quantity whose precise value has to be determined experimentally for each substance. == History == === Origin of the concept === right|thumb|Jean Perrin in 1926 The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas. The value of the Avogadro constant was chosen so that the mass of one mole of a chemical compound, expressed in grams, is approximately the number of nucleons in one constituent particle of the substance. Thus, the Avogadro constant is the proportionality factor that relates the molar mass of a substance to the average mass of one molecule. This value, the number density of particles in an ideal gas, is now called the Loschmidt constant in his honor, and is related to the Avogadro constant, , by : n_0 = \frac{p_0N_{\rm A}}{R\,T_0}, where is the pressure, is the gas constant, and is the absolute temperature. The numeric value of the Avogadro constant expressed in reciprocal moles, a dimensionless number, is called the Avogadro number. As a consequence of this definition, in the SI system the Avogadro constant had the dimension reciprocal of amount of substance rather than of a pure number, and had the approximate value . thumb|The transformation of one phase from another by the growth of nuclei forming randomly in the parent phase The Avrami equation describes how solids transform from one phase to another at constant temperature. The goal of this definition was to make the mass of a mole of a substance, in grams, be numerically equal to the mass of one molecule relative to the mass of the hydrogen atom; which, because of the law of definite proportions, was the natural unit of atomic mass, and was assumed to be 1/16 of the atomic mass of oxygen. === First measurements === right|thumb|Josef Loschmidt The value of Avogadro's number (not yet known by that name) was first obtained indirectly by Josef Loschmidt in 1865, by estimating the number of particles in a given volume of gas. In general, for uniform nucleation and growth, n = D + 1, where D is the dimensionality of space in which crystallization occurs. == Interpretation of Avrami constants == Originally, n was held to have an integer value between 1 and 4, which reflected the nature of the transformation in question. Perrin himself determined the Avogadro number by several different experimental methods. Dulong and Petit then found that when multiplied by these atomic weights, the value for the heat capacity per mole was nearly constant, and equal to a value which was later recognized to be 3R. Given the previous equations, this can be reduced to the more familiar form of the Avrami (JMAK) equation, which gives the fraction of transformed material after a hold time at a given temperature: : Y = 1 - \exp[-K\cdot t^n], where K = \pi\dot{N}\dot{G}^3/3, and n = 4. American Journal of Physics, 78 (4), 412-417 (https://doi.org/10.1119/1.3276053) and bound states in the continuum (red). ",0.000226,1.2,"""0.068""",4.85,2,C
+The half-cell potential for the reaction $\mathrm{O}_2(g)+4 \mathrm{H}^{+}(a q)+4 \mathrm{e}^{-} \rightarrow 2 \mathrm{H}_2 \mathrm{O}(l)$ is $+1.03 \mathrm{~V}$ at $298.15 \mathrm{~K}$ when $a_{\mathrm{O}_2}=1.00$. Determine $a_{\mathrm{H}^{+}}$,"The values below are standard apparent reduction potentials for electro- biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is Faraday's constant. At pH = 7, \+ → P680 ~ +1.0 Half-reaction independent of pH as no is involved in the reaction ==See also== * Nernst equation * Electron bifurcation * Pourbaix diagram * Reduction potential ** Dependency of reduction potential on pH * Standard electrode potential * Standard reduction potential * Standard reduction potential (data page) * Standard state ==References== ==Bibliography== ;Electrochemistry * ;Bio-electrochemistry * * * * * * ;Microbiology * * Category:Biochemistry Standard reduction potentials for half-reactions important in biochemistry Category:Electrochemical potentials Category:Thermodynamics databases Category:Biochemistry databases For standard conditions in electrochemistry (T = 25 °C, P = 1 atm and all concentrations being fixed at 1 mol/L, or 1 M) the standard reduction potential of hydrogen E^{\ominus}_\text{red H+} is fixed at zero by convention as it serves of reference. Definitions must be clearly expressed and carefully controlled, especially if the sources of data are different and arise from different fields (e.g., picking and directly mixing data from classical electrochemistry textbooks (E^{\ominus}_\text{red} versus SHE, pH = 0) and microbiology textbooks (E^{\ominus'}_\text{red} at pH = 7) without paying attention to the conventions on which they are based). ==Example in biochemistry== For example, in a two electrons couple like : the reduction potential becomes ~ 30 mV (or more exactly, 59.16 mV/2 = 29.6 mV) more positive for every power of ten increase in the ratio of the oxidised to the reduced form. ==Some important apparent potentials used in biochemistry== Half-reaction Δ°' (V) E' Physiological conditions References and notes −0.58 Many carboxylic acid: aldehyde redox reactions have a potential near this value 2 + 2 → −0.41 Non-zero value for the hydrogen potential because at pH = 7, [H+] = 10−7 M and not 1 M as in the standard hydrogen electrode (SHE), and that: → NADPH −0.320 −0.370 The ratio of :NADPH is maintained at around 1:50. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives: : : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 0 - \left(0.05916 \ \text{×} \ 7\right) = -0.414 \ V In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons () into hydrogen gas is no longer zero as with the standard hydrogen electrode (SHE) at 1 M (pH = 0) in classical electrochemistry, but that E_\text{red} = -0.414\mathrm V versus the standard hydrogen electrode (SHE). Te (aq) + 2 + 2 (s) + 4 1.02 2 . At pH = 7, when [] = 10−7 M, the reduction potential E_\text{red} of differs from zero because it depends on pH. Is it simply: * E_h = E_\text{red} calculated at pH 7 (with or without corrections for the activity coefficients), * E^{\ominus '}_\text{red}, a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it, * E^{\ominus '}_\text{red apparent at pH 7}, an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio \frac{h} {z} = \frac{\text{(number of involved protons)}} {\text{(number of exchanged electrons)}}. The same also applies for the reduction potential of oxygen: : For , E^{\ominus}_\text{red} = 1.229 V, so, applying the Nernst equation for pH = 7 gives: : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 1.229 - \left(0.05916 \ \text{×} \ 7\right) = 0.815 \ V For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. This is observed for the reduction of O2 into H2O, or OH−, and for reduction of H+ into H2. ==Formal standard reduction potential combined with the pH dependency== To obtain the reduction potential as a function of the measured concentrations of the redox- active species in solution, it is necessary to express the activities as a function of the concentrations. This equation predicts lower E_h at higher pH values. Fumarate + 2 + 2 → Succinate +0.03 +0.30 Formation of hydrogen peroxide from oxygen +0.82 In classical electrochemistry, E° for = +1.23 V with respect to the standard hydrogen electrode (SHE). The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . For the LJTS potential with r_\mathrm{end} = 2.5\,\sigma , the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon . At chemical equilibrium, the reaction quotient of the product activity (aRed) by the reagent activity (aOx) is equal to the equilibrium constant () of the half-reaction and in the absence of driving force () the potential () also becomes nul. This equation is the equation of a straight line for E_h as a function of pH with a slope of -0.05916\,\left(\frac{h}{z}\right) volt (pH has no units). The figure 8 shows the comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential and the 'Lennard-Jones truncated & shifted' potential. The activity coefficients \gamma_{red} and \gamma_{ox} are included in the formal potential E^{\ominus '}_\text{red}, and because they depend on experimental conditions such as temperature, ionic strength, and pH, E^{\ominus '}_\text{red} cannot be referred as an immuable standard potential but needs to be systematically determined for each specific set of experimental conditions. When the formal potential is measured under standard conditions (i.e. the activity of each dissolved species is 1 mol/L, T = 298.15 K = 25 °C = 77 °F, = 1 bar) it becomes de facto a standard potential. The properties of this ion are strongly related to the surface potential present on a corresponding solid. Without being able to correctly answering these questions, mixing data from different sources without appropriate conversion can lead to errors and confusion. ==Determination of the formal standard reduction potential when 1== The formal standard reduction potential E^{\ominus '}_\text{red} can be defined as the measured reduction potential E_\text{red} of the half-reaction at unity concentration ratio of the oxidized and reduced species (i.e., when 1) under given conditions. ",0.16,7200,"""22.2""",4.16,3.0,D
+"The partial molar volumes of water and ethanol in a solution with $x_{\mathrm{H}_2 \mathrm{O}}=0.45$ at $25^{\circ} \mathrm{C}$ are 17.0 and $57.5 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$, respectively. Calculate the volume change upon mixing sufficient ethanol with $3.75 \mathrm{~mol}$ of water to give this concentration. The densities of water and ethanol are 0.997 and $0.7893 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively, at this temperature.","It can be calculated from this table that at 25 °C, 45 g of ethanol has volume 57.3 ml, 55 g of water has volume 55.2 ml; these sum to 112.5 ml. Mixing two solutions of alcohol of different strengths usually causes a change in volume. The volume of alcohol in the solution can then be estimated. Whereas to make a 50% v/v ethanol solution, 50 ml of ethanol and 50 ml of water could be mixed but the resulting volume of solution will measure less than 100 ml due to the change of volume on mixing, and will contain a higher concentration of ethanol. This source gives density data for ethanol:water mixes by %weight ethanol in 5% increments and against temperature including at 25 °C, used here. Mixing pure water with a solution less than 24% by mass causes a slight increase in total volume, whereas the mixing of two solutions above 24% causes a decrease in volume. The International Organization of Legal Metrology has tables of density of water–ethanol mixtures at different concentrations and temperatures. For example, to make 100 ml of 50% ABV ethanol solution, water would be added to 50 ml of ethanol to make up exactly 100 ml. Therefore, one can use the following equation to convert between ABV and ABW: \text{ABV} = \text{ABW} \times \frac{\text{density of beverage}}{\text{density of alcohol}} At relatively low ABV, the alcohol percentage by weight is about 4/5 of the ABV (e.g., 3.2% ABW is about 4% ABV). In the wine/water mixing problem, one starts with two barrels, one holding wine and the other an equal volume of water. The volume of the cup is irrelevant, as is any stirring of the mixtures. == Solution == Conservation of substance implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. At 0% and 100% ABV is equal to ABW, but at values in between ABV is always higher, up to ~13% higher around 60% ABV. ==See also== *Apparent molar property *Excess molar quantity *Standard drink *Unit of alcohol *Volume fraction == Notes == == References == ==Bibliography== * * * * ==External links== * * Category:Alcohol measurement The following formulas can be used to calculate the volumes of solute (Vsolute) and solvent (Vsolvent) to be used: *Vsolute = Vtotal / F *Vsolvent = Vtotal \- Vsolute , where: *Vtotal is the desired total volume *F is the desired dilution factor number (the number in the position of F if expressed as ""1:F dilution factor"" or ""xF dilution"") However, some solutions and mixtures take up slightly less volume than their components. The density of sugar in water is greater than the density of alcohol in water. The phenomenon of volume changes due to mixing dissimilar solutions is called ""partial molar volume"". thumb|The Mollier enthalpy–entropy diagram for water and steam. However, because of the miscibility of alcohol and water, the conversion factor is not constant but rather depends upon the concentration of alcohol. The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at , which is .Haynes, William M., ed. (2011). Alcohol by volume (abbreviated as ABV, abv, or alc/vol) is a standard measure of how much alcohol (ethanol) is contained in a given volume of an alcoholic beverage (expressed as a volume percent). In some countries, e.g. France, alcohol by volume is often referred to as degrees Gay-Lussac (after the French chemist Joseph Louis Gay-Lussac), although there is a slight difference since the Gay- Lussac convention uses the International Standard Atmosphere value for temperature, . ==Volume change== thumb|upright=1.48|Change in volume with increasing ABV. Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations and as long as the molar attenuation coefficients of the two components, and are known at both wavelengths. A cup of the wine/water mixture is then returned to the wine barrel, so that the volumes in the barrels are again equal. ",1.8,-8,"""3.0""",22,0.396,B
+"If the coefficient of static friction between the block and plane in the previous example is $\mu_s=0.4$, at what angle $\theta$ will the block start sliding if it is initially at rest?","The component of the force of gravity in the direction of the incline is given by: F_g = mg\sin{\theta} The normal force (perpendicular to the surface) is given by: N = mg\cos{\theta} Therefore, since the force of friction opposes the motion of the block, F_k =\mu_k \cdot mg\cos{\theta} To find the coefficient of kinetic friction on an inclined plane, one must find the moment where the force parallel to the plane is equal to the force perpendicular; this occurs when the block is moving at a constant velocity at some angle \theta \sum F = ma = 0 F_k = F_g or \mu_k mg\cos{\theta} = mg\sin{\theta} Here it is found that: \mu_k = \frac{mg\sin{\theta}}{mg\cos{\theta}} = \tan{\theta} where \theta is the angle at which the block begins moving at a constant velocity == References == Category:Classical mechanics The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: F_{k} = \mu_\mathrm{k} F_{n}. However, the magnitude of the friction force itself depends on the normal force, and hence on the mass of the block. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: F_\text{max} = \mu_\mathrm{s} F_\text{n}. Thus, a force is required to move the back of the contact, and frictional heat is released at the front. thumb|Angle of friction, θ, when block just starts to slide. ===Angle of friction=== For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. Sliding commences only after this frictional force reaches the value F_f = \mu N. For surfaces at rest relative to each other, \mu = \mu_\mathrm{s}, where \mu_\mathrm{s} is the coefficient of static friction. The friction increases as the applied force increases until the block moves. Coefficients of friction range from near zero to greater than one. Prior to sliding, this friction force is F_f = -P_x, where P_x is the horizontal component of the external force. After the block moves, it experiences kinetic friction, which is less than the maximum static friction. It is defined as: \tan{\theta} = \mu_\mathrm{s} and thus: \theta = \arctan{\mu_\mathrm{s}} where \theta is the angle from horizontal and μs is the static coefficient of friction between the objects. If an object is on a level surface and subjected to an external force P tending to cause it to slide, then the normal force between the object and the surface is just N = mg + P_y, where mg is the block's weight and P_y is the downward component of the external force. Sliding friction is almost always less than that of static friction; this is why it is easier to move an object once it starts moving rather than to get the object to begin moving from a rest position. This is called the angle of friction or friction angle. When there is no sliding occurring, the friction force can have any value from zero up to F_\text{max}. For surfaces in relative motion \mu = \mu_\mathrm{k}, where \mu_\mathrm{k} is the coefficient of kinetic friction. In fact, the friction force always satisfies F_f\le \mu N, with equality reached only at a critical ramp angle (given by \tan^{-1}\mu) that is steep enough to initiate sliding. This formula can also be used to calculate μs from empirical measurements of the friction angle. ===Friction at the atomic level=== Determining the forces required to move atoms past each other is a challenge in designing nanomachines. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. The coefficient of friction is an empirical measurementit has to be measured experimentally, and cannot be found through calculations. The coefficient of static friction, typically denoted as μs, is usually higher than the coefficient of kinetic friction. ",-8,0.4772,"""0.02""",22,24,D
+"Halley's comet, which passed around the sun early in 1986, moves in a highly elliptical orbit with an eccentricity of 0.967 and a period of 76 years. Calculate its minimum  distances from the Sun.","The following is a list of comets with a very high eccentricity (generally 0.99 or higher) and a period of over 1,000 years that do not quite have a high enough velocity to escape the Solar System. On 23 March 2147 the comet will pass about from Earth with an uncertainty region of about ±2 million km. C/2001 OG108 (LONEOS) Closest Earth Approach on 2147-Mar-23 11:20 UT Date & time of closest approach Earth distance (AU) Sun distance (AU) Velocity wrt Earth (km/s) Velocity wrt Sun (km/s) Uncertainty region (3-sigma) Reference 2147-03-23 11:20 ± 13:38 40.3 35.3 ± 2 million km Horizons The comet has a rotational period of 2.38 ± 0.02 days (57.12 hr). The comet came to perihelion (closest approach to the Sun) on 15 March 2002. 170P/Christensen is a periodic comet in the Solar System. Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. C/ (LONEOS) is a Halley-type comet with an orbital period of 48.51 years. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. 1308 Halleria, provisional designation , is a carbonaceous Charis asteroid from the outer regions of the asteroid belt, approximately 43 kilometers in diameter. Using data from Fernandez (2004–2005) JPL lists the comet with an albedo of 0.05 and a diameter of 13.6 ± 1.0 km. 164P/Christensen is a periodic comet in the Solar System. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 164P/Christensen – Seiichi Yoshida @ aerith.net * Elements and Ephemeris for 164P/Christensen – Minor Planet Center Category:Periodic comets 0164 164P 20041221 It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 The actual orbit of these comets significantly differs from the provided coordinates. Of the short-period comets with known diameters and perihelion inside the orbit of Earth, C/ is the second largest after Comet Swift–Tuttle. In 2003, the comet was estimated to have a mean absolute V magnitude (H) of 13.05 ± 0.10, with an albedo of 0.03, giving an effective radius of 8.9 ± 0.7 km. A Solar System barycentric orbit computed at an epoch when the object is located beyond all the planets is a more accurate measurement of its long-term orbit. ==List of near-parabolic comets== Comet designation Comet name /discoverer Semimajor axis (AU) Eccentricity Inclination (°) Perihelion distance (AU) Absolute magnitude (M1/M2) Perihelion date Period (3) (years) Ref C/1680 V1 Great Comet of 1680 444.4285 0.999986 60.6784 0.006222 1680/12/18 9370 C/1769 P1 Messier 163.4554 0.999249 40.7338 0.122755 1769/10/08 2090 C/1785 E1 Méchain 120.6893 0.99646 92.639 0.42724 1785/04/08 1325 C/1807 R1 Great comet of 1807 143.2012 0.995488 63.1762 0.646124 1807/09/19 1710 C/1811 F1 Great Comet of 1811 212.3922 0.995125 106.9342 1.035412 1811/09/12 3100 C/1822 N1 Pons 310.8303 0.996316 127.3429 1.145099 1822/10/24 5480 C/1823 Y1 Great Comet of 1823 170 0.9987 103.68 0.2252 1823/12/09 2300 C/1825 K1 Gambart 246.605 0.996395 123.3414 0.889011 1825/05/31 3870 C/1825 N1 Pons 271.5793 0.995431 146.4353 1.240846 1825/12/11 4480 C/1826 P1 Pons 340.063 0.997492 25.9496 0.852878 1826/10/09 6270 C/1840 B1 Galle 180.8076 0.99325 120.7807 1.220451 1840/03/13 2430 C/1844 N1 Mauvais 3520.1687 0.999757 131.4092 0.855401 1844/10/17 208900 C/1844 Y1 Great Comet of 1844 358.9355 0.999302 45.5651 0.250537 1844/12/14 6800 C/1846 B1 de Vico 194.9063 0.992403 47.4257 1.480703 1846/01/22 2720 C/1847 C1 Hind 473.2556 0.99991 48.6636 0.042593 1847/03/30 10300 C/1847 N1 Mauvais 1251.6357 0.998589 96.5817 1.766058 1847/08/09 44280 C/1849 G1 Schweizer 568.5696 0.998427 66.9587 0.89436 1849/06/08 13560 C/1850 J1 Petersen 771.8979 0.998599 68.1848 1.081429 1850/07/24 21450 C/1854 R1 Klinkerfues 118.2650 0.993246 40.9201 0.798762 1854/10/28 1290 C/1854 Y1 Winnecke-Dien 156.4219 0.991309 14.152 1.359463 1854/12/16 1960 C/1857 Q1 Klinkerfues 182.3447 0.996913 123.9614 0.562898 1857/10/01 2460 C/1858 L1 Donati 156.132 0.996295 116.9512 0.578469 1858/09/30 1950 C/1863 G1 Klinkerfues 1269.962 0.999159 112.6209 1.068038 1863/04/05 45260 C/1863 G2 Respighi 682.7155 0.999079 85.4961 0.628781 1863/04/21 17840 C/1863 V1 Tempel 630.1632 0.998879 78.0817 0.706413 1863/11/09 15820 C/1864 N1 Tempel 249.1888 0.996351 178.1269 0.90929 1864/08/16 3930 C/1864 O1 Donati-Toussaint 1450.486 0.999358 109.7124 0.931212 1864/10/11 55240 C/1871 G1 Winnecke 299.3138 0.997814 87.6034 0.6543 1871/06/11 5180 C/1871 V1 Tempel 161.2851 0.995714 98.2992 0.691268 1871/12/20 2050 C/1873 Q1 Borrelly 225.7138 0.996482 95.9662 0.794061 1873/09/11 3390 C/1873 Q2 Henry 1425.6037 0.99973 121.4625 0.384913 1873/10/02 53830 C/1874 H1 Coggia 572.6966 0.99882 66.3439 0.675782 1874/07/07 13710 C/1874 O1 Borrelly 840.5894 0.998831 41.8266 0.982649 1874/08/27 24370 C/1877 G1 Winnecke 730.7538 0.9987 121.1548 0.94998 1877/04/18 19750 C/1877 G2 Swift 485.8223 0.997923 77.1916 1.009053 1877/04/27 10700 C/1881 K1 Great Comet of 1881 178 0.99589 63.4253 0.734547 1881/06/16 2390 C/1881 W1 Swift 195.8064 0.990169 144.8016 1.924973 1881/11/20 2740 C/1882 F1 Wells 10127.1667 0.999994 73.7977 0.060763 1882/06/11 1019140 C/1887 J1 Barnard 356.7476 0.996093 17.5479 1.393813 1887/06/17 7640 C/1888 D1 Sawerthal 169.3582 0.995874 42.2482 0.698772 1888/03/17 2200 C/1888 P1 Brooks 9806.8043 0.999908 74.1904 0.902226 1888/07/31 971160 C/1888 U1 Barnard 179.494 0.991488 56.3425 1.527853 1888/09/13 2400 C/1889 G1 Barnard 12393.3846 0.999818 163.8517 2.255596 1889/06/11 1379700 C/1889 O1 Davidson 435.2118 0.997611 65.9916 1.039721 1889/07/19 9080 C/1890 O2 Denning 1550.0898 0.999187 98.9373 1.260223 1890/09/25 61030 C/1890 V1 Zona 495.8077 0.995872 154.307 2.046694 1890/08/07 11040 C/1892 E1 Swift 809.1663 0.998731 38.7002 1.026832 1892/04/07 23020 C/1893 N1 Rordame-Quénisset 1249.1648 0.99946 159.9804 0.674549 1893/07/07 44150 C/1893 U1 Brooks 231.2706 0.996489 129.8233 0.811991 1893/09/19 3520 C/1898 V1 Chase 4634.0588 0.999507 22.5046 2.2846 1898/09/20 315460 C/1902 R1 Perrine 12533.5625 0.999968 156.3548 0.4011 1902/11/24 1403170 C/1903 A1 Giacobini 1244.1242 0.99967 30.9416 0.4106 1903/03/16 43880 C/1906 B1 Brooks 1926.3913 0.999327 126.4425 1.296 7.0 1905/12/22 84552.21 C/1907 L2 Daniel 424.6874 0.998794 8.9577 0.512173 1907/09/04 8750 C/1909 L1 Borrelly-Daniel 160.9095 0.994762 52.0803 0.842844 1909/06/05 2040 C/1910 A1 Great January comet of 1910 25795 0.999995 138.7812 0.128975 1910/01/17 4142890 C/1910 P1 Metcalf 9596.1232 0.999797 121.0556 1.948013 1910/09/16 940030 C/1911 N1 Kiess 184 0.9963 148.42 0.68383 1911/06/30 2500 C/1911 O1 Brooks 163.1454 0.997005 16.4153589 0.489429 1911/10/28 2090 C/1911 S2 Quénisset 429.6907 0.998167 108.1 0.787623 1911/11/12 8910 C/1913 J1 Schaumasse 309.1747 0.995288 152.3673 1.456831 1913/05/15 5436 C/1913 R1 Metcalf 555.7869 0.99756 143.3547 1.35612 1913/09/14 13100 C/1914 S1 Campbell? 534.2999 0.998666 77.836 0.712756 1914/08/05 12350 C/1916 G1 Wolf 2834.363 0.999405 25.6592 1.686446 1917/06/17 150900 C/1920 X1 Skjellerup 193.9334 0.994081 22.0303 1.147892 1920/12/11 2700 C/1922 B1 Reid 125.0064 0.986968 32.4456 1.629083 1921/10/28 1400 C/1922 W1 Skjellerup 147.4206 0.993735 23.3659 0.92359 1923/01/04 1790 C/1924 F1 Reid 1252.2789 0.998598 72.3273 1.755695 1924/03/13 44320 C/1925 F2 Reid 334.418 0.995116 26.9797 1.633299 1925/07/29 6120 C/1926 B1 Blathwayt 176.6645 0.992384 128.2986 1.345477 1926/01/02 2350 C/1927 E1 Stearns 2023.0104 0.998179 87.6525 3.683902 1927/03/22 90990 C/1927 X1 Skjellerup-Maristany 1100.9813 0.99984 85.1126 0.176157 1927/12/18 36530 C/1929 Y1 Wilk 691.5998 0.999028 124.5103 0.672235 1930/01/22 18190 C/1930 D1 Peltier-Schwassmann- Wachmann 581.6554 0.998131 99.883 1.087114 1930/11/15 14030 C/1931 P1 Ryves Comet 111.1632 0.999326 169.2881 0.074924 1931/08/25 1180 C/1936 K1 Peltier 133.7226 0.991775 78.5447 1.099868 1936/06/08 1550 C/1937 N1 Finsler 57516 0.999985 146.4156 0.862744 1937/08/15 13793870 C/1939 B1 Kozik-Peltier 146.3133 0.995103 63.5238 0.716496 1939/02/06 1770 C/1939 H1 Jurlof-Achmarof- Hassel 346.8588 0.998477 138.1212 0.528266 1939/04/10 6460 C/1939 V1 Friend 336.6129 0.997192 92.952 0.945209 1939/11/05 6180 C/1941 B2 de Kock- Paraskevopoulos 879.7695 0.999102 168.2039 0.790033 1941/01/27 26100 C/1942 X1 Whipple-Fedtke-Tevzadze 173.4555 0.992196 19.7127 1.353647 1943/02/06 2280 C/1944 H1 Väisälä 370.9009 0.9935 17.2882 2.410856 1945/01/04 7140 C/1947 F1 Rondanina-Bester 217.5667 0.997427 39.3015 0.559799 1947/05/20 3210 C/1947 X1-A Southern Comet of 1947 243.4336 0.999548 138.5419 0.110032 1947/12/02 3800 C/1947 X1-B Southern Comet of 1947 296.558 0.999629 138.5332 0.110023 1947/12/02 5110 C/1948 L1 Honda-Bernasconi 1661.024 0.999875 23.1489 0.207628 1948/05/15 67700 C/1948 N1 Wirtanen 3884.5787 0.999352 130.2675 2.517207 1949/05/01 242110 C/1948 V1 Eclipse Comet of 1948 2083.4 0.999935 23.117 0.135421 1948/10/27 95100 C/1948 W1 Bester 509.1675 0.997499 87.6054 1.273428 1948/10/22 11490 C/1949 N1 Bappu-Bok-Newkirk 1517.8296 0.998644 105.7686 2.058177 1949/10/26 59130 C/1951 P1 Wilson-Harrington 2836.8851 0.999739 152.5337 0.740427 1952/01/12 151100 C/1952 M1 Peltier 4605.0766 0.999739 45.5521 1.201925 1952/06/15 312500 C/1952 Q1 Harrington 407.0848 0.99591 59.1154 1.664977 1953/01/05 8210 C/1953 G1 Mrkos-Honda 391.7716 0.997391 93.8573 1.022132 1953/05/26 7750 C/1955 N1 Bakharev-Macfarlane-Krienke 244.7197 0.994167 50.0329 1.42745 1955/07/11 3830 C/1957 P1 Mrkos 558.9496 0.999365 93.9411 0.354933 1957/08/01 13210 C/1958 D1 Burnham 23205.0702 0.999943 15.7879 1.322689 1958/04/16 3534880 C/1958 R1 Burnham-Slaughter 12150.7313 0.999866 61.2576 1.628198 1959/03/11 1339380 C/1959 X1 Mrkos 4974.0634921 0.999748 19.6339 1.253464 1959/11/13 350810 C/1960 Y1 Candy 105.1101 0.9899 150.9552 1.061612 1961/02/08 1080 C/1961 O1 Wilson 1057.8684 0.999962 24.2116 0.040199 1961/07/17 34410 C/1961 R1 Humason 204.5261 0.989569 153.278 2.133412 1962/12/10 2920 C/1963 F1 Alcock 792.3201 0.99806 86.2194 1.537101 1963/05/05 22300 C/1964 L1 Tomita-Gerber-Honda 123.03 0.995933 161.8323 0.500363 1964/06/30 1360 C/1964 P1 Everhart 361.1342 0.996513 67.9689 1.259275 1964/08/23 6860 C/1965 S1-B Ikeya-Seki 103.7067 0.999925 141.861 0.007778 1965/10/21 1060 C/1966 P1 Kilston 3821.7115 0.999376 40.2648 2.384748 1966/10/28 236260 C/1966 P2 Barbon 1111.0435 0.998183 28.7058 2.018766 1966/04/17 37033 C/1967 Y1 Ikeya-Seki 2000.6851 0.999152 129.3153 1.696581 1968/02/25 89490 C/1968 H1 Tago-Honda-Yamamoto 174 0.9961 102.1698 0.680378 9.8 1968/05/16 2300 C/1968 Y1 Thomas 705.6891 0.995301 45.2291 3.316033 1969/01/12 18750 C/1969 O1-A Kohoutek 1964.6686 0.999125 86.3128 1.719085 1970/03/21 87080 C/1969 T1 Tago-Sato-Kosaka 6400 0.999926 75.81773 0.4726395 6.5 1969/12/21 508060 C/1969 Y1 Bennett 141.21513 0.996193 90.0394 0.537606 1970/03/20 1680 C/1972 E1 Bradfield 494.778 0.998126 123.693 0.927214 1972/03/27 11010 C/1972 F1 Gehrels 1071.8224 0.996943 175.616 3.276561 1971/01/06 35090 C/1972 X1 Araya 54008.3111 0.99991 113.0902 4.860748 1972/12/18 12551360 C/1973 D1 Kohoutek 1082.2388 0.998723 121.5982 1.382019 1973/06/07 35600 C/1974 C1 Bradfield 1660.6964 0.999697 61.2842 0.503191 1974/03/18 67680 C/1974 F1 Lovas 7566.4724 0.999602 50.6485 3.011456 1975/08/22 658170 C/1975 T1 Mori-Sato-Fujikawa 632 0.997461 97.6077 1.603934 5.5 1975/12/25 15880 C/1975 V1-A Comet West 6780.2069 0.999971 43.0664 0.196626 1967/02/25 558300 C/1976 D1 Bradfield 136.9866 0.993811 46.834 0.84781 1976/02/24 1600 C/1976 J1 Harlan 5143.859 0.999695 38.8063 1.568877 1976/11/03 368920 C/1977 R1 Kohler 2170 0.999543 48.71188 0.9905761 7.3 1977/11/10 101000 C/1977 V1 Tsuchinshan 9817.545 0.999633 168.5495 3.603039 1977/06/24 972760 C/1978 T1 Seargent 220 0.99832 67.828 0.36988 1978/09/14 3300 C/1980 V1 Meier 285 0.99468 100.9864 1.51956 7.2 1980/12/09 4820 C/1980 Y1 Bradfield 944.8109 0.999725 138.585 0.259823 1980/12/29 29040 C/1980 Y2 Panther 1640 0.998991 82.64774 1.657269 6.1 1981/01/27 66500 C/1981 H1 Bus 2510.8713 0.999021 160.664 2.458143 1981/07/30 125816 C/1981 M1 Gonzalez 3857.1917 0.999395 107.1467 2.333601 1981/03/25 239560 C/1982 M1 Austin 1072 0.999396 84.4951 0.6478114 8.8 1982/08/24 35100 C/1983 J1 Sugano-Saigusa- Fujikawa 4779.898 0.999901 96.623 0.471 12.3 1983/05/01 330473.13 C/1983 N1 IRAS 4168.9638 0.99942 138.8364 2.417999 1983/05/02 269180 C/1984 N1 Austin 1891.4545 0.999846 164.1533 0.291284 1984/08/12 82260 C/1984 U1 Shoemaker 1145.723 0.995209 179.2123 5.489159 1984/09/03 38780 C/1984 V1 Levy-Rudenko 1160 0.99921 65.7146 0.917949 9.4 1984/12/14 39600 C/1984 W2 Hartley 9501.7435 0.999579 89.3273 4.000234 1985/09/28 926200 C/1985 R1 Hartley-Good 5800 0.999881 79.9294 0.694577 8.4 1985/12/09 450000 C/1986 N1 Churyumov- Solodovnikov 5669.7575 0.999534 114.9293 2.642107 1986/05/06 426920 C/1986 V1 Sorrells 18913.7912 0.999909 160.5801 1.721155 1987/03/09 2601160 C/1987 B1 Nishikawa-Takamizawa-Tago 207 0.9958 172.22989 0.869589 7.4 1987/03/17 2980 C/1987 P1 Bradfield 165.2 0.99474 34.08809 0.868956 6 1987/11/07 2123 C/1987 U3 McNaught 406 0.99792 97.5751 0.84393 6.9 1987/12/02 8200 C/1988 A1 Liller 244.9295 0.996565 73.3224 0.841333 1988/03/31 3830 C/1988 F1 Levy 537.6264 0.997816 62.8074 1.174176 1987/11/29 12470 C/1988 J1 Shoemaker-Holt 541.2286 0.99783 62.8066 1.174466 1988/02/14 12590 C/1989 A1 Yanaka 1410 0.99866 52.4092 1.89458 5.1 1988/10/31 53000 C/1989 A5 Shoemaker 547.6156 0.99518 96.5548 2.639507 1989/02/26 12810 C/1989 T1 Helin-Roman-Alu 112.097 0.990657 46.0369 1.047322 1989/12/15 1190 C/1990 N1 Tsuchiya-Kiuchi 233.2246 0.995316 143.7839 1.092424 1990/09/28 3560 C/1991 A2 Masaru Arai 151.0756 0.990507 70.9783 1.434161 1990/12/10 1860 C/1991 B1 Shoemaker-Levy 348.8963 0.993508 77.2881 2.265035 1991/12/31 6520 C/1991 Q1 McNaught-Russell 589.3992 0.994581 90.5062 3.19395 1992/05/03 14310 C/1991 R1 McNaught-Russell 11160.2875 0.999374 104.5086 6.98634 1990/11/12 1179000 C/1991 T2 Shoemaker-Levy 6000 0.999860 113.49709 0.8362597 7.7 1992/07/24 4650000 C/1992 F1 Tanaka-Machholz 312.7164 0.995966 79.2924 1.261498 1992/04/22 5530 C/1992 J1 Spacewatch 77102.7179 0.999961 124.3187 3.007006 1993/09/05 21409400 C/1992 U1 Shoemaker 3928.1053 0.999411 65.9859 2.313654 1993/03/25 246190 C/1993 Y1 McNaught- Russell 134.8 0.99356 51.5866 0.8676358 12.2 1994/03/31 1564 C/1994 E2 Shoemaker-Levy 431.4296 0.997314 131.2547 1.15882 1994/05/27 8960 C/1994 G1-A Takamizawa-Levy 1549.8632 0.999123 132.8728 1.35923 1994/05/22 61020 C/1994 J2 Takamizawa 545.4374 0.996429 135.9611 1.947757 1994/06/29 12740 C/1994 T1 Machholz 3820.7081 0.999517 101.7379 1.845402 1994/10/02 236170 C/1995 O1 Comet Hale–Bopp 185.86 0.9950817 89.430154 0.9141335 2.3 1997/04/01 2534 C/1995 Q1 Bradfield 220.6208 0.998022 147.3942 0.436388 1995/08/31 3280 C/1996 B1 Szczepanski 156.9 0.99076 51.9189 1.448788 7.1 1996/02/06 1965 C/1996 B2 Comet Hyakutake 2270 0.9998987 124.92266 0.2302293 7.3 1996/05/01 108000 C/1996 Q1 Tabur 800 0.9989 73.359 0.83984 11.0 1996/11/03 22000 C/1996 R1 Hergenrother-Spahr 132 0.9856 145.8144 1.89920 5.8 1996/08/28 1510 C/1996 R3 Lagerkvist 404.015 0.987 39.2 5.24 10.5 1995/07/24 8120.91 Spacewatch 3081 0.998884 72.71704 3.436463 4.9 1999/11/27 171000 C/1997 G2 Montani 529 0.99417 69.83548 3.084966 5.3 1998/04/16 12160 C/1997 J1 Mueller 255.5 0.990991 122.96833 2.302132 8.6 1997/05/03 4085 C/1997 L1 Zhu-Balam 2420 0.99797 72.9914 4.89956 6.5 1996/11/22 119000 C/1997 T1 Utsunomiya 920 0.998523 127.99262 1.3591096 8.0 1997/12/10 27910 C/1998 H1 Stonehouse 710 0.9979 104.693 1.48729 10.0 1998/04/14 19000 C/1998 K2 LINEAR 3210 0.999276 64.45667 2.323479 8.6 1998/09/01 182000 C/1998 K3 LINEAR 1700 0.9979 160.2056 3.5463 10.0 1998/03/07 70000 C/1998 M1 LINEAR 431 0.99277 20.38455 3.11812 5.4 1998/10/28 8950 C/1998 M2 LINEAR 1215 0.997758 60.18232 2.725333 8.5 1998/08/13 42400 C/1998 M4 LINEAR 1100 0.998 154.572 2.6001 9.5 1997/12/10 30000 C/1998 M5 LINEAR 438.3 0.996025 82.22889 1.7422899 8.0 1999/01/24 9176 C/1998 M6 Montani 5400 0.9989 91.540 5.9787 7.5 1998/10/06 400000 C/1998 P1 Williams 1700 0.999325 145.72831 1.146108 8.0 1998/10/17 70000 C/1998 Q1 LINEAR 358 0.99559 32.3058 1.57788 14.0 1998/06/29 6770 C/1998 T1 LINEAR 1657 0.999114 170.15995 1.467728 9.5 1999/06/25 67400 C/1998 U5 LINEAR 102.88 0.987981 131.76474 1.2364530 10.9 1998/12/21 1043.5 C/1999 A1 Tilbrook 177 0.99587 89.481 0.730741 12.0 1999/01/29 2350 C/1999 F1 Catalina (CSS) 6700 0.999136 92.03554 5.787022 4.6 2002/02/13 548000 C/1999 F2 Dalcanton 2640 0.99821 56.42742 4.71807 7.6 1998/08/23 135000 C/1999 H1 Lee 2775 0.9997449 149.35290 0.70810722 9.4 1999/07/11 146200 C/1999 J3 LINEAR 1600 0.99939 101.6561 0.976809 11.3 1999/09/20 64000 C/1999 K2 Ferris 155 0.9658 82.191 5.2903 7.0 1999/04/10 1920 C/1999 K3 LINEAR 235 0.9918 92.274 1.92878 12.0 1999/02/27 3600 C/1999 K6 LINEAR 346.8 0.993532 46.34384 2.246976 11.3 1999/07/24 6459 C/1999 K7 LINEAR 700 0.9966 135.159 2.3227 13.0 1999/02/24 18000 C/1999 L2 LINEAR 390 0.9951 43.942 1.90476 13.0 1999/08/04 7800 C/1999 N2 Lynn 298 0.99745 111.6559 0.7612844 10.3 1999/07/23 5150 C/1999 T1 McNaught-Hartley 8100 0.999856 79.97521 1.1716989 8.6 2000/12/13 740000 C/2000 B2 LINEAR 6000 0.9994 93.647 3.7762 10.3 1999/11/10 500000 LINEAR 1916 0.998353 49.21252 3.155967 7.4 2001/06/19 83900 C/2000 K2 LINEAR 522.0 0.995332 25.63358 2.437066 9.3 2000/10/11 11930 C/2000 Y2 Skiff 490 0.99435 12.0875 2.76871 11.4 2001/03/21 10850 C/2001 A1 LINEAR 266 0.99095 59.941 2.4064 12.7 2000/09/17 4330 C/2001 A2-A LINEAR 2500 0.99969 36.487 0.779054 13 2001/05/24 130000 C/2001 A2-B LINEAR 1119 0.999304 36.47582 0.7790172 7 2001/05/24 37400 C/2001 C1 LINEAR 38000 0.99987 68.96470 5.10432 6.5 2002/03/28 7000000 LINEAR-NEAT 1193.5 0.9976606 163.212126 2.7920832 7.4 2003/07/09 41230 C/2001 K3 Skiff 2870 0.99893 52.0265 3.06012 9.4 2001/04/22 153000 C/2001 K5 LINEAR 11410 0.999546 72.590342 5.184246 4.4 2002/10/11 1220000 C/2001 O2 NEAT 2200 0.9978 90.9262 4.8194 6.6 1999/10/17 103000 C/2001 Q1 NEAT 171.2 0.96593 66.9504 5.83397 7.7 2001/09/20 2241 C/2001 U6 LINEAR 1149 0.99617 107.25550 4.40642 6.5 2002/08/08 39000 C/2001 W1 LINEAR 2100 0.9989 118.645 2.39924 13.7 2001/12/24 100000 C/2001 X1 LINEAR 570 0.99700 115.6268 1.69793 11.3 2002/01/08 13500 C/2002 B2 LINEAR 1400 0.9972 152.8726 3.8422 10.1 2002/04/06 50000 C/2002 C2 LINEAR 9000 0.99964 104.88143 3.25375 9.9 2002/04/10 860000 C/2002 F1 Utsunomiya 950 0.999539 80.8770 0.4382989 10.5 2002/04/22 29300 C/2002 H2 LINEAR 276 0.99407 110.5011 1.63484 10.5 2002/03/23 4570 C/2002 J4 NEAT 29000 0.999874 46.52550 3.633722 8.4 2003/10/03 4900000 C/2002 K1 NEAT 9000 0.9997 89.723 3.23024 11.4 2002/06/16 900000 C/2002 K2 LINEAR 763 0.99314 130.8957 5.23506 8.2 2002/06/05 21100 C/2002 L9 NEAT 4460 0.99842 68.44211 7.03301 4.7 2004/04/05 297000 C/2002 O6 SWAN 350 0.99858 58.6240 0.494648 13.0 2002/09/09 6500 C/2002 P1 NEAT 414 0.98422 34.6061 6.5302 8.2 2001/11/23 8420 C/2002 Q3-A LINEAR 465.333 0.997194 96.87858 1.30583 16.4 2002/08/19 10038.16 C/2002 V1 NEAT 1011 0.9999018 81.70600 0.0992581 10.4 2003/02/18 32100 C/2002 V2 LINEAR 5010 0.99864 166.77622 6.81203 8.4 2003/03/13 355000 LINEAR 202.14 0.966377 70.51612 6.796713 7.1 2006/02/06 2874.1 C/2002 X1 LINEAR 1376 0.998192 164.08943 2.4867001 9.8 2003/07/12 51020 C/2002 X5 Kudo-Fujikawa 1210 0.999843 94.15226 0.189935 10.6 2003/01/29 42000 C/2002 Y1 Juels-Holvorcem 250.6 0.997152 103.78154 0.7138096 9.8 2003/04/13 3967 C/2003 G2 LINEAR 440 0.9965 96.167 1.55337 16.0 2003/04/29 9000 C/2003 H1 LINEAR 2653 0.999156 138.667242 2.2396301 8.7 2004/02/22 136700 C/2003 H3 NEAT 13200 0.999780 42.81171 2.901441 9.6 2003/04/24 1510000 C/2003 J1 NEAT 577 0.99112 98.3135 5.12542 8.8 2003/10/10 13900 C/2003 L2 LINEAR 154.40 0.981446 82.05107 2.864801 9.9 2004/01/19 1918.7 C/2003 T2 LINEAR 6400 0.99972 87.5315 1.786352 9.8 2003/11/14 520000 C/2003 T3 Tabur 5730 0.999742 50.44443 1.4810758 5.8 2004/04/29 434000 C/2003 V1 LINEAR 603 0.99704 28.67513 1.78314 9.9 2003/03/11 14800 C/2004 F2 LINEAR 151.6 0.99056 104.9600 1.43044 13.2 2003/12/26 1870 C/2004 F4 Bradfield 238 0.999294 63.16456 0.168266 11.3 2004/04/17 3680 C/2004 G1 LINEAR 328.47 0.996 114.486 1.201 14.4 2004/06/04 5953.27 C/2004 K1 Catalina (CSS) 1819 0.998131 153.747521 3.399147 7.9 2005/07/05 77600 C/2004 L1 LINEAR 858 0.997615 159.36082 2.04741344 12.6 2005/03/30 25100 C/2004 L2 LINEAR 790 0.995215 62.51864 3.778629 8.3 2005/11/15 22190 C/2004 P1 NEAT 8100 0.99925 28.8163 6.01377 10.1 2003/08/08 720000 C/2004 Q1 Tucker 186.78 0.989042 56.08768 2.0467255 9.8 2004/12/06 2552.8 C/2004 Q2 Comet Machholz 2403 0.9994986 38.588963 1.2050414 9.9 2005/01/24 117800 LINEAR 700 0.997227 21.61823 1.942359 13.9 2005/03/03 18540 C/2004 T3 Siding Spring 5600 0.99842 71.9642 8.8644 6.6 2003/04/15 420000 C/2004 U1 LINEAR 3610 0.999264 130.62532 2.659321 9.0 2004/12/08 217000 C/2004 X2 LINEAR 1450 0.99738 72.118 3.79308 10.2 2004/08/24 55000 LINEAR 13000 0.99987 52.47641 1.781202 17.3 2005/03/03 1500000 C/2005 G1 LINEAR 18800 0.99974 108.41395 4.960798 7.7 2006/02/27 2600000 C/2005 L3 McNaught 13390 0.999582 139.449248 5.593622 6.4 2008/01/16 1550000 C/2005 N1 Juels-Holvorcem 729 0.998457 51.18017 1.125447 11.3 2005/08/22 19700 C/2005 R4 LINEAR 2067 0.99749 164.01260 5.188473 7.7 2006/03/08 94000 C/2005 S4 McNaught 5690 0.998972 107.95897 5.850109 7.9 2007/07/18 430000 C/2005 X1 Beshore 690 0.9958 91.944 2.8623 11.1 2005/07/05 18000 C/2005 YW LINEAR 190.4 0.989534 40.54361 1.9930109 7.4 2006/12/07 2628 C/2006 A1 Pojmański 2370 0.999765 92.73611 0.5553959 10.5 2006/02/22 115000 C/2006 A2 Catalina (CSS) 3800 0.99862 148.3226 5.3160 9.8 2005/05/20 240000 C/2006 B1 McNaught 1340 0.99776 134.28193 2.997591 10.3 2005/11/19 49100 Catalina (CSS) 216.32 0.991900 144.26278 1.7521694 12.3 2006/07/03 3182 C/2006 K4 NEAT 1818 0.998246 111.33346 3.188618 8.8 2007/11/29 77500 C/2006 L1 Garradd 551 0.997345 143.24257 1.462070 8.6 2006/10/18 12930 C/2006 M1 LINEAR 153.67 0.976859 54.87693 3.556199 9.6 2007/02/13 1905.0 C/2006 O2 Garradd 420 0.99634 43.0287 1.55479 12.7 2006/10/05 8700 C/2006 Q1 McNaught 6890 0.9995986 59.050380 2.7637144 7.0 2008/07/03 571000 C/2006 U6 Spacewatch 1931 0.998706 84.87894 2.4983978 8.8 2008/06/05 84900 C/2006 V1 Catalina (CSS) 257.6 0.989618 31.11947 2.674906 9.0 2007/11/26 4136 C/2006 W3 Christensen 17990 0.9998262 127.074692 3.1262325 6.7 2009/07/06 2410000 Lemmon 583.8 0.998987 152.70463 0.5912444 17.4 2007/04/28 14110 LINEAR 252.0 0.992839 30.62941 1.804374 7.4 2007/07/21 4000 C/2007 B2 Skiff 737.2 0.995965 27.49527 2.9749171 8.1 2008/08/20 20020 C/2007 D1 LINEAR 171366.7 0.99995 41.50701 8.793 8.9 2007/06/18 C/2007 D3 LINEAR 652 0.99201 45.92022 5.20897 9.2 2007/05/27 16650 C/2007 E2 Lovejoy 1330 0.99918 95.8830 1.092939 10.9 2007/03/27 49000 C/2007 K1 Lemmon 436 0.97880 108.4325 9.23905 8.6 2007/05/07 9100 C/2007 K6 McNaught 224 0.9847 105.064 3.4330 10.6 2007/07/01 3350 C/2007 M1 McNaught 1564 0.99522 139.72142 7.47465 5.6 2008/08/11 61900 C/2007 M2 Catalina (CSS) 5360 0.999339 80.94565 3.541050 9.0 2008/12/08 392000 C/2007 M3 LINEAR 171.45 0.979768 161.76086 3.468759 9.9 2007/09/04 2245 C/2007 N3 Lulin 72000 0.9999833 178.373611 1.21225837 9.7 2009/01/10 19500000 C/2007 T1 McNaught 4040 0.999760 117.64244 0.9685028 11.1 2007/12/12 256000 Spacewatch 12200 0.999603 86.99476 4.842732 7.1 2010/04/26 1350000 C/2007 Y2 McNaught 1210 0.99652 98.50321 4.20896 9.2 2008/04/08 42100 C/2008 C1 Chen-Gao 101627.54 0.9999876 61.7845 1.262343 11.7 2008/04/16 32398532.38 C/2008 E3 Garradd 3740 0.99852 105.07653 5.53103 5.1 2008/08/02 229000 C/2008 G1 Gibbs 365 0.98908 72.856 3.9898 10.5 2009/01/11 6980 C/2008 J1 Boattini 166.07 0.989617 61.78002 1.7242934 8.8 2008/07/13 2140.1 C/2008 L3 Hill 330 0.9939 100.201 2.0113 10.6 2008/04/22 5900 C/2008 N1 Holmes 973.1 0.997140 115.52100 2.7835117 9.9 2009/09/25 30360 C/2008 Q1 Maticic 593.6 0.995015 118.62662 2.959143 9.8 2008/12/30 14460 C/2008 Q3 Garradd 8900 0.999799 140.70663 1.7982291 6.1 2009/06/23 840000 C/2009 F1 Larson 106 0.9827 171.3755 1.8307 15.1 2009/06/25 1090 C/2009 F2 McNaught 346.1 0.98303 59.36694 5.87503 4.9 2009/11/14 6440 C/2009 F6 Yi-SWAN 512.2 0.997512 85.76481 1.274159 9.7 2009/05/07 11590 C/2009 K2 Catalina (CSS) 1460 0.997776 66.82192 3.246173 11.8 2010/02/07 55800 C/2009 O2 Catalina (CSS) 278.3 0.997501 107.96052 0.6955493 12.3 2010/03/24 4643 C/2009 T1 McNaught 3680 0.99831 89.89396 6.22041 8.5 2009/10/08 223000 C/2009 T3 LINEAR 4300 0.999470 148.74183 2.281140 13.5 2010/01/12 282000 C/2009 U3 Hill 167.88 0.991575 51.26077 1.414424 12.6 2010/03/20 2175 C/2009 U5 Grauer 10600 0.99943 25.4726 6.09424 9.1 2010/06/22 1090000 C/2009 W2 Boattini 16000 0.99956 164.49053 6.90713 6.9 2010/05/01 1900000 C/2009 Y1 Catalina (CSS) 375.4 0.993285 107.31660 2.5204945 6.5 2011/01/28 7273 C/2010 A4 Siding Spring 292.4 0.990638 96.73015 2.737999 7.4 2010/10/08 5001 C/2010 B1 Cardinal 2932 0.998997 101.97777 2.9414900 10.0 2011/02/07 158700 C/2010 D3 WISE 11600 0.99963 76.39488 4.24754 10.0 2010/09/03 1250000 C/2010 E1 Garradd 110.4 0.9759 71.698 2.66219 11.8 2009/11/07 1160 WISE-Garradd 299.3 0.990500 107.62532 2.842764 8.5 2010/11/07 5177 C/2010 G1 Boattini 480 0.9975 78.3870 1.20455 13.1 2010/04/02 10000 C/2010 G3 WISE 2630 0.99814 108.26760 4.90765 8.9 2010/04/11 135000 C/2010 H1 Garradd 8400 0.99967 36.5317 2.74555 12.4 2010/06/18 800000 C/2010 J2 McNaught 6300 0.999460 125.85156 3.386994 10.4 2010/06/03 500000 C/2010 L3 Catalina (CSS) 12800 0.99923 102.63105 9.88290 4.7 2010/11/10 1400000 C/2011 A3 Gibbs 1167 0.997992 26.07435 2.344839 9.7 2011/12/16 39900 C/2011 C1 McNaught 344.1 0.997433 16.82561 0.8833784 12.7 2011/04/18 6380 C/2011 C3 Gibbs 320 0.99527 49.3760 1.51689 14.1 2011/04/07 5700 C/2011 F1 LINEAR 2780 0.999345 56.61904 1.818266 8.3 2013/01/07 146000 C/2011 N2 McNaught 10000 0.9997 33.675 2.5634 6.3 2011/10/18 C/2011 O1 LINEAR 1210 0.996785 76.49889 3.890653 7.2 2012/08/18 42100 C/2011 Q1 PANSTARRS 3300 0.9979 94.8620 6.78009 7.5 2011/06/29 190000 C/2012 A2 LINEAR 978.2 0.996384 125.868509 3.5374738 8.4 2012/11/05 30590 C/2012 C1 McNaught 1274 0.99620 96.27770 4.837975 5.4 2013/02/04 45500 MOSS 139913.5 0.999991 27.74418 1.296092 11.1 2012/09/28 52335655.79 C/2012 E1 Hill 3760 0.99801 122.54208 7.50290 5.7 2011/07/04 231000 C/2012 E3 PANSTARRS 221 0.9827 105.658 3.8274 9.9 2011/05/12 3280 C/2012 F6 Lemmon 487.1 0.9984987 82.60885 0.7312382 5.5 2013/03/24 10750 C/2012 K5 LINEAR 774.4 0.9985256 92.848032 1.14181083 10.5 2012/11/28 21550 C/2012 K6 McNaught 4130 0.999188 135.21497 3.353033 8.8 2013/05/21 265000 C/2012 L1 LINEAR 767.3 0.997051 87.21917 2.262410 11.9 2012/12/25 21250 C/2012 L2 LINEAR 563.4 0.997322 70.98049 1.5085342 9.5 2013/05/09 13370 C/2012 L3 LINEAR 331 0.99079 134.19664 3.04503 9.0 2012/06/12 6020 Palomar 6350 0.99897 25.37958 6.53605 8.8 2015/08/16 505000 C/2012 OP Siding Spring 1054 0.99658 114.82872 3.60707 11.2 2012/12/04 34200 C/2012 S4 PANSTARRS 252223.8 0.999983 126.54131 4.34873 9.2 2013/06/28 126673944.62 C/2012 T4 McNaught 110 0.983 24.092 1.953 12.7 2012/10/10 1200 C/2012 U1 PANSTARRS 12200 0.99957 56.33902 5.26390 8.3 2014/07/04 1350000 C/2012 V1 PANSTARRS 3800 0.99945 157.8399 2.0890 11.5 2013/07/21 230000 C/2012 V2 LINEAR 616.7 0.997641 67.18470 1.4547602 8.4 2013/08/16 15320 C/2012 X1 LINEAR 156.71 0.989803 44.36218 1.597956 5.7 2014/02/21 1962 C/2013 E2 Iwamoto 233.07 0.993936 21.85771 1.413322 10.6 2013/03/09 3558 C/2013 F2 Catalina (CSS) 8400 0.99926 61.74927 6.21785 7.1 2013/04/19 770000 C/2013 F3 McNaught 759 0.99703 85.4445 2.252612 12.3 2013/05/25 20900 C/2013 G5 Catalina (CSS) 2700 0.99965 40.617 0.92894 14.5 2013/09/01 140000 C/2013 G6 Lemmon 387.1 0.994708 124.08435 2.048499 6.8 2013/07/25 7620 C/2013 G7 McNaught 2190 0.99786 105.11012 4.677404 6.2 2014/03/18 102200 C/2013 G8 PANSTARRS 3340 0.99846 27.61506 5.14118 8.4 2013/11/14 193000 C/2013 H1 La Sagra 181.5 0.98542 27.0895 2.64696 6.2 2013/05/19 2445 C/2013 J3 McNaught 1950 0.99795 118.2255 3.98869 5.8 2013/02/22 86000 C/2013 J5 Boattini 10000 0.999 136.011 4.9049 10.0 2012/11/29 C/2013 O3 McNaught 819 0.99612 102.83974 3.18010 11.1 2013/09/09 23400 C/2013 P2 PANSTARRS 2590 0.998904 125.53216 2.834925 11.8 2014/02/17 132000 C/2013 R1 Lovejoy 515.4 0.9984250 64.04094 0.81182562 11.6 2013/12/22 11702 Spacewatch 450.8 0.98707 31.40046 5.83064 6.7 2014/08/17 9570 C/2013 U2 Holvorcem 891 0.99426 43.09366 5.116745 5.3 2014/10/25 26590 C/2013 V5 Oukaimeden 488.1 0.9987183 154.88544 0.6255811 10.8 2014/09/28 10784 C/2013 Y2 PANSTARRS 219.9 0.991275 29.41474 1.919086 9.7 2014/06/13 3262 C/2014 A5 PANSTARRS 152.3 0.96848 31.9046 4.79991 11.6 2014/08/14 1879 C/2014 C3 NEOWISE 108.4 0.98283 151.7843 1.86203 12.0 2014/01/16 1129 C/2014 E2 Jacques 688 0.999035 156.392752 0.6639172 10.4 2014/07/02 18060 C/2014 F1 Hill 3600 0.9990 108.2529 3.49638 10.4 2013/10/04 210000 C/2014 F2 Tenagra 148.20 0.97089 119.06119 4.314460 5.6 2015/01/02 1804 C/2014 G1 PANSTARRS 1000 0.9943 165.6403 5.4685 6.0 2013/11/06 30000 C/2014 H1 Christensen 141 0.9849 99.936 2.1389 14.8 2014/04/15 1700 C/2014 M2 Christensen 980 0.99293 32.4062 6.9085 7.9 2014/07/18 30500 C/2014 M3 Catalina (CSS) 138 0.9824 164.90964 2.43428 12.9 2014/06/21 1630 C/2014 N2 PANSTARRS 4700 0.99954 133.0132 2.184401 12.0 2014/10/08 330000 C/2014 N3 NEOWISE 5800 0.999331 61.63825 3.882231 4.7 2015/03/13 442000 PANSTARRS 20602.48 0.99969 81.3473 6.2444 7.6 2016/12/10 2957246.18 C/2014 Q1 PANSTARRS 1129 0.999721 43.10685 0.314570 9.8 2015/07/06 38000 C/2014 Q2 Lovejoy 579.4 0.9977728 80.301302 1.2903578 7.9 2015/01/30 13946 C/2014 Q6 PANSTARRS 6883 0.999386 49.7968 4.222 6.5 2015/01/06 PANSTARRS 260 0.9913 124.818 2.2233 13.3 2014/07/09 4100 C/2014 R1 Borisov 179.4 0.992501 9.93289 1.345431 9.8 2014/11/19 2403 C/2014 R3 PANSTARRS 14434.53 0.9995 90.84 7.2756 6.3 2016/08/08 1734251.91 C/2014 R4 Gibbs 3200 0.99943 42.4116 1.81797 8.7 2014/10/21 180000 C/2014 S2 PANSTARRS 169.71 0.987622 64.67037 2.100644 5.0 2015/12/09 2210.9 C/2014 U3 Kowalski 1100 0.9976 152.9921 2.5588 12.4 2014/09/03 40000 C/2014 W2 PANSTARRS 1610 0.998341 81.998347 2.6702156 7.9 2016/03/10 64570 C/2014 W8 PANSTARRS 174.518 0.9711 42.111 5.044 10.5 2015/09/08 2305.52 PANSTARRS 902 0.99666 149.7827 3.01028 6.8 2015/04/05 27100 C/2015 C2 SWAN 471 0.99849 94.5013 0.711372 14.9 2015/03/04 10200 PANSTARRS 1484 0.999295 6.25965 1.046217 7.9 2017/05/09 57200 C/2015 F3 SWAN 232 0.99640 73.3865 0.83444 14.2 2015/03/09 3530 C/2015 F4 Jacques 116.37 0.985873 48.70495 1.6439255 11.4 2015/08/10 1255.3 C/2015 J2 PANSTARRS 246.9 0.98250 17.28183 4.32039 10.1 2015/09/08 3880 C/2015 K1 MASTER 180.6 0.98584 29.3817 2.55749 9.1 2014/10/13 2426 C/2015 K2 PANSTARRS 260 0.9944 29.110 1.45527 20.7 2015/06/08 4200 C/2015 M1 PANSTARRS 390 0.9946 57.310 2.0916 15.9 2015/05/15 8000 C/2015 M3 PANSTARRS 133.0 0.97328 65.95107 3.55241 11.5 2015/08/26 1533 C/2015 O1 PANSTARRS 651202.3 0.999994 127.211 3.7296 7.2 2018/02/19 C/2015 R3 PANSTARRS 3400 0.9985 83.6135 4.9033 5.0 2014/02/11 190000 LINEAR 1500 0.99906 11.3925 1.41314 10.8 2016/08/27 58000 C/2015 V3 PANSTARRS 822 0.99485 86.2318 4.23569 6.3 2015/11/24 23600 C/2015 WZ PANSTARRS 193.16 0.992873 134.13494 1.3766377 10.5 2016/04/15 2685 C/2015 Y1 LINEAR 292.5 0.99141 71.2196 2.514080 6.7 2016/05/15 5000 C/2016 A5 PANSTARRS 1200 0.9976 40.319 2.9469 12.8 2015/06/28 43000 C/2016 A6 PANSTARRS 217.53 0.9889 120.92 2.4124 7.8 2015/11/05 3208.44 C/2016 B1 NEOWISE 453 0.99293 50.4644 3.20625 5.9 2016/12/04 9700 C/2016 E2 Kowalski 138.88 0.992 135.95 1.074 19.5 2016/02/06 1636.74 C/2016 J2 Denneau 700 0.998 130.343 1.5184 15.3 2016/04/11 C/2016 KA Catalina (CSS) 6000 0.9990 104.6293 5.4009 8.8 2016/02/01 400000 C/2016 M1 PANSTARRS 1760 0.99875 90.99839 2.21103 8.1 2018/08/10 74000 C/2016 N4 MASTER 5315.30 0.99940 72.5573 3.19912 11.1 2017/09/16 387525 C/2016 N6 PANSTARRS 1600 0.9984 105.8345 2.6699 5.0 2018/07/18 67000 C/2016 P4 PANSTARRS 330 0.9819 29.89 5.888 10.7 2016/10/16 5900 C/2016 Q2 PANSTARRS 5467.19 0.9987 109.409 7.087 8.3 2021/05/10 404254.11 C/2016 R2 PANSTARRS 780 0.9967 58.2134 2.6020 5.1 2018/05/09 22000 C/2016 T1 Matheny 126.10 0.9818 126.095 2.3000 12.1 2017/02/01 1415.98 C/2016 T2 Matheny 101.74 0.98125 81.311 1.9078 13.8 2016/12/29 1026.30 C/2016 T3 PANSTARRS 144 0.9816 22.6727 2.6496 8.1 2017/09/06 1730 PANSTARRS 194.0 0.99531 24.0354 0.910285 18.7 2017/03/07 2700 PANSTARRS 111625.443 0.99992 32.431 9.2164 11.2 2018/02/17 37295204.74 C/2017 D2 Barros 1369.820 0.9982 31.26579 2.48587 11.1 2017/07/14 51000 C/2017 D5 PANSTARRS 112.2883 0.9806 131.03858 2.1672 14.6 2017/01/08 1200 C/2017 E4 Lovejoy 477.669 0.9989 88.1867 0.49357 15.6 2017/04/23 10000 C/2017 E5 Lemmon 388.6996 0.9954 122.6377 1.7829 12.0 2016/06/10 7600 C/2017 G3 PANSTARRS 287.3256 0.99098 159.051 2.59048 14.2 2017/04/15 4900 C/2017 K6 Jacques 1054.174 0.99810 57.2511 2.00279 10.7 2018/01/03 34000 C/2017 M3 PANSTARRS 173.8170 0.9732 77.5073 4.6561 6.2 2017/04/28 2292 C/2017 O1 ASASSN 439.1911 0.99658 39.849 1.4987 10.4 2017/10/14 9200 C/2017 P2 PANSTARRS 1210 0.997967 50.08486 2.461777 9.2 2017/12/06 42100 C/2017 T2 PANSTARRS 5007 0.99968 57.231 1.6151 10.2 2020/05/05 354300 C/2017 T3 ATLAS 1344 0.99939 88.10362 0.82522 11.1 2018/07/19 49280 C/2017 U2 Fuls 8555.39 0.99921 95.4291 6.700 8.8 2017/08/28 C/2017 Y1 PANSTARRS 3791 0.99902 55.2287 3.719 9.3 2017/08/31 234400 C/2017 Y2 PANSTARRS 2502.89 0.99841 124.67 3.957 8.0 2020/08/19 C/2018 A3 ATLAS 487.788 0.99328 139.56 3.277 9.2 2019/01/12 10773 C/2018 E2 Barros 1769.556 0.99778 97.7428 3.92 6.4 2017/12/23 74439 Lemmon 640.788 0.99757 84.694 1.55663 18.2 2018/05/23 16221.08 C/2018 F1 Grauer 322.415 0.9907 46.0706 2.993 13.7 2018/12/14 5789.36 Lemmon 833.97 0.99565 136.66655 3.627 12.2 2019/09/10 24084.39 C/2018 L2 ATLAS 246.853 0.9931 67.4235 1.712 8.1 2018/12/02 3879 C/2018 N1 NEOWISE 693.83 0.9981 159.44 1.307 15.0 2018/08/01 C/2018 R3 Lemmon 1970.10 0.99934 69.7154 1.29 11.3 2019/06/07 87446.17 C/2018 R4 Fuls 311.353 0.99451 11.68371 1.7093 11.8 2018/03/03 5494 C/2018 V4 Africano 214.059 0.98506 69.0028 3.19901 15.7 2019/03/01 3131.89 C/2018 X2 Fitzsimmons 155.971 0.9864 23.06 2.125 6.4 2019/07/08 1947.93 C/2018 Y1 Iwamoto 109.736 0.988 160.4 1.287 12.3 2019/02/07 1149.57 C/2019 B1 Africano 151.97 0.9895 123.36 1.597 14.6 2019/03/19 1873.55 C/2019 D1 Flewelling 137.6571 0.989 34.098 1.5775 11.8 2019/05/11 1615.12 C/2019 H1 NEOWISE 230.7855 0.99201 104.579 1.8448 13.5 2019/04/27 3506.07 C/2019 J2 Palomar 610.07 0.99717 105.138 1.7269 11.6 2019/07/19 ATLAS 355.31 0.99424 148.2972 2.045 15.1 2019/05/31 C/2019 K4 Ye 2370.96 0.9990 105.31 2.2594 12.8 2019/06/16 115449.93 C/2019 K5 Young 150.99 0.9865 15.315 2.035 12.3 2019/06/22 1855.41 C/2019 K8 ATLAS 1440.4914 0.998 93.222 3.195 11.3 2019/07/21 C/2019 N1 ATLAS 13156.57 0.99987 82.424 1.7047 9.0 2020/12/01 C/2019 T3 ATLAS 11484.78 0.99948 121.86 5.9468 6.6 2021/03/02 C/2019 T4 ATLAS 1007.58 0.9958 53.62 4.245 5.6 2022/06/09 31983.74 C/2019 U6 Lemmon 435.36 0.9979 61.0049 0.914 13.3 2020/06/18 9084.03 C/2019 V1 Borisov 3033.17 0.99898 61.8636 3.0968 14.5 2020/07/16 C/2019 Y1 ATLAS 231.099 0.9964 73.347 0.8378 12.4 2020/03/15 3513.22 C/2019 Y4 ATLAS 331.14 0.9992 45.380 0.253 7.9 2020/05/31 6025.89 C/2019 Y4-B ATLAS 665.948 0.99962 45.454 0.2525 15.8 2020/05/31 17185 C/2020 A2 Iwamoto 1070.02 0.9991 120.75 0.978 15.0 2020/01/08 35002.27 C/2020 A3 ATLAS 6807.33 0.9991 146.7 5.767 7.7 2019/06/29 C/2020 B3 Rankin 1919.5 0.99826 20.703 3.3446 14.5 2019/10/19 84101.96 C/2020 F3 NEOWISE 377.32 0.9992 128.937 0.295 12.3 2020/07/03 7329.46 C/2020 F6 PANSTARRS 405.3 0.99134 174.58 3.511 13.2 2020/04/11 8159.58 C/2020 F8 SWAN 6642.61 0.99994 110.80 0.430 11.6 2020/05/27 C/2020 H2 Pruyne 183.596 0.9955 125.04 0.834 19.8 2020/04/27 2487.72 C/2020 H4 Leonard 140.477 0.9933 84.320 0.9383 16.5 2020/08/29 1665.00 C/2020 H5 Robinson 2497.44 0.9963 70.204 9.3500 4.5 2020/12/05 C/2020 H7 Lemmon 1476.6 0.997 135.92 4.42 11.1 2020/06/02 56742.20 C/2020 H8 PANSTARRS 594.908 0.99214 99.65 4.6744 10.4 2020/06/04 14510 C/2020 H11 PANSTARRS Lemmon 10470 0.99927 151.41 7.631 7.4 2020/09/15 1070000 C/2020 J1 SONEAR 9376.42 0.9996 142.305 3.356 7.2 2021/04/18 C/2020 K1 PANSTARRS 3141.03 0.99902 89.646 3.078 5.6 2023/05/09 C/2020 K2 PANSTARRS 8380.85 0.99894 91.0288 8.8762 6.1 2020/08/05 C/2020 K3 Leonard 210.450 0.9924 128.72 1.593 14.8 2020/05/30 3053.03 C/2020 K6 Rankin 2876.55 0.998 103.619 5.8844 8.1 2021/09/11 C/2020 K7 PANSTARRS 108.4 0.9411 32.059 6.3847 7.9 2019/10/30 1128.70 C/2020 M5 ATLAS 4936.2 0.9994 93.223 3.005 6.9 2021/08/19 346814.64 C/2020 N2 ATLAS 108.74 0.9835 161.034 1.746 15.6 2020/08/23 1134.00 C/2020 P3 ATLAS 4910.32 0.9986 61.89 6.812 6.7 2021/04/20 C/2020 R2 ATLAS 398.332 0.9882 53.22 4.693 7.1 2022/02/24 7950.15 C/2020 R6 Rankin 451.023 0.9931 82.83 3.129 7.4 2019/09/10 C/2020 R7 ATLAS 6397.95 0.99953 114.893 2.957 10.7 2022/09/16 C/2020 S3 Erasmus 187.987 0.99788 19.861 0.3985 13.0 2020/12/12 2577.50 C/2020 S4 PANSTARRS 4962.260 0.99932 20.5750 3.3673 7.4 2023/02/09 C/2020 S8 PANSTARRS 271.2715 0.99129 108.517 2.3639 8.1 2021/04/10 4468.01 C/2020 T2 Palomar 323.34 0.99364 27.873 2.055 8.8 2021/07/11 5814.24 C/2020 T5 Lemmon 930.79 0.99797 66.604 1.889 16.2 2020/10/09 28398.06 C/2020 U5 Lemmon 79736.84 0.99995 97.280 3.756 9.7 2022/04/27 C/2020 Y2 ATLAS 1217.888 0.99743 101.281 3.132 6.4 2022/06/17 42502.95 C/2020 Y3 ATLAS 151.18 0.98678 83.097 1.999 14.6 2020/12/03 1858.91 C/2021 A2 NEOWISE 257.26 0.9945 106.978 1.413 14.7 2021/01/22 4126.29 C/2021 A6 PANSTARRS 11846.50 0.99933 75.605 7.929 7.1 2021/05/05 C/2021 A7 NEOWISE 5448.01 0.99964 78.149 1.968 13.5 2021/07/15 C/2021 B2 PANSTARRS 336.015 0.99252 38.094 2.513 4.8 2021/07/15 6159.5 C/2021 C1 Rankin 8030.77 0.99957 143.04 3.481 8.8 2020/12/07 C/2021 C4 ATLAS 4645.31 0.99903 132.84 4.504 6.9 2021/01/17 C/2021 C5 ATLAS 14042.0 0.99977 50.787 3.241 12.0 2023/02/10 C/2021 G2 ATLAS 4011.4 0.99876 48.478 4.976 5.7 2024/09/10 C/2021 N3 PANSTARRS 158.23 0.9640 26.74 5.701 7.1 2020/08/17 1990.4 C/2021 P2 PANSTARRS 2211.11 0.9977 150.02 5.072 5.4 2023/01/21 C/2021 P4 ATLAS 305.44 0.9965 56.31 1.080 8.7 2022/07/30 5338.32 C/2021 Q6 PANSTARRS 10.932 0.9992 161.85 8.716 6.9 2024/03/21 C/2021 R2 PANSTARRS 2265.45 0.9968 134.46 7.312 7.7 2021/12/25 C/2021 R7 PANSTARRS 989.93 0.9943 158.85 5.640 7.5 2021/04/14 C/2021 S3 PANSTARRS 3317.47 0.9996 58.55 1.318 6.8 2024/02/14 C/2021 S4 Tsuchinshan (*CTC) 161.91 0.9583 17.478 6.694 7.0 2023/12/31 2035.2 C/2021 T1 Lemmon 1463.83 0.9979 140.35 3.058 5.7 2021/10/14 56007.2 C/2021 T4 Lemmon 43170 0.9999 160.757 1.482 6.9 2023/07/31 8970000 C/2021 U5 Catalina 216.86 0.9891 39.05 2.363 6.9 2022/01/26 3193.50 C/2021 V1 Rankin 679.40 0.9956 71.441 3.014 15.9 2022/04/30 17709.00 C/2022 A1 Sarneczky 411.49 0.9970 116.51 1.253 19.2 2022/01/31 8347 C/2022 A3 Lemmon - ATLAS 1006.28 0.9963 88.360 3.703 5.3 2023/09/28 31921.61 C/2022 B4 382.80 0.9964 20.043 1.380 21.7 2022/01/29 C/2022 D2 Kowalski 356.11 0.9956 22.655 1.555 14.1 2022/03/27 6720 C/2022 H1 Kowalski 1158.59 0.9934 49.870 7.693 6.3 2024/01/18 C/2022 L1 Catalina 528.47 0.9970 123.468 1.591 13.3 2022/09/28 12150 C/2022 L4 PANSTARRS 254.18 0.9881 141.224 3.015 16.6 2021/12/08 C/2022 P3 ZTF 243.88 0.9894 59.519 2.561 14.5 2022/07/27 3808 C/2022 R2 ATLAS 422.92 0.9985 52.895 0.633 16.6 2022/10/25 8697 C/2022 T1 Lemmon 4655 0.9993 22.543 3.444 5.1 2024/02/17 318000 C/2022 U1 Leonard 5401 0.9992 128.126 4.203 6.7 2025/03/25 C/2022 U4 Bok 3727 0.9992 52.038 2.898 9.7 2023/08/03 227000 C/2022 W2 ATLAS 332.03 0.9906 63.533 3.123 14.1 2023/03/08 6050 C/2022 W3 Leonard 280.85 0.9950 103.560 1.398 14.0 2023/06/22 4707 C/2023 A1 Leonard 232.82 0.9921 94.744 1.835 7.8 2023/03/18 3552 C/2023 A2 SWAN 987.3 0.9990 94.708 0.948 12.5 2023/01/20 31000 C/2023 B2 ATLAS 589.595 0.9970 40.771 1.743 8.7 2023/03/10 14316 C/2023 C2 ATLAS 4219 0.9994 48.319 2.368 7.1 2024/11/16 27400 C/2023 F1 PanSTARRS 221.9 0.9923 131.744 1.708 7.8 2023/06/08 3304 C/2023 H1 PanSTARRS 1611 0.9972 21.777 4.44 13.2 2024/11/28 C/2023 H2 Lemmon 246.6 0.9963 113.75 0.894 10.0 2023/10/29 3872 Comet designation Comet name /discoverer Semimajor axis (AU) Eccentricity Inclination (°) Perihelion distance (AU) Absolute magnitude (H/M1/M2) Perihelion date Period (3) (years) Ref == See also == * List of comets by type * List of Halley-type comets * List of hyperbolic comets * List of long-period comets * List of numbered comets * List of periodic comets near parabolic Its orbit has an eccentricity of 0.01 and an inclination of 6° with respect to the ecliptic. Often, these comets, due to their extreme semimajor axes and eccentricity, will have small orbital interactions with planets and minor planets, most often ending up with the comets fluctuating significantly in their orbital path. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. Observations taken in January and February 2002 showed that the ""asteroid"" had developed a small amount of cometary activity as it approached perihelion. Damocloids have been studied as possible extinct cometary candidates due to the similarity of their orbital parameters with those of Halley-family comets. ==See also== * List of Halley-type comets == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris Category:Halley- type comets Category:Near-Earth comets Category:Damocloids 20010728 This comet probably represents the transition between typical Halley-family/long-period comets and extinct comets. The body's observation arc begins with its official discovery observation in March 1931. == Physical characteristics == Halleria is an assumed carbonaceous C-type asteroid, which agrees with the overall spectral type for members of the Charis family. === Rotation period === Between 2005 and 2011, three rotational lightcurves of Halleria were obtained from photometric observations by Donald Pray, René Roy, and Pierre Antonini (). ",0.0526315789,210,"""2.3613""",14.34457,8.8,E
+"Next, we treat projectile motion in two dimensions, first without considering air resistance. Let the muzzle velocity of the projectile be $v_0$ and the angle of elevation be $\theta$ (Figure 2-7). Calculate the projectile's range.","The Range is maximum when angle \theta = 45°, i.e. \sin 2\theta=1. ==See also== * Atlatl * Ballistics * Gunpowder * Bullet * Impact depth * Kinetic bombardment * Shell (projectile) * Projectile point * Projectile use by animals * Arrow * Dart * Missile * Sling ammunition * Spear * Torpedo * Range of a projectile * Space debris * Trajectory of a projectile ==Notes== ==References== * ==External links== * Open Source Physics computer model * Projectile Motion Applet * Another projectile Motion Applet Category:Ammunition Category:Ballistics Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. There are various calculations for projectiles at a specific angle \theta: 1\. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Mathematically, it is given as t=U \sin\theta/g where g = acceleration due to gravity (app 9.81 m/s²), U = initial velocity (m/s) and \theta = angle made by the projectile with the horizontal axis. 2\. Range (R): The Range of a projectile is the horizontal distance covered (on the x-axis) by the projectile. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The second solution is the useful one for determining the range of the projectile. A projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. In physics, a projectile launched with specific initial conditions will have a range. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. A range table was a list of angles of elevation a particular artillery gun barrel needed to be set to, to strike a target at a particular distance with a projectile of a particular weight using a propellant cartridge of a particular weight. The laser rangefinder and computer-based FCS make guns highly accurate. ==Superelevation== When firing a missile such as a MANPADS at an aircraft target, superelevation is an additional angle of elevation above the angle sighted on which corrects for the effect of gravity on the missile. ==See also== *Altitude (astronomy) *Pitching moment ==References== * Gunnery Instructions, U.S. Navy (1913), Register No. 4090 * Gunnery And Explosives For Artillery Officers (1911) * Fire Control Fundamentals, NAVPERS 91900 (1953), Part C: The Projectile in Flight - Exterior Ballistics * FM 6-40, Tactics, Techniques, and Procedures for Field Artillery Manual Cannon Gunnery (23 April 1996), Chapter 3 - Ballistics; Marine Corps Warfighting Publication No. 3-1.6.19 * FM 23-91, Mortar Gunnery (1 March 2000), Chapter 2 Fundamentals of Mortar Gunnery * Fundamentals of Naval Weapons Systems: Chapter 19 (Weapons and Systems Engineering Department United States Naval Academy) * Naval Ordnance and Gunnery (Vol.1 - Naval Ordnance) NAVPERS 10797-A (1957) * Naval Ordnance and Gunnery (Vol.2 - Fire Control) NAVPERS 10798-A (1957) * Naval Ordnance and Gunnery Category:Ballistics Category:Angle Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. thumb|A range of ""diabolo"" pellets with various nose profiles A pellet is a non-spherical projectile designed to be shot from an air gun, and an airgun that shoots such pellets is commonly known as a pellet gun. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. ",-3.5,15.425,"""72.0""",5040,556,C
+Calculate the time needed for a spacecraft to make a Hohmann transfer from Earth to Mars,"* Ls 263 (Sol 505): Earth is closest to Mars (Sep 10, 1956). This is close to the modern value of 1/154 (many sources will cite somewhat different values, such as 1/193, because even a difference of only a couple of kilometers in the values of Mars' polar and equatorial radii gives a considerably different result). Solar time is a calculation of the passage of time based on the position of the Sun in the sky. Mars Year 1 is the first year of Martian timekeeping standard developed by Clancy et al. originally for the purposes of working with the cyclical temporal variations of meteorological phenomena of Mars, but later used for general timekeeping on Mars. They occur every 26, 79 and 100 years, and every 1,000 years or so there is an extra 53rd-year transit. ==Conjunctions== Transits of Earth from Mars usually occur in pairs, with one following the other after 79 years; rarely, there are three in the series. Start and End dates of Mars Years were determined for 1607-2141 by Piqueux et al. Earth and Mars dates can be converted in the Mars Climate Database, however, the Mars Years are only rational to apply to events that take place on Mars. The Observatory, 3 (1880), 471 * * SOLEX ==External links== * Transits of Earth on Mars – Fifteen millennium catalog: 5 000 BC – 10 000 AD * JPL HORIZONS System * Near miss of the Earth-moon system (2005-11-07) Earth from Mars Category:Earth Category:Mars However, Mars Year sols may be confused with rover mission times that are also expressed in sols. This short story was first published in the January 1971 issue of Playboy magazine.'Transit Of Earth' by Arthur C. Clarke read by himself, 16 October 2017. ==Dates of transits== Transits of Earth from Mars (grouped by series) November 10, 1595 May 5, 1621 May 8, 1700 November 9, 1800 November 12, 1879 May 8, 1905 May 11, 1984 November 10, 2084 November 15, 2163 May 10, 2189 May 13, 2268 November 13, 2368 May 10, 2394 November 17, 2447 May 13, 2473 May 16, 2552 November 15, 2652 May 13, 2678 ==Grazing and simultaneous transits== Sometimes Earth only grazes the Sun during a transit. A specific time within a day, always using UTC, is specified via a decimal fraction. ==References== ==External links== * Category:Types of year Category:Time in astronomy Year thumb|Transfer orbit from Earth to Mars. The last series ending was in 1211. ==View from Mars== No one has ever seen a transit of Earth from Mars, but the next transit will take place on November 10, 2084. Scientists generally use two sub-units of the Mars Year: * the Solar Longitude (Ls) system: 360 degrees per Mars Year that represent the position of Mars in its orbit around the Sun, or * the Sol system: 668 sols per Mars Year. In astronomy, a Julian year (symbol: a or aj) is a unit of measurement of time defined as exactly 365.25 days of SI seconds each.P. Kenneth Seidelmann, ed., The equivalent on Mars is termed Mars local true solar time (LTST). When the Sun has covered exactly 15 degrees (1/24 of a circle, both angles being measured in a plane perpendicular to Earth's axis), local apparent time is 13:00 exactly; after 15 more degrees it will be 14:00 exactly. * August 25, 2005: at 15:19:32 UTC, MRO was 100 million kilometers from Mars. Mean solar time is the hour angle of the mean Sun plus 12 hours. * January 29, 2006: at 06:59:24 UTC, MRO was 10 million kilometers from Mars. Date Duration in mean solar time February 11 24 hours March 26 24 hours − 18.1 seconds May 14 24 hours June 19 24 hours + 13.1 seconds July 25/26 24 hours September 16 24 hours − 21.3 seconds November 2/3 24 hours December 22 24 hours + 29.9 seconds These lengths will change slightly in a few years and significantly in thousands of years. ==Mean solar time== thumb|right|250px|The equation of time—above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow. For example, in the next 1000 years, seven days will be dropped from the Gregorian calendar but not from 1000 Julian years, so J3000.0 will be . == Julian calendar distinguished == The Julian year, being a uniform measure of duration, should not be confused with the variable length historical years in the Julian calendar. Also, better measurements have been made by using artificial satellites that have been put into orbit around Mars, including Mariner 9, Viking 1, Viking 2, and Soviet orbiters, and the more recent orbiters that have been sent from the Earth to Mars. ==In science fiction== A science fiction short story published in 1971 by Arthur C. Clarke, called ""Transit of Earth"", depicts a doomed astronaut on Mars observing the transit in 1984. ",0.2553,30,"""7.136""",0.000216,2.24,E
+Calculate the maximum height change in the ocean tides caused by the Moon.,"File:High tide sun moon same side beginning.png|Spring tide: Sun and Moon on the same side (0°) File:Low tide sun moon 90 degrees.png|Neap tide: Sun and Moon at 90° File:High tide sun moon opposite side.png|Spring tide: Sun and Moon at opposite sides (180°) File:Low tide sun moon 270 degrees.png|Neap tide: Sun and Moon at 270° File:High tide sun moon same side end.png|Spring tide: Sun and Moon at the same side (cycle restarts) === Lunar distance === The changing distance separating the Moon and Earth also affects tide heights. The two high tides a day alternate in maximum heights: lower high (just under three feet), higher high (just over three feet), and again lower high. To calculate the actual water depth, add the charted depth to the published tide height. The alternation in high tide heights becomes smaller, until they are the same (at the lunar equinox, the Moon is above the Equator), then redevelop but with the other polarity, waxing to a maximum difference and then waning again. === Current === The tides' influence on current or flow is much more difficult to analyze, and data is much more difficult to collect. The ocean bathymetry greatly influences the tide's exact time and height at a particular coastal point. Assuming (as a crude approximation) that the deceleration rate has been constant, this would imply that 70 million years ago, day length was on the order of 1% shorter with about 4 more days per year. === Bathymetry === The shape of the shoreline and the ocean floor changes the way that tides propagate, so there is no simple, general rule that predicts the time of high water from the Moon's position in the sky. At each place of interest, the tide heights are therefore measured for a period of time sufficiently long (usually more than a year in the case of a new port not previously studied) to enable the response at each significant tide-generating frequency to be distinguished by analysis, and to extract the tidal constants for a sufficient number of the strongest known components of the astronomical tidal forces to enable practical tide prediction. The daily inequality is not consistent and is generally small when the Moon is over the Equator. === Reference levels === The following reference tide levels can be defined, from the highest level to the lowest: * Highest astronomical tide (HAT) – The highest tide which can be predicted to occur. Furthermore, the coriolis and molecular mixing terms are omitted in the equations above since they are relatively small at the temporal and spatial scale of tides in shallow waters. The graph of Cook Strait's tides shows separately the high water and low water height and time, through November 2007; these are not measured values but instead are calculated from tidal parameters derived from years-old measurements. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the Moon and ±5% for the Sun. In (The Reckoning of Time) of 725 Bede linked semidurnal tides and the phenomenon of varying tidal heights to the Moon and its phases. Using a simple harmonic fitting algorithm with a moving time window of 25 hours, the water level amplitude of different tidal constituents can be found. A tidal height is a scalar quantity and varies smoothly over a wide region. Bede then observes that the height of tides varies over the month. Tides are the rise and fall of sea levels caused by gravitational forces exerted by the Moon and Sun and by Earth's rotation. He goes on to emphasise that in two lunar months (59 days) the Moon circles the Earth 57 times and there are 114 tides. They are however only predictions, the actual time and height of the tide is affected by wind and atmospheric pressure. The difference between the height of a tide at perigean spring tide and the spring tide when the moon is at apogee depends on location but can be large as a foot higher. === Other constituents === These include solar gravitational effects, the obliquity (tilt) of the Earth's Equator and rotational axis, the inclination of the plane of the lunar orbit and the elliptical shape of the Earth's orbit of the Sun. Later the daily tides were explained more precisely by the interaction of the Moon's and the Sun's gravity. The two high waters on a given day are typically not the same height (the daily inequality); these are the higher high water and the lower high water in tide tables. 300px|thumb Tidal range is the difference in height between high tide and low tide. ",5,1.41,"""0.54""",1855,0.241,C
+A particle of mass $m$ starts at rest on top of a smooth fixed hemisphere of radius $a$. Determine the angle at which the particle leaves the hemisphere.,"thumb|upright=1.5|Spherical pendulum: angles and velocities. thumb|150px|right|Equatorial Inertial wave pulse caused patterns of fluid flow inside a steadily-rotating spherical chamber. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. Trajectory of the mass on the sphere can be obtained from the expression for the total energy :E=\underbrace{\left[\frac{1}{2}ml^2\dot\theta^2 + \frac{1}{2}ml^2\sin^2\theta \dot \phi^2\right]}_{T}+\underbrace{ \bigg[-mgl\cos\theta\bigg]}_{V} by noting that the horizontal component of angular momentum L_z = ml^2\sin^2\\!\theta \,\dot\phi is a constant of motion, independent of time. Therefore, angle AOV measures 180° − θ. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. ==Lagrangian mechanics== Routinely, in order to write down the kinetic T=\tfrac{1}{2}mv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. thumb|226px|An epispiral with equation r(θ)=2sec(2θ) The epispiral is a plane curve with polar equation :\ r=a \sec{n\theta}. The angle \theta lies between two circles of latitude, where :E>\frac{1}{2}\frac{L_z^2}{ml^2\sin^2\theta}-mgl\cos\theta. ==See also== *Foucault pendulum *Conical pendulum *Newton's three laws of motion *Pendulum *Pendulum (mathematics) *Routhian mechanics ==References== ==Further reading== * * * * * * * * Category:Pendulums That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. Arrows on this cross section show the direction and strength of flow in the equatorial plane as the sphere continues to rotate clockwise on its axis which shown at left . thumb|250px|right|The Western Hemisphere The Western Hemisphere is the half of the planet Earth that lies west of the Prime Meridian (which crosses Greenwich, London, England) and east of the 180th meridian. Angle BOA is a central angle; call it θ. Its portion lying east of the 180th meridian is the only part of the country lying in the Western Hemisphere. Therefore, : 2 \psi + 180^\circ - \theta = 180^\circ. Subtract : (180^\circ - \theta) from both sides, : 2 \psi = \theta, where θ is the central angle subtending arc AB and ψ is the inscribed angle subtending arc AB. ====Inscribed angles with the center of the circle in their interior==== thumb|Case: Center interior to angle Given a circle whose center is point O, choose three points V, C, and D on the circle. Angle DOC is a central angle, but so are angles DOE and EOC, and : \angle DOC = \angle DOE + \angle EOC. The angle θ does not change as its vertex is moved around on the circle. The last equation shows that angular momentum around the vertical axis, |\mathbf L_z| = l\sin\theta \times ml\sin\theta\,\dot\phi is conserved. Angle DOC is a central angle, but so are angles EOD and EOC, and : \angle DOC = \angle EOC - \angle EOD. In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Similarly, the Euler–Lagrange equation involving the azimuth \phi, : \frac{d}{dt}\frac{\partial}{\partial\dot\phi}L-\frac{\partial}{\partial\phi}L=0 gives : \frac{d}{dt} \left( ml^2\sin^2\theta \cdot \dot{\phi} \right) =0 . This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Combining these results with equation (4) yields : \theta_0 = 2 \psi_2 - 2 \psi_1 therefore, by equation (3), : \theta_0 = 2 \psi_0. thumb|400px|Animated gif of proof of the inscribed angle theorem. ",6.283185307,12,"""14.44""",48.189685,35.2,D
+"Consider the first stage of a Saturn $V$ rocket used for the Apollo moon program. The initial mass is $2.8 \times 10^6 \mathrm{~kg}$, and the mass of the first-stage fuel is $2.1 \times 10^6$ kg. Assume a mean thrust of $37 \times 10^6 \mathrm{~N}$. The exhaust velocity is $2600 \mathrm{~m} / \mathrm{s}$. Calculate the final speed of the first stage at burnout. ","The S-I was the first stage of the Saturn I rocket used by NASA for the Apollo program. == Design == The S-I stage was powered by eight H-1 rocket engines burning RP-1 fuel with liquid oxygen (LOX) as oxidizer. Studied with the Saturn A-1 in 1959, the Saturn A-2 was deemed more powerful than the Saturn I rocket, consisting of a first stage, which actually flew on the Saturn IB, a second stage which contains four S-3 engines that flew on the Jupiter IRBM and a Centaur high-energy liquid-fueled third stage. == References == * Koelle, Heinz Hermann, Handbook of Astronautical Engineering, McGraw-Hill, New York, 1961. {{Infobox rocket |image = |imsize = |caption = |function = Launch vehicle for Project Horizon and Apollo |manufacturer = |country-origin = United States |height = (w/o payload) |diameter = |mass = gross (to LEO) |stages = |capacities = |family = Saturn |status = Study, not developed |sites = Kennedy Space Center |payloads = |stagedata = }} The Saturn C-2 was the second rocket in the Saturn C series studied from 1959 to 1962. Studied in 1959, the Saturn B-1, was a four-stage concept rocket similar to the Jupiter-C, and consisted of a Saturn IB first stage, a cluster of four Titan I first stages used for a second stage, a S-IV third stage and a Centaur high-energy liquid-fueled fourth stage. *Free return trajectory simulation, Robert A. Braeunig, August 2008 *Encyclopedia Astronautica Saturn C-2 C2 Category:Cancelled space launch vehicles It formed the second stage of the Saturn I and was powered by a cluster of six RL-10A-3 engines. The Army's original design used the S-III stage with two J-2 engines as the second stage; after the Saturn program was transferred to NASA, the second stage was replaced with an S-II second stage using four J-2 engines. The S-IV was the second stage of the Saturn I rocket used by NASA for early flights in the Apollo program. The Saturn C-8 was the largest member of the Saturn series of rockets to be designed. This saved up to 20% of structural weight. ==References== * * Category:Apollo program Category:Rocket stages The S-IV stage was a large LOX/LH2-fueled rocket stage used for the early test flights of the Saturn I rocket. The initial launch of the Saturn I consisted of an active S-I, an inactive S-IV and inactive S-V stage. Further development of the C-2 vehicle was cancelled on 23 June 1961. ==Launch vehicle design== The original Saturn C-2 design (1959-1960) was a four-stage launch vehicle, using an S-I first stage using eight Rocketdyne H-1 engines, later flown on the Saturn I. The design was for a four-stage launch vehicle that could launch 21,500 kg (47,300 lb) to low Earth orbit and send 6,800 kg (14,900 lb) to the Moon via Trans-Lunar Injection. The S-III stage would have been added atop the S-II, to convert the C-2 into the five-stage Saturn C-3. Later, a fifth J-2 engine was added to the S-II stage to be used on the Saturn C-5, which eventually was developed as the Saturn V launch vehicle. During a discussion on the Saturn program, several major problems were brought up: * The adequacy of the Saturn C-1 launch vehicle for the orbital qualification of the complete Apollo spacecraft was in question. Excellent account of the evolution, design, and development of the Saturn launch vehicles. Excellent account of the evolution, design, and development of the Saturn launch vehicles. Excellent account of the evolution, design, and development of the Saturn launch vehicles. While this S-V/Centaur stage would never fly on any Saturn rockets, it would be used on Atlas and Titan launch vehicles. The Saturn C-8 configuration was never taken further than the design process, as it was too large and costly. ==References== *Bilstein, Roger E, Stages to Saturn, US Government Printing Office, 1980. . ",3.8,2.16,"""35.0""",3930,49,B
+How large should we take $n$ in order to guarantee that the Trapezoidal and Midpoint Rule approximations for $\int_1^2(1 / x) d x$ are accurate to within 0.0001 ?,"The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. Therefore the total error is bounded by \text{error} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right] = \frac{f(\xi)h^3N}{12}=\frac{f(\xi)(b-a)^3}{12N^2}. === Periodic and peak functions === The trapezoidal rule converges rapidly for periodic functions. As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule. thumb|Illustration of ""chained trapezoidal rule"" used on an irregularly-spaced partition of [a,b]. == History == A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic. == Numerical implementation == === Non-uniform grid === When the grid spacing is non-uniform, one can use the formula \int_{a}^{b} f(x)\, dx \approx \sum_{k=1}^N \frac{f(x_{k-1}) + f(x_k)}{2} \Delta x_k , wherein \Delta x_k = x_{k} - x_{k-1} . === Uniform grid === For a domain discretized into N equally spaced panels, considerable simplification may occur. It follows that \int_{a}^{b} f(x) \, dx \approx (b-a) \cdot \tfrac{1}{2}(f(a)+f(b)). thumb|right|An animation that shows what the trapezoidal rule is and how the error in approximation decreases as the step size decreases The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums, and is sometimes defined this way. The error in approximating an integral by Simpson's rule for n = 2 is -\frac{1}{90} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{2880} f^{(4)}(\xi), where \xi (the Greek letter xi) is some number between a and b. The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result: \text{E} = \int_a^b f(x)\,dx - \frac{b-a}{N} \left[ {f(a) + f(b) \over 2} + \sum_{k=1}^{N-1} f \left( a+k \frac{b-a}{N} \right) \right] There exists a number ξ between a and b, such that \text{E} = -\frac{(b-a)^3}{12N^2} f(\xi) It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. Interestingly, in this case the trapezoidal rule often has sharper bounds than Simpson's rule for the same number of function evaluations. == Applicability and alternatives == The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points. Namely, composite Simpson's 1/3 rule requires 1.8 times more points to achieve the same accuracy as trapezoidal rule. It follows from the above formulas for the errors of the midpoint and trapezoidal rule that the leading error term vanishes if we take the weighted average \frac{2M + T}{3}. The trapezoidal rule states that the integral on the right- hand side can be approximated as \int_{t_n}^{t_{n+1}} f(t,y(t)) \,\mathrm{d}t \approx \tfrac12 h \Big( f(t_n,y(t_n)) + f(t_{n+1},y(t_{n+1})) \Big). Let \Delta x_k = \Delta x = \frac{b-a}{N} the approximation to the integral becomes \begin{align} \int_{a}^{b} f(x)\, dx &\approx \frac{\Delta x}{2} \sum_{k=1}^{N} \left( f(x_{k-1}) + f(x_{k}) \right) \\\\[1ex] &= \frac{\Delta x}{2} \Biggl( f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + \dotsb + 2f(x_{N-1}) + f(x_N) \Biggr) \\\\[1ex] &= \Delta x \left( \sum_{k=1}^{N-1} f(x_k) + \frac{f(x_N) + f(x_0) }{2} \right). \end{align} ==Error analysis== right|thumb|An animation showing how the trapezoidal rule approximation improves with more strips for an interval with a=2 and b=8. Several techniques can be used to analyze the error, including: #Fourier series #Residue calculus #Euler–Maclaurin summation formula #Polynomial interpolation It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions. === Proof === First suppose that h=\frac{b-a}{N} and a_k=a+(k-1)h. Note since it starts and ends at zero, this approximation yields zero area. alt=Two-piece approximation|thumb|Two-piece alt=Four-piece approximation|thumb|Four-piece alt=Eight-piece approximation|thumb|Eight-piece After trapezoid rule estimates are obtained, Richardson extrapolation is applied. Number of pieces Trapezoid estimates First iteration Second iteration Third iteration (4 MA − LA)/3* (16 MA − LA)/15 (64 MA − LA)/63 1 0 (4×16 − 0)/3 = 21.333... (16×34.667 − 21.333)/15 = 35.556... (64×42.489 − 35.556)/63 = 42.599... 2 16 (4×30 − 16)/3 = 34.666... (16×42 − 34.667)/15 = 42.489... 4 30 (4×39 − 30)/3 = 42 8 39 *MA stands for more accurate, LA stands for less accurate == Example == As an example, the Gaussian function is integrated from 0 to 1, i.e. the error function erf(1) ≈ 0.842700792949715. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. The approximation becomes more accurate as the resolution of the partition increases (that is, for larger N, all \Delta x_k decrease). Using another approximation (for example, the trapezoidal rule with twice as many points), it is possible to take a suitable weighted average and eliminate another error term. Simpson's 1/3 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{1}{3} h\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\\\ &= \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right], \end{align} where h = (b - a)/2 is the step size. (This derivation is essentially a less rigorous version of the quadratic interpolation derivation, where one saves significant calculation effort by guessing the correct functional form.) === Composite Simpson's 1/3 rule === If the interval of integration [a, b] is in some sense ""small"", then Simpson's rule with n = 2 subintervals will provide an adequate approximation to the exact integral. The result is then obtained by taking the mean of the two formulas. === Simpson's rules in the case of narrow peaks === In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule. ",6,41,"""-1.0""",2.3,2.3,B
+Find the length of the cardioid $r=1+\sin \theta$.,"The foot of the perpendicular from point O on the tangent is point (r\cos \varphi, r\sin \varphi) with the still unknown distance r to the origin O. Inserting the point into the equation of the tangent yields (r\cos\varphi - 2a)\cos\varphi + r\sin^2\varphi = 2a \quad \rightarrow \quad r = 2a(1 + \cos \varphi) which is the polar equation of a cardioid. For the cardioid r(\varphi) = 2a (1 - \cos\varphi) = 4a \sin^2\left(\tfrac{\varphi}{2}\right) one gets \rho(\varphi) = \cdots = \frac{\left[16a^2\sin^2\frac{\varphi}{2}\right]^\frac{3}{2}} {24a^2 \sin^2\frac{\varphi}{2}} = \frac{8}{3}a\sin\frac{\varphi}{2} \ . }} == Properties == thumb|Chords of a cardioid === Chords through the cusp === ; C1: Chords through the cusp of the cardioid have the same length 4a. Hence the cardioid has the polar representation r(\varphi) = 1 - \cos\varphi and its inverse curve r(\varphi) = \frac{1}{1 - \cos\varphi}, which is a parabola (s. parabola in polar coordinates) with the equation x = \tfrac{1}{2}\left(y^2 - 1\right) in Cartesian coordinates. Their intersection point is x(t) = 2(1 + \cos t)\cos t,\quad y(t) = 2(1 + \cos t)\sin t, which is a point of the cardioid with polar equation r = 2(1 + \cos t). thumb|Cardioid as caustic: light source Z, light ray \vec s, reflected ray \vec r thumb|Cardioid as caustic of a circle with light source (right) on the perimeter === Cardioid as caustic of a circle === The considerations made in the previous section give a proof that the caustic of a circle with light source on the perimeter of the circle is a cardioid. In order to keep the calculations simple, the proof is given for the cardioid with polar representation r = 2(1 \mathbin{\color{red}+} \cos\varphi) (§ Cardioids in different positions). ===== Equation of the tangent of the cardioid with polar representation r = 2(1 + \cos\varphi) ===== From the parametric representation \begin{align} x(\varphi) &= 2(1 + \cos\varphi) \cos \varphi, \\\ y(\varphi) &= 2(1 + \cos\varphi) \sin \varphi \end{align} one gets the normal vector \vec n = \left(\dot y , -\dot x\right)^\mathsf{T}. For the cardioids with the equations r=2a(1-\cos\varphi) \; and r = 2b(1 + \cos\varphi)\ respectively one gets: \frac{dy_a}{dx} = \frac{\cos(\varphi) - \cos(2\varphi)}{\sin(2\varphi) - \sin(\varphi)} and \frac{dy_b}{dx} = -\frac{\cos(\varphi) + \cos(2\varphi)}{\sin(2\varphi) + \sin(\varphi)}\ . These equations describe a cardioid a third as large, rotated 180 degrees and shifted along the x-axis by -\tfrac{4}{3} a. The catacaustic of a circle with respect to a point on the circumference is a cardioid. thumb|upright=1.25|r=\frac{\sin \theta}{\theta}, -20<\theta<20 thumb|upright=1.25|cochleoid (solid) and its polar inverse (dashed) In geometry, a cochleoid is a snail-shaped curve similar to a strophoid which can be represented by the polar equation :r=\frac{a \sin \theta}{\theta}, the Cartesian equation :(x^2+y^2)\arctan\frac{y}{x}=ay, or the parametric equations :x=\frac{a\sin t\cos t}{t}, \quad y=\frac{a\sin^2 t}{t}. For suitable formulas see polar coordinate system (arc length) and polar coordinate system (area) \; d\varphi = \cdots = 8a\int_0^\pi\sqrt{\tfrac{1}{2}(1 - \cos\varphi)}\; d\varphi = 8a\int_0^\pi\sin\left(\tfrac{\varphi}{2}\right) d\varphi = 16a. }} {r(\varphi)^2 + 2 \dot r(\varphi)^2 - r(\varphi) \ddot r(\varphi)} \ . thumb|A cardioid In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. For a cardioid one gets: : The evolute of a cardioid is another cardioid, one third as large, and facing the opposite direction (s. picture). === Proof === For the cardioid with parametric representation x(\varphi) = 2a (1 - \cos\varphi)\cos\varphi = 4a \sin^2\tfrac{\varphi}{2}\cos\varphi\, , y(\varphi) = 2a (1 - \cos\varphi)\sin\varphi = 4a \sin^2\tfrac{\varphi}{2}\sin\varphi the unit normal is \vec n(\varphi) = (-\sin\tfrac{3}{2}\varphi, \cos\tfrac{3}{2}\varphi) and the radius of curvature \rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . From here one gets the parametric representation above: \begin{array}{cclcccc} x(\varphi) &=& a\;(-\cos(2\varphi) + 2\cos\varphi - 1) &=& 2a(1 - \cos\varphi)\cdot\cos\varphi & & \\\ y(\varphi) &=& a\;(-\sin(2\varphi) + 2\sin\varphi) &=& 2a(1 - \cos\varphi)\cdot\sin\varphi &.& \end{array} (The trigonometric identities e^{i\varphi} = \cos\varphi + i\sin\varphi, \ (\cos\varphi)^2 + (\sin\varphi)^2 = 1, \cos(2\varphi) = (\cos\varphi)^2 - (\sin\varphi)^2, and \sin (2\varphi) = 2\sin\varphi\cos\varphi were used.) == Metric properties == For the cardioid as defined above the following formulas hold: * area A = 6\pi a^2, * arc length L = 16 a and * radius of curvature \rho(\varphi) = \tfrac{8}{3}a\sin\tfrac{\varphi}{2} \, . The reflected ray is part of the line with equation (see previous section) \cos\left(\tfrac{3}{2}\varphi\right) x + \sin \left(\tfrac{3}{2}\varphi\right) y = 4 \left(\cos\tfrac{1}{2}\varphi\right)^3 \, , which is tangent of the cardioid with polar equation r = 2(1 + \cos\varphi) from the previous section.}} For cardioids the following is true: : The orthogonal trajectories of the pencil of cardioids with equations r=2a(1-\cos\varphi)\ , \; a>0 \ , \ are the cardioids with equations r=2b(1+\cos\varphi)\ , \; b>0 \ . Hence any secant line of the circle, defined above, is a tangent of the cardioid, too: : The cardioid is the envelope of the chords of a circle. (Trigonometric formulae were used: \sin\tfrac{3}{2}\varphi = \sin\tfrac{\varphi}{2}\cos\varphi + \cos\tfrac{\varphi}{2}\sin\varphi\ ,\ \cos\tfrac{3}{2}\varphi = \cdots, \ \sin\varphi = 2\sin\tfrac{\varphi}{2}\cos\tfrac{\varphi}{2}, \ \cos\varphi= \cdots \ . ) == Orthogonal trajectories == 300px|thumb|Orthogonal cardioids An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. Remark: If point O is not on the perimeter of the circle k, one gets a limaçon of Pascal. == The evolute of a cardioid == thumb| The evolute of a curve is the locus of centers of curvature. thumb|upright=1.0|The inscribed angle θ is half of the central angle 2θ that subtends the same arc on the circle. The proofs of these statement use in both cases the polar representation of the cardioid. # The envelope of these chords is a cardioid. thumb|Cremona's generation of a cardioid ==== Proof ==== The following consideration uses trigonometric formulae for \cos\alpha + \cos\beta, \sin\alpha + \sin\beta, 1 + \cos 2\alpha , \cos 2\alpha, and \sin 2\alpha. Hence a cardioid is a special pedal curve of a circle. ==== Proof ==== In a Cartesian coordinate system circle k may have midpoint (2a,0) and radius 2a. ",2.89,15,"""0.11""",8,4.979,D
+"Estimate the volume of the solid that lies above the square $R=[0,2] \times[0,2]$ and below the elliptic paraboloid $z=16-x^2-2 y^2$. Divide $R$ into four equal squares and choose the sample point to be the upper right corner of each square $R_{i j}$. ","The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. Hence the area of this development is thumb|cloister vault :B = \int_{0}^{\pi r} r\sin\left(\frac{\xi}{r}\right) \mathrm{d}\xi = 2r^2 and the total surface area is: :A=8\cdot B=16r^2. === Alternate proof of the volume formula === Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). This leads to :V = \int_{-r}^{r} (2x)^2 \mathrm{d}z = 4\cdot \int_{-r}^{r} x^2 \mathrm{d}z = 4\cdot \int_{-r}^{r} (r^2-z^2) \mathrm{d}z=\frac{16}{3} r^3. By the Pythagorean theorem, the radius of the cylinder is thumb|upright=1.2|Finding the measurements of the ring that is the horizontal cross-section. \sqrt{R^2 - \left(\frac{h}{2}\right)^2},\qquad\qquad(1) and the radius of the horizontal cross-section of the sphere at height y above the ""equator"" is \sqrt{R^2 - y^2}.\qquad\qquad(2) The cross-section of the band with the plane at height y is the region inside the larger circle of radius given by (2) and outside the smaller circle of radius given by (1). The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. thumb|right|150px|Animated depiction of a bicylinder == Bicylinder == 300px|thumb|The generation of a bicylinder 180px|thumb|Calculating the volume of a bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac{16}{3} r^3 and the ;surface area :A=16 r^2. The key for the determination of volume and surface area is the observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. Their intercepts with the dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times. In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. The volume of the band is : \int_{-h/2}^{h/2} (\text{area of cross-section at height }y) \, dy, and that does not depend on R. File:Sphere volume derivation using bicylinder.jpg|Zu Chongzhi's method (similar to Cavalieri's principle) for calculating a sphere's volume includes calculating the volume of a bicylinder. After integrating these two functions with the disk method we would subtract them to yield the desired volume. The cross-section's area is therefore the area of the larger circle minus the area of the smaller circle: \begin{align} & {}\quad \pi(\text{larger radius})^2 - \pi(\text{smaller radius})^2 \\\ & = \pi\left(\sqrt{R^2 - y^2}\right)^2 - \pi\left(\sqrt{R^2 - \left(\frac{h}{2}\right)^2\,{}}\,\right)^2 = \pi\left(\left(\frac{h}{2}\right)^2 - y^2\right). \end{align} The radius R does not appear in the last quantity. thumb|Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse. thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. The volume of the 8 pyramids is: \textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 , and then we can calculate that the bicylinder volume is \textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3 == Tricylinder == 450px|thumb|Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. In this case the volume of the band is the volume of the whole sphere, which matches the formula given above. thumb|300px|Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a ""napkin ring"" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole. ",0.44,0.54,"""2.3613""",10,34,E
+"Find the average value of the function $f(x)=1+x^2$ on the interval $[-1,2]$.","In mathematics and its applications, the mean square is normally defined as the arithmetic mean of the squares of a set of numbers or of a random variable. A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: . It may also be defined as the arithmetic mean of the squares of the deviations between a set of numbers and a reference value (e.g., may be a mean or an assumed mean of the data), in which case it may be known as mean square deviation. In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: : \bar{f}=\frac{1}{b-a}\int_a^bf(x)\,dx. The function provides the arithmetic mean. For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. The type of calculations used in adjusting general average gave rise to the use of ""average"" to mean ""arithmetic mean"". In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). thumb|right|A graph of the function f(x) = e^{-x^2} and the area between it and the x-axis, (i.e. the entire real line) which is equal to \sqrt{\pi}. (If there are an even number of numbers, the mean of the middle two is taken.) That is, \int_{-\infty}^{\infty} e^{-x^2} \, dx = 2\int_{0}^{\infty} e^{-x^2}\,dx. For this reason, it is recommended to avoid using the word ""average"" when discussing measures of central tendency. ==General properties== If all numbers in a list are the same number, then their average is also equal to this number. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of these two. A typical estimate for the sample variance from a set of sample values x_i uses a divisor of the number of values minus one, n-1, rather than n as in a simple quadratic mean, and this is still called the ""mean square"" (e.g. in analysis of variance): :s^2=\textstyle\frac{1}{n-1}\sum(x_i-\bar{x})^2 The second moment of a random variable, E(X^{2}) is also called the mean square. There is also a harmonic average of functions and a quadratic average (or root mean square) of functions. ==See also== *Mean Category:Means Category:Calculus ==References== By analogy, a defining property of the average value \bar{f} of a function over the interval [a,b] is that : \int_a^b\bar{f}\,dx = \int_a^bf(x)\,dx In other words, \bar{f} is the constant value which when integrated over [a,b] equals the result of integrating f(x) over [a,b]. The square root of a mean square is known as the root mean square (RMS or rms), and can be used as an estimate of the standard deviation of a random variable. ==References== Category:Means In mathematics, the mean value problem was posed by Stephen Smale in 1981. Most types of average, however, satisfy permutation- insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 6) is the same as that of (3, 2, 6, 4, 1). ==Pythagorean means== The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means. ==Statistical location== The mode, the median, and the mid- range are often used in addition to the mean as estimates of central tendency in descriptive statistics. Depending on the context, an average might be another statistic such as the median, or mode. ",0.3359,5.51,"""0.7812""",210,2,E
+Find the area of the region enclosed by the parabolas $y=x^2$ and $y=2 x-x^2$,"thumb|right|200px|Two-dimensional plot (red curve) of the algebraic equation y = x^2 - x - 2. Adding the two equations together to get: : 8x = 16 which simplifies to : x = 2. The area under the curve decreases monotonically with increasing p. == Generalization == A natural generalization for the superparabola is to relax the constraint on the power of x. thumb|right|384px|In green, confocal parabolae opening upwards, 2y = \frac {x^2}{\sigma^2}-\sigma^2 In red, confocal parabolae opening downwards, 2y =-\frac{x^2}{\tau^2}+\tau^2 Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. Using the second equation: : 2x - y = 1 Subtracting 2x from each side of the equation: : \begin{align}2x - 2x - y & = 1 - 2x \\\ \- y & = 1 - 2x \end{align} and multiplying by −1: : y = 2x - 1. In mathematics, the definite integral :\int_a^b f(x)\, dx is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total. Using the method of exhaustion, it follows that the total area of the parabolic segment is given by :\text{Area}\;=\;T \,+\, \frac14T \,+\, \frac1{4^2}T \,+\, \frac1{4^3}T \,+\, \cdots. Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord. thumb|400x300px|Superparabola functions A superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points with :\frac{y}{b} = \lbrack1-\left(\frac{x}{a}\right)^2\rbrack^p, where , , and are positive integers. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. Divide both sides by 2: \frac{2x}{2} = \frac{8}{2} 5\. This simplifies to: 2x = 8 4\. When the center of gravity of the triangle is known, the equilibrium of the lever yields the area of the parabola in terms of the area of the triangle which has the same base and equal height. The superparabola can vary in shape from a rectangular function , to a semi- ellipse (, to a parabola , to a pulse function . == Mathematical properties == thumb|400x300px| Without loss of generality we can consider the canonical form of the superparabola :f(x;p)=\left(1-x^2 \right)^p When , the function describes a continuous differentiable curve on the plane. Area function may refer to: *Inverse hyperbolic function *Antiderivative Here, however, we have the analytic solution for the area under the curve. The foci of all these parabolae are located at the origin. An interesting property is that any superparabola raised to a power n is just another superparabola; thus :\int_{-1}^{1}f^n (x) = \psi(n p) The centroid of the area under the curve is given by :C = \frac{\mathbf {i}}{A} \int_{-1}^{1} x\int_{0}^{f(x)} dydx + \frac{\mathbf {j}}{A}\ \int_{-1}^{1} \int_{0}^{f(x)}y dy dx :=\frac{\mathbf{j}}{2A}\int_{-1}^{1} f^2 (x) dx =\mathbf{j}\frac{\psi (2p)}{2\psi(p)} where the x-component is zero by virtue of symmetry. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. The indefinite and definite integrals are given by :\int f(x)dx=x \cdot_{2}F_{1} (-p, 1/q; 1+ 1/q ; x^2) :\text{Area}=\int _{-1}^{1}f(x)dx=\Psi (p,q) where \Psi is a universal function valid for all q and p>-1. Because of this a quadratic equation must contain the term ax^2, which is known as the quadratic term. The curve can be described parametrically on the complex plane as :z=\sin(u)+i\cos^{2p}(u);\quad-\tfrac{\pi}{2}\leq u\leq\tfrac{\pi}{2} Derivatives of the superparabola are given by :f'(x;p)=-2px(1-x^2)^{p-1} :\frac{\partial f}{\partial p} = (1-x^2)^p\ln(1-x^2) = f(x)\ln\lbrack f(x; 1)\rbrack The area under the curve is given by :\text{Area} = \int_{-1}^{1}\int_{0}^{f(x)}dydx = \int_{-1}^{1} (1-x^2)^p dx = \psi(p) where is a global function valid for all , :\psi( p)=\frac {\sqrt{\pi}\, \Gamma(p+1)}{\Gamma(p+\frac{3}{2})} The area under a portion of the curve requires the indefinite integral : \int (1-x^2)^p dx = x\,{_2}F{_1} (1/2, -p; 3/2; x^2) where _2F_1 is the Gaussian hypergeometric function. ",2.8,117,"""0.333333333333333""",5300,-3.5,C
+The region $\mathscr{R}$ enclosed by the curves $y=x$ and $y=x^2$ is rotated about the $x$-axis. Find the volume of the resulting solid.,"For example, the next figure shows the rotation along the -axis of the red ""leaf"" enclosed between the square- root and quadratic curves: thumb|Rotation about x-axis The volume of this solid is: :\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,. The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by :V = \pi \int_a^b \left| f(y)^2 - g(y)^2\right|\,dy\, . Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem). In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. The areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given :A_x = \int_\alpha^\beta 2 \pi r\sin{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , :A_y = \int_\alpha^\beta 2 \pi r\cos{\theta} \, \sqrt{ r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta \, , : ==See also== * Gabriel's Horn * Guldinus theorem * Pseudosphere * Surface of revolution * Ungula ==Notes== == References == * * () * Category:Integral calculus Category:Solids The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by :V = 2\pi \int_a^b x |f(x) - g(x)|\, dx\, . This works only if the axis of rotation is horizontal (example: or some other constant). ===Function of === If the function to be revolved is a function of , the following integral will obtain the volume of the solid of revolution: :\pi\int_c^d R(y)^2\,dy where is the distance between the function and the axis of rotation. The surface created by this revolution and which bounds the solid is the surface of revolution. This works only if the axis of rotation is vertical (example: or some other constant). ===Washer method=== To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness , or a cylindrical shell of width ; and then find the limiting sum of these volumes as approaches 0, a value which may be found by evaluating a suitable integral. The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. If is the value of a horizontal axis, then the volume equals :\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,. The element is created by rotating a line segment (of length ) around some axis (located units away), so that a cylindrical volume of units is enclosed. ==Finding the volume== Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. Under the same circumstances the areas of the surfaces of the solids generated by revolving the curve around the -axis or the -axis are given by :A_x = \int_a^b 2 \pi y \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, , :A_y = \int_a^b 2 \pi x \, \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt \, . == Polar form == For a polar curve r=f(\theta) where \alpha\leq \theta\leq \beta, the volumes of the solids generated by revolving the curve around the x-axis or y-axis are :V_x = \int_\alpha^\beta \left(\pi r^2\sin^2{\theta} \cos{\theta}\, \frac{dr}{d\theta}-\pi r^3\sin^3{\theta}\right)d\theta\,, :V_y = \int_\alpha^\beta \left(\pi r^2\sin{\theta} \cos^2{\theta}\, \frac{dr}{d\theta}+\pi r^3\cos^3{\theta}\right)d\theta \, . One simply must solve each equation for before one inserts them into the integration formula. ==See also== *Solid of revolution *Shell integration ==References== * * *Frank Ayres, Elliott Mendelson. In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution) that lies on the same plane. Volume solid is the term which indicates the solid proportion of the paint on a volume basis. For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. For example, to rotate the region between and along the axis , one would integrate as follows: :\pi\int_0^3\left(\left(4-\left(-2x+x^2\right)\right)^2 - (4-x)^2\right)\,dx\,. After integrating these two functions with the disk method we would subtract them to yield the desired volume. This method may be derived with the same triple integral, this time with a different order of integration: :V = \iiint_D dV = \int_a^b \int_{g(r)}^{f(r)} \int_0^{2\pi} r\,d\theta\,dz\,dr = 2\pi \int_a^b\int_{g(r)}^{f(r)} r\,dz\,dr = 2\pi\int_a^b r(f(r) - g(r))\,dr. ==Parametric form== When a curve is defined by its parametric form in some interval , the volumes of the solids generated by revolving the curve around the -axis or the -axis are given by :V_x = \int_a^b \pi y^2 \, \frac{dx}{dt} \, dt \, , :V_y = \int_a^b \pi x^2 \, \frac{dy}{dt} \, dt \, . ",+7.3,0.000216,"""0.41887902047""",3.54,2.9,C
+Use Simpson's Rule with $n=10$ to approximate $\int_1^2(1 / x) d x$.,"The error in approximating an integral by Simpson's rule for n = 2 is -\frac{1}{90} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{2880} f^{(4)}(\xi), where \xi (the Greek letter xi) is some number between a and b. Simpson's 1/3 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{1}{3} h\left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]\\\ &= \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right], \end{align} where h = (b - a)/2 is the step size. (This derivation is essentially a less rigorous version of the quadratic interpolation derivation, where one saves significant calculation effort by guessing the correct functional form.) === Composite Simpson's 1/3 rule === If the interval of integration [a, b] is in some sense ""small"", then Simpson's rule with n = 2 subintervals will provide an adequate approximation to the exact integral. This is called the trapezoidal rule \int_a^b f(x)\, dx \approx (b-a) \left(\frac{f(a) + f(b)}{2}\right). right|thumb|300px|Illustration of Simpson's rule. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, dx \approx \frac{b - a}{6} \left[f(a) + 4f\left(\frac{a + b}{2}\right) + f(b)\right]. Integration by Simpson's 1/3 rule can be represented as a weighted average with 2/3 of the value coming from integration by the trapezoidal rule with step h and 1/3 of the value coming from integration by the rectangle rule with step 2h. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error. This leads to the adaptive Simpson's method. == Simpson's 3/8 rule == Simpson's 3/8 rule, also called Simpson's second rule, is another method for numerical integration proposed by Thomas Simpson. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using Simpson's rule. If the 1/3 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 1/3 rule. The error committed by the composite Simpson's rule is -\frac{1}{180} h^4(b - a)f^{(4)}(\xi), where \xi is some number between a and b, and h = (b - a)/n is the ""step length"". thumb|right|Simpson's rule can be derived by approximating the integrand f (x) (in blue) by the quadratic interpolant P(x) (in red). thumb|right|An animation showing how Simpson's rule approximates the function with a parabola and the reduction in error with decreased step size thumb|right|An animation showing how Simpson's rule approximation improves with more strips. In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The formula above is obtained by combining the composite Simpson's 1/3 rule with the one consisting of using Simpson's 3/8 rule in the extreme subintervals and Simpson's 1/3 rule in the remaining subintervals. The 1/3 rule can be used for the remaining subintervals without changing the order of the error term (conversely, the 3/8 rule can be used with a composite 1/3 rule for odd-numbered subintervals). == Alternative extended Simpson's rule == This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding \int_a^b f(x)\, dx \approx \frac{1}{48} h\left[17f(x_0) + 59f(x_1) + 43f(x_2) + 49f(x_3) + 48 \sum_{i= 4 }^{n - 4} f(x_i) + 49f(x_{n - 3}) + 43f(x_{n - 2}) + 59f(x_{n - 1}) + 17f(x_n)\right]. If the 3/8 rule is applied to n equal subdivisions of the integration range [a, b], one obtains the composite Simpson's 3/8 rule. Simpson's 3/8 rule is as follows: \begin{align} \int_a^b f(x)\, dx &\approx \frac{3}{8} h\left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right]\\\ &= \frac{b - a}{8} \left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right], \end{align} where h = (b - a)/3 is the step size. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. See: * Simpson's rule, a method of numerical integration * Simpson's rules (ship stability) * Simpson–Kramer method In case of odd number N of subintervals, the above formula are used up to the second to last interval, and the last interval is handled separately by adding the following to the result: \alpha f_N + \beta f_{N - 1} - \eta f_{N - 2}, where \begin{align} \alpha &= \frac{2h_{N - 1}^2 + 3h_{N - 1} h_{N - 2}}{6(h_{N - 2} + h_{N - 1})},\\\\[1ex] \beta &= \frac{h_{N - 1}^2 + 3h_{N - 1} h_{N - 2}}{6h_{N - 2}},\\\\[1ex] \eta &= \frac{h_{N - 1}^3}{6 h_{N - 2}(h_{N - 2} + h_{N - 1})}. \end{align} Example implementation in Python from collections.abc import Sequence def simpson_nonuniform(x: Sequence[float], f: Sequence[float]) -> float: """""" Simpson rule for irregularly spaced data. :param x: Sampling points for the function values :param f: Function values at the sampling points :return: approximation for the integral See ``scipy.integrate.simpson`` and the underlying ``_basic_simpson`` for a more performant implementation utilizing numpy's broadcast. The result is then obtained by taking the mean of the two formulas. === Simpson's rules in the case of narrow peaks === In the task of estimation of full area of narrow peak-like functions, Simpson's rules are much less efficient than trapezoidal rule. ",0.33333333,-8,"""635013559600.0""",0.693150,3,D
+"Use the Midpoint Rule with $m=n=2$ to estimate the value of the integral $\iint_R\left(x-3 y^2\right) d A$, where $R=\{(x, y) \mid 0 \leqslant x \leqslant 2,1 \leqslant y \leqslant 2\}$.","For example, doing the previous calculation with order reversed gives the same result: : \begin{align} \int_{11}^{14} \int_{7}^{10} \, \left(x^2 + 4y\right) \, dy\, dx & = \int_{11}^{14} \Big[x^2 y + 2y^2 \Big]_{y=7}^{y=10} \, dx \\\ &= \int_{11}^{14} \, (3x^2 + 102) \, dx \\\ &= \Big[x^3 + 102x \Big]_{x=11}^{x=14} \\\ &= 1719. \end{align} === Double integral over a normal domain === thumb|160px|right|Example: double integral over the normal region D Consider the region (please see the graphic in the example): :D = \\{ (x,y) \in \mathbf{R}^2 \ : \ x \ge 0, y \le 1, y \ge x^2 \\} Calculate :\iint_D (x+y) \, dx \, dy. Let and :D = \left\\{ (x,y) \in \R^2 \ : \ 2 \le x \le 4 \ ; \ 3 > \le y \le 6 \right\\} in which case :\int_3^6 \int_2^4 \ 2 \ dx\, dy > =2\int_3^6 \int_2^4 \ 1 \ dx\, dy= 2\cdot\operatorname{area}(D) = 2 \cdot (2 > \cdot 3) = 12, since by definition we have: :\int_3^6 \int_2^4 \ 1 \ dx\, > dy=\operatorname{area}(D). ===Use of symmetry=== When the domain of integration is symmetric about the origin with respect to at least one of the variables of integration and the integrand is odd with respect to this variable, the integral is equal to zero, as the integrals over the two halves of the domain have the same absolute value but opposite signs. I = \left.\int_0^{3a}\rho^4 > d\rho = \frac{\rho^5}{5}\right\vert_0^{3a} = \frac{243}{5}a^5, II = > \int_0^\pi \sin^3\theta \, d\theta = -\int_0^\pi \sin^2\theta \, d(\cos > \theta) = \int_0^\pi (\cos^2\theta-1) \, d(\cos \theta) = > \left.\frac{\cos^3\theta}{3}\right|^\pi_0 - \left.\cos\theta\right|^\pi_0 = > \frac{4}{3}, III = \int_0^{2\pi} d \varphi = 2\pi. Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. The result of this integral, which is a function depending only on , is then integrated with respect to . :\begin{align} \int_{11}^{14} \left(x^2 + 4y\right) \, dx & = \left [\frac13 x^3 + 4yx \right]_{x=11}^{x=14} \\\ &= \frac13(14)^3 + 4y(14) - \frac13(11)^3 - 4y(11) \\\ &= 471 + 12y \end{align} We then integrate the result with respect to . :\begin{align} \int_7^{10} (471 + 12y) \ dy & = \Big[471y + 6y^2\Big]_{y=7}^{y=10} \\\ &= 471(10)+ 6(10)^2 - 471(7) - 6(7)^2 \\\ &= 1719 \end{align} In cases where the double integral of the absolute value of the function is finite, the order of integration is interchangeable, that is, integrating with respect to x first and integrating with respect to y first produce the same result. It is now possible to apply the formula: :\iint_D (x+y) \, dx \, dy = \int_0^1 dx \int_{x^2}^1 (x+y) \, dy = \int_0^1 dx \ \left[xy + \frac{y^2}{2} \right]^1_{x^2} (at first the second integral is calculated considering x as a constant). Then > we get :\begin{align} \int_0^{2\pi} d\varphi \int_0^{3a} \rho^3 d\rho > \int_{-\sqrt{9a^2 - \rho^2}}^{\sqrt{9 a^2 - \rho^2}}\, dz &= 2 \pi > \int_0^{3a} 2 \rho^3 \sqrt{9 a^2 - \rho^2} \, d\rho \\\ &= -2 \pi \int_{9 > a^2}^0 (9 a^2 - t) \sqrt{t}\, dt && t = 9 a^2 - \rho^2 \\\ &= 2 \pi > \int_0^{9 a^2} \left ( 9 a^2 \sqrt{t} - t \sqrt{t} \right ) \, dt \\\ &= 2 > \pi \left( \int_0^{9 a^2} 9 a^2 \sqrt{t} \, dt - \int_0^{9 a^2} t \sqrt{t} > \, dt\right) \\\ &= 2 \pi \left[9 a^2 \frac23 t^{ \frac32 } - \frac{2}{5} > t^{ \frac{5}{2}} \right]_0^{9 a^2} \\\ &= 2 \cdot 27 \pi a^5 \left ( 6 - > \frac{18}{5} \right ) \\\ &= \frac{648 \pi}{5} a^5. \end{align} Thanks to > the passage to cylindrical coordinates it was possible to reduce the triple > integral to an easier one-variable integral. In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). See also the differential volume entry in nabla in cylindrical and spherical coordinates. ==Examples== === Double integral over a rectangle === Let us assume that we wish to integrate a multivariable function over a region : :A = \left \\{ (x,y) \in \mathbf{R}^2 \ : \ 11 \le x \le 14 \ ; \ 7 \le y \le 10 \right \\} \mbox{ and } f(x,y) = x^2 + 4y\, From this we formulate the iterated integral :\int_7^{10} \int_{11}^{14} (x^2 + 4y) \, dx\, dy The inner integral is performed first, integrating with respect to and taking as a constant, as it is not the variable of integration. right|thumb|Illustration of the midpoint method assuming that y_n equals the exact value y(t_n). The domain is the ball with center at the origin and radius , :D > = \left \\{ x^2 + y^2 + z^2 \le 9a^2 \right \\} and is the function to > integrate. Then, by Fubini's theorem: :\iint_D f(x,y)\, dx\, dy = \int_a^b dx \int_{ \alpha (x)}^{ \beta (x)} f(x,y)\, dy. ====-axis==== If is normal with respect to the -axis and is a continuous function; then and (both of which are defined on the interval ) are the two functions that determine . Collecting all parts, > \iiint_T \rho^4 \sin^3 \theta \, d\rho\, d\theta\, d\varphi = I\cdot II\cdot > III = \frac{243}{5}a^5\cdot \frac{4}{3}\cdot 2\pi = \frac{648}{5}\pi a^5. > Alternatively, this problem can be solved by using the passage to > cylindrical coordinates. Once the intervals are known, you have :\int_0^\pi \int_2^3 \rho^2 > \cos \varphi \, d \rho \, d \varphi = \int_0^\pi \cos \varphi \ d \varphi > \left[ \frac{\rho^3}{3} \right]_2^3 = \Big[ \sin \varphi \Big]_0^\pi \ > \left(9 - \frac{8}{3} \right) = 0. ====Cylindrical coordinates==== thumb|right|190px|Cylindrical coordinates. The explicit midpoint method is given by the formula the implicit midpoint method by for n=0, 1, 2, \dots Here, h is the step size -- a small positive number, t_n=t_0 + n h, and y_n is the computed approximate value of y(t_n). If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. Using the linearity property, the > integral can be decomposed into three pieces: :\iint_T \left(2\sin x - 3y^3 > + 5\right) \, dx \, dy = \iint_T 2 \sin x \, dx \, dy - \iint_T 3y^3 \, dx > \, dy + \iint_T 5 \, dx \, dy The function is an odd function in the > variable and the disc is symmetric with respect to the -axis, so the value > of the first integral is 0. thumb|right|Integral as area between two curves. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. thumb|300px|A possible set of Jacobi coordinates for four-body problem; the Jacobi coordinates are r1, r2, r3 and the center of mass R. * Sphere: The volume of a sphere with radius can be calculated by integrating the constant function 1 over the sphere, using spherical coordinates. ::\begin{align} \text{Volume} &= \iiint_D f(x,y,z) \, dx\, dy\, dz \\\ &= \iiint_D 1 \, dV \\\ &= \iiint_S \rho^2 \sin \varphi \, d\rho\, d\theta\, d\varphi \\\ &= \int_0^{2\pi} \, d \theta \int_0^{ \pi } \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\\ &= 2 \pi \int_0^\pi \sin \varphi\, d \varphi \int_0^R \rho^2\, d \rho \\\ &= 2 \pi \int_0^\pi \sin \varphi \frac{R^3}{3 }\, d \varphi \\\ &= \frac23 \pi R^3 \Big[-\cos \varphi\Big]_0^\pi = \frac43 \pi R^3. \end{align} * Tetrahedron (triangular pyramid or 3-simplex): The volume of a tetrahedron with its apex at the origin and edges of length along the -, - and -axes can be calculated by integrating the constant function 1 over the tetrahedron. ::\begin{align} \text{Volume} &= \int_0^\ell dx \int_0^{\ell-x}\, dy \int_0^{\ell-x-y }\, dz \\\ &= \int_0^\ell dx \int_0^{\ell-x } (\ell - x - y)\, dy \\\ &= \int_0^\ell \left( l^2 - 2 \ell x + x^2 - \frac{(\ell-x)^2 }{2}\right)\, dx \\\ &= \ell^3 - \ell \ell^2 + \frac{\ell^3}{3 } - \left[\frac{\ell^2 x}{2} - \frac{ \ell x^2}{2} + \frac{x^3}{6 }\right]_0^ \ell \\\ &= \frac{\ell^3}{3} - \frac{\ell^3}{6} = \frac{ \ell^3}{6}\end{align} :This is in agreement with the formula for the volume of a pyramid ::\mathrm{Volume} = \frac13 \times \text{base area} \times \text{height} = \frac13 \times \frac{\ell^2}{2} \times \ell = \frac{ \ell^3}{6}. thumb|right|140px|Example of an improper domain. ==Multiple improper integral== In case of unbounded domains or functions not bounded near the boundary of the domain, we have to introduce the double improper integral or the triple improper integral. ==Multiple integrals and iterated integrals== Fubini's theorem states that if :\iint_{A\times B} \left|f(x,y)\right|\,d(x,y)<\infty, that is, if the integral is absolutely convergent, then the multiple integral will give the same result as either of the two iterated integrals: :\iint_{A\times B} f(x,y)\,d(x,y)=\int_A\left(\int_B f(x,y)\,dy\right)\,dx=\int_B\left(\int_A f(x,y)\,dx\right)\,dy. In the following example, the electric field produced by a distribution of charges given by the volume charge density is obtained by a triple integral of a vector function: :\vec E = \frac {1}{4 \pi \varepsilon_0} \iiint \frac {\vec r - \vec r'}{\left \| \vec r - \vec r' \right \|^3} \rho (\vec r')\, d^3 r'. ",-11.875,0.9731,"""-2.5""",0.6749,1260,A
+"The base radius and height of a right circular cone are measured as $10 \mathrm{~cm}$ and $25 \mathrm{~cm}$, respectively, with a possible error in measurement of as much as $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated volume of the cone.","For 0 \le K' \le 1 : y = R\left({2 \left({x \over L}\right) - K'\left({x \over L}\right)^2 \over 2 - K'}\right) can vary anywhere between and , but the most common values used for nose cone shapes are: Parabola Type Value Cone Half Three Quarter Full For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same and , then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated. :\rho > {R^2 + L^2 \over 2R} and \alpha = \arccos \left({\sqrt{L^2 + R^2} \over 2\rho}\right)-\arctan \left({R \over L}\right) Then the radius at any point as varies from to is: :y = \sqrt{\rho^2 - (\rho\cos(\alpha) - x)^2} - \rho\sin(\alpha) If the chosen is less than the tangent ogive and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. The radius of the circle that forms the ogive is called the ogive radius, , and it is related to the length and base radius of the nose cone as expressed by the formula: :\rho = {R^2 + L^2\over 2R} The radius at any point , as varies from to is: :y = \sqrt{\rho^2 - (L - x)^2}+R - \rho The nose cone length, , must be less than or equal to . The tangency point where the sphere meets the cone can be found from: : x_t = \frac{L^2}{R} \sqrt{ \frac{r_n^2}{R^2 + L^2} } : y_t = \frac{x_t R}{L} where is the radius of the spherical nose cap. The failure is governed by crack growth in concrete, which forms a typical cone shape having the anchor's axis as revolution axis. ==Mechanical models== ===ACI 349-85=== Under tension loading, the concrete cone failure surface has 45° inclination. For , Haack nose cones bulge to a maximum diameter greater than the base diameter. The sides of a conic profile are straight lines, so the diameter equation is simply: : y = {xR \over L} Cones are sometimes defined by their half angle, : : \phi = \arctan \left({R \over L}\right) and y = x \tan(\phi)\; ==== Spherically blunted conic ==== In practical applications such as re-entry vehicles, a conical nose is often blunted by capping it with a segment of a sphere. A conical measure is a type of laboratory glassware which consists of a conical cup with a notch on the top to allow for the easy pouring of liquids, and graduated markings on the side to allow easy and accurate measurement of volumes of liquid. Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. alt=Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.|thumb|300x300px|General parameters used for constructing nose cone profiles. For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium. == Nose cone shapes and equations == === General dimensions === In all of the following nose cone shape equations, is the overall length of the nose cone and is the radius of the base of the nose cone. is the radius at any point , as varies from , at the tip of the nose cone, to . The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = f_{ct} {A_{N}} [N] Where: f_{ct} \- tensile strength of concrete A_{N} \- Cone's projected area === Concrete capacity design (CCD) approach for fastening to concrete=== Under tension loading, the concrete capacity of a single anchor is calculated assuming an inclination between the failure surface and surface of the concrete member of about 35°. The length/diameter relation is also often called the caliber of a nose cone. The center of the spherical nose cap, , can be found from: : x_o = x_t + \sqrt{ r_n^2 - y_t^2} And the apex point, can be found from: : x_a = x_o - r_n === Bi-conic === A bi-conic nose cone shape is simply a cone with length stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with length , where the base of the upper cone is equal in radius to the top radius of the smaller frustum with base radius . The use of the conical measure usually dictates its construction material. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. Conical measures are the most commonly used item of glassware used in the preparation of extemporaneous medicaments. While the equations describe the 'perfect' shape, practical nose cones are often blunted or truncated for manufacturing, aerodynamic, or thermodynamic reasons. === Conic === A very common nose-cone shape is a simple cone. :L=L_1+L_2 :For 0 \le x \le L_1 : y = {xR_1 \over L_1} :For L_1 \le x \le L : y = R_1 + {(x - L_1)(R_2-R_1)\over L_2} Half angles: :\phi_1 = \arctan \left({R_1 \over L_1}\right) and y = x \tan(\phi_1)\; :\phi_2 = \arctan \left({R_2 - R_1 \over L_2}\right) and y = R_1 + (x - L_1) \tan(\phi_2)\; === Tangent ogive === Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. They are not as precise as graduated cylinders for measuring liquids, but make up for this in terms of easy pouring and ability to mix solutions within the measure itself. ==History== During his experiments, Abū al-Rayhān al-Bīrūnī (973-1048) invented the conical measure,Marshall Clagett (1961). ",0,5.0,"""62.8318530718""",0.333333333333333,1110,C
+A force of $40 \mathrm{~N}$ is required to hold a spring that has been stretched from its natural length of $10 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $15 \mathrm{~cm}$ to $18 \mathrm{~cm}$ ?,"Let be the amount by which the free end of the spring was displaced from its ""relaxed"" position (when it is not being stretched). An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. The force an ideal spring would exert is exactly proportional to its extension or compression. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. The work of the net force is calculated as the product of its magnitude and the particle displacement. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. The force is applied through the ends of the spring. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The work can be split into two terms \delta W = \delta W_\mathrm{s} + \delta W_\mathrm{b} where is the work done by surface forces while is the work done by body forces. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Explaining the Power of Springing Bodies, London, 1678. as: (""as the extension, so the force"" or ""the extension is proportional to the force""). Remarkably, the work of a constraint force is zero, therefore only the work of the applied forces need be considered in the work–energy principle. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. The manufacture normally specifies the spring rate. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. ",5.5,-0.0301,"""399.0""",1.56,0.7854,D
 " An automobile with a mass of $1000 \mathrm{~kg}$, including passengers, settles $1.0 \mathrm{~cm}$ closer to the road for every additional $100 \mathrm{~kg}$ of passengers. It is driven with a constant horizontal component of speed $20 \mathrm{~km} / \mathrm{h}$ over a washboard road with sinusoidal bumps. The amplitude and wavelength of the sine curve are $5.0 \mathrm{~cm}$ and $20 \mathrm{~cm}$, respectively. The distance between the front and back wheels is $2.4 \mathrm{~m}$. Find the amplitude of oscillation of the automobile, assuming it moves vertically as an undamped driven harmonic oscillator. Neglect the mass of the wheels and springs and assume that the wheels are always in contact with the road.
-","Thus, for g ≈ 9.8 ms−2, :L \approx \frac{894}{n^2} \; \text{m} | n 71 0.846 0.177 70 0.857 0.182 69 0.870 0.188 68 0.882 0.193 67 0.896 0.199 66 0.909 0.205 65 0.923 0.212 64 0.938 0.218 63 0.952 0.225 62 0.968 0.232 61 0.984 0.240 60 1.000 0.248 Parameters of the pendulum wave in the animation above ---|--- == See also == * Newton's cradle - a set of pendulums constrained to swing along the axis of the apparatus and collide with one another ==References== Category:Pendulums Category:Kinetic art Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: :x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi), with z^2=a^2+b^2 and \tan\phi = \frac{b}{a}. A vibration in a string is a wave. Constant envelope is achieved when a sinusoidal waveform reaches equilibrium in a specific system. In particular, the increase in train speeds from 140 to 180 km/h was accompanied by about tenfold increase in generated ground vibration level, which agrees with the theory. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour. ==Parameters== The parameters in the above equation are: *\delta controls the amount of damping, *\alpha controls the linear stiffness, *\beta controls the amount of non-linearity in the restoring force; if \beta=0, the Duffing equation describes a damped and driven simple harmonic oscillator, *\gamma is the amplitude of the periodic driving force; if \gamma=0 the system is without a driving force, and *\omega is the angular frequency of the periodic driving force. String no.!!Thickness [in.] (d)!!Recommended tension [lbs.] (T)!!\rho [g/cm3] |- | 1 String no. Thickness [in.] (d) Recommended tension [lbs.] (T) \rho [g/cm3] 1 0.00899 13.1 7.726 (steel alloy) 2 0.0110 11.0 "" 3 0.0160 14.7 "" 4 0.0241 15.8 6.533 (nickel-wound steel alloy) 5 0.0322 15.8 "" 6 0.0416 14.8 "" Given the above specs, what would the computed vibrational frequencies (f) of the above strings' fundamental harmonics be if the strings were strung at the tensions recommended by the manufacturer? The horizontal tensions are not well approximated by T. == Frequency of the wave == Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated. If the length of the string is L, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so L is half of the wavelength of the fundamental harmonic. String no.!!Computed frequency [Hz]!!Closest note in A440 12-TET tuning |- | 1 String no. Computed frequency [Hz] Closest note in A440 12-TET tuning 1 330 E4 (= 440 ÷ 25/12 ≈ 329.628 Hz) 2 247 B3 (= 440 ÷ 210/12 ≈ 246.942 Hz) 3 196 G3 (= 440 ÷ 214/12 ≈ 195.998 Hz) 4 147 D3 (= 440 ÷ 219/12 ≈ 146.832 Hz) 5 110 A2 (= 440 ÷ 224/12 = 110 Hz) 6 82.4 E2 (= 440 ÷ 229/12 ≈ 82.407 Hz) == See also == * Fretted instruments * Musical acoustics * Vibrations of a circular drum * Melde's experiment * 3rd bridge (harmonic resonance based on equal string divisions) * String resonance * Reflection phase change == References == * * ;Specific == External links == * ""The Vibrating String"" by Alain Goriely and Mark Robertson-Tessi, The Wolfram Demonstrations Project. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z at a given excitation frequency. Hence one obtains Mersenne's laws: :f = \frac{v}{2L} = { 1 \over 2L } \sqrt{T \over \mu} where T is the tension (in Newtons), \mu is the linear density (that is, the mass per unit length), and L is the length of the vibrating part of the string. Ground vibration boom is a phenomenon of very large increase in ground vibrations generated by high-speed railway trains travelling at speeds higher than the velocity of Rayleigh surface waves in the supporting ground. == Technical background == thumb|Swedish high-speed train X 2000 approaching Ledsgard. Therefore: * the shorter the string, the higher the frequency of the fundamental * the higher the tension, the higher the frequency of the fundamental * the lighter the string, the higher the frequency of the fundamental Moreover, if we take the nth harmonic as having a wavelength given by \lambda_n = 2L/n, then we easily get an expression for the frequency of the nth harmonic: :f_n = \frac{nv}{2L} And for a string under a tension T with linear density \mu, then :f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} == Observing string vibrations == One can see the waveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of a CRT screen such as one of a television or a computer (not of an analog oscilloscope). As a result, : \begin{align} & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma \qquad \text{and} \\\ & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0. \end{align} Squaring both equations and adding leads to the amplitude frequency response: :\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^{2}+\left(\delta\omega\right)^2\right]\,z^2=\gamma^{2}, as stated above. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. == Wave == The velocity of propagation of a wave in a string (v) is proportional to the square root of the force of tension of the string (T) and inversely proportional to the square root of the linear density (\mu) of the string: v = \sqrt{T \over \mu}. A pendulum wave is an elementary physics demonstration and kinetic art comprising a number of uncoupled simple pendulums with monotonically increasing lengths. In physics, the Toda oscillator is a special kind of nonlinear oscillator. File:Duffing frequency response.svg|alt=Frequency response as a function of for the Duffing equation, with and damping The dashed parts of the frequency response are unstable.[3]|Frequency response z/\gamma as a function of \omega/\sqrt{\alpha} for the Duffing equation, with \alpha=\gamma=1 and damping \delta=0.1. The equation is given by :\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t), where the (unknown) function x=x(t) is the displacement at time t, \dot{x} is the first derivative of x with respect to time, i.e. velocity, and \ddot{x} is the second time- derivative of x, i.e. acceleration. ",0.33333333,14.80,8.0,-0.16,0.63,D
-"Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.","In other words, the shortest path will be made by joining circular arcs of maximum curvature and straight lines. Given two points s and t, say on the surface of a convex polyhedron, the problem is to compute a shortest path that never leaves the surface and connects s with t. thumb|right|Example of a shortest path in a three-dimensional Euclidean space The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. == Two dimensions == In two dimensions, the problem can be solved in polynomial time in a model of computation allowing addition and comparisons of real numbers, despite theoretical difficulties involving the numerical precision needed to perform such calculations. We also need to know what the actual shortest path is. There are simple geometric and analytical methods to compute the optimal path. Moving along each segment in this sequence for the appropriate length will form the shortest curve that joins a starting point A to a terminal point B with the desired tangents at each endpoint and that does not exceed the given curvature. Mount Short () is a mountain, 2,110 m, standing 1 mile (1.6 km) east of Sculpture Mountain, in the upper Rennick Glacier. There are many results on computing shortest paths which stays on a polyhedral surface. Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. thumb|The shortest-path graph with t=2 In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal. If p is truly the shortest path, then it can be split into sub-paths p1 from u to w and p2 from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in Introduction to Algorithms). In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This is termed as the weighted region problem in the literature. ==See also== * Shortest path problem, in a graph of edges and vertices * Any-angle path planning, in a grid space ==Notes== ==References== *. *. *. *. *. *. *. *. *. *. *. == External links == * Implementation of Euclidean Shortest Path algorithm in Digital Geometric Kernel software Category:Geometric algorithms Category:Computational geometry In 1957, Lester Eli Dubins (1920–2010) proved using tools from analysis that any such path will consist of maximum curvature and/or straight line segments. In the following pseudocode, `n` is the size of the board, `c(i, j)` is the cost function, and `min()` returns the minimum of a number of values: function minCost(i, j) if j < 1 or j > n return infinity else if i = 1 return c(i, j) else return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j) This function only computes the path cost, not the actual path. In geometry, the term Dubins path typically refers to the shortest curve that connects two points in the two-dimensional Euclidean plane (i.e. x-y plane) with a constraint on the curvature of the path and with prescribed initial and terminal tangents to the path, and an assumption that the vehicle traveling the path can only travel forward. The tangent direction of the path at initial and final points are constrained to lie within the specified intervals. Picking the square that holds the minimum value at each rank gives us the shortest path between rank `n` and rank `1`. The Dubins' path gives the shortest path joining two oriented points that is feasible for the wheeled-robot model. The edge set of the shortest-path graph varies based on a single parameter t ≥ 1. When the weight of an edge is defined as its Euclidean length raised to the power of the parameter t ≥ 1, the edge is present in the shortest-path graph if and only if it is the least weight path between its endpoints. == Properties of shortest-path graph == When the configuration parameter t goes to infinity, shortest-path graph become the minimum spanning tree of the point set. ",14,96.4365076099,2.534324263,-0.10,6,C
+","Thus, for g ≈ 9.8 ms−2, :L \approx \frac{894}{n^2} \; \text{m} | n 71 0.846 0.177 70 0.857 0.182 69 0.870 0.188 68 0.882 0.193 67 0.896 0.199 66 0.909 0.205 65 0.923 0.212 64 0.938 0.218 63 0.952 0.225 62 0.968 0.232 61 0.984 0.240 60 1.000 0.248 Parameters of the pendulum wave in the animation above ---|--- == See also == * Newton's cradle - a set of pendulums constrained to swing along the axis of the apparatus and collide with one another ==References== Category:Pendulums Category:Kinetic art Derivation of the frequency response Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form: :x = a\, \cos(\omega t) + b\, \sin(\omega t) = z\, \cos(\omega t - \phi), with z^2=a^2+b^2 and \tan\phi = \frac{b}{a}. A vibration in a string is a wave. Constant envelope is achieved when a sinusoidal waveform reaches equilibrium in a specific system. In particular, the increase in train speeds from 140 to 180 km/h was accompanied by about tenfold increase in generated ground vibration level, which agrees with the theory. The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour. ==Parameters== The parameters in the above equation are: *\delta controls the amount of damping, *\alpha controls the linear stiffness, *\beta controls the amount of non-linearity in the restoring force; if \beta=0, the Duffing equation describes a damped and driven simple harmonic oscillator, *\gamma is the amplitude of the periodic driving force; if \gamma=0 the system is without a driving force, and *\omega is the angular frequency of the periodic driving force. String no.!!Thickness [in.] (d)!!Recommended tension [lbs.] (T)!!\rho [g/cm3] |- | 1 String no. Thickness [in.] (d) Recommended tension [lbs.] (T) \rho [g/cm3] 1 0.00899 13.1 7.726 (steel alloy) 2 0.0110 11.0 "" 3 0.0160 14.7 "" 4 0.0241 15.8 6.533 (nickel-wound steel alloy) 5 0.0322 15.8 "" 6 0.0416 14.8 "" Given the above specs, what would the computed vibrational frequencies (f) of the above strings' fundamental harmonics be if the strings were strung at the tensions recommended by the manufacturer? The horizontal tensions are not well approximated by T. == Frequency of the wave == Once the speed of propagation is known, the frequency of the sound produced by the string can be calculated. If the length of the string is L, the fundamental harmonic is the one produced by the vibration whose nodes are the two ends of the string, so L is half of the wavelength of the fundamental harmonic. String no.!!Computed frequency [Hz]!!Closest note in A440 12-TET tuning |- | 1 String no. Computed frequency [Hz] Closest note in A440 12-TET tuning 1 330 E4 (= 440 ÷ 25/12 ≈ 329.628 Hz) 2 247 B3 (= 440 ÷ 210/12 ≈ 246.942 Hz) 3 196 G3 (= 440 ÷ 214/12 ≈ 195.998 Hz) 4 147 D3 (= 440 ÷ 219/12 ≈ 146.832 Hz) 5 110 A2 (= 440 ÷ 224/12 = 110 Hz) 6 82.4 E2 (= 440 ÷ 229/12 ≈ 82.407 Hz) == See also == * Fretted instruments * Musical acoustics * Vibrations of a circular drum * Melde's experiment * 3rd bridge (harmonic resonance based on equal string divisions) * String resonance * Reflection phase change == References == * * ;Specific == External links == * ""The Vibrating String"" by Alain Goriely and Mark Robertson-Tessi, The Wolfram Demonstrations Project. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z at a given excitation frequency. Hence one obtains Mersenne's laws: :f = \frac{v}{2L} = { 1 \over 2L } \sqrt{T \over \mu} where T is the tension (in Newtons), \mu is the linear density (that is, the mass per unit length), and L is the length of the vibrating part of the string. Ground vibration boom is a phenomenon of very large increase in ground vibrations generated by high-speed railway trains travelling at speeds higher than the velocity of Rayleigh surface waves in the supporting ground. == Technical background == thumb|Swedish high-speed train X 2000 approaching Ledsgard. Therefore: * the shorter the string, the higher the frequency of the fundamental * the higher the tension, the higher the frequency of the fundamental * the lighter the string, the higher the frequency of the fundamental Moreover, if we take the nth harmonic as having a wavelength given by \lambda_n = 2L/n, then we easily get an expression for the frequency of the nth harmonic: :f_n = \frac{nv}{2L} And for a string under a tension T with linear density \mu, then :f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}} == Observing string vibrations == One can see the waveforms on a vibrating string if the frequency is low enough and the vibrating string is held in front of a CRT screen such as one of a television or a computer (not of an analog oscilloscope). As a result, : \begin{align} & -\omega^2\, a + \omega\,\delta\,b + \alpha\,a + \tfrac34\,\beta\,a^3 + \tfrac34\,\beta\,a\,b^2 = \gamma \qquad \text{and} \\\ & -\omega^2\, b - \omega\,\delta\,a + \tfrac34\,\beta\,b^3 + \alpha\,b + \tfrac34\,\beta\,a^2\,b = 0. \end{align} Squaring both equations and adding leads to the amplitude frequency response: :\left[\left(\omega^2-\alpha-\frac{3}{4}\beta z^2\right)^{2}+\left(\delta\omega\right)^2\right]\,z^2=\gamma^{2}, as stated above. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. == Wave == The velocity of propagation of a wave in a string (v) is proportional to the square root of the force of tension of the string (T) and inversely proportional to the square root of the linear density (\mu) of the string: v = \sqrt{T \over \mu}. A pendulum wave is an elementary physics demonstration and kinetic art comprising a number of uncoupled simple pendulums with monotonically increasing lengths. In physics, the Toda oscillator is a special kind of nonlinear oscillator. File:Duffing frequency response.svg|alt=Frequency response as a function of for the Duffing equation, with and damping The dashed parts of the frequency response are unstable.[3]|Frequency response z/\gamma as a function of \omega/\sqrt{\alpha} for the Duffing equation, with \alpha=\gamma=1 and damping \delta=0.1. The equation is given by :\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t), where the (unknown) function x=x(t) is the displacement at time t, \dot{x} is the first derivative of x with respect to time, i.e. velocity, and \ddot{x} is the second time- derivative of x, i.e. acceleration. ",0.33333333,14.80,"""8.0""",-0.16,0.63,D
+"Find the shortest path between the $(x, y, z)$ points $(0,-1,0)$ and $(0,1,0)$ on the conical surface $z=1-\sqrt{x^2+y^2}$. What is the length of the path? Note: this is the shortest mountain path around a volcano.","In other words, the shortest path will be made by joining circular arcs of maximum curvature and straight lines. Given two points s and t, say on the surface of a convex polyhedron, the problem is to compute a shortest path that never leaves the surface and connects s with t. thumb|right|Example of a shortest path in a three-dimensional Euclidean space The Euclidean shortest path problem is a problem in computational geometry: given a set of polyhedral obstacles in a Euclidean space, and two points, find the shortest path between the points that does not intersect any of the obstacles. == Two dimensions == In two dimensions, the problem can be solved in polynomial time in a model of computation allowing addition and comparisons of real numbers, despite theoretical difficulties involving the numerical precision needed to perform such calculations. We also need to know what the actual shortest path is. There are simple geometric and analytical methods to compute the optimal path. Moving along each segment in this sequence for the appropriate length will form the shortest curve that joins a starting point A to a terminal point B with the desired tangents at each endpoint and that does not exceed the given curvature. Mount Short () is a mountain, 2,110 m, standing 1 mile (1.6 km) east of Sculpture Mountain, in the upper Rennick Glacier. There are many results on computing shortest paths which stays on a polyhedral surface. Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. thumb|The shortest-path graph with t=2 In mathematics and geographic information science, a shortest-path graph is an undirected graph defined from a set of points in the Euclidean plane. Finding the shortest path in a graph using optimal substructure; a straight line indicates a single edge; a wavy line indicates a shortest path between the two vertices it connects (among other paths, not shown, sharing the same two vertices); the bold line is the overall shortest path from start to goal. If p is truly the shortest path, then it can be split into sub-paths p1 from u to w and p2 from w to v such that these, in turn, are indeed the shortest paths between the corresponding vertices (by the simple cut-and-paste argument described in Introduction to Algorithms). In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This is termed as the weighted region problem in the literature. ==See also== * Shortest path problem, in a graph of edges and vertices * Any-angle path planning, in a grid space ==Notes== ==References== *. *. *. *. *. *. *. *. *. *. *. == External links == * Implementation of Euclidean Shortest Path algorithm in Digital Geometric Kernel software Category:Geometric algorithms Category:Computational geometry In 1957, Lester Eli Dubins (1920–2010) proved using tools from analysis that any such path will consist of maximum curvature and/or straight line segments. In the following pseudocode, `n` is the size of the board, `c(i, j)` is the cost function, and `min()` returns the minimum of a number of values: function minCost(i, j) if j < 1 or j > n return infinity else if i = 1 return c(i, j) else return min( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j) This function only computes the path cost, not the actual path. In geometry, the term Dubins path typically refers to the shortest curve that connects two points in the two-dimensional Euclidean plane (i.e. x-y plane) with a constraint on the curvature of the path and with prescribed initial and terminal tangents to the path, and an assumption that the vehicle traveling the path can only travel forward. The tangent direction of the path at initial and final points are constrained to lie within the specified intervals. Picking the square that holds the minimum value at each rank gives us the shortest path between rank `n` and rank `1`. The Dubins' path gives the shortest path joining two oriented points that is feasible for the wheeled-robot model. The edge set of the shortest-path graph varies based on a single parameter t ≥ 1. When the weight of an edge is defined as its Euclidean length raised to the power of the parameter t ≥ 1, the edge is present in the shortest-path graph if and only if it is the least weight path between its endpoints. == Properties of shortest-path graph == When the configuration parameter t goes to infinity, shortest-path graph become the minimum spanning tree of the point set. ",14,96.4365076099,"""2.534324263""",-0.10,6,C
 "A simple pendulum of length $b$ and bob with mass $m$ is attached to a massless support moving vertically upward with constant acceleration $a$. Determine the period for small oscillations.
-","The weight of the bob itself has little effect on the period of the pendulum. The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. Note that under the small-angle approximation, the period is independent of the amplitude ; this is the property of isochronism that Galileo discovered. ===Rule of thumb for pendulum length=== T_0 = 2\pi\sqrt{\frac \ell g} gives \ell = \frac{g}{\pi^2}\frac{T_0^2} 4. In this case the pendulum's period depends on its moment of inertia around the pivot point. First start by defining the torque on the pendulum bob using the force due to gravity. \boldsymbol{ \tau } = \mathbf{l} \times \mathbf{F}_\mathrm{g} , where is the length vector of the pendulum and is the force due to gravity. The expression for is of the same form as the conventional simple pendulum and gives a period of T = 2 \pi \sqrt{\frac{I} {mgL}} And a frequency of f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{mgL}{I}} If the initial angle is taken into consideration (for large amplitudes), then the expression for \alpha becomes: \alpha = \ddot{\theta} = -\frac{mgL \sin\theta}{I} and gives a period of: T = 4 \operatorname{K}\left(\sin^2\frac{\theta_0}{2}\right) \sqrt{\frac{I}{mgL}} where is the maximum angle of oscillation (with respect to the vertical) and is the complete elliptic integral of the first kind. == Physical interpretation of the imaginary period == The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. A common weight for the bob of a one second pendulum, widely used in grandfather clocks and many others, is around 2 kilograms. == See also == * Plumb-bob ==References== Category:Pendulums Deviation of the ""true"" period of a pendulum from the small-angle approximation of the period. The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian Mechanics. Therefore, :r = L \sin \theta \, Substituting this value for r yields a formula whose only varying parameter is the suspension angle θ: For small angles θ, cos(θ) ≈ 1; in which case :t \approx 2 \pi \sqrt { \frac {L} {g} } so that for small angles the period t of a conical pendulum is equal to the period of an ordinary pendulum of the same length. A bob is the mass on the end of a pendulum found most commonly, but not exclusively, in pendulum clocks. == Reason for use == Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a rod, with the top attached to a pivot so it can swing. This yields an alternative and faster-converging formula for the period: T = \frac{2\pi}{M\left(1, \cos\frac{\theta_0} 2 \right)} \sqrt\frac\ell g. Expressing the solutions in terms of \theta_1 and \theta_2 alone: \begin{align} \theta_1&=\frac{1}{2}A\cos(\omega_1t+\alpha)+\frac{1}{2}B\cos(\omega_2t+\beta) \\\ \theta_2&=\frac{1}{2}A\cos(\omega_1t+\alpha)-\frac{1}{2}B\cos(\omega_2t+\beta) \end{align} If the bobs are not given an initial push, then the condition \dot\theta_1(0)=\dot\theta_2(0)=0 requires \alpha=\beta=0, which gives (after some rearranging): \begin{align} A&=\theta_1(0)+\theta_2(0)\\\ B&=\theta_1(0)-\theta_2(0) \end{align} ==See also== *Harmonograph *Conical pendulum *Cycloidal pendulum *Double pendulum *Inverted pendulum *Kapitza's pendulum *Rayleigh–Lorentz pendulum *Elastic pendulum *Mathieu function *Pendulum equations (software) ==References== ==Further reading== * * * ==External links== *Mathworld article on Mathieu Function Category:Differential equations Category:Dynamical systems Category:Horology Category:Mathematical physics Mathematics Since there is no acceleration in the vertical direction, the vertical component of the tension in the string is equal and opposite to the weight of the bob: :T \cos \theta = mg \, These two equations can be solved for T/m and equated, thereby eliminating T and m and yielding the centripetal acceleration: :{g\tan\theta} = \frac {v^2} {r} A little rearrangement gives: :\frac{g} {\cos\theta} = \frac {v^2} {r\sin \theta} Since the speed of the pendulum bob is constant, it can be expressed as the circumference 2πr divided by the time t required for one revolution of the bob: : v = \frac {2\pi r}{t} Substituting the right side of this equation for v in the previous equation, we find: : \frac {g} {\cos \theta} = \frac {( \frac {2 \pi r} {t} )^2} {r \sin \theta} = \frac {(2 \pi)^2 r} {t^2 \sin \theta} Using the trigonometric identity tan(θ) = sin(θ) / cos(θ) and solving for t, the time required for the bob to travel one revolution is :t = 2 \pi \sqrt {\frac {r} {g \tan \theta}} In a practical experiment, r varies and is not as easy to measure as the constant string length L. r can be eliminated from the equation by noting that r, h, and L form a right triangle, with θ being the angle between the leg h and the hypotenuse L (see diagram). Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth ( = ) at initial angle 10 degrees is 4\sqrt{\frac{1\text{ m}}{g}}\ K\left(\sin\frac{10^\circ} {2} \right)\approx 2.0102\text{ s}. If it is assumed that the pendulum is released with zero angular velocity, the solution becomes The motion is simple harmonic motion where is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). File:Pendulum_0deg.gif|Initial angle of 0°, a stable equilibrium File:Pendulum_45deg.gif|Initial angle of 45° File:Pendulum_90deg.gif|Initial angle of 90° File:Pendulum_135deg.gif|Initial angle of 135° File:Pendulum_170deg.gif|Initial angle of 170° File:Pendulum_180deg.gif|Initial angle of 180°, unstable equilibrium == Damped, Driven Pendulum == The above discussion focuses on a pendulum bob only acted upon by the force of gravity. For now just consider the magnitude of the torque on the pendulum. |\boldsymbol{\tau}| = -mg\ell\sin\theta, where is the mass of the pendulum, is the acceleration due to gravity, is the length of the pendulum, and is the angle between the length vector and the force due to gravity. The bob has mass m and is suspended by a string of length L. thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. For a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without resistance, the length of the string of a seconds pendulum is equal to L = g/π2 where g is the acceleration due to gravity, with units of length per second squared, and L is the length of the string in the same units. ",1000,6.283185307,7200.0,0.396,362880,B
-"In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \mathrm{~km} / \mathrm{hr}$ and dropped the bales from a height of $80 \mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?","Prior to rural electrification, barns were equipped with a vertical pulley and a horizontal track along which a bale of hay was guided manually. ==See also== *Baler Category:Agricultural machinery The act of throwing the bales up to a higher level is called ""bucking"". thumb|220px|right|A 1950s hay elevator A hay elevator is an elevator that hauls bales of hay or straw up to a hayloft, the section of a barn used for hay storage. thumb|Hay hooks stuck into a haystack Hay bucking, or ""bucking hay"", is a type of manual labor where rectangular hay bales, ranging in weight from about , are stacked by hand in a field, in a storage area such as a barn, or stacked on a vehicle for transportation, such as a flatbed trailer or semi truck for delivery to where the hay is needed. The work is very strenuous and physically demanding, and is dependent upon using a proper technique in order to not become fatigued and avoid injury. thumb|left|A mechanical hay stacker Large quantities of small square bales are sometimes gathered with mechanical equipment such as a hay stacker, which can hold up to about 100 hay bales. thumb|Tractor with a bale handling implement thumb|Tractor carrying bales A bale handler is a generic term describing a piece of farm implement used to transport hay or straw bales. Bale handlers with hooks are used to move large and small bales. A typical hay elevator includes an open skeletal frame, with a chain that has dull 3-inch spikes every few feet along the chain to grab bales and drag them along. Bale spears can often move both round and square large bales. The term hay elevator also includes machinery involved in the stacking and storage of bales. Because the work is so labor-intensive, many farmers have taken to making multiple ton bales that are moved with machines. Shipping live cattle by truck was much more economical, humane and offered more options in routing cattle to auctions, feeders, and processors. Hay elevators are either ramped conveyor belts that bales rest on, or a mechanized pair of chains that holds bales taut between them. Cattle trails were carefully chosen to minimize distance and maximize feed to sustain and fatten cattle. thumb|Dangerous proximity of a hot air balloon to an overhead line. They can pinch, spear, hook and fork the bales, one or several at a time. Monty Python's Cow Tossing is a catapult-physics game. Prime Cut: Livestock Raising and Meatpacking in the United States 1607–1983. Workers are usually paid by the ton or by the number of bales. An apparatus known as an elevator is used to move the bales, conveyor belt style, to levels too high to buck them. Livestock transportation is the movement of livestock, by road, rail, ship, or air. The type of bale handling attachment will be built to handle the particular size and type of bale. ",38, -1,210.0,14,1.95 ,C
+","The weight of the bob itself has little effect on the period of the pendulum. The period depends on the length of the pendulum, and also to a slight degree on its weight distribution (the moment of inertia about its own center of mass) and the amplitude (width) of the pendulum's swing. Note that under the small-angle approximation, the period is independent of the amplitude ; this is the property of isochronism that Galileo discovered. ===Rule of thumb for pendulum length=== T_0 = 2\pi\sqrt{\frac \ell g} gives \ell = \frac{g}{\pi^2}\frac{T_0^2} 4. In this case the pendulum's period depends on its moment of inertia around the pivot point. First start by defining the torque on the pendulum bob using the force due to gravity. \boldsymbol{ \tau } = \mathbf{l} \times \mathbf{F}_\mathrm{g} , where is the length vector of the pendulum and is the force due to gravity. The expression for is of the same form as the conventional simple pendulum and gives a period of T = 2 \pi \sqrt{\frac{I} {mgL}} And a frequency of f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{mgL}{I}} If the initial angle is taken into consideration (for large amplitudes), then the expression for \alpha becomes: \alpha = \ddot{\theta} = -\frac{mgL \sin\theta}{I} and gives a period of: T = 4 \operatorname{K}\left(\sin^2\frac{\theta_0}{2}\right) \sqrt{\frac{I}{mgL}} where is the maximum angle of oscillation (with respect to the vertical) and is the complete elliptic integral of the first kind. == Physical interpretation of the imaginary period == The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. A common weight for the bob of a one second pendulum, widely used in grandfather clocks and many others, is around 2 kilograms. == See also == * Plumb-bob ==References== Category:Pendulums Deviation of the ""true"" period of a pendulum from the small-angle approximation of the period. The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian Mechanics. Therefore, :r = L \sin \theta \, Substituting this value for r yields a formula whose only varying parameter is the suspension angle θ: For small angles θ, cos(θ) ≈ 1; in which case :t \approx 2 \pi \sqrt { \frac {L} {g} } so that for small angles the period t of a conical pendulum is equal to the period of an ordinary pendulum of the same length. A bob is the mass on the end of a pendulum found most commonly, but not exclusively, in pendulum clocks. == Reason for use == Although a pendulum can theoretically be any shape, any rigid object swinging on a pivot, clock pendulums are usually made of a weight or bob attached to the bottom end of a rod, with the top attached to a pivot so it can swing. This yields an alternative and faster-converging formula for the period: T = \frac{2\pi}{M\left(1, \cos\frac{\theta_0} 2 \right)} \sqrt\frac\ell g. Expressing the solutions in terms of \theta_1 and \theta_2 alone: \begin{align} \theta_1&=\frac{1}{2}A\cos(\omega_1t+\alpha)+\frac{1}{2}B\cos(\omega_2t+\beta) \\\ \theta_2&=\frac{1}{2}A\cos(\omega_1t+\alpha)-\frac{1}{2}B\cos(\omega_2t+\beta) \end{align} If the bobs are not given an initial push, then the condition \dot\theta_1(0)=\dot\theta_2(0)=0 requires \alpha=\beta=0, which gives (after some rearranging): \begin{align} A&=\theta_1(0)+\theta_2(0)\\\ B&=\theta_1(0)-\theta_2(0) \end{align} ==See also== *Harmonograph *Conical pendulum *Cycloidal pendulum *Double pendulum *Inverted pendulum *Kapitza's pendulum *Rayleigh–Lorentz pendulum *Elastic pendulum *Mathieu function *Pendulum equations (software) ==References== ==Further reading== * * * ==External links== *Mathworld article on Mathieu Function Category:Differential equations Category:Dynamical systems Category:Horology Category:Mathematical physics Mathematics Since there is no acceleration in the vertical direction, the vertical component of the tension in the string is equal and opposite to the weight of the bob: :T \cos \theta = mg \, These two equations can be solved for T/m and equated, thereby eliminating T and m and yielding the centripetal acceleration: :{g\tan\theta} = \frac {v^2} {r} A little rearrangement gives: :\frac{g} {\cos\theta} = \frac {v^2} {r\sin \theta} Since the speed of the pendulum bob is constant, it can be expressed as the circumference 2πr divided by the time t required for one revolution of the bob: : v = \frac {2\pi r}{t} Substituting the right side of this equation for v in the previous equation, we find: : \frac {g} {\cos \theta} = \frac {( \frac {2 \pi r} {t} )^2} {r \sin \theta} = \frac {(2 \pi)^2 r} {t^2 \sin \theta} Using the trigonometric identity tan(θ) = sin(θ) / cos(θ) and solving for t, the time required for the bob to travel one revolution is :t = 2 \pi \sqrt {\frac {r} {g \tan \theta}} In a practical experiment, r varies and is not as easy to measure as the constant string length L. r can be eliminated from the equation by noting that r, h, and L form a right triangle, with θ being the angle between the leg h and the hypotenuse L (see diagram). Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth ( = ) at initial angle 10 degrees is 4\sqrt{\frac{1\text{ m}}{g}}\ K\left(\sin\frac{10^\circ} {2} \right)\approx 2.0102\text{ s}. If it is assumed that the pendulum is released with zero angular velocity, the solution becomes The motion is simple harmonic motion where is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). File:Pendulum_0deg.gif|Initial angle of 0°, a stable equilibrium File:Pendulum_45deg.gif|Initial angle of 45° File:Pendulum_90deg.gif|Initial angle of 90° File:Pendulum_135deg.gif|Initial angle of 135° File:Pendulum_170deg.gif|Initial angle of 170° File:Pendulum_180deg.gif|Initial angle of 180°, unstable equilibrium == Damped, Driven Pendulum == The above discussion focuses on a pendulum bob only acted upon by the force of gravity. For now just consider the magnitude of the torque on the pendulum. |\boldsymbol{\tau}| = -mg\ell\sin\theta, where is the mass of the pendulum, is the acceleration due to gravity, is the length of the pendulum, and is the angle between the length vector and the force due to gravity. The bob has mass m and is suspended by a string of length L. thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. For a point mass on a weightless string of length L swinging with an infinitesimally small amplitude, without resistance, the length of the string of a seconds pendulum is equal to L = g/π2 where g is the acceleration due to gravity, with units of length per second squared, and L is the length of the string in the same units. ",1000,6.283185307,"""7200.0""",0.396,362880,B
+"In the blizzard of ' 88 , a rancher was forced to drop hay bales from an airplane to feed her cattle. The plane flew horizontally at $160 \mathrm{~km} / \mathrm{hr}$ and dropped the bales from a height of $80 \mathrm{~m}$ above the flat range. She wanted the bales of hay to land $30 \mathrm{~m}$ behind the cattle so as to not hit them. How far behind the cattle should she push the bales out of the airplane?","Prior to rural electrification, barns were equipped with a vertical pulley and a horizontal track along which a bale of hay was guided manually. ==See also== *Baler Category:Agricultural machinery The act of throwing the bales up to a higher level is called ""bucking"". thumb|220px|right|A 1950s hay elevator A hay elevator is an elevator that hauls bales of hay or straw up to a hayloft, the section of a barn used for hay storage. thumb|Hay hooks stuck into a haystack Hay bucking, or ""bucking hay"", is a type of manual labor where rectangular hay bales, ranging in weight from about , are stacked by hand in a field, in a storage area such as a barn, or stacked on a vehicle for transportation, such as a flatbed trailer or semi truck for delivery to where the hay is needed. The work is very strenuous and physically demanding, and is dependent upon using a proper technique in order to not become fatigued and avoid injury. thumb|left|A mechanical hay stacker Large quantities of small square bales are sometimes gathered with mechanical equipment such as a hay stacker, which can hold up to about 100 hay bales. thumb|Tractor with a bale handling implement thumb|Tractor carrying bales A bale handler is a generic term describing a piece of farm implement used to transport hay or straw bales. Bale handlers with hooks are used to move large and small bales. A typical hay elevator includes an open skeletal frame, with a chain that has dull 3-inch spikes every few feet along the chain to grab bales and drag them along. Bale spears can often move both round and square large bales. The term hay elevator also includes machinery involved in the stacking and storage of bales. Because the work is so labor-intensive, many farmers have taken to making multiple ton bales that are moved with machines. Shipping live cattle by truck was much more economical, humane and offered more options in routing cattle to auctions, feeders, and processors. Hay elevators are either ramped conveyor belts that bales rest on, or a mechanized pair of chains that holds bales taut between them. Cattle trails were carefully chosen to minimize distance and maximize feed to sustain and fatten cattle. thumb|Dangerous proximity of a hot air balloon to an overhead line. They can pinch, spear, hook and fork the bales, one or several at a time. Monty Python's Cow Tossing is a catapult-physics game. Prime Cut: Livestock Raising and Meatpacking in the United States 1607–1983. Workers are usually paid by the ton or by the number of bales. An apparatus known as an elevator is used to move the bales, conveyor belt style, to levels too high to buck them. Livestock transportation is the movement of livestock, by road, rail, ship, or air. The type of bale handling attachment will be built to handle the particular size and type of bale. ",38, -1,"""210.0""",14,1.95 ,C
 "Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to $1 / e$ of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.
-","For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the system's equation of motion is : m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt{k m} or : c_c = 2 m \sqrt{\frac{k}{m}} = 2m \omega_n where : \omega_n = \sqrt{\frac{k}{m}} is the natural frequency of the system. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. == Derivation == Using the natural frequency of a harmonic oscillator \omega_n = \sqrt{{k}/{m}} and the definition of the damping ratio above, we can rewrite this as: : \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0. The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that : \zeta = \frac{1}{2 Q} = { \alpha \over \omega_n }. If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. ==Overview== Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. * Time constant: \tau = 1 / \lambda, the time for the amplitude to decrease by the factor of e. Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. * Q factor: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio definition == thumb|400px|upright=1.3|The effect of varying damping ratio on a second-order system. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. In a mass–spring system, with mass m and spring stiffness k, the natural angular frequency can be calculated as: \omega _0 =\sqrt{\frac{k}{m}} In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term , where for a real σ, and is a constant. The cycle per second is a once-common English name for the unit of frequency now known as the hertz (Hz). * Damping ratio: \zeta is a non-dimensional characterization of the decay rate relative to the frequency, approximately \zeta = \lambda / \omega, or exactly \zeta = \lambda / \sqrt{\lambda^2 + \omega^2} < 1. For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. ==Logarithmic decrement== thumb|400px|right| For underdamped vibrations, the damping ratio is also related to the logarithmic decrement \delta. Other important parameters include: * Frequency: f = \omega / (2\pi), the number of cycles per time unit. In LC and RLC circuits, its natural angular frequency can be calculated as: \omega _0 =\frac{1}{\sqrt{LC}} ==See also== * Fundamental frequency ==Footnotes== ==References== * * * * Category:Waves Category:Oscillation The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^{-\lambda t} \cos(\omega t - \varphi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\varphi is the phase angle at ; *\omega is the angular frequency. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network. In electronics, the long-term stability of an oscillator is the degree of uniformity of frequency over time, when the frequency is measured under identical environmental conditions, such as supply voltage, load, and temperature. ",0.18,432.07,3.54,0.9992093669,1.06,D
-What is the minimum escape velocity of a spacecraft from the moon?,"Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The Apollo 17 LRV traveled a cumulative distance of approximately in a total drive time of about four hours and twenty-six minutes; the greatest distance Cernan and Schmitt traveled from the lunar module was about . The return to lunar orbit took just over seven minutes. It struck the Moon just under 87 hours into the mission, triggering the seismometers from Apollo 12, 14, 15 and 16. Apollo 17 (December 7–19, 1972) was the final mission of NASA's Apollo program, the most recent time humans have set foot on the Moon or traveled beyond low Earth orbit. This remains the furthest distance any spacefarers have ever traveled away from the safety of a pressurizable spacecraft while on a planetary body, and also during an EVA of any type. thumb|right|Concept of LESS Lunar escape systems (LESS) were a series of emergency vehicles designed for never-flown long-duration Apollo missions. The mission broke several records for crewed spaceflight, including the longest crewed lunar landing mission (12 days, 14 hours), greatest distance from a spacecraft during an extravehicular activity of any type (7.6 kilometers or 4.7 miles), longest total duration of lunar-surface extravehicular activities (22 hours, 4 minutes), largest lunar-sample return (approximately 115 kg or 254 lb), longest time in lunar orbit (6 days, 4 hours), and greatest number of lunar orbits (75). == Crew and key Mission Control personnel == In 1969, NASA announced that the backup crew of Apollo 14 would be Gene Cernan, Ronald Evans, and former X-15 pilot Joe Engle. * George J. Hurt Jr, David B. Middleton, and Marion A. Wise, Development Of A Simulator For Studying Simplified Lunar Escape Systems, April 1971 * George J. Hurt Jr and David B. Middleton, Fixed-base Simulator Investigation Of Lightweight Vehicles For Lunar Escape To Orbit With Kinesthetic Attitude Control And Simplified Manual Guidance, June 1971 * David B. Middleton and George J. Hurt Jr, A Simulation Study Of Emergency Lunar Escape To Orbit Using Several Simplified Manual Guidance And Control Techniques, October 1971 ==References== ==External links== *False Steps - LESS: The Lunar Escape System *Simulation of the LESS on YouTube Category:Apollo program hardware It was a Luna E-8-5M spacecraft, the second of three to be launched. The Apollo 17 spacecraft reentered Earth's atmosphere and splashed down safely in the Pacific Ocean at 2:25 p.m. EST, from the recovery ship, . The CMP was given information regarding the lunar features he would overfly in the CSM and which he was expected to photograph. == Mission hardware and experiments == thumb|SA-512, Apollo 17's Saturn V rocket, on the launch pad awaiting liftoff, November 1972|alt=Saturn five rocket on a launch pat at dusk while cloudy outside. === Spacecraft and launch vehicle === The Apollo 17 spacecraft comprised CSM-114 (consisting of Command Module 114 (CM-114) and Service Module 114 (SM-114)); Lunar Module 12 (LM-12); a Spacecraft-Lunar Module Adapter (SLA) numbered SLA-21; and a Launch Escape System (LES). * Apollo 17 Mission Experiments Overview at the Lunar and Planetary Institute * Apollo 17 Voice Transcript Pertaining to the Geology of the Landing Site (PDF) by N. G. Bailey and G. E. Ulrich, United States Geological Survey, 1975 * ""Apollo Program Summary Report"" (PDF), NASA, JSC-09423, April 1975 * The Apollo Spacecraft: A Chronology NASA, NASA SP-4009 * * ""The Final Flight"" – Excerpt from the September 1973 issue of National Geographic magazine Category:Gene Cernan Category:Ronald Evans (astronaut) Category:Harrison Schmitt Category:1972 in the United States Category:Apollo program missions Category:Articles containing video clips Category:Extravehicular activity Category:Lunar rovers Category:Crewed missions to the Moon Category:Sample return missions Category:Soft landings on the Moon Category:Spacecraft launched in 1972 Category:Spacecraft which reentered in 1972 Category:Last events Category:December 1972 events Category:Spacecraft launched by Saturn rockets Category:1972 on the Moon On 20 March 2013, the asteroid passed 49 lunar distances or from Earth at a relative velocity of . Launched at 12:33 a.m. Eastern Standard Time (EST) on December 7, 1972, following the only launch-pad delay in the course of the whole Apollo program that was caused by a hardware problem, Apollo 17 was a ""J-type"" mission that included three days on the lunar surface, expanded scientific capability, and the use of the third Lunar Roving Vehicle (LRV). The Lunar Roving Vehicle allowed the astronauts to travel fairly quickly over a few miles, but an improved version of the LESS could allow rapid travel over much longer distances on rocket thrust. Luna E-8-5M No.412, also known as Luna Ye-8-5M No.412, and sometimes identified by NASA as Luna 1975A, was a Soviet spacecraft which was lost in a launch failure in 1975. (7888) 1993 UC is a near-Earth minor planet in the Apollo group. At approximately 160,000 nautical miles (184,000 mi; 296,000 km) from Earth, it was the third ""deep space"" EVA in history, performed at great distance from any planetary body. For one possible solution, NASA studied a number of low-cost, low-mass lunar escape systems (LESS) which could be carried on the lunar module as a backup, rather like a lifeboat on a ship. (7341) 1991 VK is a near-Earth minor planet in the Apollo group. At 3:46 a.m. EST, the S-IVB third stage was reignited for the 351-second trans-lunar injection burn to propel the spacecraft towards the Moon. ",132.9, 35.91,2688.0,24,2380,E
-"Find the value of the integral $\int_S \mathbf{A} \cdot d \mathbf{a}$, where $\mathbf{A}=x \mathbf{i}-y \mathbf{j}+z \mathbf{k}$ and $S$ is the closed surface defined by the cylinder $c^2=x^2+y^2$. The top and bottom of the cylinder are at $z=d$ and 0 , respectively.","Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f_x, f_y and f_z. == Theorems involving surface integrals == Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. == Dependence on parametrization == Let us notice that we defined the surface integral by using a parametrization of the surface S. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s, which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. This formula defines the integral on the left (note the dot and the vector notation for the surface element). To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Then, the surface integral of f on S is given by :\iint_D \left[ f_{z} ( \mathbf{r} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{r} (s,t)) \frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{r} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt where :{\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t)}\right) is the surface element normal to S. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. This is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. The integral of v on S was defined in the previous section. The surface integral can also be expressed in the equivalent form : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt where is the determinant of the first fundamental form of the surface mapping . Then, the surface integral is given by : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). The differential vector area is , on each surface a, b and c. right|frame|Closed surface in the form of a cylinder having line charge in the center and showing differential areas of all three surfaces. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. The obvious solution is then to split that surface into several pieces, calculate the surface integral on each piece, and then add them all up. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. ", 9.73,0.5061,2.0,3.141592,3920.70763168,D
+","For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: : \zeta = \frac{c}{c_c} = \frac {\text{actual damping}} {\text{critical damping}}, where the system's equation of motion is : m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 and the corresponding critical damping coefficient is : c_c = 2 \sqrt{k m} or : c_c = 2 m \sqrt{\frac{k}{m}} = 2m \omega_n where : \omega_n = \sqrt{\frac{k}{m}} is the natural frequency of the system. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. == Derivation == Using the natural frequency of a harmonic oscillator \omega_n = \sqrt{{k}/{m}} and the definition of the damping ratio above, we can rewrite this as: : \frac{d^2x}{dt^2} + 2\zeta\omega_n\frac{dx}{dt} + \omega_n^2 x = 0. The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta), that characterizes the frequency response of a second-order ordinary differential equation. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that : \zeta = \frac{1}{2 Q} = { \alpha \over \omega_n }. If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. ==Overview== Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. * Time constant: \tau = 1 / \lambda, the time for the amplitude to decrease by the factor of e. Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. * Q factor: Q = 1 / (2 \zeta) is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio definition == thumb|400px|upright=1.3|The effect of varying damping ratio on a second-order system. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. In a mass–spring system, with mass m and spring stiffness k, the natural angular frequency can be calculated as: \omega _0 =\sqrt{\frac{k}{m}} In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term , where for a real σ, and is a constant. The cycle per second is a once-common English name for the unit of frequency now known as the hertz (Hz). * Damping ratio: \zeta is a non-dimensional characterization of the decay rate relative to the frequency, approximately \zeta = \lambda / \omega, or exactly \zeta = \lambda / \sqrt{\lambda^2 + \omega^2} < 1. For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. ==Logarithmic decrement== thumb|400px|right| For underdamped vibrations, the damping ratio is also related to the logarithmic decrement \delta. Other important parameters include: * Frequency: f = \omega / (2\pi), the number of cycles per time unit. In LC and RLC circuits, its natural angular frequency can be calculated as: \omega _0 =\frac{1}{\sqrt{LC}} ==See also== * Fundamental frequency ==Footnotes== ==References== * * * * Category:Waves Category:Oscillation The general equation for an exponentially damped sinusoid may be represented as: y(t) = A e^{-\lambda t} \cos(\omega t - \varphi) where: *y(t) is the instantaneous amplitude at time ; *A is the initial amplitude of the envelope; *\lambda is the decay rate, in the reciprocal of the time units of the independent variable ; *\varphi is the phase angle at ; *\omega is the angular frequency. The damping ratio is a system parameter, denoted by (zeta), that can vary from undamped (), underdamped () through critically damped () to overdamped (). It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network. In electronics, the long-term stability of an oscillator is the degree of uniformity of frequency over time, when the frequency is measured under identical environmental conditions, such as supply voltage, load, and temperature. ",0.18,432.07,"""3.54""",0.9992093669,1.06,D
+What is the minimum escape velocity of a spacecraft from the moon?,"Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The Apollo 17 LRV traveled a cumulative distance of approximately in a total drive time of about four hours and twenty-six minutes; the greatest distance Cernan and Schmitt traveled from the lunar module was about . The return to lunar orbit took just over seven minutes. It struck the Moon just under 87 hours into the mission, triggering the seismometers from Apollo 12, 14, 15 and 16. Apollo 17 (December 7–19, 1972) was the final mission of NASA's Apollo program, the most recent time humans have set foot on the Moon or traveled beyond low Earth orbit. This remains the furthest distance any spacefarers have ever traveled away from the safety of a pressurizable spacecraft while on a planetary body, and also during an EVA of any type. thumb|right|Concept of LESS Lunar escape systems (LESS) were a series of emergency vehicles designed for never-flown long-duration Apollo missions. The mission broke several records for crewed spaceflight, including the longest crewed lunar landing mission (12 days, 14 hours), greatest distance from a spacecraft during an extravehicular activity of any type (7.6 kilometers or 4.7 miles), longest total duration of lunar-surface extravehicular activities (22 hours, 4 minutes), largest lunar-sample return (approximately 115 kg or 254 lb), longest time in lunar orbit (6 days, 4 hours), and greatest number of lunar orbits (75). == Crew and key Mission Control personnel == In 1969, NASA announced that the backup crew of Apollo 14 would be Gene Cernan, Ronald Evans, and former X-15 pilot Joe Engle. * George J. Hurt Jr, David B. Middleton, and Marion A. Wise, Development Of A Simulator For Studying Simplified Lunar Escape Systems, April 1971 * George J. Hurt Jr and David B. Middleton, Fixed-base Simulator Investigation Of Lightweight Vehicles For Lunar Escape To Orbit With Kinesthetic Attitude Control And Simplified Manual Guidance, June 1971 * David B. Middleton and George J. Hurt Jr, A Simulation Study Of Emergency Lunar Escape To Orbit Using Several Simplified Manual Guidance And Control Techniques, October 1971 ==References== ==External links== *False Steps - LESS: The Lunar Escape System *Simulation of the LESS on YouTube Category:Apollo program hardware It was a Luna E-8-5M spacecraft, the second of three to be launched. The Apollo 17 spacecraft reentered Earth's atmosphere and splashed down safely in the Pacific Ocean at 2:25 p.m. EST, from the recovery ship, . The CMP was given information regarding the lunar features he would overfly in the CSM and which he was expected to photograph. == Mission hardware and experiments == thumb|SA-512, Apollo 17's Saturn V rocket, on the launch pad awaiting liftoff, November 1972|alt=Saturn five rocket on a launch pat at dusk while cloudy outside. === Spacecraft and launch vehicle === The Apollo 17 spacecraft comprised CSM-114 (consisting of Command Module 114 (CM-114) and Service Module 114 (SM-114)); Lunar Module 12 (LM-12); a Spacecraft-Lunar Module Adapter (SLA) numbered SLA-21; and a Launch Escape System (LES). * Apollo 17 Mission Experiments Overview at the Lunar and Planetary Institute * Apollo 17 Voice Transcript Pertaining to the Geology of the Landing Site (PDF) by N. G. Bailey and G. E. Ulrich, United States Geological Survey, 1975 * ""Apollo Program Summary Report"" (PDF), NASA, JSC-09423, April 1975 * The Apollo Spacecraft: A Chronology NASA, NASA SP-4009 * * ""The Final Flight"" – Excerpt from the September 1973 issue of National Geographic magazine Category:Gene Cernan Category:Ronald Evans (astronaut) Category:Harrison Schmitt Category:1972 in the United States Category:Apollo program missions Category:Articles containing video clips Category:Extravehicular activity Category:Lunar rovers Category:Crewed missions to the Moon Category:Sample return missions Category:Soft landings on the Moon Category:Spacecraft launched in 1972 Category:Spacecraft which reentered in 1972 Category:Last events Category:December 1972 events Category:Spacecraft launched by Saturn rockets Category:1972 on the Moon On 20 March 2013, the asteroid passed 49 lunar distances or from Earth at a relative velocity of . Launched at 12:33 a.m. Eastern Standard Time (EST) on December 7, 1972, following the only launch-pad delay in the course of the whole Apollo program that was caused by a hardware problem, Apollo 17 was a ""J-type"" mission that included three days on the lunar surface, expanded scientific capability, and the use of the third Lunar Roving Vehicle (LRV). The Lunar Roving Vehicle allowed the astronauts to travel fairly quickly over a few miles, but an improved version of the LESS could allow rapid travel over much longer distances on rocket thrust. Luna E-8-5M No.412, also known as Luna Ye-8-5M No.412, and sometimes identified by NASA as Luna 1975A, was a Soviet spacecraft which was lost in a launch failure in 1975. (7888) 1993 UC is a near-Earth minor planet in the Apollo group. At approximately 160,000 nautical miles (184,000 mi; 296,000 km) from Earth, it was the third ""deep space"" EVA in history, performed at great distance from any planetary body. For one possible solution, NASA studied a number of low-cost, low-mass lunar escape systems (LESS) which could be carried on the lunar module as a backup, rather like a lifeboat on a ship. (7341) 1991 VK is a near-Earth minor planet in the Apollo group. At 3:46 a.m. EST, the S-IVB third stage was reignited for the 351-second trans-lunar injection burn to propel the spacecraft towards the Moon. ",132.9, 35.91,"""2688.0""",24,2380,E
+"Find the value of the integral $\int_S \mathbf{A} \cdot d \mathbf{a}$, where $\mathbf{A}=x \mathbf{i}-y \mathbf{j}+z \mathbf{k}$ and $S$ is the closed surface defined by the cylinder $c^2=x^2+y^2$. The top and bottom of the cylinder are at $z=d$ and 0 , respectively.","Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f_x, f_y and f_z. == Theorems involving surface integrals == Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. == Dependence on parametrization == Let us notice that we defined the surface integral by using a parametrization of the surface S. For surface c, and will be parallel, as shown in the figure. \begin{align} \Phi_E & = \iint_a E dA\cos 90^\circ + \iint_b E d A \cos 90^\circ + \iint_c E d A\cos 0^\circ \\\ & = E \iint_c dA \end{align} The surface area of the cylinder is \iint_c dA = 2 \pi r h which implies \Phi_E = E 2 \pi r h. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s, which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. This formula defines the integral on the left (note the dot and the vector notation for the surface element). To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. If is the length of the cylinder, then the charge enclosed in the cylinder is q = \lambda h , where is the charge enclosed in the Gaussian surface. Then, the surface integral of f on S is given by :\iint_D \left[ f_{z} ( \mathbf{r} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{r} (s,t)) \frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{r} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt where :{\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t)}\right) is the surface element normal to S. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . The flux out of the spherical surface is: : The surface area of the sphere of radius is \iint_S dA = 4 \pi r^2 which implies \Phi_E = E 4\pi r^2 By Gauss's law the flux is also \Phi_E =\frac{Q_A}{\varepsilon_0} finally equating the expression for gives the magnitude of the -field at position : E 4\pi r^2 = \frac{Q_A}{\varepsilon_0} \quad \Rightarrow \quad E=\frac{Q_A}{4\pi\varepsilon_0r^2}. If the Gaussian surface is chosen such that for every point on the surface the component of the electric field along the normal vector is constant, then the calculation will not require difficult integration as the constants which arise can be taken out of the integral. This is indeed how things work, but when integrating vector fields, one needs to again be careful how to choose the normal-pointing vector for each piece of the surface, so that when the pieces are put back together, the results are consistent. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. The integral of v on S was defined in the previous section. The surface integral can also be expressed in the equivalent form : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt where is the determinant of the first fundamental form of the surface mapping . Then, the surface integral is given by : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). The differential vector area is , on each surface a, b and c. right|frame|Closed surface in the form of a cylinder having line charge in the center and showing differential areas of all three surfaces. If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. The obvious solution is then to split that surface into several pieces, calculate the surface integral on each piece, and then add them all up. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. ", 9.73,0.5061,"""2.0""",3.141592,3920.70763168,D
 "A rocket has an initial mass of $7 \times 10^4 \mathrm{~kg}$ and on firing burns its fuel at a rate of 250 $\mathrm{kg} / \mathrm{s}$. The exhaust velocity is $2500 \mathrm{~m} / \mathrm{s}$. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off? What is wrong with the design of this rocket?
-","# A two-stage rocket with a length of 6 metres and a takeoff thrust of 50 kN. # A three-stage rocket with a length of 12.8 metres, a diameter of 0.56 metres and a takeoff thrust of 50 kN. In rocketry, the Goddard problem is to optimize the peak altitude of a rocket, ascending vertically, and taking into account atmospheric drag and the gravitational field. This rocket was first launched on November 19, 1962, near Cuxhaven and reached a height of 40 km. For miniature black powder rocket motors (13 mm diameter), the maximum thrust is between 5 and 12 N, the total impulse is between .5 and 2.2 Ns, and the burn time is between .25 and 1 second. Some rockets (typically long thin rockets) are the proper proportions to safely glide to Earth tail-first. This rocket was first launched on February 7, 1963, and reached a height of 80 km. This rocket was first launched on May 2, 1963, with reduced fuel and reached an altitude of 110 km. thumb|The Rocket The Rocket (previously The Rising Sun) is a Grade II listed public house at 120 Euston Road, Euston, London NW1 2AL. thumb|250px|Picture sequence of a model rocket launch using a B4-4 engine thumb|250px|A typical model rocket during launch (16 times slower) A model rocket is a small rocket designed to reach low altitudes (e.g., for model) and be recovered by a variety of means. Many science fiction authors as well as depictions in popular culture showed rockets landing vertically, typically resting after landing on the space vehicle's fins. Kappa is a family of solid-fuel Japanese sounding rockets, which were built starting from 1956. ==Rockets== ===Kappa 1=== * Ceiling: 40 km * Takeoff thrust: 10.00 kN * Diameter: 0.13 m * Length: 2.70 m ===Kappa 2=== * Ceiling: 40 km * Mass: 300 kg * Diameter: 0.22 m * Length: 5 m ===Kappa 6 (in two stages)=== * Pay load: 20 kg * Ceiling: 60 km * Takeoff weight: 270 kg * Diameter: 0.25 m * Length: 5.61 m ===Kappa 7=== * Ceiling: 50 km * Diameter: 0.42 m * Length: 8.70 m ===Kappa 8 (in two stages)=== * Pay load: 50 kg * Ceiling: 160 km * Takeoff weight: 1500 kg * Diameter: 0.42 m * Length: 10.90 m ===Kappa 4=== * Ceiling: 80 km * Takeoff thrust: 105.00 kN * Diameter: 0.33 m * Length: 5.90 m ===Kappa 9L=== * Pay load: 15 kg * Ceiling: 350 km * Takeoff weight: 1550 kg * Diameter: 0.42 m * Length: 12.50 m ===Kappa 9M=== * Pay load: 50 kg * Ceiling: 350 km * Mass: 1500 kg * Diameter: 0.42 m * Length: 11.10 m ===Kappa 8L=== * Pay load: 25 kg * Ceiling: 200 km * Takeoff weight: 350 kg * Diameter: 0.25 m * Length: 7.30 m ===Kappa 10=== * Ceiling: 742 km ==See also== * R-25 Vulkan ==External links== * Kappa-Rocket Category:Solid- fuel rockets Category:Sounding rockets of Japan Category:Japanese inventions The spacecraft stopped mid-air again and, as the engines throttled back, began its successful vertical landing. For example, a heavier rocket would require an engine with more initial thrust to get it off of the launch pad, whereas a lighter rocket would need less initial thrust and would sustain a longer burn, reaching higher altitudes. ===Last number=== The last number is the delay in seconds between the end of the thrust phase and ignition of the ejection charge. University of New South Wales at the Australian Defence Force Academy. 2008.Measuring thrust and predicting trajectory in model rocketry M. Courtney and A. Courtney. Later with maximum fuel it reached a height of 150 km. They were # A single- stage rocket with a length of 3.4 metres and a takeoff thrust of 50 kN. *On July 20, 2021, Blue Origin's New Shepard rocket made its first-ever successful vertical landing following a crewed suborbital flight. All Seliger Rockets return to the ground by parachute. The D class 24 mm motors have a maximum thrust between 29.7 and 29.8 N, a total impulse between 16.7 and 16.85 Ns, and a burn time between 1.6 and 1.7 seconds. The E class 24 mm motors have a maximum thrust between 19.4 and 19.5 N, a total impulse between 28.45 and 28.6 Ns, and a burn time between 3 and 3.1 seconds. *On December 21, 2015, SpaceX's 20th Falcon 9 first stage made the first-ever successful vertical landing of an orbital-class booster after boosting 11 commercial satellites to low Earth orbit on Falcon 9 Flight 20. ",25,10.065778,22.0,0,0.0000092,A
-"A spacecraft of mass $10,000 \mathrm{~kg}$ is parked in a circular orbit $200 \mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\tau=24 \mathrm{hr}$ )?","The specific orbital energy associated with this orbit is −29.6MJ/kg: the potential energy is −59.2MJ/kg, and the kinetic energy 29.6MJ/kg. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced. ===Satellite=== thumb|plot of kinetic energy K, gravitational potential energy, U and mechanical energy E_\text{mechanical} versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, (by virtue of its motion) and gravitational potential energy, U, (by virtue of its position within the Earth's gravitational field; Earth's mass is M). Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r} since in circular motion, Newton's 2nd Law of motion can be taken to be G \frac{M m}{r^2}\ = \frac{m v^2}{r} ==Conversion== Today, many technological devices convert mechanical energy into other forms of energy or vice versa. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. For the Earth and a just little more than R the additional specific energy is (gR/2); which is the kinetic energy of the horizontal component of the velocity, i.e. \frac{1}{2}V^2 = \frac{1}{2}gR, V=\sqrt{gR}. ==Examples== ===ISS=== The International Space Station has an orbital period of 91.74 minutes (5504s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738km. For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy. ==Additional energy== If the central body has radius R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is -\frac{\mu}{2a}+\frac{\mu}{R} = \frac{\mu(2a-R)}{2aR} The quantity 2a-R is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divided by the reduced mass. Finally, a circularization burn is required to raise the perigee to the same altitude and remove any remaining inclination. ===Translunar or interplanetary spacecraft=== In order to reach the Moon or a planet at a desired time, the spacecraft must be launched within a limited range of times known as the launch window. A parking orbit is a temporary orbit used during the launch of a spacecraft. For an altitude of 100km (radius is 6471km): The energy is −30.8MJ/kg: the potential energy is −61.6MJ/kg, and the kinetic energy 30.8MJ/kg. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. Unless the launch site itself is quite close to the equator, it requires an impractically large amount of fuel to launch a spacecraft directly into such an orbit. Indian Mini Satellite (IMS) is a family of modular \- mini satellite buses developed by the Indian Space Research Organisation (ISRO). ==Variants== Indian Mini Satellite (Variants) Feature IMS-1 IMS-2 IMS-3 (Planned IMS-2 Derivative) Launch Mass Maximum bus mass Payload mass Propellant Design lifetime 2 years 5 years Raw bus voltage 28-33 Volts 28-33 Volts 28-42 Volts Solar Array Power 330 Watts (EOL) 675 Watts (EOL) 850 Watts (BOL) 850 Watts (BOL) Payload power 30 Watts (Continuous) 70 Watts (Duty Cycle) 250 Watts (Continuous) 600 Watts (Duty Cycle) 250 Watts (Continuous) 400 Watts (Duty Cycle) Attitude Control 3-axis stabilized Four Reaction Wheels Single 1N thruster 3-axis stabilized Four Reaction Wheels Mono-propellant RCS Four 1N thrusters Four 0.2N thrusters Pointing Accuracy ±0.1° (3σ) (all axes) ± 0.1° (all axes) ± 0.1° (all axes) SSR Storage 32 Gb 32 Gb (SDRAM) 256 Gb (Flash Memory) 32 Gb (SDRAM) 256 Gb (Flash Memory) Payload data storage ≤ 16 Gb ≤ 32 Gb Downlink ≤ 8 Mbit/s DL rate ≤ 105 Mbit/s DL rate ≤ 160 Mbit/s DL rate Missions IMS-1 Youthsat Microsat-TD SARAL \- ScatSat-1 EMISAT HySIS XPoSat (Planned) center|thumb|150x150px|IMS-1 ==See also== * Comparison of satellite buses ==References== Category:Indian Space Research Organisation Category:Satellite buses Thus, if orbital position vector (\mathbf{r}) and orbital velocity vector (\mathbf{v}) are known at one position, and \mu is known, then the energy can be computed and from that, for any other position, the orbital speed. ==Rate of change== For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is \frac{\mu}{2a^2} where * \mu={G}(m_1 + m_2) is the standard gravitational parameter; *a\,\\! is semi- major axis of the orbit. The speed is 7.8km/s, the net delta-v to reach this orbit is 8.0km/s. The average speed is 7.7km/s, the net delta-v to reach this orbit is 8.1km/s (the actual delta-v is typically 1.5–2.0km/s more for atmospheric drag and gravity drag). A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: \begin{align} \varepsilon &= \varepsilon_k + \varepsilon_p \\\ &= \frac{v^2}{2} - \frac{\mu}{r} = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right) = -\frac{\mu}{2a} \end{align} where *v is the relative orbital speed; *r is the orbital distance between the bodies; *\mu = {G}(m_1 + m_2) is the sum of the standard gravitational parameters of the bodies; *h is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; *e is the orbital eccentricity; *a is the semi- major axis. ",0.086,1068,2.57,3.00,3930,C
-A uniformly solid sphere of mass $M$ and radius $R$ is fixed a distance $h$ above a thin infinite sheet of mass density $\rho_5$ (mass/area). With what force does the sphere attract the sheet?,"This force depends on the surface separation h. Since the hydrodynamic drag of a sphere close to a planar substrate is known theoretically, the spring constant of the cantilever can be deduced. In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. The mass distribution of a solid defines its center of gravity and influences its dynamical behaviour - e.g. the oscillations and eventual rotation. ==Mathematical modelling== A mass distribution can be modeled as a measure. Similar use of the equation can be made in the settling of fine particles in water or other fluids. === Terminal velocity of sphere falling in a fluid === At terminal (or settling) velocity, the excess force due to the difference between the weight and buoyancy of the sphere (both caused by gravity) is given by: :F_g = ( \rho_p - \rho_f)\, g\, \frac{4}{3}\pi\, R^3, where (in SI units): * is the mass density of the sphere [kg/m3] * is the mass density of the fluid [kg/m3] * is the gravitational acceleration [m/s] Requiring the force balance and solving for the velocity gives the terminal velocity . A sphere of known size and density is allowed to descend through the liquid. An analytical and closed- form solution for both 2D and 3D, mechanically stable, random packings of spheres has been found by A. Zaccone in 2022 using the assumption that the most random branch of jammed states (maximally random jammed packings, extending up to the fcc closest packing) undergo crowding in a way qualitatively similar to an equilibrium liquid. Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. thumb|right|280px|Scheme of the colloidal probe technique for direct force measurements in the sphere-plane and sphere-sphere geometries. The random close packing value is significantly below the maximum possible close-packing of same-size hard spheres into a regular crystalline arrangements, which is 74.04%.Modes of wall induced granular crystallisation in vibrational packing.Granular Matter, 21(2), 26 Both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of granular crystallisation. The distance to the sphere of influence must thus satisfy \frac{m_B}{m_A} \frac{r^3}{R^3} = \frac{m_A}{m_B} \frac{R^2}{r^2} and so r = R\left(\frac{m_A}{m_B}\right)^{2/5} is the radius of the sphere of influence of body A ==See also== * Hill sphere * Sphere of influence (black hole) ==References== == General references == * * * == External links == *Project Pluto Category:Astrodynamics Category:Orbits When the force is attractive, the cantilever is attracted to the surface and may become unstable. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.Batchelor (1967), p. 233. ==Statement of the law== The force of viscosity on a small sphere moving through a viscous fluid is given by: :F_{\rm d} = 6 \pi \mu R v where (in SI units): * is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle (newtons, kg m s−2); * (some authors use the symbol ) is the dynamic viscosity (Pascal-seconds, kg m−1 s−1); * is the radius of the spherical object (meters); * is the flow velocity relative to the object (meters per second). A sphere of influence (SOI) in astrodynamics and astronomy is the oblate- spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. The problem of predicting theoretically the random close pack of spheres is difficult mainly because of the absence of a unique definition of randomness or disorder. In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. The matrix represents the identity-matrix. :\boldsymbol{\sigma} = - p \cdot \mathbf{I} + \mu \cdot \left(( abla \mathbf{u}) + ( abla \mathbf{u})^T \right) The force acting on the sphere is to calculate by surface-integral, where represents the radial unit-vector of spherical-coordinates: :\begin{align} \mathbf{F} &= \iint_{\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset \;\boldsymbol{\sigma}\cdot \text{d}\mathbf{S} \\\\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \boldsymbol{\sigma}\cdot \mathbf{e_r}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\\\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \frac{3\mu \cdot \mathbf{u}_{\infty}}{2 R}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\\\[4pt] &= 6\pi\mu R \cdot \mathbf{u}_{\infty} \end{align} === Rotational flow around a sphere === thumb|434x434px|Stokes-Flow around sphere: \boldsymbol{\omega}_R = \begin{pmatrix} 0 & 0 & 2 \end{pmatrix}^T \; \text{Hz} , \mu = 1 \; \text{mPa} \cdot \text{s}, R = 1 \; \text{m} :\begin{align} \mathbf{u}(\mathbf{x}) &= - \;R^3 \cdot \frac{ \boldsymbol{\omega}_{R} \times \mathbf{x}}{\|\mathbf{x}\|^3} \\\\[8pt] \boldsymbol{\omega}(\mathbf{x}) &= \frac{R^3 \cdot \boldsymbol{\omega}_{R}}{\|\mathbf{x}\|^3} - \frac{3 R^3 \cdot (\boldsymbol{\omega}_{R} \cdot \mathbf{x})\cdot \mathbf{x}}{\|\mathbf{x}\|^5} \\\\[8pt] p(\mathbf{x}) &= 0 \\\\[8pt] \boldsymbol{\sigma} &= - p \cdot \mathbf{I} + \mu \cdot \left( ( abla \mathbf{u}) + ( abla \mathbf{u})^T \right) \\\\[8pt] \mathbf{T} &= \iint_{\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset \mathbf{x} \times \left( \boldsymbol{\sigma} \cdot \text{d}\boldsymbol{S} \right) \\\ &= \int_{0}^{\pi} \int_{0}^{2\pi} (R \cdot \mathbf{e_r}) \times \left( \boldsymbol{\sigma} \cdot \mathbf{e_r} \cdot R^2 \sin\theta \text{d}\varphi \text{d}\theta \right) \\\ &= 8\pi\mu R^3 \cdot \boldsymbol{\omega}_{R} \end{align} ==Other types of Stokes flow== Although the liquid is static and the sphere is moving with a certain velocity, with respect to the frame of sphere, the sphere is at rest and liquid is flowing in the opposite direction to the motion of the sphere. ==See also== * Einstein relation (kinetic theory) * Scientific laws named after people * Drag equation * Viscometry * Equivalent spherical diameter * Deposition (geology) == Sources == * * Originally published in 1879, the 6th extended edition appeared first in 1932. ==References== Category:Fluid dynamics In the case of a soft repulsive force, the cantilever is repelled from the surface and only slowly approaches the constant compliance region. Experiments and computer simulations have shown that the most compact way to pack hard perfect same- size spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. It is also possible to study forces between colloidal particles by attaching another particle to the substrate and perform the measurement in the sphere-sphere geometry, see figure above. thumb|left|350px|Principle of the force measurements by the colloidal probe technique. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity in the –direction and a sphere of radius , the solution is found to beLamb (1994), §337, p. 598. : \psi(r,z) = - \frac{1}{2}\, u\, r^2\, \left[ 1 \- \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}} \+ \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\; \right]. thumb|420x420px|Stokes-Flow around sphere with parameters of Far-Field velocity \mathbf{u}_{\infty} = \begin{pmatrix} 6 & 0 & 6 \end{pmatrix}^T \text{m/s}, radius of sphere R = 1 \; \text{m}, viscosity of water (T = 20°C) \mu = 1 \; \text{mPa}\cdot \text{s} . ",418,6.283185307,1.0,209.1,8.7,B
-Consider a thin rod of length $l$ and mass $m$ pivoted about one end. Calculate the moment of inertia. ,"* The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end. * The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point P as, I_P = I_{C, \text{rod}} + M_\text{rod}\left(\frac{L}{2}\right)^2 + I_{C, \text{disc}} + M_\text{disc}(L + R)^2, where L is the length of the pendulum. The moment of inertia I_\mathbf{C} about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. The quantity I = mr^2 is the moment of inertia of this single mass around the pivot point. Description Figure Moment(s) of inertia Point mass M at a distance r from the axis of rotation. File:moment of inertia thin cylinder.png I \approx m r^2 \,\\! For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is I = \frac{L}{\omega}. File:moment of inertia thick cylinder h.svg I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right) Classical Mechanics - Moment of inertia of a uniform hollow cylinder . Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point P so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (t), to obtain I_P = \frac{mgr}{\omega_\text{n}^2} = \frac{mgrt^2}{4\pi^2}, where t is the period (duration) of oscillation (usually averaged over multiple periods). ====Center of oscillation==== A simple pendulum that has the same natural frequency as a compound pendulum defines the length L from the pivot to a point called the center of oscillation of the compound pendulum. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as, I = mr^2. File:moment of inertia solid rectangular prism.png I_h = \frac{1}{12} m \left(w^2+d^2\right) I_w = \frac{1}{12} m \left(d^2+h^2\right) I_d = \frac{1}{12} m \left(w^2+h^2\right) Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. Align the x-axis with the rod and locate the origin its center of mass at the center of the rod, then I_{C, \text{rod}} = \iiint_Q \rho\,x^2 \, dV = \int_{-\frac{\ell}{2}}^\frac{\ell}{2} \rho\,x^2 s\, dx = \left. \rho s\frac{x^3}{3}\right|_{-\frac{\ell}{2}}^\frac{\ell}{2} = \frac{\rho s}{3} \left(\frac{\ell^3}{8} + \frac{\ell^3}{8}\right) = \frac{m\ell^2}{12}, where m = \rho s \ell is the mass of the rod. Also, a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration. The moment of inertia of an arbitrarily shaped body is the sum of the values mr^2 for all of the elements of mass in the body. === Compound pendulums === A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the x-axis or y-axis depending on the load. ==== Examples ==== thumb|right The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. The moments of inertia of a mass have units of dimension ML2([mass] × [length]2). A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. frameless|upright I = M r^2 Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles. frameless|upright I = \frac{ m_1 m_2 }{ m_1 \\! + \\! m_2 } x^2 = \mu x^2 Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center. ",11000,0.66,240.0,0.33333333,4,D
+","# A two-stage rocket with a length of 6 metres and a takeoff thrust of 50 kN. # A three-stage rocket with a length of 12.8 metres, a diameter of 0.56 metres and a takeoff thrust of 50 kN. In rocketry, the Goddard problem is to optimize the peak altitude of a rocket, ascending vertically, and taking into account atmospheric drag and the gravitational field. This rocket was first launched on November 19, 1962, near Cuxhaven and reached a height of 40 km. For miniature black powder rocket motors (13 mm diameter), the maximum thrust is between 5 and 12 N, the total impulse is between .5 and 2.2 Ns, and the burn time is between .25 and 1 second. Some rockets (typically long thin rockets) are the proper proportions to safely glide to Earth tail-first. This rocket was first launched on February 7, 1963, and reached a height of 80 km. This rocket was first launched on May 2, 1963, with reduced fuel and reached an altitude of 110 km. thumb|The Rocket The Rocket (previously The Rising Sun) is a Grade II listed public house at 120 Euston Road, Euston, London NW1 2AL. thumb|250px|Picture sequence of a model rocket launch using a B4-4 engine thumb|250px|A typical model rocket during launch (16 times slower) A model rocket is a small rocket designed to reach low altitudes (e.g., for model) and be recovered by a variety of means. Many science fiction authors as well as depictions in popular culture showed rockets landing vertically, typically resting after landing on the space vehicle's fins. Kappa is a family of solid-fuel Japanese sounding rockets, which were built starting from 1956. ==Rockets== ===Kappa 1=== * Ceiling: 40 km * Takeoff thrust: 10.00 kN * Diameter: 0.13 m * Length: 2.70 m ===Kappa 2=== * Ceiling: 40 km * Mass: 300 kg * Diameter: 0.22 m * Length: 5 m ===Kappa 6 (in two stages)=== * Pay load: 20 kg * Ceiling: 60 km * Takeoff weight: 270 kg * Diameter: 0.25 m * Length: 5.61 m ===Kappa 7=== * Ceiling: 50 km * Diameter: 0.42 m * Length: 8.70 m ===Kappa 8 (in two stages)=== * Pay load: 50 kg * Ceiling: 160 km * Takeoff weight: 1500 kg * Diameter: 0.42 m * Length: 10.90 m ===Kappa 4=== * Ceiling: 80 km * Takeoff thrust: 105.00 kN * Diameter: 0.33 m * Length: 5.90 m ===Kappa 9L=== * Pay load: 15 kg * Ceiling: 350 km * Takeoff weight: 1550 kg * Diameter: 0.42 m * Length: 12.50 m ===Kappa 9M=== * Pay load: 50 kg * Ceiling: 350 km * Mass: 1500 kg * Diameter: 0.42 m * Length: 11.10 m ===Kappa 8L=== * Pay load: 25 kg * Ceiling: 200 km * Takeoff weight: 350 kg * Diameter: 0.25 m * Length: 7.30 m ===Kappa 10=== * Ceiling: 742 km ==See also== * R-25 Vulkan ==External links== * Kappa-Rocket Category:Solid- fuel rockets Category:Sounding rockets of Japan Category:Japanese inventions The spacecraft stopped mid-air again and, as the engines throttled back, began its successful vertical landing. For example, a heavier rocket would require an engine with more initial thrust to get it off of the launch pad, whereas a lighter rocket would need less initial thrust and would sustain a longer burn, reaching higher altitudes. ===Last number=== The last number is the delay in seconds between the end of the thrust phase and ignition of the ejection charge. University of New South Wales at the Australian Defence Force Academy. 2008.Measuring thrust and predicting trajectory in model rocketry M. Courtney and A. Courtney. Later with maximum fuel it reached a height of 150 km. They were # A single- stage rocket with a length of 3.4 metres and a takeoff thrust of 50 kN. *On July 20, 2021, Blue Origin's New Shepard rocket made its first-ever successful vertical landing following a crewed suborbital flight. All Seliger Rockets return to the ground by parachute. The D class 24 mm motors have a maximum thrust between 29.7 and 29.8 N, a total impulse between 16.7 and 16.85 Ns, and a burn time between 1.6 and 1.7 seconds. The E class 24 mm motors have a maximum thrust between 19.4 and 19.5 N, a total impulse between 28.45 and 28.6 Ns, and a burn time between 3 and 3.1 seconds. *On December 21, 2015, SpaceX's 20th Falcon 9 first stage made the first-ever successful vertical landing of an orbital-class booster after boosting 11 commercial satellites to low Earth orbit on Falcon 9 Flight 20. ",25,10.065778,"""22.0""",0,0.0000092,A
+"A spacecraft of mass $10,000 \mathrm{~kg}$ is parked in a circular orbit $200 \mathrm{~km}$ above Earth's surface. What is the minimum energy required (neglect the fuel mass burned) to place the satellite in a synchronous orbit (i.e., $\tau=24 \mathrm{hr}$ )?","The specific orbital energy associated with this orbit is −29.6MJ/kg: the potential energy is −59.2MJ/kg, and the kinetic energy 29.6MJ/kg. Thus, the total energy of the system remains unchanged though the mechanical energy of the system has reduced. ===Satellite=== thumb|plot of kinetic energy K, gravitational potential energy, U and mechanical energy E_\text{mechanical} versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, (by virtue of its motion) and gravitational potential energy, U, (by virtue of its position within the Earth's gravitational field; Earth's mass is M). Hence, mechanical energy E_\text{mechanical} of the satellite-Earth system is given by E_\text{mechanical} = U + K E_\text{mechanical} = - G \frac{M m}{r}\ + \frac{1}{2}\, m v^2 If the satellite is in circular orbit, the energy conservation equation can be further simplified into E_\text{mechanical} = - G \frac{M m}{2r} since in circular motion, Newton's 2nd Law of motion can be taken to be G \frac{M m}{r^2}\ = \frac{m v^2}{r} ==Conversion== Today, many technological devices convert mechanical energy into other forms of energy or vice versa. If the satellite's orbit is an ellipse the potential energy of the satellite, and its kinetic energy, both vary with time but their sum remains constant. For the Earth and a just little more than R the additional specific energy is (gR/2); which is the kinetic energy of the horizontal component of the velocity, i.e. \frac{1}{2}V^2 = \frac{1}{2}gR, V=\sqrt{gR}. ==Examples== ===ISS=== The International Space Station has an orbital period of 91.74 minutes (5504s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738km. For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy. ==Additional energy== If the central body has radius R, then the additional specific energy of an elliptic orbit compared to being stationary at the surface is -\frac{\mu}{2a}+\frac{\mu}{R} = \frac{\mu(2a-R)}{2aR} The quantity 2a-R is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth). In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divided by the reduced mass. Finally, a circularization burn is required to raise the perigee to the same altitude and remove any remaining inclination. ===Translunar or interplanetary spacecraft=== In order to reach the Moon or a planet at a desired time, the spacecraft must be launched within a limited range of times known as the launch window. A parking orbit is a temporary orbit used during the launch of a spacecraft. For an altitude of 100km (radius is 6471km): The energy is −30.8MJ/kg: the potential energy is −61.6MJ/kg, and the kinetic energy 30.8MJ/kg. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. Unless the launch site itself is quite close to the equator, it requires an impractically large amount of fuel to launch a spacecraft directly into such an orbit. Indian Mini Satellite (IMS) is a family of modular \- mini satellite buses developed by the Indian Space Research Organisation (ISRO). ==Variants== Indian Mini Satellite (Variants) Feature IMS-1 IMS-2 IMS-3 (Planned IMS-2 Derivative) Launch Mass Maximum bus mass Payload mass Propellant Design lifetime 2 years 5 years Raw bus voltage 28-33 Volts 28-33 Volts 28-42 Volts Solar Array Power 330 Watts (EOL) 675 Watts (EOL) 850 Watts (BOL) 850 Watts (BOL) Payload power 30 Watts (Continuous) 70 Watts (Duty Cycle) 250 Watts (Continuous) 600 Watts (Duty Cycle) 250 Watts (Continuous) 400 Watts (Duty Cycle) Attitude Control 3-axis stabilized Four Reaction Wheels Single 1N thruster 3-axis stabilized Four Reaction Wheels Mono-propellant RCS Four 1N thrusters Four 0.2N thrusters Pointing Accuracy ±0.1° (3σ) (all axes) ± 0.1° (all axes) ± 0.1° (all axes) SSR Storage 32 Gb 32 Gb (SDRAM) 256 Gb (Flash Memory) 32 Gb (SDRAM) 256 Gb (Flash Memory) Payload data storage ≤ 16 Gb ≤ 32 Gb Downlink ≤ 8 Mbit/s DL rate ≤ 105 Mbit/s DL rate ≤ 160 Mbit/s DL rate Missions IMS-1 Youthsat Microsat-TD SARAL \- ScatSat-1 EMISAT HySIS XPoSat (Planned) center|thumb|150x150px|IMS-1 ==See also== * Comparison of satellite buses ==References== Category:Indian Space Research Organisation Category:Satellite buses Thus, if orbital position vector (\mathbf{r}) and orbital velocity vector (\mathbf{v}) are known at one position, and \mu is known, then the energy can be computed and from that, for any other position, the orbital speed. ==Rate of change== For an elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is \frac{\mu}{2a^2} where * \mu={G}(m_1 + m_2) is the standard gravitational parameter; *a\,\\! is semi- major axis of the orbit. The speed is 7.8km/s, the net delta-v to reach this orbit is 8.0km/s. The average speed is 7.7km/s, the net delta-v to reach this orbit is 8.1km/s (the actual delta-v is typically 1.5–2.0km/s more for atmospheric drag and gravity drag). A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. thumb|250px|An example of a mechanical system: A satellite is orbiting the Earth influenced only by the conservative gravitational force; its mechanical energy is therefore conserved. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: \begin{align} \varepsilon &= \varepsilon_k + \varepsilon_p \\\ &= \frac{v^2}{2} - \frac{\mu}{r} = -\frac{1}{2} \frac{\mu^2}{h^2} \left(1 - e^2\right) = -\frac{\mu}{2a} \end{align} where *v is the relative orbital speed; *r is the orbital distance between the bodies; *\mu = {G}(m_1 + m_2) is the sum of the standard gravitational parameters of the bodies; *h is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass; *e is the orbital eccentricity; *a is the semi- major axis. ",0.086,1068,"""2.57""",3.00,3930,C
+A uniformly solid sphere of mass $M$ and radius $R$ is fixed a distance $h$ above a thin infinite sheet of mass density $\rho_5$ (mass/area). With what force does the sphere attract the sheet?,"This force depends on the surface separation h. Since the hydrodynamic drag of a sphere close to a planar substrate is known theoretically, the spring constant of the cantilever can be deduced. In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. The mass distribution of a solid defines its center of gravity and influences its dynamical behaviour - e.g. the oscillations and eventual rotation. ==Mathematical modelling== A mass distribution can be modeled as a measure. Similar use of the equation can be made in the settling of fine particles in water or other fluids. === Terminal velocity of sphere falling in a fluid === At terminal (or settling) velocity, the excess force due to the difference between the weight and buoyancy of the sphere (both caused by gravity) is given by: :F_g = ( \rho_p - \rho_f)\, g\, \frac{4}{3}\pi\, R^3, where (in SI units): * is the mass density of the sphere [kg/m3] * is the mass density of the fluid [kg/m3] * is the gravitational acceleration [m/s] Requiring the force balance and solving for the velocity gives the terminal velocity . A sphere of known size and density is allowed to descend through the liquid. An analytical and closed- form solution for both 2D and 3D, mechanically stable, random packings of spheres has been found by A. Zaccone in 2022 using the assumption that the most random branch of jammed states (maximally random jammed packings, extending up to the fcc closest packing) undergo crowding in a way qualitatively similar to an equilibrium liquid. Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. thumb|right|280px|Scheme of the colloidal probe technique for direct force measurements in the sphere-plane and sphere-sphere geometries. The random close packing value is significantly below the maximum possible close-packing of same-size hard spheres into a regular crystalline arrangements, which is 74.04%.Modes of wall induced granular crystallisation in vibrational packing.Granular Matter, 21(2), 26 Both the face-centred cubic (fcc) and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process of granular crystallisation. The distance to the sphere of influence must thus satisfy \frac{m_B}{m_A} \frac{r^3}{R^3} = \frac{m_A}{m_B} \frac{R^2}{r^2} and so r = R\left(\frac{m_A}{m_B}\right)^{2/5} is the radius of the sphere of influence of body A ==See also== * Hill sphere * Sphere of influence (black hole) ==References== == General references == * * * == External links == *Project Pluto Category:Astrodynamics Category:Orbits When the force is attractive, the cantilever is attracted to the surface and may become unstable. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.Batchelor (1967), p. 233. ==Statement of the law== The force of viscosity on a small sphere moving through a viscous fluid is given by: :F_{\rm d} = 6 \pi \mu R v where (in SI units): * is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle (newtons, kg m s−2); * (some authors use the symbol ) is the dynamic viscosity (Pascal-seconds, kg m−1 s−1); * is the radius of the spherical object (meters); * is the flow velocity relative to the object (meters per second). A sphere of influence (SOI) in astrodynamics and astronomy is the oblate- spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. The problem of predicting theoretically the random close pack of spheres is difficult mainly because of the absence of a unique definition of randomness or disorder. In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. The matrix represents the identity-matrix. :\boldsymbol{\sigma} = - p \cdot \mathbf{I} + \mu \cdot \left(( abla \mathbf{u}) + ( abla \mathbf{u})^T \right) The force acting on the sphere is to calculate by surface-integral, where represents the radial unit-vector of spherical-coordinates: :\begin{align} \mathbf{F} &= \iint_{\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset \;\boldsymbol{\sigma}\cdot \text{d}\mathbf{S} \\\\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \boldsymbol{\sigma}\cdot \mathbf{e_r}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\\\[4pt] &= \int_{0}^{\pi}\int_{0}^{2\pi} \frac{3\mu \cdot \mathbf{u}_{\infty}}{2 R}\cdot R^2 \sin\theta \text{d}\varphi\text{d}\theta \\\\[4pt] &= 6\pi\mu R \cdot \mathbf{u}_{\infty} \end{align} === Rotational flow around a sphere === thumb|434x434px|Stokes-Flow around sphere: \boldsymbol{\omega}_R = \begin{pmatrix} 0 & 0 & 2 \end{pmatrix}^T \; \text{Hz} , \mu = 1 \; \text{mPa} \cdot \text{s}, R = 1 \; \text{m} :\begin{align} \mathbf{u}(\mathbf{x}) &= - \;R^3 \cdot \frac{ \boldsymbol{\omega}_{R} \times \mathbf{x}}{\|\mathbf{x}\|^3} \\\\[8pt] \boldsymbol{\omega}(\mathbf{x}) &= \frac{R^3 \cdot \boldsymbol{\omega}_{R}}{\|\mathbf{x}\|^3} - \frac{3 R^3 \cdot (\boldsymbol{\omega}_{R} \cdot \mathbf{x})\cdot \mathbf{x}}{\|\mathbf{x}\|^5} \\\\[8pt] p(\mathbf{x}) &= 0 \\\\[8pt] \boldsymbol{\sigma} &= - p \cdot \mathbf{I} + \mu \cdot \left( ( abla \mathbf{u}) + ( abla \mathbf{u})^T \right) \\\\[8pt] \mathbf{T} &= \iint_{\partial V}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\subset\\!\supset \mathbf{x} \times \left( \boldsymbol{\sigma} \cdot \text{d}\boldsymbol{S} \right) \\\ &= \int_{0}^{\pi} \int_{0}^{2\pi} (R \cdot \mathbf{e_r}) \times \left( \boldsymbol{\sigma} \cdot \mathbf{e_r} \cdot R^2 \sin\theta \text{d}\varphi \text{d}\theta \right) \\\ &= 8\pi\mu R^3 \cdot \boldsymbol{\omega}_{R} \end{align} ==Other types of Stokes flow== Although the liquid is static and the sphere is moving with a certain velocity, with respect to the frame of sphere, the sphere is at rest and liquid is flowing in the opposite direction to the motion of the sphere. ==See also== * Einstein relation (kinetic theory) * Scientific laws named after people * Drag equation * Viscometry * Equivalent spherical diameter * Deposition (geology) == Sources == * * Originally published in 1879, the 6th extended edition appeared first in 1932. ==References== Category:Fluid dynamics In the case of a soft repulsive force, the cantilever is repelled from the surface and only slowly approaches the constant compliance region. Experiments and computer simulations have shown that the most compact way to pack hard perfect same- size spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. It is also possible to study forces between colloidal particles by attaching another particle to the substrate and perform the measurement in the sphere-sphere geometry, see figure above. thumb|left|350px|Principle of the force measurements by the colloidal probe technique. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). From the previous two equations, and with the appropriate boundary conditions, for a far-field uniform-flow velocity in the –direction and a sphere of radius , the solution is found to beLamb (1994), §337, p. 598. : \psi(r,z) = - \frac{1}{2}\, u\, r^2\, \left[ 1 \- \frac{3}{2} \frac{R}{\sqrt{r^2+z^2}} \+ \frac{1}{2} \left( \frac{R}{\sqrt{r^2+z^2}} \right)^3\; \right]. thumb|420x420px|Stokes-Flow around sphere with parameters of Far-Field velocity \mathbf{u}_{\infty} = \begin{pmatrix} 6 & 0 & 6 \end{pmatrix}^T \text{m/s}, radius of sphere R = 1 \; \text{m}, viscosity of water (T = 20°C) \mu = 1 \; \text{mPa}\cdot \text{s} . ",418,6.283185307,"""1.0""",209.1,8.7,B
+Consider a thin rod of length $l$ and mass $m$ pivoted about one end. Calculate the moment of inertia. ,"* The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end. * The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point P as, I_P = I_{C, \text{rod}} + M_\text{rod}\left(\frac{L}{2}\right)^2 + I_{C, \text{disc}} + M_\text{disc}(L + R)^2, where L is the length of the pendulum. The moment of inertia I_\mathbf{C} about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. The quantity I = mr^2 is the moment of inertia of this single mass around the pivot point. Description Figure Moment(s) of inertia Point mass M at a distance r from the axis of rotation. File:moment of inertia thin cylinder.png I \approx m r^2 \,\\! For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is I = \frac{L}{\omega}. File:moment of inertia thick cylinder h.svg I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right) Classical Mechanics - Moment of inertia of a uniform hollow cylinder . Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point P so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (t), to obtain I_P = \frac{mgr}{\omega_\text{n}^2} = \frac{mgrt^2}{4\pi^2}, where t is the period (duration) of oscillation (usually averaged over multiple periods). ====Center of oscillation==== A simple pendulum that has the same natural frequency as a compound pendulum defines the length L from the pivot to a point called the center of oscillation of the compound pendulum. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as, I = mr^2. File:moment of inertia solid rectangular prism.png I_h = \frac{1}{12} m \left(w^2+d^2\right) I_w = \frac{1}{12} m \left(d^2+h^2\right) I_d = \frac{1}{12} m \left(w^2+h^2\right) Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. Align the x-axis with the rod and locate the origin its center of mass at the center of the rod, then I_{C, \text{rod}} = \iiint_Q \rho\,x^2 \, dV = \int_{-\frac{\ell}{2}}^\frac{\ell}{2} \rho\,x^2 s\, dx = \left. \rho s\frac{x^3}{3}\right|_{-\frac{\ell}{2}}^\frac{\ell}{2} = \frac{\rho s}{3} \left(\frac{\ell^3}{8} + \frac{\ell^3}{8}\right) = \frac{m\ell^2}{12}, where m = \rho s \ell is the mass of the rod. Also, a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration. The moment of inertia of an arbitrarily shaped body is the sum of the values mr^2 for all of the elements of mass in the body. === Compound pendulums === A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the x-axis or y-axis depending on the load. ==== Examples ==== thumb|right The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. The moments of inertia of a mass have units of dimension ML2([mass] × [length]2). A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. frameless|upright I = M r^2 Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles. frameless|upright I = \frac{ m_1 m_2 }{ m_1 \\! + \\! m_2 } x^2 = \mu x^2 Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center. ",11000,0.66,"""240.0""",0.33333333,4,D
 "A clown is juggling four balls simultaneously. Students use a video tape to determine that it takes the clown $0.9 \mathrm{~s}$ to cycle each ball through his hands (including catching, transferring, and throwing) and to be ready to catch the next ball. What is the minimum vertical speed the clown must throw up each ball?
-","175px|thumb|Ladder diagram In toss juggling, a cascade is the simplest juggling pattern achievable with an odd number of props. The ball is then thrown from above the 4 diagonally downward to the opposite hand. 150px|thumb|Ladder diagram for box: (4,2x)(2x,4) In toss juggling, the box is a juggling pattern for 3 objects, most commonly balls or bean bags. ""Juggling with your arms up in the air above your head & looking up from underneath the pattern.""Darbyshire (1993), p.22. and a virtually infinite number of cascade patterns such as 522, 720, 900, 72222, and so on (see article on Siteswap notation).Beever, Ben (2001), p.15. ==Shannon's theorem== thumb|An illustration of Shannon's juggling theorem for the cascade juggling pattern 175px|thumb|Cascade ladder suggested by Shannon's formula Claude Shannon, builder of the first juggling robot, developed a juggling theorem, relating the time balls spend in the air and in the hands: (F+D)H=(V+D)N, where F = time a ball spends in the air, D = time a ball spends in a hand/time a hand is full, V = time a hand is vacant, N = number of balls, and H = number of hands. ==Number of props== ===Three-ball=== For the three-ball cascade the juggler starts with two balls in one hand and the third ball in the other hand. Before catching this ball the juggler must throw the ball in the receiving hand, in a similar arc, to the first hand. The simplest juggling pattern is the three-ball cascade,Bernstein, Nicholai A. (1996). 150px|thumb|Cascade flash: 3 throws & 3 catches 150px|thumb|Mills mess flash: 6 throws & 6 catches In toss juggling, a flash is either a form of numbers juggling where each ball in a juggling pattern is only thrown and caught once or it is a juggling trick where every prop is simultaneously in the air and both hands are empty.""Three ball flash"", TWJC.co.uk. Two balls are dedicated to a specific hand with vertical throws, and the third ball is thrown horizontally between the two hands. For some tricks the number of throws and catches to complete a juggling cycle for that trick is not simply a multiple of the number of objects being juggled. Juggling, p.23. thumb|400px|right|Racking a game of three-ball with the standard fifteen-ball triangle rack. However, in order to keep the number of props in the juggler's hands to a minimum, it is necessary to begin the pattern by throwing, from alternating hands, all but one prop (in the same hand as the first throw, which started with one more prop than the other) before any catches are made. ==Reverse cascade== thumb|right|upright|An illustration of the three-ball reverse cascade. One ball is thrown from the first hand in an arc to the other hand. One juggles, ""a cascade with two balls while the 'tennis' ball is thrown [back and forth] over the top.""Darbyshire (1993), p.23. The goal is to () the three object balls in as few shots as possible.PoolSharp's ""Three-Ball Rules"". ""The cascade is the simplest three ball juggling pattern."" Juggling, p.26. ""In the cascade...the crossing of the balls between the hands demands that one hand catches at the same rate that the other hand throws . Higher numbers require the balls to be tossed higher into the air in order to allow more time for a complete cycle of throws. Title Description Demonstration The Shuffle In a shuffle throw, the vamp ball begins above and outside the vertical path of one of the box's ""side balls"" and is thrown diagonally downward, caught below the opposite ""side ball"". Charlie Dancey's Encyclopædia of Ball Juggling p98. ",10.4,13.2,1.51,9,+11,B
-"A billiard ball of initial velocity $u_1$ collides with another billiard ball (same mass) initially at rest. The first ball moves off at $\psi=45^{\circ}$. For an elastic collision, what are the velocities of both balls after the collision? ","Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. In an elastic collision these magnitudes do not change. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. (To get the and velocities of the second ball, one needs to swap all the '1' subscripts with '2' subscripts.) In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. In the case of macroscopic bodies, perfectly elastic collisions are an ideal never fully realized, but approximated by the interactions of objects such as billiard balls. Once v_1 is determined, v_2 can be found by symmetry. ====Center of mass frame==== With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. The magnitudes of the velocities of the particles after the collision are: \begin{align} v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\\ v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}. \end{align} ===Two-dimensional collision with two moving objects=== The final x and y velocities components of the first ball can be calculated as: \begin{align} v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2}) \\\\[0.8em] v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}), \end{align} where and are the scalar sizes of the two original speeds of the objects, and are their masses, and are their movement angles, that is, v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1 (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi () is the contact angle. A useful special case of elastic collision is when the two bodies have equal mass, in which case they will simply exchange their momenta. Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. These equations may be solved directly to find v_1,v_2 when u_1,u_2 are known: \begin{array}{ccc} v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\\\[.5em] v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2. \end{array} If both masses are the same, we have a trivial solution: \begin{align} v_{1} &= u_{2} \\\ v_{2} &= u_{1}. \end{align} This simply corresponds to the bodies exchanging their initial velocities to each other. The following illustrate the case of equal mass, m_1=m_2. frame|center|Elastic collision of equal masses frame|center|Elastic collision of masses in a system with a moving frame of reference In the limiting case where m_1 is much larger than m_2, such as a ping-pong paddle hitting a ping-pong ball or an SUV hitting a trash can, the heavier mass hardly changes velocity, while the lighter mass bounces off, reversing its velocity plus approximately twice that of the heavy one. In the case of a large u_{1}, the value of v_{1} is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed. At any instant, half the collisions are, to a varying extent, inelastic collisions (the pair possesses less kinetic energy in their translational motions after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Studies of two-dimensional collisions are conducted for many bodies in the framework of a two-dimensional gas. frame|center|Two-dimensional elastic collision In a center of momentum frame at any time the velocities of the two bodies are in opposite directions, with magnitudes inversely proportional to the masses. It can be shown that v_c is given by: v_c = \frac{p_T c^2}{E} Now the velocities before the collision in the center of momentum frame u_1 ' and u_2 ' are: \begin{align} u_1' &= \frac{u_1 - v_c}{1- \frac{u_1 v_c}{c^2}} \\\ u_2' &= \frac{u_2 - v_c}{1- \frac{u_2 v_c}{c^2}} \\\ v_1' &= -u_1' \\\ v_2' &= -u_2' \\\ v_1 &= \frac{v_1' + v_c}{1+ \frac{v_1' v_c}{c^2}} \\\ v_2 &= \frac{v_2' + v_c}{1+ \frac{v_2' v_c}{c^2}} \end{align} When u_1 \ll c and u_2 \ll c\,, \begin{align} p_T &\approx m_1 u_1 + m_2 u_2 \\\ v_c &\approx \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} \\\ u_1' &\approx u_1 - v_c \approx \frac {m_1 u_1 + m_2 u_1 - m_1 u_1 - m_2 u_2}{m_1 + m_2} = \frac {m_2 (u_1 - u_2)}{m_1 + m_2} \\\ u_2' &\approx \frac {m_1 (u_2 - u_1)}{m_1 + m_2} \\\ v_1' &\approx \frac {m_2 (u_2 - u_1)}{m_1 + m_2} \\\ v_2' &\approx \frac {m_1 (u_1 - u_2)}{m_1 + m_2} \\\ v_1 &\approx v_1' + v_c \approx \frac {m_2 u_2 - m_2 u_1 + m_1 u_1 + m_2 u_2}{m_1 + m_2} = \frac{u_1 (m_1 - m_2) + 2m_2 u_2}{m_1 + m_2} \\\ v_2 &\approx \frac{u_2 (m_2 - m_1) + 2m_1 u_1}{m_1 + m_2} \end{align} Therefore, the classical calculation holds true when the speed of both colliding bodies is much lower than the speed of light (~300,000 kilometres per second). ===Relativistic derivation using hyperbolic functions=== Using the so-called parameter of velocity s (usually called the rapidity), \frac{v}{c}=\tanh(s), we get \sqrt{1-\frac{v^2}{c^2}}=\operatorname{sech}(s). While the assumptions of separate impacts is not actually valid (the balls remain in close contact with each other during most of the impact), this model will nonetheless reproduce experimental results with good agreement, and is often used to understand more complex phenomena such as the core collapse of supernovae, or gravitational slingshot manoeuvres. ==Sport regulations== Several sports governing bodies regulate the bounciness of a ball through various ways, some direct, some indirect. ",0.139,1590,0.7071067812,25,0.264,C
-"A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \mathrm{~km} / \mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\mathrm{LAB}$ angle $\psi=10^{\circ}$, what is the final speed of the deuteron?","The charge radius of the deuteron is . This suggests that the state of the deuterium is indeed to a good approximation , state, which occurs with both nucleons spinning in the same direction, but their magnetic moments subtracting because of the neutron's negative moment. While the order of magnitude is reasonable, since the deuterium radius is of order of 1 femtometer (see below) and its electric charge is e, the above model does not suffice for its computation. The Prout is an obsolete unit of energy, whose value is: 1 Prout = 2.9638 \times 10^{-14} J This is equal to one twelfth of the binding energy of the deuteron. The nucleus of a deuterium atom, called a deuteron, contains one proton and one neutron, whereas the far more common protium has no neutrons in the nucleus. In the first case the deuteron is a spin triplet, so that its total spin s is 1. The latter contribution is dominant in the absence of a pure contribution, but cannot be calculated without knowing the exact spatial form of the nucleons wavefunction inside the deuterium. In the second case the deuteron is a spin singlet, so that its total spin s is 0. For hydrogen, this amount is about , or 1.000545, and for deuterium it is even smaller: , or 1.0002725. The deuteron, composed of a proton and a neutron, is one of the simplest nuclear systems. The measured electric quadrupole of the deuterium is . The name deuterium is derived from the Greek , meaning ""second"", to denote the two particles composing the nucleus. This is a nucleus with one proton and one neutron, i.e. a deuterium nucleus. The proton and neutron making up deuterium can be dissociated through neutral current interactions with neutrinos. But the slightly lower experimental number than that which results from simple addition of proton and (negative) neutron moments shows that deuterium is actually a linear combination of mostly , state with a slight admixture of , state. The energies of spectroscopic lines for deuterium and light hydrogen (hydrogen-1) therefore differ by the ratios of these two numbers, which is 1.000272. In this case, the exchange of the two nucleons will multiply the deuterium wavefunction by (−1) from isospin exchange, (+1) from spin exchange and (+1) from parity (location exchange), for a total of (−1) as needed for antisymmetry. This is about 17% of the terrestrial deuterium-to-hydrogen ratio of 156 deuterium atoms per million hydrogen atoms. The deuteron, being an isospin singlet, is antisymmetric under nucleons exchange due to isospin, and therefore must be symmetric under the double exchange of their spin and location. In this theory, the deuterium nucleus with mass two and charge one would contain two protons and one nuclear electron. thumb|upright=0.8|The deuterium–tritium fusion reaction Deuterium–tritium fusion (sometimes abbreviated D+T) is a type of nuclear fusion in which one deuterium nucleus fuses with one tritium nucleus, giving one helium nucleus, one free neutron, and 17.6 MeV of energy. This situation is known as the deuterium bottleneck. ",85,"102,965.21",14.44,8.44,0.241,C
-"A mass $m$ moves in one dimension and is subject to a constant force $+F_0$ when $x<0$ and to a constant force $-F_0$ when $x>0$. Describe the motion by constructing a phase diagram. Calculate the period of the motion in terms of $m, F_0$, and the amplitude $A$ (disregard damping) .","The phase is zero at the start of each period; that is :\phi(t_0 + kT) = 0\quad\quad{} for any integer k. The initial conditions are x(0)=0 and \dot{x}(0)=0. *Damped harmonic motion, see animation (right). thumb|Phase portrait of damped oscillator, with increasing damping strength. The equation of motion is \ddot x + 2\gamma \dot x + \omega^2 x = 0. A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables. ==Examples== 400px|thumbnail|Illustration of how a phase portrait would be constructed for the motion of a simple pendulum. A motion diagram represents the motion of an object by displaying its location at various equally spaced times on the same diagram. The phase for each argument value, relative to the start of the cycle, is shown at the bottom, in degrees from 0° to 360° and in radians from 0 to 2π. In physics and mathematics, the phase of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. Then, F is said to be ""at the same phase"" at two argument values t_1 and t_2 (that is, \phi(t_1) = \phi(t_2)) if the difference between them is a whole number of periods. thumb|350px|right|A plot of f(y) (left) and its phase line (right). The term phase can refer to several different things: * It can refer to a specified reference, such as \textstyle \cos(2 \pi f t), in which case we would say the phase of \textstyle x(t) is \textstyle \varphi, and the phase of \textstyle y(t) is \textstyle \varphi - \frac{\pi}{2}. The red dots in the phase portraits are at times t which are an integer multiple of the period T=2\pi/\omega.Based on the examples shown in . ==References== ===Inline=== ===Historical=== * ===Other=== *. *. *. *. ==External links== *Duffing oscillator on Scholarpedia *MathWorld page * Category:Ordinary differential equations Category:Chaotic maps Category:Nonlinear systems Category:Articles containing video clips Motion diagrams. Motion diagrams. To get the phase as an angle between -\pi and +\pi, one uses instead :\phi(t) = 2\pi\left(\left[\\!\\!\left[\frac{t - t_0}{T} + \frac{1}{2}\right]\\!\\!\right] - \frac{1}{2}\right) The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with ""360°"" in place of ""2π"". ===Consequences=== With any of the above definitions, the phase \phi(t) of a periodic signal is periodic too, with the same period T: :\phi(t + T) = \phi(t)\quad\quad{} for all t. *Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. ==Formula for phase of an oscillation or a periodic signal== The phase of an oscillation or signal refers to a sinusoidal function such as the following: :\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \tfrac{\pi}{2}\right) \end{align} where \textstyle A, \textstyle f, and \textstyle \varphi are constant parameters called the amplitude, frequency, and phase of the sinusoid. In mathematics, a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. A Campbell diagram plot represents a system's response spectrum as a function of its oscillation regime. The formula above gives the phase as an angle in radians between 0 and 2\pi. ",-0.5,12,1.154700538,4,2.8108,D
-"A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears the water splash $4.021 \mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \mathrm{~m} / \mathrm{s}$, find the height of the building, neglecting air resistance.","Integrating the internal air pressure over one hemisphere of the balloon then gives : P_\mathrm{in} - P_\mathrm{out} \equiv P = \frac{f_t}{\pi r^2} = \frac{C}{r_0^2r} \left[1-\left(\frac{r_0}{r}\right)^6 \right] where r0 is the balloon's uninflated radius. When air is first added to the balloon, the pressure rises rapidly to a peak. This is easy to verify by squeezing the air back and forth between two interconnected balloons. ==Non-ideal balloons== At large extensions, the pressure inside a natural rubber balloon once again goes up. The air flow ceases when the two balloons have equal pressure, with one on the left branch of the pressure curve (r < rp) and one on the right branch (r > rp). This result is surprising, since most people assume that the two balloons will have equal sizes after exchanging air. If the total quantity of air in both balloons is less than Np, defined as the number of molecules in both balloons if they both sit at the peak of the pressure curve, then both balloons settle down to the left of the pressure peak with the same radius, r < rp. thumb|upright=1.25|The pub in 2007 The Air Balloon was a public house and road junction at Birdlip, Gloucestershire, England and closed in 2022 as part of road improvements. Pressure curve for an ideal rubber balloon. It becomes smaller, and the larger balloon becomes larger. Two balloons are connected via a hollow tube. For many starting conditions, the smaller balloon then gets smaller and the balloon with the larger diameter inflates even more. Two identical balloons are inflated to different diameters and connected by means of a tube. The lower pressure balloon will expand. thumb|upright=1.5|Observation balloon being shot down by a German biplane Balloon busters were military pilots known for destroying enemy observation balloons. Amer., 62, 1129-35. ,and Mackenzie.Mackenzie, K.V. (1981) Nine-term equation for sound speed in the oceans. So, when the valve is opened, the smaller balloon pushes air into the larger balloon. The simplest way to do this is to imagine that the balloon is made up of a large number of small rubber patches, and to analyze how the size of a patch is affected by the force acting on it. Bio-physical models suggest that this process is effectively similar to the behavior of the balloons in the two-balloon experiment . For a balloon of radius r, a fixed volume of rubber means that r2t is constant, or equivalently : t \propto \frac{1}{r^2} hence : \frac{t}{t_0} = \left(\frac{r_0}{r}\right)^2 and the radial force equation becomes : p = \frac{1}{C_2} \left(\frac{r_0}{r}\right)^4 The equation for the tangential force ft (where Lt \propto r) then becomes : f_t \propto (r/r_0^2)\left[1-(r_0/r)^6\right]. thumb|left|300px|Fig. thumb|Dangerous proximity of a hot air balloon to an overhead line. Figure 2 (above left) shows a typical initial configuration: The smaller balloon has the higher pressure because of the sum of pressure of elastic force Fe which is proportional to pressure (P=Fe/S) plus air pressure in small balloon is greater than air pressure in big balloon. Although balloons were occasionally shot down by small-arms fire, generally it was difficult to shoot down a balloon with solid bullets, particularly at the distances and altitude involved. ",71,6.283185307,4.946,4,-0.0301,A
-"A thin disk of mass $M$ and radius $R$ lies in the $(x, y)$ plane with the $z$-axis passing through the center of the disk. Calculate the gravitational potential $\Phi(z)$.","For this the gravitational force, i.e. the gradient of the potential, must be computed. For a non-pointlike object of continuous mass distribution, each mass element dm can be treated as mass distributed over a small volume, so the volume integral over the extent of object 2 gives: = - Gm_1 \int\limits_V \frac{\rho_2 }{r^2}\mathbf{\hat{r}}\,dx\,dy\,dz |}} with corresponding gravitational potential where ρ = ρ(x, y, z) is the mass density at the volume element and of the direction from the volume element to point mass 1. u is the gravitational potential energy per unit mass. ===The case of a homogeneous sphere=== In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(s); i.e., density depends only on the radial distance :s = \sqrt{x^2 + y^2 + z^2} \,. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. This leaves an ordinary differential equation in terms only of the radius, r, which determines the eigenstates for the particular potential, V(r). == Structure of the eigenfunctions == The eigenstates of the system have the form: \psi(r, \theta, \phi) = R(r)\Theta(\theta)\Phi(\phi) in which the spherical angles \theta and \phi represent the polar and azimuthal angle, respectively. In the literature it is common to introduce some arbitrary ""reference radius"" R close to Earth's radius and to work with the dimensionless coefficients :\begin{align} \tilde{J_n} &= -\frac{J_n}{\mu\ R^n}, & \tilde{C_{n}^m} &= -\frac{C_{n}^m}{\mu\ R^n}, & \tilde{S_{n}^m} &= -\frac{S_{n}^m}{\mu\ R^n} \end{align} and to write the potential as {{NumBlk|:| u = -\frac{\mu }{r} \left(1 + \sum_{n=2}^{N_z} \frac{\tilde{J_n} P^0_n(\sin\theta) }{{(\frac{r}{R})}^n} + \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) (\tilde{C_{n}^m} \cos m\varphi + \tilde{S_{n}^m} \sin m\varphi)}{{(\frac{r}{R})}^n}\right) |}} ===Largest terms=== The dominating term (after the term −μ/r) in () is the ""J2 coefficient"", representing the oblateness of Earth: :u = \frac{J_2\ P^0_2(\sin\theta)}{r^3} = J_2 \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta -1) = J_2 \frac{1}{r^5} \frac{1}{2} (3 z^2 -r^2) Relative the coordinate system thumb|right|Figure 1: The unit vectors. If this shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), the integrals () and () could be evaluated with numerical methods to find a more accurate model for Earth's gravitational field. From the definition of φ as a spherical coordinate it is clear that Φ(φ) must be periodic with the period 2π and one must therefore have that and for some integer m as the family of solutions to () then are With the variable substitution :x=\sin \theta equation () takes the form From () follows that in order to have a solution \phi with :R(r) = \frac{1}{r^{n+1}} one must have that :\lambda = n (n + 1) If Pn(x) is a solution to the differential equation one therefore has that the potential corresponding to m = 0 :\phi = \frac{1}{r^{n+1}}\ P_n(\sin\theta) which is rotationally symmetric around the z-axis is a harmonic function If P_{n}^{m}(x) is a solution to the differential equation {dx}\right)\ +\ \left(n(n + 1) - \frac{m^2}{1 - x^2} \right)\ P_{n}^{m}\ =\ 0 |}} with m ≥ 1 one has the potential \ P_{n}^{m}(\sin\theta)\ (a\ \cos m\varphi\ +\ b\ \sin m\varphi) |}} where a and b are arbitrary constants is a harmonic function that depends on φ and therefore is not rotationally symmetric around the z-axis The differential equation () is the Legendre differential equation for which the Legendre polynomials defined are the solutions. The differential equation which characterizes the function R(r) is called the radial equation. == Derivation of the radial equation == The kinetic energy operator in spherical polar coordinates is:\frac{\hat{p}^2}{2m_0} = -\frac{\hbar^2}{2m_0} abla^2 = \- \frac{\hbar^2}{2m_0\,r^2} \left[ \frac{\partial}{\partial r} \left(r^2 \frac{\partial}{\partial r}\right) - \hat{L}^2 \right].The spherical harmonics satisfy \hat{L}^2 Y_{\ell m}(\theta,\phi)\equiv \left\\{ -\frac{1}{\sin^2\theta} \left[ \sin\theta \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right) +\frac{\partial^2}{\partial \phi^2}\right]\right\\} Y_{\ell m}(\theta,\phi) = \ell(\ell+1)Y_{\ell m}(\theta,\phi). There should be a theta, not lambda \hat{\varphi}\ ,\ \hat{\theta}\ ,\ \hat{r} illustrated in figure 1 the components of the force caused by the ""J2 term"" are In the rectangular coordinate system (x, y, z) with unit vectors (x̂ ŷ ẑ) the force components are: The components of the force corresponding to the ""J3 term"" :u = \frac{J_3 P^0_3(\sin\theta) }{r^4} = J_3 \frac{1}{r^4} \frac{1}{2} \sin\theta \left(5\sin^2\theta - 3\right) = J_3 \frac{1}{r^7} \frac{1}{2} z \left(5 z^2 - 3 r^2\right) are and The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly. This is the shell theorem saying that in this case: with corresponding potential where M = ∫Vρ(s)dxdydz is the total mass of the sphere. ==Spherical harmonics representation== In reality, Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. By straightforward calculations one gets that for any function f Introducing the expression () in () one gets that As the term :\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) only depends on the variable r and the sum :\frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right) + \frac{1}{\Phi\cos^2\theta}\frac{d^2\Phi}{d\varphi^2} only depends on the variables θ and φ. thumb|right|300px|A contour plot of the effective potential of a two-body system due to gravity and inertia at one point in time. For the same reason, the solution will be of this kind inside the sphere:R(r) = A j_\ell\left(\sqrt{\frac{2 m_0 (E-V_0)}{\hbar^2}}r\right), \qquad r < r_0.Note that for bound states, V_0 < E < 0. Several other definitions are in use, and so care must be taken in comparing different sources.Wolfram Mathworld, spherical coordinates == Cylindrical coordinate system == === Vector fields === Vectors are defined in cylindrical coordinates by (ρ, φ, z), where * ρ is the length of the vector projected onto the xy- plane, * φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), * z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: \begin{bmatrix} \rho \\\ \phi \\\ z \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\\ \operatorname{arctan}(y / x) \\\ z \end{bmatrix},\ \ \ 0 \le \phi < 2\pi, thumb or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} \rho\cos\phi \\\ \rho\sin\phi \\\ z \end{bmatrix}. Let the points have position vectors \textbf{r} and \textbf{r}' , then the Laplace expansion is : \frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} (-1)^m \frac{r_^\ell }{r_{\scriptscriptstyle>}^{\ell+1} } Y^{-m}_\ell(\theta, \varphi) Y^m_\ell(\theta', \varphi'). The first spherical harmonics with n = 0, 1, 2, 3 are presented in the table below. : n Spherical harmonics 0 \frac{1}{r} 1 \frac{1}{r^2} P^0_1(\sin\theta) = \frac{1}{r^2} \sin\theta \frac{1}{r^2} P^1_1(\sin\theta) \cos\varphi= \frac{1}{r^2} \cos\theta \cos\varphi \frac{1}{r^2} P^1_1(\sin\theta) \sin\varphi= \frac{1}{r^2} \cos\theta \sin\varphi 2 \frac{1}{r^3} P^0_2(\sin\theta) = \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta - 1) \frac{1}{r^3} P^1_2(\sin\theta) \cos\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta\ \cos\varphi \frac{1}{r^3} P^1_2(\sin\theta) \sin\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta \sin\varphi \frac{1}{r^3} P^2_2(\sin\theta) \cos2\varphi = \frac{1}{r^3} 3 \cos^2 \theta\ \cos2\varphi \frac{1}{r^3} P^2_2(\sin\theta) \sin2\varphi = \frac{1}{r^3} 3 \cos^2 \theta \sin 2\varphi 3 \frac{1}{r^4} P^0_3(\sin\theta) = \frac{1}{r^4} \frac{1}{2} \sin\theta\ (5\sin^2\theta -3) \frac{1}{r^4} P^1_3(\sin\theta) \cos\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \cos\varphi \frac{1}{r^4} P^1_3(\sin\theta) \sin\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \sin\varphi \frac{1}{r^4} P^2_3(\sin\theta) \cos 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \cos 2\varphi \frac{1}{r^4} P^2_3(\sin\theta) \sin 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \sin 2\varphi \frac{1}{r^4} P^3_3(\sin\theta) \cos 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \cos 3\varphi \frac{1}{r^4} P^3_3(\sin\theta) \sin 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \sin 3\varphi ===Application=== The model for Earth's gravitational potential is a sum \+ \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) \left(C_n^m \cos m\varphi + S_n^m \sin m\varphi\right)}{r^{n+1}} |}} where \mu = GM and the coordinates () are relative to the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with z-axis in the direction of the polar axis. In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance ( 1/r ), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanics, a particle in a spherically symmetric potential is a system with a potential that depends only on the distance between the particle and a center. Inclination to the invariable plane for the giant planets Year Jupiter Saturn Uranus Neptune 2009 0.32° 0.93° 1.02° 0.72° 142400 (produced with Solex 10) 0.48° 0.79° 1.04° 0.55° 168000 (produced with Solex 10) 0.23° 1.01° 1.12° 0.55° The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter (center of mass) perpendicular to its angular momentum vector. From () then follows that :\frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right)\ + \lambda\ \cos^2\theta\ +\ \frac{1}{\Phi}\frac{d^2\Phi}{d\varphi^2}\ =\ 0 The first two terms only depend on the variable \theta and the third only on the variable \varphi. They take the forms: P^m_n(\sin \theta) \cos m\varphi \,,& 0 &\le m \le n \,,& n &= 0, 1, 2, \dots \\\ h(x, y, z) &= \frac{1}{r^{n+1}} P^m_n(\sin \theta) \sin m\varphi \,,& 1 &\le m \le n \,,& n &= 1, 2, \dots \end{align}|}} where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference: also P0n are the Legendre polynomials and Pmn for are the associated Legendre functions. ",-994.3,1.5,-2.0,804.62,1.91,C
-"A steel ball of velocity $5 \mathrm{~m} / \mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?","The motion of a ball is generally described by projectile motion (which can be affected by gravity, drag, the Magnus effect, and buoyancy), while its impact is usually characterized through the coefficient of restitution (which can be affected by the nature of the ball, the nature of the impacting surface, the impact velocity, rotation, and local conditions such as temperature and pressure). A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. This implies that the ball would bounce to 9 times its original height.Since conservation of mechanical energy implies \textstyle \frac{1}{2}mv_\text{f}^2 = mgH_\text{f}, then \textstyle H_\text{f} is proportional to v^2_\text{f}. Composites Science and Technology 64:35-54. . is as follows: V_b=\frac{\pi\,\Gamma\,\sqrt{\rho_t\,\sigma_e}\,D^2\,T}{4\,m} \left [1+\sqrt{1+\frac{8\,m}{\pi\,\Gamma^2\,\rho_t\,D^2\,T}}\, \right ] where *V_b\, is the ballistic limit *\Gamma\, is a projectile constant determined experimentally *\rho_t\, is the density of the laminate *\sigma_e\, is the static linear elastic compression limit *D\, is the diameter of the projectile *T\, is the thickness of the laminate *m\, is the mass of the projectile Additionally, the ballistic limit for small-caliber into homogeneous armor by TM5-855-1 is: V_1= 19.72 \left [ \frac{7800 d^3 \left [ \left ( \frac{e_h}{d} \right) \sec \theta \right ]^{1.6}}{W_T} \right ]^{0.5} where *V_1 is the ballistic limit velocity in fps *d is the caliber of the projectile, in inches *e_h is the thickness of the homogeneous armor (valid from BHN 360 - 440) in inches *\theta is the angle of obliquity *W_T is the weight of the projectile, in lbs == References == Category:Ballistics A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by \left(\frac{R_1}{R_2}\right)^{{3}/{8}} Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material. * e = coefficient of restitution * Sy = dynamic yield strength (dynamic ""elastic limit"") * E′ = effective elastic modulus * ρ = density * v = velocity at impact * μ = Poisson's ratio e = 3.1 \left(\frac{S_\text{y}}{1}\right)^{5/8} \left(\frac{1}{E'}\right)^{1/2} \left(\frac{1}{v}\right)^{1/4} \left(\frac{1}{\rho}\right)^{1/8} E' = \frac{E}{1-\mu^2} This equation overestimates the actual COR. The International Table Tennis Federation specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.887 to 0.923. The ball's angular velocity will be reduced after impact, but its horizontal velocity will be increased. The ball's angular velocity will be increased after impact, but its horizontal velocity will be decreased. To analyze the vertical and horizontal components of the motion, the COR is sometimes split up into a normal COR (ey), and tangential COR (ex), defined as :e_\text{y} = -\frac{v_\text{yf} - u_\text{yf}}{v_\text{yi} - u_\text{yi}}, :e_\text{x} = -\frac{(v_\text{xf}-r\omega_\text{f})-(u_\text{xf}-R\Omega_\text{f})}{(v_\text{xi}-r\omega_\text{i})-(u_\text{xi}-R\Omega_\text{i})}, where r and ω denote the radius and angular velocity of the ball, while R and Ω denote the radius and angular velocity the impacting surface (such as a baseball bat). It gives the following theoretical coefficient of restitution for solid spheres dropped 1 meter (v = 4.5 m/s). Depending on the ball's alignment at impact, the normal force can act ahead or behind the centre of mass of the ball, and friction from the ground will depend on the alignment of the ball, as well as its rotation, spin, and impact velocity. The ball's angular velocity will be reduced after impact, as will its horizontal velocity, and the ball is propelled upwards, possibly even exceeding its original height. A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). === Predicting from material properties === The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. More specifically, if the ball is bounced at an angle θ with the ground, the motion in the x- and y-axes (representing horizontal and vertical motion, respectively) is described by x-axis y-axis :\begin{align} a_\text{x} & = 0, \\\ v_\text{x} & = v_0 \cos \left(\theta \right), \\\ x & = x_0 + v_0 \cos \left( \theta \right) t, \end{align} :\begin{align} a_\text{y} & = -g, \\\ v_\text{y} & = v_0 \sin \left(\theta \right) -gt, \\\ y & = y_0 + v_0 \sin \left( \theta \right) t -\frac{1}{2}gt^2. \end{align} The equations imply that the maximum height (H) and range (R) and time of flight (T) of a ball bouncing on a flat surface are given by :\begin{align} H & = \frac{v_0^2}{2g}\sin^2\left(\theta\right), \\\ R &= \frac{v_0^2}{g}\sin\left(2\theta\right),~\text{and} \\\ T &= \frac{2v_0}{g} \sin \left(\theta \right). \end{align} Further refinements to the motion of the ball can be made by taking into account air resistance (and related effects such as drag and wind), the Magnus effect, and buoyancy. For an object bouncing off a stationary target, C_R is defined as the ratio of the object's speed after the impact to that prior to impact: C_R = \frac{v}{u}, where *v is the speed of the object after impact *u is the speed of the object before impact In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to: C_R = \sqrt{\frac{h}{H}}, where *h is the bounce height *H is the drop height The coefficient of restitution can be thought of as a measure of the extent to which mechanical energy is conserved when an object bounces off a surface. In particular rω is the tangential velocity of the ball's surface, while RΩ is the tangential velocity of the impacting surface. According to one article (addressing COR in tennis racquets), ""[f]or the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution."" If the ball moves horizontally at impact, friction will have a 'translational' component in the direction opposite to the ball's motion. This roughly corresponds to a COR of 0.727 to 0.806.Calculated using \textstyle e = \sqrt{\frac{H_\text{f}}{H_\text{i}}} and (if applicable) the diameter of the ball. For a straight drop on the ground with no rotation, with only the force of gravity acting on the ball, the COR can be related to several other quantities by: :e = \left|\frac{v_\text{f}}{v_\text{i}}\right| = \sqrt{\frac{K_\text{f}}{K_\text{i}}} = \sqrt{\frac{U_\text{f}}{U_\text{i}}} = \sqrt{\frac{H_\text{f}}{H_\text{i}}} = \frac{T_\text{f}}{T_\text{i}} =\sqrt{\frac{gT^2_\text{f}}{8H_\text{i}}}. In general, the ball will deform more at higher impact velocities and will accordingly lose more of its energy, decreasing its COR. ===Spin and angle of impact=== Upon impacting the ground, some translational kinetic energy can be converted to rotational kinetic energy and vice versa depending on the ball's impact angle and angular velocity. This energy loss is usually characterized (indirectly) through the coefficient of restitution (or COR, denoted e):Here, v and u are not just the magnitude of velocities, but include also their direction (sign). :e = -\frac{v_\text{f} - u_\text{f}}{v_\text{i} - u_\text{i}}, where vf and vi are the final and initial velocities of the ball, and uf and ui are the final and initial velocities impacting surface, respectively. ",96.4365076099,0.36,4.3,41.40,0.6321205588,C
-"Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \mathrm{~cm}$ and the mass be $200 \mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence. ","The study concludes that, assuming average observed values for lift coefficient, a 65mph rise ball must have at least a three degree launch angle in order to pass the strike zone at a point higher than the release point (the bottom of the strike zone and release point are the same at 1.5 feet).Clark, J.M., Greer, M.L. & Semon, M.D. Modeling pitch trajectories in fastpitch softball. The most appropriate are the Reynolds number, given by \mathrm{Re} = \frac{u\sqrt{A}}{ u} and the drag coefficient, given by c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. Another study utilizes a theoretical physics approach to modelling the trajectories of various softball pitches, including the rise ball. Because the only unknown in the above equation is the drag force Fd, it is possible to express it as \begin{align} \frac{F_{\rm d}}{\frac12 \rho A u^2} &= f_c\left(\frac{u \sqrt{A}}{ u} \right) \\\ F_{\rm d} &= \tfrac12 \rho A u^2 f_c(\mathrm{Re}) \\\ c_{\rm d} &= f_c(\mathrm{Re}) \end{align} Thus the force is simply ½ ρ A u2 times some (as-yet-unknown) function fc of the Reynolds number Re – a considerably simpler system than the original five-argument function given above. The general rules for the softball throw parallel those of the javelin throw when conducted in a formal environment, but the implement being thrown is a standard softball, which resembles the size of a standard shot but is considerably lighter. One image appears to show that the ball follows an increasingly upward trajectory; however, this image was taken of a particular type of training ball known as a JUGS LITE-FLITE ball, which has “one third of the mass (59.5g) of a regulation softball (181.71g)”. A similar image shown of a regulation softball pitched at the same speed (70mph) seems to show a decreasing upward trajectory, although the author describes the outcome nebulously as “the rise is not apparent”. Alongside the Olympic discipline of fastpitch softball, which is the most popular variation of softball, there is also modified fastpitch softball and slow-pitch softball. === Baseball5 === thumb|A B5 batter hitting the ball into play. In softball, a pitch is the act of throwing a ball underhand by using a windmill motion. The authors consider the effects of gravity, drag and the Magnus Effect using Newton’s laws of motion to calculate the position of the ball at different points in time, allowing them to model the trajectory of the ball in 3 dimensions. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac{\rho d}{\mu} V. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). Air is 1.293 kg/m3 at 0°C and 1 atmosphere *u is the flow velocity relative to the object, *A is the reference area, and *c_{\rm d} is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). That this is so becomes apparent when the drag force Fd is expressed as part of a function of the other variables in the problem: f_a(F_{\rm d}, u, A, \rho, u) = 0. The softball throw is a track and field event used as a substitute for more technical throwing events in competitions involving Youth, Paralympic, Special Olympics and Senior competitors. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. ",35.2, 11.58,4.68,322,2.89,A
-"A child slides a block of mass $2 \mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\mathrm{m} / \mathrm{s}$ and the block hits a spring with spring constant $6 \mathrm{~N} / \mathrm{m}$, what is the maximum compression of the spring? ","If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. The effective mass of the spring can be determined by finding its kinetic energy. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. The force an ideal spring would exert is exactly proportional to its extension or compression. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). ; Compression spring: Designed to operate with a compression load, so the spring gets shorter as the load is applied to it. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. The manufacture normally specifies the spring rate. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. These springs are compression springs and can differ greatly in strength and in size depending on application. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. Springs can store energy when compressed. In a real spring–mass system, the spring has a non-negligible mass m. ",8.99,-1.5,2.19,2.3,-114.40,D
-"If the field vector is independent of the radial distance within a sphere, find the function describing the density $\rho=\rho(r)$ of the sphere.","In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. For three dimensions, this normalization is the number density of the system ( \rho ) multiplied by the volume of the spherical shell, which symbolically can be expressed as \rho \, 4\pi r^2 dr. Taking particle 0 as fixed at the origin of the coordinates, \textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r}) is the average number of particles (among the remaining N-1) to be found in the volume \textstyle d^3r around the position \textstyle \mathbf{r}. thumb|250px|right|calculation of g(r) In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. In probability and statistics, a spherical contact distribution function, first contact distribution function,D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein-Zernike equation or structure-optimized potential refinement. ==Definition== Consider a system of N particles in a volume V (for an average number density \rho =N/V) and at a temperature T (let us also define \textstyle \beta = \frac{1}{kT}). In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper- sphere of radius r. ===Contact distribution function=== The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in \textstyle \textbf{R}^{ d}. That is, ƒ is radial if and only if :f\circ \rho = f\, for all , the special orthogonal group in n dimensions. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. The spherical contact function is also referred to as the contact distribution function, but some authors define the contact distribution function in relation to a more general set, and not simply a sphere as in the case of the spherical contact distribution function. In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. For a non-interacting gas, it is independent of the position \textstyle \mathbf{r}_1 and equal to the overall number density, \rho, of the system. In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. For distances r such that u(r) is significant, the mean local density will differ from the mean density \rho, depending on the sign of u(r) (higher for negative interaction energy and lower for positive u(r)). For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting, \begin{align} \rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n) &=\frac{1}{(N-n)!}\left(\prod_{i=n+1}^N\int\mathrm{d}^3\mathbf{r}_i\right)\sum_{\pi\in S_N} P^{(N)}(\mathbf{r}_{\pi (1)},\ldots,\mathbf{r}_{\pi (N)}) \\\ \end{align} This quantity is called the n-particle density function. To wit, :\phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx')\,dx' where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and , . Spherical contact distribution functions are used in the study of point processesD. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. ",0.1591549431,30,3.29527,-273,15,A
-"An Earth satellite has a perigee of $300 \mathrm{~km}$ and an apogee of $3,500 \mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\circ}$ around Earth from perigee?","Objects orbiting the Earth must be within this radius, otherwise, they may become unbound by the gravitational perturbation of the Sun. Orbital characteristics epoch J2000.0 aphelion 1.0167 AU perihelion 0.98329 AU semimajor axis 1.0000010178 AU eccentricity 0.0167086 inclination 7.155° to Sun's equator 1.578690° to invariable plane longitude of the ascending node 174.9° longitude of perihelion 102.9° argument of periapsis 288.1° period daysThe figure appears in multiple references, and is derived from the VSOP87 elements from section 5.8.3, p. 675 of the following: average orbital speed speed at aphelion speed at perihelion The following diagram shows the relation between the line of the solstice and the line of apsides of Earth's elliptical orbit. Theta Persei (Theta Per, θ Persei, θ Per) is a star system 37 light years away from Earth, in the constellation Perseus. A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. thumb|upright=1.5|Earth at seasonal points in its orbit (not to scale) thumb|Earth orbit (yellow) compared to a circle (gray) Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. A full orbit has 360°. Beta angle can be controlled to keep a satellite as cool as possible (for instruments that require low temperatures, such as infrared cameras) by keeping the beta angle as close to zero as possible, or, conversely, to keep a satellite in sunlight as much as possible (for conversion of sunlight by its solar panels, for solar stability of sensors, or to study the Sun) by maintaining a beta angle as close to +90 or -90 as possible. ==Determination and application of beta angles== The value of a solar beta angle for a satellite in Earth orbit can be found using the equation \beta=\sin^{-1}[\cos(\Gamma)\sin(\Omega)\sin(i)-\sin(\Gamma)\cos(\epsilon)\cos(\Omega)\sin(i)+\sin(\Gamma)\sin(\epsilon)\cos(i)] where \Gamma is the ecliptic true solar longitude, \Omega is the right ascension of ascending node (RAAN), i is the orbit's inclination, and \epsilon is the obliquity of the ecliptic (approximately 23.45 degrees for Earth at present). thumb|300px|Beta angle (\boldsymbol{\beta}) In orbital spaceflight, the beta angle (\boldsymbol{\beta}) is the angle between a satellite's orbital plane around Earth and the geocentric position of the sun. 9 Persei is a single variable star in the northern constellation Perseus, located around 4,300 light years away from the Sun. At a LEO of 280 kilometers, the object is in sunlight through 59% of its orbit (approximately 53 minutes in Sunlight, and 37 minutes in shadow.) The Satellite () is a small rock peak rising to 1,100 m, protruding slightly above the ice sheet 3 nautical miles (6 km) southwest of Pearce Peak and 8 nautical miles (15 km) east of Baillieu Peak. The changing Earth-Sun distance results in an increase of about 7% in total solar energy reaching the Earth at perihelion relative to aphelion. This angle is called the orbit's inclination. The beta angle varies between +90° and −90°, and the direction in which the satellite orbits its primary body determines whether the beta angle sign is positive or negative. It is radiating over 12,000 times the luminosity of the Sun from its swollen photosphere at an effective temperature of 9,840 K. 9 Persei has one visual companion, designated component B, at an angular separation of and magnitude 12.0. One complete orbit takes days (1 sidereal year), during which time Earth has traveled 940 million km (584 million mi).Jean Meeus, Astronomical Algorithms 2nd ed, (Richmond, VA: Willmann-Bell, 1998) 238. Since this value is close to zero, the center of the orbit is relatively close to the center of the Sun (relative to the size of the orbit). That same satellite also will have a beta angle with respect to the Sun, and in fact it has a beta angle for any celestial object one might wish to calculate one for: any satellite orbiting a body (i.e. the Earth) will be in that body's shadow with respect to a given celestial object (like a star) some of the time, and in its line-of-sight the rest of the time. A satellite in such an orbit spends at least 59% of its orbital period in sunlight. ==Light and shadow== The degree of orbital shadowing an object in LEO experiences is determined by that object's beta angle. The above discussion defines the beta angle of satellites orbiting the Earth, but a beta angle can be calculated for any orbiting three body system: the same definition can be applied to give the beta angle of other objects. The Hill sphere (gravitational sphere of influence) of the Earth is about 1,500,000 kilometers (0.01 AU) in radius, or approximately four times the average distance to the Moon.For the Earth, the Hill radius is :R_H = a \left(\frac{m}{3M}\right)^{1/3}, where m is the mass of the Earth, a is an astronomical unit, and M is the mass of the Sun. The orbital ellipse goes through each of the six Earth images, which are sequentially the perihelion (periapsis—nearest point to the Sun) on anywhere from January 2 to January 5, the point of March equinox on March 19, 20, or 21, the point of June solstice on June 20, 21, or 22, the aphelion (apoapsis—the farthest point from the Sun) on anywhere from July 3 to July 5, the September equinox on September 22, 23, or 24, and the December solstice on December 21, 22, or 23. So the radius in AU is about \left(\frac{1}{3 \cdot 332\,946}\right)^{1/3} \approx 0.01. ",2380,28,0.332,9,1590,E
-Two masses $m_1=100 \mathrm{~g}$ and $m_2=200 \mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \mathrm{~N} / \mathrm{m}$. Find the frequency of oscillatory motion for this system.,"As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. Many clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal. ==Description== The natural frequency of a simple mechanical system consisting of a weight suspended by a spring is: :f = {1\over 2 \pi} \sqrt {k\over m} where m is the mass and k is the spring constant. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. The effective mass of the spring can be determined by finding its kinetic energy. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. The resonance frequency of a pendulum, the only frequency at which it will vibrate, is given approximately, for small displacements, by the equation:Mechanical resonance :f = {1\over 2 \pi} \sqrt {g\over L} where g is the acceleration due to gravity (about 9.8 m/s2 near the surface of Earth), and L is the length from the pivot point to the center of mass. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Note that, in this approximation, the frequency does not depend on mass. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. To relate v to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring: :T=\frac{1}{2}m(\dot x^2+(l_0+x)^2\dot \theta^2) So the Lagrangian becomes: :L = T -V_k - V_g :L[x,\dot x,\theta, \dot \theta] = \frac{1}{2}m(\dot x^2+(l_0+x)^2\dot \theta^2) -\frac{1}{2}kx^2 + gm(l_0+x)\cos \theta ===Equations of motion=== With two degrees of freedom, for x and \theta, the equations of motion can be found using two Euler-Lagrange equations: :{\partial L\over\partial x}-{\operatorname d \over \operatorname dt} {\partial L\over\partial \dot x}=0 :{\partial L\over\partial \theta}-{\operatorname d \over \operatorname dt} {\partial L\over\partial \dot \theta}=0 For x: :m(l_0+x)\dot \theta^2 -kx + gm\cos \theta-m \ddot x=0 \ddot x isolated: :\ddot x =(l_0+x)\dot \theta^2 -\frac{k}{m}x + g\cos \theta And for \theta: :-gm(l_0+x)\sin \theta - m(l_0+x)^2\ddot \theta- 2m(l_0+x)\dot x \dot \theta=0 \ddot \theta isolated: :\ddot \theta=-\frac{g}{l_0+x}\sin \theta-\frac{2\dot x}{l_0+x}\dot \theta The elastic pendulum is now described with two coupled ordinary differential equations. In a real spring–mass system, the spring has a non-negligible mass m. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. If the system is excited (pushed) with a period between pushes equal to the inverse of the pendulum's natural frequency, the swing will swing higher and higher, but if excited at a different frequency, it will be difficult to move. For a hardening spring oscillator (\alpha>0 and large enough positive \beta>\beta_{c+}>0) the frequency response overhangs to the high- frequency side, and to the low-frequency side for the softening spring oscillator (\alpha>0 and \beta<\beta_{c-}<0). The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order in this system. ==See also== * Double pendulum * Duffing oscillator * Pendulum (mathematics) * Spring-mass system == References == == Further reading == * * ==External links== * Holovatsky V., Holovatska Y. (2019) ""Oscillations of an elastic pendulum"" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019. ",556,1.8763,0.444444444444444,1.81,2.74,E
-A particle moves with $v=$ const. along the curve $r=k(1+\cos \theta)$ (a cardioid). Find $\ddot{\mathbf{r}} \cdot \mathbf{e}_r=\mathbf{a} \cdot \mathbf{e}_r$.,"Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously as opposed to the eccentricity and argument of periapsis parameters for which eccentricity zero (circular orbit) corresponds to a singularity. ==Calculation== The eccentricity vector \mathbf{e} \, is: : \mathbf{e} = {\mathbf{v}\times\mathbf{h}\over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}} = \left ( {\mathbf{\left |v \right |}^2 \over {\mu} }- {1 \over{\left|\mathbf{r}\right|}} \right ) \mathbf{r} - {\mathbf{r} \cdot \mathbf{v} \over{\mu}} \mathbf{v} which follows immediately from the vector identity: : \mathbf{v}\times \left ( \mathbf{r}\times \mathbf{v} \right ) = \left ( \mathbf{v} \cdot \mathbf{v} \right ) \mathbf{r} - \left ( \mathbf{r} \cdot \mathbf{v} \right ) \mathbf{v} where: *\mathbf{r}\,\\! is position vector *\mathbf{v}\,\\! is velocity vector *\mathbf{h}\,\\! is specific angular momentum vector (equal to \mathbf{r}\times\mathbf{v}) *\mu\,\\! is standard gravitational parameter ==See also== *Kepler orbit *Orbit *Eccentricity *Laplace–Runge–Lenz vector ==References== Category:Orbits Category:Vectors (mathematics and physics) Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Then, A is a vector potential for , that is, abla \times \mathbf{A} =\mathbf{v}. By analogy with Biot-Savart's law, the following \boldsymbol{A}(\textbf{x}) is also qualify as a vector potential for v. :\boldsymbol{A}(\textbf{x}) =\int_\Omega \frac{\boldsymbol{v}(\boldsymbol{y}) \times (\boldsymbol{x} - \boldsymbol{y})}{4 \pi |\boldsymbol{x} - \boldsymbol{y}|^3} d^3 \boldsymbol{y} Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. That is, A' below is also a vector potential of v; \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ abla_y \times \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}. For this purpose Newton's notation will be used for the time derivative (\dot{\mathbf{A}}). The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The direction from r′ to r does not enter into the equation. For Kepler orbits the eccentricity vector is a constant of motion. * The position of r, the point at which values for ϕ and A are found, only enters the equation as part of the scalar distance from r′ to r. * The equation for A is a vector equation. In vector calculus, a vector potential is a vector field whose curl is a given vector field. Substituting curl[v] for the current density j of the retarded potential, you will get this formula. The drag curve or drag polar is the relationship between the drag on an aircraft and other variables, such as lift, the coefficient of lift, angle-of- attack or speed. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: abla \times \mathbf{A} = \mathbf{B}. They are given by: \begin{align} \boldsymbol{\dot{\hat r}} &= \dot\theta \boldsymbol{\hat\theta} + \dot\phi\sin\theta \boldsymbol{\hat\phi} \\\ \boldsymbol{\dot{\hat\theta}} &= - \dot\theta \boldsymbol{\hat r} + \dot\phi\cos\theta \boldsymbol{\hat\phi} \\\ \boldsymbol{\dot{\hat\phi}} &= - \dot\phi\sin\theta \boldsymbol{\hat{r}} - \dot\phi\cos\theta \boldsymbol{\hat\theta} \end{align} Thus the time derivative becomes: \mathbf{\dot A} = \boldsymbol{\hat r} \left(\dot A_r - A_\theta \dot\theta - A_\phi \dot\phi \sin\theta \right) \+ \boldsymbol{\hat\theta} \left(\dot A_\theta + A_r \dot\theta - A_\phi \dot\phi \cos\theta\right) \+ \boldsymbol{\hat\phi} \left(\dot A_\phi + A_r \dot\phi \sin\theta + A_\theta \dot\phi \cos\theta\right) == See also == * Del in cylindrical and spherical coordinates for the specification of gradient, divergence, curl, and Laplacian in various coordinate systems. ==References== Category:Vector calculus Category:Coordinate systems This means that \mathbf{A} = \mathbf{P} = \rho \mathbf{\hat \rho} + z \mathbf{\hat z}. Hence the charge and current densities affecting the electric and magnetic potential at r and t, from remote location r′ must also be at some prior time t′. The forces involved are obtained from the coefficients by multiplication with , where ρ is the density of the atmosphere at the flight altitude, is the wing area and is the speed. The vector potential A is used when studying the Lagrangian in classical mechanics and in quantum mechanics (see Schrödinger equation for charged particles, Dirac equation, Aharonov–Bohm effect). ",4.85,-0.75,0.405,0.70710678,1.06,B
-Calculate the minimum $\Delta v$ required to place a satellite already in Earth's heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth). Consider only the gravitational attraction of the Sun. ,"The formula by Ramanujan is accurate enough. giving an average orbital speed of . ==Conjunctions and transits== When the geocentric ecliptic longitude of Venus coincides with that of the Sun, it is in conjunction with the Sun – inferior if Venus is nearer and superior if farther. That said, Venus and Earth still have the lowest gravitational potential difference between them than to any other planet, needing the lowest delta-v to transfer between them, than to any other planet from them. A heliocentric orbit (also called circumsolar orbit) is an orbit around the barycenter of the Solar System, which is usually located within or very near the surface of the Sun. It is necessary for a spacecraft, like Gaia, to follow a Lissajous orbit or a halo orbit around L2 in order for its solar panels to get full sun. ===L3=== The location of L3 is the solution to the following equation, gravitation providing the centripetal force: \frac{M_1}{\left(R-r\right)^2}+\frac{M_2}{\left(2R-r\right)^2}=\left(\frac{M_2}{M_1+M_2}R+R-r\right)\frac{M_1+M_2}{R^3} with parameters M1, M2, and R defined as for the L1 and L2 cases, and r now indicates the distance of L3 from the position of the smaller object, if it were rotated 180 degrees about the larger object, while positive r implying L3 is closer to the larger object than the smaller object. Because the range of heliocentric distances is greater for the Earth than for Venus, the closest approaches come near Earth's perihelion. The 3.4° inclination of Venus's orbit is great enough to usually prevent the inferior planet from passing directly between the Sun and Earth at inferior conjunction. thumb|right|300 px|Representation of Venus (yellow) and Earth (blue) circling around the Sun. Venus has an orbit with a semi-major axis of , and an eccentricity of 0.007.Jean Meeus, Astronomical Formulæ for Calculators, by Jean Meeus. Elements by Simon Newcomb The low eccentricity and comparatively small size of its orbit give Venus the least range in distance between perihelion and aphelion of the planets: 1.46 million km. The orbit is now known to sub-kilometer accuracy. ==Table of orbital parameters== No more than five significant figures are presented here, and to this level of precision the numbers match very well the VSOP87 elements and calculations derived from them, Standish's (of JPL) 250-year best fit,Standish and Williams(2012) p 27 Newcomb's, and calculations using the actual positions of Venus over time. distances au Million km semimajor axis 0.72333 108.21 perihelion 0.71843 107.48 aphelion 0.7282 108.94 averageAverage distance over times. Sun orbit may refer to: * Heliocentric orbit, around the sun * Orbit of the sun around the Galactic Center The longitudes of perihelion were only 29 degrees apart at J2000, so the smallest distances, which come when inferior conjunction happens near Earth's perihelion, occur when Venus is near perihelion. A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. It is at the point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit. === and points=== thumb|right|200px|Gravitational accelerations at The and points lie at the third vertices of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of () or behind () the smaller mass with regard to its orbit around the larger mass. ===Stability=== The triangular points ( and ) are stable equilibria, provided that the ratio of is greater than 24.96.Actually (25 + 3)/2 ≈ This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. Now the orbit estimates are dominated by observations of the Venus Express spacecraft. The first spacecraft to be put in a heliocentric orbit was Luna 1 in 1959. The heliocentric longitude of Earth advances by 0.9856° per day, and after 2919.6 days, it has advanced by 2878°, only 2° short of eight revolutions (2880°). Two important Lagrange points in the Sun-Earth system are L1, between the Sun and Earth, and L2, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Lagrangian points in Solar System Body pair Semimajor axis, SMA (×109 m) L1 (×109 m) 1 − L1/SMA (%) L2 (×109 m) L2/SMA − 1 (%) L3 (×109 m) 1 + L3/SMA (%) Earth–Moon Sun–Mercury Sun–Venus Sun–Earth Sun–Mars Sun–Jupiter Sun–Saturn Sun–Uranus Sun–Neptune ==Spaceflight applications== ===Sun–Earth=== Sun–Earth is suited for making observations of the Sun–Earth system. This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M2 in the absence of M1, is that of M2 around M1, divided by ≈ 1.73: T_{s,M_2}(r) = \frac{T_{M_2,M_1}(R)}{\sqrt{3}}. ====== The location of L2 is the solution to the following equation, gravitation providing the centripetal force: \frac{M_1}{(R+r)^2}+\frac{M_2}{r^2}=\left(\frac{M_1}{M_1+M_2}R+r\right)\frac{M_1+M_2}{R^3} with parameters defined as for the L1 case. Such solar transits of Venus rarely occur, but with great predictability and interest.Venus transit page. by Aldo Vitagliano, creator of SolexWilliam Sheehan, John Westfall The Transits of Venus (Prometheus Books, 2004) ==Close approaches to Earth and Mercury== In this current era, the nearest that Venus comes to Earth is just under 40 million km. The distance between Venus and Earth varies from about 42 million km (at inferior conjunction) to about 258 million km (at superior conjunction). ",+0.60,0.88,5275.0,30,1590,C
-Find the ratio of the radius $R$ to the height $H$ of a right-circular cylinder of fixed volume $V$ that minimizes the surface area $A$.,"Since the area of a circle of radius r\,, which is the base of the cylinder, is given by B = \pi r^2 it follows that: * V = \pi r^2 h or even * V = \pi r^2 g . == Equilateral cylinder == thumb|Illustration of a cylinder circumscribed by a sphere of radius r. Where: * L\,represents the lateral surface area of the cylinder; * \pi\,is approximately 3.14; * r\,is the distance between the lateral surface of the cylinder and the axis, i.e. it is the value of the radius of the base; * h\,is the height of the cylinder; * 2 \pi r is the length of the circumference of the base, since \pi = \frac{C}{2r}, that is, C = 2\pi r. Simply substitute the radius and height measurements defined earlier into the volume formula for a straight circular cylinder: * V = \pi r^2 \cdot h \Rightarrow V = \pi r^2 \cdot 2r \Rightarrow V = 2\pi r^3 == Meridian section == It is the intersection between a plane containing the axis of the cylinder and the cylinder. For a right circular cylinder of radius and height , the lateral area is the area of the side surface of the cylinder: . To calculate the total area of a right circular cylinder, you simply add the lateral area to the area of the two bases: * A = L + 2 \cdot B. Replacing L = 2 \pi r h and B = \pi r^2, we have: * A=2\pi rh + 2\pi r^2 \Rightarrow A = 2 \pi r (h + r) or even * A = 2 \pi r (g + r) . == Volume == thumb|Illustration of a cylinder and a prism, both with height h. Then, assuming that the radius of the base of an equilateral cylinder is r\, then the diameter of the base of this cylinder is 2r\, and its height is 2r\,. Its lateral area can be obtained by replacing the height value by 2r: * L = 2 \pi r \cdot 2r \Rightarrow L = 4 \pi r^2 . The result can be obtained in a similar way for the total area: * T = 2 \pi r (h + r) \Rightarrow T = 2 \pi r (2r + r) \Rightarrow T = 2 \pi r \cdot 3 r \Rightarrow T = 6 \pi r^2 . Therefore, the lateral surface area is given by: * L=2\pi rh. For the equilateral cylinder it is possible to obtain a simpler formula to calculate the volume. Note that in the case of the right circular cylinder, the height and the generatrix have the same measure, so the lateral area can also be given by: * L = 2 \pi r g . Therefore, simply multiply the area of the base by the height: * V = B \cdot h. The area of the base of a cylinder is the area of a circle (in this case we define that the circle has a radius with measure r): * B = \pi r^2. Note that the cylinder is equilateral. Fixing g as the side on which the revolution takes place, we obtain that the side r, perpendicular to g, will be the measure of the radius of the cylinder. It can be obtained by the product between the length of the circumference of the base and the height of the cylinder. However, the head of the radius is not perfectly cylindrical but slightly oval. Through Cavalieri's principle, which defines that if two solids of the same height, with congruent base areas, are positioned on the same plane, such that any other plane parallel to this plane sections both solids, determining from this section two polygons with the same area, then the volume of the two solids will be the same, we can determine the volume of the cylinder. This lateral surface area can be calculated by multiplying the perimeter of the base by the height of the prism. The surface to volume ratio for this cube is thus :\mbox{SA:V} = \frac{6~\mbox{cm}^2}{1~\mbox{cm}^3} = 6~\mbox{cm}^{-1}. This is because the volume of a cylinder can be obtained in the same way as the volume of a prism with the same height and the same area of the base. For example, the volume of the torus with minor radius r and major radius R is V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2. ",0.5768,54.7,2.567,1.7,0.5,E
-Find the dimension of the parallelepiped of maximum volume circumscribed by a sphere of radius $R$.,"If the radius of the sphere is called , the radii of the spherical segment bases are and and the height of the segment (the distance from one parallel plane to the other) called , then the volume of the spherical segment is : V = \frac{\pi h}{6} \left(3 r_1^2 + 3 r_2^2 + h^2\right). The volume can also be expressed in terms of A_n, the area of the unit -sphere. == Formulas == The first volumes are as follows: Dimension Volume of a ball of radius Radius of a ball of volume 0 1 (all 0-balls have volume 1) 1 2R \frac{V}{2}=0.5\times V 2 \pi R^2 \approx 3.142\times R^2 \frac{V^{1/2}}{\sqrt{\pi}}\approx 0.564\times V^{\frac{1}{2}} 3 \frac{4\pi}{3} R^3 \approx 4.189\times R^3 \left(\frac{3V}{4\pi}\right)^{1/3}\approx 0.620\times V^{1/3} 4 \frac{\pi^2}{2} R^4 \approx 4.935\times R^4 \frac{(2V)^{1/4}}{\sqrt{\pi}}\approx 0.671\times V^{1/4} 5 \frac{8\pi^2}{15} R^5\approx 5.264\times R^5 \left(\frac{15V}{8\pi^2}\right)^{1/5}\approx 0.717\times V^{1/5} 6 \frac{\pi^3}{6} R^6 \approx 5.168 \times R^6 \frac{(6V)^{1/6}}{\sqrt{\pi}}\approx 0.761\times V^{1/6} 7 \frac{16\pi^3}{105} R^7 \approx 4.725\times R^7 \left(\frac{105V}{16\pi^3}\right)^{1/7}\approx 0.801\times V^{1/7} 8 \frac{\pi^4}{24} R^8 \approx 4.059\times R^8 \frac{(24V)^{1/8}}{\sqrt{\pi}}\approx 0.839\times V^{1/8} 9 \frac{32\pi^4}{945} R^9 \approx 3.299\times R^9 \left(\frac{945V}{32\pi^4}\right)^{1/9}\approx 0.876\times V^{1/9} 10 \frac{\pi^5}{120} R^{10} \approx 2.550\times R^{10} \frac{(120V)^{1/10}}{\sqrt{\pi}}\approx 0.911\times V^{1/10} 11 \frac{64\pi^5}{10395} R^{11} \approx 1.884\times R^{11} \left(\frac{10395V}{64\pi^5}\right)^{1/11}\approx 0.944\times V^{1/11} 12 \frac{\pi^6}{720} R^{12} \approx 1.335\times R^{12} \frac{(720V)^{1/12}}{\sqrt{\pi}}\approx 0.976\times V^{1/12} 13 \frac{128\pi^6}{135135} R^{13} \approx 0.911\times R^{13} \left(\frac{135135V}{128\pi^6}\right)^{1/13}\approx 1.007\times V^{1/13} 14 \frac{\pi^7}{5040} R^{14} \approx 0.599\times R^{14} \frac{(5040V)^{1/14}}{\sqrt{\pi}}\approx 1.037\times V^{1/14} 15 \frac{256\pi^7}{2027025} R^{15} \approx 0.381\times R^{15} \left(\frac{2027025V}{256\pi^7}\right)^{1/15}\approx 1.066\times V^{1/15} n Vn(R) Rn(V) === Two-dimension recurrence relation === As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation: : V_n(R) = \begin{cases} 1 &\text{if } n=0,\\\\[0.5ex] 2R &\text{if } n=1,\\\\[0.5ex] \dfrac{2\pi}{n}R^2 \times V_{n-2}(R) &\text{otherwise}. \end{cases} This allows computation of in approximately steps. === Closed form === The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06. It is the three-dimensional analogue of the sector of a circle. ==Volume== If the radius of the sphere is denoted by and the height of the cap by , the volume of the spherical sector is V = \frac{2\pi r^2 h}{3}\,. The volume of an odd-dimensional ball is :V_{2k+1}(R) = \frac{2(2\pi)^k}{(2k + 1)!!}R^{2k+1}. The volume of the sector is related to the area of the cap by: V = \frac{rA}{3}\,. ==Area== The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2\pi rh\,. The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then: :A_{n-1}(r) = r^{n-1} A_{n-1}(1). It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation : :V_n(R) = \int_0^R \frac{2\pi^{n/2}}{\Gamma\bigl(\tfrac n2\bigr)} \,r^{n-1}\,dr = \frac{2\pi^{n/2}}{n\,\Gamma\bigl(\tfrac n2\bigr)}R^n = \frac{\pi^{n/2}}{\Gamma\bigl(\tfrac n2 + 1\bigr)}R^n. === Geometric proof === The relations V_{n+1}(R) = \frac{R}{n+1}A_n(R) and A_{n+1}(R) = (2\pi R)V_n(R) and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. The volume of a -ball of radius is R^nV_n, where V_n is the volume of the unit -ball, the -ball of radius . The volume of an ball of radius is :V^p_n(R) = \frac{\Bigl(2\,\Gamma\bigl(\tfrac1p + 1\bigr)\Bigr)^n}{\Gamma\bigl(\tfrac np + 1\bigr)}R^n. The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by: :A_{n-1}(R) = \frac{d}{dR} V_{n}(R) = \frac{n}{R}V_{n}(R). Let denote the distance between a point in the plane and the center of the sphere, and let denote the azimuth. Also, A_{n-1}(R) = \frac{dV_n(R)}{dR} because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. The volume of the ball can therefore be written as an iterated integral of the volumes of the -balls over the possible radii and azimuths: :V_n(R) = \int_0^{2\pi} \int_0^R V_{n-2}\\!\left(\sqrt{R^2 - r^2}\right) r\,dr\,d\theta, The azimuthal coordinate can be immediately integrated out. As with the two-dimension recursive formula, the same technique can be used to give an inductive proof of the volume formula. === Direct integration in spherical coordinates === The volume of the n-ball V_n(R) can be computed by integrating the volume element in spherical coordinates. The spherical volume element is: :dV = r^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2) \cdots \sin(\varphi_{n-2})\,dr\,d\varphi_1\,d\varphi_2 \cdots d\varphi_{n-1}, and the volume is the integral of this quantity over between 0 and and all possible angles: :V_n(R) = \int_0^R \int_0^\pi \cdots \int_0^{2\pi} r^{n-1}\sin^{n-2}(\varphi_1) \cdots \sin(\varphi_{n-2})\,d\varphi_{n-1} \cdots d\varphi_1\,dr. Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume: :\begin{align} R_n(V) &= \frac{\Gamma\bigl(\tfrac n2 + 1\bigr)^{1/n}}{\sqrt{\pi}}V^{1/n} \\\ &= \left(\frac{n!! These are: :\begin{align} V_{2k}(R) &= \frac{\pi^k}{k!}R^{2k}, \\\ V_{2k+1}(R) &= \frac{2(k!)(4\pi)^k}{(2k + 1)!}R^{2k+1}. \end{align} The volume can also be expressed in terms of double factorials. thumb|A spherical segment In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. In computational geometry, the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d-dimensional space whose interior does not overlap with any given obstacles. ==Two dimensions== The largest empty circle problem is the problem of finding a circle of largest radius in the plane whose interior does not overlap with any given obstacles. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. ",2.3,1.154700538,-2.0,0.5,15.757,B
-"A potato of mass $0.5 \mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. (a) Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \mathrm{~s}^{-1}$. (b) Find the maximum height of the potato if it has the same value of $k$, but it is initially shot directly upward with a student-made potato gun with an initial velocity of $120 \mathrm{~m} / \mathrm{s}$.","Note that d has its maximum value when : \sin 2\theta=1 , which necessarily corresponds to : 2\theta=90^\circ , or : \theta=45^\circ . thumb|350px|Trajectories of projectiles launched at different elevation angles but the same speed of 10 m/s in a vacuum and uniform downward gravity field of 10 m/s2. In air, which has a kinematic viscosity around 0.15\,\mathrm{cm^2/s}, this means that the drag force becomes quadratic in v when the product of speed and diameter is more than about 0.015\,\mathrm{m^2/s}, which is typically the case for projectiles. Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). The magnitude of the velocity (under the Pythagorean theorem, also known as the triangle law): : v = \sqrt{v_x^2 + v_y^2 } . === Displacement === thumb|250px|Displacement and coordinates of parabolic throwing At any time t , the projectile's horizontal and vertical displacement are: : x = v_0 t \cos(\theta) , : y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 . If the starting point is at height y0 with respect to the point of impact, the time of flight is: : t = \frac{d}{v \cos\theta} = \frac{v \sin \theta + \sqrt{(v \sin \theta)^2 + 2gy_0}}{g} As above, this expression can be reduced to : t = \frac{v\sin{\theta} + \sqrt{(v\sin{\theta})^{2}}}{g} = \frac{v\sin{\theta} + v\sin{\theta}}{g} = \frac{2v\sin{\theta}}{g} = \frac{2v\sin{(45)}}{g} = \frac{2v\frac{\sqrt{2}}{2}}{g} = \frac{\sqrt{2}v}{g} if θ is 45° and y0 is 0. === Time of flight to the target's position === As shown above in the Displacement section, the horizontal and vertical velocity of a projectile are independent of each other. The following assumptions are made: * Constant gravitational acceleration * Air resistance is given by the following drag formula, ::\mathbf{F_D} = -\tfrac{1}{2} c \rho A\, v\,\mathbf{v} ::Where: ::*FD is the drag force ::*c is the drag coefficient ::*ρ is the air density ::*A is the cross sectional area of the projectile ::*μ = k/m = cρA/(2m) ==== Special cases ==== Even though the general case of a projectile with Newton drag cannot be solved analytically, some special cases can. If h = R : \theta = \arctan(4)\approx 76.0^\circ === Maximum distance of projectile === thumb|250px|The maximum distance of projectile The range and the maximum height of the projectile does not depend upon its mass. Mathematically, it is given as t=U \sin\theta/g where g = acceleration due to gravity (app 9.81 m/s²), U = initial velocity (m/s) and \theta = angle made by the projectile with the horizontal axis. 2\. *Vertical motion downward: ::\dot{v}_y(t) = -g+\mu\,v_y^2(t) ::v_y(t) = -v_\infty \tanh\frac{t-t_{\mathrm{peak}}}{t_f} ::y(t) = y_{\mathrm{peak}} - \frac{1}{\mu}\ln\left(\cosh\frac{t-t_{\mathrm{peak}}}{t_f}\right) :After a time t_f, the projectile reaches almost terminal velocity -v_\infty. ==== Numerical solution ==== A projectile motion with drag can be computed generically by numerical integration of the ordinary differential equation, for instance by applying a reduction to a first-order system. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. * Stokes drag: \mathbf{F_{air}} = -k_{\mathrm{Stokes}}\cdot\mathbf{v}\qquad (for Re \lesssim 1000) * Newton drag: \mathbf{F_{air}} = -k\,|\mathbf{v}|\cdot\mathbf{v}\qquad (for Re \gtrsim 1000) right|thumb|320px|Free body diagram of a body on which only gravity and air resistance acts The free body diagram on the right is for a projectile that experiences air resistance and the effects of gravity. The vertical motion of the projectile is the motion of a particle during its free fall. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. Hence range and maximum height are equal for all bodies that are thrown with the same velocity and direction. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Because of this, we can find the time to reach a target using the displacement formula for the horizontal velocity: x = v_0 t \cos(\theta) \frac{x}{t}=v_0\cos(\theta) t=\frac{x}{v_0\cos(\theta)} This equation will give the total time t the projectile must travel for to reach the target's horizontal displacement, neglecting air resistance. === Maximum height of projectile === thumb|250px|Maximum height of projectile The greatest height that the object will reach is known as the peak of the object's motion. Attack of the Killer Potatoes is a 1997 science-fiction children's story by Peter Lerangis. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. The mass of the projectile will be denoted by m, and \mu:=k/m. For the vertical displacement of the maximum height of the projectile: : h = v_0 t_h \sin(\theta) - \frac{1}{2} gt^2_h : h = \frac{v_0^2 \sin^2(\theta)}{2g} The maximum reachable height is obtained for θ=90°: : h_{\mathrm{max}} = \frac{v_0^2}{2g} If the projectile's position (x,y) and launch angle (θ) are known, the maximum height can be found by solving for h in the following equation: :h=\frac{(x\tan\theta)^2}{4(x\tan\theta-y)}. === Relation between horizontal range and maximum height === The relation between the range d on the horizontal plane and the maximum height h reached at \frac{t_d}{2} is: : h = \frac{d\tan\theta}{4} h = \frac{v_0^2\sin^2\theta}{2g} : d = \frac{v_0^2\sin2\theta}{g} : \frac{h}{d} = \frac{v_0^2\sin^2\theta}{2g} × \frac{g}{v_0^2\sin2\theta} : \frac{h}{d} = \frac{\sin^2\theta}{4\sin\theta\cos\theta} h = \frac{d\tan\theta}{4} . The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. ",2.00,46.7,1000.0,-2,37,C
-"A particle of mass $m$ and velocity $u_1$ makes a head-on collision with another particle of mass $2 m$ at rest. If the coefficient of restitution is such to make the loss of total kinetic energy a maximum, what are the velocities $v_1$ after the collision?","It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text{Coefficient of restitution } (e) = \frac{\left | \text{Relative velocity after collision} \right |}{\left | \text{Relative velocity before collision}\right |} The mathematics were developed by Sir Isaac Newton in 1687. The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. Since the total energy and momentum of the system are conserved and their rest masses do not change, it is shown that the momentum of the colliding body is decided by the rest masses of the colliding bodies, total energy and the total momentum. Likewise, the conservation of the total kinetic energy is expressed by: \tfrac12 m_1u_1^2+\tfrac12 m_2u_2^2 \ =\ \tfrac12 m_1v_1^2 +\tfrac12 m_2v_2^2. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. Let v_1, v_2 be the final velocity of object 1 and object 2 respectively. \begin{cases} \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 \\\ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \end{cases} From the first equation, m_1 \left(u_1^2 - v_1^2\right) = m_2 \left(v_2^2 - u_2^2\right) m_1 \left(u_1 + v_1\right) \left(u_1 - v_1\right) = m_2 \left(v_2 + u_2\right) \left(v_2 - u_2\right) From the second equation, m_1 \left(u_1 - v_1\right) = m_2 \left(v_2 - u_2\right) After division, u_1+v_1=v_2+u_2 u_1-u_2 = -(v_1-v_2) \frac{\left | v_1-v_2 \right |}{\left | u_1-u_2 \right |} = 1 The equation above is the restitution equation, and the coefficient of restitution is 1, which is a perfectly elastic collision. ===Sports equipment=== Thin-faced golf club drivers utilize a ""trampoline effect"" that creates drives of a greater distance as a result of the flexing and subsequent release of stored energy which imparts greater impulse to the ball. In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. When vehicles collide, the damage increases with the relative velocity of the vehicles, the damage increasing as the square of the velocity since it is the impact kinetic energy (1/2 mv2) which is the variable of importance. The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. During the collision of small objects, kinetic energy is first converted to potential energy associated with a repulsive or attractive force between the particles (when the particles move against this force, i.e. the angle between the force and the relative velocity is obtuse), then this potential energy is converted back to kinetic energy (when the particles move with this force, i.e. the angle between the force and the relative velocity is acute). Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. Using the notation from above where u represents the velocity before the collision and v after, yields: \begin{align} & m_\text{a} u_\text{a} + m_\text{b} u_\text{b} = m_\text{a} v_\text{a} + m_\text{b} v_\text{b} \\\ & C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |} \\\ \end{align} Solving the momentum conservation equation for v_\text{a} and the definition of the coefficient of restitution for v_\text{b} yields: \begin{align} & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} - m_\text{b} v_\text{b}}{m_\text{a}} = v_\text{a} \\\ & v_\text{b} = C_R(u_\text{a} - u_\text{b}) + v_\text{a} \\\ \end{align} Next, substitution into the first equation for v_\text{b} and then resolving for v_\text{a} gives: \begin{align} & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} - m_\text{b} C_R(u_\text{a} - u_\text{b}) - m_\text{b} v_\text{a}}{m_\text{a}} = v_\text{a} \\\ & \\\ & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b} - u_\text{a})}{m_\text{a}} = v_\text{a} \left[ 1 + \frac{m_\text{b}}{m_\text{a}} \right] \\\ & \\\ & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b} - u_\text{a})}{m_\text{a} + m_\text{b}} = v_\text{a} \\\ \end{align} A similar derivation yields the formula for v_\text{b}. === COR variation due to object shape and off-center collisions === When colliding objects do not have a direction of motion that is in-line with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the post-collision velocity difference will be lost to rotation and friction. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). === Predicting from material properties === The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. For two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact, i.e. along the line of impact. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. In the case of a large u_{1}, the value of v_{1} is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed. ",9.13,4.68,0.33333333,167,54.7,C
-"The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?","Nimbus Hills () is a rugged line of hills and peaks about 14 nautical miles (26 km) long, forming the southeast part of Pioneer Heights in the Heritage Range, Ellsworth Mountains. Target Hill () is a prominent hill which rises 1,010 m above the level of Larsen Ice Shelf. Gerdkooh ancient hills (Persian: تپه باستان گردکوه) consists of three hills, the tallest of which is 26 m in height. Vantages Hill () (Adam Hayat) is a flat-topped hill, over 2,000 m above sea level and 300 m above the surrounding plateau, standing 10 nautical miles (18 km) southwest of Mount Henderson in the western part of Britannia Range. Sistenup Peak () is a low peak at the northeast end of the Kirwan Escarpment, about 5 nautical miles (9 km) north of Sistefjell Mountain, in Queen Maud Land. Powell Hill () is a rounded, ice-covered prominence 6 nautical miles (11 km) west-southwest of Mount Christmas, overlooking the head of Algie Glacier. Named by Advisory Committee on Antarctic Names (US-ACAN) after the National Aeronautics and Space Administration weather satellite, Nimbus, which took photographs of Antarctica (including the Ellsworth Mountains) from approximately 500 nautical miles (900 km) above earth on September 13, 1964. ==See also== Geographical features include: ===Samuel Nunataks=== ===Other features=== * Flanagan Glacier * Mount Capley * Warren Nunatak Category:Hills of Ellsworth Land The hill was the most westerly point reached by the Falkland Islands Dependencies Survey (FIDS) survey party in 1955; it was visible to the party as a target upon which to steer from the summit of Richthofen Pass. Category:Hills of Oates Land The hills are located in Qaem Shahr in Mazandaran Province. It stands 6 nautical miles (11 km) west of Mount Fritsche on the south flank of Leppard Glacier in eastern Graham Land. There is evidence that the hills at the time of the Sasanian Empire and Muslim conquest of Persia were part of a Castle. == See also == *Castles in Iran *Tepe Sialk ==References== == External links == * video link * Persian Wikipedia article Category:Archaeological sites in Iran Category:Geography of Mazandaran Province Category:Castles in Iran Category:Buildings and structures in Mazandaran Province Category:Tourist attractions in Mazandaran Province Category:National works of Iran Category:Mountains of Queen Maud Land Category:Princess Martha Coast Category:Hills of Graham Land Category:Oscar II Coast Mapped by Norwegian cartographers from surveys and air photos by Norwegian-British-Swedish Antarctic Expedition (NBSAE) (1949–52) and air photos by the Norwegian exp (1958–59) and named Sistenup (last peak). Category:Hills of the Ross Dependency Category:Shackleton Coast Mapped by United States Geological Survey (USGS) from ground surveys and U.S. Navy air photos, 1961–66. This is the most southerly point reached by the Darwin Glacier Party of the Commonwealth Trans-Antarctic Expedition (1957–58), who gave it this name because of the splendid view it afforded. Named by Advisory Committee on Antarctic Names (US-ACAN) for Lieutenant Commander James A. Powell, U.S. Navy, communications officer at McMurdo Station during U.S. Navy Operation Deepfreeze 1963 and 1964. In exploring this area, a 4500-year-old grave has been found, as well as objects such as disposable tableware dishes related to the Parthian Empire and Sasanian Empire. Their history has been estimated to date back to the Iron Age. There is evidence that the hills at the time of the Sasanian Empire and Muslim conquest of Persia were part of a Castle. == See also == *Castles in Iran *Tepe Sialk ==References== == External links == * video link * Persian Wikipedia article Category:Archaeological sites in Iran Category:Geography of Mazandaran Province Category:Castles in Iran Category:Buildings and structures in Mazandaran Province Category:Tourist attractions in Mazandaran Province Category:National works of Iran ",-0.041,2,-2.0,3,0,C
-"Shot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\circ} \mathrm{N}$, and the lead fell a distance of $27 \mathrm{~m}$. In what direction and how far did the lead bullets land from the direct vertical?","Large shot which could not be made by the shot tower was made by tumbling pieces of cut lead sheet in a barrel until round.. Molten lead at the top of the tower was poured through a sieve or mesh, forming uniform spherical shot before falling into a large vat of water at the bottom of the tower. A shot tower with a 40-meter drop can produce up to #6 shot (nominally 2.4mm in diameter) while an 80-meter drop can produce #2 shot (nominally 3.8mm in diameter). thumb|100px|How a shot tower works A shot tower is a tower designed for the production of small-diameter shot balls by free fall of molten lead, which is then caught in a water basin. Shot towers work on the principle that molten lead forms perfectly round balls when poured from a high place. The ""wind tower"" method, which used a blast of cold air to dramatically shorten the drop necessary and was patented in 1848 by the T.O LeRoy Company of New York City,, Lynne Belluscio, LeRoy Penny Saver NewsHistory of the American Shot Tower meant that tall shot towers became unnecessary, but many were still constructed into the late 1880s, and two surviving examples date from 1916 and 1969. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. It can be seen looking west from I-95. 1870 Panorama from Sparks Shot Tower.jpg|1870 photo from the top of the tower toward the Delaware River 1880 survey Sparks Shot Tower.png|1880 Hexamer General Survey page on the tower Sparks Shot Tower2.jpg|1973 Historic American Buildings Survey photo Sparks Shot Tower Historical Marker 129-131 Carpenter St Philadelphia PA (DSC 3814).jpg|Historical Marker ==See also== *Lead shot *Phoenix Shot Tower *Shotgun shell ==References== ==External links== *Listing and photographs at the Historic American Buildings Survey *Sparks Shot Tower at USHistory.org *Waymark *Listing and photograph at Philadelphia Buildings and Architects Category:Industrial buildings completed in 1808 Category:Towers completed in 1808 Category:Buildings and structures in Philadelphia Category:Shot towers Category:South Philadelphia The shot is primarily used for projectiles in shotguns, and for ballast, radiation shielding, and other applications for which small lead balls are useful. == Shot making == === Process === In a shot tower, lead is heated until molten, then dropped through a copper sieve high in the tower. Use of shot towers replaced earlier techniques of casting shot in moulds, which was expensive, or of dripping molten lead into water barrels, which produced insufficiently spherical balls. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Originally used to produce shot for hunters, the tower produced ammunition during the War of 1812 and the Civil War. The Sparks Shot Tower is a historic shot tower located at 129-131 Carpenter Street in Philadelphia, Pennsylvania. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. thumb|upright|Shortly after completion thumb|upright|Shortly before demolition The Tower Building was a structure in the Financial District of Manhattan, New York City, located at 50-52 Broadway on a lot that extended east to New Street. Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. thumb|right|360px|Lamoka projectile points from central New York State. At the start of the War of 1812, the federal government became their major customer, buying war munitions, and Quaker John Bishop sold his part of the company to Thomas Sparks.Sparks Shot Tower, 1808, in John Mayer, Workshop of the World (Oliver Evans Press, 1990) Before the use of shot towers, shot was made in wooden molds, which resulted in unevenly formed, low quality shot. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. ",30,131,-167.0,54.394,2.26,E
-"Perform an explicit calculation of the time average (i.e., the average over one complete period) of the potential energy for a particle moving in an elliptical orbit in a central inverse-square-law force field. Express the result in terms of the force constant of the field and the semimajor axis of the ellipse. ","Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Using the virial theorem we find: *the time-average of the specific potential energy is equal to −2ε **the time-average of r−1 is a−1 *the time-average of the specific kinetic energy is equal to ε === Energy in terms of semi major axis === It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The empty focus (\mathbf{F2} = \left(f_x,f_y\right)) can be found by first determining the Eccentricity vector: :\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu} Where \mathbf{h} is the specific angular momentum of the orbiting body: :\mathbf{h} = \mathbf{r} \times \mathbf{v} Then :\mathbf{F2} = -2a\mathbf{e} ====Using XY Coordinates==== This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2 + y^2 } = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt{r_x^2 + r_y^2} \quad initial distance from F1 (at the origin) :a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad the semi-major axis length :e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad the Eccentricity vector coordinates :e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values fx, fy and a can be applied to the general ellipse equation above. == Orbital parameters == The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. The solution of this equation is u(\varphi) = -\frac{\alpha}{mh^{2}} \left[ 1 + e \cos \left( \varphi - \varphi_{0}\right) \right] which shows that the orbit is a conic section of eccentricity e; here, φ0 is the initial angle, and the center of force is at the focus of the conic section. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. Adopting the radial distance r and the azimuthal angle φ as the coordinates, the Hamilton-Jacobi equation for a central-force problem can be written \frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + \frac{1}{2m r^{2}} \left( \frac{dS_{\varphi}}{d\varphi} \right)^{2} + U(r) = E_{\mathrm{tot}} where S = Sφ(φ) + Sr(r) − Etott is Hamilton's principal function, and Etot and t represent the total energy and time, respectively. Assume that a particle is moving under an arbitrary central force F1(r), and let its radius r and azimuthal angle φ be denoted as r(t) and φ1(t) as a function of time t. Four line segments go out from the left focus to the ellipse, forming two shaded pseudo-triangles with two straight sides and the third side made from the curved segment of the intervening ellipse.|As for all central forces, the particle in the Kepler problem sweeps out equal areas in equal times, as illustrated by the two blue elliptical sectors. A central-force problem is said to be ""integrable"" if this final integration can be solved in terms of known functions. ===Orbit of the particle=== The total energy of the system Etot equals the sum of the potential energy and the kinetic energyGoldstein, p. For an attractive force (α < 0), the orbit is an ellipse, a hyperbola or parabola, depending on whether u1 is positive, negative, or zero, respectively; this corresponds to an eccentricity e less than one, greater than one, or equal to one. The eccentricity e is related to the total energy E (cf. the Laplace–Runge–Lenz vector) : e = \sqrt{1 + \frac{2EL^2}{k^2 m}} Comparing these formulae shows that E<0 corresponds to an ellipse (all solutions which are closed orbits are ellipses), E=0 corresponds to a parabola, and E>0 corresponds to a hyperbola. In particular, E=-\frac{k^2 m}{2L^2} for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius). If the force is the sum of an inverse quadratic law and a linear term, i.e., if F(r) = \frac{a}{r^2} + cr, the problem also is solved explicitly in terms of Weierstrass elliptic functions.Izzo and Biscani ==References== ==Bibliography== * * Category:Classical mechanics If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Thus, the areal velocity is constant for a particle acted upon by any type of central force; this is Kepler's second law.Goldstein, p. 73; Landau and Lifshitz, p. 31; Sommerfeld, p. 39; Symon, p. 135. 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. This includes the radial elliptic orbit, with eccentricity equal to 1. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. The center of force is located at one of the foci of the elliptical orbit. where u1 and u2 are constants, with u2 larger than u1. In that case, :n = \frac{d}{2\pi}\sqrt{\frac{ G( M + m ) }{a^3}} = d\sqrt{\frac{ G( M + m ) }{4\pi^2 a^3}}\,\\! where *d is the quantity of time in a day, *G is the gravitational constant, *M and m are the masses of the orbiting bodies, *a is the length of the semi-major axis. By Newton's second law of motion, the magnitude of F equals the mass m of the particle times the magnitude of its radial accelerationGoldstein, p. ",7166.67,0.6749,-1.0,0.75,0,C
-A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \mathrm{dyne} / \mathrm{cm}$. The mass is displaced $3 \mathrm{~cm}$ and released from rest. Calculate the natural frequency $\nu_0$.,"As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). In a mass–spring system, with mass m and spring stiffness k, the natural angular frequency can be calculated as: \omega _0 =\sqrt{\frac{k}{m}} In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term , where for a real σ, and is a constant. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. ==Overview== Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. In LC and RLC circuits, its natural angular frequency can be calculated as: \omega _0 =\frac{1}{\sqrt{LC}} ==See also== * Fundamental frequency ==Footnotes== ==References== * * * * Category:Waves Category:Oscillation The effective mass of the spring can be determined by finding its kinetic energy. thumb|Scale of harmonics on C. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Natural vibrations are different from forced vibrations which happen at the frequency of an applied force (forced frequency). thumb|119x119px|Energy level scheme of half-harmonic generation process. In a real spring–mass system, the spring has a non-negligible mass m. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. For instance: the frequency ratio 5:4 is equal to of the string length and is the complement of , the position of the fifth harmonic (and the fourth overtone). It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network. ",-8,15.757,1.56,6.9,35.2,D
- Find the center of mass of a uniformly solid cone of base diameter $2a$ and height  $h$,"If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same and , then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated. :\rho > {R^2 + L^2 \over 2R} and \alpha = \arccos \left({\sqrt{L^2 + R^2} \over 2\rho}\right)-\arctan \left({R \over L}\right) Then the radius at any point as varies from to is: :y = \sqrt{\rho^2 - (\rho\cos(\alpha) - x)^2} - \rho\sin(\alpha) If the chosen is less than the tangent ogive and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. The radius of the circle that forms the ogive is called the ogive radius, , and it is related to the length and base radius of the nose cone as expressed by the formula: :\rho = {R^2 + L^2\over 2R} The radius at any point , as varies from to is: :y = \sqrt{\rho^2 - (L - x)^2}+R - \rho The nose cone length, , must be less than or equal to . The center of the spherical nose cap, , can be found from: : x_o = x_t + \sqrt{ r_n^2 - y_t^2} And the apex point, can be found from: : x_a = x_o - r_n === Bi-conic === A bi-conic nose cone shape is simply a cone with length stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with length , where the base of the upper cone is equal in radius to the top radius of the smaller frustum with base radius . For 0 \le K' \le 1 : y = R\left({2 \left({x \over L}\right) - K'\left({x \over L}\right)^2 \over 2 - K'}\right) can vary anywhere between and , but the most common values used for nose cone shapes are: Parabola Type Value Cone Half Three Quarter Full For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. The failure is governed by crack growth in concrete, which forms a typical cone shape having the anchor's axis as revolution axis. ==Mechanical models== ===ACI 349-85=== Under tension loading, the concrete cone failure surface has 45° inclination. The tangency point where the sphere meets the cone can be found from: : x_t = \frac{L^2}{R} \sqrt{ \frac{r_n^2}{R^2 + L^2} } : y_t = \frac{x_t R}{L} where is the radius of the spherical nose cap. :L=L_1+L_2 :For 0 \le x \le L_1 : y = {xR_1 \over L_1} :For L_1 \le x \le L : y = R_1 + {(x - L_1)(R_2-R_1)\over L_2} Half angles: :\phi_1 = \arctan \left({R_1 \over L_1}\right) and y = x \tan(\phi_1)\; :\phi_2 = \arctan \left({R_2 - R_1 \over L_2}\right) and y = R_1 + (x - L_1) \tan(\phi_2)\; === Tangent ogive === Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. alt=Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.|thumb|300x300px|General parameters used for constructing nose cone profiles. Finally, the apex point can be found from: : x_a = x_o - r_n === Secant ogive === The profile of this shape is also formed by a segment of a circle, but the base of the shape is not on the radius of the circle defined by the ogive radius. For , Haack nose cones bulge to a maximum diameter greater than the base diameter. The full body of revolution of the nose cone is formed by rotating the profile around the centerline . For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium. == Nose cone shapes and equations == === General dimensions === In all of the following nose cone shape equations, is the overall length of the nose cone and is the radius of the base of the nose cone. is the radius at any point , as varies from , at the tip of the nose cone, to . The sides of a conic profile are straight lines, so the diameter equation is simply: : y = {xR \over L} Cones are sometimes defined by their half angle, : : \phi = \arctan \left({R \over L}\right) and y = x \tan(\phi)\; ==== Spherically blunted conic ==== In practical applications such as re-entry vehicles, a conical nose is often blunted by capping it with a segment of a sphere. Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. right|thumb|Concrete Cone Model Concrete cone is one of the failure modes of anchors in concrete, loaded by a tensile force. The length/diameter relation is also often called the caliber of a nose cone. Wallis's Conical Edge with right|thumb|600px| Figure 2. thumb|right|400px|The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = k \sqrt{f_{cc}} {h_{ef}}^{1.5} [N] , Where: k \- 13.5 for post-installed fasteners, 15.5 for cast-in-site fasteners f_{cc} \- Concrete compressive strength measured on cubes [MPa] {h_{ef}} \- Embedment depth of the anchor [mm] The model is based on fracture mechanics theory and takes into account the size effect, particularly for the factor {h_{ef}}^{1.5} which differentiates from {h_{ef}}^{2} expected from the first model. The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = f_{ct} {A_{N}} [N] Where: f_{ct} \- tensile strength of concrete A_{N} \- Cone's projected area === Concrete capacity design (CCD) approach for fastening to concrete=== Under tension loading, the concrete capacity of a single anchor is calculated assuming an inclination between the failure surface and surface of the concrete member of about 35°. The rocket body will not be tangent to the curve of the nose at its base. ",0.75,272.8,22.0,1,+65.49,A
-A particle is projected with an initial velocity $v_0$ up a slope that makes an angle $\alpha$ with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for $v_0=2.4 \mathrm{~m} / \mathrm{s}$ and $\alpha=26^{\circ}$.,"Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = at+v_0 & [1]\\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\\ \end{align} where: * is the particle's initial position * is the particle's final position * is the particle's initial velocity * is the particle's final velocity * is the particle's acceleration * is the time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; \begin{align} \mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\\ \end{align} in magnitudes, \begin{align} v & = at+v_0 \,, & [1] \\\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. thumb|360px|v vs t graph for a moving particle under a non-uniform acceleration a. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary, \begin{align} \omega & = \omega_0 + \alpha t \\\ \theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\\ \theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\\ \omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\\ \theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\\ \end{align} where is the constant angular acceleration, is the angular velocity, is the initial angular velocity, is the angle turned through (angular displacement), is the initial angle, and is the time taken to rotate from the initial state to the final state. ===General planar motion=== These are the kinematic equations for a particle traversing a path in a plane, described by position . Suppose that C is the curve traced out by P and s is the arc length of C corresponding to time t. The velocity vector of the particle is : \mathbf{v} = \frac{d \mathbf{r}}{dt} = \dot{s}\mathbf{e}_t = v\mathbf{e}_t , where et is the unit tangent vector to C. Define the angular momentum of P as : \mathbf{h} = \mathbf{r} \times m\mathbf{v} = h\mathbf{k}, where k = i x j. Then the acceleration vector of P can be expressed as : \mathbf{a} = -\frac{\kappa v^2r}{p} \mathbf{e}_r + \left( v \frac{dv}{ds} + \frac{\kappa v^2q}{p} \right) \mathbf{e}_t . Given initial velocity , one can calculate how high the ball will travel before it begins to fall. According to Siacci's theorem, the acceleration a of P can be expressed as : \mathbf{a} = -\frac{\kappa v^2r}{p} \mathbf{e}_r + \frac{(h^2)'}{2p^2} \mathbf{e}_t = S_r \mathbf{e}_r + S_t \mathbf{e}_t . where the prime denotes differentiation with respect to the arc length s, and κ is the curvature function of the curve C. Let a particle P of mass m move in a two-dimensional Euclidean space (planar motion). The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. Let there be m variables that govern the forward- kinematics equation, i.e. the position function. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the polar unit vectors. right|thumb|200px|The forward kinematics equations define the trajectory of the end-effector of a PUMA robot reaching for parts. Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Only Domingo de Soto, a Spanish theologian, in his commentary on Aristotle's Physics published in 1545, after defining ""uniform difform"" motion (which is uniformly accelerated motion) – the word velocity was not used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. Let p_0 = p(x_0) give the initial position of the system, and :p_1 = p(x_0 + \Delta x) be the goal position of the system. In kinematics, the acceleration of a particle moving along a curve in space is the time derivative of its velocity. Differentiating with respect to time again obtains the acceleration \mathbf{a} =\left ( \frac{d^2 r}{dt^2} - r\omega^2\right )\mathbf{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{dr}{dt} \right )\mathbf{\hat{e}}_\theta which breaks into the radial acceleration , centripetal acceleration , Coriolis acceleration , and angular acceleration . The initial position, initial velocity, and acceleration vectors need not be collinear, and take an almost identical form. The solution to the equation of motion, with specified initial values, describes the system for all times after . Thus, let C be a space curve traced out by P and s is the arc length of C corresponding to time t. From the instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions; \mathbf{v} = \frac{d \mathbf{r}}{d t} \,, \quad \mathbf{a} = \frac{d \mathbf{v}}{d t} = \frac{d^2 \mathbf{r}}{d t^2} Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. ",2,273,-6.8,362880,2.8108,A
-" Use the function described in Example 4.3, $x_{n+1}=\alpha x_n\left(1-x_n^2\right)$ where $\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Make a plot of $x_n$ versus $n$ for the two starting values and determine the lowest value of $n$ for which the two values diverge by more than $30 \%$.","In mathematics, the reciprocal difference of a finite sequence of numbers (x_0, x_1, ..., x_n) on a function f(x) is defined inductively by the following formulas: :\rho_1(x_1, x_2) = \frac{x_1 - x_2}{f(x_1) - f(x_2)} :\rho_2(x_1, x_2, x_3) = \frac{x_1 - x_3}{\rho_1(x_1, x_2) - \rho_1(x_2, x_3)} + f(x_2) :\rho_n(x_1,x_2,\ldots,x_{n+1})=\frac{x_1-x_{n+1}}{\rho_{n-1}(x_1,x_2,\ldots,x_{n})-\rho_{n-1}(x_2,x_3,\ldots,x_{n+1})}+\rho_{n-2}(x_2,\ldots,x_{n}) ==See also== *Divided differences ==References== * * Category:Finite differences Finally, the sequence :(d_k) = \left\\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots, \frac{1}{k + 1}, \ldots \right\\} converges sublinearly and logarithmically. thumb|alt=Plot showing the different rates of convergence for the sequences ak, bk, ck and dk.|Linear, linear, superlinear (quadratic), and sublinear rates of convergence|600px|center ==Convergence speed for discretization methods== A similar situation exists for discretization methods designed to approximate a function y = f(x), which might be an integral being approximated by numerical quadrature, or the solution of an ordinary differential equation (see example below). More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative argument, which reaches to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap. In particular, iterating a point x0 in [0, 1] gives rise to a sequence x_n: :x_{n+1} = f_\mu(x_n) = \begin{cases} \mu x_n & \mathrm{for}~~ x_n < \frac{1}{2} \\\ \mu (1-x_n) & \mathrm{for}~~ \frac{1}{2} \le x_n \end{cases} where μ is a positive real constant. In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. From top to bottom. 10, 100, 1000, 10000 points. ===Additive recurrence=== For any irrational \alpha, the sequence : s_n = \\{s_0 + n\alpha\\} has discrepancy tending to 1/N. Note that the sequence can be defined recursively by : s_{n+1} = (s_n + \alpha)\bmod 1 \;. Convergence with order * q = 1 is called linear convergence if \mu \in (0, 1), and the sequence is said to converge Q-linearly to L. * q = 2 is called quadratic convergence. * q = 3 is called cubic convergence. * etc. ==== Order estimation ==== A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to q: :q \approx \frac{\log \left|\frac{x_{k+1} - x_k}{x_k - x_{k-1}}\right|}{\log \left|\frac{x_k - x_{k-1}}{x_{k-1} - x_{k-2}}\right|}. ==== Q-convergence definitions ==== In addition to the previously defined Q-linear convergence, a few other Q-convergence definitions exist. The difference of the approximations, 2\,\text{cm}, is in error by 100% of the magnitude of the difference of the true values, 1\,\text{cm}. Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of an equidistributed sequence. thumb|right|Graph of tent map function 300px|thumb|right|Example of iterating the initial condition x0 = 0.4 over the tent map with μ = 1.9. The value of c with lowest discrepancy is the fractional part of the golden ratio: : c = \frac{\sqrt{5}-1}{2} = \varphi - 1 \approx 0.618034. We can solve this equation using the Forward Euler scheme for numerical discretization: : \frac{y_{n+1} - y_n}{h} = -\kappa y_{n}, which generates the sequence : y_{n+1} = y_n(1 - h\kappa). thumb|320px|right|Standard logistic function where L=1,k=1,x_0=0 A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac{L}{1 + e^{-k(x-x_0)}}, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the graph of f approaching L as x approaches +\infty and approaching zero as x approaches -\infty. If the points are chosen as elements of a low-discrepancy sequence, this is the quasi-Monte Carlo method. Category:Numerical analysis Convergence This means running a Monte-Carlo analysis with e.g. s=20 variables and N=1000 points from a low-discrepancy sequence generator may offer only a very minor accuracy improvement. ===Random numbers=== Sequences of quasirandom numbers can be generated from random numbers by imposing a negative correlation on those random numbers. The important parameter here for the convergence speed to y = f(x) is the grid spacing h, inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of x. In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. The relative errors of `x` from 1.000000000000001 and of `y` from 1.000000000000002 are both below 10^{-15} = 0.0000000000001\%, and the floating-point subtraction `y - x` is computed exactly by the Sterbenz lemma. This sequence converges with order 1 according to the convention for discretization methods. ",1.11,2,-2.0,8,30,E
-A gun fires a projectile of mass $10 \mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \mathrm{~m} / \mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \mathrm{~m}$ away? Compare the results with those for the case of no retardation.,"The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. The projectile trajectory is affected by atmospheric conditions, the velocity of the projectile, the difference in altitude between the firer and the target, and other factors. The second solution is the useful one for determining the range of the projectile. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. The laser rangefinder and computer-based FCS make guns highly accurate. ==Superelevation== When firing a missile such as a MANPADS at an aircraft target, superelevation is an additional angle of elevation above the angle sighted on which corrects for the effect of gravity on the missile. ==See also== *Altitude (astronomy) *Pitching moment ==References== * Gunnery Instructions, U.S. Navy (1913), Register No. 4090 * Gunnery And Explosives For Artillery Officers (1911) * Fire Control Fundamentals, NAVPERS 91900 (1953), Part C: The Projectile in Flight - Exterior Ballistics * FM 6-40, Tactics, Techniques, and Procedures for Field Artillery Manual Cannon Gunnery (23 April 1996), Chapter 3 - Ballistics; Marine Corps Warfighting Publication No. 3-1.6.19 * FM 23-91, Mortar Gunnery (1 March 2000), Chapter 2 Fundamentals of Mortar Gunnery * Fundamentals of Naval Weapons Systems: Chapter 19 (Weapons and Systems Engineering Department United States Naval Academy) * Naval Ordnance and Gunnery (Vol.1 - Naval Ordnance) NAVPERS 10797-A (1957) * Naval Ordnance and Gunnery (Vol.2 - Fire Control) NAVPERS 10798-A (1957) * Naval Ordnance and Gunnery Category:Ballistics Category:Angle Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. thumb|right|upright=1.35|Cutaway view of M1128 round The M1128 ""Insensitive Munition High Explosive Base Burn Projectile"" is a 155 mm boosted artillery round designed to achieve a maximum range of 30–40 km. The surface of the projectile also must be considered: a smooth projectile will face less air resistance than a rough-surfaced one, and irregularities on the surface of a projectile may change its trajectory if they create more drag on one side of the projectile than on the other. There are two dimensions in aiming a weapon: * In the horizontal plane (azimuth); and * In the vertical plane (elevation), which is governed by the distance (range) to the target and the energy of the propelling charge. Projectile and propellant gases act on barrel along barrel centerline A. Forces are resisted by shooter contact with gun at grips and stock B. Height difference between barrel centerline and average point of contact is height C. Forces A and B operating over moment arm / height C create torque or moment D, which rotates the firearm's muzzle up as illustrated at E. Muzzle rise, muzzle flip or muzzle climb refers to the tendency of a firearm's or airgun's muzzle (front end of the barrel) to rise up after firing.Recoil management: how you hold makes all the difference, Guns Magazine, Oct 2006 by Dave Anderson It more specifically refers to the seemingly unpredictable ""jump"" of the firearm's muzzle, caused by combined recoil from multiple shots being fired in quick succession. thumb|upright=1.35|Indirect fire trajectories for rockets, howitzers, field guns and mortars Indirect fire is aiming and firing a projectile without relying on a direct line of sight between the gun and its target, as in the case of direct fire. Originally ""zero"", meaning 6400 mils, 360 degrees or their equivalent, was set at whatever the direction the oriented gun was pointed. File:AKM and MP5K.JPEG|An AKM assault rifle asymmetric slant cut muzzle fixture designed to counteract muzzle rise (and muzzle climb) during (automatic) firing. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. ",47,2,17.4,-131.1,1.61,C
-A spacecraft is placed in orbit $200 \mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth. ,"For the Earth at perihelion, the value is: : \sqrt {1.327 \times 10^{20} ~\text{m}^3 \text{s}^{-2} \cdot \left({2 \over 1.471 \times 10^{11} ~\text{m}} - {1 \over 1.496 \times 10^{11} ~\text{m}}\right)} \approx 30,300 ~\text{m}/\text{s} which is slightly faster than Earth's average orbital speed of , as expected from Kepler's 2nd Law. == Planets == The closer an object is to the Sun the faster it needs to move to maintain the orbit. This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac{Mm}{r_1 + r_2} Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result. ==Flight path angle== The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Orbital velocities of the Planets Planet Orbital velocity Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Halley's Comet on an eccentric orbit that reaches beyond Neptune will be moving 54.6 km/s when from the Sun, 41.5 km/s when 1 AU from the Sun (passing Earth's orbit), and roughly 1 km/s at aphelion from the Sun., where r is the distance from the Sun, and a is the major semi-axis. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Objects passing Earth's orbit going faster than 42.1 km/s have achieved escape velocity and will be ejected from the Solar System if not slowed down by a gravitational interaction with a planet. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. When one of the bodies is not of considerably lesser mass see: Gravitational two-body problem So, when one of the masses is almost negligible compared to the other mass, as the case for Earth and Sun, one can approximate the orbit velocity v_o as: :v_o \approx \sqrt{\frac{GM}{r}} or assuming equal to the radius of the orbit :v_o \approx \frac{v_e}{\sqrt{2}} Where is the (greater) mass around which this negligible mass or body is orbiting, and is the escape velocity. This can be used to obtain a more accurate estimate of the average orbital speed: : v_o = \frac{2\pi a}{T}\left[1-\frac{1}{4}e^2-\frac{3}{64}e^4 -\frac{5}{256}e^6 -\frac{175}{16384}e^8 - \cdots \right] The mean orbital speed decreases with eccentricity. ==Instantaneous orbital speed== For the instantaneous orbital speed of a body at any given point in its trajectory, both the mean distance and the instantaneous distance are taken into account: : v = \sqrt {\mu \left({2 \over r} - {1 \over a}\right)} where is the standard gravitational parameter of the orbited body, is the distance at which the speed is to be calculated, and is the length of the semi-major axis of the elliptical orbit. Being closest to the Sun and having the most eccentric orbit, Mercury's orbital speed varies from about 59 km/s at perihelion to 39 km/s at aphelion. Under standard assumptions of the conservation of angular momentum the flight path angle \phi satisfies the equation: :h\, = r\, v\, \cos \phi where: * h\, is the specific relative angular momentum of the orbit, * v\, is the orbital speed of the orbiting body, * r\, is the radial distance of the orbiting body from the central body, * \phi \, is the flight path angle \psi is the angle between the orbital velocity vector and the semi-major axis. u is the local true anomaly. \phi = u + \frac{\pi}{2} - \psi, therefore, :\cos \phi = \sin(\psi - u) = \sin\psi\cos u - \cos\psi\sin u = \frac{1 + e\cos u}{\sqrt{1 + e^2 + 2e\cos u}} :\tan \phi = \frac{e\sin u}{1 + e\cos u} where e is the eccentricity. Velocities of better-known numbered objects that have perihelion close to the Sun Object Velocity at perihelion Velocity at 1 AU (passing Earth's orbit) 322P/SOHO 181 km/s @ 0.0537 AU 37.7 km/s 96P/Machholz 118 km/s @ 0.124 AU 38.5 km/s 3200 Phaethon 109 km/s @ 0.140 AU 32.7 km/s 1566 Icarus 93.1 km/s @ 0.187 AU 30.9 km/s 66391 Moshup 86.5 km/s @ 0.200 AU 19.8 km/s 1P/Halley 54.6 km/s @ 0.586 AU 41.5 km/s ==See also== *Escape velocity *Delta-v budget *Hohmann transfer orbit *Bi-elliptic transfer ==References== Category:Orbits hu:Kozmikus sebességek#Szökési sebességek Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. This is an approximation that only holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero. The maximum (instantaneous) orbital speed occurs at periapsis (perigee, perihelion, etc.), while the minimum speed for objects in closed orbits occurs at apoapsis (apogee, aphelion, etc.). Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). The empty focus (\mathbf{F2} = \left(f_x,f_y\right)) can be found by first determining the Eccentricity vector: :\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu} Where \mathbf{h} is the specific angular momentum of the orbiting body: :\mathbf{h} = \mathbf{r} \times \mathbf{v} Then :\mathbf{F2} = -2a\mathbf{e} ====Using XY Coordinates==== This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2 + y^2 } = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt{r_x^2 + r_y^2} \quad initial distance from F1 (at the origin) :a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad the semi-major axis length :e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad the Eccentricity vector coordinates :e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values fx, fy and a can be applied to the general ellipse equation above. == Orbital parameters == The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. See radial hyperbolic trajectory * If the total energy is zero, (Ek = Ep): the orbit is a parabola with focus at the other body. For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above: :# The central body's position is at the origin and is the primary focus (\mathbf{F1}) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass) :# The central body's mass (m1) is known :# The orbiting body's initial position(\mathbf{r}) and velocity(\mathbf{v}) are known :# The ellipse lies within the XY-plane The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, and tundra orbit. ==Velocity== Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, the orbital speed (v\,) of one body traveling along an elliptic orbit can be computed from the vis-viva equation as: :v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)} where: *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\\! is the length of the semi-major axis. ",-383,3.23,4.09,273,7.654,B
-"Find the value of the integral $\int_S(\nabla \times \mathbf{A}) \cdot d \mathbf{a}$ if the vector $\mathbf{A}=y \mathbf{i}+z \mathbf{j}+x \mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \geq 0$.","Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f_x, f_y and f_z. == Theorems involving surface integrals == Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. == Dependence on parametrization == Let us notice that we defined the surface integral by using a parametrization of the surface S. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses. This formula defines the integral on the left (note the dot and the vector notation for the surface element). Then, the surface integral of f on S is given by :\iint_D \left[ f_{z} ( \mathbf{r} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{r} (s,t)) \frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{r} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt where :{\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t)}\right) is the surface element normal to S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. We find the formula :\begin{align} \iint_S {\mathbf v}\cdot\mathrm d{\mathbf {s}} &= \iint_S \left({\mathbf v}\cdot {\mathbf n}\right)\,\mathrm ds\\\ &{}= \iint_T \left({\mathbf v}(\mathbf{r}(s, t)) \cdot {\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t} \over \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\|}\right) \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\| \mathrm ds\, \mathrm dt\\\ &{}=\iint_T {\mathbf v}(\mathbf{r}(s, t))\cdot \left(\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right) \mathrm ds\, \mathrm dt. \end{align} The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrisation. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s, which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface. Then, the surface integral is given by : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). A natural question is then whether the definition of the surface integral depends on the chosen parametrization. Such a surface is called non-orientable, and on this kind of surface, one cannot talk about integrating vector fields. == See also == * Divergence theorem * Stokes' theorem * Line integral * Volume element * Volume integral * Cartesian coordinate system * Volume and surface area elements in spherical coordinate systems * Volume and surface area elements in cylindrical coordinate systems * Holstein–Herring method ==References== == External links == * Surface Integral — from MathWorld * Surface Integral — Theory and exercises Category:Multivariable calculus Category:Area Category:Surfaces If a vector field \mathbf{F}(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z)) is defined and has continuous first order partial derivatives in a region containing \Sigma, then \iint_\Sigma ( abla \times \mathbf{F}) \cdot \mathrm{d} \mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{\Gamma}. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . That is, A' below is also a vector potential of v; \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ abla_y \times \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}. By analogy with Biot-Savart's law, the following \boldsymbol{A}(\textbf{x}) is also qualify as a vector potential for v. :\boldsymbol{A}(\textbf{x}) =\int_\Omega \frac{\boldsymbol{v}(\boldsymbol{y}) \times (\boldsymbol{x} - \boldsymbol{y})}{4 \pi |\boldsymbol{x} - \boldsymbol{y}|^3} d^3 \boldsymbol{y} Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law. The surface integral can also be expressed in the equivalent form : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt where is the determinant of the first fundamental form of the surface mapping . If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. The circulation of a vector field around a closed curve is the line integral: \Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}. ",1.39,-3.141592,0.123,4.68,1.2,B
-A skier weighing $90 \mathrm{~kg}$ starts from rest down a hill inclined at $17^{\circ}$. He skis $100 \mathrm{~m}$ down the hill and then coasts for $70 \mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. ,"Kinetic (or dynamic) friction occurs when the ski is moving over the snow. One type of friction acting on the skier is the kinetic friction between the skis and snow. The coefficient of kinetic friction, \mu_\mathrm{k}, is less than the coefficient of static friction for both ice and snow. The motion of a skier is determined by the physical principles of the conservation of energy and the frictional forces acting on the body. The second type of frictional force acting on a skier is drag. The kinetic friction can be reduced by applying wax to the bottom of the skis which reduces the coefficient of friction. The necessary speed required to keep the skier upright varies by the weight of the barefooter and can be approximated by the following formula: (W / 10) + 20, where W is the skier's weight in pounds and the result is in miles per hour. The Physics of Skiing. The force required for sliding on snow is the product of the coefficient of kinetic friction and the normal force: F_{k} = \mu_\mathrm{k} F_{n}\,. However, the heat generated by friction can be lost by conduction to a cold ski, thereby diminishing the production of the melt layer. The ability of a ski or other runner to slide over snow depends on both the properties of the snow and the ski to result in an optimum amount of lubrication from melting the snow by friction with the ski—too little and the ski interacts with solid snow crystals, too much and capillary attraction of meltwater retards the ski. ===Friction=== Before a ski can slide, it must overcome the maximum value static friction, F_{max} = \mu_\mathrm{s} F_{n}\,, for the ski/snow contact, where \mu_\mathrm{s} is the coefficient of static friction and F_{n}\, is the normal force of the ski on snow. *Moisture content: The percentage of mass that is liquid water and may create suction friction with the base of the ski as it slides. A skier with skis pointed perpendicular to the fall line, across the hill instead of down it, will accelerate more slowly. The physics of skiing refers to the analysis of the forces acting on a person while skiing. right|thumb|300x300px|The texture of this top layer dependent on the weather history. Kuzmin and Fuss suggest that the most favorable combination of ski base material properties to minimize ski sliding friction on snow include: increased hardness and lowered thermal conductivity of the base material to promote meltwater generation for lubrication, wear resistance in cold snow, and hydrophobicity to minimize capillary suction. The shape and construction material of a ski can also greatly impact the forces acting on a skier.D. A. Lind and S. P. Sanders. Typically, a sliding ski melts a thin and transitory film of lubricating layer of water, caused by the heat of friction between the ski and the snow in its passing. Additionally, the skier can use the same techniques to turn the ski away from the direction of movement, generating skidding forces between the skis and snow which further slow the descent. Wax is adjusted for hardness to minimize sliding friction as a function of snow properties, which include the effects of: *Age: Reflects the metamorphism of snow crystals that are sharp and well- defined, when new, but with aging become broken or truncated with wind action or rounded into ice granules with freeze-thaw, all of which affects a ski's coefficient of friction. The material of Mr. Snow is claimed to have very good sliding capacities, is predictable in all climates and does not harm the ski or sliding surface. In doing so, the snow resists passage of the stemmed ski, creating a force that retards downhill speed and sustains a turn in the opposite direction. Too little melting and sharp edges of snow crystals or too much suction impede the passage of the ski. ",+37,7.25,6.0,0.4772,0.18,E
+","175px|thumb|Ladder diagram In toss juggling, a cascade is the simplest juggling pattern achievable with an odd number of props. The ball is then thrown from above the 4 diagonally downward to the opposite hand. 150px|thumb|Ladder diagram for box: (4,2x)(2x,4) In toss juggling, the box is a juggling pattern for 3 objects, most commonly balls or bean bags. ""Juggling with your arms up in the air above your head & looking up from underneath the pattern.""Darbyshire (1993), p.22. and a virtually infinite number of cascade patterns such as 522, 720, 900, 72222, and so on (see article on Siteswap notation).Beever, Ben (2001), p.15. ==Shannon's theorem== thumb|An illustration of Shannon's juggling theorem for the cascade juggling pattern 175px|thumb|Cascade ladder suggested by Shannon's formula Claude Shannon, builder of the first juggling robot, developed a juggling theorem, relating the time balls spend in the air and in the hands: (F+D)H=(V+D)N, where F = time a ball spends in the air, D = time a ball spends in a hand/time a hand is full, V = time a hand is vacant, N = number of balls, and H = number of hands. ==Number of props== ===Three-ball=== For the three-ball cascade the juggler starts with two balls in one hand and the third ball in the other hand. Before catching this ball the juggler must throw the ball in the receiving hand, in a similar arc, to the first hand. The simplest juggling pattern is the three-ball cascade,Bernstein, Nicholai A. (1996). 150px|thumb|Cascade flash: 3 throws & 3 catches 150px|thumb|Mills mess flash: 6 throws & 6 catches In toss juggling, a flash is either a form of numbers juggling where each ball in a juggling pattern is only thrown and caught once or it is a juggling trick where every prop is simultaneously in the air and both hands are empty.""Three ball flash"", TWJC.co.uk. Two balls are dedicated to a specific hand with vertical throws, and the third ball is thrown horizontally between the two hands. For some tricks the number of throws and catches to complete a juggling cycle for that trick is not simply a multiple of the number of objects being juggled. Juggling, p.23. thumb|400px|right|Racking a game of three-ball with the standard fifteen-ball triangle rack. However, in order to keep the number of props in the juggler's hands to a minimum, it is necessary to begin the pattern by throwing, from alternating hands, all but one prop (in the same hand as the first throw, which started with one more prop than the other) before any catches are made. ==Reverse cascade== thumb|right|upright|An illustration of the three-ball reverse cascade. One ball is thrown from the first hand in an arc to the other hand. One juggles, ""a cascade with two balls while the 'tennis' ball is thrown [back and forth] over the top.""Darbyshire (1993), p.23. The goal is to () the three object balls in as few shots as possible.PoolSharp's ""Three-Ball Rules"". ""The cascade is the simplest three ball juggling pattern."" Juggling, p.26. ""In the cascade...the crossing of the balls between the hands demands that one hand catches at the same rate that the other hand throws . Higher numbers require the balls to be tossed higher into the air in order to allow more time for a complete cycle of throws. Title Description Demonstration The Shuffle In a shuffle throw, the vamp ball begins above and outside the vertical path of one of the box's ""side balls"" and is thrown diagonally downward, caught below the opposite ""side ball"". Charlie Dancey's Encyclopædia of Ball Juggling p98. ",10.4,13.2,"""1.51""",9,+11,B
+"A billiard ball of initial velocity $u_1$ collides with another billiard ball (same mass) initially at rest. The first ball moves off at $\psi=45^{\circ}$. For an elastic collision, what are the velocities of both balls after the collision? ","Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. In an elastic collision these magnitudes do not change. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. (To get the and velocities of the second ball, one needs to swap all the '1' subscripts with '2' subscripts.) In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. In the case of macroscopic bodies, perfectly elastic collisions are an ideal never fully realized, but approximated by the interactions of objects such as billiard balls. Once v_1 is determined, v_2 can be found by symmetry. ====Center of mass frame==== With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. The magnitudes of the velocities of the particles after the collision are: \begin{align} v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\\ v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}. \end{align} ===Two-dimensional collision with two moving objects=== The final x and y velocities components of the first ball can be calculated as: \begin{align} v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2}) \\\\[0.8em] v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}), \end{align} where and are the scalar sizes of the two original speeds of the objects, and are their masses, and are their movement angles, that is, v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1 (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi () is the contact angle. A useful special case of elastic collision is when the two bodies have equal mass, in which case they will simply exchange their momenta. Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. These equations may be solved directly to find v_1,v_2 when u_1,u_2 are known: \begin{array}{ccc} v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\\\[.5em] v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2. \end{array} If both masses are the same, we have a trivial solution: \begin{align} v_{1} &= u_{2} \\\ v_{2} &= u_{1}. \end{align} This simply corresponds to the bodies exchanging their initial velocities to each other. The following illustrate the case of equal mass, m_1=m_2. frame|center|Elastic collision of equal masses frame|center|Elastic collision of masses in a system with a moving frame of reference In the limiting case where m_1 is much larger than m_2, such as a ping-pong paddle hitting a ping-pong ball or an SUV hitting a trash can, the heavier mass hardly changes velocity, while the lighter mass bounces off, reversing its velocity plus approximately twice that of the heavy one. In the case of a large u_{1}, the value of v_{1} is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed. At any instant, half the collisions are, to a varying extent, inelastic collisions (the pair possesses less kinetic energy in their translational motions after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Studies of two-dimensional collisions are conducted for many bodies in the framework of a two-dimensional gas. frame|center|Two-dimensional elastic collision In a center of momentum frame at any time the velocities of the two bodies are in opposite directions, with magnitudes inversely proportional to the masses. It can be shown that v_c is given by: v_c = \frac{p_T c^2}{E} Now the velocities before the collision in the center of momentum frame u_1 ' and u_2 ' are: \begin{align} u_1' &= \frac{u_1 - v_c}{1- \frac{u_1 v_c}{c^2}} \\\ u_2' &= \frac{u_2 - v_c}{1- \frac{u_2 v_c}{c^2}} \\\ v_1' &= -u_1' \\\ v_2' &= -u_2' \\\ v_1 &= \frac{v_1' + v_c}{1+ \frac{v_1' v_c}{c^2}} \\\ v_2 &= \frac{v_2' + v_c}{1+ \frac{v_2' v_c}{c^2}} \end{align} When u_1 \ll c and u_2 \ll c\,, \begin{align} p_T &\approx m_1 u_1 + m_2 u_2 \\\ v_c &\approx \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2} \\\ u_1' &\approx u_1 - v_c \approx \frac {m_1 u_1 + m_2 u_1 - m_1 u_1 - m_2 u_2}{m_1 + m_2} = \frac {m_2 (u_1 - u_2)}{m_1 + m_2} \\\ u_2' &\approx \frac {m_1 (u_2 - u_1)}{m_1 + m_2} \\\ v_1' &\approx \frac {m_2 (u_2 - u_1)}{m_1 + m_2} \\\ v_2' &\approx \frac {m_1 (u_1 - u_2)}{m_1 + m_2} \\\ v_1 &\approx v_1' + v_c \approx \frac {m_2 u_2 - m_2 u_1 + m_1 u_1 + m_2 u_2}{m_1 + m_2} = \frac{u_1 (m_1 - m_2) + 2m_2 u_2}{m_1 + m_2} \\\ v_2 &\approx \frac{u_2 (m_2 - m_1) + 2m_1 u_1}{m_1 + m_2} \end{align} Therefore, the classical calculation holds true when the speed of both colliding bodies is much lower than the speed of light (~300,000 kilometres per second). ===Relativistic derivation using hyperbolic functions=== Using the so-called parameter of velocity s (usually called the rapidity), \frac{v}{c}=\tanh(s), we get \sqrt{1-\frac{v^2}{c^2}}=\operatorname{sech}(s). While the assumptions of separate impacts is not actually valid (the balls remain in close contact with each other during most of the impact), this model will nonetheless reproduce experimental results with good agreement, and is often used to understand more complex phenomena such as the core collapse of supernovae, or gravitational slingshot manoeuvres. ==Sport regulations== Several sports governing bodies regulate the bounciness of a ball through various ways, some direct, some indirect. ",0.139,1590,"""0.7071067812""",25,0.264,C
+"A deuteron (nucleus of deuterium atom consisting of a proton and a neutron) with speed $14.9 \mathrm{~km} / \mathrm{s}$ collides elastically with a neutron at rest. Use the approximation that the deuteron is twice the mass of the neutron. If the deuteron is scattered through a $\mathrm{LAB}$ angle $\psi=10^{\circ}$, what is the final speed of the deuteron?","The charge radius of the deuteron is . This suggests that the state of the deuterium is indeed to a good approximation , state, which occurs with both nucleons spinning in the same direction, but their magnetic moments subtracting because of the neutron's negative moment. While the order of magnitude is reasonable, since the deuterium radius is of order of 1 femtometer (see below) and its electric charge is e, the above model does not suffice for its computation. The Prout is an obsolete unit of energy, whose value is: 1 Prout = 2.9638 \times 10^{-14} J This is equal to one twelfth of the binding energy of the deuteron. The nucleus of a deuterium atom, called a deuteron, contains one proton and one neutron, whereas the far more common protium has no neutrons in the nucleus. In the first case the deuteron is a spin triplet, so that its total spin s is 1. The latter contribution is dominant in the absence of a pure contribution, but cannot be calculated without knowing the exact spatial form of the nucleons wavefunction inside the deuterium. In the second case the deuteron is a spin singlet, so that its total spin s is 0. For hydrogen, this amount is about , or 1.000545, and for deuterium it is even smaller: , or 1.0002725. The deuteron, composed of a proton and a neutron, is one of the simplest nuclear systems. The measured electric quadrupole of the deuterium is . The name deuterium is derived from the Greek , meaning ""second"", to denote the two particles composing the nucleus. This is a nucleus with one proton and one neutron, i.e. a deuterium nucleus. The proton and neutron making up deuterium can be dissociated through neutral current interactions with neutrinos. But the slightly lower experimental number than that which results from simple addition of proton and (negative) neutron moments shows that deuterium is actually a linear combination of mostly , state with a slight admixture of , state. The energies of spectroscopic lines for deuterium and light hydrogen (hydrogen-1) therefore differ by the ratios of these two numbers, which is 1.000272. In this case, the exchange of the two nucleons will multiply the deuterium wavefunction by (−1) from isospin exchange, (+1) from spin exchange and (+1) from parity (location exchange), for a total of (−1) as needed for antisymmetry. This is about 17% of the terrestrial deuterium-to-hydrogen ratio of 156 deuterium atoms per million hydrogen atoms. The deuteron, being an isospin singlet, is antisymmetric under nucleons exchange due to isospin, and therefore must be symmetric under the double exchange of their spin and location. In this theory, the deuterium nucleus with mass two and charge one would contain two protons and one nuclear electron. thumb|upright=0.8|The deuterium–tritium fusion reaction Deuterium–tritium fusion (sometimes abbreviated D+T) is a type of nuclear fusion in which one deuterium nucleus fuses with one tritium nucleus, giving one helium nucleus, one free neutron, and 17.6 MeV of energy. This situation is known as the deuterium bottleneck. ",85,"102,965.21","""14.44""",8.44,0.241,C
+"A mass $m$ moves in one dimension and is subject to a constant force $+F_0$ when $x<0$ and to a constant force $-F_0$ when $x>0$. Describe the motion by constructing a phase diagram. Calculate the period of the motion in terms of $m, F_0$, and the amplitude $A$ (disregard damping) .","The phase is zero at the start of each period; that is :\phi(t_0 + kT) = 0\quad\quad{} for any integer k. The initial conditions are x(0)=0 and \dot{x}(0)=0. *Damped harmonic motion, see animation (right). thumb|Phase portrait of damped oscillator, with increasing damping strength. The equation of motion is \ddot x + 2\gamma \dot x + \omega^2 x = 0. A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables. ==Examples== 400px|thumbnail|Illustration of how a phase portrait would be constructed for the motion of a simple pendulum. A motion diagram represents the motion of an object by displaying its location at various equally spaced times on the same diagram. The phase for each argument value, relative to the start of the cycle, is shown at the bottom, in degrees from 0° to 360° and in radians from 0 to 2π. In physics and mathematics, the phase of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. Then, F is said to be ""at the same phase"" at two argument values t_1 and t_2 (that is, \phi(t_1) = \phi(t_2)) if the difference between them is a whole number of periods. thumb|350px|right|A plot of f(y) (left) and its phase line (right). The term phase can refer to several different things: * It can refer to a specified reference, such as \textstyle \cos(2 \pi f t), in which case we would say the phase of \textstyle x(t) is \textstyle \varphi, and the phase of \textstyle y(t) is \textstyle \varphi - \frac{\pi}{2}. The red dots in the phase portraits are at times t which are an integer multiple of the period T=2\pi/\omega.Based on the examples shown in . ==References== ===Inline=== ===Historical=== * ===Other=== *. *. *. *. ==External links== *Duffing oscillator on Scholarpedia *MathWorld page * Category:Ordinary differential equations Category:Chaotic maps Category:Nonlinear systems Category:Articles containing video clips Motion diagrams. Motion diagrams. To get the phase as an angle between -\pi and +\pi, one uses instead :\phi(t) = 2\pi\left(\left[\\!\\!\left[\frac{t - t_0}{T} + \frac{1}{2}\right]\\!\\!\right] - \frac{1}{2}\right) The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with ""360°"" in place of ""2π"". ===Consequences=== With any of the above definitions, the phase \phi(t) of a periodic signal is periodic too, with the same period T: :\phi(t + T) = \phi(t)\quad\quad{} for all t. *Simple harmonic oscillator where the phase portrait is made up of ellipses centred at the origin, which is a fixed point. In this case the phase difference is increasing, indicating that the test signal is lower in frequency than the reference. ==Formula for phase of an oscillation or a periodic signal== The phase of an oscillation or signal refers to a sinusoidal function such as the following: :\begin{align} x(t) &= A\cdot \cos( 2 \pi f t + \varphi ) \\\ y(t) &= A\cdot \sin( 2 \pi f t + \varphi ) = A\cdot \cos\left( 2 \pi f t + \varphi - \tfrac{\pi}{2}\right) \end{align} where \textstyle A, \textstyle f, and \textstyle \varphi are constant parameters called the amplitude, frequency, and phase of the sinusoid. In mathematics, a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. A Campbell diagram plot represents a system's response spectrum as a function of its oscillation regime. The formula above gives the phase as an angle in radians between 0 and 2\pi. ",-0.5,12,"""1.154700538""",4,2.8108,D
+"A student drops a water-filled balloon from the roof of the tallest building in town trying to hit her roommate on the ground (who is too quick). The first student ducks back but hears the water splash $4.021 \mathrm{~s}$ after dropping the balloon. If the speed of sound is $331 \mathrm{~m} / \mathrm{s}$, find the height of the building, neglecting air resistance.","Integrating the internal air pressure over one hemisphere of the balloon then gives : P_\mathrm{in} - P_\mathrm{out} \equiv P = \frac{f_t}{\pi r^2} = \frac{C}{r_0^2r} \left[1-\left(\frac{r_0}{r}\right)^6 \right] where r0 is the balloon's uninflated radius. When air is first added to the balloon, the pressure rises rapidly to a peak. This is easy to verify by squeezing the air back and forth between two interconnected balloons. ==Non-ideal balloons== At large extensions, the pressure inside a natural rubber balloon once again goes up. The air flow ceases when the two balloons have equal pressure, with one on the left branch of the pressure curve (r < rp) and one on the right branch (r > rp). This result is surprising, since most people assume that the two balloons will have equal sizes after exchanging air. If the total quantity of air in both balloons is less than Np, defined as the number of molecules in both balloons if they both sit at the peak of the pressure curve, then both balloons settle down to the left of the pressure peak with the same radius, r < rp. thumb|upright=1.25|The pub in 2007 The Air Balloon was a public house and road junction at Birdlip, Gloucestershire, England and closed in 2022 as part of road improvements. Pressure curve for an ideal rubber balloon. It becomes smaller, and the larger balloon becomes larger. Two balloons are connected via a hollow tube. For many starting conditions, the smaller balloon then gets smaller and the balloon with the larger diameter inflates even more. Two identical balloons are inflated to different diameters and connected by means of a tube. The lower pressure balloon will expand. thumb|upright=1.5|Observation balloon being shot down by a German biplane Balloon busters were military pilots known for destroying enemy observation balloons. Amer., 62, 1129-35. ,and Mackenzie.Mackenzie, K.V. (1981) Nine-term equation for sound speed in the oceans. So, when the valve is opened, the smaller balloon pushes air into the larger balloon. The simplest way to do this is to imagine that the balloon is made up of a large number of small rubber patches, and to analyze how the size of a patch is affected by the force acting on it. Bio-physical models suggest that this process is effectively similar to the behavior of the balloons in the two-balloon experiment . For a balloon of radius r, a fixed volume of rubber means that r2t is constant, or equivalently : t \propto \frac{1}{r^2} hence : \frac{t}{t_0} = \left(\frac{r_0}{r}\right)^2 and the radial force equation becomes : p = \frac{1}{C_2} \left(\frac{r_0}{r}\right)^4 The equation for the tangential force ft (where Lt \propto r) then becomes : f_t \propto (r/r_0^2)\left[1-(r_0/r)^6\right]. thumb|left|300px|Fig. thumb|Dangerous proximity of a hot air balloon to an overhead line. Figure 2 (above left) shows a typical initial configuration: The smaller balloon has the higher pressure because of the sum of pressure of elastic force Fe which is proportional to pressure (P=Fe/S) plus air pressure in small balloon is greater than air pressure in big balloon. Although balloons were occasionally shot down by small-arms fire, generally it was difficult to shoot down a balloon with solid bullets, particularly at the distances and altitude involved. ",71,6.283185307,"""4.946""",4,-0.0301,A
+"A thin disk of mass $M$ and radius $R$ lies in the $(x, y)$ plane with the $z$-axis passing through the center of the disk. Calculate the gravitational potential $\Phi(z)$.","For this the gravitational force, i.e. the gradient of the potential, must be computed. For a non-pointlike object of continuous mass distribution, each mass element dm can be treated as mass distributed over a small volume, so the volume integral over the extent of object 2 gives: = - Gm_1 \int\limits_V \frac{\rho_2 }{r^2}\mathbf{\hat{r}}\,dx\,dy\,dz |}} with corresponding gravitational potential where ρ = ρ(x, y, z) is the mass density at the volume element and of the direction from the volume element to point mass 1. u is the gravitational potential energy per unit mass. ===The case of a homogeneous sphere=== In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(s); i.e., density depends only on the radial distance :s = \sqrt{x^2 + y^2 + z^2} \,. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. This leaves an ordinary differential equation in terms only of the radius, r, which determines the eigenstates for the particular potential, V(r). == Structure of the eigenfunctions == The eigenstates of the system have the form: \psi(r, \theta, \phi) = R(r)\Theta(\theta)\Phi(\phi) in which the spherical angles \theta and \phi represent the polar and azimuthal angle, respectively. In the literature it is common to introduce some arbitrary ""reference radius"" R close to Earth's radius and to work with the dimensionless coefficients :\begin{align} \tilde{J_n} &= -\frac{J_n}{\mu\ R^n}, & \tilde{C_{n}^m} &= -\frac{C_{n}^m}{\mu\ R^n}, & \tilde{S_{n}^m} &= -\frac{S_{n}^m}{\mu\ R^n} \end{align} and to write the potential as {{NumBlk|:| u = -\frac{\mu }{r} \left(1 + \sum_{n=2}^{N_z} \frac{\tilde{J_n} P^0_n(\sin\theta) }{{(\frac{r}{R})}^n} + \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) (\tilde{C_{n}^m} \cos m\varphi + \tilde{S_{n}^m} \sin m\varphi)}{{(\frac{r}{R})}^n}\right) |}} ===Largest terms=== The dominating term (after the term −μ/r) in () is the ""J2 coefficient"", representing the oblateness of Earth: :u = \frac{J_2\ P^0_2(\sin\theta)}{r^3} = J_2 \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta -1) = J_2 \frac{1}{r^5} \frac{1}{2} (3 z^2 -r^2) Relative the coordinate system thumb|right|Figure 1: The unit vectors. If this shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), the integrals () and () could be evaluated with numerical methods to find a more accurate model for Earth's gravitational field. From the definition of φ as a spherical coordinate it is clear that Φ(φ) must be periodic with the period 2π and one must therefore have that and for some integer m as the family of solutions to () then are With the variable substitution :x=\sin \theta equation () takes the form From () follows that in order to have a solution \phi with :R(r) = \frac{1}{r^{n+1}} one must have that :\lambda = n (n + 1) If Pn(x) is a solution to the differential equation one therefore has that the potential corresponding to m = 0 :\phi = \frac{1}{r^{n+1}}\ P_n(\sin\theta) which is rotationally symmetric around the z-axis is a harmonic function If P_{n}^{m}(x) is a solution to the differential equation {dx}\right)\ +\ \left(n(n + 1) - \frac{m^2}{1 - x^2} \right)\ P_{n}^{m}\ =\ 0 |}} with m ≥ 1 one has the potential \ P_{n}^{m}(\sin\theta)\ (a\ \cos m\varphi\ +\ b\ \sin m\varphi) |}} where a and b are arbitrary constants is a harmonic function that depends on φ and therefore is not rotationally symmetric around the z-axis The differential equation () is the Legendre differential equation for which the Legendre polynomials defined are the solutions. The differential equation which characterizes the function R(r) is called the radial equation. == Derivation of the radial equation == The kinetic energy operator in spherical polar coordinates is:\frac{\hat{p}^2}{2m_0} = -\frac{\hbar^2}{2m_0} abla^2 = \- \frac{\hbar^2}{2m_0\,r^2} \left[ \frac{\partial}{\partial r} \left(r^2 \frac{\partial}{\partial r}\right) - \hat{L}^2 \right].The spherical harmonics satisfy \hat{L}^2 Y_{\ell m}(\theta,\phi)\equiv \left\\{ -\frac{1}{\sin^2\theta} \left[ \sin\theta \frac{\partial}{\partial\theta} \left(\sin\theta \frac{\partial}{\partial\theta}\right) +\frac{\partial^2}{\partial \phi^2}\right]\right\\} Y_{\ell m}(\theta,\phi) = \ell(\ell+1)Y_{\ell m}(\theta,\phi). There should be a theta, not lambda \hat{\varphi}\ ,\ \hat{\theta}\ ,\ \hat{r} illustrated in figure 1 the components of the force caused by the ""J2 term"" are In the rectangular coordinate system (x, y, z) with unit vectors (x̂ ŷ ẑ) the force components are: The components of the force corresponding to the ""J3 term"" :u = \frac{J_3 P^0_3(\sin\theta) }{r^4} = J_3 \frac{1}{r^4} \frac{1}{2} \sin\theta \left(5\sin^2\theta - 3\right) = J_3 \frac{1}{r^7} \frac{1}{2} z \left(5 z^2 - 3 r^2\right) are and The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly. This is the shell theorem saying that in this case: with corresponding potential where M = ∫Vρ(s)dxdydz is the total mass of the sphere. ==Spherical harmonics representation== In reality, Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. By straightforward calculations one gets that for any function f Introducing the expression () in () one gets that As the term :\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) only depends on the variable r and the sum :\frac{1}{\Theta\cos\theta}\frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right) + \frac{1}{\Phi\cos^2\theta}\frac{d^2\Phi}{d\varphi^2} only depends on the variables θ and φ. thumb|right|300px|A contour plot of the effective potential of a two-body system due to gravity and inertia at one point in time. For the same reason, the solution will be of this kind inside the sphere:R(r) = A j_\ell\left(\sqrt{\frac{2 m_0 (E-V_0)}{\hbar^2}}r\right), \qquad r < r_0.Note that for bound states, V_0 < E < 0. Several other definitions are in use, and so care must be taken in comparing different sources.Wolfram Mathworld, spherical coordinates == Cylindrical coordinate system == === Vector fields === Vectors are defined in cylindrical coordinates by (ρ, φ, z), where * ρ is the length of the vector projected onto the xy- plane, * φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), * z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by: \begin{bmatrix} \rho \\\ \phi \\\ z \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2} \\\ \operatorname{arctan}(y / x) \\\ z \end{bmatrix},\ \ \ 0 \le \phi < 2\pi, thumb or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} \rho\cos\phi \\\ \rho\sin\phi \\\ z \end{bmatrix}. Let the points have position vectors \textbf{r} and \textbf{r}' , then the Laplace expansion is : \frac{1}{|\mathbf{r}-\mathbf{r}'|} = \sum_{\ell=0}^\infty \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{\ell} (-1)^m \frac{r_^\ell }{r_{\scriptscriptstyle>}^{\ell+1} } Y^{-m}_\ell(\theta, \varphi) Y^m_\ell(\theta', \varphi'). The first spherical harmonics with n = 0, 1, 2, 3 are presented in the table below. : n Spherical harmonics 0 \frac{1}{r} 1 \frac{1}{r^2} P^0_1(\sin\theta) = \frac{1}{r^2} \sin\theta \frac{1}{r^2} P^1_1(\sin\theta) \cos\varphi= \frac{1}{r^2} \cos\theta \cos\varphi \frac{1}{r^2} P^1_1(\sin\theta) \sin\varphi= \frac{1}{r^2} \cos\theta \sin\varphi 2 \frac{1}{r^3} P^0_2(\sin\theta) = \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta - 1) \frac{1}{r^3} P^1_2(\sin\theta) \cos\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta\ \cos\varphi \frac{1}{r^3} P^1_2(\sin\theta) \sin\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta \sin\varphi \frac{1}{r^3} P^2_2(\sin\theta) \cos2\varphi = \frac{1}{r^3} 3 \cos^2 \theta\ \cos2\varphi \frac{1}{r^3} P^2_2(\sin\theta) \sin2\varphi = \frac{1}{r^3} 3 \cos^2 \theta \sin 2\varphi 3 \frac{1}{r^4} P^0_3(\sin\theta) = \frac{1}{r^4} \frac{1}{2} \sin\theta\ (5\sin^2\theta -3) \frac{1}{r^4} P^1_3(\sin\theta) \cos\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \cos\varphi \frac{1}{r^4} P^1_3(\sin\theta) \sin\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \sin\varphi \frac{1}{r^4} P^2_3(\sin\theta) \cos 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \cos 2\varphi \frac{1}{r^4} P^2_3(\sin\theta) \sin 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \sin 2\varphi \frac{1}{r^4} P^3_3(\sin\theta) \cos 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \cos 3\varphi \frac{1}{r^4} P^3_3(\sin\theta) \sin 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \sin 3\varphi ===Application=== The model for Earth's gravitational potential is a sum \+ \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) \left(C_n^m \cos m\varphi + S_n^m \sin m\varphi\right)}{r^{n+1}} |}} where \mu = GM and the coordinates () are relative to the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with z-axis in the direction of the polar axis. In physics, the Laplace expansion of potentials that are directly proportional to the inverse of the distance ( 1/r ), such as Newton's gravitational potential or Coulomb's electrostatic potential, expresses them in terms of the spherical Legendre polynomials. In quantum mechanics, a particle in a spherically symmetric potential is a system with a potential that depends only on the distance between the particle and a center. Inclination to the invariable plane for the giant planets Year Jupiter Saturn Uranus Neptune 2009 0.32° 0.93° 1.02° 0.72° 142400 (produced with Solex 10) 0.48° 0.79° 1.04° 0.55° 168000 (produced with Solex 10) 0.23° 1.01° 1.12° 0.55° The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter (center of mass) perpendicular to its angular momentum vector. From () then follows that :\frac{1}{\Theta}\ \cos\theta\ \frac{d}{d\theta}\left(\cos\theta \frac{d\Theta}{d\theta}\right)\ + \lambda\ \cos^2\theta\ +\ \frac{1}{\Phi}\frac{d^2\Phi}{d\varphi^2}\ =\ 0 The first two terms only depend on the variable \theta and the third only on the variable \varphi. They take the forms: P^m_n(\sin \theta) \cos m\varphi \,,& 0 &\le m \le n \,,& n &= 0, 1, 2, \dots \\\ h(x, y, z) &= \frac{1}{r^{n+1}} P^m_n(\sin \theta) \sin m\varphi \,,& 1 &\le m \le n \,,& n &= 1, 2, \dots \end{align}|}} where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference: also P0n are the Legendre polynomials and Pmn for are the associated Legendre functions. ",-994.3,1.5,"""-2.0""",804.62,1.91,C
+"A steel ball of velocity $5 \mathrm{~m} / \mathrm{s}$ strikes a smooth, heavy steel plate at an angle of $30^{\circ}$ from the normal. If the coefficient of restitution is 0.8 , at what velocity does the steel ball bounce off the plate?","The motion of a ball is generally described by projectile motion (which can be affected by gravity, drag, the Magnus effect, and buoyancy), while its impact is usually characterized through the coefficient of restitution (which can be affected by the nature of the ball, the nature of the impacting surface, the impact velocity, rotation, and local conditions such as temperature and pressure). A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. This implies that the ball would bounce to 9 times its original height.Since conservation of mechanical energy implies \textstyle \frac{1}{2}mv_\text{f}^2 = mgH_\text{f}, then \textstyle H_\text{f} is proportional to v^2_\text{f}. Composites Science and Technology 64:35-54. . is as follows: V_b=\frac{\pi\,\Gamma\,\sqrt{\rho_t\,\sigma_e}\,D^2\,T}{4\,m} \left [1+\sqrt{1+\frac{8\,m}{\pi\,\Gamma^2\,\rho_t\,D^2\,T}}\, \right ] where *V_b\, is the ballistic limit *\Gamma\, is a projectile constant determined experimentally *\rho_t\, is the density of the laminate *\sigma_e\, is the static linear elastic compression limit *D\, is the diameter of the projectile *T\, is the thickness of the laminate *m\, is the mass of the projectile Additionally, the ballistic limit for small-caliber into homogeneous armor by TM5-855-1 is: V_1= 19.72 \left [ \frac{7800 d^3 \left [ \left ( \frac{e_h}{d} \right) \sec \theta \right ]^{1.6}}{W_T} \right ]^{0.5} where *V_1 is the ballistic limit velocity in fps *d is the caliber of the projectile, in inches *e_h is the thickness of the homogeneous armor (valid from BHN 360 - 440) in inches *\theta is the angle of obliquity *W_T is the weight of the projectile, in lbs == References == Category:Ballistics A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by \left(\frac{R_1}{R_2}\right)^{{3}/{8}} Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material. * e = coefficient of restitution * Sy = dynamic yield strength (dynamic ""elastic limit"") * E′ = effective elastic modulus * ρ = density * v = velocity at impact * μ = Poisson's ratio e = 3.1 \left(\frac{S_\text{y}}{1}\right)^{5/8} \left(\frac{1}{E'}\right)^{1/2} \left(\frac{1}{v}\right)^{1/4} \left(\frac{1}{\rho}\right)^{1/8} E' = \frac{E}{1-\mu^2} This equation overestimates the actual COR. The International Table Tennis Federation specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.887 to 0.923. The ball's angular velocity will be reduced after impact, but its horizontal velocity will be increased. The ball's angular velocity will be increased after impact, but its horizontal velocity will be decreased. To analyze the vertical and horizontal components of the motion, the COR is sometimes split up into a normal COR (ey), and tangential COR (ex), defined as :e_\text{y} = -\frac{v_\text{yf} - u_\text{yf}}{v_\text{yi} - u_\text{yi}}, :e_\text{x} = -\frac{(v_\text{xf}-r\omega_\text{f})-(u_\text{xf}-R\Omega_\text{f})}{(v_\text{xi}-r\omega_\text{i})-(u_\text{xi}-R\Omega_\text{i})}, where r and ω denote the radius and angular velocity of the ball, while R and Ω denote the radius and angular velocity the impacting surface (such as a baseball bat). It gives the following theoretical coefficient of restitution for solid spheres dropped 1 meter (v = 4.5 m/s). Depending on the ball's alignment at impact, the normal force can act ahead or behind the centre of mass of the ball, and friction from the ground will depend on the alignment of the ball, as well as its rotation, spin, and impact velocity. The ball's angular velocity will be reduced after impact, as will its horizontal velocity, and the ball is propelled upwards, possibly even exceeding its original height. A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). === Predicting from material properties === The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. More specifically, if the ball is bounced at an angle θ with the ground, the motion in the x- and y-axes (representing horizontal and vertical motion, respectively) is described by x-axis y-axis :\begin{align} a_\text{x} & = 0, \\\ v_\text{x} & = v_0 \cos \left(\theta \right), \\\ x & = x_0 + v_0 \cos \left( \theta \right) t, \end{align} :\begin{align} a_\text{y} & = -g, \\\ v_\text{y} & = v_0 \sin \left(\theta \right) -gt, \\\ y & = y_0 + v_0 \sin \left( \theta \right) t -\frac{1}{2}gt^2. \end{align} The equations imply that the maximum height (H) and range (R) and time of flight (T) of a ball bouncing on a flat surface are given by :\begin{align} H & = \frac{v_0^2}{2g}\sin^2\left(\theta\right), \\\ R &= \frac{v_0^2}{g}\sin\left(2\theta\right),~\text{and} \\\ T &= \frac{2v_0}{g} \sin \left(\theta \right). \end{align} Further refinements to the motion of the ball can be made by taking into account air resistance (and related effects such as drag and wind), the Magnus effect, and buoyancy. For an object bouncing off a stationary target, C_R is defined as the ratio of the object's speed after the impact to that prior to impact: C_R = \frac{v}{u}, where *v is the speed of the object after impact *u is the speed of the object before impact In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to: C_R = \sqrt{\frac{h}{H}}, where *h is the bounce height *H is the drop height The coefficient of restitution can be thought of as a measure of the extent to which mechanical energy is conserved when an object bounces off a surface. In particular rω is the tangential velocity of the ball's surface, while RΩ is the tangential velocity of the impacting surface. According to one article (addressing COR in tennis racquets), ""[f]or the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution."" If the ball moves horizontally at impact, friction will have a 'translational' component in the direction opposite to the ball's motion. This roughly corresponds to a COR of 0.727 to 0.806.Calculated using \textstyle e = \sqrt{\frac{H_\text{f}}{H_\text{i}}} and (if applicable) the diameter of the ball. For a straight drop on the ground with no rotation, with only the force of gravity acting on the ball, the COR can be related to several other quantities by: :e = \left|\frac{v_\text{f}}{v_\text{i}}\right| = \sqrt{\frac{K_\text{f}}{K_\text{i}}} = \sqrt{\frac{U_\text{f}}{U_\text{i}}} = \sqrt{\frac{H_\text{f}}{H_\text{i}}} = \frac{T_\text{f}}{T_\text{i}} =\sqrt{\frac{gT^2_\text{f}}{8H_\text{i}}}. In general, the ball will deform more at higher impact velocities and will accordingly lose more of its energy, decreasing its COR. ===Spin and angle of impact=== Upon impacting the ground, some translational kinetic energy can be converted to rotational kinetic energy and vice versa depending on the ball's impact angle and angular velocity. This energy loss is usually characterized (indirectly) through the coefficient of restitution (or COR, denoted e):Here, v and u are not just the magnitude of velocities, but include also their direction (sign). :e = -\frac{v_\text{f} - u_\text{f}}{v_\text{i} - u_\text{i}}, where vf and vi are the final and initial velocities of the ball, and uf and ui are the final and initial velocities impacting surface, respectively. ",96.4365076099,0.36,"""4.3""",41.40,0.6321205588,C
+"Include air resistance proportional to the square of the ball's speed in the previous problem. Let the drag coefficient be $c_W=0.5$, the softball radius be $5 \mathrm{~cm}$ and the mass be $200 \mathrm{~g}$. Find the initial speed of the softball needed now to clear the fence. ","The study concludes that, assuming average observed values for lift coefficient, a 65mph rise ball must have at least a three degree launch angle in order to pass the strike zone at a point higher than the release point (the bottom of the strike zone and release point are the same at 1.5 feet).Clark, J.M., Greer, M.L. & Semon, M.D. Modeling pitch trajectories in fastpitch softball. The most appropriate are the Reynolds number, given by \mathrm{Re} = \frac{u\sqrt{A}}{ u} and the drag coefficient, given by c_{\rm d} = \frac{F_{\rm d}}{\frac12 \rho A u^2}. Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. Another study utilizes a theoretical physics approach to modelling the trajectories of various softball pitches, including the rise ball. Because the only unknown in the above equation is the drag force Fd, it is possible to express it as \begin{align} \frac{F_{\rm d}}{\frac12 \rho A u^2} &= f_c\left(\frac{u \sqrt{A}}{ u} \right) \\\ F_{\rm d} &= \tfrac12 \rho A u^2 f_c(\mathrm{Re}) \\\ c_{\rm d} &= f_c(\mathrm{Re}) \end{align} Thus the force is simply ½ ρ A u2 times some (as-yet-unknown) function fc of the Reynolds number Re – a considerably simpler system than the original five-argument function given above. The general rules for the softball throw parallel those of the javelin throw when conducted in a formal environment, but the implement being thrown is a standard softball, which resembles the size of a standard shot but is considerably lighter. One image appears to show that the ball follows an increasingly upward trajectory; however, this image was taken of a particular type of training ball known as a JUGS LITE-FLITE ball, which has “one third of the mass (59.5g) of a regulation softball (181.71g)”. A similar image shown of a regulation softball pitched at the same speed (70mph) seems to show a decreasing upward trajectory, although the author describes the outcome nebulously as “the rise is not apparent”. Alongside the Olympic discipline of fastpitch softball, which is the most popular variation of softball, there is also modified fastpitch softball and slow-pitch softball. === Baseball5 === thumb|A B5 batter hitting the ball into play. In softball, a pitch is the act of throwing a ball underhand by using a windmill motion. The authors consider the effects of gravity, drag and the Magnus Effect using Newton’s laws of motion to calculate the position of the ball at different points in time, allowing them to model the trajectory of the ball in 3 dimensions. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. From Stokes' solution, the drag force acting on the sphere of diameter d can be obtained as where the Reynolds number, Re = \frac{\rho d}{\mu} V. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). Air is 1.293 kg/m3 at 0°C and 1 atmosphere *u is the flow velocity relative to the object, *A is the reference area, and *c_{\rm d} is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. As the speed of an object increases, so does the drag force acting on it, which also depends on the substance it is passing through (for example air or water). That this is so becomes apparent when the drag force Fd is expressed as part of a function of the other variables in the problem: f_a(F_{\rm d}, u, A, \rho, u) = 0. The softball throw is a track and field event used as a substitute for more technical throwing events in competitions involving Youth, Paralympic, Special Olympics and Senior competitors. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. ",35.2, 11.58,"""4.68""",322,2.89,A
+"A child slides a block of mass $2 \mathrm{~kg}$ along a slick kitchen floor. If the initial speed is 4 $\mathrm{m} / \mathrm{s}$ and the block hits a spring with spring constant $6 \mathrm{~N} / \mathrm{m}$, what is the maximum compression of the spring? ","If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. The effective mass of the spring can be determined by finding its kinetic energy. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. The force an ideal spring would exert is exactly proportional to its extension or compression. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). ; Compression spring: Designed to operate with a compression load, so the spring gets shorter as the load is applied to it. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. The manufacture normally specifies the spring rate. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. These springs are compression springs and can differ greatly in strength and in size depending on application. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. Springs can store energy when compressed. In a real spring–mass system, the spring has a non-negligible mass m. ",8.99,-1.5,"""2.19""",2.3,-114.40,D
+"If the field vector is independent of the radial distance within a sphere, find the function describing the density $\rho=\rho(r)$ of the sphere.","In the plane \textstyle \textbf{R}^2, this expression simplifies to : H_s(r)=1-e^{-\lambda \pi r^2}. ==Relationship to other functions== ===Nearest neighbour function=== In general, the spherical contact distribution function and the corresponding nearest neighbour function are not equal. For three dimensions, this normalization is the number density of the system ( \rho ) multiplied by the volume of the spherical shell, which symbolically can be expressed as \rho \, 4\pi r^2 dr. Taking particle 0 as fixed at the origin of the coordinates, \textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r}) is the average number of particles (among the remaining N-1) to be found in the volume \textstyle d^3r around the position \textstyle \mathbf{r}. thumb|250px|right|calculation of g(r) In statistical mechanics, the radial distribution function, (or pair correlation function) g(r) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. More specifically, a spherical contact distribution function is defined as probability distribution of the radius of a sphere when it first encounters or makes contact with a point in a point process. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. In probability and statistics, a spherical contact distribution function, first contact distribution function,D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein-Zernike equation, using approximative closure relations like the Percus-Yevick approximation or the Hypernetted Chain Theory. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein-Zernike equation or structure-optimized potential refinement. ==Definition== Consider a system of N particles in a volume V (for an average number density \rho =N/V) and at a temperature T (let us also define \textstyle \beta = \frac{1}{kT}). In other words, spherical contact distribution function is the probability there are no points from the point process located in a hyper- sphere of radius r. ===Contact distribution function=== The spherical contact distribution function can be generalized for sets other than the (hyper-)sphere in \textstyle \textbf{R}^{ d}. That is, ƒ is radial if and only if :f\circ \rho = f\, for all , the special orthogonal group in n dimensions. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. The spherical contact function is also referred to as the contact distribution function, but some authors define the contact distribution function in relation to a more general set, and not simply a sphere as in the case of the spherical contact distribution function. In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. For a non-interacting gas, it is independent of the position \textstyle \mathbf{r}_1 and equal to the overall number density, \rho, of the system. In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals to charged colloids. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. For distances r such that u(r) is significant, the mean local density will differ from the mean density \rho, depending on the sign of u(r) (higher for negative interaction energy and lower for positive u(r)). For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting, \begin{align} \rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n) &=\frac{1}{(N-n)!}\left(\prod_{i=n+1}^N\int\mathrm{d}^3\mathbf{r}_i\right)\sum_{\pi\in S_N} P^{(N)}(\mathbf{r}_{\pi (1)},\ldots,\mathbf{r}_{\pi (N)}) \\\ \end{align} This quantity is called the n-particle density function. To wit, :\phi(x) = \frac{1}{\omega_{n-1}}\int_{S^{n-1}} f(rx')\,dx' where ωn−1 is the surface area of the (n−1)-sphere Sn−1, and , . Spherical contact distribution functions are used in the study of point processesD. The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. ",0.1591549431,30,"""3.29527""",-273,15,A
+"An Earth satellite has a perigee of $300 \mathrm{~km}$ and an apogee of $3,500 \mathrm{~km}$ above Earth's surface. How far is the satellite above Earth when it has rotated $90^{\circ}$ around Earth from perigee?","Objects orbiting the Earth must be within this radius, otherwise, they may become unbound by the gravitational perturbation of the Sun. Orbital characteristics epoch J2000.0 aphelion 1.0167 AU perihelion 0.98329 AU semimajor axis 1.0000010178 AU eccentricity 0.0167086 inclination 7.155° to Sun's equator 1.578690° to invariable plane longitude of the ascending node 174.9° longitude of perihelion 102.9° argument of periapsis 288.1° period daysThe figure appears in multiple references, and is derived from the VSOP87 elements from section 5.8.3, p. 675 of the following: average orbital speed speed at aphelion speed at perihelion The following diagram shows the relation between the line of the solstice and the line of apsides of Earth's elliptical orbit. Theta Persei (Theta Per, θ Persei, θ Per) is a star system 37 light years away from Earth, in the constellation Perseus. A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. thumb|upright=1.5|Earth at seasonal points in its orbit (not to scale) thumb|Earth orbit (yellow) compared to a circle (gray) Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi) in a counterclockwise direction as viewed from above the Northern Hemisphere. A full orbit has 360°. Beta angle can be controlled to keep a satellite as cool as possible (for instruments that require low temperatures, such as infrared cameras) by keeping the beta angle as close to zero as possible, or, conversely, to keep a satellite in sunlight as much as possible (for conversion of sunlight by its solar panels, for solar stability of sensors, or to study the Sun) by maintaining a beta angle as close to +90 or -90 as possible. ==Determination and application of beta angles== The value of a solar beta angle for a satellite in Earth orbit can be found using the equation \beta=\sin^{-1}[\cos(\Gamma)\sin(\Omega)\sin(i)-\sin(\Gamma)\cos(\epsilon)\cos(\Omega)\sin(i)+\sin(\Gamma)\sin(\epsilon)\cos(i)] where \Gamma is the ecliptic true solar longitude, \Omega is the right ascension of ascending node (RAAN), i is the orbit's inclination, and \epsilon is the obliquity of the ecliptic (approximately 23.45 degrees for Earth at present). thumb|300px|Beta angle (\boldsymbol{\beta}) In orbital spaceflight, the beta angle (\boldsymbol{\beta}) is the angle between a satellite's orbital plane around Earth and the geocentric position of the sun. 9 Persei is a single variable star in the northern constellation Perseus, located around 4,300 light years away from the Sun. At a LEO of 280 kilometers, the object is in sunlight through 59% of its orbit (approximately 53 minutes in Sunlight, and 37 minutes in shadow.) The Satellite () is a small rock peak rising to 1,100 m, protruding slightly above the ice sheet 3 nautical miles (6 km) southwest of Pearce Peak and 8 nautical miles (15 km) east of Baillieu Peak. The changing Earth-Sun distance results in an increase of about 7% in total solar energy reaching the Earth at perihelion relative to aphelion. This angle is called the orbit's inclination. The beta angle varies between +90° and −90°, and the direction in which the satellite orbits its primary body determines whether the beta angle sign is positive or negative. It is radiating over 12,000 times the luminosity of the Sun from its swollen photosphere at an effective temperature of 9,840 K. 9 Persei has one visual companion, designated component B, at an angular separation of and magnitude 12.0. One complete orbit takes days (1 sidereal year), during which time Earth has traveled 940 million km (584 million mi).Jean Meeus, Astronomical Algorithms 2nd ed, (Richmond, VA: Willmann-Bell, 1998) 238. Since this value is close to zero, the center of the orbit is relatively close to the center of the Sun (relative to the size of the orbit). That same satellite also will have a beta angle with respect to the Sun, and in fact it has a beta angle for any celestial object one might wish to calculate one for: any satellite orbiting a body (i.e. the Earth) will be in that body's shadow with respect to a given celestial object (like a star) some of the time, and in its line-of-sight the rest of the time. A satellite in such an orbit spends at least 59% of its orbital period in sunlight. ==Light and shadow== The degree of orbital shadowing an object in LEO experiences is determined by that object's beta angle. The above discussion defines the beta angle of satellites orbiting the Earth, but a beta angle can be calculated for any orbiting three body system: the same definition can be applied to give the beta angle of other objects. The Hill sphere (gravitational sphere of influence) of the Earth is about 1,500,000 kilometers (0.01 AU) in radius, or approximately four times the average distance to the Moon.For the Earth, the Hill radius is :R_H = a \left(\frac{m}{3M}\right)^{1/3}, where m is the mass of the Earth, a is an astronomical unit, and M is the mass of the Sun. The orbital ellipse goes through each of the six Earth images, which are sequentially the perihelion (periapsis—nearest point to the Sun) on anywhere from January 2 to January 5, the point of March equinox on March 19, 20, or 21, the point of June solstice on June 20, 21, or 22, the aphelion (apoapsis—the farthest point from the Sun) on anywhere from July 3 to July 5, the September equinox on September 22, 23, or 24, and the December solstice on December 21, 22, or 23. So the radius in AU is about \left(\frac{1}{3 \cdot 332\,946}\right)^{1/3} \approx 0.01. ",2380,28,"""0.332""",9,1590,E
+Two masses $m_1=100 \mathrm{~g}$ and $m_2=200 \mathrm{~g}$ slide freely in a horizontal frictionless track and are connected by a spring whose force constant is $k=0.5 \mathrm{~N} / \mathrm{m}$. Find the frequency of oscillatory motion for this system.,"As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. Many clocks keep time by mechanical resonance in a balance wheel, pendulum, or quartz crystal. ==Description== The natural frequency of a simple mechanical system consisting of a weight suspended by a spring is: :f = {1\over 2 \pi} \sqrt {k\over m} where m is the mass and k is the spring constant. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. The effective mass of the spring can be determined by finding its kinetic energy. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. thumb|The second pendulum, with a period of two seconds so each swing takes one second A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.Seconds pendulum == Pendulum == A pendulum is a weight suspended from a pivot so that it can swing freely. The resonance frequency of a pendulum, the only frequency at which it will vibrate, is given approximately, for small displacements, by the equation:Mechanical resonance :f = {1\over 2 \pi} \sqrt {g\over L} where g is the acceleration due to gravity (about 9.8 m/s2 near the surface of Earth), and L is the length from the pivot point to the center of mass. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The frequency response of this oscillator describes the amplitude z of steady state response of the equation (i.e. x(t)) at a given frequency of excitation \omega. Note that, in this approximation, the frequency does not depend on mass. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. To relate v to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring: :T=\frac{1}{2}m(\dot x^2+(l_0+x)^2\dot \theta^2) So the Lagrangian becomes: :L = T -V_k - V_g :L[x,\dot x,\theta, \dot \theta] = \frac{1}{2}m(\dot x^2+(l_0+x)^2\dot \theta^2) -\frac{1}{2}kx^2 + gm(l_0+x)\cos \theta ===Equations of motion=== With two degrees of freedom, for x and \theta, the equations of motion can be found using two Euler-Lagrange equations: :{\partial L\over\partial x}-{\operatorname d \over \operatorname dt} {\partial L\over\partial \dot x}=0 :{\partial L\over\partial \theta}-{\operatorname d \over \operatorname dt} {\partial L\over\partial \dot \theta}=0 For x: :m(l_0+x)\dot \theta^2 -kx + gm\cos \theta-m \ddot x=0 \ddot x isolated: :\ddot x =(l_0+x)\dot \theta^2 -\frac{k}{m}x + g\cos \theta And for \theta: :-gm(l_0+x)\sin \theta - m(l_0+x)^2\ddot \theta- 2m(l_0+x)\dot x \dot \theta=0 \ddot \theta isolated: :\ddot \theta=-\frac{g}{l_0+x}\sin \theta-\frac{2\dot x}{l_0+x}\dot \theta The elastic pendulum is now described with two coupled ordinary differential equations. In a real spring–mass system, the spring has a non-negligible mass m. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. If the system is excited (pushed) with a period between pushes equal to the inverse of the pendulum's natural frequency, the swing will swing higher and higher, but if excited at a different frequency, it will be difficult to move. For a hardening spring oscillator (\alpha>0 and large enough positive \beta>\beta_{c+}>0) the frequency response overhangs to the high- frequency side, and to the low-frequency side for the softening spring oscillator (\alpha>0 and \beta<\beta_{c-}<0). The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order in this system. ==See also== * Double pendulum * Duffing oscillator * Pendulum (mathematics) * Spring-mass system == References == == Further reading == * * ==External links== * Holovatsky V., Holovatska Y. (2019) ""Oscillations of an elastic pendulum"" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019. ",556,1.8763,"""0.444444444444444""",1.81,2.74,E
+A particle moves with $v=$ const. along the curve $r=k(1+\cos \theta)$ (a cardioid). Find $\ddot{\mathbf{r}} \cdot \mathbf{e}_r=\mathbf{a} \cdot \mathbf{e}_r$.,"Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously as opposed to the eccentricity and argument of periapsis parameters for which eccentricity zero (circular orbit) corresponds to a singularity. ==Calculation== The eccentricity vector \mathbf{e} \, is: : \mathbf{e} = {\mathbf{v}\times\mathbf{h}\over{\mu}} - {\mathbf{r}\over{\left|\mathbf{r}\right|}} = \left ( {\mathbf{\left |v \right |}^2 \over {\mu} }- {1 \over{\left|\mathbf{r}\right|}} \right ) \mathbf{r} - {\mathbf{r} \cdot \mathbf{v} \over{\mu}} \mathbf{v} which follows immediately from the vector identity: : \mathbf{v}\times \left ( \mathbf{r}\times \mathbf{v} \right ) = \left ( \mathbf{v} \cdot \mathbf{v} \right ) \mathbf{r} - \left ( \mathbf{r} \cdot \mathbf{v} \right ) \mathbf{v} where: *\mathbf{r}\,\\! is position vector *\mathbf{v}\,\\! is velocity vector *\mathbf{h}\,\\! is specific angular momentum vector (equal to \mathbf{r}\times\mathbf{v}) *\mu\,\\! is standard gravitational parameter ==See also== *Kepler orbit *Orbit *Eccentricity *Laplace–Runge–Lenz vector ==References== Category:Orbits Category:Vectors (mathematics and physics) Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Then, A is a vector potential for , that is, abla \times \mathbf{A} =\mathbf{v}. By analogy with Biot-Savart's law, the following \boldsymbol{A}(\textbf{x}) is also qualify as a vector potential for v. :\boldsymbol{A}(\textbf{x}) =\int_\Omega \frac{\boldsymbol{v}(\boldsymbol{y}) \times (\boldsymbol{x} - \boldsymbol{y})}{4 \pi |\boldsymbol{x} - \boldsymbol{y}|^3} d^3 \boldsymbol{y} Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law. The net electrostatic force acting on a charged particle with index i contained within a collection of particles is given as: \mathbf{F}(\mathbf{r}) = \sum_{j e i}F(r)\mathbf{\hat{r}} \, \, \,; \, \, F(r) = \frac{q_{i}q_{j}}{4\pi\varepsilon_0 r^{2}} where \mathbf{r} is the spatial coordinate, j is a particle index, r is the separation distance between particles i and j, \mathbf{\hat{r}} is the unit vector from particle j to particle i, F(r) is the force magnitude, and q_{i} and q_{j} are the charges of particles i and j, respectively. That is, A' below is also a vector potential of v; \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ abla_y \times \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}. For this purpose Newton's notation will be used for the time derivative (\dot{\mathbf{A}}). The Cartesian unit vectors are thus related to the spherical unit vectors by: \begin{bmatrix}\mathbf{\hat x} \\\ \mathbf{\hat y} \\\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\\ \cos\theta & -\sin\theta & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{\hat{r}} \\\ \boldsymbol{\hat\theta} \\\ \boldsymbol{\hat\phi} \end{bmatrix} === Time derivative of a vector field === To find out how the vector field A changes in time, the time derivatives should be calculated. The direction from r′ to r does not enter into the equation. For Kepler orbits the eccentricity vector is a constant of motion. * The position of r, the point at which values for ϕ and A are found, only enters the equation as part of the scalar distance from r′ to r. * The equation for A is a vector equation. In vector calculus, a vector potential is a vector field whose curl is a given vector field. Substituting curl[v] for the current density j of the retarded potential, you will get this formula. The drag curve or drag polar is the relationship between the drag on an aircraft and other variables, such as lift, the coefficient of lift, angle-of- attack or speed. After substituting, the result is given: \ddot\mathbf{P} = \mathbf{\hat \rho} \left(\ddot \rho - \rho \dot\phi^2\right) \+ \boldsymbol{\hat\phi} \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) \+ \mathbf{\hat z} \ddot z In mechanics, the terms of this expression are called: \begin{align} \ddot \rho \mathbf{\hat \rho} &= \text{central outward acceleration} \\\ -\rho \dot\phi^2 \mathbf{\hat \rho} &= \text{centripetal acceleration} \\\ \rho \ddot\phi \boldsymbol{\hat\phi} &= \text{angular acceleration} \\\ 2 \dot \rho \dot\phi \boldsymbol{\hat\phi} &= \text{Coriolis effect} \\\ \ddot z \mathbf{\hat z} &= \text{z-acceleration} \end{align} == Spherical coordinate system == === Vector fields === Vectors are defined in spherical coordinates by (r, θ, φ), where * r is the length of the vector, * θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and * φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π). (r, θ, φ) is given in Cartesian coordinates by: \begin{bmatrix}r \\\ \theta \\\ \phi \end{bmatrix} = \begin{bmatrix} \sqrt{x^2 + y^2 + z^2} \\\ \arccos(z / \sqrt{x^2 + y^2 + z^2}) \\\ \arctan(y / x) \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, or inversely by: \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} r\sin\theta\cos\phi \\\ r\sin\theta\sin\phi \\\ r\cos\theta\end{bmatrix}. In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: abla \times \mathbf{A} = \mathbf{B}. They are given by: \begin{align} \boldsymbol{\dot{\hat r}} &= \dot\theta \boldsymbol{\hat\theta} + \dot\phi\sin\theta \boldsymbol{\hat\phi} \\\ \boldsymbol{\dot{\hat\theta}} &= - \dot\theta \boldsymbol{\hat r} + \dot\phi\cos\theta \boldsymbol{\hat\phi} \\\ \boldsymbol{\dot{\hat\phi}} &= - \dot\phi\sin\theta \boldsymbol{\hat{r}} - \dot\phi\cos\theta \boldsymbol{\hat\theta} \end{align} Thus the time derivative becomes: \mathbf{\dot A} = \boldsymbol{\hat r} \left(\dot A_r - A_\theta \dot\theta - A_\phi \dot\phi \sin\theta \right) \+ \boldsymbol{\hat\theta} \left(\dot A_\theta + A_r \dot\theta - A_\phi \dot\phi \cos\theta\right) \+ \boldsymbol{\hat\phi} \left(\dot A_\phi + A_r \dot\phi \sin\theta + A_\theta \dot\phi \cos\theta\right) == See also == * Del in cylindrical and spherical coordinates for the specification of gradient, divergence, curl, and Laplacian in various coordinate systems. ==References== Category:Vector calculus Category:Coordinate systems This means that \mathbf{A} = \mathbf{P} = \rho \mathbf{\hat \rho} + z \mathbf{\hat z}. Hence the charge and current densities affecting the electric and magnetic potential at r and t, from remote location r′ must also be at some prior time t′. The forces involved are obtained from the coefficients by multiplication with , where ρ is the density of the atmosphere at the flight altitude, is the wing area and is the speed. The vector potential A is used when studying the Lagrangian in classical mechanics and in quantum mechanics (see Schrödinger equation for charged particles, Dirac equation, Aharonov–Bohm effect). ",4.85,-0.75,"""0.405""",0.70710678,1.06,B
+Calculate the minimum $\Delta v$ required to place a satellite already in Earth's heliocentric orbit (assumed circular) into the orbit of Venus (also assumed circular and coplanar with Earth). Consider only the gravitational attraction of the Sun. ,"The formula by Ramanujan is accurate enough. giving an average orbital speed of . ==Conjunctions and transits== When the geocentric ecliptic longitude of Venus coincides with that of the Sun, it is in conjunction with the Sun – inferior if Venus is nearer and superior if farther. That said, Venus and Earth still have the lowest gravitational potential difference between them than to any other planet, needing the lowest delta-v to transfer between them, than to any other planet from them. A heliocentric orbit (also called circumsolar orbit) is an orbit around the barycenter of the Solar System, which is usually located within or very near the surface of the Sun. It is necessary for a spacecraft, like Gaia, to follow a Lissajous orbit or a halo orbit around L2 in order for its solar panels to get full sun. ===L3=== The location of L3 is the solution to the following equation, gravitation providing the centripetal force: \frac{M_1}{\left(R-r\right)^2}+\frac{M_2}{\left(2R-r\right)^2}=\left(\frac{M_2}{M_1+M_2}R+R-r\right)\frac{M_1+M_2}{R^3} with parameters M1, M2, and R defined as for the L1 and L2 cases, and r now indicates the distance of L3 from the position of the smaller object, if it were rotated 180 degrees about the larger object, while positive r implying L3 is closer to the larger object than the smaller object. Because the range of heliocentric distances is greater for the Earth than for Venus, the closest approaches come near Earth's perihelion. The 3.4° inclination of Venus's orbit is great enough to usually prevent the inferior planet from passing directly between the Sun and Earth at inferior conjunction. thumb|right|300 px|Representation of Venus (yellow) and Earth (blue) circling around the Sun. Venus has an orbit with a semi-major axis of , and an eccentricity of 0.007.Jean Meeus, Astronomical Formulæ for Calculators, by Jean Meeus. Elements by Simon Newcomb The low eccentricity and comparatively small size of its orbit give Venus the least range in distance between perihelion and aphelion of the planets: 1.46 million km. The orbit is now known to sub-kilometer accuracy. ==Table of orbital parameters== No more than five significant figures are presented here, and to this level of precision the numbers match very well the VSOP87 elements and calculations derived from them, Standish's (of JPL) 250-year best fit,Standish and Williams(2012) p 27 Newcomb's, and calculations using the actual positions of Venus over time. distances au Million km semimajor axis 0.72333 108.21 perihelion 0.71843 107.48 aphelion 0.7282 108.94 averageAverage distance over times. Sun orbit may refer to: * Heliocentric orbit, around the sun * Orbit of the sun around the Galactic Center The longitudes of perihelion were only 29 degrees apart at J2000, so the smallest distances, which come when inferior conjunction happens near Earth's perihelion, occur when Venus is near perihelion. A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. Calculations assume the two bodies orbit in a perfect circle with separation equal to the semimajor axis and no other bodies are nearby. It is at the point that the combined pull of Earth and Sun causes the object to orbit with the same period as Earth, in effect orbiting an Earth+Sun mass with the Earth-Sun barycenter at one focus of its orbit. === and points=== thumb|right|200px|Gravitational accelerations at The and points lie at the third vertices of the two equilateral triangles in the plane of orbit whose common base is the line between the centers of the two masses, such that the point lies 60° ahead of () or behind () the smaller mass with regard to its orbit around the larger mass. ===Stability=== The triangular points ( and ) are stable equilibria, provided that the ratio of is greater than 24.96.Actually (25 + 3)/2 ≈ This is the case for the Sun–Earth system, the Sun–Jupiter system, and, by a smaller margin, the Earth–Moon system. Now the orbit estimates are dominated by observations of the Venus Express spacecraft. The first spacecraft to be put in a heliocentric orbit was Luna 1 in 1959. The heliocentric longitude of Earth advances by 0.9856° per day, and after 2919.6 days, it has advanced by 2878°, only 2° short of eight revolutions (2880°). Two important Lagrange points in the Sun-Earth system are L1, between the Sun and Earth, and L2, on the same line at the opposite side of the Earth; both are well outside the Moon's orbit. Lagrangian points in Solar System Body pair Semimajor axis, SMA (×109 m) L1 (×109 m) 1 − L1/SMA (%) L2 (×109 m) L2/SMA − 1 (%) L3 (×109 m) 1 + L3/SMA (%) Earth–Moon Sun–Mercury Sun–Venus Sun–Earth Sun–Mars Sun–Jupiter Sun–Saturn Sun–Uranus Sun–Neptune ==Spaceflight applications== ===Sun–Earth=== Sun–Earth is suited for making observations of the Sun–Earth system. This distance can be described as being such that the orbital period, corresponding to a circular orbit with this distance as radius around M2 in the absence of M1, is that of M2 around M1, divided by ≈ 1.73: T_{s,M_2}(r) = \frac{T_{M_2,M_1}(R)}{\sqrt{3}}. ====== The location of L2 is the solution to the following equation, gravitation providing the centripetal force: \frac{M_1}{(R+r)^2}+\frac{M_2}{r^2}=\left(\frac{M_1}{M_1+M_2}R+r\right)\frac{M_1+M_2}{R^3} with parameters defined as for the L1 case. Such solar transits of Venus rarely occur, but with great predictability and interest.Venus transit page. by Aldo Vitagliano, creator of SolexWilliam Sheehan, John Westfall The Transits of Venus (Prometheus Books, 2004) ==Close approaches to Earth and Mercury== In this current era, the nearest that Venus comes to Earth is just under 40 million km. The distance between Venus and Earth varies from about 42 million km (at inferior conjunction) to about 258 million km (at superior conjunction). ",+0.60,0.88,"""5275.0""",30,1590,C
+Find the ratio of the radius $R$ to the height $H$ of a right-circular cylinder of fixed volume $V$ that minimizes the surface area $A$.,"Since the area of a circle of radius r\,, which is the base of the cylinder, is given by B = \pi r^2 it follows that: * V = \pi r^2 h or even * V = \pi r^2 g . == Equilateral cylinder == thumb|Illustration of a cylinder circumscribed by a sphere of radius r. Where: * L\,represents the lateral surface area of the cylinder; * \pi\,is approximately 3.14; * r\,is the distance between the lateral surface of the cylinder and the axis, i.e. it is the value of the radius of the base; * h\,is the height of the cylinder; * 2 \pi r is the length of the circumference of the base, since \pi = \frac{C}{2r}, that is, C = 2\pi r. Simply substitute the radius and height measurements defined earlier into the volume formula for a straight circular cylinder: * V = \pi r^2 \cdot h \Rightarrow V = \pi r^2 \cdot 2r \Rightarrow V = 2\pi r^3 == Meridian section == It is the intersection between a plane containing the axis of the cylinder and the cylinder. For a right circular cylinder of radius and height , the lateral area is the area of the side surface of the cylinder: . To calculate the total area of a right circular cylinder, you simply add the lateral area to the area of the two bases: * A = L + 2 \cdot B. Replacing L = 2 \pi r h and B = \pi r^2, we have: * A=2\pi rh + 2\pi r^2 \Rightarrow A = 2 \pi r (h + r) or even * A = 2 \pi r (g + r) . == Volume == thumb|Illustration of a cylinder and a prism, both with height h. Then, assuming that the radius of the base of an equilateral cylinder is r\, then the diameter of the base of this cylinder is 2r\, and its height is 2r\,. Its lateral area can be obtained by replacing the height value by 2r: * L = 2 \pi r \cdot 2r \Rightarrow L = 4 \pi r^2 . The result can be obtained in a similar way for the total area: * T = 2 \pi r (h + r) \Rightarrow T = 2 \pi r (2r + r) \Rightarrow T = 2 \pi r \cdot 3 r \Rightarrow T = 6 \pi r^2 . Therefore, the lateral surface area is given by: * L=2\pi rh. For the equilateral cylinder it is possible to obtain a simpler formula to calculate the volume. Note that in the case of the right circular cylinder, the height and the generatrix have the same measure, so the lateral area can also be given by: * L = 2 \pi r g . Therefore, simply multiply the area of the base by the height: * V = B \cdot h. The area of the base of a cylinder is the area of a circle (in this case we define that the circle has a radius with measure r): * B = \pi r^2. Note that the cylinder is equilateral. Fixing g as the side on which the revolution takes place, we obtain that the side r, perpendicular to g, will be the measure of the radius of the cylinder. It can be obtained by the product between the length of the circumference of the base and the height of the cylinder. However, the head of the radius is not perfectly cylindrical but slightly oval. Through Cavalieri's principle, which defines that if two solids of the same height, with congruent base areas, are positioned on the same plane, such that any other plane parallel to this plane sections both solids, determining from this section two polygons with the same area, then the volume of the two solids will be the same, we can determine the volume of the cylinder. This lateral surface area can be calculated by multiplying the perimeter of the base by the height of the prism. The surface to volume ratio for this cube is thus :\mbox{SA:V} = \frac{6~\mbox{cm}^2}{1~\mbox{cm}^3} = 6~\mbox{cm}^{-1}. This is because the volume of a cylinder can be obtained in the same way as the volume of a prism with the same height and the same area of the base. For example, the volume of the torus with minor radius r and major radius R is V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2. ",0.5768,54.7,"""2.567""",1.7,0.5,E
+Find the dimension of the parallelepiped of maximum volume circumscribed by a sphere of radius $R$.,"If the radius of the sphere is called , the radii of the spherical segment bases are and and the height of the segment (the distance from one parallel plane to the other) called , then the volume of the spherical segment is : V = \frac{\pi h}{6} \left(3 r_1^2 + 3 r_2^2 + h^2\right). The volume can also be expressed in terms of A_n, the area of the unit -sphere. == Formulas == The first volumes are as follows: Dimension Volume of a ball of radius Radius of a ball of volume 0 1 (all 0-balls have volume 1) 1 2R \frac{V}{2}=0.5\times V 2 \pi R^2 \approx 3.142\times R^2 \frac{V^{1/2}}{\sqrt{\pi}}\approx 0.564\times V^{\frac{1}{2}} 3 \frac{4\pi}{3} R^3 \approx 4.189\times R^3 \left(\frac{3V}{4\pi}\right)^{1/3}\approx 0.620\times V^{1/3} 4 \frac{\pi^2}{2} R^4 \approx 4.935\times R^4 \frac{(2V)^{1/4}}{\sqrt{\pi}}\approx 0.671\times V^{1/4} 5 \frac{8\pi^2}{15} R^5\approx 5.264\times R^5 \left(\frac{15V}{8\pi^2}\right)^{1/5}\approx 0.717\times V^{1/5} 6 \frac{\pi^3}{6} R^6 \approx 5.168 \times R^6 \frac{(6V)^{1/6}}{\sqrt{\pi}}\approx 0.761\times V^{1/6} 7 \frac{16\pi^3}{105} R^7 \approx 4.725\times R^7 \left(\frac{105V}{16\pi^3}\right)^{1/7}\approx 0.801\times V^{1/7} 8 \frac{\pi^4}{24} R^8 \approx 4.059\times R^8 \frac{(24V)^{1/8}}{\sqrt{\pi}}\approx 0.839\times V^{1/8} 9 \frac{32\pi^4}{945} R^9 \approx 3.299\times R^9 \left(\frac{945V}{32\pi^4}\right)^{1/9}\approx 0.876\times V^{1/9} 10 \frac{\pi^5}{120} R^{10} \approx 2.550\times R^{10} \frac{(120V)^{1/10}}{\sqrt{\pi}}\approx 0.911\times V^{1/10} 11 \frac{64\pi^5}{10395} R^{11} \approx 1.884\times R^{11} \left(\frac{10395V}{64\pi^5}\right)^{1/11}\approx 0.944\times V^{1/11} 12 \frac{\pi^6}{720} R^{12} \approx 1.335\times R^{12} \frac{(720V)^{1/12}}{\sqrt{\pi}}\approx 0.976\times V^{1/12} 13 \frac{128\pi^6}{135135} R^{13} \approx 0.911\times R^{13} \left(\frac{135135V}{128\pi^6}\right)^{1/13}\approx 1.007\times V^{1/13} 14 \frac{\pi^7}{5040} R^{14} \approx 0.599\times R^{14} \frac{(5040V)^{1/14}}{\sqrt{\pi}}\approx 1.037\times V^{1/14} 15 \frac{256\pi^7}{2027025} R^{15} \approx 0.381\times R^{15} \left(\frac{2027025V}{256\pi^7}\right)^{1/15}\approx 1.066\times V^{1/15} n Vn(R) Rn(V) === Two-dimension recurrence relation === As is proved below using a vector-calculus double integral in polar coordinates, the volume of an -ball of radius can be expressed recursively in terms of the volume of an -ball, via the interleaved recurrence relation: : V_n(R) = \begin{cases} 1 &\text{if } n=0,\\\\[0.5ex] 2R &\text{if } n=1,\\\\[0.5ex] \dfrac{2\pi}{n}R^2 \times V_{n-2}(R) &\text{otherwise}. \end{cases} This allows computation of in approximately steps. === Closed form === The -dimensional volume of a Euclidean ball of radius in -dimensional Euclidean space is:Equation 5.19.4, NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/5.19#E4, Release 1.0.6 of 2013-05-06. It is the three-dimensional analogue of the sector of a circle. ==Volume== If the radius of the sphere is denoted by and the height of the cap by , the volume of the spherical sector is V = \frac{2\pi r^2 h}{3}\,. The volume of an odd-dimensional ball is :V_{2k+1}(R) = \frac{2(2\pi)^k}{(2k + 1)!!}R^{2k+1}. The volume of the sector is related to the area of the cap by: V = \frac{rA}{3}\,. ==Area== The curved surface area of the spherical sector (on the surface of the sphere, excluding the cone surface) is A = 2\pi rh\,. The surface area of the sphere satisfies a proportionality equation similar to the one for the volume of a ball: If is the surface area of an -sphere of radius , then: :A_{n-1}(r) = r^{n-1} A_{n-1}(1). It is the region of a ball between two concentric spheres of differing radii. ==Volume== The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere: : V=\frac{4}{3}\pi R^3- \frac{4}{3}\pi r^3 : V=\frac{4}{3}\pi \left(R^3-r^3\right) : V=\frac{4}{3}\pi (R-r)\left(R^2+Rr+r^2\right) where is the radius of the inner sphere and is the radius of the outer sphere. ==Approximation== An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness of the shell: : V \approx 4 \pi r^2 t, when is very small compared to (t \ll r). To derive the volume of an -ball of radius from this formula, integrate the surface area of a sphere of radius for and apply the functional equation : :V_n(R) = \int_0^R \frac{2\pi^{n/2}}{\Gamma\bigl(\tfrac n2\bigr)} \,r^{n-1}\,dr = \frac{2\pi^{n/2}}{n\,\Gamma\bigl(\tfrac n2\bigr)}R^n = \frac{\pi^{n/2}}{\Gamma\bigl(\tfrac n2 + 1\bigr)}R^n. === Geometric proof === The relations V_{n+1}(R) = \frac{R}{n+1}A_n(R) and A_{n+1}(R) = (2\pi R)V_n(R) and thus the volumes of n-balls and areas of n-spheres can also be derived geometrically. The volume of a -ball of radius is R^nV_n, where V_n is the volume of the unit -ball, the -ball of radius . The volume of an ball of radius is :V^p_n(R) = \frac{\Bigl(2\,\Gamma\bigl(\tfrac1p + 1\bigr)\Bigr)^n}{\Gamma\bigl(\tfrac np + 1\bigr)}R^n. The total surface area of the spherical shell is 4 \pi r^2. == See also == * Spherical pressure vessel * Ball * Solid torus * Bubble * Sphere == References == Category:Elementary geometry Category:Geometric shapes Category:Spherical geometry The -sphere is the -dimensional boundary (surface) of the -dimensional ball of radius , and the sphere's hypervolume and the ball's hypervolume are related by: :A_{n-1}(R) = \frac{d}{dR} V_{n}(R) = \frac{n}{R}V_{n}(R). Let denote the distance between a point in the plane and the center of the sphere, and let denote the azimuth. Also, A_{n-1}(R) = \frac{dV_n(R)}{dR} because a ball is a union of concentric spheres and increasing radius by ε corresponds to a shell of thickness ε. The volume of the ball can therefore be written as an iterated integral of the volumes of the -balls over the possible radii and azimuths: :V_n(R) = \int_0^{2\pi} \int_0^R V_{n-2}\\!\left(\sqrt{R^2 - r^2}\right) r\,dr\,d\theta, The azimuthal coordinate can be immediately integrated out. As with the two-dimension recursive formula, the same technique can be used to give an inductive proof of the volume formula. === Direct integration in spherical coordinates === The volume of the n-ball V_n(R) can be computed by integrating the volume element in spherical coordinates. The spherical volume element is: :dV = r^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2) \cdots \sin(\varphi_{n-2})\,dr\,d\varphi_1\,d\varphi_2 \cdots d\varphi_{n-1}, and the volume is the integral of this quantity over between 0 and and all possible angles: :V_n(R) = \int_0^R \int_0^\pi \cdots \int_0^{2\pi} r^{n-1}\sin^{n-2}(\varphi_1) \cdots \sin(\varphi_{n-2})\,d\varphi_{n-1} \cdots d\varphi_1\,dr. Instead of expressing the volume of the ball in terms of its radius , the formulas can be inverted to express the radius as a function of the volume: :\begin{align} R_n(V) &= \frac{\Gamma\bigl(\tfrac n2 + 1\bigr)^{1/n}}{\sqrt{\pi}}V^{1/n} \\\ &= \left(\frac{n!! These are: :\begin{align} V_{2k}(R) &= \frac{\pi^k}{k!}R^{2k}, \\\ V_{2k+1}(R) &= \frac{2(k!)(4\pi)^k}{(2k + 1)!}R^{2k+1}. \end{align} The volume can also be expressed in terms of double factorials. thumb|A spherical segment In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes. In computational geometry, the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d-dimensional space whose interior does not overlap with any given obstacles. ==Two dimensions== The largest empty circle problem is the problem of finding a circle of largest radius in the plane whose interior does not overlap with any given obstacles. 300px|thumb|spherical shell, right: two halves In geometry, a spherical shell is a generalization of an annulus to three dimensions. ",2.3,1.154700538,"""-2.0""",0.5,15.757,B
+"A potato of mass $0.5 \mathrm{~kg}$ moves under Earth's gravity with an air resistive force of $-k m v$. (a) Find the terminal velocity if the potato is released from rest and $k=$ $0.01 \mathrm{~s}^{-1}$. (b) Find the maximum height of the potato if it has the same value of $k$, but it is initially shot directly upward with a student-made potato gun with an initial velocity of $120 \mathrm{~m} / \mathrm{s}$.","Note that d has its maximum value when : \sin 2\theta=1 , which necessarily corresponds to : 2\theta=90^\circ , or : \theta=45^\circ . thumb|350px|Trajectories of projectiles launched at different elevation angles but the same speed of 10 m/s in a vacuum and uniform downward gravity field of 10 m/s2. In air, which has a kinematic viscosity around 0.15\,\mathrm{cm^2/s}, this means that the drag force becomes quadratic in v when the product of speed and diameter is more than about 0.015\,\mathrm{m^2/s}, which is typically the case for projectiles. Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid (air is the most common example). The magnitude of the velocity (under the Pythagorean theorem, also known as the triangle law): : v = \sqrt{v_x^2 + v_y^2 } . === Displacement === thumb|250px|Displacement and coordinates of parabolic throwing At any time t , the projectile's horizontal and vertical displacement are: : x = v_0 t \cos(\theta) , : y = v_0 t \sin(\theta) - \frac{1}{2}gt^2 . If the starting point is at height y0 with respect to the point of impact, the time of flight is: : t = \frac{d}{v \cos\theta} = \frac{v \sin \theta + \sqrt{(v \sin \theta)^2 + 2gy_0}}{g} As above, this expression can be reduced to : t = \frac{v\sin{\theta} + \sqrt{(v\sin{\theta})^{2}}}{g} = \frac{v\sin{\theta} + v\sin{\theta}}{g} = \frac{2v\sin{\theta}}{g} = \frac{2v\sin{(45)}}{g} = \frac{2v\frac{\sqrt{2}}{2}}{g} = \frac{\sqrt{2}v}{g} if θ is 45° and y0 is 0. === Time of flight to the target's position === As shown above in the Displacement section, the horizontal and vertical velocity of a projectile are independent of each other. The following assumptions are made: * Constant gravitational acceleration * Air resistance is given by the following drag formula, ::\mathbf{F_D} = -\tfrac{1}{2} c \rho A\, v\,\mathbf{v} ::Where: ::*FD is the drag force ::*c is the drag coefficient ::*ρ is the air density ::*A is the cross sectional area of the projectile ::*μ = k/m = cρA/(2m) ==== Special cases ==== Even though the general case of a projectile with Newton drag cannot be solved analytically, some special cases can. If h = R : \theta = \arctan(4)\approx 76.0^\circ === Maximum distance of projectile === thumb|250px|The maximum distance of projectile The range and the maximum height of the projectile does not depend upon its mass. Mathematically, it is given as t=U \sin\theta/g where g = acceleration due to gravity (app 9.81 m/s²), U = initial velocity (m/s) and \theta = angle made by the projectile with the horizontal axis. 2\. *Vertical motion downward: ::\dot{v}_y(t) = -g+\mu\,v_y^2(t) ::v_y(t) = -v_\infty \tanh\frac{t-t_{\mathrm{peak}}}{t_f} ::y(t) = y_{\mathrm{peak}} - \frac{1}{\mu}\ln\left(\cosh\frac{t-t_{\mathrm{peak}}}{t_f}\right) :After a time t_f, the projectile reaches almost terminal velocity -v_\infty. ==== Numerical solution ==== A projectile motion with drag can be computed generically by numerical integration of the ordinary differential equation, for instance by applying a reduction to a first-order system. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. * Stokes drag: \mathbf{F_{air}} = -k_{\mathrm{Stokes}}\cdot\mathbf{v}\qquad (for Re \lesssim 1000) * Newton drag: \mathbf{F_{air}} = -k\,|\mathbf{v}|\cdot\mathbf{v}\qquad (for Re \gtrsim 1000) right|thumb|320px|Free body diagram of a body on which only gravity and air resistance acts The free body diagram on the right is for a projectile that experiences air resistance and the effects of gravity. The vertical motion of the projectile is the motion of a particle during its free fall. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. Hence range and maximum height are equal for all bodies that are thrown with the same velocity and direction. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Because of this, we can find the time to reach a target using the displacement formula for the horizontal velocity: x = v_0 t \cos(\theta) \frac{x}{t}=v_0\cos(\theta) t=\frac{x}{v_0\cos(\theta)} This equation will give the total time t the projectile must travel for to reach the target's horizontal displacement, neglecting air resistance. === Maximum height of projectile === thumb|250px|Maximum height of projectile The greatest height that the object will reach is known as the peak of the object's motion. Attack of the Killer Potatoes is a 1997 science-fiction children's story by Peter Lerangis. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. The mass of the projectile will be denoted by m, and \mu:=k/m. For the vertical displacement of the maximum height of the projectile: : h = v_0 t_h \sin(\theta) - \frac{1}{2} gt^2_h : h = \frac{v_0^2 \sin^2(\theta)}{2g} The maximum reachable height is obtained for θ=90°: : h_{\mathrm{max}} = \frac{v_0^2}{2g} If the projectile's position (x,y) and launch angle (θ) are known, the maximum height can be found by solving for h in the following equation: :h=\frac{(x\tan\theta)^2}{4(x\tan\theta-y)}. === Relation between horizontal range and maximum height === The relation between the range d on the horizontal plane and the maximum height h reached at \frac{t_d}{2} is: : h = \frac{d\tan\theta}{4} h = \frac{v_0^2\sin^2\theta}{2g} : d = \frac{v_0^2\sin2\theta}{g} : \frac{h}{d} = \frac{v_0^2\sin^2\theta}{2g} × \frac{g}{v_0^2\sin2\theta} : \frac{h}{d} = \frac{\sin^2\theta}{4\sin\theta\cos\theta} h = \frac{d\tan\theta}{4} . The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. ",2.00,46.7,"""1000.0""",-2,37,C
+"A particle of mass $m$ and velocity $u_1$ makes a head-on collision with another particle of mass $2 m$ at rest. If the coefficient of restitution is such to make the loss of total kinetic energy a maximum, what are the velocities $v_1$ after the collision?","It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity. \text{Coefficient of restitution } (e) = \frac{\left | \text{Relative velocity after collision} \right |}{\left | \text{Relative velocity before collision}\right |} The mathematics were developed by Sir Isaac Newton in 1687. The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. Since the total energy and momentum of the system are conserved and their rest masses do not change, it is shown that the momentum of the colliding body is decided by the rest masses of the colliding bodies, total energy and the total momentum. Likewise, the conservation of the total kinetic energy is expressed by: \tfrac12 m_1u_1^2+\tfrac12 m_2u_2^2 \ =\ \tfrac12 m_1v_1^2 +\tfrac12 m_2v_2^2. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. A basketball's COR is designated by requiring that the ball shall rebound to a height of between 960 and 1160 mm when dropped from a height of 1800 mm, resulting in a COR between 0.73–0.80. == Equations == In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by: C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |}, where: *v_\text{a} is the final speed of object A after impact *v_\text{b} is the final speed of object B after impact *u_\text{a} is the initial speed of object A before impact *u_\text{b} is the initial speed of object B before impact Though C_R does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. Let v_1, v_2 be the final velocity of object 1 and object 2 respectively. \begin{cases} \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 \\\ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \end{cases} From the first equation, m_1 \left(u_1^2 - v_1^2\right) = m_2 \left(v_2^2 - u_2^2\right) m_1 \left(u_1 + v_1\right) \left(u_1 - v_1\right) = m_2 \left(v_2 + u_2\right) \left(v_2 - u_2\right) From the second equation, m_1 \left(u_1 - v_1\right) = m_2 \left(v_2 - u_2\right) After division, u_1+v_1=v_2+u_2 u_1-u_2 = -(v_1-v_2) \frac{\left | v_1-v_2 \right |}{\left | u_1-u_2 \right |} = 1 The equation above is the restitution equation, and the coefficient of restitution is 1, which is a perfectly elastic collision. ===Sports equipment=== Thin-faced golf club drivers utilize a ""trampoline effect"" that creates drives of a greater distance as a result of the flexing and subsequent release of stored energy which imparts greater impulse to the ball. In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. When vehicles collide, the damage increases with the relative velocity of the vehicles, the damage increasing as the square of the velocity since it is the impact kinetic energy (1/2 mv2) which is the variable of importance. The coefficient of restitution (COR, also denoted by e), is the ratio of the final to initial relative speed between two objects after they collide. During the collision of small objects, kinetic energy is first converted to potential energy associated with a repulsive or attractive force between the particles (when the particles move against this force, i.e. the angle between the force and the relative velocity is obtuse), then this potential energy is converted back to kinetic energy (when the particles move with this force, i.e. the angle between the force and the relative velocity is acute). Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. Using the notation from above where u represents the velocity before the collision and v after, yields: \begin{align} & m_\text{a} u_\text{a} + m_\text{b} u_\text{b} = m_\text{a} v_\text{a} + m_\text{b} v_\text{b} \\\ & C_R = \frac{\left | v_\text{b} - v_\text{a} \right |}{\left | u_\text{a} - u_\text{b} \right |} \\\ \end{align} Solving the momentum conservation equation for v_\text{a} and the definition of the coefficient of restitution for v_\text{b} yields: \begin{align} & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} - m_\text{b} v_\text{b}}{m_\text{a}} = v_\text{a} \\\ & v_\text{b} = C_R(u_\text{a} - u_\text{b}) + v_\text{a} \\\ \end{align} Next, substitution into the first equation for v_\text{b} and then resolving for v_\text{a} gives: \begin{align} & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} - m_\text{b} C_R(u_\text{a} - u_\text{b}) - m_\text{b} v_\text{a}}{m_\text{a}} = v_\text{a} \\\ & \\\ & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b} - u_\text{a})}{m_\text{a}} = v_\text{a} \left[ 1 + \frac{m_\text{b}}{m_\text{a}} \right] \\\ & \\\ & \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b} - u_\text{a})}{m_\text{a} + m_\text{b}} = v_\text{a} \\\ \end{align} A similar derivation yields the formula for v_\text{b}. === COR variation due to object shape and off-center collisions === When colliding objects do not have a direction of motion that is in-line with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the post-collision velocity difference will be lost to rotation and friction. In this case, the change in Ek is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus: C_R = \sqrt{\frac{E_\text{p, (at bounce height)}}{E_\text{p, (at drop height)}}} = \sqrt{\frac{mgh}{mgH}} = \sqrt{\frac{h}{H}} == Speeds after impact == The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between. v_\text{a} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{b} C_R(u_\text{b}-u_\text{a})}{m_\text{a}+m_\text{b}} and v_\text{b} = \frac{m_\text{a} u_\text{a} + m_\text{b} u_\text{b} + m_\text{a} C_R(u_\text{a}-u_\text{b})}{m_\text{a}+m_\text{b}} where *v_\text{a} is the final velocity of the first object after impact *v_\text{b} is the final velocity of the second object after impact *u_\text{a} is the initial velocity of the first object before impact *u_\text{b} is the initial velocity of the second object before impact *m_\text{a} is the mass of the first object *m_\text{b} is the mass of the second object === Derivation === The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). A comprehensive study of coefficients of restitution in dependence on material properties (elastic moduli, rheology), direction of impact, coefficient of friction and adhesive properties of impacting bodies can be found in Willert (2020). === Predicting from material properties === The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. For two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact, i.e. along the line of impact. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. In the case of a large u_{1}, the value of v_{1} is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed. ",9.13,4.68,"""0.33333333""",167,54.7,C
+"The height of a hill in meters is given by $z=2 x y-3 x^2-4 y^2-18 x+28 y+12$, where $x$ is the distance east and $y$ is the distance north of the origin. What is the $x$ distance of the top of the hill?","Nimbus Hills () is a rugged line of hills and peaks about 14 nautical miles (26 km) long, forming the southeast part of Pioneer Heights in the Heritage Range, Ellsworth Mountains. Target Hill () is a prominent hill which rises 1,010 m above the level of Larsen Ice Shelf. Gerdkooh ancient hills (Persian: تپه باستان گردکوه) consists of three hills, the tallest of which is 26 m in height. Vantages Hill () (Adam Hayat) is a flat-topped hill, over 2,000 m above sea level and 300 m above the surrounding plateau, standing 10 nautical miles (18 km) southwest of Mount Henderson in the western part of Britannia Range. Sistenup Peak () is a low peak at the northeast end of the Kirwan Escarpment, about 5 nautical miles (9 km) north of Sistefjell Mountain, in Queen Maud Land. Powell Hill () is a rounded, ice-covered prominence 6 nautical miles (11 km) west-southwest of Mount Christmas, overlooking the head of Algie Glacier. Named by Advisory Committee on Antarctic Names (US-ACAN) after the National Aeronautics and Space Administration weather satellite, Nimbus, which took photographs of Antarctica (including the Ellsworth Mountains) from approximately 500 nautical miles (900 km) above earth on September 13, 1964. ==See also== Geographical features include: ===Samuel Nunataks=== ===Other features=== * Flanagan Glacier * Mount Capley * Warren Nunatak Category:Hills of Ellsworth Land The hill was the most westerly point reached by the Falkland Islands Dependencies Survey (FIDS) survey party in 1955; it was visible to the party as a target upon which to steer from the summit of Richthofen Pass. Category:Hills of Oates Land The hills are located in Qaem Shahr in Mazandaran Province. It stands 6 nautical miles (11 km) west of Mount Fritsche on the south flank of Leppard Glacier in eastern Graham Land. There is evidence that the hills at the time of the Sasanian Empire and Muslim conquest of Persia were part of a Castle. == See also == *Castles in Iran *Tepe Sialk ==References== == External links == * video link * Persian Wikipedia article Category:Archaeological sites in Iran Category:Geography of Mazandaran Province Category:Castles in Iran Category:Buildings and structures in Mazandaran Province Category:Tourist attractions in Mazandaran Province Category:National works of Iran Category:Mountains of Queen Maud Land Category:Princess Martha Coast Category:Hills of Graham Land Category:Oscar II Coast Mapped by Norwegian cartographers from surveys and air photos by Norwegian-British-Swedish Antarctic Expedition (NBSAE) (1949–52) and air photos by the Norwegian exp (1958–59) and named Sistenup (last peak). Category:Hills of the Ross Dependency Category:Shackleton Coast Mapped by United States Geological Survey (USGS) from ground surveys and U.S. Navy air photos, 1961–66. This is the most southerly point reached by the Darwin Glacier Party of the Commonwealth Trans-Antarctic Expedition (1957–58), who gave it this name because of the splendid view it afforded. Named by Advisory Committee on Antarctic Names (US-ACAN) for Lieutenant Commander James A. Powell, U.S. Navy, communications officer at McMurdo Station during U.S. Navy Operation Deepfreeze 1963 and 1964. In exploring this area, a 4500-year-old grave has been found, as well as objects such as disposable tableware dishes related to the Parthian Empire and Sasanian Empire. Their history has been estimated to date back to the Iron Age. There is evidence that the hills at the time of the Sasanian Empire and Muslim conquest of Persia were part of a Castle. == See also == *Castles in Iran *Tepe Sialk ==References== == External links == * video link * Persian Wikipedia article Category:Archaeological sites in Iran Category:Geography of Mazandaran Province Category:Castles in Iran Category:Buildings and structures in Mazandaran Province Category:Tourist attractions in Mazandaran Province Category:National works of Iran ",-0.041,2,"""-2.0""",3,0,C
+"Shot towers were popular in the eighteenth and nineteenth centuries for dropping melted lead down tall towers to form spheres for bullets. The lead solidified while falling and often landed in water to cool the lead bullets. Many such shot towers were built in New York State. Assume a shot tower was constructed at latitude $42^{\circ} \mathrm{N}$, and the lead fell a distance of $27 \mathrm{~m}$. In what direction and how far did the lead bullets land from the direct vertical?","Large shot which could not be made by the shot tower was made by tumbling pieces of cut lead sheet in a barrel until round.. Molten lead at the top of the tower was poured through a sieve or mesh, forming uniform spherical shot before falling into a large vat of water at the bottom of the tower. A shot tower with a 40-meter drop can produce up to #6 shot (nominally 2.4mm in diameter) while an 80-meter drop can produce #2 shot (nominally 3.8mm in diameter). thumb|100px|How a shot tower works A shot tower is a tower designed for the production of small-diameter shot balls by free fall of molten lead, which is then caught in a water basin. Shot towers work on the principle that molten lead forms perfectly round balls when poured from a high place. The ""wind tower"" method, which used a blast of cold air to dramatically shorten the drop necessary and was patented in 1848 by the T.O LeRoy Company of New York City,, Lynne Belluscio, LeRoy Penny Saver NewsHistory of the American Shot Tower meant that tall shot towers became unnecessary, but many were still constructed into the late 1880s, and two surviving examples date from 1916 and 1969. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. It can be seen looking west from I-95. 1870 Panorama from Sparks Shot Tower.jpg|1870 photo from the top of the tower toward the Delaware River 1880 survey Sparks Shot Tower.png|1880 Hexamer General Survey page on the tower Sparks Shot Tower2.jpg|1973 Historic American Buildings Survey photo Sparks Shot Tower Historical Marker 129-131 Carpenter St Philadelphia PA (DSC 3814).jpg|Historical Marker ==See also== *Lead shot *Phoenix Shot Tower *Shotgun shell ==References== ==External links== *Listing and photographs at the Historic American Buildings Survey *Sparks Shot Tower at USHistory.org *Waymark *Listing and photograph at Philadelphia Buildings and Architects Category:Industrial buildings completed in 1808 Category:Towers completed in 1808 Category:Buildings and structures in Philadelphia Category:Shot towers Category:South Philadelphia The shot is primarily used for projectiles in shotguns, and for ballast, radiation shielding, and other applications for which small lead balls are useful. == Shot making == === Process === In a shot tower, lead is heated until molten, then dropped through a copper sieve high in the tower. Use of shot towers replaced earlier techniques of casting shot in moulds, which was expensive, or of dripping molten lead into water barrels, which produced insufficiently spherical balls. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Originally used to produce shot for hunters, the tower produced ammunition during the War of 1812 and the Civil War. The Sparks Shot Tower is a historic shot tower located at 129-131 Carpenter Street in Philadelphia, Pennsylvania. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. thumb|upright|Shortly after completion thumb|upright|Shortly before demolition The Tower Building was a structure in the Financial District of Manhattan, New York City, located at 50-52 Broadway on a lot that extended east to New Street. Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. thumb|right|360px|Lamoka projectile points from central New York State. At the start of the War of 1812, the federal government became their major customer, buying war munitions, and Quaker John Bishop sold his part of the company to Thomas Sparks.Sparks Shot Tower, 1808, in John Mayer, Workshop of the World (Oliver Evans Press, 1990) Before the use of shot towers, shot was made in wooden molds, which resulted in unevenly formed, low quality shot. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. ",30,131,"""-167.0""",54.394,2.26,E
+"Perform an explicit calculation of the time average (i.e., the average over one complete period) of the potential energy for a particle moving in an elliptical orbit in a central inverse-square-law force field. Express the result in terms of the force constant of the field and the semimajor axis of the ellipse. ","Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Using the virial theorem we find: *the time-average of the specific potential energy is equal to −2ε **the time-average of r−1 is a−1 *the time-average of the specific kinetic energy is equal to ε === Energy in terms of semi major axis === It can be helpful to know the energy in terms of the semi major axis (and the involved masses). The empty focus (\mathbf{F2} = \left(f_x,f_y\right)) can be found by first determining the Eccentricity vector: :\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu} Where \mathbf{h} is the specific angular momentum of the orbiting body: :\mathbf{h} = \mathbf{r} \times \mathbf{v} Then :\mathbf{F2} = -2a\mathbf{e} ====Using XY Coordinates==== This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2 + y^2 } = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt{r_x^2 + r_y^2} \quad initial distance from F1 (at the origin) :a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad the semi-major axis length :e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad the Eccentricity vector coordinates :e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values fx, fy and a can be applied to the general ellipse equation above. == Orbital parameters == The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. The solution of this equation is u(\varphi) = -\frac{\alpha}{mh^{2}} \left[ 1 + e \cos \left( \varphi - \varphi_{0}\right) \right] which shows that the orbit is a conic section of eccentricity e; here, φ0 is the initial angle, and the center of force is at the focus of the conic section. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. Adopting the radial distance r and the azimuthal angle φ as the coordinates, the Hamilton-Jacobi equation for a central-force problem can be written \frac{1}{2m} \left( \frac{dS_{r}}{dr} \right)^{2} + \frac{1}{2m r^{2}} \left( \frac{dS_{\varphi}}{d\varphi} \right)^{2} + U(r) = E_{\mathrm{tot}} where S = Sφ(φ) + Sr(r) − Etott is Hamilton's principal function, and Etot and t represent the total energy and time, respectively. Assume that a particle is moving under an arbitrary central force F1(r), and let its radius r and azimuthal angle φ be denoted as r(t) and φ1(t) as a function of time t. Four line segments go out from the left focus to the ellipse, forming two shaded pseudo-triangles with two straight sides and the third side made from the curved segment of the intervening ellipse.|As for all central forces, the particle in the Kepler problem sweeps out equal areas in equal times, as illustrated by the two blue elliptical sectors. A central-force problem is said to be ""integrable"" if this final integration can be solved in terms of known functions. ===Orbit of the particle=== The total energy of the system Etot equals the sum of the potential energy and the kinetic energyGoldstein, p. For an attractive force (α < 0), the orbit is an ellipse, a hyperbola or parabola, depending on whether u1 is positive, negative, or zero, respectively; this corresponds to an eccentricity e less than one, greater than one, or equal to one. The eccentricity e is related to the total energy E (cf. the Laplace–Runge–Lenz vector) : e = \sqrt{1 + \frac{2EL^2}{k^2 m}} Comparing these formulae shows that E<0 corresponds to an ellipse (all solutions which are closed orbits are ellipses), E=0 corresponds to a parabola, and E>0 corresponds to a hyperbola. In particular, E=-\frac{k^2 m}{2L^2} for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius). If the force is the sum of an inverse quadratic law and a linear term, i.e., if F(r) = \frac{a}{r^2} + cr, the problem also is solved explicitly in terms of Weierstrass elliptic functions.Izzo and Biscani ==References== ==Bibliography== * * Category:Classical mechanics If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Thus, the areal velocity is constant for a particle acted upon by any type of central force; this is Kepler's second law.Goldstein, p. 73; Landau and Lifshitz, p. 31; Sommerfeld, p. 39; Symon, p. 135. 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. This includes the radial elliptic orbit, with eccentricity equal to 1. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. The center of force is located at one of the foci of the elliptical orbit. where u1 and u2 are constants, with u2 larger than u1. In that case, :n = \frac{d}{2\pi}\sqrt{\frac{ G( M + m ) }{a^3}} = d\sqrt{\frac{ G( M + m ) }{4\pi^2 a^3}}\,\\! where *d is the quantity of time in a day, *G is the gravitational constant, *M and m are the masses of the orbiting bodies, *a is the length of the semi-major axis. By Newton's second law of motion, the magnitude of F equals the mass m of the particle times the magnitude of its radial accelerationGoldstein, p. ",7166.67,0.6749,"""-1.0""",0.75,0,C
+A simple harmonic oscillator consists of a 100-g mass attached to a spring whose force constant is $10^4 \mathrm{dyne} / \mathrm{cm}$. The mass is displaced $3 \mathrm{~cm}$ and released from rest. Calculate the natural frequency $\nu_0$.,"As such, m cannot be simply added to M to determine the frequency of oscillation, and the effective mass of the spring is defined as the mass that needs to be added to M to correctly predict the behavior of the system. ==Ideal uniform spring== right|frame|vertical spring-mass system The effective mass of the spring in a spring-mass system when using an ideal spring of uniform linear density is 1/3 of the mass of the spring and is independent of the direction of the spring- mass system (i.e., horizontal, vertical, and oblique systems all have the same effective mass). In a mass–spring system, with mass m and spring stiffness k, the natural angular frequency can be calculated as: \omega _0 =\sqrt{\frac{k}{m}} In an electrical network, ω is a natural angular frequency of a response function f(t) if the Laplace transform F(s) of f(t) includes the term , where for a real σ, and is a constant. By differentiation of the equation with respect to time, the equation of motion is: :\left( \frac{-m}{3}-M \right) \ a = kx -\tfrac{1}{2} mg - Mg The equilibrium point x_{\mathrm{eq}} can be found by letting the acceleration be zero: :x_{\mathrm{eq}} = \frac{1}{k}\left(\tfrac{1}{2}mg + Mg \right) Defining \bar{x} = x - x_{\mathrm{eq}}, the equation of motion becomes: :\left( \frac{m}{3}+M \right) \ a = -k\bar{x} This is the equation for a simple harmonic oscillator with period: :\tau = 2 \pi \left( \frac{M + m/3}{k} \right)^{1/2} So the effective mass of the spring added to the mass of the load gives us the ""effective total mass"" of the system that must be used in the standard formula 2 \pi\sqrt{\frac{m}{k}} to determine the period of oscillation. ==General case == As seen above, the effective mass of a spring does not depend upon ""external"" factors such as the acceleration of gravity along it. *Reduced mass ==References== ==External links== *http://tw.knowledge.yahoo.com/question/question?qid=1405121418180 *http://tw.knowledge.yahoo.com/question/question?qid=1509031308350 *https://web.archive.org/web/20110929231207/http://hk.knowledge.yahoo.com/question/article?qid=6908120700201 *https://web.archive.org/web/20080201235717/http://www.goiit.com/posts/list/mechanics- effective-mass-of-spring-40942.htm *http://www.juen.ac.jp/scien/sadamoto_base/spring.html *""The Effective Mass of an Oscillating Spring"" Am. J. Phys., 38, 98 (1970) *""Effective Mass of an Oscillating Spring"" The Physics Teacher, 45, 100 (2007) Category:Mechanical vibrations Category:Mass This requires adding all the mass elements' kinetic energy, and requires the following integral, where u is the velocity of mass element: :K =\int_m\tfrac{1}{2}u^2\,dm Since the spring is uniform, dm=\left(\frac{dy}{L}\right)m, where L is the length of the spring at the time of measuring the speed. If the forced frequency is equal to the natural frequency, the vibrations' amplitude increases manyfold. If the oscillating system is driven by an external force at the frequency at which the amplitude of its motion is greatest (close to a natural frequency of the system), this frequency is called resonant frequency. ==Overview== Free vibrations of an elastic body are called natural vibrations and occur at a frequency called the natural frequency. In LC and RLC circuits, its natural angular frequency can be calculated as: \omega _0 =\frac{1}{\sqrt{LC}} ==See also== * Fundamental frequency ==Footnotes== ==References== * * * * Category:Waves Category:Oscillation The effective mass of the spring can be determined by finding its kinetic energy. thumb|Scale of harmonics on C. This unexpected behavior of the effective mass can be explained in terms of the elastic after- effect (which is the spring's not returning to its original length after the load is removed). ==See also== *Simple harmonic motion (SHM) examples. Using this result, the total energy of system can be written in terms of the displacement x from the spring's unstretched position (ignoring constant potential terms and taking the upwards direction as positive): : T(Total energy of system) ::= \tfrac{1}{2}(\frac{m}{3})\ v^2 + \tfrac{1}{2}M v^2 + \tfrac{1}{2} k x^2 - \tfrac{1}{2}m g x - M g x Note that g here is the acceleration of gravity along the spring. Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. Since not all of the spring's length moves at the same velocity v as the suspended mass M, its kinetic energy is not equal to \tfrac{1}{2} m v^2. Natural vibrations are different from forced vibrations which happen at the frequency of an applied force (forced frequency). thumb|119x119px|Energy level scheme of half-harmonic generation process. In a real spring–mass system, the spring has a non-negligible mass m. Jun-ichi Ueda and Yoshiro Sadamoto have found that as M/m increases beyond 7, the effective mass of a spring in a vertical spring-mass system becomes smaller than Rayleigh's value m/3 and eventually reaches negative values. Hence, :K = \int_0^L\tfrac{1}{2}u^2\left(\frac{dy}{L}\right)m\\! ::=\tfrac{1}{2}\frac{m}{L}\int_0^L u^2\,dy The velocity of each mass element of the spring is directly proportional to length from the position where it is attached (if near to the block then more velocity and if near to the ceiling then less velocity), i.e. u=\frac{vy}{L}, from which it follows: :K =\tfrac{1}{2}\frac{m}{L}\int_0^L\left(\frac{vy}{L}\right)^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\int_0^L y^2\,dy :=\tfrac{1}{2}\frac{m}{L^3}v^2\left[\frac{y^3}{3}\right]_0^L :=\tfrac{1}{2}\frac{m}{3}v^2 Comparing to the expected original kinetic energy formula \tfrac{1}{2}mv^2, the effective mass of spring in this case is m/3. In fact, for a non-uniform spring, the effective mass solely depends on its linear density \rho(x) along its length: ::K = \int_m\tfrac{1}{2}u^2\,dm ::: = \int_0^L\tfrac{1}{2}u^2 \rho(x) \,dx ::: = \int_0^L\tfrac{1}{2}\left(\frac{v x}{L} \right)^2 \rho(x) \,dx ::: = \tfrac{1}{2} \left[ \int_0^L \frac{x^2}{L^2} \rho(x) \,dx \right] v^2 So the effective mass of a spring is: :m_{\mathrm{eff}} = \int_0^L \frac{x^2}{L^2} \rho(x) \,dx This result also shows that m_{\mathrm{eff}} \le m, with m_{\mathrm{eff}} = m occurring in the case of an unphysical spring whose mass is located purely at the end farthest from the support. ==Real spring== The above calculations assume that the stiffness coefficient of the spring does not depend on its length. For instance: the frequency ratio 5:4 is equal to of the string length and is the complement of , the position of the fifth harmonic (and the fourth overtone). It can be shown that the set of natural frequencies in a network can be obtained by calculating the poles of all impedance and admittance functions of the network. ",-8,15.757,"""1.56""",6.9,35.2,D
+ Find the center of mass of a uniformly solid cone of base diameter $2a$ and height  $h$,"If the chosen ogive radius of a secant ogive is greater than the ogive radius of a tangent ogive with the same and , then the resulting secant ogive appears as a tangent ogive with a portion of the base truncated. :\rho > {R^2 + L^2 \over 2R} and \alpha = \arccos \left({\sqrt{L^2 + R^2} \over 2\rho}\right)-\arctan \left({R \over L}\right) Then the radius at any point as varies from to is: :y = \sqrt{\rho^2 - (\rho\cos(\alpha) - x)^2} - \rho\sin(\alpha) If the chosen is less than the tangent ogive and greater than half the length of the nose cone, then the result will be a secant ogive that bulges out to a maximum diameter that is greater than the base diameter. The profile of this shape is formed by a segment of a circle such that the rocket body is tangent to the curve of the nose cone at its base, and the base is on the radius of the circle. The radius of the circle that forms the ogive is called the ogive radius, , and it is related to the length and base radius of the nose cone as expressed by the formula: :\rho = {R^2 + L^2\over 2R} The radius at any point , as varies from to is: :y = \sqrt{\rho^2 - (L - x)^2}+R - \rho The nose cone length, , must be less than or equal to . The center of the spherical nose cap, , can be found from: : x_o = x_t + \sqrt{ r_n^2 - y_t^2} And the apex point, can be found from: : x_a = x_o - r_n === Bi-conic === A bi-conic nose cone shape is simply a cone with length stacked on top of a frustum of a cone (commonly known as a conical transition section shape) with length , where the base of the upper cone is equal in radius to the top radius of the smaller frustum with base radius . For 0 \le K' \le 1 : y = R\left({2 \left({x \over L}\right) - K'\left({x \over L}\right)^2 \over 2 - K'}\right) can vary anywhere between and , but the most common values used for nose cone shapes are: Parabola Type Value Cone Half Three Quarter Full For the case of the full parabola () the shape is tangent to the body at its base, and the base is on the axis of the parabola. The failure is governed by crack growth in concrete, which forms a typical cone shape having the anchor's axis as revolution axis. ==Mechanical models== ===ACI 349-85=== Under tension loading, the concrete cone failure surface has 45° inclination. The tangency point where the sphere meets the cone can be found from: : x_t = \frac{L^2}{R} \sqrt{ \frac{r_n^2}{R^2 + L^2} } : y_t = \frac{x_t R}{L} where is the radius of the spherical nose cap. :L=L_1+L_2 :For 0 \le x \le L_1 : y = {xR_1 \over L_1} :For L_1 \le x \le L : y = R_1 + {(x - L_1)(R_2-R_1)\over L_2} Half angles: :\phi_1 = \arctan \left({R_1 \over L_1}\right) and y = x \tan(\phi_1)\; :\phi_2 = \arctan \left({R_2 - R_1 \over L_2}\right) and y = R_1 + (x - L_1) \tan(\phi_2)\; === Tangent ogive === Next to a simple cone, the tangent ogive shape is the most familiar in hobby rocketry. alt=Two-dimensional drawing of an elliptical nose cone with dimensions added to show how L is the total length of the nose cone, R is the radius at the base, and y is the radius at a point x distance from the tip.|thumb|300x300px|General parameters used for constructing nose cone profiles. Finally, the apex point can be found from: : x_a = x_o - r_n === Secant ogive === The profile of this shape is also formed by a segment of a circle, but the base of the shape is not on the radius of the circle defined by the ogive radius. For , Haack nose cones bulge to a maximum diameter greater than the base diameter. The full body of revolution of the nose cone is formed by rotating the profile around the centerline . For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium. == Nose cone shapes and equations == === General dimensions === In all of the following nose cone shape equations, is the overall length of the nose cone and is the radius of the base of the nose cone. is the radius at any point , as varies from , at the tip of the nose cone, to . The sides of a conic profile are straight lines, so the diameter equation is simply: : y = {xR \over L} Cones are sometimes defined by their half angle, : : \phi = \arctan \left({R \over L}\right) and y = x \tan(\phi)\; ==== Spherically blunted conic ==== In practical applications such as re-entry vehicles, a conical nose is often blunted by capping it with a segment of a sphere. Given the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. right|thumb|Concrete Cone Model Concrete cone is one of the failure modes of anchors in concrete, loaded by a tensile force. The length/diameter relation is also often called the caliber of a nose cone. Wallis's Conical Edge with right|thumb|600px| Figure 2. thumb|right|400px|The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = k \sqrt{f_{cc}} {h_{ef}}^{1.5} [N] , Where: k \- 13.5 for post-installed fasteners, 15.5 for cast-in-site fasteners f_{cc} \- Concrete compressive strength measured on cubes [MPa] {h_{ef}} \- Embedment depth of the anchor [mm] The model is based on fracture mechanics theory and takes into account the size effect, particularly for the factor {h_{ef}}^{1.5} which differentiates from {h_{ef}}^{2} expected from the first model. The concrete cone failure load N_0 of a single anchor in uncracked concrete unaffected by edge influences or overlapping cones of neighboring anchors is given by: N_0 = f_{ct} {A_{N}} [N] Where: f_{ct} \- tensile strength of concrete A_{N} \- Cone's projected area === Concrete capacity design (CCD) approach for fastening to concrete=== Under tension loading, the concrete capacity of a single anchor is calculated assuming an inclination between the failure surface and surface of the concrete member of about 35°. The rocket body will not be tangent to the curve of the nose at its base. ",0.75,272.8,"""22.0""",1,+65.49,A
+A particle is projected with an initial velocity $v_0$ up a slope that makes an angle $\alpha$ with the horizontal. Assume frictionless motion and find the time required for the particle to return to its starting position. Find the time for $v_0=2.4 \mathrm{~m} / \mathrm{s}$ and $\alpha=26^{\circ}$.,"Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. \begin{align} v & = at+v_0 & [1]\\\ r & = r_0 + v_0 t + \tfrac12 {a}t^2 & [2]\\\ r & = r_0 + \tfrac12 \left( v+v_0 \right )t & [3]\\\ v^2 & = v_0^2 + 2a\left( r - r_0 \right) & [4]\\\ r & = r_0 + vt - \tfrac12 {a}t^2 & [5]\\\ \end{align} where: * is the particle's initial position * is the particle's final position * is the particle's initial velocity * is the particle's final velocity * is the particle's acceleration * is the time interval Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ; \begin{align} \mathbf{v} & = \int \mathbf{a} dt = \mathbf{a}t+\mathbf{v}_0 \,, & [1] \\\ \mathbf{r} & = \int (\mathbf{a}t+\mathbf{v}_0) dt = \frac{\mathbf{a}t^2}{2}+\mathbf{v}_0t +\mathbf{r}_0 \,, & [2] \\\ \end{align} in magnitudes, \begin{align} v & = at+v_0 \,, & [1] \\\ r & = \frac{{a}t^2}{2}+v_0t +r_0 \,. thumb|360px|v vs t graph for a moving particle under a non-uniform acceleration a. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary, \begin{align} \omega & = \omega_0 + \alpha t \\\ \theta &= \theta_0 + \omega_0t + \tfrac12\alpha t^2 \\\ \theta & = \theta_0 + \tfrac12(\omega_0 + \omega)t \\\ \omega^2 & = \omega_0^2 + 2\alpha(\theta - \theta_0) \\\ \theta & = \theta_0 + \omega t - \tfrac12\alpha t^2 \\\ \end{align} where is the constant angular acceleration, is the angular velocity, is the initial angular velocity, is the angle turned through (angular displacement), is the initial angle, and is the time taken to rotate from the initial state to the final state. ===General planar motion=== These are the kinematic equations for a particle traversing a path in a plane, described by position . Suppose that C is the curve traced out by P and s is the arc length of C corresponding to time t. The velocity vector of the particle is : \mathbf{v} = \frac{d \mathbf{r}}{dt} = \dot{s}\mathbf{e}_t = v\mathbf{e}_t , where et is the unit tangent vector to C. Define the angular momentum of P as : \mathbf{h} = \mathbf{r} \times m\mathbf{v} = h\mathbf{k}, where k = i x j. Then the acceleration vector of P can be expressed as : \mathbf{a} = -\frac{\kappa v^2r}{p} \mathbf{e}_r + \left( v \frac{dv}{ds} + \frac{\kappa v^2q}{p} \right) \mathbf{e}_t . Given initial velocity , one can calculate how high the ball will travel before it begins to fall. According to Siacci's theorem, the acceleration a of P can be expressed as : \mathbf{a} = -\frac{\kappa v^2r}{p} \mathbf{e}_r + \frac{(h^2)'}{2p^2} \mathbf{e}_t = S_r \mathbf{e}_r + S_t \mathbf{e}_t . where the prime denotes differentiation with respect to the arc length s, and κ is the curvature function of the curve C. Let a particle P of mass m move in a two-dimensional Euclidean space (planar motion). The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. Let there be m variables that govern the forward- kinematics equation, i.e. the position function. The position of the particle is \mathbf{r} =\mathbf{r}\left ( r(t),\theta(t) \right ) = r \mathbf{\hat{e}}_r where and are the polar unit vectors. right|thumb|200px|The forward kinematics equations define the trajectory of the end-effector of a PUMA robot reaching for parts. Using equation [4] in the set above, we have: s= \frac{v^2 - u^2}{-2g}. Only Domingo de Soto, a Spanish theologian, in his commentary on Aristotle's Physics published in 1545, after defining ""uniform difform"" motion (which is uniformly accelerated motion) – the word velocity was not used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. Let p_0 = p(x_0) give the initial position of the system, and :p_1 = p(x_0 + \Delta x) be the goal position of the system. In kinematics, the acceleration of a particle moving along a curve in space is the time derivative of its velocity. Differentiating with respect to time again obtains the acceleration \mathbf{a} =\left ( \frac{d^2 r}{dt^2} - r\omega^2\right )\mathbf{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{dr}{dt} \right )\mathbf{\hat{e}}_\theta which breaks into the radial acceleration , centripetal acceleration , Coriolis acceleration , and angular acceleration . The initial position, initial velocity, and acceleration vectors need not be collinear, and take an almost identical form. The solution to the equation of motion, with specified initial values, describes the system for all times after . Thus, let C be a space curve traced out by P and s is the arc length of C corresponding to time t. From the instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions; \mathbf{v} = \frac{d \mathbf{r}}{d t} \,, \quad \mathbf{a} = \frac{d \mathbf{v}}{d t} = \frac{d^2 \mathbf{r}}{d t^2} Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. ",2,273,"""-6.8""",362880,2.8108,A
+" Use the function described in Example 4.3, $x_{n+1}=\alpha x_n\left(1-x_n^2\right)$ where $\alpha=2.5$. Consider two starting values of $x_1$ that are similar, 0.9000000 and 0.9000001 . Make a plot of $x_n$ versus $n$ for the two starting values and determine the lowest value of $n$ for which the two values diverge by more than $30 \%$.","In mathematics, the reciprocal difference of a finite sequence of numbers (x_0, x_1, ..., x_n) on a function f(x) is defined inductively by the following formulas: :\rho_1(x_1, x_2) = \frac{x_1 - x_2}{f(x_1) - f(x_2)} :\rho_2(x_1, x_2, x_3) = \frac{x_1 - x_3}{\rho_1(x_1, x_2) - \rho_1(x_2, x_3)} + f(x_2) :\rho_n(x_1,x_2,\ldots,x_{n+1})=\frac{x_1-x_{n+1}}{\rho_{n-1}(x_1,x_2,\ldots,x_{n})-\rho_{n-1}(x_2,x_3,\ldots,x_{n+1})}+\rho_{n-2}(x_2,\ldots,x_{n}) ==See also== *Divided differences ==References== * * Category:Finite differences Finally, the sequence :(d_k) = \left\\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots, \frac{1}{k + 1}, \ldots \right\\} converges sublinearly and logarithmically. thumb|alt=Plot showing the different rates of convergence for the sequences ak, bk, ck and dk.|Linear, linear, superlinear (quadratic), and sublinear rates of convergence|600px|center ==Convergence speed for discretization methods== A similar situation exists for discretization methods designed to approximate a function y = f(x), which might be an integral being approximated by numerical quadrature, or the solution of an ordinary differential equation (see example below). More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative argument, which reaches to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap. In particular, iterating a point x0 in [0, 1] gives rise to a sequence x_n: :x_{n+1} = f_\mu(x_n) = \begin{cases} \mu x_n & \mathrm{for}~~ x_n < \frac{1}{2} \\\ \mu (1-x_n) & \mathrm{for}~~ \frac{1}{2} \le x_n \end{cases} where μ is a positive real constant. In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. From top to bottom. 10, 100, 1000, 10000 points. ===Additive recurrence=== For any irrational \alpha, the sequence : s_n = \\{s_0 + n\alpha\\} has discrepancy tending to 1/N. Note that the sequence can be defined recursively by : s_{n+1} = (s_n + \alpha)\bmod 1 \;. Convergence with order * q = 1 is called linear convergence if \mu \in (0, 1), and the sequence is said to converge Q-linearly to L. * q = 2 is called quadratic convergence. * q = 3 is called cubic convergence. * etc. ==== Order estimation ==== A practical method to calculate the order of convergence for a sequence is to calculate the following sequence, which converges to q: :q \approx \frac{\log \left|\frac{x_{k+1} - x_k}{x_k - x_{k-1}}\right|}{\log \left|\frac{x_k - x_{k-1}}{x_{k-1} - x_{k-2}}\right|}. ==== Q-convergence definitions ==== In addition to the previously defined Q-linear convergence, a few other Q-convergence definitions exist. The difference of the approximations, 2\,\text{cm}, is in error by 100% of the magnitude of the difference of the true values, 1\,\text{cm}. Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of an equidistributed sequence. thumb|right|Graph of tent map function 300px|thumb|right|Example of iterating the initial condition x0 = 0.4 over the tent map with μ = 1.9. The value of c with lowest discrepancy is the fractional part of the golden ratio: : c = \frac{\sqrt{5}-1}{2} = \varphi - 1 \approx 0.618034. We can solve this equation using the Forward Euler scheme for numerical discretization: : \frac{y_{n+1} - y_n}{h} = -\kappa y_{n}, which generates the sequence : y_{n+1} = y_n(1 - h\kappa). thumb|320px|right|Standard logistic function where L=1,k=1,x_0=0 A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with equation f(x) = \frac{L}{1 + e^{-k(x-x_0)}}, where For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is obtained, with the graph of f approaching L as x approaches +\infty and approaching zero as x approaches -\infty. If the points are chosen as elements of a low-discrepancy sequence, this is the quasi-Monte Carlo method. Category:Numerical analysis Convergence This means running a Monte-Carlo analysis with e.g. s=20 variables and N=1000 points from a low-discrepancy sequence generator may offer only a very minor accuracy improvement. ===Random numbers=== Sequences of quasirandom numbers can be generated from random numbers by imposing a negative correlation on those random numbers. The important parameter here for the convergence speed to y = f(x) is the grid spacing h, inversely proportional to the number of grid points, i.e. the number of points in the sequence required to reach a given value of x. In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x1, ..., xN has a low discrepancy. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. The relative errors of `x` from 1.000000000000001 and of `y` from 1.000000000000002 are both below 10^{-15} = 0.0000000000001\%, and the floating-point subtraction `y - x` is computed exactly by the Sterbenz lemma. This sequence converges with order 1 according to the convention for discretization methods. ",1.11,2,"""-2.0""",8,30,E
+A gun fires a projectile of mass $10 \mathrm{~kg}$ of the type to which the curves of Figure 2-3 apply. The muzzle velocity is $140 \mathrm{~m} / \mathrm{s}$. Through what angle must the barrel be elevated to hit a target on the same horizontal plane as the gun and $1000 \mathrm{~m}$ away? Compare the results with those for the case of no retardation.,"The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. In ballistics, the elevation is the angle between the horizontal plane and the axial direction of the barrel of a gun, mortar or heavy artillery. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. With advances in the 21st century, it has become easy to determine how much elevation a gun needed to hit a target. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. The projectile trajectory is affected by atmospheric conditions, the velocity of the projectile, the difference in altitude between the firer and the target, and other factors. The second solution is the useful one for determining the range of the projectile. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. The laser rangefinder and computer-based FCS make guns highly accurate. ==Superelevation== When firing a missile such as a MANPADS at an aircraft target, superelevation is an additional angle of elevation above the angle sighted on which corrects for the effect of gravity on the missile. ==See also== *Altitude (astronomy) *Pitching moment ==References== * Gunnery Instructions, U.S. Navy (1913), Register No. 4090 * Gunnery And Explosives For Artillery Officers (1911) * Fire Control Fundamentals, NAVPERS 91900 (1953), Part C: The Projectile in Flight - Exterior Ballistics * FM 6-40, Tactics, Techniques, and Procedures for Field Artillery Manual Cannon Gunnery (23 April 1996), Chapter 3 - Ballistics; Marine Corps Warfighting Publication No. 3-1.6.19 * FM 23-91, Mortar Gunnery (1 March 2000), Chapter 2 Fundamentals of Mortar Gunnery * Fundamentals of Naval Weapons Systems: Chapter 19 (Weapons and Systems Engineering Department United States Naval Academy) * Naval Ordnance and Gunnery (Vol.1 - Naval Ordnance) NAVPERS 10797-A (1957) * Naval Ordnance and Gunnery (Vol.2 - Fire Control) NAVPERS 10798-A (1957) * Naval Ordnance and Gunnery Category:Ballistics Category:Angle Originally, elevation was a linear measure of how high the gunners had to physically lift the muzzle of a gun up from the gun carriage to compensate for projectile drop and hit targets at a certain distance. ==Until WWI== Though early 20th-century firearms were relatively easy to fire, artillery was not. thumb|right|upright=1.35|Cutaway view of M1128 round The M1128 ""Insensitive Munition High Explosive Base Burn Projectile"" is a 155 mm boosted artillery round designed to achieve a maximum range of 30–40 km. The surface of the projectile also must be considered: a smooth projectile will face less air resistance than a rough-surfaced one, and irregularities on the surface of a projectile may change its trajectory if they create more drag on one side of the projectile than on the other. There are two dimensions in aiming a weapon: * In the horizontal plane (azimuth); and * In the vertical plane (elevation), which is governed by the distance (range) to the target and the energy of the propelling charge. Projectile and propellant gases act on barrel along barrel centerline A. Forces are resisted by shooter contact with gun at grips and stock B. Height difference between barrel centerline and average point of contact is height C. Forces A and B operating over moment arm / height C create torque or moment D, which rotates the firearm's muzzle up as illustrated at E. Muzzle rise, muzzle flip or muzzle climb refers to the tendency of a firearm's or airgun's muzzle (front end of the barrel) to rise up after firing.Recoil management: how you hold makes all the difference, Guns Magazine, Oct 2006 by Dave Anderson It more specifically refers to the seemingly unpredictable ""jump"" of the firearm's muzzle, caused by combined recoil from multiple shots being fired in quick succession. thumb|upright=1.35|Indirect fire trajectories for rockets, howitzers, field guns and mortars Indirect fire is aiming and firing a projectile without relying on a direct line of sight between the gun and its target, as in the case of direct fire. Originally ""zero"", meaning 6400 mils, 360 degrees or their equivalent, was set at whatever the direction the oriented gun was pointed. File:AKM and MP5K.JPEG|An AKM assault rifle asymmetric slant cut muzzle fixture designed to counteract muzzle rise (and muzzle climb) during (automatic) firing. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. ",47,2,"""17.4""",-131.1,1.61,C
+A spacecraft is placed in orbit $200 \mathrm{~km}$ above Earth in a circular orbit. Calculate the minimum escape speed from Earth. ,"For the Earth at perihelion, the value is: : \sqrt {1.327 \times 10^{20} ~\text{m}^3 \text{s}^{-2} \cdot \left({2 \over 1.471 \times 10^{11} ~\text{m}} - {1 \over 1.496 \times 10^{11} ~\text{m}}\right)} \approx 30,300 ~\text{m}/\text{s} which is slightly faster than Earth's average orbital speed of , as expected from Kepler's 2nd Law. == Planets == The closer an object is to the Sun the faster it needs to move to maintain the orbit. This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac{Mm}{r_1 + r_2} Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result. ==Flight path angle== The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Orbital velocities of the Planets Planet Orbital velocity Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Halley's Comet on an eccentric orbit that reaches beyond Neptune will be moving 54.6 km/s when from the Sun, 41.5 km/s when 1 AU from the Sun (passing Earth's orbit), and roughly 1 km/s at aphelion from the Sun., where r is the distance from the Sun, and a is the major semi-axis. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. Objects passing Earth's orbit going faster than 42.1 km/s have achieved escape velocity and will be ejected from the Solar System if not slowed down by a gravitational interaction with a planet. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. When one of the bodies is not of considerably lesser mass see: Gravitational two-body problem So, when one of the masses is almost negligible compared to the other mass, as the case for Earth and Sun, one can approximate the orbit velocity v_o as: :v_o \approx \sqrt{\frac{GM}{r}} or assuming equal to the radius of the orbit :v_o \approx \frac{v_e}{\sqrt{2}} Where is the (greater) mass around which this negligible mass or body is orbiting, and is the escape velocity. This can be used to obtain a more accurate estimate of the average orbital speed: : v_o = \frac{2\pi a}{T}\left[1-\frac{1}{4}e^2-\frac{3}{64}e^4 -\frac{5}{256}e^6 -\frac{175}{16384}e^8 - \cdots \right] The mean orbital speed decreases with eccentricity. ==Instantaneous orbital speed== For the instantaneous orbital speed of a body at any given point in its trajectory, both the mean distance and the instantaneous distance are taken into account: : v = \sqrt {\mu \left({2 \over r} - {1 \over a}\right)} where is the standard gravitational parameter of the orbited body, is the distance at which the speed is to be calculated, and is the length of the semi-major axis of the elliptical orbit. Being closest to the Sun and having the most eccentric orbit, Mercury's orbital speed varies from about 59 km/s at perihelion to 39 km/s at aphelion. Under standard assumptions of the conservation of angular momentum the flight path angle \phi satisfies the equation: :h\, = r\, v\, \cos \phi where: * h\, is the specific relative angular momentum of the orbit, * v\, is the orbital speed of the orbiting body, * r\, is the radial distance of the orbiting body from the central body, * \phi \, is the flight path angle \psi is the angle between the orbital velocity vector and the semi-major axis. u is the local true anomaly. \phi = u + \frac{\pi}{2} - \psi, therefore, :\cos \phi = \sin(\psi - u) = \sin\psi\cos u - \cos\psi\sin u = \frac{1 + e\cos u}{\sqrt{1 + e^2 + 2e\cos u}} :\tan \phi = \frac{e\sin u}{1 + e\cos u} where e is the eccentricity. Velocities of better-known numbered objects that have perihelion close to the Sun Object Velocity at perihelion Velocity at 1 AU (passing Earth's orbit) 322P/SOHO 181 km/s @ 0.0537 AU 37.7 km/s 96P/Machholz 118 km/s @ 0.124 AU 38.5 km/s 3200 Phaethon 109 km/s @ 0.140 AU 32.7 km/s 1566 Icarus 93.1 km/s @ 0.187 AU 30.9 km/s 66391 Moshup 86.5 km/s @ 0.200 AU 19.8 km/s 1P/Halley 54.6 km/s @ 0.586 AU 41.5 km/s ==See also== *Escape velocity *Delta-v budget *Hohmann transfer orbit *Bi-elliptic transfer ==References== Category:Orbits hu:Kozmikus sebességek#Szökési sebességek Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. This is an approximation that only holds true when the orbiting body is of considerably lesser mass than the central one, and eccentricity is close to zero. The maximum (instantaneous) orbital speed occurs at periapsis (perigee, perihelion, etc.), while the minimum speed for objects in closed orbits occurs at apoapsis (apogee, aphelion, etc.). Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). The empty focus (\mathbf{F2} = \left(f_x,f_y\right)) can be found by first determining the Eccentricity vector: :\mathbf{e} = \frac{\mathbf{r}}{|\mathbf{r}|} - \frac{\mathbf{v}\times \mathbf{h}}{\mu} Where \mathbf{h} is the specific angular momentum of the orbiting body: :\mathbf{h} = \mathbf{r} \times \mathbf{v} Then :\mathbf{F2} = -2a\mathbf{e} ====Using XY Coordinates==== This can be done in cartesian coordinates using the following procedure: The general equation of an ellipse under the assumptions above is: : \sqrt{ \left(f_x - x\right)^2 + \left(f_y - y\right)^2} + \sqrt{ x^2 + y^2 } = 2a \qquad\mid z=0 Given: :r_x, r_y \quad the initial position coordinates :v_x, v_y \quad the initial velocity coordinates and :\mu = Gm_1 \quad the gravitational parameter Then: :h = r_x v_y - r_y v_x \quad specific angular momentum :r = \sqrt{r_x^2 + r_y^2} \quad initial distance from F1 (at the origin) :a = \frac{\mu r}{2\mu - r \left(v_x^2 + v_y^2 \right)} \quad the semi-major axis length :e_x = \frac{r_x}{r} - \frac{h v_y}{\mu} \quad the Eccentricity vector coordinates :e_y = \frac{r_y}{r} + \frac{h v_x}{\mu} \quad Finally, the empty focus coordinates :f_x = - 2 a e_x \quad :f_y = - 2 a e_y \quad Now the result values fx, fy and a can be applied to the general ellipse equation above. == Orbital parameters == The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. See radial hyperbolic trajectory * If the total energy is zero, (Ek = Ep): the orbit is a parabola with focus at the other body. For this case it is convenient to use the following assumptions which differ somewhat from the standard assumptions above: :# The central body's position is at the origin and is the primary focus (\mathbf{F1}) of the ellipse (alternatively, the center of mass may be used instead if the orbiting body has a significant mass) :# The central body's mass (m1) is known :# The orbiting body's initial position(\mathbf{r}) and velocity(\mathbf{v}) are known :# The ellipse lies within the XY-plane The fourth assumption can be made without loss of generality because any three points (or vectors) must lie within a common plane. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, and tundra orbit. ==Velocity== Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, the orbital speed (v\,) of one body traveling along an elliptic orbit can be computed from the vis-viva equation as: :v = \sqrt{\mu\left({2\over{r}} - {1\over{a}}\right)} where: *\mu\, is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other. *r\, is the distance between the orbiting body and center of mass. *a\,\\! is the length of the semi-major axis. ",-383,3.23,"""4.09""",273,7.654,B
+"Find the value of the integral $\int_S(\nabla \times \mathbf{A}) \cdot d \mathbf{a}$ if the vector $\mathbf{A}=y \mathbf{i}+z \mathbf{j}+x \mathbf{k}$ and $S$ is the surface defined by the paraboloid $z=1-x^2-y^2$, where $z \geq 0$.","Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f_x, f_y and f_z. == Theorems involving surface integrals == Various useful results for surface integrals can be derived using differential geometry and vector calculus, such as the divergence theorem, and its generalization, Stokes' theorem. == Dependence on parametrization == Let us notice that we defined the surface integral by using a parametrization of the surface S. For integrals of scalar fields, the answer to this question is simple; the value of the surface integral will be the same no matter what parametrization one uses. This formula defines the integral on the left (note the dot and the vector notation for the surface element). Then, the surface integral of f on S is given by :\iint_D \left[ f_{z} ( \mathbf{r} (s,t)) \frac{\partial(x,y)}{\partial(s,t)} + f_{x} ( \mathbf{r} (s,t)) \frac{\partial(y,z)}{\partial(s,t)} + f_{y} ( \mathbf{r} (s,t))\frac{\partial(z,x)}{\partial(s,t)} \right]\, \mathrm ds\, \mathrm dt where :{\partial \mathbf{r} \over \partial s}\times {\partial \mathbf{r} \over \partial t}=\left(\frac{\partial(y,z)}{\partial(s,t)}, \frac{\partial(z,x)}{\partial(s,t)}, \frac{\partial(x,y)}{\partial(s,t)}\right) is the surface element normal to S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. We find the formula :\begin{align} \iint_S {\mathbf v}\cdot\mathrm d{\mathbf {s}} &= \iint_S \left({\mathbf v}\cdot {\mathbf n}\right)\,\mathrm ds\\\ &{}= \iint_T \left({\mathbf v}(\mathbf{r}(s, t)) \cdot {\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t} \over \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\|}\right) \left\|\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right\| \mathrm ds\, \mathrm dt\\\ &{}=\iint_T {\mathbf v}(\mathbf{r}(s, t))\cdot \left(\frac{\partial \mathbf{r}}{\partial s}\times \frac{\partial \mathbf{r}}{\partial t}\right) \mathrm ds\, \mathrm dt. \end{align} The cross product on the right-hand side of this expression is a (not necessarily unital) surface normal determined by the parametrisation. In other words, we have to integrate v with respect to the vector surface element \mathrm{d}\mathbf s = {\mathbf n} \mathrm{d}s, which is the vector normal to S at the given point, whose magnitude is \mathrm{d}s = \|\mathrm{d}{\mathbf s}\|. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. Based on this reasoning, to find the flux, we need to take the dot product of v with the unit surface normal n to S at each point, which will give us a scalar field, and integrate the obtained field as above. Suppose now that it is desired to integrate only the normal component of the vector field over the surface, the result being a scalar, usually called the flux passing through the surface. Then, the surface integral is given by : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \left\|{\partial \mathbf{r} \over \partial s} \times {\partial \mathbf{r} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of , and is known as the surface element (which would, for example, yield a smaller value near the poles of a sphere. where the lines of longitude converge more dramatically, and latitudinal coordinates are more compactly spaced). A natural question is then whether the definition of the surface integral depends on the chosen parametrization. Such a surface is called non-orientable, and on this kind of surface, one cannot talk about integrating vector fields. == See also == * Divergence theorem * Stokes' theorem * Line integral * Volume element * Volume integral * Cartesian coordinate system * Volume and surface area elements in spherical coordinate systems * Volume and surface area elements in cylindrical coordinate systems * Holstein–Herring method ==References== == External links == * Surface Integral — from MathWorld * Surface Integral — Theory and exercises Category:Multivariable calculus Category:Area Category:Surfaces If a vector field \mathbf{F}(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z)) is defined and has continuous first order partial derivatives in a region containing \Sigma, then \iint_\Sigma ( abla \times \mathbf{F}) \cdot \mathrm{d} \mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{\Gamma}. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . That is, A' below is also a vector potential of v; \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ abla_y \times \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d^3\mathbf{y}. By analogy with Biot-Savart's law, the following \boldsymbol{A}(\textbf{x}) is also qualify as a vector potential for v. :\boldsymbol{A}(\textbf{x}) =\int_\Omega \frac{\boldsymbol{v}(\boldsymbol{y}) \times (\boldsymbol{x} - \boldsymbol{y})}{4 \pi |\boldsymbol{x} - \boldsymbol{y}|^3} d^3 \boldsymbol{y} Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law. The surface integral can also be expressed in the equivalent form : \iint_S f \,\mathrm dS = \iint_T f(\mathbf{r}(s, t)) \sqrt{g} \, \mathrm ds\, \mathrm dt where is the determinant of the first fundamental form of the surface mapping . If, however, the normals for these parametrizations point in opposite directions, the value of the surface integral obtained using one parametrization is the negative of the one obtained via the other parametrization. The circulation of a vector field around a closed curve is the line integral: \Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}. ",1.39,-3.141592,"""0.123""",4.68,1.2,B
+A skier weighing $90 \mathrm{~kg}$ starts from rest down a hill inclined at $17^{\circ}$. He skis $100 \mathrm{~m}$ down the hill and then coasts for $70 \mathrm{~m}$ along level snow until he stops. Find the coefficient of kinetic friction between the skis and the snow. ,"Kinetic (or dynamic) friction occurs when the ski is moving over the snow. One type of friction acting on the skier is the kinetic friction between the skis and snow. The coefficient of kinetic friction, \mu_\mathrm{k}, is less than the coefficient of static friction for both ice and snow. The motion of a skier is determined by the physical principles of the conservation of energy and the frictional forces acting on the body. The second type of frictional force acting on a skier is drag. The kinetic friction can be reduced by applying wax to the bottom of the skis which reduces the coefficient of friction. The necessary speed required to keep the skier upright varies by the weight of the barefooter and can be approximated by the following formula: (W / 10) + 20, where W is the skier's weight in pounds and the result is in miles per hour. The Physics of Skiing. The force required for sliding on snow is the product of the coefficient of kinetic friction and the normal force: F_{k} = \mu_\mathrm{k} F_{n}\,. However, the heat generated by friction can be lost by conduction to a cold ski, thereby diminishing the production of the melt layer. The ability of a ski or other runner to slide over snow depends on both the properties of the snow and the ski to result in an optimum amount of lubrication from melting the snow by friction with the ski—too little and the ski interacts with solid snow crystals, too much and capillary attraction of meltwater retards the ski. ===Friction=== Before a ski can slide, it must overcome the maximum value static friction, F_{max} = \mu_\mathrm{s} F_{n}\,, for the ski/snow contact, where \mu_\mathrm{s} is the coefficient of static friction and F_{n}\, is the normal force of the ski on snow. *Moisture content: The percentage of mass that is liquid water and may create suction friction with the base of the ski as it slides. A skier with skis pointed perpendicular to the fall line, across the hill instead of down it, will accelerate more slowly. The physics of skiing refers to the analysis of the forces acting on a person while skiing. right|thumb|300x300px|The texture of this top layer dependent on the weather history. Kuzmin and Fuss suggest that the most favorable combination of ski base material properties to minimize ski sliding friction on snow include: increased hardness and lowered thermal conductivity of the base material to promote meltwater generation for lubrication, wear resistance in cold snow, and hydrophobicity to minimize capillary suction. The shape and construction material of a ski can also greatly impact the forces acting on a skier.D. A. Lind and S. P. Sanders. Typically, a sliding ski melts a thin and transitory film of lubricating layer of water, caused by the heat of friction between the ski and the snow in its passing. Additionally, the skier can use the same techniques to turn the ski away from the direction of movement, generating skidding forces between the skis and snow which further slow the descent. Wax is adjusted for hardness to minimize sliding friction as a function of snow properties, which include the effects of: *Age: Reflects the metamorphism of snow crystals that are sharp and well- defined, when new, but with aging become broken or truncated with wind action or rounded into ice granules with freeze-thaw, all of which affects a ski's coefficient of friction. The material of Mr. Snow is claimed to have very good sliding capacities, is predictable in all climates and does not harm the ski or sliding surface. In doing so, the snow resists passage of the stemmed ski, creating a force that retards downhill speed and sustains a turn in the opposite direction. Too little melting and sharp edges of snow crystals or too much suction impede the passage of the ski. ",+37,7.25,"""6.0""",0.4772,0.18,E
 "Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\operatorname{Sun}$ is $\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\beta<1$, show that the time the comet spends within the orbit of Earth is given by
 $$
 \sqrt{2(1-\beta)} \cdot(1+2 \beta) / 3 \pi \times 1 \text { year }
 $$
-If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?","209P/LINEAR is a periodic comet with an orbital period of 5.1 years. 71P/Clark is a periodic comet in the Solar System with an orbital period of 5.5 years. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. Prediscovery images of the comet, dating back to December 2003, were found during 2009. 209P/LINEAR came to perihelion (closest approach to the Sun) on 6 May 2014. The 2014 Earth approach was the 9th closest known comet approach to Earth. 170P/Christensen is a periodic comet in the Solar System. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 164P/Christensen is a periodic comet in the Solar System. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 164P/Christensen – Seiichi Yoshida @ aerith.net * Elements and Ephemeris for 164P/Christensen – Minor Planet Center Category:Periodic comets 0164 164P 20041221 The nucleus of the comet has a radius of 0.68 ± 0.04 kilometers, assuming a geometric albedo of 0.04, based on observations by Hubble Space Telescope, while observations by Keck indicate a radius of 1.305 km. ==See also== * List of numbered comets == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 71P/Clark – Seiichi Yoshida @ aerith.net Category:Periodic comets 0071 Category:Comets in 2011 Category:Comets in 2017 19730609 The radar imaging showed the comet nucleus is elongated and about 2.4 km by 3 km in size, later refined to 3.9 × 2.7 × 2.6 km. On 29 May 2014 the comet passed from Earth, but only brightened to about apparent magnitude 12. It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 All the trails from the comet from 1803 through 1924 were expected to intersect Earth's orbit during May 2014. The comet has extremely low activity for its size and is probably in the process of evolving into an extinct comet. == Observational history == The comet discovered on 3 February 2004 by Lincoln Near-Earth Asteroid Research (LINEAR) using a reflector. Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). The peak activity was expected to occur around 24 May 2014 7h UT when dust trails produced from past returns of the comet could pass from Earth. 2005 HC4 is the asteroid with the smallest known perihelion of any known object orbiting the Sun (except sungrazing comets). The close approach allowed the comet nucleus to be imaged by Arecibo, producing the most detailed radar image of a comet nucleus to that date. It was given the permanent number 209P on 12 December 2008 as it was the second observed appearance of the comet. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. The comet also had very low water production, mol/s, from an active area measuring just 0.007 km². 209P/LINEAR was recovered on 31 December 2018 at magnitude 19.2 by Hidetaka Sato. ==Associated meteor showers== Preliminary results by Esko Lyytinen and Peter Jenniskens, later confirmed by other researchers, predicted 209P/LINEAR might a big meteor shower which would come from the constellation Camelopardalis on the night of 23/24 May 2014. ", 0.01961,-191.2,54.7,76,21,D
-An automobile drag racer drives a car with acceleration $a$ and instantaneous velocity $v$. The tires (of radius $r_0$ ) are not slipping. For what initial velocity in the rotating system will the hockey puck appear to be subsequently motionless in the fixed system? ,"For a swept angle the change in is a vector at right angles to and of magnitude , which in turn means that the magnitude of the acceleration is given by a_c = v \frac{d\theta}{dt} = v\omega = \frac{v^2}{r} Centripetal acceleration for some values of radius and magnitude of velocity colspan=""2"" rowspan=""2"" 1 m/s 3.6 km/h 2.2 mph 2 m/s 7.2 km/h 4.5 mph 5 m/s 18 km/h 11 mph 10 m/s 36 km/h 22 mph 20 m/s 72 km/h 45 mph 50 m/s 180 km/h 110 mph 100 m/s 360 km/h 220 mph Slow walk Bicycle City car Aerobatics 10 cm 3.9 in Laboratory centrifuge 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 4.0 km/s2 410 g 25 km/s2 2500 g 100 km/s2 10000 g 20 cm 7.9 in 5.0 m/s2 0.51 g 20 m/s2 2.0 g 130 m/s2 13 g 500 m/s2 51 g 2.0 km/s2 200 g 13 km/s2 1300 g 50 km/s2 5100 g 50 cm 1.6 ft 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 800 m/s2 82 g 5.0 km/s2 510 g 20 km/s2 2000 g 1 m 3.3 ft Playground carousel 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 400 m/s2 41 g 2.5 km/s2 250 g 10 km/s2 1000 g 2 m 6.6 ft 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 200 m/s2 20 g 1.3 km/s2 130 g 5.0 km/s2 510 g 5 m 16 ft 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 80 m/s2 8.2 g 500 m/s2 51 g 2.0 km/s2 200 g 10 m 33 ft Roller-coaster vertical loop 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 20 m 66 ft 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 1.3 m/s2 0.13 g 5.0 m/s2 0.51 g 20 m/s2 2 g 130 m/s2 13 g 500 m/s2 51 g 50 m 160 ft 20 mm/s2 0.0020 g 80 mm/s2 0.0082 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 100 m 330 ft Freeway on-ramp 10 mm/s2 0.0010 g 40 mm/s2 0.0041 g 250 mm/s2 0.025 g 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 200 m 660 ft 5.0 mm/s2 0.00051 g 20 mm/s2 0.0020 g 130 m/s2 0.013 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 500 m 1600 ft 2.0 mm/s2 0.00020 g 8.0 mm/s2 0.00082 g 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 1 km 3300 ft High-speed railway 1.0 mm/s2 0.00010 g 4.0 mm/s2 0.00041 g 25 mm/s2 0.0025 g 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g ==Non-uniform== right|293 px|frameless In a non-uniform circular motion, an object is moving in a circular path with a varying speed. * The period of the motion is 2 seconds per turn. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion. ==Uniform circular motion== thumb|upright=0.82|Figure 1: Velocity and acceleration in uniform circular motion at angular rate ; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation. thumb|upright=1.14|right|Figure 2: The velocity vectors at time and time are moved from the orbit on the left to new positions where their tails coincide, on the right. Rotation with no velocity, r_e \Omega e 0 and v = 0, means that \text{slip} = \infty. ==Lateral slip== The lateral slip of a tire is the angle between the direction it is moving and the direction it is pointing. This root sum of squares of separate radial and tangential accelerations is only correct for circular motion; for general motion within a plane with polar coordinates (r, \theta), the Coriolis term a_c = 2 \left(\frac{dr}{dt}\right)\left(\frac{d\theta}{dt}\right) should be added to a_t, whereas radial acceleration then becomes a_r = \frac{-v^2}{r} + \frac{d^2 r}{dt^2}. ==See also== * Angular momentum * Equations of motion for circular motion * * Fictitious force * Geostationary orbit * Geosynchronous orbit * Pendulum (mathematics) * Reactive centrifugal force * Reciprocating motion * * Sling (weapon) ==References== ==External links== * Physclips: Mechanics with animations and video clips from the University of New South Wales * Circular Motion – a chapter from an online textbook * Circular Motion Lecture – a video lecture on CM * – an online textbook with different analysis for circular motion Category:Rotation Category:Classical mechanics Category:Motion (physics) The speed of the object traveling the circle is: v = \frac{2 \pi r}{T} = \omega r The angle swept out in a time is: \theta = 2 \pi \frac{t}{T} = \omega t The angular acceleration, , of the particle is: \alpha = \frac{d\omega}{dt} In the case of uniform circular motion, will be zero. Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. In a uniform circular motion, the total acceleration of an object in a circular path is equal to the radial acceleration. The acceleration due to change in the direction is: a_c = \frac{v^2}{r} = \omega^2 r The centripetal and centrifugal force can also be found using acceleration: F_c = \dot{p} \mathrel\overset{\dot{m} = 0}{=} ma_c = \frac{mv^2}{r} The vector relationships are shown in Figure 1. To find the total acceleration of an object in a non uniform circular, find the vector sum of the tangential acceleration and the radial acceleration. \sqrt{a_r^2 + a_t^2} = a Radial acceleration is still equal to \frac{v^2}{r}. Locked brakes, r_e \Omega = 0, means that \text{slip} = -1 = -100\% and sliding without rotating. Therefore, the speed of travel around the orbit is v = r \frac{d\theta}{dt} = r\omega , where the angular rate of rotation is . Because the radius of the circle is constant, the radial component of the velocity is zero. In a non-uniform circular motion, the net acceleration (a) is along the direction of , which is directed inside the circle but does not pass through its center (see figure). The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. With this notation, the velocity becomes: v = \dot{z} = \frac{d}{dt}\left(R e^{i\theta[t]}\right) = R \frac{d}{dt}\left(e^{i\theta[t]}\right) = R e^{i\theta(t)} \frac{d}{dt} \left(i \theta[t] \right) = iR\dot{\theta}(t) e^{i\theta(t)} = i\omega R e^{i\theta(t)} = i\omega z and the acceleration becomes: \begin{align} a &= \dot{v} = i\dot{\omega} z + i\omega\dot{z} = \left(i\dot{\omega} - \omega^2\right)z \\\ &= \left(i\dot{\omega} - \omega^2 \right) R e^{i\theta(t)} \\\ &= -\omega^2 R e^{i\theta(t)} + \dot{\omega} e^{i\frac{\pi}{2}} R e^{i\theta(t)} \, . \end{align} The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before. ====Velocity==== Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Thus, is a constant, and the velocity vector also rotates with constant magnitude , at the same angular rate . ====Relativistic circular motion==== In this case, the three- acceleration vector is perpendicular to the three-velocity vector, \mathbf{u} \cdot \mathbf{a} = 0. and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames, \alpha^2 = \gamma^4 a^2 + \gamma^6 \left(\mathbf{u} \cdot \mathbf{a}\right)^2, becomes the expression for circular motion, \alpha^2 = \gamma^4 a^2. or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for circular motion: \alpha = \gamma^2 \frac{v^2}{r}. ====Acceleration==== The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. As , the acceleration vector becomes perpendicular to , which means it points toward the center of the orbit in the circle on the left. The only acceleration responsible for keeping an object moving in a circle is the radial acceleration. The rotor is rigid if R is independent of time. Consequently, the acceleration is: \begin{align} \mathbf{a}(t) &= R \left( \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) + \omega \frac{d \hat\mathbf{u}_\theta}{dt} \right) \\\ &= R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) - \omega^2 R \hat\mathbf{u}_R(t) \,. \end{align} The centripetal acceleration is the radial component, which is directed radially inward: \mathbf{a}_R(t) = -\omega^2 R \hat\mathbf{u}_R(t) \, , while the tangential component changes the magnitude of the velocity: \mathbf{a}_\theta(t) = R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d R \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d \left|\mathbf{v}(t)\right|}{dt} \hat\mathbf{u}_\theta(t) \, . ====Using complex numbers==== Circular motion can be described using complex numbers. The equations of motion describe the movement of the center of mass of a body. ",0.0547,0.5,6.0,2.00,2.534324263,B
-" A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\circ} \mathrm{S}$. If the shells are fired at $37^{\circ}$ elevation with a speed of $800 \mathrm{~m} / \mathrm{s}$, by how much do the shells miss their target and in what direction? Ignore air resistance.","Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. The target range at the time of projectile impact can be estimated using Equation 1, which is illustrated in Figure 3. The Ballistic Trajectory Extended Range Munition (BTERM) was a failed program to develop a precision guided rocket-assisted 127 mm (5-inch) artillery shell for the U.S. Navy. The higher altitude readings were needed for firings of the coast defense mortars, which sent their shells on very high trajectories. In naval gunnery, when long-range guns became available, an enemy ship would move some distance after the shells were fired. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The impact point of a projectile is a function of many variables: :* Air temperature :* Air density :* Wind :* Range :* Earth rotation :* Projectile, fuze, weapon characteristics :* Muzzle velocity :* Propellant temperature :* Drift :* Parallax between the guns and the rangefinders and radar systems :* Elevation difference between target and artillery piece The firing tables provide data for an artillery piece firing under standardized conditions and the corrections required to determine the point of impact under actual conditions. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The M110 155mm Projectile is an artillery shell used by the U.S. Army and U.S. Marine Corps. Adjustments were usually made by observing and plotting the fall (splashes) of the shells fired and reporting by how much they were left or right in azimuth or over or under in range.FM 4-15, Ch. 4 ===Factors influencing corrections=== Corrections could be made for the following factors: # Variations in muzzle velocity (including the results of variations in temperature of powder) # Variations in atmospheric density # Variations in atmospheric temperature # Height of site (taking account of the level of the tide) # Variations in weight of projectile # Travel of the target during the time of the projectile's flight # Wind # Rotation of the earth (for long range guns) # Drift The uncorrected firing data, to which such corrections were applied, were those derived, for instance, from using a plotting board to track the position of an observed target (e.g., a ship) and the range and azimuth to that target from the guns of a battery. ===Implementing corrections=== ====Meteorological data==== Several of the common corrections depended on meteorological data. The second solution is the useful one for determining the range of the projectile. (Equation 1) \begin{array}{lcr} R_{TP}&=& R_{T} + \frac{dR_{T}}{dt} \cdot t_{TOF'}\\\ &=& R_{T} + \frac{dR_{T}}{dt} \cdot \left( t_{TOF}+t_{Delay}\right) \end{array} where :* R_{TP}\, is the range to the target at the time of projectile impact. The projectile's advantages in terms of speed and rate of fire make ranging shots possible. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. For example, a torpedo's time of flight is much longer than that of battleship's main gun projectile. The exact prediction of the target range at the time of projectile impact is difficult because it requires knowing the projectile time of flight, which is a function of the projected target position. Special devices, like the ""deflection board"" (for corrections in azimuth) or the ""range correction board"" (for corrections in range) were used to produce corrected firing data (described below).Coast Artillery Journal index at sill- www.army.mil The final stage (the red ""3"" in the diagram at right) had to do with using feedback from the battery's observers, who spotted the fall of the projectiles (over or under range, left or right in azimuth, or on target) and telephoned their data to the plotting room so that the aim of the guns could be corrected for future salvos.""Seacoast Artillery Firing,"" Coast Artillery Journal, Vol. 63, No. 4, October, 1925, pp. 375–391 ==Fire control timing== thumb|left|This example shows the relationship of steps in the fire control process playing out over time. The projectile in this case would have had a time of flight of ~40 seconds (based on the 16 inch guns of the Iowa class). During World War I the Germans created an exceptionally large cannon, the Paris Gun, which could fire a shell more than 80 miles (130 km). ",0.925,14.5115,2.81,0.245,260,E
+If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?","209P/LINEAR is a periodic comet with an orbital period of 5.1 years. 71P/Clark is a periodic comet in the Solar System with an orbital period of 5.5 years. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. Prediscovery images of the comet, dating back to December 2003, were found during 2009. 209P/LINEAR came to perihelion (closest approach to the Sun) on 6 May 2014. The 2014 Earth approach was the 9th closest known comet approach to Earth. 170P/Christensen is a periodic comet in the Solar System. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 164P/Christensen is a periodic comet in the Solar System. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 164P/Christensen – Seiichi Yoshida @ aerith.net * Elements and Ephemeris for 164P/Christensen – Minor Planet Center Category:Periodic comets 0164 164P 20041221 The nucleus of the comet has a radius of 0.68 ± 0.04 kilometers, assuming a geometric albedo of 0.04, based on observations by Hubble Space Telescope, while observations by Keck indicate a radius of 1.305 km. ==See also== * List of numbered comets == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 71P/Clark – Seiichi Yoshida @ aerith.net Category:Periodic comets 0071 Category:Comets in 2011 Category:Comets in 2017 19730609 The radar imaging showed the comet nucleus is elongated and about 2.4 km by 3 km in size, later refined to 3.9 × 2.7 × 2.6 km. On 29 May 2014 the comet passed from Earth, but only brightened to about apparent magnitude 12. It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 All the trails from the comet from 1803 through 1924 were expected to intersect Earth's orbit during May 2014. The comet has extremely low activity for its size and is probably in the process of evolving into an extinct comet. == Observational history == The comet discovered on 3 February 2004 by Lincoln Near-Earth Asteroid Research (LINEAR) using a reflector. Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). The peak activity was expected to occur around 24 May 2014 7h UT when dust trails produced from past returns of the comet could pass from Earth. 2005 HC4 is the asteroid with the smallest known perihelion of any known object orbiting the Sun (except sungrazing comets). The close approach allowed the comet nucleus to be imaged by Arecibo, producing the most detailed radar image of a comet nucleus to that date. It was given the permanent number 209P on 12 December 2008 as it was the second observed appearance of the comet. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. The comet also had very low water production, mol/s, from an active area measuring just 0.007 km². 209P/LINEAR was recovered on 31 December 2018 at magnitude 19.2 by Hidetaka Sato. ==Associated meteor showers== Preliminary results by Esko Lyytinen and Peter Jenniskens, later confirmed by other researchers, predicted 209P/LINEAR might a big meteor shower which would come from the constellation Camelopardalis on the night of 23/24 May 2014. ", 0.01961,-191.2,"""54.7""",76,21,D
+An automobile drag racer drives a car with acceleration $a$ and instantaneous velocity $v$. The tires (of radius $r_0$ ) are not slipping. For what initial velocity in the rotating system will the hockey puck appear to be subsequently motionless in the fixed system? ,"For a swept angle the change in is a vector at right angles to and of magnitude , which in turn means that the magnitude of the acceleration is given by a_c = v \frac{d\theta}{dt} = v\omega = \frac{v^2}{r} Centripetal acceleration for some values of radius and magnitude of velocity colspan=""2"" rowspan=""2"" 1 m/s 3.6 km/h 2.2 mph 2 m/s 7.2 km/h 4.5 mph 5 m/s 18 km/h 11 mph 10 m/s 36 km/h 22 mph 20 m/s 72 km/h 45 mph 50 m/s 180 km/h 110 mph 100 m/s 360 km/h 220 mph Slow walk Bicycle City car Aerobatics 10 cm 3.9 in Laboratory centrifuge 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 4.0 km/s2 410 g 25 km/s2 2500 g 100 km/s2 10000 g 20 cm 7.9 in 5.0 m/s2 0.51 g 20 m/s2 2.0 g 130 m/s2 13 g 500 m/s2 51 g 2.0 km/s2 200 g 13 km/s2 1300 g 50 km/s2 5100 g 50 cm 1.6 ft 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 800 m/s2 82 g 5.0 km/s2 510 g 20 km/s2 2000 g 1 m 3.3 ft Playground carousel 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 400 m/s2 41 g 2.5 km/s2 250 g 10 km/s2 1000 g 2 m 6.6 ft 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 200 m/s2 20 g 1.3 km/s2 130 g 5.0 km/s2 510 g 5 m 16 ft 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 80 m/s2 8.2 g 500 m/s2 51 g 2.0 km/s2 200 g 10 m 33 ft Roller-coaster vertical loop 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g 40 m/s2 4.1 g 250 m/s2 25 g 1.0 km/s2 100 g 20 m 66 ft 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 1.3 m/s2 0.13 g 5.0 m/s2 0.51 g 20 m/s2 2 g 130 m/s2 13 g 500 m/s2 51 g 50 m 160 ft 20 mm/s2 0.0020 g 80 mm/s2 0.0082 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 8.0 m/s2 0.82 g 50 m/s2 5.1 g 200 m/s2 20 g 100 m 330 ft Freeway on-ramp 10 mm/s2 0.0010 g 40 mm/s2 0.0041 g 250 mm/s2 0.025 g 1.0 m/s2 0.10 g 4.0 m/s2 0.41 g 25 m/s2 2.5 g 100 m/s2 10 g 200 m 660 ft 5.0 mm/s2 0.00051 g 20 mm/s2 0.0020 g 130 m/s2 0.013 g 500 mm/s2 0.051 g 2.0 m/s2 0.20 g 13 m/s2 1.3 g 50 m/s2 5.1 g 500 m 1600 ft 2.0 mm/s2 0.00020 g 8.0 mm/s2 0.00082 g 50 mm/s2 0.0051 g 200 mm/s2 0.020 g 800 mm/s2 0.082 g 5.0 m/s2 0.51 g 20 m/s2 2.0 g 1 km 3300 ft High-speed railway 1.0 mm/s2 0.00010 g 4.0 mm/s2 0.00041 g 25 mm/s2 0.0025 g 100 mm/s2 0.010 g 400 mm/s2 0.041 g 2.5 m/s2 0.25 g 10 m/s2 1.0 g ==Non-uniform== right|293 px|frameless In a non-uniform circular motion, an object is moving in a circular path with a varying speed. * The period of the motion is 2 seconds per turn. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion. ==Uniform circular motion== thumb|upright=0.82|Figure 1: Velocity and acceleration in uniform circular motion at angular rate ; the speed is constant, but the velocity is always tangent to the orbit; the acceleration has constant magnitude, but always points toward the center of rotation. thumb|upright=1.14|right|Figure 2: The velocity vectors at time and time are moved from the orbit on the left to new positions where their tails coincide, on the right. Rotation with no velocity, r_e \Omega e 0 and v = 0, means that \text{slip} = \infty. ==Lateral slip== The lateral slip of a tire is the angle between the direction it is moving and the direction it is pointing. This root sum of squares of separate radial and tangential accelerations is only correct for circular motion; for general motion within a plane with polar coordinates (r, \theta), the Coriolis term a_c = 2 \left(\frac{dr}{dt}\right)\left(\frac{d\theta}{dt}\right) should be added to a_t, whereas radial acceleration then becomes a_r = \frac{-v^2}{r} + \frac{d^2 r}{dt^2}. ==See also== * Angular momentum * Equations of motion for circular motion * * Fictitious force * Geostationary orbit * Geosynchronous orbit * Pendulum (mathematics) * Reactive centrifugal force * Reciprocating motion * * Sling (weapon) ==References== ==External links== * Physclips: Mechanics with animations and video clips from the University of New South Wales * Circular Motion – a chapter from an online textbook * Circular Motion Lecture – a video lecture on CM * – an online textbook with different analysis for circular motion Category:Rotation Category:Classical mechanics Category:Motion (physics) The speed of the object traveling the circle is: v = \frac{2 \pi r}{T} = \omega r The angle swept out in a time is: \theta = 2 \pi \frac{t}{T} = \omega t The angular acceleration, , of the particle is: \alpha = \frac{d\omega}{dt} In the case of uniform circular motion, will be zero. Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. In a uniform circular motion, the total acceleration of an object in a circular path is equal to the radial acceleration. The acceleration due to change in the direction is: a_c = \frac{v^2}{r} = \omega^2 r The centripetal and centrifugal force can also be found using acceleration: F_c = \dot{p} \mathrel\overset{\dot{m} = 0}{=} ma_c = \frac{mv^2}{r} The vector relationships are shown in Figure 1. To find the total acceleration of an object in a non uniform circular, find the vector sum of the tangential acceleration and the radial acceleration. \sqrt{a_r^2 + a_t^2} = a Radial acceleration is still equal to \frac{v^2}{r}. Locked brakes, r_e \Omega = 0, means that \text{slip} = -1 = -100\% and sliding without rotating. Therefore, the speed of travel around the orbit is v = r \frac{d\theta}{dt} = r\omega , where the angular rate of rotation is . Because the radius of the circle is constant, the radial component of the velocity is zero. In a non-uniform circular motion, the net acceleration (a) is along the direction of , which is directed inside the circle but does not pass through its center (see figure). The acceleration points radially inwards (centripetally) and is perpendicular to the velocity. With this notation, the velocity becomes: v = \dot{z} = \frac{d}{dt}\left(R e^{i\theta[t]}\right) = R \frac{d}{dt}\left(e^{i\theta[t]}\right) = R e^{i\theta(t)} \frac{d}{dt} \left(i \theta[t] \right) = iR\dot{\theta}(t) e^{i\theta(t)} = i\omega R e^{i\theta(t)} = i\omega z and the acceleration becomes: \begin{align} a &= \dot{v} = i\dot{\omega} z + i\omega\dot{z} = \left(i\dot{\omega} - \omega^2\right)z \\\ &= \left(i\dot{\omega} - \omega^2 \right) R e^{i\theta(t)} \\\ &= -\omega^2 R e^{i\theta(t)} + \dot{\omega} e^{i\frac{\pi}{2}} R e^{i\theta(t)} \, . \end{align} The first term is opposite in direction to the displacement vector and the second is perpendicular to it, just like the earlier results shown before. ====Velocity==== Figure 1 illustrates velocity and acceleration vectors for uniform motion at four different points in the orbit. Thus, is a constant, and the velocity vector also rotates with constant magnitude , at the same angular rate . ====Relativistic circular motion==== In this case, the three- acceleration vector is perpendicular to the three-velocity vector, \mathbf{u} \cdot \mathbf{a} = 0. and the square of proper acceleration, expressed as a scalar invariant, the same in all reference frames, \alpha^2 = \gamma^4 a^2 + \gamma^6 \left(\mathbf{u} \cdot \mathbf{a}\right)^2, becomes the expression for circular motion, \alpha^2 = \gamma^4 a^2. or, taking the positive square root and using the three-acceleration, we arrive at the proper acceleration for circular motion: \alpha = \gamma^2 \frac{v^2}{r}. ====Acceleration==== The left-hand circle in Figure 2 is the orbit showing the velocity vectors at two adjacent times. As , the acceleration vector becomes perpendicular to , which means it points toward the center of the orbit in the circle on the left. The only acceleration responsible for keeping an object moving in a circle is the radial acceleration. The rotor is rigid if R is independent of time. Consequently, the acceleration is: \begin{align} \mathbf{a}(t) &= R \left( \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) + \omega \frac{d \hat\mathbf{u}_\theta}{dt} \right) \\\ &= R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) - \omega^2 R \hat\mathbf{u}_R(t) \,. \end{align} The centripetal acceleration is the radial component, which is directed radially inward: \mathbf{a}_R(t) = -\omega^2 R \hat\mathbf{u}_R(t) \, , while the tangential component changes the magnitude of the velocity: \mathbf{a}_\theta(t) = R \frac{d \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d R \omega}{dt} \hat\mathbf{u}_\theta(t) = \frac{d \left|\mathbf{v}(t)\right|}{dt} \hat\mathbf{u}_\theta(t) \, . ====Using complex numbers==== Circular motion can be described using complex numbers. The equations of motion describe the movement of the center of mass of a body. ",0.0547,0.5,"""6.0""",2.00,2.534324263,B
+" A British warship fires a projectile due south near the Falkland Islands during World War I at latitude $50^{\circ} \mathrm{S}$. If the shells are fired at $37^{\circ}$ elevation with a speed of $800 \mathrm{~m} / \mathrm{s}$, by how much do the shells miss their target and in what direction? Ignore air resistance.","Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. The target range at the time of projectile impact can be estimated using Equation 1, which is illustrated in Figure 3. The Ballistic Trajectory Extended Range Munition (BTERM) was a failed program to develop a precision guided rocket-assisted 127 mm (5-inch) artillery shell for the U.S. Navy. The higher altitude readings were needed for firings of the coast defense mortars, which sent their shells on very high trajectories. In naval gunnery, when long-range guns became available, an enemy ship would move some distance after the shells were fired. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. The impact point of a projectile is a function of many variables: :* Air temperature :* Air density :* Wind :* Range :* Earth rotation :* Projectile, fuze, weapon characteristics :* Muzzle velocity :* Propellant temperature :* Drift :* Parallax between the guns and the rangefinders and radar systems :* Elevation difference between target and artillery piece The firing tables provide data for an artillery piece firing under standardized conditions and the corrections required to determine the point of impact under actual conditions. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. The M110 155mm Projectile is an artillery shell used by the U.S. Army and U.S. Marine Corps. Adjustments were usually made by observing and plotting the fall (splashes) of the shells fired and reporting by how much they were left or right in azimuth or over or under in range.FM 4-15, Ch. 4 ===Factors influencing corrections=== Corrections could be made for the following factors: # Variations in muzzle velocity (including the results of variations in temperature of powder) # Variations in atmospheric density # Variations in atmospheric temperature # Height of site (taking account of the level of the tide) # Variations in weight of projectile # Travel of the target during the time of the projectile's flight # Wind # Rotation of the earth (for long range guns) # Drift The uncorrected firing data, to which such corrections were applied, were those derived, for instance, from using a plotting board to track the position of an observed target (e.g., a ship) and the range and azimuth to that target from the guns of a battery. ===Implementing corrections=== ====Meteorological data==== Several of the common corrections depended on meteorological data. The second solution is the useful one for determining the range of the projectile. (Equation 1) \begin{array}{lcr} R_{TP}&=& R_{T} + \frac{dR_{T}}{dt} \cdot t_{TOF'}\\\ &=& R_{T} + \frac{dR_{T}}{dt} \cdot \left( t_{TOF}+t_{Delay}\right) \end{array} where :* R_{TP}\, is the range to the target at the time of projectile impact. The projectile's advantages in terms of speed and rate of fire make ranging shots possible. The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. For example, a torpedo's time of flight is much longer than that of battleship's main gun projectile. The exact prediction of the target range at the time of projectile impact is difficult because it requires knowing the projectile time of flight, which is a function of the projected target position. Special devices, like the ""deflection board"" (for corrections in azimuth) or the ""range correction board"" (for corrections in range) were used to produce corrected firing data (described below).Coast Artillery Journal index at sill- www.army.mil The final stage (the red ""3"" in the diagram at right) had to do with using feedback from the battery's observers, who spotted the fall of the projectiles (over or under range, left or right in azimuth, or on target) and telephoned their data to the plotting room so that the aim of the guns could be corrected for future salvos.""Seacoast Artillery Firing,"" Coast Artillery Journal, Vol. 63, No. 4, October, 1925, pp. 375–391 ==Fire control timing== thumb|left|This example shows the relationship of steps in the fire control process playing out over time. The projectile in this case would have had a time of flight of ~40 seconds (based on the 16 inch guns of the Iowa class). During World War I the Germans created an exceptionally large cannon, the Paris Gun, which could fire a shell more than 80 miles (130 km). ",0.925,14.5115,"""2.81""",0.245,260,E
 "Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?
-","The four stars are each about half the mass of the Sun and are approximately 500 million years old. The system is unusual in how closely the four stars are orbiting each other; one pair has an orbital separation of at most .04 astronomical units and an orbital period of about two days, the other pair has a separation of at most .26 astronomical units and a period of about 55 days, and the two pairs are separated by 5.8 AU and have an orbital period of less than nine years. Kepler-84 is a Sun-like star 4,700 light-years from the Sun. The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. For celestial objects in general, the orbital period is determined by a 360° revolution of one body around its primary, e.g. Earth around the Sun. Periods in astronomy are expressed in units of time, usually hours, days, or years. ==Small body orbiting a central body== thumb|upright=1.2|The semi-major axis (a) and semi-minor axis (b) of an ellipse According to Kepler's Third Law, the orbital period T of two point masses orbiting each other in a circular or elliptic orbit is: :T = 2\pi\sqrt{\frac{a^3}{GM}} where: * a is the orbit's semi-major axis * G is the gravitational constant, * M is the mass of the more massive body. Kepler-444 (or KOI-3158, KIC 6278762, 2MASS J19190052+4138043, BD+41°3306) is a triple star system, estimated to be 11.2 billion years old (more than 80% of the age of the universe), approximately away from Earth in the constellation Lyra. At that conference, the star was known as KOI-3158. ==Characteristics== The star, Kepler-444, is approximately 11.2 billion years old, whereas the Sun is only 4.6 billion years old. The star is believed to have 2 M dwarfs in orbit around it with > the fainter companion 1.8 arc-seconds from the main star. ==Stellar system== The Kepler-444 system consists of the planet hosting primary and a pair of M-dwarf stars. Gliese 623 is a dim double star 25.6 light years from Earth in the constellation Hercules. BD−22 5866 is a quadruple-star system located 166 light years from Earth. The age is that of Kepler-444 A, an orange main sequence star of spectral type K0. Another (which is a background star with a probability 0.5%) is a yellow star of mass 0.855 on projected separations of 0.18″ or 0.26″ (213.6 AU). ==Planetary system== Kepler-84 is orbited by five known planets, four small gas giants and a Super- Earth. The rotation period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) may refer to its sidereal rotation period, i.e. the time that the object takes to complete a single revolution around its axis of rotation relative to the background stars, measured in sidereal time. This period differs from the sidereal period because both the orbital plane of the object and the plane of the ecliptic precess with respect to the fixed stars, so their intersection, the line of nodes, also precesses with respect to the fixed stars. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry. ===Synodic period=== One of the observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods, is their synodic period, which is the time between conjunctions. Kepler-444 is the > densest star with detected solar-like oscillations. There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics, particularly they must not be confused with other revolving periods like rotational periods. For example, Hyperion, a moon of Saturn, exhibits this behaviour, and its rotation period is described as chaotic. ==Rotation period of selected objects== Celestial objects Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Synodic rotation period (mean Solar day) Apparent rotational period viewed from Earth Sun* 25.379995 days (Carrington rotation) 35 days (high latitude) 25d 9h 7m 11.6s 35d ~28 days (equatorial) Mercury 58.6462 days 58d 15h 30m 30s 176 days Venus −243.0226 daysThis rotation is negative because the pole which points north of the invariable plane rotates in the opposite direction to most other planets. −243d 0h 33m −116.75 days Earth 0.99726968 daysReference adds about 1 ms to Earth's stellar day given in mean solar time to account for the length of Earth's mean solar day in excess of 86400 SI seconds. 0d 23h 56m 4.0910s 1.00 days (24h 00m 00s) Moon 27.321661 days (equal to sidereal orbital period due to spin-orbit locking, a sidereal lunar month) 27d 7h 43m 11.5s 29.530588 days (equal to synodic orbital period, due to spin-orbit locking, a synodic lunar month) none (due to spin-orbit locking) Mars 1.02595675 days 1d 0h 37m 22.663s 1.02749125 days Ceres 0.37809 days 0d 9h 4m 27.0s 0.37818 days Jupiter 0.41354 days(average) 0.4135344 days (deep interiorRotation period of the deep interior is that of the planet's magnetic field.) 0.41007 days (equatorial) 0.4136994 days (high latitude) 0d 9h 55m 30s 0d 9h 55m 29.37s 0d 9h 50m 30s 0d 9h 55m 43.63s (9 h 55 m 33 s) (average) Saturn days (average, deep interiorFound through examination of Saturn's C Ring) 0.44401 days (deep interior) 0.4264 days (equatorial) 0.44335 days (high latitude) 0d 10h 39m 22.4s 0d 10h 13m 59s 0d 10h 38m 25.4s (10 h 32 m 36 s) Uranus −0.71833 days −0d 17h 14m 24s (−17 h 14 m 23 s) Neptune 0.67125 days 0d 16h 6m 36s (16 h 6 m 36 s) Pluto −6.38718 days (synchronous with Charon) –6d 9h 17m 32s (–6d 9h 17m 0s) Haumea 0.1631458 ±0.0000042 days 0d 3h 56m 43.80 ±0.36s 0.1631461 ±0.0000042 days Makemake 0.9511083 ±0.0000042 days 22h 49m 35.76 ±0.36s 0.9511164 ±0.0000042 days Eris ~1.08 days 25h ~54m ~1.08 days * See Solar rotation for more detail. == See also == * Apparent retrograde motion * List of slow rotators (minor planets) * List of fast rotators (minor planets) * Retrograde motion * Rotational speed * Synodic day ==References== ==External links== * Note, the rotation periods for Mercury and Earth in this work may be inaccurate. We use asteroseismology > to directly measure a precise age of 11.2+/-1.0 Gyr for the host star, > indicating that Kepler-444 formed when the Universe was less than 20% of its > current age and making it the oldest known system of terrestrial-size > planets. AM Canum Venaticorum 17.146 minutes Beta Lyrae AB 12.9075 days Alpha Centauri AB 79.91 years Proxima Centauri – Alpha Centauri AB 500,000 years or more ==See also== * Geosynchronous orbit derivation * Rotation period – time that it takes to complete one revolution around its axis of rotation * Satellite revisit period * Sidereal time * Sidereal year * Opposition (astronomy) * List of periodic comets ==Notes== ==Bibliography== * == External links == Category:Time in astronomy Period Category:Kinematic properties Period Category:Time in astronomy The age of Kepler-444 not only > suggests that thick-disk stars were among the hosts to the first Galactic > planets, but may also help to pinpoint the beginning of the era of planet > formation."" ",2688,1.45,418.0,30,9,E
-"To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \mathrm{~m} / \mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?","The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The lunar GM = 4902.8001 km3/s2 from GRAIL analyses. Instead, after it arrested its velocity at an altitude of 3.4m it simply fell to the lunar surface. Since rocketry is used for descent and landing, the Moon's gravity necessitates the use of more fuel than is needed for asteroid landing. thumb|Maglev hover car A hover car is a personal vehicle that flies at a constant altitude of up to a few meters (yards) above the ground and used for personal transportation in the same way a modern automobile is employed. The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. Orbital speed around the Moon can, depending on altitude, exceed 1500 m/s. A lunar lander or Moon lander is a spacecraft designed to land on the surface of the Moon. thumb| Project Horizon Lunar Landing-and-Return Vehicle. In comparison, the much lighter (292 kg) Surveyor 3 landed on the Moon in 1967 using nearly 700 kg of fuel. All lunar landers require rocket engines for descent. The relatively high gravity (higher than all known asteroids, but lower than all solar system planets) and lack of lunar atmosphere negates the use of aerobraking, so a lander must use propulsion to decelerate and achieve a soft landing. ==History== The Luna program was a series of robotic impactors, flybys, orbiters, and landers flown by the Soviet Union between 1958 and 1976. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. The idea is that engine exhaust and lunar regolith can cause problems if they were to be kicked back from the surface to the spacecraft, and thus the engines cut off just before touchdown. Over the entire surface, the variation in gravitational acceleration is about 0.0253 m/s2 (1.6% of the acceleration due to gravity). This was in lieu of a 12 million- pound thrust superbooster required for a direct-ascent lunar flight, which could not possibly be developed in time for the 1966 deployment target. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. For example, the 900-kg Curiosity rover was landed on Mars by a craft having a mass (at the time of Mars atmospheric entry) of 2400 kg, of which only 390 kg was fuel. The landing gear was designed to withstand landings with engine cut-out at up to of height, though it was intended for descent engine shutdown to commence when one of the probes touched the surface. The design requirements for these landers depend on factors imposed by the payload, flight rate, propulsive requirements, and configuration constraints.Lunar Lander Stage Requirements Based on the Civil Needs Data Base (PDF). During this period the rockets would transport some 220 tonnes of useful cargo to the Moon. Higher speeds can be attained if the skydiver pulls in their limbs (see also freeflying). ",0.5,-191.2,273.0,20.2,117,C
-"In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\mathbf{u}_1$ and $\mathbf{u}_2=\alpha \mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\alpha$ is positive. ","The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. However, in the case of the proton, the evidence suggested three distinct concentrations of charge (quarks) and not one. ==Formula== The formula for the velocities after a one-dimensional collision is: \begin{align} v_a &= \frac{C_R m_b (u_b - u_a) + m_a u_a + m_b u_b} {m_a+m_b} \\\ v_b &= \frac{C_R m_a (u_a - u_b) + m_a u_a + m_b u_b} {m_a+m_b} \end{align} where *va is the final velocity of the first object after impact *vb is the final velocity of the second object after impact *ua is the initial velocity of the first object before impact *ub is the initial velocity of the second object before impact *ma is the mass of the first object *mb is the mass of the second object *CR is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a perfectly inelastic collision, see below. Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. These equations may be solved directly to find v_1,v_2 when u_1,u_2 are known: \begin{array}{ccc} v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\\\[.5em] v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2. \end{array} If both masses are the same, we have a trivial solution: \begin{align} v_{1} &= u_{2} \\\ v_{2} &= u_{1}. \end{align} This simply corresponds to the bodies exchanging their initial velocities to each other. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. A useful special case of elastic collision is when the two bodies have equal mass, in which case they will simply exchange their momenta. In this example, momentum of the system is conserved because there is no friction between the sliding bodies and the surface. m_a u_a + m_b u_b = \left( m_a + m_b \right) v where v is the final velocity, which is hence given by v=\frac{m_a u_a + m_b u_b}{m_a + m_b} Another perfectly inelastic collision|frame|center The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. Assuming no friction, this gives the velocity updates: \begin{align} \Delta \vec{v_{a}} &= \frac{J_{n}}{m_{a}} \vec{n} \\\ \Delta \vec{v_{b}} &= -\frac{J_{n}}{m_{b}} \vec{n} \end{align} ==Perfectly inelastic collision== A completely inelastic collision between equal masses|frame|center A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. The reduction of kinetic energy E_r is hence: E_r = \frac{1}{2}\frac{m_a m_b}{m_a + m_b}|u_a - u_b|^2 With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation). ==Partially inelastic collisions== Partially inelastic collisions are the most common form of collisions in the real world. The magnitudes of the velocities of the particles after the collision are: \begin{align} v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\\ v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}. \end{align} ===Two-dimensional collision with two moving objects=== The final x and y velocities components of the first ball can be calculated as: \begin{align} v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2}) \\\\[0.8em] v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}), \end{align} where and are the scalar sizes of the two original speeds of the objects, and are their masses, and are their movement angles, that is, v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1 (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi () is the contact angle. Once v_1 is determined, v_2 can be found by symmetry. ====Center of mass frame==== With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. This is why a neutron moderator (a medium which slows down fast neutrons, thereby turning them into thermal neutrons capable of sustaining a chain reaction) is a material full of atoms with light nuclei which do not easily absorb neutrons: the lightest nuclei have about the same mass as a neutron. ====Derivation of solution==== To derive the above equations for v_1,v_2, rearrange the kinetic energy and momentum equations: \begin{align} m_1(v_1^2-u_1^2) &= m_2(u_2^2-v_2^2) \\\ m_1(v_1-u_1) &= m_2(u_2-v_2) \end{align} Dividing each side of the top equation by each side of the bottom equation, and using \tfrac{a^2-b^2}{(a-b)} = a+b, gives: v_1+u_1=u_2+v_2 \quad\Rightarrow\quad v_1-v_2 = u_2-u_1. In an elastic collision these magnitudes do not change. In such a collision, kinetic energy is lost by bonding the two bodies together. An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. When considering energies, possible rotational energy before and/or after a collision may also play a role. ==Equations== ===One-dimensional Newtonian=== In an elastic collision, both momentum and kinetic energy are conserved. ",1.6,5.828427125,650000.0,420,0.6749,B
-"Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \mathrm{~m} / \mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \mathrm{~kg}$, including a pressurized tank of mass $10 \mathrm{~kg}$. The tank includes only $2 \mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \mathrm{~m} / \mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?","Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. The formula for terminal velocity (V)] appears on p. [52], equation (127). ""Escape Velocity"" is the fourth episode in the fourth season of the science fiction television series Battlestar Galactica. Substitution of equations (–) in equation () and solving for terminal velocity, V_t to yield the following expression In equation (), it is assumed that the object is denser than the fluid. So instead of m use the reduced mass m_r = m-\rho V in this and subsequent formulas. Dividing both sides by m gives \frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right). The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. Escape Velocity may also refer to: ==Books== * Escape Velocity (Doctor Who), a Doctor Who novel * Escape Velocity: Cyberculture at the End of the Century, a nonfiction book by Mark Dery * Escape Velocity, prequel to the Warlock series by Christopher Stasheff ==Video games== * Escape Velocity (video game) * Escape Velocity Override, its sequel * Escape Velocity Nova, the most recent title in the Escape Velocity franchise, along with an expandable card-driven board game based on it ==Music== * ""Escape Velocity"" (song), a 2010 song by The Chemical Brothers * Escape Velocity, an album by The Phenomenauts ==Film and television== * ""Escape Velocity"" (Battlestar Galactica), an episode of the TV show Battlestar Galactica * Escape Velocity (film), a 1998 Canadian thriller film ==See also== thumb|upright=1.3|A flight envelope diagram showing VS (Stall speed at 1G), VC (Corner/Maneuvering speed) and VD (Dive speed) thumb|upright=1.3|Vg diagram. Note the 1g stall speed, and the Maneuvering Speed (Corner Speed) for both positive and negative g. After a lengthy investigation, Velocity found and solved the cause of these stalls. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. Note that this is a different concept than design maneuvering speed. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). ",2.74,2.3,170.0,1.88,11,E
-"Use the $D_0$ value of $\mathrm{H}_2(4.478 \mathrm{eV})$ and the $D_0$ value of $\mathrm{H}_2^{+}(2.651 \mathrm{eV})$ to calculate the first ionization energy of $\mathrm{H}_2$ (that is, the energy needed to remove an electron from $\mathrm{H}_2$ ).","The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or ""HOMO"" and the lowest unoccupied molecular orbital or ""LUMO"", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. There are two main ways in which ionization energy is calculated. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. The 2s electrons then shield the 2p electron from the nucleus to some extent, and it is easier to remove the 2p electron from boron than to remove a 2s electron from beryllium, resulting in a lower ionization energy for B. In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. To convert from ""value of ionization energy"" to the corresponding ""value of molar ionization energy"", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. As can be seen in the above graph for ionization energies, the sharp rise in IE values from (: 3.89 eV) to (: 5.21 eV) is followed by a small increase (with some fluctuations) as the f-block proceeds from to . The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. Since the ionization cross section depends on the chemical nature of the sample and the energy of ionizing electrons a standard value of 70 eV is used. That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph). ===Solid surfaces: work function=== Work function is the minimum amount of energy required to remove an electron from a solid surface, where the work function for a given surface is defined by the difference :W = -e\phi - E_{\rm F}, where is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level (electrochemical potential of electrons) inside the material. ==Note== ==See also== * Rydberg equation, a calculation that could determine the ionization energies of hydrogen and hydrogen-like elements. * Electron pairing energies: Half-filled subshells usually result in higher ionization energies. When the next ionization energy involves removing an electron from the same electron shell, the increase in ionization energy is primarily due to the increased net charge of the ion from which the electron is being removed. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable. ==Analogs of ionization energy to other systems== While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems. ===Electron binding energy=== thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. ""use"" and ""WEL"" give ionization energy in the unit kJ/mol; ""CRC"" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. The term ionization potential is an older and obsolete term for ionization energy, because the oldest method of measuring ionization energy was based on ionizing a sample and accelerating the electron removed using an electrostatic potential. == Determination of ionization energies == thumb|304x304px|Ionization energy measurement apparatus. |alt= The ionization energy of atoms, denoted Ei, is measured by finding the minimal energy of light quanta (photons) or electrons accelerated to a known energy that will kick out the least bound atomic electrons. This in turn makes its ionization energies increase by 18 kJ/mol−1. The kinetic energy of the bombarding electrons should have higher energy than the ionization energy of the sample molecule. ",650000,2.9,15.0, 11.58,15.425,E
-Calculate the energy of one mole of UV photons of wavelength $300 \mathrm{~nm}$ and compare it with a typical single-bond energy of $400 \mathrm{~kJ} / \mathrm{mol}$.,"To find the photon energy in electronvolts using the wavelength in micrometres, the equation is approximately :E\text{ (eV)} = \frac{1.2398}{\lambda\text{ (μm)}} since hc/e=1.239 \; 841 \; 984... \times 10^{-6} eVm where h is Planck's constant, c is the speed of light in m/sec, and e is the electron charge. Additionally, E = \frac{hc}{\lambda} where *E is photon energy *λ is the photon's wavelength *c is the speed of light in vacuum *h is the Planck constant The photon energy at 1 Hz is equal to 6.62607015 × 10−34 J That is equal to 4.135667697 × 10−15 eV === Electronvolt === Energy is often measured in electronvolts. The photon energy of near infrared radiation at 1 μm wavelength is approximately 1.2398 eV. ==Examples== An FM radio station transmitting at 100 MHz emits photons with an energy of about 4.1357 × 10−7 eV. These wavelengths correspond to photon energies of down to . The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. During photosynthesis, specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm in the photosystem I, corresponding to an energy of each photon of ≈ 2 eV ≈ 3 × 10−19 J ≈ 75 kBT, where kBT denotes the thermal energy. Photon energy is the energy carried by a single photon. Equivalently, the longer the photon's wavelength, the lower its energy. Very-high-energy gamma rays have photon energies of 100 GeV to over 1 PeV (1011 to 1015 electronvolts) or 16 nanojoules to 160 microjoules. Photon energy can be expressed using any unit of energy. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. The higher the photon's frequency, the higher its energy. Spectral irradiance of wavelengths in the solar spectrum. As one joule equals 6.24 × 1018 eV, the larger units may be more useful in denoting the energy of photons with higher frequency and higher energy, such as gamma rays, as opposed to lower energy photons as in the optical and radio frequency regions of the electromagnetic spectrum. ==Formulas== === Physics === Photon energy is directly proportional to frequency. The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5 × 10−18 J) with a maximal energy conversion efficiency of 35%. ==See also== *Photon *Electromagnetic radiation *Electromagnetic spectrum *Planck constant *Planck–Einstein relation *Soft photon ==References== Category:Foundational quantum physics Category:Electromagnetic spectrum Category:Photons Ultra-high-energy gamma rays are gamma rays with photon energies higher than 100 TeV (0.1 PeV). UV-B lamps are lamps that emit a spectrum of ultraviolet light with wavelengths ranging from 290–320 nanometers. This minuscule amount of energy is approximately 8 × 10−13 times the electron's mass (via mass-energy equivalence). These tables list values of molar ionization energies, measured in kJ⋅mol−1. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. Among the units commonly used to denote photon energy are the electronvolt (eV) and the joule (as well as its multiples, such as the microjoule). ",0.54,399,2.0,6.0,0.118,B
-Calculate the magnitude of the spin angular momentum of a proton. Give a numerical answer. ,"The sum of the proton and neutron magnetic moments gives 0.879 µN, which is within 3% of the measured value 0.857 µN. The result is larger than μ by a factor equal to the ratio of the proton to electron mass, or about a factor of 1836. ==See also== * Nucleon magnetic moment ==References== ==External links== *. The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn. The resulting value was not zero and had a sign opposite to that of the proton. In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment. == Nature of the nucleon magnetic moments == thumb|upright|A magnetic dipole moment can be created by either a current loop (top; Ampèrian) or by two magnetic monopoles (bottom; Gilbertian). The key question is how the nucleon's spin, whose magnitude is 1/2ħ, is carried by its constituent partons (quarks and gluons). The g-factor for the proton is 5.6, and the chargeless neutron, which should have no magnetic moment at all, has a g-factor of −3.8. The CODATA recommended value for the magnetic moment of the proton is or The best available measurement for the value of the magnetic moment of the neutron is Here, μN is the nuclear magneton, a standard unit for the magnetic moments of nuclear components, and μB is the Bohr magneton, both being physical constants. Nucleon spin structure describes the partonic structure of nucleon (proton and neutron) intrinsic angular momentum (spin). In SI units, these values are and A magnetic moment is a vector quantity, and the direction of the nucleon's magnetic moment is determined by its spin. The nuclear magneton is the spin magnetic moment of a Dirac particle, a charged, spin-1/2 elementary particle, with a proton's mass p, in which anomalous corrections are ignored. Since for the neutron the sign of γn is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field. ===Proton nuclear magnetic resonance=== Nuclear magnetic resonance employing the magnetic moments of protons is used for nuclear magnetic resonance (NMR) spectroscopy. The value of nuclear magneton System of units Value Unit SI J·T CGS Erg·G eV eV·T MHz/T (per h) MHz/T The nuclear magneton (symbol μ) is a physical constant of magnetic moment, defined in SI units by: :\mu_\text{N} = {{e \hbar} \over {2 m_\text{p}}} and in Gaussian CGS units by: :\mu_\text{N} = {{e \hbar} \over{2 m_\text{p}c}} where: :e is the elementary charge, :ħ is the reduced Planck constant, :m is the proton rest mass, and :c is the speed of light In SI units, its value is approximately: :μ = In Gaussian CGS units, its value can be given in convenient units as :μ = The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei. Baryon Magnetic moment of quark model Computed (\mu_\text{N}) Observed (\mu_\text{N}) p u − d 2.79 2.793 n d − u −1.86 −1.913 The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be the mass of a nucleon. Thus, in units of nuclear magneton, for the neutron and for the proton. The nuclear magneton is \mu_\text{N} = \frac{e \hbar}{2 m_\text{p}}, where is the elementary charge, and is the reduced Planck constant. Nucleon magnetic moments have been successfully computed from first principles, requiring significant computing resources. ==See also== * Neutron triple-axis spectrometry * LARMOR neutron microscope * Neutron electric dipole moment * Aharonov–Casher effect ==References== ==Bibliography== * S. W. Lovesey (1986). The magnetic moment of such a particle is parallel to its spin. The nuclear magnetic moment also includes contributions from the orbital motion of the charged protons. The calculation assumes that the quarks behave like pointlike Dirac particles, each having their own magnetic moment, as computed using an expression similar to the one above for the nuclear magneton: \ \mu_\text{q} = \frac{\ e_\text{q} \hbar\ }{2 m_\text{q}}\ , where the q-subscripted variables refer to quark magnetic moment, charge, or mass. For a nucleus of which the numbers of protons and of neutrons are both even in its ground state (i.e. lowest energy state), the nuclear spin and magnetic moment are both always zero. The magnetic moment is calculated through j, l and s of the unpaired nucleon, but nuclei are not in states of well defined l and s. ",30,nan,0.21,9.13,14.5115,D
-"The ${ }^7 \mathrm{Li}^1 \mathrm{H}$ ground electronic state has $D_0=2.4287 \mathrm{eV}, \nu_e / c=1405.65 \mathrm{~cm}^{-1}$, and $\nu_e x_e / c=23.20 \mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\omega_e$ and $\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \mathrm{Li}^1 \mathrm{H}$.","The calculated abundance and ratio of 1H and 4He is in agreement with data from observations of young stars. ===The P-P II branch=== In stars, lithium-7 is made in a proton-proton chain reaction. thumb|Proton–proton II chain reaction :{| border=""0"" |- style=""height:2em;"" | ||+ || ||→ || ||+ || |- style=""height:2em;"" | ||+ || ||→ || ||+ || ||+ || ||/ || |- style=""height:2em;"" | ||+ || ||→ ||2 |} The P-P II branch is dominant at temperatures of 14 to . thumb|right|400px|Stable nuclides of the first few elements ==Observed abundance of lithium== Despite the low theoretical abundance of lithium, the actual observable amount is less than the calculated amount by a factor of 3–4. thumb|250px|7Li NMR spectrum of LiCl (1M) in D2O. However, they didn't use the Mössbauer effect but made magnetic resonance measurements of the nucleus of lithium-7, whose ground state possesses a spin of . E7, E07, E-7 or E7 may refer to: ==Science and engineering== * E7 liquid crystal mixture * E7, the Lie group in mathematics * E7 polytope, in geometry * E7 papillomavirus protein * E7 European long distance path ==Transport== * EMD E7, a diesel locomotive * European route E07, an international road * Peugeot E7, a hackney cab * PRR E7, a steam locomotive * Carbon Motors E7,a police car * E7 series, a Japanese high-speed train * Nihonkai-Tōhoku Expressway and Akita Expressway (between Kawabe JCT and Kosaka JCT), route E7 in Japan * Cheras–Kajang Expressway, route E7 in Malaysia ==Other uses== * Boeing E-7, either: ** Boeing E-7 ARIA, the original designation assigned by the United States Air Force under the Mission Designation System to the EC-18B Advanced Range Instrumentation Aircraft. The SD7 is a model of 6-axle diesel locomotive built by General Motors Electro-Motive Division between May 1951 and November 1953. The molecular formula C7H7I (molar mass: 218.03 g/mol) may refer to: * Benzyl iodide * Iodotoluene Other isotopes including 2H, 3H, 3He, 6Li, 7Li, and 7Be are much rarer; the estimated abundance of primordial lithium is 10−10 relative to hydrogen. Though it transmutes into two atoms of helium due to collision with a proton at temperatures above 2.4 million degrees Celsius (most stars easily attain this temperature in their interiors), lithium is more abundant than current computations would predict in later-generation stars. The discrepancy is highlighted in a so-called ""Schramm plot"", named in honor of astrophysicist David Schramm, which depicts these primordial abundances as a function of cosmic baryon content from standard BBN predictions. ==Origin of lithium== Minutes after the Big Bang, the universe was made almost entirely of hydrogen and helium, with trace amounts of lithium and beryllium, and negligibly small abundances of all heavier elements. ===Lithium synthesis in the Big Bang=== Big Bang nucleosynthesis produced both lithium-7 and beryllium-7, and indeed the latter dominates the primordial synthesis of mass 7 nuclides. In summary, accurate measurements of the primordial lithium abundance is the current focus of progress, and it could be possible that the final answer does not lie in astrophysical solutions. The ground state is split into four equally spaced magnetic energy levels when measured in a magnetic field in accordance with its allowed magnetic quantum number. They test the framework of Tsallis non-extensive statistics.Their result suggest that 1.069 === Nuclear physics solutions === When one considers the possibility that the measured primordial lithium abundance is correct and based on the Standard Model of particle physics and the standard cosmology, the lithium problem implies errors in the BBN light element predictions. Namely, the most widely accepted models of the Big Bang suggest that three times as much primordial lithium, in particular lithium-7, should exist. EMD ended production in November 1953 and began producing the SD7's successor, the SD9, in January 1954. == Original buyers == Owner Quantity Numbers Notes Electro-Motive Division 2 990 to Southern Pacific 5308 then 2715 to 1415 ne 1518 Electro-Motive Division 2 991 to Baltimore and Ohio 760 Baltimore and Ohio Railroad 4 761–764 These units were built with the 567BC engine. Since a series of coefficients are tested in those experiments, only the value of maximal sensitivity is given (for precise data, see the individual articles): Author Year SME constraints SME constraints SME constraints Description Author Year Proton Neutron Electron Description Prestage et al. 1985 10−27 Comparing the nuclear spin-flip transition of (stored in a penning trap) with a hydrogen maser transition. EMD SD7 Original Owners. Retrieved on August 27, 2006 * Diesel Era Volume 6 Number 6 November/December 1995, ""EMD's SD7"" by Paul K. Withers pp 5-20; 47-50. == External links == * Locomotive Truck EMD FlexiCoil C SD07 Category:C-C locomotives Category:Diesel-electric locomotives of the United States Category:Railway locomotives introduced in 1952 Category:Freight locomotives Category:Standard gauge locomotives of the United States Spectroscopic observations of stars in NGC 6397, a metal-poor globular cluster, are consistent with an inverse relation between lithium abundance and age, but a theoretical mechanism for diffusion has not been formalized. EMD produced its first examples of the SD7 in May 1951, using the 567B engine. Furthermore, more observations on lithium depletion remain important since present lithium levels might not reflect the initial abundance in the star. This was the first model in EMD's SD (Special Duty) series of locomotives, a lengthened B-B GP7 with a C-C truck arrangement. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. ",260,2.5151,96.4365076099,3.0,-1368,B
-"The positron has charge $+e$ and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an ""atom"" that consists of a positron and an electron.","While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. So finally, the energy levels of positronium are given by E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}. Positronium can also be considered by a particular form of the two-body Dirac equation; Two particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of Janine Shertzer. The electron is commonly symbolized by , and the positron is symbolized by . However, because of the reduced mass, the frequencies of the spectral lines are less than half of those for the corresponding hydrogen lines. ==States== The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. The antiparticle of the electron is called the positron; it is identical to the electron, except that it carries electrical charge of the opposite sign. Approximately: * ~60% of positrons will directly annihilate with an electron without forming positronium. In this approximation, the energy levels are different because of a different effective mass, μ, in the energy equation (see electron energy levels for a derivation): E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2}, where: * is the charge magnitude of the electron (same as the positron), * is the Planck constant, * is the electric constant (otherwise known as the permittivity of free space), * is the reduced mass: \mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2}, where and are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles). The lowest energy level of positronium () is . Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. In 2012, Cassidy et al. were able to produce the excited molecular positronium L = 1 angular momentum state. ==See also== *Hydrogen molecule *Hydrogen molecular ion *Positronium *Protonium *Exotic atom ==References== ==External links== *Molecules of Positronium Observed in the Laboratory for the First Time, press release, University of California, Riverside, September 12, 2007. Upon slowing down in the silica, the positrons captured ordinary electrons to form positronium atoms. The electron's mass is approximately 1/1836 that of the proton. If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons totalling 1.022 MeV. thumb|350px|Naturally occurring electron-positron annihilation as a result of beta plus decay Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. The lowest energy orbital state of positronium is 1S, and like with hydrogen, it has a hyperfine structure arising from the relative orientations of the spins of the electron and the positron. The resulting weakly bound electron and positron spiral inwards and eventually annihilate, with an estimated lifetime of years. == See also == * Breit equation * Antiprotonic helium * Di-positronium * Quantum electrodynamics * Protonium * Two-body Dirac equations == References == == External links == * The annihilation of positronium - The Feynman Lectures on Physics * The Search for Positronium * Obituary of Martin Deutsch, discoverer of Positronium Category:Molecular physics Category:Quantum electrodynamics Category:Spintronics Category:Onia Category:Antimatter Category:Substances discovered in the 1950s The electron (symbol e) is on the left. The opposite is also true: the antiparticle of the positron is the electron. For example, the antiparticle of the electron is the positron (also known as an antielectron). Positron paths in a cloud- chamber trace the same helical path as an electron but rotate in the opposite direction with respect to the magnetic field direction due to their having the same magnitude of charge-to-mass ratio but with opposite charge and, therefore, opposite signed charge-to-mass ratios. ",-0.28, -6.8,50.7,0.9974,0.14,B
-What is the value of the angular-momentum quantum number $l$ for a $t$ orbital?,"Here is the total orbital angular momentum quantum number. An atomic electron's angular momentum, L, is related to its quantum number ℓ by the following equation: \mathbf{L}^2\Psi = \hbar^2 \ell(\ell + 1) \Psi, where ħ is the reduced Planck's constant, L2 is the orbital angular momentum operator and \Psi is the wavefunction of the electron. When referring to angular momentum, it is better to simply use the quantum number ℓ. The associated quantum number is the main total angular momentum quantum number j. Each orbital is characterized by its number , where takes integer values from to , and its angular momentum number , where takes integer values from to . It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as (pronounced ell). == Derivation == Connected with the energy states of the atom's electrons are four quantum numbers: n, ℓ, mℓ, and ms. The quantum number ℓ is always a non-negative integer: 0, 1, 2, 3, etc. L has no real meaning except in its use as the angular momentum operator. It can take the following range of values, jumping only in integer steps: \vert \ell - s\vert \le j \le \ell + s where ℓ is the azimuthal quantum number (parameterizing the orbital angular momentum) and s is the spin quantum number (parameterizing the spin). In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). In general, the values of range from to , where is the spin quantum number, associated with the particle's intrinsic spin angular momentum: :. Furthermore, the eigenvectors of j, s, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of ℓ, s, mℓ and ms. == List of angular momentum quantum numbers == * Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number * orbital angular momentum quantum number (the subject of this article) * magnetic quantum number, related to the orbital momentum quantum number * total angular momentum quantum number == History == The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld. This number gives the information about the direction of spinning of the electron present in any orbital. Shape of orbital is also given by azimuthal quantum number. ====Magnetic quantum number==== The magnetic quantum number describes the specific orbital (or ""cloud"") within that subshell, and yields the projection of the orbital angular momentum along a specified axis: : The values of range from to , with integer intervals. Simultaneous measurement of electron energy and orbital angular momentum is allowed because the Hamiltonian commutes with the angular momentum operator related to L_z. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is \| \bold{s} \| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\ \hbar ~. The value of is the component of spin angular momentum, in units of the reduced Planck constant , parallel to a given direction (conventionally labelled the –axis). Depending on the value of n, there is an angular momentum quantum number ℓ and the following series. The value of ranges from 0 to , so the first p orbital () appears in the second electron shell (), the first d orbital () appears in the third shell (), and so on: : A quantum number beginning in = 3, = 0, describes an electron in the s orbital of the third electron shell of an atom. If s is the particle's spin angular momentum and ℓ its orbital angular momentum vector, the total angular momentum j is \mathbf j = \mathbf s + \boldsymbol {\ell} ~. These were identified as, respectively, the electron ""shell"" number , the ""orbital"" number , and the ""orbital angular momentum"" number . The azimuthal quantum number can also denote the number of angular nodes present in an orbital. Each of the different angular momentum states can take 2(2ℓ + 1) electrons. ",292,14,0.0,0.70710678,11,B
-How many states belong to the carbon configurations $1 s^2 2 s^2 2 p^2$?,"thumb|upright=1.3|Carbon dioxide pressure-temperature phase diagram Supercritical carbon dioxide (s) is a fluid state of carbon dioxide where it is held at or above its critical temperature and critical pressure. secondary Carbon 150x150px Structural formula of propane (secondary carbon is highlighted red) A secondary carbon is a carbon atom bound to two other carbon atoms. Franckeite, chemical formula Pb5Sn3Sb2S14, belongs to a family of complex sulfide minerals. thumb|Schematic of a binary star system with one planet on an S-type orbit and one on a P-type orbit. quaternary carbon 150x150px Structural formula of neopentane (quaternary carbon is highlighted red) A quaternary carbon is a carbon atom bound to four other carbon atoms. For this reason, quaternary carbon atoms are found only in hydrocarbons having at least five carbon atoms. Quaternary carbon atoms can occur in branched alkanes, but not in linear alkanes. primary carbon secondary carbon tertiary carbon quaternary carbon General structure (R = Organyl group) frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px Partial Structural formula frameless=1.0|96x96px frameless=1.0|102x102px frameless=1.0|98x98px frameless=1.0|105x105px == Synthesis == The formation of chiral quaternary carbon centers has been a synthetic challenge. In unbranched alkanes, the inner carbon atoms are always secondary carbon atoms (see figure). primary carbon secondary carbon tertiary carbon quaternary carbon General structure (R = Organyl group) frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px Partial Structural formula frameless=1.0|96x96px frameless=1.0|102x102px frameless=1.0|98x98px frameless=1.0|105x105px == References == Category:Chemical nomenclature Category:Organic chemistry For this reason, secondary carbon atoms are found in all hydrocarbons having at least three carbon atoms. The limits of stability for S-type and P-type orbits within binary as well as trinary stellar systems have been established as a function of the orbital characteristics of the stars, for both prograde and retrograde motions of stars and planets. ==See also== *Astrobiology *Circumstellar habitable zone *Habitability of yellow dwarf systems *Planetary habitability *Circumbinary planet ==References== Binary star systems Category:Binary stars Red atoms are oxygens. thumb|upright=1.3|TEM images of amorphous HBS Two-dimensional silica (2D silica) is a layered polymorph of silicon dioxide. Planets that orbit just one star in a binary pair are said to have ""S-type"" orbits, whereas those that orbit around both stars have ""P-type"" or ""circumbinary"" orbits. Typical estimates often suggest that 50% or more of all star systems are binary systems. Habitability of binary star systems is determined by many factors from a variety of sources. The planets have semi-major axes that lie between 1.09 and 1.46 times this critical radius. Carbon dioxide usually behaves as a gas in air at standard temperature and pressure (STP), or as a solid called dry ice when cooled and/or pressurised sufficiently. This would have implications for bulk thermal and nuclear generation of electricity, because the supercritical properties of carbon dioxide at above 500 °C and 20 MPa enable thermal efficiencies approaching 45 percent. thumb|upright=1.3|Top and side views of graphene (left) and HBS structures (right). The minimum stable star-to-circumbinary- planet separation is about 2–4 times the binary star separation, or orbital period about 3–8 times the binary period. Volume 2001, Issue 40 , Pages 2482–2486 Heck reaction, Enyne cyclization, cycloaddition reactions, Quasdorf, K.W.; Overman, L. E. Nature Volume 2014, Volume 516, Pages 181 C–H activation, Allylic substitution, Pauson–Khand reaction, Ishizaki, M.; Niimi, Y.; Hoshino, O.; Hara, H.; Takahashi, T. Tetrahedron Volume 2001, Issue 61, Pages 4053–4065 etc. to construct asymmetric quaternary carbon atoms. == References == Category:Chemical nomenclature Category:Organic chemistry For example, Kepler-47c is a gas giant in the circumbinary habitable zone of the Kepler-47 system. It was shown to be a member of the auxetics materials family with a negative Poisson's ratio. ==References== Category:Two-dimensional nanomaterials Category:Silicon dioxide Category:Silica polymorphs ",50.7,0.38,15.0,12,7,C
-Calculate the energy needed to compress three carbon-carbon single bonds and stretch three carbon-carbon double bonds to the benzene bond length $1.397 Å$. Assume a harmonicoscillator potential-energy function for bond stretching and compression. Typical carboncarbon single- and double-bond lengths are 1.53 and $1.335 Å$; typical stretching force constants for carbon-carbon single and double bonds are 500 and $950 \mathrm{~N} / \mathrm{m}$.,"Bond energy (BE) is the average of all bond-dissociation energies of a single type of bond in a given molecule.Madhusha (2017), Difference Between Bond Energy and Bond Dissociation Energy, Pediaa, Difference Between Bond Energy and Bond Dissociation Energy The bond-dissociation energies of several different bonds of the same type can vary even within a single molecule. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually at a temperature of 298.15 K) for all bonds of the same type within the same chemical species. The molecule has eight bond lengths ranging between 0.137 and 0.146 nm. right|thumb|illustrative example of C-C length molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C angle molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C-C torsion molecular energy dependence, numerical accuracy is not guaranteed A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration. Applying a harmonic approximation to the potential minimum (at V(r_m) = -\varepsilon), the exponent n and the energy parameter \varepsilon can be related to the harmonic spring constant: k = 2 \varepsilon \left(\frac{n}{r_0}\right)^2, from which n can be calculated if k is known. In chemistry, bond energy (BE), also called the mean bond enthalpyClark, J (2013), BOND ENTHALPY (BOND ENERGY), Chemguide, BOND ENTHALPY (BOND ENERGY) or average bond enthalpy is a measure of bond strength in a chemical bond. Most authors prefer to use the BDE values at 298.15 K.Luo, Yu-Ran and Jin-Pei Cheng ""Bond Dissociation Energies"". The first reduction requires around 1.0 V (Fc/), indicating that C70 is an electron acceptor. === Solution === Saturated solubility of C70 (S, mg/mL) Solvent S (mg/mL) 1,2-dichlorobenzene 36.2 carbon disulfide 9.875 xylene 3.985 toluene 1.406 benzene 1.3 carbon tetrachloride 0.121 n-hexane 0.013 cyclohexane 0.08 pentane 0.002 octane 0.042 decane 0.053 dodecane 0.098 heptane 0.047 isopropanol 0.0021 mesitylene 1.472 dichloromethane 0.080 Fullerenes are sparingly soluble in many aromatic solvents such as toluene and others like carbon disulfide, but not in water. C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. This step yields a solution containing up to 70% of C60 and 15% of C70, as well as other fullerenes. For example, the carbon–hydrogen bond energy in methane BE(C–H) is the enthalpy change (∆H) of breaking one molecule of methane into a carbon atom and four hydrogen radicals, divided by four. Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. In addition, the force needed to draw molecular string to its maximum length could be impractically high - comparable to the tensile strength of particular polymer molecule (~100GPa for some carbon compounds) == See also == *Ultra high molecular weight polyethylene *Carbon nanotube *Carbon nanotube springs ==References == * Stretching molecular springs:elasticity of titin filaments in vertebrate striated muscle, W.A. Linke, Institute of Physiology II, University of Heidelberg, Heidelberg, Germany Category:Nanotechnology Category:Molecular physics Valence bond (VB) computer programs for modern valence bond calculations:- * CRUNCH, by Gordon A. Gallup and his group. The bond energy for H2O is the average of energy required to break each of the two O–H bonds in sequence: : \begin{array}{lcl} \mathrm{H-O-H} & \rightarrow & \mathrm{H\cdot + \cdot O-H} & , D_1 \\\ \mathrm{\cdot O-H} & \rightarrow & \mathrm{\cdot O\cdot + \cdot H} & , D_2 \\\ \mathrm{H-O-H} & \rightarrow & \mathrm{H\cdot + \cdot O\cdot + \cdot H} & , D =(D_1 + D_2)/2 \\\ \end{array} Although the two bonds are the equivalent in the original symmetric molecule, the bond-dissociation energy of an oxygen–hydrogen bond varies slightly depending on whether or not there is another hydrogen atom bonded to the oxygen atom. Each carbon atom in the structure is bonded covalently with 3 others. thumb|left|The structure of C70 molecule. A related fullerene molecule, named buckminsterfullerene (C60 fullerene), consists of 60 carbon atoms. The amount of energy storable in molecular spring is limited by the value of deformation the molecule can withstand until it undergoes chemical change. The resulting structural unit [-C≡(-CH2-)3≡C-] is a rigid cage, consisting of two carbon atoms joined by three methylene bridges; therefore the joined units are constrained to lie on a straight line. The bond dissociation energy (enthalpy) is also referred to as bond disruption energy, bond energy, bond strength, or binding energy (abbreviation: BDE, BE, or D). * Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites (same \varepsilon, \sigma, m) that are bonded as a rigid body. For the LJTS potential with r_\mathrm{end} = 2.5\,\sigma , the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon . ",30,27,1.6,-20,22.2036033112,B
-"When a particle of mass $9.1 \times 10^{-28} \mathrm{~g}$ in a certain one-dimensional box goes from the $n=5$ level to the $n=2$ level, it emits a photon of frequency $6.0 \times 10^{14} \mathrm{~s}^{-1}$. Find the length of the box.","In particle physics, the radiation length is a characteristic of a material, related to the energy loss of high energy particles electromagnetically interacting with it. The radiation length for a given material consisting of a single type of nucleus can be approximated by the following expression: (http://pdg.lbl.gov/) X_0 = 716.4 \text{ g cm}^{-2} \frac{A}{Z (Z+1) \ln{\frac{287}{\sqrt{Z}}}} = 1433 \text{ g cm}^{-2} \frac{A}{Z (Z+1) (11.319 - \ln{Z})}, where is the atomic number and is mass number of the nucleus. This length is useful for renormalizing a non-isotropic scattering problem into an isotropic one in order to use classical diffusion laws (Fick law and Brownian motion). The four-frequency of a massless particle, such as a photon, is a four-vector defined by :N^a = \left( u, u \hat{\mathbf{n}} \right) where u is the photon's frequency and \hat{\mathbf{n}} is a unit vector in the direction of the photon's motion. The characteristic amount of matter traversed for these related interactions is called the radiation length , usually measured in g·cm−2. The parameter a_s of dimension length is defined as the scattering length. The transport length in a strongly diffusing medium (noted l*) is the length over which the direction of propagation of the photon is randomized. The scattering length in quantum mechanics describes low-energy scattering. To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section \sigma. It is also the appropriate length scale for describing high-energy electromagnetic cascades. Comprehensive tables for radiation lengths and other properties of materials are available from the Particle Data Group. ==See also== * Mean free path * Attenuation length * Attenuation coefficient * Attenuation * Range (particle radiation) * Stopping power (particle radiation) * Electron energy loss spectroscopy ==References== Category:Experimental particle physics The 10.5 cm leFH 18M ( ""light field howitzer"") was a German light howitzer used in the Second World War. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the multipole expansion in classical electrodynamics), where one expands in the angular momentum components of the outgoing wave. It is defined as the mean length (in cm) into the material at which the energy of an electron is reduced by the factor 1/e. ==Definition== In materials of high atomic number (e.g. tungsten, uranium, plutonium) the electrons of energies >~10 MeV predominantly lose energy by , and high-energy photons by pair production. The transport length might be measured by transmission experiments and backscattering experiments. For , a good approximation is \frac{1}{X_0} = 4 \left( \frac{\hbar}{m_\mathrm{e} c} \right)^2 Z(Z+1) \alpha^3 n_\mathrm{a} \log\left(\frac{183}{Z^{1/3}}\right), where * is the number density of the nucleus, *\hbar denotes the reduced Planck constant, * is the electron rest mass, * is the speed of light, * is the fine-structure constant. thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. For our potential we have therefore a=r_0, in other words the scattering length for a hard sphere is just the radius. The concept of the scattering length can also be extended to potentials that decay slower than 1/r^3 as r\to \infty. It is both the mean distance over which a high- energy electron loses all but of its energy by , and of the mean free path for pair production by a high-energy photon. An observer moving with four-velocity V^b will observe a frequency :\frac{1}{c}\eta\left(N^a, V^b\right) Where \eta is the Minkowski inner-product (+−−−) Closely related to the four-frequency is the four-wavevector defined by :K^a = \left(\frac{\omega}{c}, \mathbf{k}\right) where \omega = 2 \pi u, c is the speed of light and \mathbf{k} = \frac{2 \pi}{\lambda}\hat{\mathbf{n}} and \lambda is the wavelength of the photon. Sterling Publishing Company, Inc., 2002, p.144 ==History== The 10.5 cm leFH 18M superseded the 10.5 cm leFH 18 as the standard German divisional field howitzer used during the Second World War. ",-11.2,0.6321205588,1.8,2.3613,0.4908,C
-Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the $v=0$ state.,"\\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. {\pi R^2} + \frac{\arcsin\\!\left(\frac{x}{R}\right)}{\pi}\\! for -R\leq x \leq R| mean =0\,| median =0\,| mode =0\,| variance =\frac{R^2}{4}\\!| skewness =0\,| kurtosis =-1\,| entropy =\ln (\pi R) - \frac12 \,| mgf =2\,\frac{I_1(R\,t)}{R\,t}| char =2\,\frac{J_1(R\,t)}{R\,t}| }} The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)={2 \over \pi R^2}\sqrt{R^2-x^2\,}\, for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Phys. 2009, 39, 337–356 The normal distribution is recovered as q → 1\. If we denote by u_{0\ell} the wave function subject to the given potential with total energy E=0 and azimuthal quantum number \ell, the Sturm Oscillation Theorem implies that N_\ell equals the number of nodes of u_{0\ell}. The entropy is calculated as H_{N}(n)=\int_{-1}^{+1} f_{X}(x;n)\ln (f_{X}(x;n))dx The first 5 moments (n=-1 to 3), such that R=1 are \ -\ln(2/\pi) ; n=-1 \ -\ln(2) ;n=0 \ -1/2+\ln(\pi) ;n=1 \ 5/3-\ln(3) ;n=2 \ -7/4-\ln(1/3\pi) ; n=3 == N-sphere Wigner distribution with odd symmetry applied == The marginal PDF distribution with odd symmetry is f{_X}(x;n) ={(1-x^2)^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt{\pi}} \Gamma((n+1)/2)}\sgn(x)\, ; such that R=1 Hence, the CF is expressed in terms of Struve functions CF(t;n) ={ \Gamma(n/2+1) H_{n/2}(t)/(t/2)^{(n/2)} }\, \urcorner (n>=-1); ""The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by"" Z= { \rho c \pi a^2 [R_1 (2ka)-i X_1 (2 ka)], } R_1 ={1-{2 J_1(x) \over 2x} , } X_1 ={{2 H_1(x) \over x} , } == Example (Normalized Received Signal Strength): quadrature terms == The normalized received signal strength is defined as |R| ={{1 \over N} | }\sum_{k=1}^N \exp [i x_n t]| and using standard quadrature terms x ={{1 \over N} }\sum_{k=1}^N \cos ( x_n t) y ={{1 \over N} }\sum_{k=1}^N \sin ( x_n t) Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining {\sqrt{x^2+y^2}}=x+{3 \over 2}y^2-{3 \over 2}xy^2+{1 \over 2}x^2y^2 + O(y^3) +O(y^3)(x-1) +O(y^3)(x-1)^2 +O(x-1)^3 The expanded form of the Characteristic function of the received signal strength becomes E[x] = {1\over N }CF(t;n) E[y^2] ={1\over 2 N}(1 - CF(2t;n)) E[x^2] ={1\over 2N}(1 + CF(2t;n)) E[xy^2] = {t^2 \over 3N^2} CF(t;n)^3+({N-1 \over 2N^2})(1-t CF(2t;n))CF(t;n) E[x^2y^2] = {1\over 8N^3} (1-CF(4t;n))+({N-1 \over 4N^3})(1-CF(2t;n)^2) +({N-1 \over 3N^3})t^2CF(t;n)^4 +({(N-1)(N-2)\over N^3})CF(t;n)^2(1-CF(2t;n)) == See also == * Wigner surmise * The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity. thumb|right|Figure 1 Part of a semi-Markovian discrete system in one dimension with directional jumping time probability density functions (JT-PDFs), including ""death"" terms (the JT-PDFs from state i in state I). Let V:\mathbb{R}^3\to\mathbb{R}:\mathbf{r}\mapsto V(r) be a spherically symmetric potential, such that it is piecewise continuous in r, V(r)=O(1/r^a) for r\to0 and V(r)=O(1/r^b) for r\to+\infty, where a\in(2,+\infty) and b\in(-\infty,2). If :\int_0^{+\infty}r|V(r)|dr<+\infty, then the number of bound states N_\ell with azimuthal quantum number \ell for a particle of mass m obeying the corresponding Schrödinger equation, is bounded from above by :N_\ell<\frac{1}{2\ell+1}\frac{2m}{\hbar^2}\int_0^{+\infty}r|V(r)|dr. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. It takes the form :N_\ell < \frac{1}{2\ell+1} \frac{2m}{\hbar^2} \int_0^\infty r |V(r)|\, dr This limit is the best possible upper bound in such a way that for a given \ell, one can always construct a potential V_\ell for which N_\ell is arbitrarily close to this upper bound. \right] for q < 1 | pdf ={\sqrt{\beta} \over C_q} e_q({-\beta x^2}) | cdf = | mean =0\text{ for }q<2, otherwise undefined| median =0| mode =0| variance = { 1 \over {\beta (5-3q)}} \text{ for } q < {5 \over 3} \infty \text{ for } {5 \over 3} \le q < 2 \text{Undefined for }2 \le q <3| skewness = 0 \text{ for } q < {3 \over 2} | kurtosis = 6{q-1 \over 7-5q} \text{ for } q < {7 \over 5} | entropy =| mgf =| cf =| }} The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number N_\ell of bound states with azimuthal quantum number \ell in a system with central potential V. Groeneboom (1989) shows that : f_Z (z) \sim \frac{1}{2} \frac{4^{4/3} |z|}{\operatorname{Ai}' (\tilde{a}_1)} \exp \left( - \frac{2}{3} |z|^3 + 2^{1/3} \tilde{a}_1 |z| \right) \text{ as }z \rightarrow \infty where \tilde{a}_1 \approx -2.3381 is the largest zero of the Airy function Ai and where \operatorname{Ai}' (\tilde{a}_1 ) \approx 0.7022. The choice of the q-Gaussian form may arise if the system is non-extensive, or if there is lack of a connection to small samples sizes. ==Characterization== ===Probability density function=== The standard q-Gaussian has the probability density function : f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) where :e_q(x) = [1+(1-q)x]_+^{1 \over 1-q} is the q-exponential and the normalization factor C_q is given by :C_q = {{2 \sqrt{\pi} \Gamma\left({1 \over 1-q}\right)} \over {(3-q) \sqrt{1-q} \Gamma\left({3-q \over 2(1-q)}\right)}} \text{ for } -\infty < q < 1 : C_q = \sqrt{\pi} \text{ for } q = 1 \, :C_q = { {\sqrt{\pi} \Gamma\left({3-q \over 2(q-1)}\right)} \over {\sqrt{q-1} \Gamma\left({1 \over q-1}\right)}} \text{ for }1 < q < 3 . The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals. == Wigner n-sphere distribution == The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0): f_n(x;n)={(1-x^2)^{(n-1)/2}\Gamma (1+n/2) \over \sqrt{\pi} \Gamma((n+1)/2)}\, (n>= -1) , for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1\. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution. == Related distributions == === Wigner (spherical) parabolic distribution === The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0): f(x)={3 \over \ 4 R^3}{(R^2-x^2)}\, for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Example. If : V(a,c) = \underset{s \in \mathbf{R}}{\operatorname{argmax}} \ (W(s) - c(s-a)^2), then V(0, c) has density : f_c(t) = \frac{1}{2} g_c(t) g_c(-t) where gc has Fourier transform given by : \hat{g}_c (s) = \frac{(2/c)^{1/3}}{\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \ \ \ s \in \mathbf{R} and where Ai is the Airy function. The solution is based on the path representation of the Green's function, calculated when including all the path probability density functions of all lengths: Here, : \bar{\Psi}_{i}(s) =\sum_j \bar{\Psi}_{ij}(s) and : \bar{\Psi}_{ij}(s)=\frac{1-\bar{\psi}_{ij}(s)}{s}. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). For the corresponding wave function with total energy E=0 and azimuthal quantum number \ell, denoted by \phi_{0\ell}, the radial Schrödinger equation becomes :\frac{d^{2}}{d r^{2}} \phi_{0\ell}(r)-\frac{\ell(\ell+1)}{r^{2}} \phi_{0\ell}(r)=-W(r) \phi_{0\ell}(r), with W=2m|V|/\hbar^2. The following formula will generate deviates from a q-Gaussian with specified parameter q and \beta = {1 \over {3-q}} :Z = \sqrt{-2 \text{ ln}_{q'}(U_1)} \text{ cos}(2 \pi U_2) where \text{ ln}_q is the q-logarithm and q' = { {1+q} \over {3-q}} These deviates can be transformed to generate deviates from an arbitrary q-Gaussian by : Z' = \mu + {Z \over \sqrt{\beta (3-q)}} == Applications == === Physics === It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q-Gaussian. ",0.11,0.16,-87.8,0.85,11,B
-"Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.)","Thus the uncertainty in position must be greater than half of the reduced Compton wavelength . ==Relationship to other constants== Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron {2\pi}\simeq 386~\textrm{fm})}} and the electromagnetic fine-structure constant The Bohr radius is related to the Compton wavelength by: a_0 = \frac{1}{\alpha}\left(\frac{\lambda_\text{e}}{2\pi}\right) = \frac{\bar{\lambda}_\text{e}}{\alpha} \simeq 137\times\bar{\lambda}_\text{e}\simeq 5.29\times 10^4~\textrm{fm} The classical electron radius is about 3 times larger than the proton radius, and is written: r_\text{e} = \alpha\left(\frac{\lambda_\text{e}}{2\pi}\right) = \alpha\bar{\lambda}_\text{e} \simeq\frac{\bar{\lambda}_\text{e}}{137}\simeq 2.82~\textrm{fm} The Rydberg constant, having dimensions of linear wavenumber, is written: \frac{1}{R_\infty}=\frac{2\lambda_\text{e}}{\alpha^2} \simeq 91.1~\textrm{nm} \frac{1}{2\pi R_\infty} = \frac{2}{\alpha^2}\left(\frac{\lambda_\text{e}}{2\pi}\right) = 2 \frac{\bar{\lambda}_\text{e}}{\alpha^2} \simeq 14.5~\textrm{nm} This yields the sequence: r_{\text{e}} = \alpha \bar{\lambda}_{\text{e}} = \alpha^2 a_0 = \alpha^3 \frac{1}{4\pi R_\infty}. The standard Compton wavelength of a particle is given by \lambda = \frac{h}{m c}, while its frequency is given by f = \frac{m c^2}{h}, where is the Planck constant, is the particle's proper mass, and is the speed of light. The CODATA 2018 value for the Compton wavelength of the electron is .CODATA 2018 value for Compton wavelength for the electron from NIST. The Compton wavelength for this particle is the wavelength of a photon of the same energy. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. For photons of frequency , energy is given by E = h f = \frac{h c}{\lambda} = m c^2, which yields the Compton wavelength formula if solved for . ==Limitation on measurement== The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: \mathbf{ abla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi. The Fermi space observatory detected a gamma-ray with an energy of at least 94 billion electron volts. The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: \sqrt{g_{kk}}=\lambda_c ==See also== * de Broglie wavelength * Planck–Einstein relation ==References== ==External links== * Length Scales in Physics: the Compton Wavelength Category:Atomic physics Category:Foundational quantum physics de:Compton-Effekt#Compton-Wellenlänge The Planck mass and length are defined by: m_{\rm P} = \sqrt{\hbar c/G} l_{\rm P} = \sqrt{\hbar G /c^3}. ==Geometrical interpretation== A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. thumb|left|upright=1.0|Spectrum of WR 137 showing the prominent emission lines of ionised Carbon and Helium WR 137 is a variable Wolf-Rayet star located around 6,000 light years away from Earth in the constellation of Cygnus. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Relativistic electron beams are streams of electrons moving at relativistic speeds. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. Equations that pertain to the wavelengths of photons interacting with mass use the non-reduced Compton wavelength. The Planck mass is the order of mass for which the Compton wavelength and the Schwarzschild radius r_{\rm S} = 2 G M /c^2 are the same, when their value is close to the Planck length (l_{\rm P}). Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Other particles have different Compton wavelengths. == Reduced Compton wavelength == The reduced Compton wavelength (barred lambda, denoted below by \bar\lambda) is defined as the Compton wavelength divided by : : \bar\lambda = \frac{\lambda}{2 \pi} = \frac{\hbar}{m c}, where is the reduced Planck constant. ==Role in equations for massive particles== The inverse reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. ",1260,1590,2.0,0.332,12,D
-Calculate the angle that the spin vector $S$ makes with the $z$ axis for an electron with spin function $\alpha$.,"The component of the spin along a specified axis is given by the spin magnetic quantum number, conventionally written . * The spin value of an electron, proton, neutron is . The direct observation of the electron's intrinsic angular momentum was achieved in the Stern–Gerlach experiment. === Stern–Gerlach experiment === The theory of spatial quantization of the spin moment of the momentum of electrons of atoms situated in the magnetic field needed to be proved experimentally. The component of nuclear spin parallel to the –axis can have (2 + 1) values , –1, ..., . Given an arbitrary direction (usually determined by an external magnetic field) the spin -projection is given by :s_z = m_s \, \hbar where is the secondary spin quantum number, ranging from − to + in steps of one. The direction of spin is described by spin quantum number. The electron spin magnetic moment is given by the formula: \ \boldsymbol{\mu}_s = -\frac{e}{\ 2m\ }\ g\ \mathbf{s}\ where : is the charge of the electron : is the Landé g-factor and by the equation: \ \mu_z = \pm \frac{1}{2}\ g\ \mu_\mathsf{B}\ where \ \mu_\mathsf{B}\ is the Bohr magneton. In nuclear magnetic resonance spectroscopy and magnetic resonance imaging, the Ernst angle is the flip angle (a.k.a. ""tip"" or ""nutation"" angle) for excitation of a particular spin that gives the maximal signal intensity in the least amount of time when signal averaging over many transients. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is \| \bold{s} \| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\ \hbar ~. The name ""spin"" comes from a physical spinning of the electron about an axis, as proposed by Uhlenbeck and Goudsmit. The value of is the component of spin angular momentum, in units of the reduced Planck constant , parallel to a given direction (conventionally labelled the –axis). This number gives the information about the direction of spinning of the electron present in any orbital. Nuclear-spin quantum numbers are conventionally written for spin, and or for the -axis component. * The magnitude spin quantum number of an electron cannot be changed. Here is the total orbital angular momentum quantum number. In the electron, the two different spin orientations are sometimes called ""spin-up"" or ""spin-down"". In physics, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. Spin engineering describes the control and manipulation of quantum spin systems to develop devices and materials. In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. First of all, spin satisfies the fundamental commutation relation: \ [S_i, S_j ] = i\ \hbar\ \epsilon_{ijk}\ S_k\ , \ \left[S_i, S^2 \right] = 0\ where \ \epsilon_{ijk}\ is the (antisymmetric) Levi-Civita symbol. thumb|upright=1.35|Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation to that of its immediate neighbor(s). ",54.7,0.2553,0.9992093669,92,48,A
-"The AM1 valence electronic energies of the atoms $\mathrm{H}$ and $\mathrm{O}$ are $-11.396 \mathrm{eV}$ and $-316.100 \mathrm{eV}$, respectively. For $\mathrm{H}_2 \mathrm{O}$ at its AM1-calculated equilibrium geometry, the AM1 valence electronic energy (core-core repulsion omitted) is $-493.358 \mathrm{eV}$ and the AM1 core-core repulsion energy is $144.796 \mathrm{eV}$. For $\mathrm{H}(g)$ and $\mathrm{O}(g), \Delta H_{f, 298}^{\circ}$ values are 52.102 and $59.559 \mathrm{kcal} / \mathrm{mol}$, respectively. Find the AM1 prediction of $\Delta H_{f, 298}^{\circ}$ of $\mathrm{H}_2 \mathrm{O}(g)$.","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. Indeed, AM1* is an extension of AM1 molecular orbital theory and uses AM1 parameters and theory unchanged for the elements H, C, N, O and F. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ×10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. AM1* is a semiempirical molecular orbital technique in computational chemistry. But, other elements have been parameterized using an additional set of d-orbitals in the basis set and with two-center core–core parameters, rather than the Gaussian functions used to modify the core–core potential in AM1. Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). AM1* parameters are now available for H, C, N, O, F, Al, Si, P, S, Cl, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Br, Zr, Mo, Pd, Ag, I and Au. Additionally, for transition metal-hydrogen interactions, a distance dependent term is used to calculate core-core potentials rather than the constant term. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? AM1* is implemented in VAMP 10.0 Clark T, Alex A, Beck B, Chandrasekhar J, Gedeck P, Horn AHC, Hutter M, Martin B, Rauhut G, Sauer W, Schindler T, Steinke T (2005) Computer- Chemie-Centrum. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Element Atomic number Abundance in urban soils Ag 47 0.37 Al 13 38200 As 33 15.9 B 5 45 Ba 56 853.12 Be 4 3.3 Bi 83 1.12 C 6 45100 Ca 20 53800 Cd 48 0.9 Cl 17 285 Co 27 14.1 Cr 24 80 Cs 55 5.0 Cu 29 39 Fe 26 22300 Ga 31 16.2 Ge 32 1.8 H 1 15000 Hg 80 0.88 K 19 13400 La 57 34 Li 3 49.5 Mg 12 7900 Mn 25 729 Mo 42 2.4 N 7 10000 Na 11 5800 Nb 41 15.7 Ni 28 33 O 8 490000 P 15 1200 Pb 82 54.5 Rb 37 58 S 16 1200 Sb 51 1.0 Sc 21 9.4 Si 14 289000 Sn 50 6.8 Sr 38 458 Ta 73 1.5 Ti 22 4758 Tl 81 1.1 V 23 104.9 W 74 2.9 Y 39 23.4 Yb 70 2.4 Zn 30 158 Zr 40 255.6 ==Sea water== *W1 — CRC Handbook *W2 — Kaye & Laby Mass per volume fraction, in kg/L. ",-2,3.00,1.2,-59.24,2598960,D
-"Given that $D_e=4.75 \mathrm{eV}$ and $R_e=0.741 Å$ for the ground electronic state of $\mathrm{H}_2$, find $U\left(R_e\right)$ for this state.","The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . The International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS), is a biannual event, where over hundred scientists meet for the presentation of new developments on the special field of two-dimensional electron systems in semiconductors. * After dividing by the number of electrons, the standard potential E° is related to the standard Gibbs free energy of formation ΔGf° by: E = \frac{\sum \Delta G_\text{left}-\sum \Delta G_\text{right}}{F} where F is the Faraday constant. E2D may refer to: * Estradiol decanoate * trans-4,5-Epoxy-(E)-2-decenal * E2-D the fourth variant of the Northrop Grumman E-2 Hawkeye reconnaissance plane. ER2, ER-2, ER II etc. may refer to: * Elizabeth II's royal cypher E II R (sometimes written as ER II) for Elizabeth II Regina (Elizabeth II, Queen) *""ER2"" (Kanjani Eight song), a single by Japanese boy band Kanjani Eight *ER2 electric trainset, an electric passenger railcar built in Latvia and Russia from 1962 to 1984 *NASA ER-2, ""Earth Resources 2"", an American very high- altitude civilian atmospheric research fixed-wing aircraft based on the Lockheed U-2 reconnaissance aircraft In the weakened potential at the surface, new electronic states can be formed, so called surface states. ==Origin at condensed matter interfaces== thumbnail|350px|Figure 1. Te (aq) + 2 + 2 (s) + 4 1.02 2 . Since the potential is periodic deep inside the crystal, the electronic wave functions must be Bloch waves here. This means that the US Government's use of E85 is effectively doubled as of August 8, 2005 with the signing into law of the Energy Policy Act of 2005. Surface states are electronic states found at the surface of materials. E85 is an abbreviation for an ethanol fuel blend of between 51% and 83% denatured ethanol fuel and gasoline or other hydrocarbon (HC) by volume. ==Availability== All data August 2014 from the Department of Energy, e85prices.com, and E85refueling.com.http://www.e85refueling.com Links go to each state's list of stations; see notes below for caveats. It can be shown that the energies of these states all lie within the band gap. The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations. This theory provides some fundamental new understandings of those electronic states, including surface states. The Shockley states are then found as solutions to the one- dimensional single electron Schrödinger equation : \begin{align} \left[-\frac{\hbar^2}{2m}\frac{d^2}{dz^2}+V(z)\right]\Psi(z) &=& E\Psi(z), \end{align} with the periodic potential : \begin{align} V(z)=\left\\{ \begin{array}{cc} P\delta(z+la),& \textrm{for}\quad z<0 \\\ V_0,&\textrm{for} \quad z>0 \end{array}\right., \end{align} where l is an integer, and P is the normalization factor. For example: : \+ Cu(s) ( = +0.520 V) Cu + 2 Cu(s) ( = +0.337 V) Cu + ( = +0.159 V) :Calculating the potential using Gibbs free energy ( = 2 – ) gives the potential for as 0.154 V, not the experimental value of 0.159 V. : __TOC__ ==Table of standard electrode potentials== Legend: (s) - solid; (l) - liquid; (g) - gas; (aq) - aqueous (default for all charged species); (Hg) - amalgam; bold - water electrolysis equations. Shockley states are thus states that arise due to the change in the electron potential associated solely with the crystal termination. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. All of the reactions should be divided by the stoichiometric coefficient for the electron to get the corresponding corrected reaction equation. For example, the equation Fe + 2 Fe(s) (–0.44 V) means that it requires 2 × 0.44 eV = 0.88 eV of energy to be absorbed (hence the minus sign) in order to create one neutral atom of Fe(s) from one Fe ion and two electrons, or 0.44 eV per electron, which is 0.44 J/C of electrons, which is 0.44 V. The energy levels of such states are expected to significantly shift from the bulk values. The nearly free electron approximation can be used to derive the basic properties of surface states for narrow gap semiconductors. ",-8,0,-5.0,0.14, -31.95,E
-"For $\mathrm{NaCl}, R_e=2.36 Å$. The ionization energy of $\mathrm{Na}$ is $5.14 \mathrm{eV}$, and the electron affinity of $\mathrm{Cl}$ is $3.61 \mathrm{eV}$. Use the simple model of $\mathrm{NaCl}$ as a pair of spherical ions in contact to estimate $D_e$. [One debye (D) is $3.33564 \times 10^{-30} \mathrm{C} \mathrm{m}$.]","Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. If the distances are normalized to the nearest neighbor distance , the potential may be written :V_i = \frac{e}{4 \pi \varepsilon_0 r_0 } \sum_{j} \frac{z_j r_0}{r_{ij}} = \frac{e}{4 \pi \varepsilon_0 r_0 } M_i with being the (dimensionless) Madelung constant of the th ion :M_i = \sum_{j} \frac{z_j}{r_{ij}/r_0}. Comparison between observed and calculated ion separations (in pm) MX Observed Soft-sphere model LiCl 257.0 257.2 LiBr 275.1 274.4 NaCl 282.0 281.9 NaBr 298.7 298.2 In the soft-sphere model, k has a value between 1 and 2. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. In this way values for the radii of 8 ions were determined. Pauling used effective nuclear charge to proportion the distance between ions into anionic and a cationic radii.Pauling, L. (1960). For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). One approach to improving the calculated accuracy is to model ions as ""soft spheres"" that overlap in the crystal. In the next step, D&H; assume that there is a certain radius a_i, beyond which no ions in the atmosphere may approach the (charge) center of the singled out ion. The electrostatic energy of the ion at site then is the product of its charge with the potential acting at its site :E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i. D&H; say that this approximation holds at large distances between ions, which is the same as saying that the concentration is low. _Effective_ ionic radii in pm of elements as a function of ionic charge and spin (ls = low spin, hs = high spin). All compounds crystallize in the NaCl structure. thumb|300 px|Relative radii of atoms and ions. For ions on lower-symmetry sites significant deviations of their electron density from a spherical shape may occur. These tables list values of molar ionization energies, measured in kJ⋅mol−1. The iodide ions nearly touch (but don't quite), indicating that Landé's assumption is fairly good. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as ""effective"" ionic radii. For table salt in 0.01 M solution at 25 °C, a typical value of (\kappa a)^2 is 0.0005636, while a typical value of Z_0 is 7.017, highlighting the fact that, in low concentrations, (\kappa a)^2 is a target for a zero order of magnitude approximation such as perturbation analysis. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants. == Formula == A fast converging formula for the Madelung constant of NaCl is :12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right) == Generalization == It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge. ", 4.56,3.0,0.1353,30,3.61,A
-Find the number of CSFs in a full CI calculation of $\mathrm{CH}_2 \mathrm{SiHF}$ using a 6-31G** basis set.,"The molecular formula C6H8S (molar mass: 112.19 g/mol, exact mass: 112.0347 u) may refer to: * 2,3-Dihydrothiepine * 2,7-Dihydrothiepine * 2,5-Dimethylthiophene Category:Molecular formulas Caesium fluoride or cesium fluoride is an inorganic compound with the formula CsF and it is a hygroscopic white salt. Caltech Intermediate Form (CIF) is a file format for describing integrated circuits. The molecular formula C6H6O2S (molar mass: 142.18 g/mol, exact mass: 142.0089 u) may refer to: * 3,4-Ethylenedioxythiophene (EDOT) * Phenylsulfinic acid * 3-Thiophene acetic acid * Thiophene-2-acetic acid All numbers in CIF are integers that refer to centimicrons of distance, unless subroutine scaling is specified (described later). CsF reaches a vapor pressure of 1 kilopascal at 825 °C, 10 kPa at 999 °C, and 100 kPa at 1249 °C. The molecular formula C8H6S (molar mass: 134.20 g/mol, exact mass: 134.0190 u) may refer to: *Benzo[c]thiophene *Benzothiophene Category:Molecular formulas The molecular formula C3H2F6O (molar mass: 168.038 g/mol, exact mass: 168.0010 u) may refer to: * Desflurane * Hexafluoro-2-propanol (HFIP) CsF is an alternative to tetra-n-butylammonium fluoride (TBAF) and TAS-fluoride (TASF). ===As a base=== As with other soluble fluorides, CsF is moderately basic, because HF is a weak acid. Extensions to CIF can be done with the numeric statements `0` through `9`. CsF gives higher yields in Knoevenagel condensation reactions than KF or NaF. ===Formation of Cs-F bonds=== Caesium fluoride serves as a source of fluoride in organofluorine chemistry. CsF chains with a thickness as small as one or two atoms can be grown inside carbon nanotubes. ==Structure== Caesium fluoride has the halite structure, which means that the Cs+ and F− pack in a cubic closest packed array as do Na+ and Cl− in sodium chloride. ==Applications in organic synthesis== Being highly dissociated, CsF is a more reactive source of fluoride than related salts. The reaction is shown below: :Cs2CO3 \+ 2 HF → 2 CsF + H2O + CO2 CsF is more soluble than sodium fluoride or potassium fluoride in organic solvents. FIGURE B.5 Typical user extensions to CIF. The final statement in a CIF file is the `END` statement (or the letter `E`). The reaction is shown below: :CsOH + HF → CsF + H2O Using the same reaction, another way to create caesium fluoride is to treat caesium carbonate (Cs2CO3) with hydrofluoric acid and again, the resulting salt can then be purified by recrystallization. Solutions of caesium fluoride in THF or DMF attack a wide variety of organosilicon compounds to produce an organosilicon fluoride and a carbanion, which can then react with electrophiles, for example: :500px ==Precautions== Like other soluble fluorides, CsF is moderately toxic.MSDS Listing for cesium fluoride . www.hazard.com . Similarly to potassium fluoride, CsF reacts with hexafluoroacetone to form a stable perfluoroalkoxide salt. FIGURE B.1 CIF layer names for MOS processes. CIF provides a limited set of graphics primitives that are useful for describing the two-dimensional shapes on the different layers of a chip. Note that the magnitude of this rotation vector has no meaning. thumb|333px|right|FIGURE B.2 A sample CIF ""wire"" statement. Caesium also has the highest electropositivity of all known elements and fluorine has the highest electronegativity of all known elements. ==Synthesis and properties== Caesium fluoride can be prepared by the reaction of caesium hydroxide (CsOH) with hydrofluoric acid (HF) and the resulting salt can then be purified by recrystallization. ",1.86,1855,0.686,62.8318530718,93.4,A
-Calculate the ratio of the electrical and gravitational forces between a proton and an electron.,"However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. In January 2013, an updated value for the charge radius of a proton——was published. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). The proton radius puzzle is an unanswered problem in physics relating to the size of the proton. The result is again ~5% smaller than the previously-accepted proton radius. Their measurement of the root-mean-square charge radius of a proton is "", which differs by 5.0 standard deviations from the CODATA value of "". Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). The radius of the proton is linked to the form factor and momentum-transfer cross section. However, in such an association with an electron, the character of the bound proton is not changed, and it remains a proton. Physical parameter 1H 16O relative atomic mass of the XZ+ ion relative atomic mass of the Z electrons correction for the binding energy relative atomic mass of the neutral atom The principle can be shown by the determination of the electron relative atomic mass by Farnham et al. at the University of Washington (1995). The internationally accepted value of a proton's charge radius is . It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. The proton radius was a puzzle as of 2017. Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius. === 2019 measurements === In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton-to- electron mass ratio). ",0,14.44,479.0,2,22,D
+","The four stars are each about half the mass of the Sun and are approximately 500 million years old. The system is unusual in how closely the four stars are orbiting each other; one pair has an orbital separation of at most .04 astronomical units and an orbital period of about two days, the other pair has a separation of at most .26 astronomical units and a period of about 55 days, and the two pairs are separated by 5.8 AU and have an orbital period of less than nine years. Kepler-84 is a Sun-like star 4,700 light-years from the Sun. The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. For celestial objects in general, the orbital period is determined by a 360° revolution of one body around its primary, e.g. Earth around the Sun. Periods in astronomy are expressed in units of time, usually hours, days, or years. ==Small body orbiting a central body== thumb|upright=1.2|The semi-major axis (a) and semi-minor axis (b) of an ellipse According to Kepler's Third Law, the orbital period T of two point masses orbiting each other in a circular or elliptic orbit is: :T = 2\pi\sqrt{\frac{a^3}{GM}} where: * a is the orbit's semi-major axis * G is the gravitational constant, * M is the mass of the more massive body. Kepler-444 (or KOI-3158, KIC 6278762, 2MASS J19190052+4138043, BD+41°3306) is a triple star system, estimated to be 11.2 billion years old (more than 80% of the age of the universe), approximately away from Earth in the constellation Lyra. At that conference, the star was known as KOI-3158. ==Characteristics== The star, Kepler-444, is approximately 11.2 billion years old, whereas the Sun is only 4.6 billion years old. The star is believed to have 2 M dwarfs in orbit around it with > the fainter companion 1.8 arc-seconds from the main star. ==Stellar system== The Kepler-444 system consists of the planet hosting primary and a pair of M-dwarf stars. Gliese 623 is a dim double star 25.6 light years from Earth in the constellation Hercules. BD−22 5866 is a quadruple-star system located 166 light years from Earth. The age is that of Kepler-444 A, an orange main sequence star of spectral type K0. Another (which is a background star with a probability 0.5%) is a yellow star of mass 0.855 on projected separations of 0.18″ or 0.26″ (213.6 AU). ==Planetary system== Kepler-84 is orbited by five known planets, four small gas giants and a Super- Earth. The rotation period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) may refer to its sidereal rotation period, i.e. the time that the object takes to complete a single revolution around its axis of rotation relative to the background stars, measured in sidereal time. This period differs from the sidereal period because both the orbital plane of the object and the plane of the ecliptic precess with respect to the fixed stars, so their intersection, the line of nodes, also precesses with respect to the fixed stars. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry. ===Synodic period=== One of the observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods, is their synodic period, which is the time between conjunctions. Kepler-444 is the > densest star with detected solar-like oscillations. There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics, particularly they must not be confused with other revolving periods like rotational periods. For example, Hyperion, a moon of Saturn, exhibits this behaviour, and its rotation period is described as chaotic. ==Rotation period of selected objects== Celestial objects Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Rotation period with respect to distant stars, the sidereal rotation period (compared to Earth's mean Solar days) Synodic rotation period (mean Solar day) Apparent rotational period viewed from Earth Sun* 25.379995 days (Carrington rotation) 35 days (high latitude) 25d 9h 7m 11.6s 35d ~28 days (equatorial) Mercury 58.6462 days 58d 15h 30m 30s 176 days Venus −243.0226 daysThis rotation is negative because the pole which points north of the invariable plane rotates in the opposite direction to most other planets. −243d 0h 33m −116.75 days Earth 0.99726968 daysReference adds about 1 ms to Earth's stellar day given in mean solar time to account for the length of Earth's mean solar day in excess of 86400 SI seconds. 0d 23h 56m 4.0910s 1.00 days (24h 00m 00s) Moon 27.321661 days (equal to sidereal orbital period due to spin-orbit locking, a sidereal lunar month) 27d 7h 43m 11.5s 29.530588 days (equal to synodic orbital period, due to spin-orbit locking, a synodic lunar month) none (due to spin-orbit locking) Mars 1.02595675 days 1d 0h 37m 22.663s 1.02749125 days Ceres 0.37809 days 0d 9h 4m 27.0s 0.37818 days Jupiter 0.41354 days(average) 0.4135344 days (deep interiorRotation period of the deep interior is that of the planet's magnetic field.) 0.41007 days (equatorial) 0.4136994 days (high latitude) 0d 9h 55m 30s 0d 9h 55m 29.37s 0d 9h 50m 30s 0d 9h 55m 43.63s (9 h 55 m 33 s) (average) Saturn days (average, deep interiorFound through examination of Saturn's C Ring) 0.44401 days (deep interior) 0.4264 days (equatorial) 0.44335 days (high latitude) 0d 10h 39m 22.4s 0d 10h 13m 59s 0d 10h 38m 25.4s (10 h 32 m 36 s) Uranus −0.71833 days −0d 17h 14m 24s (−17 h 14 m 23 s) Neptune 0.67125 days 0d 16h 6m 36s (16 h 6 m 36 s) Pluto −6.38718 days (synchronous with Charon) –6d 9h 17m 32s (–6d 9h 17m 0s) Haumea 0.1631458 ±0.0000042 days 0d 3h 56m 43.80 ±0.36s 0.1631461 ±0.0000042 days Makemake 0.9511083 ±0.0000042 days 22h 49m 35.76 ±0.36s 0.9511164 ±0.0000042 days Eris ~1.08 days 25h ~54m ~1.08 days * See Solar rotation for more detail. == See also == * Apparent retrograde motion * List of slow rotators (minor planets) * List of fast rotators (minor planets) * Retrograde motion * Rotational speed * Synodic day ==References== ==External links== * Note, the rotation periods for Mercury and Earth in this work may be inaccurate. We use asteroseismology > to directly measure a precise age of 11.2+/-1.0 Gyr for the host star, > indicating that Kepler-444 formed when the Universe was less than 20% of its > current age and making it the oldest known system of terrestrial-size > planets. AM Canum Venaticorum 17.146 minutes Beta Lyrae AB 12.9075 days Alpha Centauri AB 79.91 years Proxima Centauri – Alpha Centauri AB 500,000 years or more ==See also== * Geosynchronous orbit derivation * Rotation period – time that it takes to complete one revolution around its axis of rotation * Satellite revisit period * Sidereal time * Sidereal year * Opposition (astronomy) * List of periodic comets ==Notes== ==Bibliography== * == External links == Category:Time in astronomy Period Category:Kinematic properties Period Category:Time in astronomy The age of Kepler-444 not only > suggests that thick-disk stars were among the hosts to the first Galactic > planets, but may also help to pinpoint the beginning of the era of planet > formation."" ",2688,1.45,"""418.0""",30,9,E
+"To perform a rescue, a lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of $g / 6$. The exhaust velocity is $2000 \mathrm{~m} / \mathrm{s}$, but fuel amounting to only 20 percent of the total mass may be used. How long can the landing craft hover?","The acceleration due to gravity on the surface of the Moon is approximately 1.625 m/s2, about 16.6% that on Earth's surface or 0.166 . The lunar GM = 4902.8001 km3/s2 from GRAIL analyses. Instead, after it arrested its velocity at an altitude of 3.4m it simply fell to the lunar surface. Since rocketry is used for descent and landing, the Moon's gravity necessitates the use of more fuel than is needed for asteroid landing. thumb|Maglev hover car A hover car is a personal vehicle that flies at a constant altitude of up to a few meters (yards) above the ground and used for personal transportation in the same way a modern automobile is employed. The lunar GM is 1/81.30057 of the Earth's GM. == Theory == For the lunar gravity field, it is conventional to use an equatorial radius of R = 1738.0 km. Orbital speed around the Moon can, depending on altitude, exceed 1500 m/s. A lunar lander or Moon lander is a spacecraft designed to land on the surface of the Moon. thumb| Project Horizon Lunar Landing-and-Return Vehicle. In comparison, the much lighter (292 kg) Surveyor 3 landed on the Moon in 1967 using nearly 700 kg of fuel. All lunar landers require rocket engines for descent. The relatively high gravity (higher than all known asteroids, but lower than all solar system planets) and lack of lunar atmosphere negates the use of aerobraking, so a lander must use propulsion to decelerate and achieve a soft landing. ==History== The Luna program was a series of robotic impactors, flybys, orbiters, and landers flown by the Soviet Union between 1958 and 1976. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. The idea is that engine exhaust and lunar regolith can cause problems if they were to be kicked back from the surface to the spacecraft, and thus the engines cut off just before touchdown. Over the entire surface, the variation in gravitational acceleration is about 0.0253 m/s2 (1.6% of the acceleration due to gravity). This was in lieu of a 12 million- pound thrust superbooster required for a direct-ascent lunar flight, which could not possibly be developed in time for the 1966 deployment target. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. For example, the 900-kg Curiosity rover was landed on Mars by a craft having a mass (at the time of Mars atmospheric entry) of 2400 kg, of which only 390 kg was fuel. The landing gear was designed to withstand landings with engine cut-out at up to of height, though it was intended for descent engine shutdown to commence when one of the probes touched the surface. The design requirements for these landers depend on factors imposed by the payload, flight rate, propulsive requirements, and configuration constraints.Lunar Lander Stage Requirements Based on the Civil Needs Data Base (PDF). During this period the rockets would transport some 220 tonnes of useful cargo to the Moon. Higher speeds can be attained if the skydiver pulls in their limbs (see also freeflying). ",0.5,-191.2,"""273.0""",20.2,117,C
+"In an elastic collision of two particles with masses $m_1$ and $m_2$, the initial velocities are $\mathbf{u}_1$ and $\mathbf{u}_2=\alpha \mathbf{u}_1$. If the initial kinetic energies of the two particles are equal, find the conditions on $u_1 / u_2$ such that $m_1$ is at rest after the collision and $\alpha$ is positive. ","The conservation of the total momentum before and after the collision is expressed by: m_{1}u_{1}+m_{2}u_{2} \ =\ m_{1}v_{1} + m_{2}v_{2}. Consider particles 1 and 2 with masses m1, m2, and velocities u1, u2 before collision, v1, v2 after collision. In the center of momentum frame where the total momentum equals zero, \begin{align} p_1 &= - p_2 \\\ p_1^2 &= p_2^2 \\\ E &= \sqrt {m_1^2c^4 + p_1^2c^2} + \sqrt {m_2^2c^4 + p_2^2c^2} = E \\\ p_1 &= \pm \frac{\sqrt{E^4 - 2E^2m_1^2c^4 - 2E^2m_2^2c^4 + m_1^4c^8 - 2m_1^2m_2^2c^8 + m_2^4c^8}}{2cE} \\\ u_1 &= -v_1. \end{align} Here m_1, m_2 represent the rest masses of the two colliding bodies, u_1, u_2 represent their velocities before collision, v_1, v_2 their velocities after collision, p_1, p_2 their momenta, c is the speed of light in vacuum, and E denotes the total energy, the sum of rest masses and kinetic energies of the two bodies. Indeed, to derive the equations, one may first change the frame of reference so that one of the known velocities is zero, determine the unknown velocities in the new frame of reference, and convert back to the original frame of reference. ====Examples==== ;Before collision: :Ball 1: mass = 3 kg, velocity = 4 m/s :Ball 2: mass = 5 kg, velocity = −6 m/s ;After collision: :Ball 1: velocity = −8.5 m/s :Ball 2: velocity = 1.5 m/s Another situation: frame|center|Elastic collision of unequal masses. However, in the case of the proton, the evidence suggested three distinct concentrations of charge (quarks) and not one. ==Formula== The formula for the velocities after a one-dimensional collision is: \begin{align} v_a &= \frac{C_R m_b (u_b - u_a) + m_a u_a + m_b u_b} {m_a+m_b} \\\ v_b &= \frac{C_R m_a (u_a - u_b) + m_a u_a + m_b u_b} {m_a+m_b} \end{align} where *va is the final velocity of the first object after impact *vb is the final velocity of the second object after impact *ua is the initial velocity of the first object before impact *ub is the initial velocity of the second object before impact *ma is the mass of the first object *mb is the mass of the second object *CR is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a perfectly inelastic collision, see below. Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, \theta_1 and \theta_2, are related to the angle of deflection \theta in the system of the center of mass by \tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad \theta_2=\frac{{\pi}-{\theta}}{2}. In physics, an elastic collision is an encounter (collision) between two bodies in which the total kinetic energy of the two bodies remains the same. To see this, consider the center of mass at time t before collision and time t' after collision: \begin{align} \bar{x}(t) &= \frac{m_{1} x_{1}(t)+m_{2} x_{2}(t)}{m_{1}+m_{2}} \\\ \bar{x}(t') &= \frac{m_{1} x_{1}(t')+m_{2} x_{2}(t')}{m_{1}+m_{2}}. \end{align} Hence, the velocities of the center of mass before and after collision are: \begin{align} v_{ \bar{x} } &= \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1}+m_{2}} \\\ v_{ \bar{x} }' &= \frac{m_{1}v_{1}+m_{2}v_{2}}{m_{1}+m_{2}}. \end{align} The numerators of v_{ \bar{x} } and v_{ \bar{x} }' are the total momenta before and after collision. These equations may be solved directly to find v_1,v_2 when u_1,u_2 are known: \begin{array}{ccc} v_1 &=& \dfrac{m_1-m_2}{m_1+m_2} u_1 + \dfrac{2m_2}{m_1+m_2} u_2 \\\\[.5em] v_2 &=& \dfrac{2m_1}{m_1+m_2} u_1 + \dfrac{m_2-m_1}{m_1+m_2} u_2. \end{array} If both masses are the same, we have a trivial solution: \begin{align} v_{1} &= u_{2} \\\ v_{2} &= u_{1}. \end{align} This simply corresponds to the bodies exchanging their initial velocities to each other. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. A useful special case of elastic collision is when the two bodies have equal mass, in which case they will simply exchange their momenta. In this example, momentum of the system is conserved because there is no friction between the sliding bodies and the surface. m_a u_a + m_b u_b = \left( m_a + m_b \right) v where v is the final velocity, which is hence given by v=\frac{m_a u_a + m_b u_b}{m_a + m_b} Another perfectly inelastic collision|frame|center The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. Assuming no friction, this gives the velocity updates: \begin{align} \Delta \vec{v_{a}} &= \frac{J_{n}}{m_{a}} \vec{n} \\\ \Delta \vec{v_{b}} &= -\frac{J_{n}}{m_{b}} \vec{n} \end{align} ==Perfectly inelastic collision== A completely inelastic collision between equal masses|frame|center A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision. The reduction of kinetic energy E_r is hence: E_r = \frac{1}{2}\frac{m_a m_b}{m_a + m_b}|u_a - u_b|^2 With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation). ==Partially inelastic collisions== Partially inelastic collisions are the most common form of collisions in the real world. The magnitudes of the velocities of the particles after the collision are: \begin{align} v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\\ v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}. \end{align} ===Two-dimensional collision with two moving objects=== The final x and y velocities components of the first ball can be calculated as: \begin{align} v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2}) \\\\[0.8em] v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}), \end{align} where and are the scalar sizes of the two original speeds of the objects, and are their masses, and are their movement angles, that is, v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1 (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi () is the contact angle. Once v_1 is determined, v_2 can be found by symmetry. ====Center of mass frame==== With respect to the center of mass, both velocities are reversed by the collision: a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed. This is why a neutron moderator (a medium which slows down fast neutrons, thereby turning them into thermal neutrons capable of sustaining a chain reaction) is a material full of atoms with light nuclei which do not easily absorb neutrons: the lightest nuclei have about the same mass as a neutron. ====Derivation of solution==== To derive the above equations for v_1,v_2, rearrange the kinetic energy and momentum equations: \begin{align} m_1(v_1^2-u_1^2) &= m_2(u_2^2-v_2^2) \\\ m_1(v_1-u_1) &= m_2(u_2-v_2) \end{align} Dividing each side of the top equation by each side of the bottom equation, and using \tfrac{a^2-b^2}{(a-b)} = a+b, gives: v_1+u_1=u_2+v_2 \quad\Rightarrow\quad v_1-v_2 = u_2-u_1. In an elastic collision these magnitudes do not change. In such a collision, kinetic energy is lost by bonding the two bodies together. An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction. When considering energies, possible rotational energy before and/or after a collision may also play a role. ==Equations== ===One-dimensional Newtonian=== In an elastic collision, both momentum and kinetic energy are conserved. ",1.6,5.828427125,"""650000.0""",420,0.6749,B
+"Astronaut Stumblebum wanders too far away from the space shuttle orbiter while repairing a broken communications satellite. Stumblebum realizes that the orbiter is moving away from him at $3 \mathrm{~m} / \mathrm{s}$. Stumblebum and his maneuvering unit have a mass of $100 \mathrm{~kg}$, including a pressurized tank of mass $10 \mathrm{~kg}$. The tank includes only $2 \mathrm{~kg}$ of gas that is used to propel him in space. The gas escapes with a constant velocity of $100 \mathrm{~m} / \mathrm{s}$. With what velocity will Stumblebum have to throw the empty tank away to reach the orbiter?","Escape velocity is the minimum speed an object without propulsion needs to have to move away indefinitely from the source of the gravity field. The terminal velocity in such cases will have a negative value, corresponding to the rate of rising up. ==See also== * Stokes's law * Terminal ballistics ==References== ==External links== *Terminal Velocity - NASA site *Onboard video of Space Shuttle Solid Rocket Boosters rapidly decelerating to terminal velocity on entry to the thicker atmosphere, from at 5:15 in the video, to 220 mph at 6:45 when the parachutes are deployed 90 seconds later—NASA video and sound, @ io9.com. *Terminal settling velocity of a sphere at all realistic Reynolds Numbers, by Heywood Tables approach. Assuming that g is positive (which it was defined to be), and substituting α back in, the speed v becomes v = \sqrt\frac{2mg}{\rho A C_d} \tanh \left(t \sqrt{\frac{g \rho A C_d }{2m}}\right). Using the formula for terminal velocity V_t = \sqrt\frac{2mg}{\rho A C_d} the equation can be rewritten as v = V_t \tanh \left(t \frac{g}{V_t}\right). Solving for Vt yields Derivation of the solution for the velocity v as a function of time t The drag equation is—assuming ρ, g and Cd to be constants: m a = m \frac{\mathrm{d}v}{\mathrm{d}t} = m g - \frac{1}{2} \rho v^2 A C_d. For objects falling through the atmosphere, for every of fall, the terminal speed decreases 1%. The formula for terminal velocity (V)] appears on p. [52], equation (127). ""Escape Velocity"" is the fourth episode in the fourth season of the science fiction television series Battlestar Galactica. Substitution of equations (–) in equation () and solving for terminal velocity, V_t to yield the following expression In equation (), it is assumed that the object is denser than the fluid. So instead of m use the reduced mass m_r = m-\rho V in this and subsequent formulas. Dividing both sides by m gives \frac{\mathrm{d}v}{\mathrm{d}t} = g \left( 1 - \alpha^2 v^2 \right). The biologist J. B. S. Haldane wrote, ==Physics== Using mathematical terms, terminal speed—without considering buoyancy effects—is given by V_t= \sqrt\frac{2 m g}{\rho A C_d} where *V_t represents terminal velocity, *m is the mass of the falling object, *g is the acceleration due to gravity, *C_d is the drag coefficient, *\rho is the density of the fluid through which the object is falling, and *A is the projected area of the object. At equilibrium, the net force is zero (Fnet = 0) and the velocity becomes the terminal velocity : m g - {1 \over 2} \rho V_t^2 A C_d = 0. Escape Velocity may also refer to: ==Books== * Escape Velocity (Doctor Who), a Doctor Who novel * Escape Velocity: Cyberculture at the End of the Century, a nonfiction book by Mark Dery * Escape Velocity, prequel to the Warlock series by Christopher Stasheff ==Video games== * Escape Velocity (video game) * Escape Velocity Override, its sequel * Escape Velocity Nova, the most recent title in the Escape Velocity franchise, along with an expandable card-driven board game based on it ==Music== * ""Escape Velocity"" (song), a 2010 song by The Chemical Brothers * Escape Velocity, an album by The Phenomenauts ==Film and television== * ""Escape Velocity"" (Battlestar Galactica), an episode of the TV show Battlestar Galactica * Escape Velocity (film), a 1998 Canadian thriller film ==See also== thumb|upright=1.3|A flight envelope diagram showing VS (Stall speed at 1G), VC (Corner/Maneuvering speed) and VD (Dive speed) thumb|upright=1.3|Vg diagram. Note the 1g stall speed, and the Maneuvering Speed (Corner Speed) for both positive and negative g. After a lengthy investigation, Velocity found and solved the cause of these stalls. An object with a large projected area relative to its mass, such as a parachute, has a lower terminal velocity than one with a small projected area relative to its mass, such as a dart. The same terminal speed is reached for a typical .30-06 bullet dropping downwards—when it is returning to the ground having been fired upwards, or dropped from a tower—according to a 1920 U.S. Army Ordnance study. the current record is held by Felix Baumgartner who jumped from an altitude of and reached , though he achieved this speed at high altitude where the density of the air is much lower than at the Earth's surface, producing a correspondingly lower drag force. Note that this is a different concept than design maneuvering speed. If the falling object is spherical in shape, the expression for the three forces are given below: where *d is the diameter of the spherical object, *g is the gravitational acceleration, *\rho is the density of the fluid, *\rho_s is the density of the object, *A = \frac{1}{4} \pi d^2 is the projected area of the sphere, *C_d is the drag coefficient, and *V is the characteristic velocity (taken as terminal velocity, V_t ). At some speed, the drag or force of resistance will equal the gravitational pull on the object (buoyancy is considered below). ",2.74,2.3,"""170.0""",1.88,11,E
+"Use the $D_0$ value of $\mathrm{H}_2(4.478 \mathrm{eV})$ and the $D_0$ value of $\mathrm{H}_2^{+}(2.651 \mathrm{eV})$ to calculate the first ionization energy of $\mathrm{H}_2$ (that is, the energy needed to remove an electron from $\mathrm{H}_2$ ).","The second way of calculating ionization energies is mainly used at the lowest level of approximation, where the ionization energy is provided by Koopmans' theorem, which involves the highest occupied molecular orbital or ""HOMO"" and the lowest unoccupied molecular orbital or ""LUMO"", and states that the ionization energy of an atom or molecule is equal to the negative value of energy of the orbital from which the electron is ejected. There are two main ways in which ionization energy is calculated. For hydrogen in the ground state Z=1 and n=1 so that the energy of the atom before ionization is simply E = - 13.6\ \mathrm{eV} After ionization, the energy is zero for a motionless electron infinitely far from the proton, so that the ionization energy is : I = E(\mathrm{H}^+) - E(\mathrm{H}) = +13.6\ \mathrm{eV}. The first ionization energy is quantitatively expressed as :X(g) + energy ⟶ X+(g) + e− where X is any atom or molecule, X+ is the resultant ion when the original atom was stripped of a single electron, and e− is the removed electron. In physics, ionization energy is usually expressed in electronvolts (eV) or joules (J). For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). The ionization energy will be the energy of photons hνi (h is the Planck constant) that caused a steep rise in the current: Ei = hνi. The 2s electrons then shield the 2p electron from the nucleus to some extent, and it is easier to remove the 2p electron from boron than to remove a 2s electron from beryllium, resulting in a lower ionization energy for B. In physics and chemistry, ionization energy (IE) (American English spelling), ionisation energy (British English spelling) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, positive ion, or molecule. To convert from ""value of ionization energy"" to the corresponding ""value of molar ionization energy"", the conversion is: *:1 eV = 96.48534 kJ/mol *:1 kJ/mol = 0.0103642688 eV == References == === WEL (Webelements) === As quoted at http://www.webelements.com/ from these sources: * J.E. Huheey, E.A. Keiter, and R.L. Keiter in Inorganic Chemistry : Principles of Structure and Reactivity, 4th edition, HarperCollins, New York, USA, 1993. As can be seen in the above graph for ionization energies, the sharp rise in IE values from (: 3.89 eV) to (: 5.21 eV) is followed by a small increase (with some fluctuations) as the f-block proceeds from to . The second, third, etc., molar ionization energy applies to the further removal of an electron from a singly, doubly, etc., charged ion. Since the ionization cross section depends on the chemical nature of the sample and the energy of ionizing electrons a standard value of 70 eV is used. That 2p electron is much closer to the nucleus than the 3s electrons removed previously. thumb|350x350px|Ionization energies peak in noble gases at the end of each period in the periodic table of elements and, as a rule, dip when a new shell is starting to fill. The ionization energy is the lowest binding energy for a particular atom (although these are not all shown in the graph). ===Solid surfaces: work function=== Work function is the minimum amount of energy required to remove an electron from a solid surface, where the work function for a given surface is defined by the difference :W = -e\phi - E_{\rm F}, where is the charge of an electron, is the electrostatic potential in the vacuum nearby the surface, and is the Fermi level (electrochemical potential of electrons) inside the material. ==Note== ==See also== * Rydberg equation, a calculation that could determine the ionization energies of hydrogen and hydrogen-like elements. * Electron pairing energies: Half-filled subshells usually result in higher ionization energies. When the next ionization energy involves removing an electron from the same electron shell, the increase in ionization energy is primarily due to the increased net charge of the ion from which the electron is being removed. However, due to experimental limitations, the adiabatic ionization energy is often difficult to determine, whereas the vertical detachment energy is easily identifiable and measurable. ==Analogs of ionization energy to other systems== While the term ionization energy is largely used only for gas-phase atomic, cationic, or molecular species, there are a number of analogous quantities that consider the amount of energy required to remove an electron from other physical systems. ===Electron binding energy=== thumb|500px|Binding energies of specific atomic orbitals as a function of the atomic number. == Numerical values == For each atom, the column marked 1 is the first ionization energy to ionize the neutral atom, the column marked 2 is the second ionization energy to remove a second electron from the +1 ion, the column marked 3 is the third ionization energy to remove a third electron from the +2 ion, and so on. ""use"" and ""WEL"" give ionization energy in the unit kJ/mol; ""CRC"" gives atomic ionization energy in the unit eV. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 H hydrogen use 1312.0 WEL 1312.0 CRC 13.59844 2 He helium use 2372.3 5250.5 WEL 2372.3 5250.5 CRC 24.58738 54.41776 3 Li lithium use 520.2 7298.1 11815.0 WEL 520.2 7298.1 11815.0 CRC 5.39171 75.64009 122.45435 4 Be beryllium use 899.5 1757.1 14848.7 21006.6 WEL 899.5 1757.1 14848.7 21006.6 CRC 9.32269 18.21115 153.89620 217.71858 5 B boron use 800.6 2427.1 3659.7 25025.8 32826.7 WEL 800.6 2427.1 3659.7 25025.8 32826.7 CRC 8.29803 25.15484 37.93064 259.37521 340.22580 6 C carbon use 1086.5 2352.6 4620.5 6222.7 37831 47277.0 WEL 1086.5 2352.6 4620.5 6222.7 37831 47277.0 CRC 11.26030 24.38332 47.8878 64.4939 392.087 489.99334 7 N nitrogen use 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 WEL 1402.3 2856 4578.1 7475.0 9444.9 53266.6 64360 CRC 14.53414 29.6013 47.44924 77.4735 97.8902 552.0718 667.046 8 O oxygen use 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 WEL 1313.9 3388.3 5300.5 7469.2 10989.5 13326.5 71330 84078.0 CRC 13.61806 35.11730 54.9355 77.41353 113.8990 138.1197 739.29 871.4101 9 F fluorine use 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 WEL 1681.0 3374.2 6050.4 8407.7 11022.7 15164.1 17868 92038.1 106434.3 CRC 17.42282 34.97082 62.7084 87.1398 114.2428 157.1651 185.186 953.9112 1103.1176 10 Ne neon use 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 WEL 2080.7 3952.3 6122 9371 12177 15238 19999.0 23069.5 115379.5 131432 CRC 21.5646 40.96328 63.45 97.12 126.21 157.93 207.2759 239.0989 1195.8286 1362.1995 11 Na sodium use 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 159076 WEL 495.8 4562 6910.3 9543 13354 16613 20117 25496 28932 141362 CRC 5.13908 47.2864 71.6200 98.91 138.40 172.18 208.50 264.25 299.864 1465.121 1648.702 12 Mg magnesium use 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 169988 189368 WEL 737.7 1450.7 7732.7 10542.5 13630 18020 21711 25661 31653 35458 CRC 7.64624 15.03528 80.1437 109.2655 141.27 186.76 225.02 265.96 328.06 367.50 1761.805 1962.6650 13 Al aluminium use 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 42647 201266 222316 WEL 577.5 1816.7 2744.8 11577 14842 18379 23326 27465 31853 38473 CRC 5.98577 18.82856 28.44765 119.992 153.825 190.49 241.76 284.66 330.13 398.75 442.00 2085.98 2304.1410 14 Si silicon use 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 45962 50502 235196 257923 WEL 786.5 1577.1 3231.6 4355.5 16091 19805 23780 29287 33878 38726 CRC 8.15169 16.34585 33.49302 45.14181 166.767 205.27 246.5 303.54 351.12 401.37 476.36 523.42 2437.63 2673.182 15 P phosphorus use 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 46261 54110 59024 271791 296195 WEL 1011.8 1907 2914.1 4963.6 6273.9 21267 25431 29872 35905 40950 CRC 10.48669 19.7694 30.2027 51.4439 65.0251 220.421 263.57 309.60 372.13 424.4 479.46 560.8 611.74 2816.91 3069.842 16 S sulfur use 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 48710 54460 62930 68216 311048 337138 WEL 999.6 2252 3357 4556 7004.3 8495.8 27107 31719 36621 43177 CRC 10.36001 23.3379 34.79 47.222 72.5945 88.0530 280.948 328.75 379.55 447.5 504.8 564.44 652.2 707.01 3223.78 3494.1892 17 Cl chlorine use 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 51068 57119 63363 72341 78095 352994 380760 WEL 1251.2 2298 3822 5158.6 6542 9362 11018 33604 38600 43961 CRC 12.96764 23.814 39.61 53.4652 67.8 97.03 114.1958 348.28 400.06 455.63 529.28 591.99 656.71 749.76 809.40 3658.521 3946.2960 18 Ar argon use 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 52002 59653 66199 72918 82473 88576 397605 427066 WEL 1520.6 2665.8 3931 5771 7238 8781 11995 13842 40760 46186 CRC 15.75962 27.62967 40.74 59.81 75.02 91.009 124.323 143.460 422.45 478.69 538.96 618.26 686.10 755.74 854.77 918.03 4120.8857 4426.2296 19 K potassium use 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 54490 60730 68950 75900 83080 93400 99710 444880 476063 WEL 418.8 3052 4420 5877 7975 9590 11343 14944 16963.7 48610 CRC 4.34066 31.63 45.806 60.91 82.66 99.4 117.56 154.88 175.8174 503.8 564.7 629.4 714.6 786.6 861.1 968 1033.4 4610.8 4934.046 20 Ca calcium use 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 63410 70110 78890 86310 94000 104900 111711 494850 527762 WEL 589.8 1145.4 4912.4 6491 8153 10496 12270 14206 18191 20385 57110 CRC 6.11316 11.87172 50.9131 67.27 84.50 108.78 127.2 147.24 188.54 211.275 591.9 657.2 726.6 817.6 894.5 974 1087 1157.8 5128.8 5469.864 21 Sc scandium use 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 WEL 633.1 1235.0 2388.6 7090.6 8843 10679 13310 15250 17370 21726 24102 66320 73010 80160 89490 97400 105600 117000 124270 547530 582163 CRC 6.5615 12.79967 24.75666 73.4894 91.65 110.68 138.0 158.1 180.03 225.18 249.798 687.36 756.7 830.8 927.5 1009 1094 1213 1287.97 5674.8 6033.712 22 Ti titanium use 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 639294 WEL 658.8 1309.8 2652.5 4174.6 9581 11533 13590 16440 18530 20833 25575 28125 76015 83280 90880 100700 109100 117800 129900 137530 602930 CRC 6.8281 13.5755 27.4917 43.2672 99.30 119.53 140.8 170.4 192.1 215.92 265.07 291.500 787.84 863.1 941.9 1044 1131 1221 1346 1425.4 6249.0 6625.82 23 V vanadium use 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 661050 699144 WEL 650.9 1414 2830 4507 6298.7 12363 14530 16730 19860 22240 24670 29730 32446 86450 94170 102300 112700 121600 130700 143400 151440 CRC 6.7462 14.66 29.311 46.709 65.2817 128.13 150.6 173.4 205.8 230.5 255.7 308.1 336.277 896.0 976 1060 1168 1260 1355 1486 1569.6 6851.3 7246.12 24 Cr chromium use 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 166090 721870 761733 WEL 652.9 1590.6 2987 4743 6702 8744.9 15455 17820 20190 23580 26130 28750 34230 37066 97510 105800 114300 125300 134700 144300 157700 CRC 6.7665 16.4857 30.96 49.16 69.46 90.6349 160.18 184.7 209.3 244.4 270.8 298.0 354.8 384.168 1010.6 1097 1185 1299 1396 1496 1634 1721.4 7481.7 7894.81 25 Mn manganese use 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 172500 181380 785450 827067 WEL 717.3 1509.0 3248 4940 6990 9220 11500 18770 21400 23960 27590 30330 33150 38880 41987 109480 118100 127100 138600 148500 158600 CRC 7.43402 15.63999 33.668 51.2 72.4 95.6 119.203 194.5 221.8 248.3 286.0 314.4 343.6 403.0 435.163 1134.7 1224 1317 1437 1539 1644 1788 1879.9 8140.6 8571.94 26 Fe iron use 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 173600 188100 195200 851800 895161 WEL 762.5 1561.9 2957 5290 7240 9560 12060 14580 22540 25290 28000 31920 34830 37840 44100 47206 122200 131000 140500 152600 163000 CRC 7.9024 16.1878 30.652 54.8 75.0 99.1 124.98 151.06 233.6 262.1 290.2 330.8 361.0 392.2 457 489.256 1266 1358 1456 1582 1689 1799 1950 2023 8828 9277.69 27 Co cobalt use 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 178100 189300 204500 214100 920870 966023 WEL 760.4 1648 3232 4950 7670 9840 12440 15230 17959 26570 29400 32400 36600 39700 42800 49396 52737 134810 145170 154700 167400 CRC 7.8810 17.083 33.50 51.3 79.5 102.0 128.9 157.8 186.13 275.4 305 336 379 411 444 511.96 546.58 1397.2 1504.6 1603 1735 1846 1962 2119 2219.0 9544.1 10012.12 28 Ni nickel use 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 182700 194000 205600 221400 231490 992718 1039668 WEL 737.1 1753.0 3395 5300 7339 10400 12800 15600 18600 21670 30970 34000 37100 41500 44800 48100 55101 58570 148700 159000 169400 CRC 7.6398 18.16884 35.19 54.9 76.06 108 133 162 193 224.6 321.0 352 384 430 464 499 571.08 607.06 1541 1648 1756 1894 2011 2131 2295 2399.2 10288.8 10775.40 29 Cu copper use 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 184900 198800 210500 222700 239100 249660 1067358 1116105 WEL 745.5 1957.9 3555 5536 7700 9900 13400 16000 19200 22400 25600 35600 38700 42000 46700 50200 53700 61100 64702 163700 174100 CRC 7.72638 20.29240 36.841 57.38 79.8 103 139 166 199 232 265.3 369 401 435 484 520 557 633 670.588 1697 1804 1916 2060 2182 2308 2478 2587.5 11062.38 11567.617 30 Zn zinc use 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 WEL 906.4 1733.3 3833 5731 7970 10400 12900 16800 19600 23000 26400 29990 40490 43800 47300 52300 55900 59700 67300 71200 179100 CRC 9.3942 17.96440 39.723 59.4 82.6 108 134 174 203 238 274 310.8 419.7 454 490 542 579 619 698 738 1856 31 Ga gallium use 578.8 1979.3 2963 6180 WEL 578.8 1979.3 2963 6180 CRC 5.99930 20.5142 30.71 64 32 Ge germanium use 762 1537.5 3302.1 4411 9020 WEL 762 1537.5 3302.1 4411 9020 CRC 7.8994 15.93462 34.2241 45.7131 93.5 33 As arsenic use 947.0 1798 2735 4837 6043 12310 WEL 947.0 1798 2735 4837 6043 12310 CRC 9.7886 18.633 28.351 50.13 62.63 127.6 34 Se selenium use 941.0 2045 2973.7 4144 6590 7880 14990 WEL 941.0 2045 2973.7 4144 6590 7880 14990 CRC 9.75238 21.19 30.8204 42.9450 68.3 81.7 155.4 35 Br bromine use 1139.9 2103 3470 4560 5760 8550 9940 18600 WEL 1139.9 2103 3470 4560 5760 8550 9940 18600 CRC 11.81381 21.8 36 47.3 59.7 88.6 103.0 192.8 36 Kr krypton use 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 90400 96300 101400 111100 116290 282500 296200 311400 326200 WEL 1350.8 2350.4 3565 5070 6240 7570 10710 12138 22274 25880 29700 33800 37700 43100 47500 52200 57100 61800 75800 80400 85300 CRC 13.99961 24.35985 36.950 52.5 64.7 78.5 111.0 125.802 230.85 268.2 308 350 391 447 492 541 592 641 786 833 884 937 998 1051 1151 1205.3 2928 3070 3227 3381 37 Rb rubidium use 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 WEL 403.0 2633 3860 5080 6850 8140 9570 13120 14500 26740 CRC 4.17713 27.285 40 52.6 71.0 84.4 99.2 136 150 277.1 38 Sr strontium use 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 31270 WEL 549.5 1064.2 4138 5500 6910 8760 10230 11800 15600 17100 CRC 5.6949 11.03013 42.89 57 71.6 90.8 106 122.3 162 177 324.1 39 Y yttrium use 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 19900 36090 WEL 600 1180 1980 5847 7430 8970 11190 12450 14110 18400 CRC 6.2171 12.24 20.52 60.597 77.0 93.0 116 129 146.2 191 206 374.0 40 Zr zirconium use 640.1 1270 2218 3313 7752 9500 WEL 640.1 1270 2218 3313 7752 9500 CRC 6.63390 13.13 22.99 34.34 80.348 41 Nb niobium use 652.1 1380 2416 3700 4877 9847 12100 WEL 652.1 1380 2416 3700 4877 9847 12100 CRC 6.75885 14.32 25.04 38.3 50.55 102.057 125 42 Mo molybdenum use 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 93400 98420 104400 121900 127700 133800 139800 148100 154500 WEL 684.3 1560 2618 4480 5257 6640.8 12125 13860 15835 17980 20190 22219 26930 29196 52490 55000 61400 67700 74000 80400 87000 CRC 7.09243 16.16 27.13 46.4 54.49 68.8276 125.664 143.6 164.12 186.4 209.3 230.28 279.1 302.60 544.0 570 636 702 767 833 902 968 1020 1082 1263 1323 1387 1449 1535 1601 43 Tc technetium use 702 1470 2850 WEL 702 1470 2850 CRC 7.28 15.26 29.54 44 Ru ruthenium use 710.2 1620 2747 WEL 710.2 1620 2747 CRC 7.36050 16.76 28.47 45 Rh rhodium use 719.7 1740 2997 WEL 719.7 1740 2997 CRC 7.45890 18.08 31.06 46 Pd palladium use 804.4 1870 3177 WEL 804.4 1870 3177 CRC 8.3369 19.43 32.93 47 Ag silver use 731.0 2070 3361 WEL 731.0 2070 3361 CRC 7.5762 21.49 34.83 48 Cd cadmium use 867.8 1631.4 3616 WEL 867.8 1631.4 3616 CRC 8.9938 16.90832 37.48 49 In indium use 558.3 1820.7 2704 5210 WEL 558.3 1820.7 2704 5210 CRC 5.78636 18.8698 28.03 54 50 Sn tin use 708.6 1411.8 2943.0 3930.3 7456 WEL 708.6 1411.8 2943.0 3930.3 7456 CRC 7.3439 14.63225 30.50260 40.73502 72.28 51 Sb antimony use 834 1594.9 2440 4260 5400 10400 WEL 834 1594.9 2440 4260 5400 10400 CRC 8.6084 16.53051 25.3 44.2 56 108 52 Te tellurium use 869.3 1790 2698 3610 5668 6820 13200 WEL 869.3 1790 2698 3610 5668 6820 13200 CRC 9.0096 18.6 27.96 37.41 58.75 70.7 137 53 I iodine use 1008.4 1845.9 3180 WEL 1008.4 1845.9 3180 CRC 10.45126 19.1313 33 54 Xe xenon use 1170.4 2046.4 3099.4 WEL 1170.4 2046.4 3099.4 CRC 12.1298 21.20979 32.1230 55 Cs caesium use 375.7 2234.3 3400 WEL 375.7 2234.3 3400 CRC 3.89390 23.15745 56 Ba barium use 502.9 965.2 3600 WEL 502.9 965.2 3600 CRC 5.21170 10.00390 57 La lanthanum use 538.1 1067 1850.3 4819 5940 WEL 538.1 1067 1850.3 4819 5940 CRC 5.5769 11.060 19.1773 49.95 61.6 58 Ce cerium use 534.4 1050 1949 3547 6325 7490 WEL 534.4 1050 1949 3547 6325 7490 CRC 5.5387 10.85 20.198 36.758 65.55 77.6 59 Pr praseodymium use 527 1020 2086 3761 5551 WEL 527 1020 2086 3761 5551 CRC 5.473 10.55 21.624 38.98 57.53 60 Nd neodymium use 533.1 1040 2130 3900 WEL 533.1 1040 2130 3900 CRC 5.5250 10.73 22.1 40.41 61 Pm promethium use 540 1050 2150 3970 WEL 540 1050 2150 3970 CRC 5.582 10.90 22.3 41.1 62 Sm samarium use 544.5 1070 2260 3990 WEL 544.5 1070 2260 3990 CRC 5.6436 11.07 23.4 41.4 63 Eu europium use 547.1 1085 2404 4120 WEL 547.1 1085 2404 4120 CRC 5.6704 11.241 24.92 42.7 64 Gd gadolinium use 593.4 1170 1990 4250 WEL 593.4 1170 1990 4250 CRC 6.1501 12.09 20.63 44.0 65 Tb terbium use 565.8 1110 2114 3839 WEL 565.8 1110 2114 3839 CRC 5.8638 11.52 21.91 39.79 66 Dy dysprosium use 573.0 1130 2200 3990 WEL 573.0 1130 2200 3990 CRC 5.9389 11.67 22.8 41.47 67 Ho holmium use 581.0 1140 2204 4100 WEL 581.0 1140 2204 4100 CRC 6.0215 11.80 22.84 42.5 68 Er erbium use 589.3 1150 2194 4120 WEL 589.3 1150 2194 4120 CRC 6.1077 11.93 22.74 42.7 69 Tm thulium use 596.7 1160 2285 4120 WEL 596.7 1160 2285 4120 CRC 6.18431 12.05 23.68 42.7 70 Yb ytterbium use 603.4 1174.8 2417 4203 WEL 603.4 1174.8 2417 4203 CRC 6.25416 12.1761 25.05 43.56 71 Lu lutetium use 523.5 1340 2022.3 4370 6445 WEL 523.5 1340 2022.3 4370 6445 CRC 5.4259 13.9 20.9594 45.25 66.8 72 Hf hafnium use 658.5 1440 2250 3216 WEL 658.5 1440 2250 3216 CRC 6.82507 14.9 23.3 33.33 73 Ta tantalum use 761 1500 WEL 761 1500 CRC 7.5496 74 W tungsten use 770 1700 WEL 770 1700 CRC 7.8640 75 Re rhenium use 760 1260 2510 3640 WEL 760 1260 2510 3640 CRC 7.8335 76 Os osmium use 840 1600 WEL 840 1600 CRC 8.4382 77 Ir iridium use 880 1600 WEL 880 1600 CRC 8.9670 78 Pt platinum use 870 1791 WEL 870 1791 CRC 8.9587 18.563 28https://dept.astro.lsa.umich.edu/~cowley/ionen.htm archived 79 Au gold use 890.1 1980 WEL 890.1 1980 CRC 9.2255 20.5 30 80 Hg mercury use 1007.1 1810 3300 WEL 1007.1 1810 3300 CRC 10.43750 18.756 34.2 81 Tl thallium use 589.4 1971 2878 WEL 589.4 1971 2878 CRC 6.1082 20.428 29.83 82 Pb lead use 715.6 1450.5 3081.5 4083 6640 WEL 715.6 1450.5 3081.5 4083 6640 CRC 7.41666 15.0322 31.9373 42.32 68.8 83 Bi bismuth use 703 1610 2466 4370 5400 8520 WEL 703 1610 2466 4370 5400 8520 CRC 7.2856 16.69 25.56 45.3 56.0 88.3 84 Po polonium use 812.1 WEL 812.1 CRC 8.417 85 At astatine use 899.003 WEL 920 CRC talk Originally quoted as 9.31751(8) eV. 86 Rn radon use 1037 WEL 1037 CRC 10.74850 87 Fr francium use 380 WEL 380 CRC 4.0727 talk give 4.0712±0.00004 eV (392.811(4) kJ/mol) 88 Ra radium use 509.3 979.0 WEL 509.3 979.0 CRC 5.2784 10.14716 89 Ac actinium use 499 1170 WEL 499 1170 CRC 5.17 12.1 90 Th thorium use 587 1110 1930 2780 WEL 587 1110 1930 2780 CRC 6.3067 11.5 20.0 28.8 91 Pa protactinium use 568 WEL 568 CRC 5.89 92 U uranium use 597.6 1420 WEL 597.6 1420 CRC 6.19405 14.72 93 Np neptunium use 604.5 WEL 604.5 CRC 6.2657 94 Pu plutonium use 584.7 WEL 584.7 CRC 6.0262 95 Am americium use 578 WEL 578 CRC 5.9738 96 Cm curium use 581 WEL 581 CRC 5.9915 97 Bk berkelium use 601 WEL 601 CRC 6.1979 98 Cf californium use 608 WEL 608 CRC 6.2817 99 Es einsteinium use 619 WEL 619 CRC 6.42 100 Fm fermium use 627 WEL 627 CRC 6.50 101 Md mendelevium use 635 WEL 635 CRC 6.58 102 No nobelium use 642 WEL CRC 6.65 103 Lr lawrencium use 470 WEL CRC 4.9 104 Rf rutherfordium use 580 WEL CRC 6.0 == Notes == * Values from CRC are ionization energies given in the unit eV; other values are molar ionization energies given in the unit kJ/mol. The term ionization potential is an older and obsolete term for ionization energy, because the oldest method of measuring ionization energy was based on ionizing a sample and accelerating the electron removed using an electrostatic potential. == Determination of ionization energies == thumb|304x304px|Ionization energy measurement apparatus. |alt= The ionization energy of atoms, denoted Ei, is measured by finding the minimal energy of light quanta (photons) or electrons accelerated to a known energy that will kick out the least bound atomic electrons. This in turn makes its ionization energies increase by 18 kJ/mol−1. The kinetic energy of the bombarding electrons should have higher energy than the ionization energy of the sample molecule. ",650000,2.9,"""15.0""", 11.58,15.425,E
+Calculate the energy of one mole of UV photons of wavelength $300 \mathrm{~nm}$ and compare it with a typical single-bond energy of $400 \mathrm{~kJ} / \mathrm{mol}$.,"To find the photon energy in electronvolts using the wavelength in micrometres, the equation is approximately :E\text{ (eV)} = \frac{1.2398}{\lambda\text{ (μm)}} since hc/e=1.239 \; 841 \; 984... \times 10^{-6} eVm where h is Planck's constant, c is the speed of light in m/sec, and e is the electron charge. Additionally, E = \frac{hc}{\lambda} where *E is photon energy *λ is the photon's wavelength *c is the speed of light in vacuum *h is the Planck constant The photon energy at 1 Hz is equal to 6.62607015 × 10−34 J That is equal to 4.135667697 × 10−15 eV === Electronvolt === Energy is often measured in electronvolts. The photon energy of near infrared radiation at 1 μm wavelength is approximately 1.2398 eV. ==Examples== An FM radio station transmitting at 100 MHz emits photons with an energy of about 4.1357 × 10−7 eV. These wavelengths correspond to photon energies of down to . The amount of energy is directly proportional to the photon's electromagnetic frequency and thus, equivalently, is inversely proportional to the wavelength. During photosynthesis, specific chlorophyll molecules absorb red-light photons at a wavelength of 700 nm in the photosystem I, corresponding to an energy of each photon of ≈ 2 eV ≈ 3 × 10−19 J ≈ 75 kBT, where kBT denotes the thermal energy. Photon energy is the energy carried by a single photon. Equivalently, the longer the photon's wavelength, the lower its energy. Very-high-energy gamma rays have photon energies of 100 GeV to over 1 PeV (1011 to 1015 electronvolts) or 16 nanojoules to 160 microjoules. Photon energy can be expressed using any unit of energy. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. The higher the photon's frequency, the higher its energy. Spectral irradiance of wavelengths in the solar spectrum. As one joule equals 6.24 × 1018 eV, the larger units may be more useful in denoting the energy of photons with higher frequency and higher energy, such as gamma rays, as opposed to lower energy photons as in the optical and radio frequency regions of the electromagnetic spectrum. ==Formulas== === Physics === Photon energy is directly proportional to frequency. The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. A minimum of 48 photons is needed for the synthesis of a single glucose molecule from CO2 and water (chemical potential difference 5 × 10−18 J) with a maximal energy conversion efficiency of 35%. ==See also== *Photon *Electromagnetic radiation *Electromagnetic spectrum *Planck constant *Planck–Einstein relation *Soft photon ==References== Category:Foundational quantum physics Category:Electromagnetic spectrum Category:Photons Ultra-high-energy gamma rays are gamma rays with photon energies higher than 100 TeV (0.1 PeV). UV-B lamps are lamps that emit a spectrum of ultraviolet light with wavelengths ranging from 290–320 nanometers. This minuscule amount of energy is approximately 8 × 10−13 times the electron's mass (via mass-energy equivalence). These tables list values of molar ionization energies, measured in kJ⋅mol−1. This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. Among the units commonly used to denote photon energy are the electronvolt (eV) and the joule (as well as its multiples, such as the microjoule). ",0.54,399,"""2.0""",6.0,0.118,B
+Calculate the magnitude of the spin angular momentum of a proton. Give a numerical answer. ,"The sum of the proton and neutron magnetic moments gives 0.879 µN, which is within 3% of the measured value 0.857 µN. The result is larger than μ by a factor equal to the ratio of the proton to electron mass, or about a factor of 1836. ==See also== * Nucleon magnetic moment ==References== ==External links== *. The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn. The resulting value was not zero and had a sign opposite to that of the proton. In this calculation, the spins of the nucleons are aligned, but their magnetic moments offset because of the neutron's negative magnetic moment. == Nature of the nucleon magnetic moments == thumb|upright|A magnetic dipole moment can be created by either a current loop (top; Ampèrian) or by two magnetic monopoles (bottom; Gilbertian). The key question is how the nucleon's spin, whose magnitude is 1/2ħ, is carried by its constituent partons (quarks and gluons). The g-factor for the proton is 5.6, and the chargeless neutron, which should have no magnetic moment at all, has a g-factor of −3.8. The CODATA recommended value for the magnetic moment of the proton is or The best available measurement for the value of the magnetic moment of the neutron is Here, μN is the nuclear magneton, a standard unit for the magnetic moments of nuclear components, and μB is the Bohr magneton, both being physical constants. Nucleon spin structure describes the partonic structure of nucleon (proton and neutron) intrinsic angular momentum (spin). In SI units, these values are and A magnetic moment is a vector quantity, and the direction of the nucleon's magnetic moment is determined by its spin. The nuclear magneton is the spin magnetic moment of a Dirac particle, a charged, spin-1/2 elementary particle, with a proton's mass p, in which anomalous corrections are ignored. Since for the neutron the sign of γn is negative, the neutron's spin angular momentum precesses counterclockwise about the direction of the external magnetic field. ===Proton nuclear magnetic resonance=== Nuclear magnetic resonance employing the magnetic moments of protons is used for nuclear magnetic resonance (NMR) spectroscopy. The value of nuclear magneton System of units Value Unit SI J·T CGS Erg·G eV eV·T MHz/T (per h) MHz/T The nuclear magneton (symbol μ) is a physical constant of magnetic moment, defined in SI units by: :\mu_\text{N} = {{e \hbar} \over {2 m_\text{p}}} and in Gaussian CGS units by: :\mu_\text{N} = {{e \hbar} \over{2 m_\text{p}c}} where: :e is the elementary charge, :ħ is the reduced Planck constant, :m is the proton rest mass, and :c is the speed of light In SI units, its value is approximately: :μ = In Gaussian CGS units, its value can be given in convenient units as :μ = The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei. Baryon Magnetic moment of quark model Computed (\mu_\text{N}) Observed (\mu_\text{N}) p u − d 2.79 2.793 n d − u −1.86 −1.913 The results of this calculation are encouraging, but the masses of the up or down quarks were assumed to be the mass of a nucleon. Thus, in units of nuclear magneton, for the neutron and for the proton. The nuclear magneton is \mu_\text{N} = \frac{e \hbar}{2 m_\text{p}}, where is the elementary charge, and is the reduced Planck constant. Nucleon magnetic moments have been successfully computed from first principles, requiring significant computing resources. ==See also== * Neutron triple-axis spectrometry * LARMOR neutron microscope * Neutron electric dipole moment * Aharonov–Casher effect ==References== ==Bibliography== * S. W. Lovesey (1986). The magnetic moment of such a particle is parallel to its spin. The nuclear magnetic moment also includes contributions from the orbital motion of the charged protons. The calculation assumes that the quarks behave like pointlike Dirac particles, each having their own magnetic moment, as computed using an expression similar to the one above for the nuclear magneton: \ \mu_\text{q} = \frac{\ e_\text{q} \hbar\ }{2 m_\text{q}}\ , where the q-subscripted variables refer to quark magnetic moment, charge, or mass. For a nucleus of which the numbers of protons and of neutrons are both even in its ground state (i.e. lowest energy state), the nuclear spin and magnetic moment are both always zero. The magnetic moment is calculated through j, l and s of the unpaired nucleon, but nuclei are not in states of well defined l and s. ",30,,"""0.21""",9.13,14.5115,D
+"The ${ }^7 \mathrm{Li}^1 \mathrm{H}$ ground electronic state has $D_0=2.4287 \mathrm{eV}, \nu_e / c=1405.65 \mathrm{~cm}^{-1}$, and $\nu_e x_e / c=23.20 \mathrm{~cm}^{-1}$, where $c$ is the speed of light. (These last two quantities are usually designated $\omega_e$ and $\omega_e x_e$ in the literature.) Calculate $D_e$ for ${ }^7 \mathrm{Li}^1 \mathrm{H}$.","The calculated abundance and ratio of 1H and 4He is in agreement with data from observations of young stars. ===The P-P II branch=== In stars, lithium-7 is made in a proton-proton chain reaction. thumb|Proton–proton II chain reaction :{| border=""0"" |- style=""height:2em;"" | ||+ || ||→ || ||+ || |- style=""height:2em;"" | ||+ || ||→ || ||+ || ||+ || ||/ || |- style=""height:2em;"" | ||+ || ||→ ||2 |} The P-P II branch is dominant at temperatures of 14 to . thumb|right|400px|Stable nuclides of the first few elements ==Observed abundance of lithium== Despite the low theoretical abundance of lithium, the actual observable amount is less than the calculated amount by a factor of 3–4. thumb|250px|7Li NMR spectrum of LiCl (1M) in D2O. However, they didn't use the Mössbauer effect but made magnetic resonance measurements of the nucleus of lithium-7, whose ground state possesses a spin of . E7, E07, E-7 or E7 may refer to: ==Science and engineering== * E7 liquid crystal mixture * E7, the Lie group in mathematics * E7 polytope, in geometry * E7 papillomavirus protein * E7 European long distance path ==Transport== * EMD E7, a diesel locomotive * European route E07, an international road * Peugeot E7, a hackney cab * PRR E7, a steam locomotive * Carbon Motors E7,a police car * E7 series, a Japanese high-speed train * Nihonkai-Tōhoku Expressway and Akita Expressway (between Kawabe JCT and Kosaka JCT), route E7 in Japan * Cheras–Kajang Expressway, route E7 in Malaysia ==Other uses== * Boeing E-7, either: ** Boeing E-7 ARIA, the original designation assigned by the United States Air Force under the Mission Designation System to the EC-18B Advanced Range Instrumentation Aircraft. The SD7 is a model of 6-axle diesel locomotive built by General Motors Electro-Motive Division between May 1951 and November 1953. The molecular formula C7H7I (molar mass: 218.03 g/mol) may refer to: * Benzyl iodide * Iodotoluene Other isotopes including 2H, 3H, 3He, 6Li, 7Li, and 7Be are much rarer; the estimated abundance of primordial lithium is 10−10 relative to hydrogen. Though it transmutes into two atoms of helium due to collision with a proton at temperatures above 2.4 million degrees Celsius (most stars easily attain this temperature in their interiors), lithium is more abundant than current computations would predict in later-generation stars. The discrepancy is highlighted in a so-called ""Schramm plot"", named in honor of astrophysicist David Schramm, which depicts these primordial abundances as a function of cosmic baryon content from standard BBN predictions. ==Origin of lithium== Minutes after the Big Bang, the universe was made almost entirely of hydrogen and helium, with trace amounts of lithium and beryllium, and negligibly small abundances of all heavier elements. ===Lithium synthesis in the Big Bang=== Big Bang nucleosynthesis produced both lithium-7 and beryllium-7, and indeed the latter dominates the primordial synthesis of mass 7 nuclides. In summary, accurate measurements of the primordial lithium abundance is the current focus of progress, and it could be possible that the final answer does not lie in astrophysical solutions. The ground state is split into four equally spaced magnetic energy levels when measured in a magnetic field in accordance with its allowed magnetic quantum number. They test the framework of Tsallis non-extensive statistics.Their result suggest that 1.069 === Nuclear physics solutions === When one considers the possibility that the measured primordial lithium abundance is correct and based on the Standard Model of particle physics and the standard cosmology, the lithium problem implies errors in the BBN light element predictions. Namely, the most widely accepted models of the Big Bang suggest that three times as much primordial lithium, in particular lithium-7, should exist. EMD ended production in November 1953 and began producing the SD7's successor, the SD9, in January 1954. == Original buyers == Owner Quantity Numbers Notes Electro-Motive Division 2 990 to Southern Pacific 5308 then 2715 to 1415 ne 1518 Electro-Motive Division 2 991 to Baltimore and Ohio 760 Baltimore and Ohio Railroad 4 761–764 These units were built with the 567BC engine. Since a series of coefficients are tested in those experiments, only the value of maximal sensitivity is given (for precise data, see the individual articles): Author Year SME constraints SME constraints SME constraints Description Author Year Proton Neutron Electron Description Prestage et al. 1985 10−27 Comparing the nuclear spin-flip transition of (stored in a penning trap) with a hydrogen maser transition. EMD SD7 Original Owners. Retrieved on August 27, 2006 * Diesel Era Volume 6 Number 6 November/December 1995, ""EMD's SD7"" by Paul K. Withers pp 5-20; 47-50. == External links == * Locomotive Truck EMD FlexiCoil C SD07 Category:C-C locomotives Category:Diesel-electric locomotives of the United States Category:Railway locomotives introduced in 1952 Category:Freight locomotives Category:Standard gauge locomotives of the United States Spectroscopic observations of stars in NGC 6397, a metal-poor globular cluster, are consistent with an inverse relation between lithium abundance and age, but a theoretical mechanism for diffusion has not been formalized. EMD produced its first examples of the SD7 in May 1951, using the 567B engine. Furthermore, more observations on lithium depletion remain important since present lithium levels might not reflect the initial abundance in the star. This was the first model in EMD's SD (Special Duty) series of locomotives, a lengthened B-B GP7 with a C-C truck arrangement. == Heat of vaporization == kJ/mol 1 H hydrogen (H2) use (H2) 0.904 CRC (H2) 0.90 LNG 0.904 2 He helium use 0.0829 CRC 0.08 LNG 0.0829 WEL 0.083 3 Li lithium use 136 LNG 147.1 WEL 147 Zhang et al. 136 4 Be beryllium use 292 LNG 297 WEL 297 Zhang et al. 292 5 B boron use 508 CRC 480 LNG 480 WEL 507 Zhang et al. 508 6 C carbon use WEL (sublimation) 715 7 N nitrogen (N2) use (N2) 5.57 CRC (N2) 5.57 LNG (N2) 5.577 WEL (per mole N atoms) 2.79 Zhang et al. 5.58 8 O oxygen (O2) use (O2) 6.82 CRC (O2) 6.82 LNG (O2) 6.820 WEL (per mole O atoms) 3.41 Zhang et al. 6.82 9 F fluorine (F2) use (F2) 6.62 CRC (F2) 6.62 LNG (F2) 6.62 WEL (per mole F atoms) 3.27 Zhang et al. 6.32 10 Ne neon use 1.71 CRC 1.71 LNG 1.71 WEL 1.75 Zhang et al. 1.9 11 Na sodium use 97.42 LNG 97.42 WEL 97.7 Zhang et al. 97.4 12 Mg magnesium use 128 LNG 128 WEL 128 Zhang et al. 132 13 Al aluminium use 284 CRC 294 LNG 294.0 WEL 293 Zhang et al. 284 14 Si silicon use 359 LNG 359 WEL 359 Zhang et al. 383 15 P phosphorus use 12.4 CRC 12.4 LNG 12.4 WEL 12.4 Zhang et al. 51.9 (white) 16 S sulfur use (mono) 45 CRC 45 LNG (mono) 45 WEL 9.8 Zhang et al. 45 17 Cl chlorine (Cl2) use (Cl2) 20.41 CRC (Cl2) 20.41 LNG (Cl2) 20.41 WEL (per mole Cl atoms) 10.2 Zhang et al. 20.4 18 Ar argon use 6.53 CRC 6.43 LNG 6.43 WEL 6.5 Zhang et al. 6.53 19 K potassium use 76.90 LNG 76.90 WEL 76.9 Zhang et al. 79.1 20 Ca calcium use 154.7 LNG 154.7 WEL 155 Zhang et al. 153 21 Sc scandium use 332.7 LNG 332.7 WEL 318 Zhang et al. 310 22 Ti titanium use 425 LNG 425 WEL 425 Zhang et al. 427 23 V vanadium use 444 LNG 459 WEL 453 Zhang et al. 451 24 Cr chromium use 339.5 LNG 339.5 WEL 339 Zhang et al. 347 25 Mn manganese use 221 LNG 221 WEL 220 Zhang et al. 225 26 Fe iron use 340 LNG 340 WEL 347 Zhang et al. 354 27 Co cobalt use 377 LNG 377 WEL 375 Zhang et al. 390 28 Ni nickel use 379 LNG 377.5 WEL 378 Zhang et al. 379 29 Cu copper use 300.4 LNG 300.4 WEL 300 Zhang et al. 305 30 Zn zinc use 115 LNG 123.6 WEL 119 Zhang et al. 115 31 Ga gallium use 256 CRC 254 LNG 254 WEL 256 Zhang et al. 256 32 Ge germanium use 334 CRC 334 LNG 334 WEL 334 Zhang et al. 330 33 As arsenic use 32.4 WEL 32.4 (sublimation) Zhang et al. 32.4 34 Se selenium use 95.48 CRC 95.48 LNG 95.48 WEL 26 Zhang et al. 95.5 35 Br bromine (Br2) use (Br2) 29.96 CRC (Br2) 29.96 LNG (Br2) 29.96 WEL (per mole Br atoms) 14.8 Zhang et al. 30 36 Kr krypton use 9.08 CRC 9.08 LNG 9.08 WEL 9.02 Zhang et al. 9.03 37 Rb rubidium use 75.77 LNG 75.77 WEL 72 Zhang et al. 69 38 Sr strontium use 141 LNG 136.9 WEL 137 Zhang et al. 141 39 Y yttrium use 390 LNG 365 WEL 380 Zhang et al. 363 40 Zr zirconium use 573 LNG 573 WEL 580 Zhang et al. 591 41 Nb niobium use 689.9 LNG 689.9 WEL 690 Zhang et al. 682 42 Mo molybdenum use 617 LNG 617 WEL 600 Zhang et al. 617 43 Tc technetium use 585.2 LNG 585.2 WEL 550 44 Ru ruthenium use 619 LNG 591.6 WEL 580 Zhang et al. 619 45 Rh rhodium use 494 LNG 494 WEL 495 Zhang et al. 493 46 Pd palladium use 358 LNG 362 WEL 380 Zhang et al. 358 47 Ag silver use 254 LNG 258 WEL 255 Zhang et al. 254 48 Cd cadmium use 99.87 CRC 99.87 LNG 99.9 WEL 100 Zhang et al. 100 49 In indium use 231.8 LNG 231.8 WEL 230 Zhang et al. 225 50 Sn tin use (white) 296.1 LNG (white) 296.1 WEL 290 Zhang et al. 296 51 Sb antimony use 193.43 LNG 193.43 WEL 68 Zhang et al. 193 52 Te tellurium use 114.1 CRC 114.1 LNG 114.1 WEL 48 Zhang et al. 114 53 I iodine (I2) use (I2) 41.57 CRC (I2) 41.57 LNG (I2) 41.6 WEL (per mole I atoms) 20.9 Zhang et al. 41.6 54 Xe xenon use 12.64 CRC 12.57 LNG 12.64 WEL 12.64 Zhang et al. 12.6 55 Cs caesium use 63.9 LNG 63.9 WEL 65 Zhang et al. 66.1 56 Ba barium use 140.3 CRC 140 LNG 140.3 WEL 140 Zhang et al. 142 57 La lanthanum use 400 LNG 402.1 WEL 400 Zhang et al. 400 58 Ce cerium use 398 LNG 398 WEL 350 Zhang et al. 398 59 Pr praseodymium use 331 LNG 331 WEL 330 Zhang et al. 331 60 Nd neodymium use 289 LNG 289 WEL 285 Zhang et al. 289 61 Pm promethium use 289 LNG 289 WEL 290 62 Sm samarium use 192 LNG 165 WEL 175 Zhang et al. 192 63 Eu europium use 176 LNG 176 WEL 175 Zhang et al. 176 64 Gd gadolinium use 301.3 LNG 301.3 WEL 305 Zhang et al. 301 65 Tb terbium use 391 LNG 293 WEL 295 Zhang et al. 391 66 Dy dysprosium use 280 LNG 280 WEL 280 Zhang et al. 280 67 Ho holmium use 251 LNG 71 WEL 265 Zhang et al. 251 68 Er erbium use 280 LNG 280 WEL 285 Zhang et al. 280 69 Tm thulium use 191 LNG 247 WEL 250 Zhang et al. 191 70 Yb ytterbium use 129 LNG 159 WEL 160 Zhang et al. 129 71 Lu lutetium use 414 LNG 414 WEL 415 Zhang et al. 414 72 Hf hafnium use 648 LNG 571 WEL 630 Zhang et al. 648 73 Ta tantalum use 732.8 LNG 732.8 WEL 735 Zhang et al. 753 74 W tungsten use 806.7 LNG 806.7 WEL 800 Zhang et al. 774 75 Re rhenium use 704 LNG 704 WEL 705 Zhang et al. 707 76 Os osmium use 678 LNG 738 WEL 630 Zhang et al. 678 77 Ir iridium use 564 LNG 231.8 WEL 560 Zhang et al. 564 78 Pt platinum use 510 LNG 469 WEL 490 Zhang et al. 510 79 Au gold use 342 CRC 324 LNG 324 WEL 330 Zhang et al. 342 80 Hg mercury use 59.11 CRC 59.11 LNG 59.1 WEL 59.2 Zhang et al. 58.2 81 Tl thallium use 165 LNG 165 WEL 165 Zhang et al. 162 82 Pb lead use 179.5 CRC 179.5 LNG 179.5 WEL 178 Zhang et al. 177 83 Bi bismuth use 179 CRC 151 LNG 151 WEL 160 Zhang et al. 179 84 Po polonium use 102.91 LNG 102.91 WEL about 100 85 At astatine use 54.39 (At2) GME 54.39 (At2) WEL about 40 86 Rn radon use 18.10 LNG 18.10 WEL 17 87 Fr francium use ca. 65 WEL about 65 88 Ra radium use 113 LNG 113 WEL about 125 89 Ac actinium use 400 WEL 400 90 Th thorium use 514 LNG 514 WEL 530 91 Pa protactinium use 481 LNG 481 WEL 470 92 U uranium use 417.1 LNG 417.1 WEL 420 93 Np neptunium use 336 LNG 336 WEL 335 94 Pu plutonium use 333.5 LNG 333.5 WEL 325 == Notes == * Values refer to the enthalpy change in the conversion of liquid to gas at the boiling point (normal, 101.325 kPa). == References == ===Zhang et al.=== ===CRC=== As quoted from various sources in an online version of: * David R. Lide (ed.), CRC Handbook of Chemistry and Physics, 84th Edition. ",260,2.5151,"""96.4365076099""",3.0,-1368,B
+"The positron has charge $+e$ and mass equal to the electron mass. Calculate in electronvolts the ground-state energy of positronium-an ""atom"" that consists of a positron and an electron.","While the electron has a negative electric charge, the positron has a positive electric charge, and is produced naturally in certain types of radioactive decay. Thus, for positronium, its reduced mass only differs from the electron by a factor of 2. So finally, the energy levels of positronium are given by E_n = -\frac{1}{2} \frac{m_\mathrm{e} q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2} = \frac{-6.8~\mathrm{eV}}{n^2}. Positronium can also be considered by a particular form of the two-body Dirac equation; Two particles with a Coulomb interaction can be exactly separated in the (relativistic) center-of-momentum frame and the resulting ground-state energy has been obtained very accurately using finite element methods of Janine Shertzer. The electron is commonly symbolized by , and the positron is symbolized by . However, because of the reduced mass, the frequencies of the spectral lines are less than half of those for the corresponding hydrogen lines. ==States== The mass of positronium is 1.022 MeV, which is twice the electron mass minus the binding energy of a few eV. The antiparticle of the electron is called the positron; it is identical to the electron, except that it carries electrical charge of the opposite sign. Approximately: * ~60% of positrons will directly annihilate with an electron without forming positronium. In this approximation, the energy levels are different because of a different effective mass, μ, in the energy equation (see electron energy levels for a derivation): E_n = -\frac{\mu q_\mathrm{e}^4}{8 h^2 \varepsilon_0^2} \frac{1}{n^2}, where: * is the charge magnitude of the electron (same as the positron), * is the Planck constant, * is the electric constant (otherwise known as the permittivity of free space), * is the reduced mass: \mu = \frac{m_\mathrm{e} m_\mathrm{p}}{m_\mathrm{e} + m_\mathrm{p}} = \frac{m_\mathrm{e}^2}{2m_\mathrm{e}} = \frac{m_\mathrm{e}}{2}, where and are, respectively, the mass of the electron and the positron (which are the same by definition as antiparticles). The lowest energy level of positronium () is . Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. In 2012, Cassidy et al. were able to produce the excited molecular positronium L = 1 angular momentum state. ==See also== *Hydrogen molecule *Hydrogen molecular ion *Positronium *Protonium *Exotic atom ==References== ==External links== *Molecules of Positronium Observed in the Laboratory for the First Time, press release, University of California, Riverside, September 12, 2007. Upon slowing down in the silica, the positrons captured ordinary electrons to form positronium atoms. The electron's mass is approximately 1/1836 that of the proton. If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons totalling 1.022 MeV. thumb|350px|Naturally occurring electron-positron annihilation as a result of beta plus decay Electron–positron annihilation occurs when an electron () and a positron (, the electron's antiparticle) collide. The lowest energy orbital state of positronium is 1S, and like with hydrogen, it has a hyperfine structure arising from the relative orientations of the spins of the electron and the positron. The resulting weakly bound electron and positron spiral inwards and eventually annihilate, with an estimated lifetime of years. == See also == * Breit equation * Antiprotonic helium * Di-positronium * Quantum electrodynamics * Protonium * Two-body Dirac equations == References == == External links == * The annihilation of positronium - The Feynman Lectures on Physics * The Search for Positronium * Obituary of Martin Deutsch, discoverer of Positronium Category:Molecular physics Category:Quantum electrodynamics Category:Spintronics Category:Onia Category:Antimatter Category:Substances discovered in the 1950s The electron (symbol e) is on the left. The opposite is also true: the antiparticle of the positron is the electron. For example, the antiparticle of the electron is the positron (also known as an antielectron). Positron paths in a cloud- chamber trace the same helical path as an electron but rotate in the opposite direction with respect to the magnetic field direction due to their having the same magnitude of charge-to-mass ratio but with opposite charge and, therefore, opposite signed charge-to-mass ratios. ",-0.28, -6.8,"""50.7""",0.9974,0.14,B
+What is the value of the angular-momentum quantum number $l$ for a $t$ orbital?,"Here is the total orbital angular momentum quantum number. An atomic electron's angular momentum, L, is related to its quantum number ℓ by the following equation: \mathbf{L}^2\Psi = \hbar^2 \ell(\ell + 1) \Psi, where ħ is the reduced Planck's constant, L2 is the orbital angular momentum operator and \Psi is the wavefunction of the electron. When referring to angular momentum, it is better to simply use the quantum number ℓ. The associated quantum number is the main total angular momentum quantum number j. Each orbital is characterized by its number , where takes integer values from to , and its angular momentum number , where takes integer values from to . It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as (pronounced ell). == Derivation == Connected with the energy states of the atom's electrons are four quantum numbers: n, ℓ, mℓ, and ms. The quantum number ℓ is always a non-negative integer: 0, 1, 2, 3, etc. L has no real meaning except in its use as the angular momentum operator. It can take the following range of values, jumping only in integer steps: \vert \ell - s\vert \le j \le \ell + s where ℓ is the azimuthal quantum number (parameterizing the orbital angular momentum) and s is the spin quantum number (parameterizing the spin). In quantum mechanics, the total angular momentum quantum number parametrises the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin). In general, the values of range from to , where is the spin quantum number, associated with the particle's intrinsic spin angular momentum: :. Furthermore, the eigenvectors of j, s, mj and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of ℓ, s, mℓ and ms. == List of angular momentum quantum numbers == * Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number * orbital angular momentum quantum number (the subject of this article) * magnetic quantum number, related to the orbital momentum quantum number * total angular momentum quantum number == History == The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld. This number gives the information about the direction of spinning of the electron present in any orbital. Shape of orbital is also given by azimuthal quantum number. ====Magnetic quantum number==== The magnetic quantum number describes the specific orbital (or ""cloud"") within that subshell, and yields the projection of the orbital angular momentum along a specified axis: : The values of range from to , with integer intervals. Simultaneous measurement of electron energy and orbital angular momentum is allowed because the Hamiltonian commutes with the angular momentum operator related to L_z. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is \| \bold{s} \| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\ \hbar ~. The value of is the component of spin angular momentum, in units of the reduced Planck constant , parallel to a given direction (conventionally labelled the –axis). Depending on the value of n, there is an angular momentum quantum number ℓ and the following series. The value of ranges from 0 to , so the first p orbital () appears in the second electron shell (), the first d orbital () appears in the third shell (), and so on: : A quantum number beginning in = 3, = 0, describes an electron in the s orbital of the third electron shell of an atom. If s is the particle's spin angular momentum and ℓ its orbital angular momentum vector, the total angular momentum j is \mathbf j = \mathbf s + \boldsymbol {\ell} ~. These were identified as, respectively, the electron ""shell"" number , the ""orbital"" number , and the ""orbital angular momentum"" number . The azimuthal quantum number can also denote the number of angular nodes present in an orbital. Each of the different angular momentum states can take 2(2ℓ + 1) electrons. ",292,14,"""0.0""",0.70710678,11,B
+How many states belong to the carbon configurations $1 s^2 2 s^2 2 p^2$?,"thumb|upright=1.3|Carbon dioxide pressure-temperature phase diagram Supercritical carbon dioxide (s) is a fluid state of carbon dioxide where it is held at or above its critical temperature and critical pressure. secondary Carbon 150x150px Structural formula of propane (secondary carbon is highlighted red) A secondary carbon is a carbon atom bound to two other carbon atoms. Franckeite, chemical formula Pb5Sn3Sb2S14, belongs to a family of complex sulfide minerals. thumb|Schematic of a binary star system with one planet on an S-type orbit and one on a P-type orbit. quaternary carbon 150x150px Structural formula of neopentane (quaternary carbon is highlighted red) A quaternary carbon is a carbon atom bound to four other carbon atoms. For this reason, quaternary carbon atoms are found only in hydrocarbons having at least five carbon atoms. Quaternary carbon atoms can occur in branched alkanes, but not in linear alkanes. primary carbon secondary carbon tertiary carbon quaternary carbon General structure (R = Organyl group) frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px Partial Structural formula frameless=1.0|96x96px frameless=1.0|102x102px frameless=1.0|98x98px frameless=1.0|105x105px == Synthesis == The formation of chiral quaternary carbon centers has been a synthetic challenge. In unbranched alkanes, the inner carbon atoms are always secondary carbon atoms (see figure). primary carbon secondary carbon tertiary carbon quaternary carbon General structure (R = Organyl group) frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px frameless=1.0|85x85px Partial Structural formula frameless=1.0|96x96px frameless=1.0|102x102px frameless=1.0|98x98px frameless=1.0|105x105px == References == Category:Chemical nomenclature Category:Organic chemistry For this reason, secondary carbon atoms are found in all hydrocarbons having at least three carbon atoms. The limits of stability for S-type and P-type orbits within binary as well as trinary stellar systems have been established as a function of the orbital characteristics of the stars, for both prograde and retrograde motions of stars and planets. ==See also== *Astrobiology *Circumstellar habitable zone *Habitability of yellow dwarf systems *Planetary habitability *Circumbinary planet ==References== Binary star systems Category:Binary stars Red atoms are oxygens. thumb|upright=1.3|TEM images of amorphous HBS Two-dimensional silica (2D silica) is a layered polymorph of silicon dioxide. Planets that orbit just one star in a binary pair are said to have ""S-type"" orbits, whereas those that orbit around both stars have ""P-type"" or ""circumbinary"" orbits. Typical estimates often suggest that 50% or more of all star systems are binary systems. Habitability of binary star systems is determined by many factors from a variety of sources. The planets have semi-major axes that lie between 1.09 and 1.46 times this critical radius. Carbon dioxide usually behaves as a gas in air at standard temperature and pressure (STP), or as a solid called dry ice when cooled and/or pressurised sufficiently. This would have implications for bulk thermal and nuclear generation of electricity, because the supercritical properties of carbon dioxide at above 500 °C and 20 MPa enable thermal efficiencies approaching 45 percent. thumb|upright=1.3|Top and side views of graphene (left) and HBS structures (right). The minimum stable star-to-circumbinary- planet separation is about 2–4 times the binary star separation, or orbital period about 3–8 times the binary period. Volume 2001, Issue 40 , Pages 2482–2486 Heck reaction, Enyne cyclization, cycloaddition reactions, Quasdorf, K.W.; Overman, L. E. Nature Volume 2014, Volume 516, Pages 181 C–H activation, Allylic substitution, Pauson–Khand reaction, Ishizaki, M.; Niimi, Y.; Hoshino, O.; Hara, H.; Takahashi, T. Tetrahedron Volume 2001, Issue 61, Pages 4053–4065 etc. to construct asymmetric quaternary carbon atoms. == References == Category:Chemical nomenclature Category:Organic chemistry For example, Kepler-47c is a gas giant in the circumbinary habitable zone of the Kepler-47 system. It was shown to be a member of the auxetics materials family with a negative Poisson's ratio. ==References== Category:Two-dimensional nanomaterials Category:Silicon dioxide Category:Silica polymorphs ",50.7,0.38,"""15.0""",12,7,C
+Calculate the energy needed to compress three carbon-carbon single bonds and stretch three carbon-carbon double bonds to the benzene bond length $1.397 Å$. Assume a harmonicoscillator potential-energy function for bond stretching and compression. Typical carboncarbon single- and double-bond lengths are 1.53 and $1.335 Å$; typical stretching force constants for carbon-carbon single and double bonds are 500 and $950 \mathrm{~N} / \mathrm{m}$.,"Bond energy (BE) is the average of all bond-dissociation energies of a single type of bond in a given molecule.Madhusha (2017), Difference Between Bond Energy and Bond Dissociation Energy, Pediaa, Difference Between Bond Energy and Bond Dissociation Energy The bond-dissociation energies of several different bonds of the same type can vary even within a single molecule. IUPAC defines bond energy as the average value of the gas-phase bond-dissociation energy (usually at a temperature of 298.15 K) for all bonds of the same type within the same chemical species. The molecule has eight bond lengths ranging between 0.137 and 0.146 nm. right|thumb|illustrative example of C-C length molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C angle molecular energy dependence, numerical accuracy is not guaranteed right|thumb|illustrative example of C-C-C-C torsion molecular energy dependence, numerical accuracy is not guaranteed A Molecular spring is a device or part of a biological system based on molecular mechanics and is associated with molecular vibration. Applying a harmonic approximation to the potential minimum (at V(r_m) = -\varepsilon), the exponent n and the energy parameter \varepsilon can be related to the harmonic spring constant: k = 2 \varepsilon \left(\frac{n}{r_0}\right)^2, from which n can be calculated if k is known. In chemistry, bond energy (BE), also called the mean bond enthalpyClark, J (2013), BOND ENTHALPY (BOND ENERGY), Chemguide, BOND ENTHALPY (BOND ENERGY) or average bond enthalpy is a measure of bond strength in a chemical bond. Most authors prefer to use the BDE values at 298.15 K.Luo, Yu-Ran and Jin-Pei Cheng ""Bond Dissociation Energies"". The first reduction requires around 1.0 V (Fc/), indicating that C70 is an electron acceptor. === Solution === Saturated solubility of C70 (S, mg/mL) Solvent S (mg/mL) 1,2-dichlorobenzene 36.2 carbon disulfide 9.875 xylene 3.985 toluene 1.406 benzene 1.3 carbon tetrachloride 0.121 n-hexane 0.013 cyclohexane 0.08 pentane 0.002 octane 0.042 decane 0.053 dodecane 0.098 heptane 0.047 isopropanol 0.0021 mesitylene 1.472 dichloromethane 0.080 Fullerenes are sparingly soluble in many aromatic solvents such as toluene and others like carbon disulfide, but not in water. C70 fullerene is the fullerene molecule consisting of 70 carbon atoms. This step yields a solution containing up to 70% of C60 and 15% of C70, as well as other fullerenes. For example, the carbon–hydrogen bond energy in methane BE(C–H) is the enthalpy change (∆H) of breaking one molecule of methane into a carbon atom and four hydrogen radicals, divided by four. Graph of the Lennard-Jones potential function: Intermolecular potential energy as a function of the distance of a pair of particles. In addition, the force needed to draw molecular string to its maximum length could be impractically high - comparable to the tensile strength of particular polymer molecule (~100GPa for some carbon compounds) == See also == *Ultra high molecular weight polyethylene *Carbon nanotube *Carbon nanotube springs ==References == * Stretching molecular springs:elasticity of titin filaments in vertebrate striated muscle, W.A. Linke, Institute of Physiology II, University of Heidelberg, Heidelberg, Germany Category:Nanotechnology Category:Molecular physics Valence bond (VB) computer programs for modern valence bond calculations:- * CRUNCH, by Gordon A. Gallup and his group. The bond energy for H2O is the average of energy required to break each of the two O–H bonds in sequence: : \begin{array}{lcl} \mathrm{H-O-H} & \rightarrow & \mathrm{H\cdot + \cdot O-H} & , D_1 \\\ \mathrm{\cdot O-H} & \rightarrow & \mathrm{\cdot O\cdot + \cdot H} & , D_2 \\\ \mathrm{H-O-H} & \rightarrow & \mathrm{H\cdot + \cdot O\cdot + \cdot H} & , D =(D_1 + D_2)/2 \\\ \end{array} Although the two bonds are the equivalent in the original symmetric molecule, the bond-dissociation energy of an oxygen–hydrogen bond varies slightly depending on whether or not there is another hydrogen atom bonded to the oxygen atom. Each carbon atom in the structure is bonded covalently with 3 others. thumb|left|The structure of C70 molecule. A related fullerene molecule, named buckminsterfullerene (C60 fullerene), consists of 60 carbon atoms. The amount of energy storable in molecular spring is limited by the value of deformation the molecule can withstand until it undergoes chemical change. The resulting structural unit [-C≡(-CH2-)3≡C-] is a rigid cage, consisting of two carbon atoms joined by three methylene bridges; therefore the joined units are constrained to lie on a straight line. The bond dissociation energy (enthalpy) is also referred to as bond disruption energy, bond energy, bond strength, or binding energy (abbreviation: BDE, BE, or D). * Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites (same \varepsilon, \sigma, m) that are bonded as a rigid body. For the LJTS potential with r_\mathrm{end} = 2.5\,\sigma , the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: V_\mathrm{LJ}(r_\mathrm{end} = 2.5\,\sigma) = -0.0163\,\varepsilon . ",30,27,"""1.6""",-20,22.2036033112,B
+"When a particle of mass $9.1 \times 10^{-28} \mathrm{~g}$ in a certain one-dimensional box goes from the $n=5$ level to the $n=2$ level, it emits a photon of frequency $6.0 \times 10^{14} \mathrm{~s}^{-1}$. Find the length of the box.","In particle physics, the radiation length is a characteristic of a material, related to the energy loss of high energy particles electromagnetically interacting with it. The radiation length for a given material consisting of a single type of nucleus can be approximated by the following expression: (http://pdg.lbl.gov/) X_0 = 716.4 \text{ g cm}^{-2} \frac{A}{Z (Z+1) \ln{\frac{287}{\sqrt{Z}}}} = 1433 \text{ g cm}^{-2} \frac{A}{Z (Z+1) (11.319 - \ln{Z})}, where is the atomic number and is mass number of the nucleus. This length is useful for renormalizing a non-isotropic scattering problem into an isotropic one in order to use classical diffusion laws (Fick law and Brownian motion). The four-frequency of a massless particle, such as a photon, is a four-vector defined by :N^a = \left( u, u \hat{\mathbf{n}} \right) where u is the photon's frequency and \hat{\mathbf{n}} is a unit vector in the direction of the photon's motion. The characteristic amount of matter traversed for these related interactions is called the radiation length , usually measured in g·cm−2. The parameter a_s of dimension length is defined as the scattering length. The transport length in a strongly diffusing medium (noted l*) is the length over which the direction of propagation of the photon is randomized. The scattering length in quantum mechanics describes low-energy scattering. To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section \sigma. It is also the appropriate length scale for describing high-energy electromagnetic cascades. Comprehensive tables for radiation lengths and other properties of materials are available from the Particle Data Group. ==See also== * Mean free path * Attenuation length * Attenuation coefficient * Attenuation * Range (particle radiation) * Stopping power (particle radiation) * Electron energy loss spectroscopy ==References== Category:Experimental particle physics The 10.5 cm leFH 18M ( ""light field howitzer"") was a German light howitzer used in the Second World War. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the multipole expansion in classical electrodynamics), where one expands in the angular momentum components of the outgoing wave. It is defined as the mean length (in cm) into the material at which the energy of an electron is reduced by the factor 1/e. ==Definition== In materials of high atomic number (e.g. tungsten, uranium, plutonium) the electrons of energies >~10 MeV predominantly lose energy by , and high-energy photons by pair production. The transport length might be measured by transmission experiments and backscattering experiments. For , a good approximation is \frac{1}{X_0} = 4 \left( \frac{\hbar}{m_\mathrm{e} c} \right)^2 Z(Z+1) \alpha^3 n_\mathrm{a} \log\left(\frac{183}{Z^{1/3}}\right), where * is the number density of the nucleus, *\hbar denotes the reduced Planck constant, * is the electron rest mass, * is the speed of light, * is the fine-structure constant. thumb|230px|The three leading Slepian sequences for T=1000 and 2WT=6. For our potential we have therefore a=r_0, in other words the scattering length for a hard sphere is just the radius. The concept of the scattering length can also be extended to potentials that decay slower than 1/r^3 as r\to \infty. It is both the mean distance over which a high- energy electron loses all but of its energy by , and of the mean free path for pair production by a high-energy photon. An observer moving with four-velocity V^b will observe a frequency :\frac{1}{c}\eta\left(N^a, V^b\right) Where \eta is the Minkowski inner-product (+−−−) Closely related to the four-frequency is the four-wavevector defined by :K^a = \left(\frac{\omega}{c}, \mathbf{k}\right) where \omega = 2 \pi u, c is the speed of light and \mathbf{k} = \frac{2 \pi}{\lambda}\hat{\mathbf{n}} and \lambda is the wavelength of the photon. Sterling Publishing Company, Inc., 2002, p.144 ==History== The 10.5 cm leFH 18M superseded the 10.5 cm leFH 18 as the standard German divisional field howitzer used during the Second World War. ",-11.2,0.6321205588,"""1.8""",2.3613,0.4908,C
+Use the normalized Numerov-method harmonic-oscillator wave functions found by going from -5 to 5 in steps of 0.1 to estimate the probability of being in the classically forbidden region for the $v=0$ state.,"\\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. {\pi R^2} + \frac{\arcsin\\!\left(\frac{x}{R}\right)}{\pi}\\! for -R\leq x \leq R| mean =0\,| median =0\,| mode =0\,| variance =\frac{R^2}{4}\\!| skewness =0\,| kurtosis =-1\,| entropy =\ln (\pi R) - \frac12 \,| mgf =2\,\frac{I_1(R\,t)}{R\,t}| char =2\,\frac{J_1(R\,t)}{R\,t}| }} The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)={2 \over \pi R^2}\sqrt{R^2-x^2\,}\, for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Phys. 2009, 39, 337–356 The normal distribution is recovered as q → 1\. If we denote by u_{0\ell} the wave function subject to the given potential with total energy E=0 and azimuthal quantum number \ell, the Sturm Oscillation Theorem implies that N_\ell equals the number of nodes of u_{0\ell}. The entropy is calculated as H_{N}(n)=\int_{-1}^{+1} f_{X}(x;n)\ln (f_{X}(x;n))dx The first 5 moments (n=-1 to 3), such that R=1 are \ -\ln(2/\pi) ; n=-1 \ -\ln(2) ;n=0 \ -1/2+\ln(\pi) ;n=1 \ 5/3-\ln(3) ;n=2 \ -7/4-\ln(1/3\pi) ; n=3 == N-sphere Wigner distribution with odd symmetry applied == The marginal PDF distribution with odd symmetry is f{_X}(x;n) ={(1-x^2)^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt{\pi}} \Gamma((n+1)/2)}\sgn(x)\, ; such that R=1 Hence, the CF is expressed in terms of Struve functions CF(t;n) ={ \Gamma(n/2+1) H_{n/2}(t)/(t/2)^{(n/2)} }\, \urcorner (n>=-1); ""The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by"" Z= { \rho c \pi a^2 [R_1 (2ka)-i X_1 (2 ka)], } R_1 ={1-{2 J_1(x) \over 2x} , } X_1 ={{2 H_1(x) \over x} , } == Example (Normalized Received Signal Strength): quadrature terms == The normalized received signal strength is defined as |R| ={{1 \over N} | }\sum_{k=1}^N \exp [i x_n t]| and using standard quadrature terms x ={{1 \over N} }\sum_{k=1}^N \cos ( x_n t) y ={{1 \over N} }\sum_{k=1}^N \sin ( x_n t) Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining {\sqrt{x^2+y^2}}=x+{3 \over 2}y^2-{3 \over 2}xy^2+{1 \over 2}x^2y^2 + O(y^3) +O(y^3)(x-1) +O(y^3)(x-1)^2 +O(x-1)^3 The expanded form of the Characteristic function of the received signal strength becomes E[x] = {1\over N }CF(t;n) E[y^2] ={1\over 2 N}(1 - CF(2t;n)) E[x^2] ={1\over 2N}(1 + CF(2t;n)) E[xy^2] = {t^2 \over 3N^2} CF(t;n)^3+({N-1 \over 2N^2})(1-t CF(2t;n))CF(t;n) E[x^2y^2] = {1\over 8N^3} (1-CF(4t;n))+({N-1 \over 4N^3})(1-CF(2t;n)^2) +({N-1 \over 3N^3})t^2CF(t;n)^4 +({(N-1)(N-2)\over N^3})CF(t;n)^2(1-CF(2t;n)) == See also == * Wigner surmise * The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity. thumb|right|Figure 1 Part of a semi-Markovian discrete system in one dimension with directional jumping time probability density functions (JT-PDFs), including ""death"" terms (the JT-PDFs from state i in state I). Let V:\mathbb{R}^3\to\mathbb{R}:\mathbf{r}\mapsto V(r) be a spherically symmetric potential, such that it is piecewise continuous in r, V(r)=O(1/r^a) for r\to0 and V(r)=O(1/r^b) for r\to+\infty, where a\in(2,+\infty) and b\in(-\infty,2). If :\int_0^{+\infty}r|V(r)|dr<+\infty, then the number of bound states N_\ell with azimuthal quantum number \ell for a particle of mass m obeying the corresponding Schrödinger equation, is bounded from above by :N_\ell<\frac{1}{2\ell+1}\frac{2m}{\hbar^2}\int_0^{+\infty}r|V(r)|dr. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. It takes the form :N_\ell < \frac{1}{2\ell+1} \frac{2m}{\hbar^2} \int_0^\infty r |V(r)|\, dr This limit is the best possible upper bound in such a way that for a given \ell, one can always construct a potential V_\ell for which N_\ell is arbitrarily close to this upper bound. \right] for q < 1 | pdf ={\sqrt{\beta} \over C_q} e_q({-\beta x^2}) | cdf = | mean =0\text{ for }q<2, otherwise undefined| median =0| mode =0| variance = { 1 \over {\beta (5-3q)}} \text{ for } q < {5 \over 3} \infty \text{ for } {5 \over 3} \le q < 2 \text{Undefined for }2 \le q <3| skewness = 0 \text{ for } q < {3 \over 2} | kurtosis = 6{q-1 \over 7-5q} \text{ for } q < {7 \over 5} | entropy =| mgf =| cf =| }} The q-Gaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. In quantum mechanics, Bargmann's limit, named for Valentine Bargmann, provides an upper bound on the number N_\ell of bound states with azimuthal quantum number \ell in a system with central potential V. Groeneboom (1989) shows that : f_Z (z) \sim \frac{1}{2} \frac{4^{4/3} |z|}{\operatorname{Ai}' (\tilde{a}_1)} \exp \left( - \frac{2}{3} |z|^3 + 2^{1/3} \tilde{a}_1 |z| \right) \text{ as }z \rightarrow \infty where \tilde{a}_1 \approx -2.3381 is the largest zero of the Airy function Ai and where \operatorname{Ai}' (\tilde{a}_1 ) \approx 0.7022. The choice of the q-Gaussian form may arise if the system is non-extensive, or if there is lack of a connection to small samples sizes. ==Characterization== ===Probability density function=== The standard q-Gaussian has the probability density function : f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) where :e_q(x) = [1+(1-q)x]_+^{1 \over 1-q} is the q-exponential and the normalization factor C_q is given by :C_q = {{2 \sqrt{\pi} \Gamma\left({1 \over 1-q}\right)} \over {(3-q) \sqrt{1-q} \Gamma\left({3-q \over 2(1-q)}\right)}} \text{ for } -\infty < q < 1 : C_q = \sqrt{\pi} \text{ for } q = 1 \, :C_q = { {\sqrt{\pi} \Gamma\left({3-q \over 2(q-1)}\right)} \over {\sqrt{q-1} \Gamma\left({1 \over q-1}\right)}} \text{ for }1 < q < 3 . The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals. == Wigner n-sphere distribution == The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0): f_n(x;n)={(1-x^2)^{(n-1)/2}\Gamma (1+n/2) \over \sqrt{\pi} \Gamma((n+1)/2)}\, (n>= -1) , for −1 ≤ x ≤ 1, and f(x) = 0 if |x| > 1\. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution. == Related distributions == === Wigner (spherical) parabolic distribution === The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0): f(x)={3 \over \ 4 R^3}{(R^2-x^2)}\, for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. Example. If : V(a,c) = \underset{s \in \mathbf{R}}{\operatorname{argmax}} \ (W(s) - c(s-a)^2), then V(0, c) has density : f_c(t) = \frac{1}{2} g_c(t) g_c(-t) where gc has Fourier transform given by : \hat{g}_c (s) = \frac{(2/c)^{1/3}}{\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \ \ \ s \in \mathbf{R} and where Ai is the Airy function. The solution is based on the path representation of the Green's function, calculated when including all the path probability density functions of all lengths: Here, : \bar{\Psi}_{i}(s) =\sum_j \bar{\Psi}_{ij}(s) and : \bar{\Psi}_{ij}(s)=\frac{1-\bar{\psi}_{ij}(s)}{s}. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). For the corresponding wave function with total energy E=0 and azimuthal quantum number \ell, denoted by \phi_{0\ell}, the radial Schrödinger equation becomes :\frac{d^{2}}{d r^{2}} \phi_{0\ell}(r)-\frac{\ell(\ell+1)}{r^{2}} \phi_{0\ell}(r)=-W(r) \phi_{0\ell}(r), with W=2m|V|/\hbar^2. The following formula will generate deviates from a q-Gaussian with specified parameter q and \beta = {1 \over {3-q}} :Z = \sqrt{-2 \text{ ln}_{q'}(U_1)} \text{ cos}(2 \pi U_2) where \text{ ln}_q is the q-logarithm and q' = { {1+q} \over {3-q}} These deviates can be transformed to generate deviates from an arbitrary q-Gaussian by : Z' = \mu + {Z \over \sqrt{\beta (3-q)}} == Applications == === Physics === It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a q-Gaussian. ",0.11,0.16,"""-87.8""",0.85,11,B
+"Calculate the de Broglie wavelength of an electron moving at 1/137th the speed of light. (At this speed, the relativistic correction to the mass is negligible.)","Thus the uncertainty in position must be greater than half of the reduced Compton wavelength . ==Relationship to other constants== Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron {2\pi}\simeq 386~\textrm{fm})}} and the electromagnetic fine-structure constant The Bohr radius is related to the Compton wavelength by: a_0 = \frac{1}{\alpha}\left(\frac{\lambda_\text{e}}{2\pi}\right) = \frac{\bar{\lambda}_\text{e}}{\alpha} \simeq 137\times\bar{\lambda}_\text{e}\simeq 5.29\times 10^4~\textrm{fm} The classical electron radius is about 3 times larger than the proton radius, and is written: r_\text{e} = \alpha\left(\frac{\lambda_\text{e}}{2\pi}\right) = \alpha\bar{\lambda}_\text{e} \simeq\frac{\bar{\lambda}_\text{e}}{137}\simeq 2.82~\textrm{fm} The Rydberg constant, having dimensions of linear wavenumber, is written: \frac{1}{R_\infty}=\frac{2\lambda_\text{e}}{\alpha^2} \simeq 91.1~\textrm{nm} \frac{1}{2\pi R_\infty} = \frac{2}{\alpha^2}\left(\frac{\lambda_\text{e}}{2\pi}\right) = 2 \frac{\bar{\lambda}_\text{e}}{\alpha^2} \simeq 14.5~\textrm{nm} This yields the sequence: r_{\text{e}} = \alpha \bar{\lambda}_{\text{e}} = \alpha^2 a_0 = \alpha^3 \frac{1}{4\pi R_\infty}. The standard Compton wavelength of a particle is given by \lambda = \frac{h}{m c}, while its frequency is given by f = \frac{m c^2}{h}, where is the Planck constant, is the particle's proper mass, and is the speed of light. The CODATA 2018 value for the Compton wavelength of the electron is .CODATA 2018 value for Compton wavelength for the electron from NIST. The Compton wavelength for this particle is the wavelength of a photon of the same energy. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. For photons of frequency , energy is given by E = h f = \frac{h c}{\lambda} = m c^2, which yields the Compton wavelength formula if solved for . ==Limitation on measurement== The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: \mathbf{ abla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi. The Fermi space observatory detected a gamma-ray with an energy of at least 94 billion electron volts. The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: \sqrt{g_{kk}}=\lambda_c ==See also== * de Broglie wavelength * Planck–Einstein relation ==References== ==External links== * Length Scales in Physics: the Compton Wavelength Category:Atomic physics Category:Foundational quantum physics de:Compton-Effekt#Compton-Wellenlänge The Planck mass and length are defined by: m_{\rm P} = \sqrt{\hbar c/G} l_{\rm P} = \sqrt{\hbar G /c^3}. ==Geometrical interpretation== A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. thumb|left|upright=1.0|Spectrum of WR 137 showing the prominent emission lines of ionised Carbon and Helium WR 137 is a variable Wolf-Rayet star located around 6,000 light years away from Earth in the constellation of Cygnus. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. Relativistic electron beams are streams of electrons moving at relativistic speeds. Species Mass (kg) \lambda_{\mathrm{th}} (m) Electron Photon 0 H2 O2 ==References== * Vu-Quoc, L., Configuration integral (statistical mechanics), 2008. this wiki site is down; see this article in the web archive on 2012 April 28. Equations that pertain to the wavelengths of photons interacting with mass use the non-reduced Compton wavelength. The Planck mass is the order of mass for which the Compton wavelength and the Schwarzschild radius r_{\rm S} = 2 G M /c^2 are the same, when their value is close to the Planck length (l_{\rm P}). Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Other particles have different Compton wavelengths. == Reduced Compton wavelength == The reduced Compton wavelength (barred lambda, denoted below by \bar\lambda) is defined as the Compton wavelength divided by : : \bar\lambda = \frac{\lambda}{2 \pi} = \frac{\hbar}{m c}, where is the reduced Planck constant. ==Role in equations for massive particles== The inverse reduced Compton wavelength is a natural representation for mass on the quantum scale, and as such, it appears in many of the fundamental equations of quantum mechanics. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. ",1260,1590,"""2.0""",0.332,12,D
+Calculate the angle that the spin vector $S$ makes with the $z$ axis for an electron with spin function $\alpha$.,"The component of the spin along a specified axis is given by the spin magnetic quantum number, conventionally written . * The spin value of an electron, proton, neutron is . The direct observation of the electron's intrinsic angular momentum was achieved in the Stern–Gerlach experiment. === Stern–Gerlach experiment === The theory of spatial quantization of the spin moment of the momentum of electrons of atoms situated in the magnetic field needed to be proved experimentally. The component of nuclear spin parallel to the –axis can have (2 + 1) values , –1, ..., . Given an arbitrary direction (usually determined by an external magnetic field) the spin -projection is given by :s_z = m_s \, \hbar where is the secondary spin quantum number, ranging from − to + in steps of one. The direction of spin is described by spin quantum number. The electron spin magnetic moment is given by the formula: \ \boldsymbol{\mu}_s = -\frac{e}{\ 2m\ }\ g\ \mathbf{s}\ where : is the charge of the electron : is the Landé g-factor and by the equation: \ \mu_z = \pm \frac{1}{2}\ g\ \mu_\mathsf{B}\ where \ \mu_\mathsf{B}\ is the Bohr magneton. In nuclear magnetic resonance spectroscopy and magnetic resonance imaging, the Ernst angle is the flip angle (a.k.a. ""tip"" or ""nutation"" angle) for excitation of a particular spin that gives the maximal signal intensity in the least amount of time when signal averaging over many transients. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is \| \bold{s} \| = \hbar\sqrt{s(s+1)} = \frac{\sqrt{3}}{2}\ \hbar ~. The name ""spin"" comes from a physical spinning of the electron about an axis, as proposed by Uhlenbeck and Goudsmit. The value of is the component of spin angular momentum, in units of the reduced Planck constant , parallel to a given direction (conventionally labelled the –axis). This number gives the information about the direction of spinning of the electron present in any orbital. Nuclear-spin quantum numbers are conventionally written for spin, and or for the -axis component. * The magnitude spin quantum number of an electron cannot be changed. Here is the total orbital angular momentum quantum number. In the electron, the two different spin orientations are sometimes called ""spin-up"" or ""spin-down"". In physics, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. Spin engineering describes the control and manipulation of quantum spin systems to develop devices and materials. In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. First of all, spin satisfies the fundamental commutation relation: \ [S_i, S_j ] = i\ \hbar\ \epsilon_{ijk}\ S_k\ , \ \left[S_i, S^2 \right] = 0\ where \ \epsilon_{ijk}\ is the (antisymmetric) Levi-Civita symbol. thumb|upright=1.35|Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation to that of its immediate neighbor(s). ",54.7,0.2553,"""0.9992093669""",92,48,A
+"The AM1 valence electronic energies of the atoms $\mathrm{H}$ and $\mathrm{O}$ are $-11.396 \mathrm{eV}$ and $-316.100 \mathrm{eV}$, respectively. For $\mathrm{H}_2 \mathrm{O}$ at its AM1-calculated equilibrium geometry, the AM1 valence electronic energy (core-core repulsion omitted) is $-493.358 \mathrm{eV}$ and the AM1 core-core repulsion energy is $144.796 \mathrm{eV}$. For $\mathrm{H}(g)$ and $\mathrm{O}(g), \Delta H_{f, 298}^{\circ}$ values are 52.102 and $59.559 \mathrm{kcal} / \mathrm{mol}$, respectively. Find the AM1 prediction of $\Delta H_{f, 298}^{\circ}$ of $\mathrm{H}_2 \mathrm{O}(g)$.","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. Indeed, AM1* is an extension of AM1 molecular orbital theory and uses AM1 parameters and theory unchanged for the elements H, C, N, O and F. ==Earth bulk continental crust and upper continental crust== *C1 — Crust: CRC Handbook *C2 — Crust: Kaye and Laby *C3 — Crust: Greenwood *C4 — Crust: Ahrens (Taylor) *C5 — Crust: Ahrens (Wänke) *C6 — Crust: Ahrens (Weaver) *U1 — Upper crust: Ahrens (Taylor) *U2 — Upper crust: Ahrens (Shaw) Element C1 C2 C3 C4 C5 C6 U1 U2 01 H hydrogen 1.40×10−3 1.52 _0_ ×10−3 02 He helium 8×10−9 03 Li lithium 2.0×10−5 2.0×10−5 1.8×10−5 1.3×10−5 1.37×10−5 2. _0_ ×10−5 2.2×10−5 04 Be beryllium 2.8×10−6 2.0×10−6 2×10−6 1.5 _00_ ×10−6 3. _000_ ×10−6 05 B boron 1.0×10−5 7.0×10−6 9×10−6 1. _0000_ ×10−5 1.5 _000_ ×10−5 06 C carbon 2.00×10−4 1.8 _0_ ×10−4 3.76×10−3 07 N nitrogen 1.9×10−5 2.0×10−5 1.9×10−5 08 O oxygen 4.61×10−1 3.7×10−1 4.55 _000_ ×10−1 09 F fluorine 5.85×10−4 4.6×10−4 5.44×10−4 5.25×10−4 10 Ne neon 5×10−9 11 Na sodium 2.36×10−2 2.3×10−2 2.27 _00_ ×10−2 2.3 _000_ ×10−2 2.44 _00_ ×10−2 3.1 _000_ ×10−2 2.89×10−2 2.57×10−2 12 Mg magnesium 2.33×10−2 2.8×10−2 2.764 _0_ ×10−2 3.20×10−2 2.37×10−2 1.69×10−2 1.33×10−2 1.35×10−2 13 Al aluminium 8.23×10−2 8.0×10−2 8.3 _000_ ×10−2 8.41 _00_ ×10−2 8.305 _0_ ×10−2 8.52 _00_ ×10−2 8.04 _00_ ×10−2 7.74 _00_ ×10−2 14 Si silicon 2.82×10−1 2.7×10−1 2.72 _000_ ×10−1 2.677×10−1 2.81×10−1 2.95×10−1 3.08×10−1 3.04×10−1 15 P phosphorus 1.05×10−3 1.0×10−3 1.12 _0_ ×10−3 7.63×10−4 8.3 _0_ ×10−4 16 S sulfur 3.50×10−4 3.0×10−4 3.4 _0_ ×10−4 8.81×10−4 17 Cl chlorine 1.45×10−4 1.9×10−4 1.26×10−4 1.9 _00_ ×10−3 18 Ar argon 3.5×10−6 19 K potassium 2.09×10−2 1.7×10−2 1.84 _00_ ×10−2 9.1 _00_ ×10−3 1.76 _00_ ×10−2 1.7 _000_ ×10−2 2.8 _000_ ×10−2 2.57 _00_ ×10−2 20 Ca calcium 4.15×10−2 5.1×10−2 4.66 _00_ ×10−2 5.29 _00_ ×10−2 4.92 _00_ ��10−2 3.4 _000_ ×10−2 3. _0000_ ×10−2 2.95 _00_ ×10−2 21 Sc scandium 2.2×10−5 2.2×10−5 2.5×10−5 3. _0_ ×10−5 2.14×10−5 1.1×10−5 7×10−6 22 Ti titanium 5.65×10−3 8.6×10−3 6.32 _0_ ×10−3 5.4 _00_ ×10−3 5.25 _0_ ×10−3 3.6 _00_ ×10−3 3. _000_ ×10−3 3.12 _0_ ×10−3 23 V vanadium 1.20×10−4 1.7×10−4 1.36×10−4 2.3 _0_ ×10−4 1.34×10−4 6. _0_ ×10−5 5.3×10−5 24 Cr chromium 1.02×10−4 9.6×10−5 1.22×10−4 1.85×10−4 1.46×10−4 5.6×10−5 3.5×10−5 3.5×10−5 25 Mn manganese 9.50×10−4 1.0×10−3 1.06 _0_ ×10−3 1.4 _00_ ×10−3 8.47×10−4 1. _000_ ×10−3 6. _00_ ×10−4 5.27×10−4 26 Fe iron 5.63×10−2 5.8×10−2 6.2 _000_ ×10−2 7.07×10−2 4.92×10−2 3.8×10−2 3.50×10−2 3.09×10−2 27 Co cobalt 2.5×10−5 2.8×10−5 2.9×10−5 2.9×10−5 2.54×10−5 1. _0_ ×10−5 1.2×10−5 28 Ni nickel 8.4×10−5 7.2×10−5 9.9×10−5 1.05×10−4 6.95×10−5 3.5×10−5 2×10−5 1.9×10−5 29 Cu copper 6.0×10−5 5.8×10−5 6.8×10−5 7.5×10−5 4.7×10−5 2.5×10−5 1.4×10−5 30 Zn zinc 7.0×10−5 8.2×10−5 7.6×10−5 8. _0_ ×10−5 7.6×10−5 7.1×10−5 5.2×10−5 31 Ga gallium 1.9×10−5 1.7×10−5 1.9×10−5 1.8×10−5 1.86×10−5 1.7×10−5 1.4×10−5 32 Ge germanium 1.5×10−6 1.3×10−6 1.5×10−6 1.6×10−6 1.32×10−6 1.6×10−6 33 As arsenic 1.8×10−6 2.0×10−6 1.8×10−6 1.0×10−6 2.03×10−6 1.5×10−6 34 Se selenium 5×10−8 5×10−8 5×10−8 5×10−8 1.53×10−7 5×10−8 35 Br bromine 2.4×10−6 4.0×10−6 2.5×10−6 6.95×10−6 36 Kr krypton 1×10−10 37 Rb rubidium 9.0×10−5 7.0×10−5 7.8×10−5 3.2×10−5 7.90×10−5 6.1×10−5 1.12×10−4 1.1 _0_ ×10−4 38 Sr strontium 3.70×10−4 4.5×10−4 3.84×10−4 2.6 _0_ ×10−4 2.93×10−4 5.03×10−4 3.5 _0_ ×10−4 3.16×10−4 39 Y yttrium 3.3×10−5 3.5×10−7 3.1×10−5 2. _0_ ×10−5 1.4×10−5 2.2×10−5 2.1×10−5 40 Zr zirconium 1.65×10−4 1.4×10−4 1.62×10−4 1. _00_ ×10−4 2.1 _0_ ×10−4 1.9 _0_ ×10−4 2.4 _0_ ×10−4 41 Nb niobium 2.0×10−5 2.0×10−5 2. _0_ ×10−5 1.1 _000_ ×10−5 1.3 _000_ ×10−5 2.5 _000_ ×10−5 2.6 _000_ ×10−5 42 Mo molybdenum 1.2×10−6 1.2×10−6 1.2×10−6 1. _000_ ×10−6 1.5 _00_ ×10−6 43 Tc technetium 44 Ru ruthenium 1×10−9 1×10−10 45 Rh rhodium 1×10−9 1×10−10 46 Pd palladium 1.5×10−8 3×10−9 1.5×10−8 1.0×10−9 5×10−10 47 Ag silver 7.5×10−8 8×10−8 8×10−8 8. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 48 Cd cadmium 1.5×10−7 1.8×10−7 1.6×10−7 9.8×10−8 1. _00_ ×10−7 9.8×10−8 49 In indium 2.5×10−7 2×10−7 2.4×10−7 5. _0_ ×10−8 6.95×10−8 5. _0_ ×10−8 50 Sn tin 2.3×10−6 1.5×10−6 2.1×10−6 2.5 _00_ ×10−6 5.5 _00_ ×10−6 51 Sb antimony 2×10−7 2×10−7 2×10−7 2. _00_ ×10−7 2.03×10−7 2. _00_ ×10−7 52 Te tellurium 1×10−9 1×10−9 2.03×10−9 53 I iodine 4.5×10−7 5×10−7 4.6×10−7 1.54 _0_ ×10−6 54 Xe xenon 3×10−11 55 Cs caesium 3×10−6 1.6×10−6 2.6×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 3.7 _00_ ×10−6 56 Ba barium 4.25×10−4 3.8×10−4 3.9 _0_ ×10−4 2.5 _0000_ ×10−4 5.42 _000_ ×10−4 7.07 _000_ ×10−4 5.5 _0000_ ×10−4 1.07 _0000_ ×10−3 57 La lanthanum 3.9×10−5 5.0×10−5 3.5×10−5 1.6 _000_ ×10−5 2.9 _000_ ×10−5 2.8 _000_ ×10−5 3. _0000_ ×10−5 3.2 _00_ ×10−6 58 Ce cerium 6.65×10−5 8.3×10−5 6.6×10−5 3.3 _000_ ×10−5 5.42 _00_ ×10−5 5.7 _000_ ×10−5 6.4 _000_ ×10−5 6.5 _000_ ×10−5 59 Pr praseodymium 9.2×10−6 1.3×10−5 9.1×10−6 3.9 _00_ ×10−6 7.1 _00_ ×10−6 60 Nd neodymium 4.15×10−5 4.4×10−5 4. _0_ ×10−5 1.6 _000_ ×10−5 2.54 _00_ ×10−5 2.3 _000_ ×10−5 2.6 _000_ ×10−5 2.6 _000_ ×10−5 61 Pm promethium 62 Sm samarium 7.05×10−6 7.7×10−6 7.0×10−6 3.5 _00_ ×10−6 5.59 _0_ ×10−6 4.1 _00_ ×10−6 4.5 _00_ ×10−6 4.5 _00_ ×10−6 63 Eu europium 2.0×10−6 2.2×10−6 2.1×10−6 1.1 _00_ ×10−6 1.407×10−6 1.09 _0_ ×10−6 8.8 _0_ ×10−7 9.4 _0_ ×10−7 64 Gd gadolinium 6.2×10−6 6.3×10−6 6.1×10−6 3.3 _00_ ×10−6 8.14 _0_ ×10−6 3.8 _00_ ×10−6 2.8 _00_ ×10−6 65 Tb terbium 1.2×10−6 1.0×10−6 1.2×10−6 6. _00_ ×10−7 1.02 _0_ ×10−6 5.3 _0_ ×10−7 6.4 _0_ ×10−7 4.8 _0_ ×10−7 66 Dy dysprosium 5.2×10−6 8.5×10−6 3.7 _00_ ×10−6 6.102×10−6 3.5 _00_ ×10−6 67 Ho holmium 1.3×10−6 1.6×10−6 1.3×10−6 7.8 _0_ ×10−7 1.86 _0_ ×10−6 8. _00_ ×10−7 6.2 _0_ ×10−7 68 Er erbium 3.5×10−6 3.6×10−6 3.5×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 2.3 _00_ ×10−6 69 Tm thulium 5.2×10−7 5.2×10−7 5×10−7 3.2 _0_ ×10−7 2.4 _0_ ×10−7 3.3 _0_ ×10−7 70 Yb ytterbium 3.2×10−6 3.4×10−6 3.1×10−6 2.2 _00_ ×10−6 3.39 _0_ ×10−6 1.53 _0_ ×10−6 2.2 _00_ ×10−6 1.5 _00_ ×10−6 71 Lu lutetium 8×10−7 8×10−7 3. _00_ ×10−7 5.76×10−7 2.3 _0_ ×10−7 3.2 _0_ ×10−7 2.3 _0_ ×10−7 72 Hf hafnium 3.0×10−6 4×10−6 2.8×10−6 3. _000_ ×10−6 3.46 _0_ ×10−6 4.7 _00_ ×10−6 5.8 _00_ ×10−6 5.8 _00_ ×10−6 73 Ta tantalum 2.0×10−6 2.4×10−6 1.7×10−6 1. _000_ ×10−6 2.203×10−6 2.2 _00_ ×10−6 74 W tungsten 1.25×10−6 1.0×10−6 1.2×10−6 1. _000_ ×10−6 1.31 _0_ ×10−6 2. _000_ ×10−6 75 Re rhenium 7×10−10 4×10−10 7×10−10 5×10−10 1.02×10−9 5×10−10 76 Os osmium 1.5×10−9 2×10−10 5×10−9 1.02×10−9 77 Ir iridium 1×10−9 2×10−10 1×10−9 1×10−10 1.02×10−9 2×10−11 78 Pt platinum 5×10−9 1×10−8 79 Au gold 4×10−9 2×10−9 4×10−9 3.0×10−9 4.07×10−9 1.8×10−9 80 Hg mercury 8.5×10−8 2×10−8 8×10−8 81 Tl thallium 8.5×10−7 4.7×10−7 7×10−7 3.6 _0_ ×10−7 7.5 _0_ ×10−7 5.2 _0_ ×10−7 82 Pb lead 1.4×10−5 1.0×10−5 1.3×10−5 8. _000_ ×10−6 1.5 _000_ ×10−5 2. _0000_ ×10−5 1.7 _000_ ×10−5 83 Bi bismuth 8.5×10−9 4×10−9 8×10−9 6. _0_ ×10−8 1.27×10−7 84 Po polonium 2×10−16 85 At astatine 86 Rn radon 4×10−19 87 Fr francium 88 Ra radium 9×10−13 89 Ac actinium 5.5×10−16 90 Th thorium 9.6×10−6 5.8×10−6 8.1×10−6 3.5 _00_ ×10−6 5.7 _00_ ×10−6 1.07 _00_ ×10−5 1. _0000_ ×10−5 91 Pa protactinium 1.4×10−12 92 U uranium 2.7×10−6 1.6×10−6 2.3×10−6 9.1 _0_ ×10−7 1.2 _00_ ×10−6 1.3 _00_ ×10−6 2.8 _00_ ×10−6 2.5 _00_ ×10−6 93 Np neptunium 94 Pu plutonium ==Urban soils== The established abundances of chemical elements in urban soils can be considered a geochemical (ecological and geochemical) characteristic, the accumulated impact of technogenic and natural processes at the beginning of the 21st century. AM1* is a semiempirical molecular orbital technique in computational chemistry. But, other elements have been parameterized using an additional set of d-orbitals in the basis set and with two-center core–core parameters, rather than the Gaussian functions used to modify the core–core potential in AM1. Element S1 Y1 Y2 01 H hydrogen 2.8×104 2.8×104* 2.79×104 02 He helium 2.7×103 2.7×103* 2.72×103 03 Li lithium 4.0×10−7 5.7×10−5 5.71×10−5 (9.2%) 04 Be beryllium 4.0×10−7 7.0×10−7 7.30×10−7 (9.5%) 05 B boron 1.1×10−5 2.1×10−5 2.12×10−5 (10%) 06 C carbon 1.0×101 1.0×101* 1.01×101 07 N nitrogen 3.1×100 3.1×100* 3.13×100 08 O oxygen 2.4×101 2.4×101* 2.38×101 (10%) 09 F fluorine about 1.0×10−3 8.5×10−4 8.43×10−4 (15%) 10 Ne neon 3.0×100 3.0×100* 3.44×100 (14%) 11 Na sodium 6.0×10−2 5.7×10−2 5.74×10−2 (7.1%) 12 Mg magnesium 1.0×100 1.1×100 1.074×100 (3.8%) 13 Al aluminium 8.3×10−2 8.5×10−2 8.49×10−2 (3.6%) 14 Si silicon 1.0×100 1.0×100 1.0×100 (4.4%) 15 P phosphorus 8.0×10−3 1.0×10−2 1.04×10−2 (10%) 16 S sulfur 4.5×10−1 5.2×10−1 5.15×10−1 (13%) 17 Cl chlorine about 9.0×10−3 5.2×10−3 5.24×10−3 (15%) 18 Ar argon 1.0×10−1* 1.0×10−1* 1.01×10−1 (6%) 19 K potassium 3.7×10−3 3.8×10−3 3.77×10−3 (7.7%) 20 Ca calcium 6.4×10−2 6.1×10−2 6.11×10−2 (7.1%) 21 Sc scandium 3.5×10−5 3.4×10−5 3.42×10−5 (8.6%) 22 Ti titanium 2.7×10−3 2.4×10−3 2.40×10−3 (5.0%) 23 V vanadium 2.8×10−4 2.9×10−4 2.93×10−4 (5.1%) 24 Cr chromium 1.3×10−2 1.3×10−2 1.35×10−2 (7.6%) 25 Mn manganese 6.9×10−3 9.5×10−3 9.55×10−3 (9.6%) 26 Fe iron 9.0×10−1 9.0×10−1 9.00×10−1 (2.7%) 27 Co cobalt 2.3×10−3 2.3×10−3 2.25×10−3 (6.6%) 28 Ni nickel 5.0×10−2 5.0×10−2 4.93×10−2 (5.1%) 29 Cu copper 4.5×10−4 5.2×10−4 5.22×10−4 (11%) 30 Zn zinc 1.1×10−3 1.3×10−3 1.26×10−3 (4.4%) 31 Ga gallium 2.1×10−5 3.8×10−5 3.78×10−5 (6.9%) 32 Ge germanium 7.2×10−5 1.2×10−4 1.19×10−4 (9.6%) 33 As arsenic 6.6×10−6 6.56×10−6 (12%) 34 Se selenium 6.3×10−5 6.21×10−5 (6.4%) 35 Br bromine 1.2×10−5 1.18×10−5 (19%) 36 Kr krypton 4.8×10−5 4.50×10−5 (18%) 37 Rb rubidium 1.1×10−5 7.0×10−6 7.09×10−6 (6.6%) 38 Sr strontium 2.2×10−5 2.4×10−5 2.35×10−5 (8.1%) 39 Y yttrium 4.9×10−6 4.6×10−6 4.64×10−6 (6.0%) 40 Zr zirconium 1.12×10−5 1.14×10−5 1.14×10−5 (6.4%) 41 Nb niobium 7.0×10−7 7.0×10−7 6.98×10−7 (1.4%) 42 Mo molybdenum 2.3×10−6 2.6×10−6 2.55×10−6 (5.5%) 43 Tc technetium 44 Ru ruthenium 1.9×10−6 1.9×10−6 1.86×10−6 (5.4%) 45 Rh rhodium 4.0×10−7 3.4×10−7 3.44×10−7 (8%) 46 Pd palladium 1.4×10−6 1.4×10−6 1.39×10−6 (6.6%) 47 Ag silver about 2.0×10−7 4.9×10−7 4.86×10−7 (2.9%) 48 Cd cadmium 2.0×10−6 1.6×10−6 1.61×10−6 (6.5%) 49 In indium about 1.3×10−6 1.9×10−7 1.84×10−7 (6.4%) 50 Sn tin about 3.0×10−6 3.9×10−6 3.82×10−6 (9.4%) 51 Sb antimony about 3.0×10−7 3.1×10−7 3.09×10−7 (18%) 52 Te tellurium 4.9×10−6 4.81×10−6 (10%) 53 I iodine 9.0×10−7 9.00×10−7 (21%) 54 Xe xenon 4.8×10−6 4.70×10−6 (20%) 55 Cs caesium 3.7×10−7 3.72×10−7 (5.6%) 56 Ba barium 3.8×10−6 4.5×10−6 4.49×10−6 (6.3%) 57 La lanthanum 5.0×10−7 4.4×10−7 4.46×10−7 (2.0%) 58 Ce cerium 1.0×10−6 1.1×10−6 1.136×10−6 (1.7%) 59 Pr praseodymium 1.4×10−7 1.7×10−7 1.669×10−7 (2.4%) 60 Nd neodymium 9.0×10−7 8.3×10−7 8.279×10−7 (1.3%) 61 Pm promethium 62 Sm samarium 3.0×10−7 2.6×10−7 2.582×10−7 (1.3%) 63 Eu europium 9.0×10−8 9.7×10−8 9.73×10−8 (1.6%) 64 Gd gadolinium 3.7×10−7 3.3×10−7 3.30×10−7 (1.4%) 65 Tb terbium about 2.0×10−8 6.0×10−8 6.03×10−8 (2.2%) 66 Dy dysprosium 3.5×10−7 4.0×10−7 3.942×10−7 (1.4%) 67 Ho holmium about 5.0×10−8 8.9×10−8 8.89×10−8 (2.4%) 68 Er erbium 2.4×10−7 2.5×10−7 2.508×10−7 (1.3%) 69 Tm thulium about 3.0×10−8 3.8×10−8 3.78×10−8 (2.3%) 70 Yb ytterbium 3.4×10−7 2.5×10−7 2.479×10−7 (1.6%) 71 Lu lutetium about 1.5×10−7 3.7×10−8 3.67×10−8 (1.3%) 72 Hf hafnium 2.1×10−7 1.5×10−7 1.54×10−7 (1.9%) 73 Ta tantalum 3.8×10−8 2.07×10−8 (1.8%) 74 W tungsten about 3.6×10−7 1.3×10−7 1.33×10−7 (5.1%) 75 Re rhenium 5.0×10−8 5.17×10−8 (9.4%) 76 Os osmium 8.0×10−7 6.7×10−7 6.75×10−7 (6.3%) 77 Ir iridium 6.0×10−7 6.6×10−7 6.61×10−7 (6.1%) 78 Pt platinum about 1.8×10−6 1.34×10−6 1.34×10−6 (7.4%) 79 Au gold about 3.0×10−7 1.9×10−7 1.87×10−7 (15%) 80 Hg mercury 3.4×10−7 3.40×10−7 (12%) 81 Tl thallium about 2.0×10−7 1.9×10−7 1.84×10−7 (9.4%) 82 Pb lead 2.0×10−6 3.1×10−6 3.15×10−6 (7.8%) 83 Bi bismuth 1.4×10−7 1.44×10−7 (8.2%) 84 Po polonium 85 At astatine 86 Rn radon 87 Fr francium 88 Ra radium 89 Ac actinium 90 Th thorium 5.0×10−8 4.5×10−8 3.35×10−8 (5.7%) 91 Pa protactinium 92 U uranium 1.8×10−8 9.00×10−9 (8.4%) 93 Np neptunium 94 Pu plutonium ==See also== *Chemical elements data references ==Notes== Due to the estimate nature of these values, no single recommendations are given. ==Thermal conductivity== 1 H hydrogen use 180.5 mW/(m·K) CR2 (P=0) 186.9 mW/(m·K) LNG 0.1805 W/(m·K) WEL 0.1805 W/(m·K) 2 He helium use 151.3 mW/(m·K) CR2 (P=0) 156.7 mW/(m·K) LNG 0.1513 W/(m·K) WEL 0.1513 W/(m·K) 3 Li lithium use 84.8 W/(m·K) CRC 0.847 W/(cm·K) LNG 84.8 W/(m·K) WEL 85 W/(m·K) 4 Be beryllium use 200 W/(m·K) CRC 2.00 W/(cm·K) LNG 200 W/(m·K) WEL 190 W/(m·K) 5 B boron use 27.4 W/(m·K) LNG 27.4 W/(m·K) WEL 27 W/(m·K) 6 C carbon (graphite) use (graphite) (119-165) W/(m·K) LNG (119-165) W/(m·K) LNG (amorphous) 1.59 W/(m·K) WEL 140 W/(m·K) CRC pyrolytic graphite (parallel to the layer planes) 19.5 W/(cm·K) (perpendicular) 0.0570 W/(cm·K) 6 C carbon (diamond) use (diamond) (900-2320) W/(m·K) LNG (900-2320) W/(m·K) CRC (type I) 8.95 W/(cm·K) (type IIa) 23.0 W/(cm·K) (type IIb) 13.5 W/(cm·K) 7 N nitrogen use 25.83 mW/(m·K) CR2 26.0 mW/(m·K) LNG 0.02583 W/(m·K) WEL 0.02583 W/(m·K) 8 O oxygen use 26.58 mW/(m·K) CR2 26.3 mW/(m·K) LNG (gas) 0.02658 W/(m·K) LNG (liquid) 0.149 W/(m·K) WEL 0.02658 W/(m·K) 9 F fluorine use 27.7 mW/(m·K) LNG 0.0277 W/(m·K) WEL 0.0277 W/(m·K) 10 Ne neon use 49.1 mW/(m·K) CR2 (P=0) 49.8 mW/(m·K) LNG 0.0491 W/(m·K) WEL 0.0491 W/(m·K) 11 Na sodium use 142 W/(m·K) CRC 1.41 W/(cm·K) LNG 142 W/(m·K) WEL 140 W/(m·K) 12 Mg magnesium use 156 W/(m·K) CRC 1.56 W/(cm·K) LNG 156 W/(m·K) WEL 160 W/(m·K) 13 Al aluminium use 237 W/(m·K) CRC 2.37 W/(cm·K) LNG 237 W/(m·K) WEL 235 W/(m·K) 14 Si silicon use 149 W/(m·K) LNG 149 W/(m·K) WEL 150 W/(m·K) 15 P phosphorus use (white) 0.236 W/(m·K) LNG (white) 0.23617 W/(m·K) [sic] WEL 0.236 W/(m·K) 16 S sulfur use (amorphous) 0.205 W/(m·K) LNG (amorphous) 0.205 W/(m·K) WEL 0.205 W/(m·K) 17 Cl chlorine use 8.9 mW/(m·K) LNG 0.0089 W/(m·K) WEL 0.0089 W/(m·K) 18 Ar argon use 17.72 mW/(m·K) CR2 17.9 mW/(m·K) LNG 0.01772 W/(m·K) WEL 0.01772 W/(m·K) 19 K potassium use 102.5 W/(m·K) CRC 1.024 W/(cm·K) LNG 102.5 W/(m·K) WEL 100 W/(m·K) 20 Ca calcium use 201 W/(m·K) CRC 2.00 W/(cm·K) LNG 201 W/(m·K) WEL 200 W/(m·K) 21 Sc scandium use 15.8 W/(m·K) CRC 0.158 W/(cm·K) LNG 15.8 W/(m·K) WEL 16 W/(m·K) 22 Ti titanium use 21.9 W/(m·K) CRC 0.219 W/(cm·K) LNG 21.9 W/(m·K) WEL 22 W/(m·K) 23 V vanadium use 30.7 W/(m·K) CRC 0.307 W/(cm·K) LNG 30.7 W/(m·K) WEL 31 W/(m·K) 24 Cr chromium use 93.9 W/(m·K) CRC 0.937 W/(cm·K) LNG 93.9 W/(m·K) WEL 94 W/(m·K) 25 Mn manganese use 7.81 W/(m·K) CRC 0.0782 W/(cm·K) LNG 7.81 W/(m·K) WEL 7.8 W/(m·K) 26 Fe iron use 80.4 W/(m·K) CRC 0.802 W/(cm·K) LNG 80.4 W/(m·K) WEL 80 W/(m·K) 27 Co cobalt use 100 W/(m·K) CRC 1.00 W/(cm·K) LNG 100 W/(m·K) WEL 100 W/(m·K) 28 Ni nickel use 90.9 W/(m·K) CRC 0.907 W/(cm·K) LNG 90.9 W/(m·K) WEL 91 W/(m·K) 29 Cu copper use 401 W/(m·K) CRC 4.01 W/(cm·K) LNG 401 W/(m·K) WEL 400 W/(m·K) 30 Zn zinc use 116 W/(m·K) CRC 1.16 W/(cm·K) LNG 116 W/(m·K) WEL 120 W/(m·K) 31 Ga gallium use 40.6 W/(m·K) CRC 0.406 W/(cm·K) LNG (liquid) 29.4 W/(m·K) LNG (solid) 40.6 W/(m·K) WEL 29 W/(m·K) 32 Ge germanium use 60.2 W/(m·K) LNG 60.2 W/(m·K) WEL 60 W/(m·K) 33 As arsenic use 50.2 W/(m·K) LNG 50.2 W/(m·K) WEL 50 W/(m·K) 34 Se selenium use (amorphous) 0.519 W/(m·K) LNG (amorphous) 0.519 W/(m·K) WEL 0.52 W/(m·K) 35 Br bromine use 0.122 W/(m·K) LNG 0.122 W/(m·K) WEL 0.12 W/(m·K) 36 Kr krypton use 9.43 mW/(m·K) CR2 (P=0) 9.5 mW/(m·K) LNG 9.43 W/(m·K) [sic] WEL 0.00943 W/(m·K) 37 Rb rubidium use 58.2 W/(m·K) CRC 0.582 W/(cm·K) LNG 58.2 W/(m·K) WEL 58 W/(m·K) 38 Sr strontium use 35.4 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.4 W/(m·K) WEL 35 W/(m·K) 39 Y yttrium use 17.2 W/(m·K) CRC 0.172 W/(cm·K) LNG 17.2 W/(m·K) WEL 17 W/(m·K) 40 Zr zirconium use 22.6 W/(m·K) CRC 0.227 W/(cm·K) LNG 22.6 W/(m·K) WEL 23 W/(m·K) 41 Nb niobium use 53.7 W/(m·K) CRC 0.537 W/(cm·K) LNG 53.7 W/(m·K) WEL 54 W/(m·K) 42 Mo molybdenum use 138 W/(m·K) CRC 1.38 W/(cm·K) LNG 138 W/(m·K) WEL 139 W/(m·K) 43 Tc technetium use 50.6 W/(m·K) CRC 0.506 W/(cm·K) LNG 50.6 W/(m·K) WEL 51 W/(m·K) 44 Ru ruthenium use 117 W/(m·K) CRC 1.17 W/(cm·K) LNG 117 W/(m·K) WEL 120 W/(m·K) 45 Rh rhodium use 150 W/(m·K) CRC 1.50 W/(cm·K) LNG 150 W/(m·K) WEL 150 W/(m·K) 46 Pd palladium use 71.8 W/(m·K) CRC 0.718 W/(cm·K) LNG 71.8 W/(m·K) WEL 72 W/(m·K) 47 Ag silver use 429 W/(m·K) CRC 4.29 W/(cm·K) LNG 429 W/(m·K) WEL 430 W/(m·K) 48 Cd cadmium use 96.6 W/(m·K) CRC 0.968 W/(cm·K) LNG 96.6 W/(m·K) WEL 97 W/(m·K) 49 In indium use 81.8 W/(m·K) CRC 0.816 W/(cm·K) LNG 81.8 W/(m·K) WEL 82 W/(m·K) 50 Sn tin use 66.8 W/(m·K) CRC 0.666 W/(cm·K) LNG 66.8 W/(m·K) WEL 67 W/(m·K) 51 Sb antimony use 24.4 W/(m·K) CRC 0.243 W/(cm·K) LNG 24.4 W/(m·K) WEL 24 W/(m·K) 52 Te tellurium use (1.97-3.38) W/(m·K) LNG (1.97-3.38) W/(m·K) WEL 3 W/(m·K) 53 I iodine use 0.449 W/(m·K) CRC (300 K) 0.45 W/(m·K) LNG 449 W/(m·K) [sic] WEL 0.449 W/(m·K) 54 Xe xenon use 5.65 mW/(m·K) CR2 (P=0) 5.5 mW/(m·K) LNG 0.00565 W/(m·K) WEL 0.00565 W/(m·K) 55 Cs caesium use 35.9 W/(m·K) CRC 0.359 W/(cm·K) LNG 35.9 W/(m·K) WEL 36 W/(m·K) 56 Ba barium use 18.4 W/(m·K) CRC 0.184 W/(cm·K) LNG 18.4 W/(m·K) WEL 18 W/(m·K) 57 La lanthanum use 13.4 W/(m·K) CRC 0.134 W/(cm·K) LNG 13.4 W/(m·K) WEL 13 W/(m·K) 58 Ce cerium use 11.3 W/(m·K) CRC 0.113 W/(cm·K) LNG 11.3 W/(m·K) WEL 11 W/(m·K) 59 Pr praseodymium use 12.5 W/(m·K) CRC 0.125 W/(cm·K) LNG 12.5 W/(m·K) WEL 13 W/(m·K) 60 Nd neodymium use 16.5 W/(m·K) CRC 0.165 W/(cm·K) LNG 16.5 W/(m·K) WEL 17 W/(m·K) 61 Pm promethium use 17.9 W/(m·K) CRC (est.) 0.15 W/(cm·K) LNG 17.9 W/(m·K) WEL 15 W/(m·K) 62 Sm samarium use 13.3 W/(m·K) CRC 0.133 W/(cm·K) LNG 13.3 W/(m·K) WEL 13 W/(m·K) 63 Eu europium use (est.) 13.9 W/(m·K) CRC (est.) 0.139 W/(cm·K) LNG 13.9 W/(m·K) WEL 14 W/(m·K) 64 Gd gadolinium use 10.6 W/(m·K) (300 K) 10.6 W/(m·K) CRC (100 °C) 0.105 W/(cm·K) LNG 10.5 W/(m·K) WEL 11 W/(m·K) 65 Tb terbium use 11.1 W/(m·K) CRC 0.111 W/(cm·K) LNG 11.1 W/(m·K) WEL 11 W/(m·K) 66 Dy dysprosium use 10.7 W/(m·K) CRC 0.107 W/(cm·K) LNG 10.7 W/(m·K) WEL 11 W/(m·K) 67 Ho holmium use 16.2 W/(m·K) CRC 0.162 W/(cm·K) LNG 16.2 W/(m·K) WEL 16 W/(m·K) 68 Er erbium use 14.5 W/(m·K) CRC 0.145 W/(cm·K) LNG 14.5 W/(m·K) WEL 15 W/(m·K) 69 Tm thulium use 16.9 W/(m·K) CRC 0.169 W/(cm·K) LNG 16.9 W/(m·K) WEL 17 W/(m·K) 70 Yb ytterbium use 38.5 W/(m·K) CRC 0.385 W/(cm·K) LNG 38.5 W/(m·K) WEL 39 W/(m·K) 71 Lu lutetium use 16.4 W/(m·K) CRC 0.164 W/(cm·K) LNG 16.4 W/(m·K) WEL 16 W/(m·K) 72 Hf hafnium use 23.0 W/(m·K) CRC 0.230 W/(cm·K) LNG 23.0 W/(m·K) WEL 23 W/(m·K) 73 Ta tantalum use 57.5 W/(m·K) CRC 0.575 W/(cm·K) LNG 57.5 W/(m·K) WEL 57 W/(m·K) 74 W tungsten use 173 W/(m·K) CRC 1.74 W/(cm·K) LNG 173 W/(m·K) WEL 170 W/(m·K) 75 Re rhenium use 48.0 W/(m·K) CRC 0.479 W/(cm·K) LNG 48.0 W/(m·K) WEL 48 W/(m·K) 76 Os osmium use 87.6 W/(m·K) CRC 0.876 W/(cm·K) LNG 87.6 W/(m·K) WEL 88 W/(m·K) 77 Ir iridium use 147 W/(m·K) CRC 1.47 W/(cm·K) LNG 147 W/(m·K) WEL 150 W/(m·K) 78 Pt platinum use 71.6 W/(m·K) CRC 0.716 W/(cm·K) LNG 71.6 W/(m·K) WEL 72 W/(m·K) 79 Au gold use 318 W/(m·K) CRC 3.17 W/(cm·K) LNG 318 W/(m·K) WEL 320 W/(m·K) 80 Hg mercury use 8.30 W/(m·K) CRC 0.0834 W/(cm·K) LNG 8.30 W/(m·K) WEL 8.3 W/(m·K) 81 Tl thallium use 46.1 W/(m·K) CRC 0.461 W/(cm·K) LNG 46.1 W/(m·K) WEL 46 W/(m·K) 82 Pb lead use 35.3 W/(m·K) CRC 0.353 W/(cm·K) LNG 35.3 W/(m·K) WEL 35 W/(m·K) 83 Bi bismuth use 7.97 W/(m·K) CRC 0.0787 W/(cm·K) LNG 7.97 W/(m·K) WEL 8 W/(m·K) 84 Po polonium use CRC 0.20 W/(cm·K) LNG 0.2 W/(m·K) 85 At astatine use 1.7 W/(m·K) LNG 1.7 W/(m·K) WEL 2 W/(m·K) 86 Rn radon use 3.61 mW/(m·K) LNG 0.00361 W/(m·K) WEL 0.00361 W/(m·K) 88 Ra radium use 18.6 W/(m·K) LNG 18.6 W/(m·K) WEL 19 W/(m·K) 89 Ac actinium use 12 W/(m·K) LNG 12 W/(m·K) WEL 12 W/(m·K) 90 Th thorium use 54.0 W/(m·K) CRC 0.540 W/(cm·K) LNG 54.0 W/(m·K) WEL 54 W/(m·K) 91 Pa protactinium use 47 W/(m·K) LNG 47 W/(m·K) WEL 47 W/(m·K) 92 U uranium use 27.5 W/(m·K) CRC 0.276 W/(cm·K) LNG 27.5 W/(m·K) WEL 27 W/(m·K) 93 Np neptunium use 6.3 W/(m·K) CRC 0.063 W/(cm·K) LNG 6.3 W/(m·K) WEL 6 W/(m·K) 94 Pu plutonium use 6.74 W/(m·K) CRC 0.0674 W/(cm·K) LNG 6.74 W/(m·K) WEL 6 W/(m·K) 95 Am americium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) 97 Bk berkelium use 10 W/(m·K) LNG 10 W/(m·K) WEL 10 W/(m·K) ==Notes== * Ref. CRC: Values refer to 27 °C unless noted. :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). AM1* parameters are now available for H, C, N, O, F, Al, Si, P, S, Cl, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Br, Zr, Mo, Pd, Ag, I and Au. Additionally, for transition metal-hydrogen interactions, a distance dependent term is used to calculate core-core potentials rather than the constant term. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? AM1* is implemented in VAMP 10.0 Clark T, Alex A, Beck B, Chandrasekhar J, Gedeck P, Horn AHC, Hutter M, Martin B, Rauhut G, Sauer W, Schindler T, Steinke T (2005) Computer- Chemie-Centrum. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? (The average density of sea water in the surface is 1.025 kg/L) Element W1 W2 01 H hydrogen 1.08×10−1 1.1×10−1 02 He helium 7×10−12 7.2×10−12 03 Li lithium 1.8×10−7 1.7×10−7 04 Be beryllium 5.6×10−12 6×10−13 05 B boron 4.44×10−6 4.4×10−6 06 C carbon 2.8×10−5 2.8×10−5 07 N nitrogen 5×10−7 1.6×10−5 08 O oxygen 8.57×10−1 8.8×10−1 09 F fluorine 1.3×10−6 1.3×10−6 10 Ne neon 1.2×10−10 1.2×10−10 11 Na sodium 1.08×10−2 1.1×10−2 12 Mg magnesium 1.29×10−3 1.3×10−3 13 Al aluminium 2×10−9 1×10−9 14 Si silicon 2.2×10−6 2.9×10−6 15 P phosphorus 6×10−8 8.8×10−8 16 S sulfur 9.05×10−4 9.0×10−4 17 Cl chlorine 1.94×10−2 1.9×10−2 18 Ar argon 4.5×10−7 4.5×10−7 19 K potassium 3.99×10−4 3.9×10−4 20 Ca calcium 4.12×10−4 4.1×10−4 21 Sc scandium 6×10−13 < 4×10−12 22 Ti titanium 1×10−9 1×10−9 23 V vanadium 2.5×10−9 1.9×10−9 24 Cr chromium 3×10−10 2×10−10 25 Mn manganese 2×10−10 1.9×10−9 26 Fe iron 2×10−9 3.4×10−9 27 Co cobalt 2×10−11 3.9×10−10 28 Ni nickel 5.6×10−10 6.6×10−9 29 Cu copper 2.5×10−10 2.3×10−8 30 Zn zinc 4.9×10−9 1.1×10−8 31 Ga gallium 3×10−11 3×10−11 32 Ge germanium 5×10−11 6×10−11 33 As arsenic 3.7×10−9 2.6×10−9 34 Se selenium 2×10−10 9.0×10−11 35 Br bromine 6.73×10−5 6.7×10−5 36 Kr krypton 2.1×10−10 2.1×10−10 37 Rb rubidium 1.2×10−7 1.2×10−7 38 Sr strontium 7.9×10−6 8.1×10−6 39 Y yttrium 1.3×10−11 1.3×10−12 40 Zr zirconium 3×10−11 2.6×10−11 41 Nb niobium 1×10−11 1.5×10−11 42 Mo molybdenum 1×10−8 1.0×10−8 43 Tc technetium 44 Ru ruthenium 7×10−13 45 Rh rhodium 46 Pd palladium 47 Ag silver 4×10−11 2.8×10−10 48 Cd cadmium 1.1×10−10 1.1×10−10 49 In indium 2×10−8 50 Sn tin 4×10−12 8.1×10−10 51 Sb antimony 2.4×10−10 3.3×10−10 52 Te tellurium 53 I iodine 6×10−8 6.4×10−8 54 Xe xenon 5×10−11 4.7×10−11 55 Cs caesium 3×10−10 3.0×10−10 56 Ba barium 1.3×10−8 2.1×10−8 57 La lanthanum 3.4×10−12 3.4×10−12 58 Ce cerium 1.2×10−12 1.2×10−12 59 Pr praseodymium 6.4×10−13 6.4×10−13 60 Nd neodymium 2.8×10−12 2.8×10−12 61 Pm promethium 62 Sm samarium 4.5×10−13 4.5×10−13 63 Eu europium 1.3×10−13 1.3×10−13 64 Gd gadolinium 7×10−13 7.0×10−13 65 Tb terbium 1.4×10−13 1.4×10−12 66 Dy dysprosium 9.1×10−13 9.1×10−13 67 Ho holmium 2.2×10−13 2.2×10−13 68 Er erbium 8.7×10−13 8.7×10−12 69 Tm thulium 1.7×10−13 1.7×10−13 70 Yb ytterbium 8.2×10−13 8.2×10−13 71 Lu lutetium 1.5×10−13 1.5×10−13 72 Hf hafnium 7×10−12 < 8×10−12 73 Ta tantalum 2×10−12 < 2.5×10−12 74 W tungsten 1×10−10 < 1×10−12 75 Re rhenium 4×10−12 76 Os osmium 77 Ir iridium 78 Pt platinum 79 Au gold 4×10−12 1.1×10−11 80 Hg mercury 3×10−11 1.5×10−10 81 Tl thallium 1.9×10−11 82 Pb lead 3×10−11 3×10−11 83 Bi bismuth 2×10−11 2×10−11 84 Po polonium 1.5×10−20 85 At astatine 86 Rn radon 6×10−22 87 Fr francium 88 Ra radium 8.9×10−17 89 Ac actinium 90 Th thorium 1×10−12 1.5×10−12 91 Pa protactinium 5×10−17 92 U uranium 3.2×10−9 3.3×10−9 93 Np neptunium 94 Pu plutonium ==Sun and solar system== *S1 — Sun: Kaye & Laby *Y1 — Solar system: Kaye & Laby *Y2 — Solar system: Ahrens, with uncertainty s (%) Atom mole fraction relative to silicon = 1. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Boca Raton, Florida, 2003; Section 6, Fluid Properties; Thermal Conductivity of Gases ===LNG=== As quoted from this source in an online version of: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 4; Table 4.1, Electronic Configuration and Properties of the Elements * Ho, C. Y., Powell, R. W., and Liley, P. E., J. Phys. Chem. Ref. Data 3:Suppl. 1 (1974) ===WEL=== As quoted at http://www.webelements.com/ from these sources: * G.W.C. Kaye and T.H. Laby in Tables of physical and chemical constants, Longman, London, UK, 15th edition, 1993. Element Atomic number Abundance in urban soils Ag 47 0.37 Al 13 38200 As 33 15.9 B 5 45 Ba 56 853.12 Be 4 3.3 Bi 83 1.12 C 6 45100 Ca 20 53800 Cd 48 0.9 Cl 17 285 Co 27 14.1 Cr 24 80 Cs 55 5.0 Cu 29 39 Fe 26 22300 Ga 31 16.2 Ge 32 1.8 H 1 15000 Hg 80 0.88 K 19 13400 La 57 34 Li 3 49.5 Mg 12 7900 Mn 25 729 Mo 42 2.4 N 7 10000 Na 11 5800 Nb 41 15.7 Ni 28 33 O 8 490000 P 15 1200 Pb 82 54.5 Rb 37 58 S 16 1200 Sb 51 1.0 Sc 21 9.4 Si 14 289000 Sn 50 6.8 Sr 38 458 Ta 73 1.5 Ti 22 4758 Tl 81 1.1 V 23 104.9 W 74 2.9 Y 39 23.4 Yb 70 2.4 Zn 30 158 Zr 40 255.6 ==Sea water== *W1 — CRC Handbook *W2 — Kaye & Laby Mass per volume fraction, in kg/L. ",-2,3.00,"""1.2""",-59.24,2598960,D
+"Given that $D_e=4.75 \mathrm{eV}$ and $R_e=0.741 Å$ for the ground electronic state of $\mathrm{H}_2$, find $U\left(R_e\right)$ for this state.","The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . The International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS), is a biannual event, where over hundred scientists meet for the presentation of new developments on the special field of two-dimensional electron systems in semiconductors. * After dividing by the number of electrons, the standard potential E° is related to the standard Gibbs free energy of formation ΔGf° by: E = \frac{\sum \Delta G_\text{left}-\sum \Delta G_\text{right}}{F} where F is the Faraday constant. E2D may refer to: * Estradiol decanoate * trans-4,5-Epoxy-(E)-2-decenal * E2-D the fourth variant of the Northrop Grumman E-2 Hawkeye reconnaissance plane. ER2, ER-2, ER II etc. may refer to: * Elizabeth II's royal cypher E II R (sometimes written as ER II) for Elizabeth II Regina (Elizabeth II, Queen) *""ER2"" (Kanjani Eight song), a single by Japanese boy band Kanjani Eight *ER2 electric trainset, an electric passenger railcar built in Latvia and Russia from 1962 to 1984 *NASA ER-2, ""Earth Resources 2"", an American very high- altitude civilian atmospheric research fixed-wing aircraft based on the Lockheed U-2 reconnaissance aircraft In the weakened potential at the surface, new electronic states can be formed, so called surface states. ==Origin at condensed matter interfaces== thumbnail|350px|Figure 1. Te (aq) + 2 + 2 (s) + 4 1.02 2 . Since the potential is periodic deep inside the crystal, the electronic wave functions must be Bloch waves here. This means that the US Government's use of E85 is effectively doubled as of August 8, 2005 with the signing into law of the Energy Policy Act of 2005. Surface states are electronic states found at the surface of materials. E85 is an abbreviation for an ethanol fuel blend of between 51% and 83% denatured ethanol fuel and gasoline or other hydrocarbon (HC) by volume. ==Availability== All data August 2014 from the Department of Energy, e85prices.com, and E85refueling.com.http://www.e85refueling.com Links go to each state's list of stations; see notes below for caveats. It can be shown that the energies of these states all lie within the band gap. The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations. This theory provides some fundamental new understandings of those electronic states, including surface states. The Shockley states are then found as solutions to the one- dimensional single electron Schrödinger equation : \begin{align} \left[-\frac{\hbar^2}{2m}\frac{d^2}{dz^2}+V(z)\right]\Psi(z) &=& E\Psi(z), \end{align} with the periodic potential : \begin{align} V(z)=\left\\{ \begin{array}{cc} P\delta(z+la),& \textrm{for}\quad z<0 \\\ V_0,&\textrm{for} \quad z>0 \end{array}\right., \end{align} where l is an integer, and P is the normalization factor. For example: : \+ Cu(s) ( = +0.520 V) Cu + 2 Cu(s) ( = +0.337 V) Cu + ( = +0.159 V) :Calculating the potential using Gibbs free energy ( = 2 – ) gives the potential for as 0.154 V, not the experimental value of 0.159 V. : __TOC__ ==Table of standard electrode potentials== Legend: (s) - solid; (l) - liquid; (g) - gas; (aq) - aqueous (default for all charged species); (Hg) - amalgam; bold - water electrolysis equations. Shockley states are thus states that arise due to the change in the electron potential associated solely with the crystal termination. :The Nernst equation will then give potentials at concentrations, pressures, and temperatures other than standard. All of the reactions should be divided by the stoichiometric coefficient for the electron to get the corresponding corrected reaction equation. For example, the equation Fe + 2 Fe(s) (–0.44 V) means that it requires 2 × 0.44 eV = 0.88 eV of energy to be absorbed (hence the minus sign) in order to create one neutral atom of Fe(s) from one Fe ion and two electrons, or 0.44 eV per electron, which is 0.44 J/C of electrons, which is 0.44 V. The energy levels of such states are expected to significantly shift from the bulk values. The nearly free electron approximation can be used to derive the basic properties of surface states for narrow gap semiconductors. ",-8,0,"""-5.0""",0.14, -31.95,E
+"For $\mathrm{NaCl}, R_e=2.36 Å$. The ionization energy of $\mathrm{Na}$ is $5.14 \mathrm{eV}$, and the electron affinity of $\mathrm{Cl}$ is $3.61 \mathrm{eV}$. Use the simple model of $\mathrm{NaCl}$ as a pair of spherical ions in contact to estimate $D_e$. [One debye (D) is $3.33564 \times 10^{-30} \mathrm{C} \mathrm{m}$.]","Therefore, the distance between the Na+ and Cl− ions is half of 564.02 pm, which is 282.01 pm. right|thumb|The Madelung constant being calculated for the NaCl ion labeled 0 in the expanding spheres method. For example, the inter-ionic distance in RbI is 356 pm, giving 142 pm for the ionic radius of Rb+. If the distances are normalized to the nearest neighbor distance , the potential may be written :V_i = \frac{e}{4 \pi \varepsilon_0 r_0 } \sum_{j} \frac{z_j r_0}{r_{ij}} = \frac{e}{4 \pi \varepsilon_0 r_0 } M_i with being the (dimensionless) Madelung constant of the th ion :M_i = \sum_{j} \frac{z_j}{r_{ij}/r_0}. Comparison between observed and calculated ion separations (in pm) MX Observed Soft-sphere model LiCl 257.0 257.2 LiBr 275.1 274.4 NaCl 282.0 281.9 NaBr 298.7 298.2 In the soft-sphere model, k has a value between 1 and 2. The Debye–Hückel limiting law enables one to determine the activity coefficient of an ion in a dilute solution of known ionic strength. In this way values for the radii of 8 ions were determined. Pauling used effective nuclear charge to proportion the distance between ions into anionic and a cationic radii.Pauling, L. (1960). For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). One approach to improving the calculated accuracy is to model ions as ""soft spheres"" that overlap in the crystal. In the next step, D&H; assume that there is a certain radius a_i, beyond which no ions in the atmosphere may approach the (charge) center of the singled out ion. The electrostatic energy of the ion at site then is the product of its charge with the potential acting at its site :E_{el,i} = z_ieV_i = \frac{e^2}{4 \pi \varepsilon_0 r_0 } z_i M_i. D&H; say that this approximation holds at large distances between ions, which is the same as saying that the concentration is low. _Effective_ ionic radii in pm of elements as a function of ionic charge and spin (ls = low spin, hs = high spin). All compounds crystallize in the NaCl structure. thumb|300 px|Relative radii of atoms and ions. For ions on lower-symmetry sites significant deviations of their electron density from a spherical shape may occur. These tables list values of molar ionization energies, measured in kJ⋅mol−1. The iodide ions nearly touch (but don't quite), indicating that Landé's assumption is fairly good. To be consistent with Pauling's radii, Shannon has used a value of rion(O2−) = 140 pm; data using that value are referred to as ""effective"" ionic radii. For table salt in 0.01 M solution at 25 °C, a typical value of (\kappa a)^2 is 0.0005636, while a typical value of Z_0 is 7.017, highlighting the fact that, in low concentrations, (\kappa a)^2 is a target for a zero order of magnitude approximation such as perturbation analysis. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. == Formal expression == The Madelung constant allows for the calculation of the electric potential of all ions of the lattice felt by the ion at position :V_i = \frac{e}{4 \pi \varepsilon_0 } \sum_{j eq i} \frac{z_j}{r_{ij}}\,\\! where r_{ij} = |r_i-r_j| is the distance between the th and the th ion. Note also how the fluorite structure being intermediate between the caesium chloride and sphalerite structures is reflected in the Madelung constants. == Formula == A fast converging formula for the Madelung constant of NaCl is :12 \, \pi \sum_{m, n \geq 1, \, \mathrm{odd}} \operatorname{sech}^2\left(\frac{\pi}{2}(m^2+n^2)^{1/2}\right) == Generalization == It is assumed for the calculation of Madelung constants that an ion's charge density may be approximated by a point charge. ", 4.56,3.0,"""0.1353""",30,3.61,A
+Find the number of CSFs in a full CI calculation of $\mathrm{CH}_2 \mathrm{SiHF}$ using a 6-31G** basis set.,"The molecular formula C6H8S (molar mass: 112.19 g/mol, exact mass: 112.0347 u) may refer to: * 2,3-Dihydrothiepine * 2,7-Dihydrothiepine * 2,5-Dimethylthiophene Category:Molecular formulas Caesium fluoride or cesium fluoride is an inorganic compound with the formula CsF and it is a hygroscopic white salt. Caltech Intermediate Form (CIF) is a file format for describing integrated circuits. The molecular formula C6H6O2S (molar mass: 142.18 g/mol, exact mass: 142.0089 u) may refer to: * 3,4-Ethylenedioxythiophene (EDOT) * Phenylsulfinic acid * 3-Thiophene acetic acid * Thiophene-2-acetic acid All numbers in CIF are integers that refer to centimicrons of distance, unless subroutine scaling is specified (described later). CsF reaches a vapor pressure of 1 kilopascal at 825 °C, 10 kPa at 999 °C, and 100 kPa at 1249 °C. The molecular formula C8H6S (molar mass: 134.20 g/mol, exact mass: 134.0190 u) may refer to: *Benzo[c]thiophene *Benzothiophene Category:Molecular formulas The molecular formula C3H2F6O (molar mass: 168.038 g/mol, exact mass: 168.0010 u) may refer to: * Desflurane * Hexafluoro-2-propanol (HFIP) CsF is an alternative to tetra-n-butylammonium fluoride (TBAF) and TAS-fluoride (TASF). ===As a base=== As with other soluble fluorides, CsF is moderately basic, because HF is a weak acid. Extensions to CIF can be done with the numeric statements `0` through `9`. CsF gives higher yields in Knoevenagel condensation reactions than KF or NaF. ===Formation of Cs-F bonds=== Caesium fluoride serves as a source of fluoride in organofluorine chemistry. CsF chains with a thickness as small as one or two atoms can be grown inside carbon nanotubes. ==Structure== Caesium fluoride has the halite structure, which means that the Cs+ and F− pack in a cubic closest packed array as do Na+ and Cl− in sodium chloride. ==Applications in organic synthesis== Being highly dissociated, CsF is a more reactive source of fluoride than related salts. The reaction is shown below: :Cs2CO3 \+ 2 HF → 2 CsF + H2O + CO2 CsF is more soluble than sodium fluoride or potassium fluoride in organic solvents. FIGURE B.5 Typical user extensions to CIF. The final statement in a CIF file is the `END` statement (or the letter `E`). The reaction is shown below: :CsOH + HF → CsF + H2O Using the same reaction, another way to create caesium fluoride is to treat caesium carbonate (Cs2CO3) with hydrofluoric acid and again, the resulting salt can then be purified by recrystallization. Solutions of caesium fluoride in THF or DMF attack a wide variety of organosilicon compounds to produce an organosilicon fluoride and a carbanion, which can then react with electrophiles, for example: :500px ==Precautions== Like other soluble fluorides, CsF is moderately toxic.MSDS Listing for cesium fluoride . www.hazard.com . Similarly to potassium fluoride, CsF reacts with hexafluoroacetone to form a stable perfluoroalkoxide salt. FIGURE B.1 CIF layer names for MOS processes. CIF provides a limited set of graphics primitives that are useful for describing the two-dimensional shapes on the different layers of a chip. Note that the magnitude of this rotation vector has no meaning. thumb|333px|right|FIGURE B.2 A sample CIF ""wire"" statement. Caesium also has the highest electropositivity of all known elements and fluorine has the highest electronegativity of all known elements. ==Synthesis and properties== Caesium fluoride can be prepared by the reaction of caesium hydroxide (CsOH) with hydrofluoric acid (HF) and the resulting salt can then be purified by recrystallization. ",1.86,1855,"""0.686""",62.8318530718,93.4,A
+Calculate the ratio of the electrical and gravitational forces between a proton and an electron.,"However, the Coulomb force between protons has a much greater range as it varies as the inverse square of the charge separation, and Coulomb repulsion thus becomes the only significant force between protons when their separation exceeds about 2 to 2.5 fm. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. Together with the mass-to-charge ratio, the electron mass was determined with reasonable precision. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. In January 2013, an updated value for the charge radius of a proton——was published. Since the ratio doesn't vary for resting electrons, the data points should be on a single horizontal line (see Fig. 6). Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). The proton radius puzzle is an unanswered problem in physics relating to the size of the proton. The result is again ~5% smaller than the previously-accepted proton radius. Their measurement of the root-mean-square charge radius of a proton is "", which differs by 5.0 standard deviations from the CODATA value of "". Because protons are not fundamental particles, they possess a measurable size; the root mean square charge radius of a proton is about 0.84–0.87 fm ( = ). The radius of the proton is linked to the form factor and momentum-transfer cross section. However, in such an association with an electron, the character of the bound proton is not changed, and it remains a proton. Physical parameter 1H 16O relative atomic mass of the XZ+ ion relative atomic mass of the Z electrons correction for the binding energy relative atomic mass of the neutral atom The principle can be shown by the determination of the electron relative atomic mass by Farnham et al. at the University of Washington (1995). The internationally accepted value of a proton's charge radius is . It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. The proton radius was a puzzle as of 2017. Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius. === 2019 measurements === In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. Its mass is slightly less than that of a neutron and 1,836 times the mass of an electron (the proton-to- electron mass ratio). ",0,14.44,"""479.0""",2,22,D
 "A one-particle, one-dimensional system has the state function
 $$
 \Psi=(\sin a t)\left(2 / \pi c^2\right)^{1 / 4} e^{-x^2 / c^2}+(\cos a t)\left(32 / \pi c^6\right)^{1 / 4} x e^{-x^2 / c^2}
 $$
-where $a$ is a constant and $c=2.000 Å$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 Å$ and $2.001 Å$.","Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. A quantum particle is in a bound state if it is never found “too far away from any finite region R\subseteq X”, i.e. using a wavefunction representation, \begin{align} 0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\\ &= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)} \end{align} Consequently, \int_X{|\psi(x)|^{2}\,d\mu(x)} is finite. If a state has energy E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}, then the wavefunction satisfies, for some X > 0 :\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X so that is exponentially suppressed at large . Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an ""eigenstate of position"", meaning that its position has a known value, an eigenvalue of the eigenstate of position. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. *If the state evolution of ""moves this wave package constantly to the right"", e.g. if [t-1,t+1] \in \mathrm{Supp}(\rho(t)) for all t \geq 0, then is not bound state with respect to position. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. thumb|CEP concept and hit probability. 0.2% outside the outmost circle. In probability theory, the spectral expansion solution method is a technique for computing the stationary probability distribution of a continuous-time Markov chain whose state space is a semi-infinite lattice strip. In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. To obtain g(0), the following limit is applied, g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \mathrm{rect}(\frac{x}{a}) and this can be written in terms of the Dirac delta function as, g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).The Fourier transform of the Dirac delta function \delta (t) is \delta (f) = \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \mathrm{rect}(\frac{t}{a})\cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \mathrm{sinc}{(a f)}. where the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at f = 1 / a and a goes to infinity, the Fourier transform of \delta (t) is \delta (f) = 1, means that the frequency spectrum of the Dirac delta function is infinitely broad. The approximation error is the gap between the curves, and it increases for x values further from 0. In order for the first bound state to exist at all, D\gtrsim 0.8. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. The characteristic function is \varphi(k) = \frac{\sin(k/2)}{k/2}, and its moment-generating function is M(k) = \frac{\sinh(k/2)}{k/2}, where \sinh(t) is the hyperbolic sine function. ==Rational approximation== The pulse function may also be expressed as a limit of a rational function: \Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}. ===Demonstration of validity=== First, we consider the case where |t|<\frac{1}{2}. Its periodic version is called a rectangular wave. ==History== The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively. ==Relation to the boxcar function== The rectangular function is a special case of the more general boxcar function: \operatorname{rect}\left(\frac{t-X}{Y} \right) = H(t - (X - Y/2)) - H(t - (X + Y/2)) = H(t - X + Y/2) - H(t - X - Y/2) where H(x) is the Heaviside step function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2. ==Fourier transform of the rectangular function== thumb|400px|right|Plot of normalized \mathrm{sinc}(x) function (i.e. \mathrm{sinc}(\pi x)) with its spectral frequency components. If v e 0, the relative error is : \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, and the percent error (an expression of the relative error) is :\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. The approximation error in a data value is the discrepancy between an exact value and some approximation to it. ",7,59.4,0.000216,3.0,4943,C
+where $a$ is a constant and $c=2.000 Å$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 Å$ and $2.001 Å$.","Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. A quantum particle is in a bound state if it is never found “too far away from any finite region R\subseteq X”, i.e. using a wavefunction representation, \begin{align} 0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\\ &= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)} \end{align} Consequently, \int_X{|\psi(x)|^{2}\,d\mu(x)} is finite. If a state has energy E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}, then the wavefunction satisfies, for some X > 0 :\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X so that is exponentially suppressed at large . Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an ""eigenstate of position"", meaning that its position has a known value, an eigenvalue of the eigenstate of position. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. *If the state evolution of ""moves this wave package constantly to the right"", e.g. if [t-1,t+1] \in \mathrm{Supp}(\rho(t)) for all t \geq 0, then is not bound state with respect to position. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. thumb|CEP concept and hit probability. 0.2% outside the outmost circle. In probability theory, the spectral expansion solution method is a technique for computing the stationary probability distribution of a continuous-time Markov chain whose state space is a semi-infinite lattice strip. In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. To obtain g(0), the following limit is applied, g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \mathrm{rect}(\frac{x}{a}) and this can be written in terms of the Dirac delta function as, g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).The Fourier transform of the Dirac delta function \delta (t) is \delta (f) = \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \mathrm{rect}(\frac{t}{a})\cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \mathrm{sinc}{(a f)}. where the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at f = 1 / a and a goes to infinity, the Fourier transform of \delta (t) is \delta (f) = 1, means that the frequency spectrum of the Dirac delta function is infinitely broad. The approximation error is the gap between the curves, and it increases for x values further from 0. In order for the first bound state to exist at all, D\gtrsim 0.8. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. The characteristic function is \varphi(k) = \frac{\sin(k/2)}{k/2}, and its moment-generating function is M(k) = \frac{\sinh(k/2)}{k/2}, where \sinh(t) is the hyperbolic sine function. ==Rational approximation== The pulse function may also be expressed as a limit of a rational function: \Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}. ===Demonstration of validity=== First, we consider the case where |t|<\frac{1}{2}. Its periodic version is called a rectangular wave. ==History== The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively. ==Relation to the boxcar function== The rectangular function is a special case of the more general boxcar function: \operatorname{rect}\left(\frac{t-X}{Y} \right) = H(t - (X - Y/2)) - H(t - (X + Y/2)) = H(t - X + Y/2) - H(t - X - Y/2) where H(x) is the Heaviside step function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2. ==Fourier transform of the rectangular function== thumb|400px|right|Plot of normalized \mathrm{sinc}(x) function (i.e. \mathrm{sinc}(\pi x)) with its spectral frequency components. If v e 0, the relative error is : \eta = \frac{\epsilon}{|v|} = \left| \frac{v-v_\text{approx}}{v} \right| = \left| 1 - \frac{v_\text{approx}}{v} \right|, and the percent error (an expression of the relative error) is :\delta = 100\%\times\eta = 100\%\times\left| \frac{v-v_\text{approx}}{v} \right|. The approximation error in a data value is the discrepancy between an exact value and some approximation to it. ",7,59.4,"""0.000216""",3.0,4943,C
 "A one-particle, one-dimensional system has the state function
 $$
 \Psi=(\sin a t)\left(2 / \pi c^2\right)^{1 / 4} e^{-x^2 / c^2}+(\cos a t)\left(32 / \pi c^6\right)^{1 / 4} x e^{-x^2 / c^2}
 $$
-where $a$ is a constant and $c=2.000 Å$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 Å$ and $2.001 Å$.","Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. A quantum particle is in a bound state if it is never found “too far away from any finite region R\subseteq X”, i.e. using a wavefunction representation, \begin{align} 0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\\ &= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)} \end{align} Consequently, \int_X{|\psi(x)|^{2}\,d\mu(x)} is finite. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. Groeneboom (1989) shows that : f_Z (z) \sim \frac{1}{2} \frac{4^{4/3} |z|}{\operatorname{Ai}' (\tilde{a}_1)} \exp \left( - \frac{2}{3} |z|^3 + 2^{1/3} \tilde{a}_1 |z| \right) \text{ as }z \rightarrow \infty where \tilde{a}_1 \approx -2.3381 is the largest zero of the Airy function Ai and where \operatorname{Ai}' (\tilde{a}_1 ) \approx 0.7022. If a state has energy E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}, then the wavefunction satisfies, for some X > 0 :\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X so that is exponentially suppressed at large . 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. *If the state evolution of ""moves this wave package constantly to the right"", e.g. if [t-1,t+1] \in \mathrm{Supp}(\rho(t)) for all t \geq 0, then is not bound state with respect to position. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an ""eigenstate of position"", meaning that its position has a known value, an eigenvalue of the eigenstate of position. In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. thumb|CEP concept and hit probability. 0.2% outside the outmost circle. If : V(a,c) = \underset{s \in \mathbf{R}}{\operatorname{argmax}} \ (W(s) - c(s-a)^2), then V(0, c) has density : f_c(t) = \frac{1}{2} g_c(t) g_c(-t) where gc has Fourier transform given by : \hat{g}_c (s) = \frac{(2/c)^{1/3}}{\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \ \ \ s \in \mathbf{R} and where Ai is the Airy function. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. To obtain g(0), the following limit is applied, g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \mathrm{rect}(\frac{x}{a}) and this can be written in terms of the Dirac delta function as, g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).The Fourier transform of the Dirac delta function \delta (t) is \delta (f) = \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \mathrm{rect}(\frac{t}{a})\cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \mathrm{sinc}{(a f)}. where the sinc function here is the normalized sinc function. In order for the first bound state to exist at all, D\gtrsim 0.8. thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. Because the first zero of the sinc function is at f = 1 / a and a goes to infinity, the Fourier transform of \delta (t) is \delta (f) = 1, means that the frequency spectrum of the Dirac delta function is infinitely broad. The characteristic function is \varphi(k) = \frac{\sin(k/2)}{k/2}, and its moment-generating function is M(k) = \frac{\sinh(k/2)}{k/2}, where \sinh(t) is the hyperbolic sine function. ==Rational approximation== The pulse function may also be expressed as a limit of a rational function: \Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}. ===Demonstration of validity=== First, we consider the case where |t|<\frac{1}{2}. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. The approximation error is the gap between the curves, and it increases for x values further from 0. Its periodic version is called a rectangular wave. ==History== The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively. ==Relation to the boxcar function== The rectangular function is a special case of the more general boxcar function: \operatorname{rect}\left(\frac{t-X}{Y} \right) = H(t - (X - Y/2)) - H(t - (X + Y/2)) = H(t - X + Y/2) - H(t - X - Y/2) where H(x) is the Heaviside step function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2. ==Fourier transform of the rectangular function== thumb|400px|right|Plot of normalized \mathrm{sinc}(x) function (i.e. \mathrm{sinc}(\pi x)) with its spectral frequency components. ",117,0,-273.0,4.946,0.000216,E
-"The $J=2$ to 3 rotational transition in a certain diatomic molecule occurs at $126.4 \mathrm{GHz}$, where $1 \mathrm{GHz} \equiv 10^9 \mathrm{~Hz}$. Find the frequency of the $J=5$ to 6 absorption in this molecule.","As 1 GHz = 109 Hz, the numerical conversion can be expressed as :\tilde u / \text{cm}^{-1} \approx \frac{ u / \text{GHz}}{30}. ===Effect of vibration on rotation=== The population of vibrationally excited states follows a Boltzmann distribution, so low- frequency vibrational states are appreciably populated even at room temperatures. For the so-called R branch of the spectrum, J' = J + 1 so that there is simultaneous excitation of both vibration and rotation. However, since only integer values of J are allowed, the maximum line intensity is observed for a neighboring integer J. :J = \sqrt{\frac{kT}{2hcB}} - \frac{1}{2} The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it. ====Centrifugal distortion==== When a molecule rotates, the centrifugal force pulls the atoms apart. The energy difference between successive J terms in any of these triplets is about 2 cm−1 (60 GHz), with the single exception of J = 1←0 difference which is about 4 cm−1. For a linear molecule, analysis of the rotational spectrum provides values for the rotational constantThis article uses the molecular spectroscopist's convention of expressing the rotational constant B in cm−1. Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., \Delta J = J^{\prime} - J^{\prime\prime} = \pm 1 . Constants of Diatomic Molecules, by K. P. Huber and Gerhard Herzberg (Van nostrand Reinhold company, New York, 1979, ), is a classic comprehensive multidisciplinary reference text contains a critical compilation of available data for all diatomic molecules and ions known at the time of publication - over 900 diatomic species in all - including electronic energies, vibrational and rotational constants, and observed transitions. Under the rigid rotor model, the rotational energy levels, F(J), of the molecule can be expressed as, : F\left( J \right) = B J \left( J+1 \right) \qquad J = 0,1,2,... where B is the rotational constant of the molecule and is related to the moment of inertia of the molecule. J is the quantum number of the lower rotational state. The Schumann–Runge bands are a set of absorption bands of molecular oxygen that occur at wavelengths between 176 and 192.6 nanometres. Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. This spectrum is also interesting because it shows clear evidence of Coriolis coupling in the asymmetric structure of the band. ===Linear molecules=== right|thumb|300px|Energy levels and line positions calculated in the rigid rotor approximation The rigid rotor is a good starting point from which to construct a model of a rotating molecule. The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. The number of molecules in an excited state with quantum number J, relative to the number of molecules in the ground state, NJ/N0 is given by the Boltzmann distribution as :\frac{N_J}{N_0} = e^{-\frac{E_J}{kT}} = e^{-\frac {BhcJ(J+1)}{kT}}, where k is the Boltzmann constant and T the absolute temperature. The Rydberg–Klein–Rees method is a procedure used in the analysis of rotational-vibrational spectra of diatomic molecules to obtain a potential energy curve from the experimentally-known line positions. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule. In this approximation, the vibration-rotation wavenumbers of transitions are :\tilde u = \tilde u_\text{vib} + BJ(J + 1) - B'J'(J' + 1), where B and B' are rotational constants for the upper and lower vibrational state respectively, while J and J' are the rotational quantum numbers of the upper and lower levels. Adjacent J^{\prime\prime}{\leftarrow}J^{\prime} transitions are separated by 2B in the observed spectrum. This gives the transition wavenumbers as : \tilde u_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) = 2 B \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,... which is the same as in the case of a linear molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, I_B = I_C, I_A=0 , so : B = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}} For a diatomic molecule : I=\frac{m_1m_2}{m_1 +m_2}d^2 where m1 and m2 are the masses of the atoms and d is the distance between them. The first study of the microwave spectrum of a molecule () was performed by Cleeton & Williams in 1934. J band may refer to: * J band (infrared), an atmospheric transmission window centred on 1.25 μm * J band (JRC), radio frequency bands from 139.5 to 140.5 and 148 to 149 MHz * J band (NATO), a radio frequency band from 10 to 20 GHz ", 252.8,4,-0.38,0.19,14.5115,A
+where $a$ is a constant and $c=2.000 Å$. If the particle's position is measured at $t=0$, estimate the probability that the result will lie between $2.000 Å$ and $2.001 Å$.","Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value. A quantum particle is in a bound state if it is never found “too far away from any finite region R\subseteq X”, i.e. using a wavefunction representation, \begin{align} 0 &= \lim_{R\to\infty}{\mathbb{P}(\text{particle measured inside }X\setminus R)} \\\ &= \lim_{R\to\infty}{\int_{X\setminus R}|\psi(x)|^2\,d\mu(x)} \end{align} Consequently, \int_X{|\psi(x)|^{2}\,d\mu(x)} is finite. Initial attempts at approximating Chernoff's distribution via solving the heat equation, however, did not achieve satisfactory precision due to the nature of the boundary conditions. Groeneboom (1989) shows that : f_Z (z) \sim \frac{1}{2} \frac{4^{4/3} |z|}{\operatorname{Ai}' (\tilde{a}_1)} \exp \left( - \frac{2}{3} |z|^3 + 2^{1/3} \tilde{a}_1 |z| \right) \text{ as }z \rightarrow \infty where \tilde{a}_1 \approx -2.3381 is the largest zero of the Airy function Ai and where \operatorname{Ai}' (\tilde{a}_1 ) \approx 0.7022. If a state has energy E < \max{\left(\lim_{x\to\infty}{V(x)}, \lim_{x\to-\infty}{V(x)}\right)}, then the wavefunction satisfies, for some X > 0 :\frac{\psi^{\prime\prime}}{\psi}=\frac{2m}{\hbar^2}(V(x)-E) > 0\text{ for }x > X so that is exponentially suppressed at large . 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. *If the state evolution of ""moves this wave package constantly to the right"", e.g. if [t-1,t+1] \in \mathrm{Supp}(\rho(t)) for all t \geq 0, then is not bound state with respect to position. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an electron, given that its position has been determined at a given instant. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an ""eigenstate of position"", meaning that its position has a known value, an eigenvalue of the eigenstate of position. In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. thumb|CEP concept and hit probability. 0.2% outside the outmost circle. If : V(a,c) = \underset{s \in \mathbf{R}}{\operatorname{argmax}} \ (W(s) - c(s-a)^2), then V(0, c) has density : f_c(t) = \frac{1}{2} g_c(t) g_c(-t) where gc has Fourier transform given by : \hat{g}_c (s) = \frac{(2/c)^{1/3}}{\operatorname{Ai} (i (2c^2)^{-1/3} s)}, \ \ \ s \in \mathbf{R} and where Ai is the Airy function. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. To obtain g(0), the following limit is applied, g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \mathrm{rect}(\frac{x}{a}) and this can be written in terms of the Dirac delta function as, g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).The Fourier transform of the Dirac delta function \delta (t) is \delta (f) = \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \mathrm{rect}(\frac{t}{a})\cdot e^{-i 2\pi f t} \, dt = \lim_{a \to 0} \mathrm{sinc}{(a f)}. where the sinc function here is the normalized sinc function. In order for the first bound state to exist at all, D\gtrsim 0.8. thumb|Graph of f(x) = e^x (blue) with its linear approximation P_1(x) = 1 + x (red) at a = 0. Because the first zero of the sinc function is at f = 1 / a and a goes to infinity, the Fourier transform of \delta (t) is \delta (f) = 1, means that the frequency spectrum of the Dirac delta function is infinitely broad. The characteristic function is \varphi(k) = \frac{\sin(k/2)}{k/2}, and its moment-generating function is M(k) = \frac{\sinh(k/2)}{k/2}, where \sinh(t) is the hyperbolic sine function. ==Rational approximation== The pulse function may also be expressed as a limit of a rational function: \Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}. ===Demonstration of validity=== First, we consider the case where |t|<\frac{1}{2}. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. The approximation error is the gap between the curves, and it increases for x values further from 0. Its periodic version is called a rectangular wave. ==History== The rect function has been introduced by Woodward in as an ideal cutout operator, together with the sinc function as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively. ==Relation to the boxcar function== The rectangular function is a special case of the more general boxcar function: \operatorname{rect}\left(\frac{t-X}{Y} \right) = H(t - (X - Y/2)) - H(t - (X + Y/2)) = H(t - X + Y/2) - H(t - X - Y/2) where H(x) is the Heaviside step function; the function is centered at X and has duration Y, from X-Y/2 to X+Y/2. ==Fourier transform of the rectangular function== thumb|400px|right|Plot of normalized \mathrm{sinc}(x) function (i.e. \mathrm{sinc}(\pi x)) with its spectral frequency components. ",117,0,"""-273.0""",4.946,0.000216,E
+"The $J=2$ to 3 rotational transition in a certain diatomic molecule occurs at $126.4 \mathrm{GHz}$, where $1 \mathrm{GHz} \equiv 10^9 \mathrm{~Hz}$. Find the frequency of the $J=5$ to 6 absorption in this molecule.","As 1 GHz = 109 Hz, the numerical conversion can be expressed as :\tilde u / \text{cm}^{-1} \approx \frac{ u / \text{GHz}}{30}. ===Effect of vibration on rotation=== The population of vibrationally excited states follows a Boltzmann distribution, so low- frequency vibrational states are appreciably populated even at room temperatures. For the so-called R branch of the spectrum, J' = J + 1 so that there is simultaneous excitation of both vibration and rotation. However, since only integer values of J are allowed, the maximum line intensity is observed for a neighboring integer J. :J = \sqrt{\frac{kT}{2hcB}} - \frac{1}{2} The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it. ====Centrifugal distortion==== When a molecule rotates, the centrifugal force pulls the atoms apart. The energy difference between successive J terms in any of these triplets is about 2 cm−1 (60 GHz), with the single exception of J = 1←0 difference which is about 4 cm−1. For a linear molecule, analysis of the rotational spectrum provides values for the rotational constantThis article uses the molecular spectroscopist's convention of expressing the rotational constant B in cm−1. Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity; i.e., \Delta J = J^{\prime} - J^{\prime\prime} = \pm 1 . Constants of Diatomic Molecules, by K. P. Huber and Gerhard Herzberg (Van nostrand Reinhold company, New York, 1979, ), is a classic comprehensive multidisciplinary reference text contains a critical compilation of available data for all diatomic molecules and ions known at the time of publication - over 900 diatomic species in all - including electronic energies, vibrational and rotational constants, and observed transitions. Under the rigid rotor model, the rotational energy levels, F(J), of the molecule can be expressed as, : F\left( J \right) = B J \left( J+1 \right) \qquad J = 0,1,2,... where B is the rotational constant of the molecule and is related to the moment of inertia of the molecule. J is the quantum number of the lower rotational state. The Schumann–Runge bands are a set of absorption bands of molecular oxygen that occur at wavelengths between 176 and 192.6 nanometres. Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. This spectrum is also interesting because it shows clear evidence of Coriolis coupling in the asymmetric structure of the band. ===Linear molecules=== right|thumb|300px|Energy levels and line positions calculated in the rigid rotor approximation The rigid rotor is a good starting point from which to construct a model of a rotating molecule. The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. The number of molecules in an excited state with quantum number J, relative to the number of molecules in the ground state, NJ/N0 is given by the Boltzmann distribution as :\frac{N_J}{N_0} = e^{-\frac{E_J}{kT}} = e^{-\frac {BhcJ(J+1)}{kT}}, where k is the Boltzmann constant and T the absolute temperature. The Rydberg–Klein–Rees method is a procedure used in the analysis of rotational-vibrational spectra of diatomic molecules to obtain a potential energy curve from the experimentally-known line positions. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule. In this approximation, the vibration-rotation wavenumbers of transitions are :\tilde u = \tilde u_\text{vib} + BJ(J + 1) - B'J'(J' + 1), where B and B' are rotational constants for the upper and lower vibrational state respectively, while J and J' are the rotational quantum numbers of the upper and lower levels. Adjacent J^{\prime\prime}{\leftarrow}J^{\prime} transitions are separated by 2B in the observed spectrum. This gives the transition wavenumbers as : \tilde u_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) = 2 B \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,... which is the same as in the case of a linear molecule. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, I_B = I_C, I_A=0 , so : B = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}} For a diatomic molecule : I=\frac{m_1m_2}{m_1 +m_2}d^2 where m1 and m2 are the masses of the atoms and d is the distance between them. The first study of the microwave spectrum of a molecule () was performed by Cleeton & Williams in 1934. J band may refer to: * J band (infrared), an atmospheric transmission window centred on 1.25 μm * J band (JRC), radio frequency bands from 139.5 to 140.5 and 148 to 149 MHz * J band (NATO), a radio frequency band from 10 to 20 GHz ", 252.8,4,"""-0.38""",0.19,14.5115,A
 "Assume that the charge of the proton is distributed uniformly throughout the volume of a sphere of radius $10^{-13} \mathrm{~cm}$. Use perturbation theory to estimate the shift in the ground-state hydrogen-atom energy due to the finite proton size. The potential energy experienced by the electron when it has penetrated the nucleus and is at distance $r$ from the nuclear center is $-e Q / 4 \pi \varepsilon_0 r$, where $Q$ is the amount of proton charge within the sphere of radius $r$. The evaluation of the integral is simplified by noting that the exponential factor in $\psi$ is essentially equal to 1 within the nucleus.
-","Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius. === 2019 measurements === In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen. Measurements of hydrogen's energy levels are now so precise that the accuracy of the proton radius is the limiting factor when comparing experimental results to theoretical calculations. His personal assumption is that past measurements have misgauged the Rydberg constant and that the current official proton size is inaccurate. ===Quantum chromodynamic calculation=== In a paper by Belushkin et al. (2007), including different constraints and perturbative quantum chromodynamics, a smaller proton radius than the then-accepted 0.877 femtometres was predicted. ===Proton radius extrapolation=== Papers from 2016 suggested that the problem was with the extrapolations that had typically been used to extract the proton radius from the electron scattering data though these explanation would require that there was also a problem with the atomic Lamb shift measurements. ===Data analysis method=== In one of the attempts to resolve the puzzle without new physics, Alarcón et al. (2018) of Jefferson Lab have proposed that a different technique to fit the experimental scattering data, in a theoretically as well as analytically justified manner, produces a proton charge radius from the existing electron scattering data that is consistent with the muonic hydrogen measurement. The result is again ~5% smaller than the previously-accepted proton radius. By measuring the energy required to excite hydrogen atoms from the 2S to the 2P state, the Rydberg constant could be calculated, and from this the proton radius inferred. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. The form of the potential, in terms of the distance r from the center of nucleus, is: V(r) = -\frac{V_0}{1+\exp({r-R\over a})} where V0 (having dimension of energy) represents the potential well depth, a is a length representing the ""surface thickness"" of the nucleus, and R = r_0 A^{1/3} is the nuclear radius where and A is the mass number. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. Effectively, this approach attributes the cause of the proton radius puzzle to a failure to use a theoretically motivated function for the extraction of the proton charge radius from the experimental data. Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). He also noted that in the asymptotic limit (far away from the nucleus), his approximate form coincides with the exact hydrogen-like wave function in the presence of a nuclear charge of Z-s and in the state with a principal quantum number n equal to his effective quantum number n*. this opinion is not yet universally held. ==Problem== Prior to 2010, the proton charge radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering. ===Spectroscopy method=== The spectroscopy method uses the energy levels of electrons orbiting the nucleus. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. thumb|right|300px|Woods–Saxon potential for , relative to V0 with a and R=4.6 fm The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. thumb|300px|Comparison of a wavefunction in the Coulomb potential of the nucleus (blue) to the one in the pseudopotential (red). This method produces a proton radius of about , with approximately 1% relative uncertainty. ===Nuclear scattering=== The nuclear method is similar to Rutherford's scattering experiments that established the existence of the nucleus. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. ",6,7.136,-0.28,1.2,-131.1,D
-"An electron in a three-dimensional rectangular box with dimensions of $5.00 Å, 3.00 Å$, and $6.00 Å$ makes a radiative transition from the lowest-lying excited state to the ground state. Calculate the frequency of the photon emitted.","We may derive the two- photon Rabi frequency by returning to the equations \begin{align} i \dot{c}_1(t) &= \frac{\Omega_{1i} c_2}{2} e^{i\Delta t}\\\ i \dot{c}_i(t) &= \frac{\Omega^*_{1i} c_1}{2} e^{-i\Delta t} \end{align} which now describe excitation between the ground and intermediate states. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. Photoexcitation is the production of an excited state of a quantum system by photon absorption. Transition frequency may refer to: *A measure of the high-frequency operating characteristics of a transistor, usually symbolized as *A characteristic of spectral lines *The frequency of the radiation associated with a transition between hyperfine structure energy states of an atom *Turnover frequency in enzymology In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. Also note that as the incident light frequency shifts further from the transition frequency, the amplitude of the Rabi oscillation decreases, as is illustrated by the dashed envelope in the above plot. == Two-Photon Rabi Frequency == Coherent Rabi oscillations may also be driven by two-photon transitions. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. After excitation the atom may return to the ground state or a lower excited state, by emitting a photon with a characteristic energy. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. The excited state originates from the interaction between a photon and the quantum system. A two-photon transition is not the same as excitation from the ground to intermediate state, and then out of the intermediate state to the excited state. In this case we consider a system with three atomic energy levels, |1\rangle , |i\rangle , and |2\rangle , where |i\rangle represents a so-called intermediate state with corresponding frequency \omega_i , and an electromagnetic field with two frequency components: \hat{V}(t) = e\mathbf{r} \cdot \mathbf{E}_{L1} \cos(\omega_{L1} t) + e\mathbf{r} \cdot \mathbf{E}_{L2} \cos(\omega_{L2} t) Now, \omega_i may be much greater than both \omega_1 and \omega_2 , or \omega_2 > \omega_i > \omega_1 , as illustrated in the figure on the right. thumb|Two photon excitation schema. \omega_i >> \omega_2 > \omega_1 is shown on the left, while \omega_2 > \omega_i > \omega_1 is shown on the right. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). Emission of photons from atoms in various excited states leads to an electromagnetic spectrum showing a series of characteristic emission lines (including, in the case of the hydrogen atom, the Lyman, Balmer, Paschen and Brackett series). The Rabi frequency is a semiclassical concept since it treats the atom as an object with quantized energy levels and the electromagnetic field as a continuous wave. A plot as a function of detuning and ramping the time from 0 to t = \frac{\pi}{\Omega} gives: Animation of optical resonance, frequency domain We see that for \delta = 0 the population will oscillate between the two states at the Rabi frequency. == Generalized Rabi frequency == The quantity \sqrt{\Omega^2 + \delta^2} is commonly referred to as the ""generalized Rabi frequency."" The energy released is equal to the difference in energy levels between the electron energy states. The absorption of the photon takes place in accordance with Planck's quantum theory. The next rule follows from the Frank-Condon Principle, which states that the absorption of a photon by an electron and the subsequent jump in energy levels is near-instantaneous. ",7.58,0.4772,24.0,655,0.11,A
-Do $\mathrm{HF} / 6-31 \mathrm{G}^*$ geometry optimizations on one conformers of $\mathrm{HCOOH}$ with $\mathrm{OCOH}$ dihedral angle of $0^{\circ}$. Calculate the dipole moment.,"Order-6 hexagonal tiling honeycomb Order-6 hexagonal tiling honeycomb 320px Perspective projection view from center of Poincaré disk model 320px Perspective projection view from center of Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {6,3,6} {6,3[3]} Coxeter diagram ↔ ↔ Cells {6,3} 40px Faces hexagon {6} Edge figure hexagon {6} Vertex figure {3,6} or {3[3]} 40px 40px Dual Self-dual Coxeter group \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Regular, quasiregular In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. Order-6 dodecahedral honeycomb Order-6 dodecahedral honeycomb 320px Perspective projection view within Poincaré disk model 320px Perspective projection view within Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {5,3,6} {5,3[3]} Coxeter diagram ↔ Cells {5,3} 40px Faces pentagon {5} Edge figure hexagon {6} Vertex figure 80px 80px triangular tiling Dual Order-5 hexagonal tiling honeycomb Coxeter group \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Regular, quasiregular The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. Order-5 hexagonal tiling honeycomb Order-5 hexagonal tiling honeycomb 320px Perspective projection view from center of Poincaré disk model 320px Perspective projection view from center of Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {6,3,5} Coxeter-Dynkin diagrams 80px ↔ Cells {6,3} 40px Faces hexagon {6} Edge figure pentagon {5} Vertex figure icosahedron Dual Order-6 dodecahedral honeycomb Coxeter group \overline{HV}_3, [5,3,6] Properties Regular In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). alt=|thumb|300x300px|The flow structure of the Lamb-Chaplygin dipole The Lamb–Chaplygin dipole model is a mathematical description for a particular inviscid and steady dipolar vortex flow. This dipole is the two-dimensional analogue of Hill's spherical vortex. The Lamb–Chaplygin model follows from demanding the following characteristics: * The dipole has a circular atmosphere/separatrix with radius R: \psi\left(r = R\right) = 0. It is a quasiregular honeycomb. === Cantic order-5 hexagonal tiling honeycomb === Cantic order-5 hexagonal tiling honeycomb Cantic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h2{6,3,5} Coxeter diagram ↔ Cells h2{6,3} 40px t{3,5} 40px r{5,3} 40px Faces triangle {3} pentagon {5} hexagon {6} Vertex figure 80px triangular prism Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex- transitive The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔ , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure. === Runcic order-5 hexagonal tiling honeycomb === Runcic order-5 hexagonal tiling honeycomb Runcic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h3{6,3,5} Coxeter diagram ↔ Cells {3[3]} 40px rr{5,3} 40px {5,3} 40px {}x{3} 40px Faces triangle {3} square {4} pentagon {5} Vertex figure 80px triangular cupola Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔ , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure. === Runcicantic order-5 hexagonal tiling honeycomb === Runcicantic order-5 hexagonal tiling honeycomb Runcicantic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h2,3{6,3,5} Coxeter diagram ↔ Cells h2{6,3} 40px tr{5,3} 40px t{5,3} 40px {}x{3} 40px Faces triangle {3} square {4} hexagon {6} decagon {10} Vertex figure 80px rectangular pyramid Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔ , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . All vertices are on the ideal surface. : 180px === Truncated order-5 hexagonal tiling honeycomb === Truncated order-5 hexagonal tiling honeycomb Truncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t{6,3,5} or t0,1{6,3,5} Coxeter diagram Cells {3,5} 40px t{6,3} 40px Faces triangle {3} dodecagon {12} Vertex figure 80px pentagonal pyramid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure. 480px === Bitruncated order-5 hexagonal tiling honeycomb === Bitruncated order-5 hexagonal tiling honeycomb Bitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol 2t{6,3,5} or t1,2{6,3,5} Coxeter diagram ↔ Cells t{3,6} 40px t{3,5} 40px Faces pentagon {5} hexagon {6} Vertex figure 80px digonal disphenoid Coxeter groups {\overline{HV}}_3, [5,3,6] {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure. 480px === Cantellated order-5 hexagonal tiling honeycomb === Cantellated order-5 hexagonal tiling honeycomb Cantellated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol rr{6,3,5} or t0,2{6,3,5} Coxeter diagram Cells r{3,5} 40px rr{6,3} 40px {}x{5} 40px Faces triangle {3} square {4} pentagon {5} hexagon {6} Vertex figure 80px wedge Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure. 480px === Cantitruncated order-5 hexagonal tiling honeycomb === Cantitruncated order-5 hexagonal tiling honeycomb Cantitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol tr{6,3,5} or t0,1,2{6,3,5} Coxeter diagram Cells t{3,5} 40px tr{6,3} 40px {}x{5} 40px Faces square {4} pentagon {5} hexagon {6} dodecagon {12} Vertex figure 80px mirrored sphenoid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex- transitive The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure. 480px === Runcinated order-5 hexagonal tiling honeycomb === Runcinated order-5 hexagonal tiling honeycomb Runcinated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,3{6,3,5} Coxeter diagram Cells {6,3} 40px {5,3} 40px {}x{6} 40px {}x{5} 40px Faces square {4} pentagon {5} hexagon {6} Vertex figure 80px irregular triangular antiprism Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure. 480px === Runcitruncated order-5 hexagonal tiling honeycomb === Runcitruncated order-5 hexagonal tiling honeycomb Runcitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,3{6,3,5} Coxeter diagram Cells t{6,3} 40px rr{5,3} 40px {}x{5} 40px {}x{12} 40px Faces triangle {3} square {4} pentagon {5} dodecagon {12} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure. 480px === Runcicantellated order-5 hexagonal tiling honeycomb === The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb. === Omnitruncated order-5 hexagonal tiling honeycomb === Omnitruncated order-5 hexagonal tiling honeycomb Omnitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,2,3{6,3,5} Coxeter diagram Cells tr{6,3} 40px tr{5,3} 40px {}x{10} 40px {}x{12} 40px Faces square {4} hexagon {6} decagon {10} dodecagon {12} Vertex figure 80px irregular tetrahedron Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure. 480px === Alternated order-5 hexagonal tiling honeycomb === Alternated order-5 hexagonal tiling honeycomb Alternated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Semiregular honeycomb Schläfli symbol h{6,3,5} Coxeter diagram ↔ Cells {3[3]} 40px {3,5} 40px Faces triangle {3} Vertex figure 40px truncated icosahedron Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive, edge-transitive, quasiregular The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by: :T_{abcd}=C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} + \frac{1}{4}\epsilon_{ae}{}^{hi} \epsilon_{b}{}^{ej}{}_{k} C_{hicf} C_{j}{}^{k}{}_{d}{}^{f} Alternatively, :T_{abcd} = C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} - \frac{3}{2} g_{a[b} C_{jk]cf} C^{jk}{}_{d}{}^{f} where C_{abcd} is the Weyl tensor. It contains triangular tiling facets in a hexagonal tiling vertex figure. === Cantic order-6 hexagonal tiling honeycomb === Cantic order-6 hexagonal tiling honeycomb Cantic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h2{6,3,6} Coxeter diagrams ↔ Cells t{3,6} 40px r{6,3} 40px h2{6,3} 40px Faces triangle {3} hexagon {6} Vertex figure 80px triangular prism Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex- transitive, edge-transitive The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, ↔ , with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure. ===Runcic order-6 hexagonal tiling honeycomb=== Runcic order-6 hexagonal tiling honeycomb Runcic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h3{6,3,6} Coxeter diagrams ↔ Cells rr{3,6} 40px {6,3} 40px {3[3]} 40px {3}x{} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px triangular cupola Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex- transitive The runcic hexagonal tiling honeycomb, h3{6,3,6}, , or , has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure. ===Runicantic order-6 hexagonal tiling honeycomb=== Runcicantic order-6 hexagonal tiling honeycomb Runcicantic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h2,3{6,3,6} Coxeter diagrams ↔ Cells tr{6,3} 40px t{6,3} 40px h2{6,3} 40px {}x{3} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px rectangular pyramid Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The runcicantic order-6 hexagonal tiling honeycomb, h2,3{6,3,6}, , or , contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells: === Rectified order-6 dodecahedral honeycomb === Rectified order-6 dodecahedral honeycomb \--> Type Paracompact uniform honeycomb Schläfli symbols r{5,3,6} t1{5,3,6} Coxeter diagrams ↔ Cells r{5,3} 40px {3,6} 40px Faces triangle {3} pentagon {5} Vertex figure 80px hexagonal prism Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive, edge-transitive The rectified order-6 dodecahedral honeycomb, t1{5,3,6} has icosidodecahedron and triangular tiling cells connected in a hexagonal prism vertex figure. : 480px Perspective projection view within Poincaré disk model It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces. : 180px === Truncated order-6 dodecahedral honeycomb === Truncated order-6 dodecahedral honeycomb \--> Type Paracompact uniform honeycomb Schläfli symbols t{5,3,6} t0,1{5,3,6} Coxeter diagrams ↔ Cells t{5,3} 40px {3,6} 40px Faces triangle {3} decagon {10} Vertex figure 80px hexagonal pyramid Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The truncated order-6 dodecahedral honeycomb, t0,1{5,3,6} has truncated dodecahedron and triangular tiling cells connected in a hexagonal pyramid vertex figure. 480px === Bitruncated order-6 dodecahedral honeycomb === The bitruncated order-6 dodecahedral honeycomb is the same as the bitruncated order-5 hexagonal tiling honeycomb. === Cantellated order-6 dodecahedral honeycomb === Cantellated order-6 dodecahedral honeycomb Cantellated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols rr{5,3,6} t0,2{5,3,6} Coxeter diagrams ↔ Cells rr{5,3} 40px rr{6,3} 40px {}x{6} 40px Faces triangle {3} square {4} pentagon {5} hexagon {6} Vertex figure 80px wedge Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The cantellated order-6 dodecahedral honeycomb, t0,2{5,3,6}, has rhombicosidodecahedron, trihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure. 480px === Cantitruncated order-6 dodecahedral honeycomb === Cantitruncated order-6 dodecahedral honeycomb Cantitruncated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols tr{5,3,6} t0,1,2{5,3,6} Coxeter diagrams ↔ Cells tr{5,3} 40px t{3,6} 40px {}x{6} 40px Faces square {4} hexagon {6} decagon {10} Vertex figure 80px mirrored sphenoid Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has truncated icosidodecahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure. 480px === Runcinated order-6 dodecahedral honeycomb === The runcinated order-6 dodecahedral honeycomb is the same as the runcinated order-5 hexagonal tiling honeycomb. === Runcitruncated order-6 dodecahedral honeycomb === Runcitruncated order-6 dodecahedral honeycomb Runcitruncated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols t0,1,3{5,3,6} Coxeter diagrams Cells t{5,3} 40px rr{6,3} 40px {}x{10} 40px {}x{6} 40px Faces square {4} hexagon {6} decagon {10} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups \overline{HV}_3, [5,3,6] Properties Vertex-transitive The runcitruncated order-6 dodecahedral honeycomb, t0,1,3{5,3,6} has truncated dodecahedron, rhombitrihexagonal tiling, decagonal prism, and hexagonal prism facets, with an isosceles- trapezoidal pyramid vertex figure. 480px === Runcicantellated order-6 dodecahedral honeycomb === The runcicantellated order-6 dodecahedral honeycomb is the same as the runcitruncated order-5 hexagonal tiling honeycomb. === Omnitruncated order-6 dodecahedral honeycomb === The omnitruncated order-6 dodecahedral honeycomb is the same as the omnitruncated order-5 hexagonal tiling honeycomb. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III == Related tilings == The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px It contains and that tile 2-hypercycle surfaces, which are similar to the paracompact tilings and (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively): : 120px 120px == Symmetry == 120px|thumb|left|Subgroup relations: ↔ The order-6 hexagonal tiling honeycomb has a half-symmetry construction: . The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface. : 240px == Related polytopes and honeycombs == The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. The polar circles of the triangles of a complete quadrilateral form a coaxal system. The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells: It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures: === Rectified order-6 hexagonal tiling honeycomb === Rectified order-6 hexagonal tiling honeycomb Rectified order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols r{6,3,6} or t1{6,3,6} Coxeter diagrams ↔ ↔ ↔ ↔ Cells {3,6} 40px r{6,3} 40px Faces triangle {3} hexagon {6} Vertex figure 80px hexagonal prism Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] \overline{PP}_3, [3[3,3]] Properties Vertex-transitive, edge-transitive The rectified order-6 hexagonal tiling honeycomb, t1{6,3,6}, has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure. it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, ↔ . 480px It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px ==== Related honeycombs==== The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures: It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q} === Truncated order-6 hexagonal tiling honeycomb === Truncated order-6 hexagonal tiling honeycomb Truncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t{6,3,6} or t0,1{6,3,6} Coxeter diagram ↔ Cells {3,6} 40px t{6,3} 40px Faces triangle {3} dodecagon {12} Vertex figure 80px hexagonal pyramid Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.Twitter Rotation around 3 fold axis 480px === Bitruncated order-6 hexagonal tiling honeycomb === Bitruncated order-6 hexagonal tiling honeycomb Bitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol bt{6,3,6} or t1,2{6,3,6} Coxeter diagram ↔ Cells t{3,6} 40px Faces hexagon {6} Vertex figure 80px tetrahedron Coxeter groups 2\times\overline{Z}_3, 6,3,6 \overline{VP}_3, [6,3[3]] \overline{V}_3, [3,3,6] Properties Regular The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, ↔ . Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III == Symmetry== A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches. == Images== The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex. : 180px == Related polytopes and honeycombs == The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. It contains hexagonal tiling facets, with a tetrahedron vertex figure. 480px === Cantellated order-6 hexagonal tiling honeycomb === Cantellated order-6 hexagonal tiling honeycomb Cantellated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol rr{6,3,6} or t0,2{6,3,6} Coxeter diagram ↔ Cells r{3,6} 40px rr{6,3} 40px {}x{6} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px wedge Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The cantellated order-6 hexagonal tiling honeycomb, t0,2{6,3,6}, has trihexagonal tiling, rhombitrihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure. 480px === Cantitruncated order-6 hexagonal tiling honeycomb === Cantitruncated order-6 hexagonal tiling honeycomb Cantitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol tr{6,3,6} or t0,1,2{6,3,6} Coxeter diagram ↔ Cells tr{3,6} 40px t{3,6} 40px {}x{6} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px mirrored sphenoid Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The cantitruncated order-6 hexagonal tiling honeycomb, t0,1,2{6,3,6}, has hexagonal tiling, truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure. 480px === Runcinated order-6 hexagonal tiling honeycomb === Runcinated order-6 hexagonal tiling honeycomb Runcinated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,3{6,3,6} Coxeter diagram ↔ Cells {6,3} 40px40px {}×{6} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px triangular antiprism Coxeter groups 2\times\overline{Z}_3, 6,3,6 Properties Vertex-transitive, edge-transitive The runcinated order-6 hexagonal tiling honeycomb, t0,3{6,3,6}, has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure. 480px It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, with square and hexagonal faces: : 240px === Runcitruncated order-6 hexagonal tiling honeycomb === Runcitruncated order-6 hexagonal tiling honeycomb Runcitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,3{6,3,6} Coxeter diagram Cells t{6,3} 40px rr{6,3} 40px {}x{6}40px {}x{12} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups \overline{Z}_3, [6,3,6] Properties Vertex-transitive The runcitruncated order-6 hexagonal tiling honeycomb, t0,1,3{6,3,6}, has truncated hexagonal tiling, rhombitrihexagonal tiling, hexagonal prism, and dodecagonal prism cells, with an isosceles- trapezoidal pyramid vertex figure. 480px === Omnitruncated order-6 hexagonal tiling honeycomb === Omnitruncated order-6 hexagonal tiling honeycomb Omnitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,2,3{6,3,6} Coxeter diagram Cells tr{6,3} 40px {}x{12} 40px Faces square {4} hexagon {6} dodecagon {12} Vertex figure 80px phyllic disphenoid Coxeter groups 2\times\overline{Z}_3, 6,3,6 Properties Vertex-transitive The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3{6,3,6}, has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure. 480px ===Alternated order-6 hexagonal tiling honeycomb=== Alternated order-6 hexagonal tiling honeycomb Alternated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h{6,3,6} Coxeter diagrams ↔ Cells {3,6} 40px {3[3]} 40px Faces triangle {3} Vertex figure 80px hexagonal tiling Coxeter groups \overline{VP}_3, [6,3[3]] Properties Regular, quasiregular The alternated order-6 hexagonal tiling honeycomb is a lower- symmetry construction of the regular triangular tiling honeycomb, ↔ . A less favorable property is that the second derivative of the flow field at the dipole's edge is not continuous. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: . == Related polytopes and honeycombs == The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space. ",1.41,4.76,0.87,-1270,-0.55,A
-Frozen-core $\mathrm{SCF} / \mathrm{DZP}$ and CI-SD/DZP calculations on $\mathrm{H}_2 \mathrm{O}$ at its equilibrium geometry gave energies of -76.040542 and -76.243772 hartrees. Application of the Davidson correction brought the energy to -76.254549 hartrees. Find the coefficient of $\Phi_0$ in the normalized CI-SD wave function.,"As with other perturbative approaches, the Davidson correction is not reliable when the electronic structure of CISD and the reference Hartree–Fock wave functions are significantly different (i.e. when a_0^2 is not close to 1). A solution of these equations yields the Hartree–Fock wave function and energy of the system. The Davidson correction is an energy correction often applied in calculations using the method of truncated configuration interaction, which is one of several post-Hartree-Fock ab initio quantum chemistry methods in the field of computational chemistry. It uses the formula :\Delta E_Q = (1 - a_0^2)(E_{\rm CISD} - E_{\rm HF}), \ :E_{\rm CISDTQ} \approx E_{\rm CISD} + \Delta E_Q, \ where a0 is the coefficient of the Hartree-Fock wavefunction in the CISD expansion, ECISD and EHF are the energies of the CISD and Hartree-Fock wavefunctions respectively, and ΔEQ is the correction to estimate ECISDTQ, the energy of the CISDTQ wavefunction. Developing post-Hartree–Fock methods based on a ROHF wave function is inherently more difficult than using a UHF wave function, due to the lack of a unique set of molecular orbitals. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state. In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Davidson correction does not give information about the wave function. An alternative to Hartree–Fock calculations used in some cases is density functional theory, which treats both exchange and correlation energies, albeit approximately. Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation : \hat F(1)\phi_i(1) = \epsilon_i \phi_i(1), where \phi_i(1) are a set of one-electron wave functions, called the Hartree–Fock molecular orbitals. === Linear combination of atomic orbitals === Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a linear combination of atomic orbitals. They add electron correlation which is a more accurate way of including the repulsions between electrons than in the Hartree–Fock method where repulsions are only averaged. == Details == In general, the SCF procedure makes several assumptions about the nature of the multi-body Schrödinger equation and its set of solutions: * For molecules, the Born–Oppenheimer approximation is inherently assumed. Charlotte Froese Fischer (born 1929) is a Canadian-American applied mathematician and computer scientist noted for the development and implementation of the Multi-Configurational Hartree–Fock (MCHF) approach to atomic-structure calculations and its application to the description of atomic structure and spectra. In computational chemistry, post–Hartree–Fock (post-HF) methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as spin-orbitals. The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. However, different choices of reference orbitals have shown to provide similar results, and thus many different post-Hartree–Fock methods have been implemented in a variety of electronic structure packages. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). Another option is to use modern valence bond methods. == Software packages == For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the list of quantum chemistry and solid state physics software. == See also == Related fields * Quantum chemistry * Molecular physics * Quantum chemistry computer programs * Fock symmetry Concepts * Roothaan equations * Koopmans' theorem * Post-Hartree–Fock * Direct inversion of iterative subspace People * Vladimir Aleksandrovich Fock * Clemens Roothaan * George G. Hall * John Pople * Reinhart Ahlrichs == References == == Sources == * * * == External links == * The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. However, neither Davidson correction itself nor the corrected energies are size-consistent or size-extensive. ",0.2244,30,-2.0,0.9731,35.2,D
-Let $w$ be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find $\langle w\rangle$.,"thumb|upright=1.5|Coin of Epander. thumb|upright=1.5|Coin of Bhradrayasha. thumb|upright=1.35|Coin of Tennes. thumb|upright=1.2|Coin of Sabaces. It was designed by Nico de Haas, a Dutch national-socialist, and struck in 1941 and 1942. ==Mintage== Year Mintage Notes 1941 27,600,000 1942 ==References== Category:Netherlands in World War II Category:Coins of the Netherlands Category:Modern obsolete currencies Category:Currencies of Europe Category:Zinc and aluminum coins minted in Germany and occupied territories during World War II Japanese coins from this period are read clockwise from right to left: :""Year"" ← ""Number representing year of reign"" ← ""Emperors name"" (Ex: 年 ← 五 ← 和昭) Year of reign Japanese date Gregorian date Mintage 5th 五 1930 6th 六 1931 7th 七 1932 Unknown ==Collecting== The value of any given coin is determined by survivability rate and condition as collectors in general prefer uncleaned appealing coins. The -cent coin minted in the Netherlands during World War II was made of zinc, and worth , or .025, of the Dutch guilder. Japanese coins from this period are read clockwise from right to left: :""Year"" ← ""Number representing year of reign"" ← ""Emperors name"" (Ex: 年 ← 六 ← 正大) Year of reign Japanese date Gregorian date Mintage 1st 元 1912 2nd 二 1913 3rd 三 1914 4th 四 1915 5th 五 1916 6th 六 1917 7th 七 1918 8th 八 1919 9th 九 1920 === Shōwa === The following are mintage figures for coins minted between the 5th and the 7th year of Emperor Shōwa's reign. This system was officially put into place on May 10, 1871 setting standards for the 20 yen coin. Japanese coins from this period are read clockwise from right to left :""Year"" ← ""Number representing year of reign"" ← ""Emperors name"" (Ex: 年 ← 七十三 ← 治明) thumb|right|20 yen coin from 1870 (year 3) Design 1 - (1870 - 1892) thumb|right|20 yen coin from 1897 (year 30) Design 2 - (1897 - 1912) Year of reign Japanese date Gregorian date Mintage 3rd 三 1870 5th 五 1872 6th 六 1873 9th 九 1876 10th 十 1877 13th 三十 1880 25th 五十二 1892 Not circulated 30th 十三 1897 37th 七十三 1904 38th 八十三 1905 39th 九十三 1906 40th 十四 1907 41st 一十四 1908 42nd 二十四 1909 43rd 三十四 1910 44th 四十四 1911 45th 五十四 1912 ===Taishō=== The following are mintage figures for the coins that were minted from the 1st to the 9th year of Taishō's reign. Many of these coins were then melted or destroyed as a result of the wars between 1931 and 1945. An auction held in 2011 featuring one of these coins sold it for $230,000 (USD). Some of these coins were kept away in bank vaults for decades before being released as part of a hoard in the mid 2000s. ==Weight and size== Image Minted Size Weight 150px 1870–1880 35.06mm 33.33g 150px 1897–1932 28.78mm 16.66g ==Circulation figures== ===Meiji=== The following are mintage figures for the coins that were minted between the 3rd and 45th (last) year of Meiji's reign. Gold coins of the 20 yen denomination were last minted in 1932, it is unknown how many Shōwa era coins were later melted. Twenty yen coins dated 1877 (year 10) have an extremely low mintage of just 29 coins struck. These coins which are dated from 1870 to 1876 (year 3 to 9) are all priced in five digit dollar amounts (USD) in average condition. These new standards lowered both the size and weight of the coin, the new diameter was set at 28.78mm (previously 35.06mm), and the weight was lowered from 33.3g down to 16.6g. Coinage of the 20 yen piece had all but stopped by 1877, and those struck in 1880 were only done so as part of presentation sets for visiting dignitaries and heads of state. For this denomination all 20 yen coins are scarce as the amount remaining today are dependent on how many were saved or kept away. Twenty Yen coins spanned three different Imperial eras before mintage was halted in 1932. The was a denomination of Japanese yen. These coins were minted in gold, and during their lifespan were the highest denomination of coin that circulated in the country. ",-233,-2,8.3147,0.4207,1,E
-Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 Å$.,"Initially the alpha particles are at a very large distance from the nucleus. :\frac{1}{2} mv^2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 q_2}{r_\text{min}} Rearranging: :r_\text{min} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2 q_1 q_2}{mv^2} For an alpha particle: * (mass) = = * (for helium) = 2 × = * (for gold) = 79 × = * (initial velocity) = (for this example) Substituting these in gives the value of about , or 27 fm. The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear center, as noted, when the actual radius of gold is 7.3 fm. Rutherford realized this, and also realized that actual impact of the alphas on gold causing any force-deviation from that of the coulomb potential would change the form of his scattering curve at high scattering angles (the smallest impact parameters) from a hyperbola to something else. This was not seen, indicating that the surface of the gold nucleus had not been ""touched"" so that Rutherford also knew the gold nucleus (or the sum of the gold and alpha radii) was smaller than 27 fm. == Extension to situations with relativistic particles and target recoil == The extension of low-energy Rutherford-type scattering to relativistic energies and particles that have intrinsic spin is beyond the scope of this article. In Rutherford's gold foil experiment conducted by his students Hans Geiger and Ernest Marsden, a narrow beam of alpha particles was established, passing through very thin (a few hundred atoms thick) gold foil. Applying the inverse-square law between the charges on the alpha particle and nucleus, one can write: Assumptions: 1\. The distance from the center of the alpha particle to the center of the nucleus () at this point is an upper limit for the nuclear radius, if it is evident from the experiment that the scattering process obeys the cross section formula given above. The classical Rutherford scattering process of alpha particles against gold nuclei is an example of ""elastic scattering"" because neither the alpha particles nor the gold nuclei are internally excited. For any central potential, the differential cross- section in the lab frame is related to that in the center-of-mass frame by \frac{d\sigma}{d\Omega}_L=\frac{(1+2s\cos\Theta+s^2)^{3/2}}{1+s\cos\Theta} \frac{d\sigma}{d\Omega} To give a sense of the importance of recoil, we evaluate the head-on energy ratio F for an incident alpha particle (mass number \approx 4) scattering off a gold nucleus (mass number \approx 197): F \approx 0.0780. This same result can be expressed alternatively as : \frac{d\sigma}{d\Omega} = \left( \frac{ Z_1 Z_2 \alpha (\hbar c)} {4 E_{\mathrm{K}10} \sin^2 \frac{\Theta}{2} } \right)^2, where is the dimensionless fine structure constant, is the initial non-relativistic kinetic energy of particle 1 in MeV, and . ==Details of calculating maximal nuclear size== For head-on collisions between alpha particles and the nucleus (with zero impact parameter), all the kinetic energy of the alpha particle is turned into potential energy and the particle is at rest. The Rutherford formula (see below) further neglects the recoil kinetic energy of the massive target nucleus. When a (positive) alpha particle approached sufficiently close to the nucleus, it was repelled strongly enough to rebound at high angles. In classical physics, alpha particles do not have enough energy to escape the potential well from the strong force inside the nucleus (this well involves escaping the strong force to go up one side of the well, which is followed by the electromagnetic force causing a repulsive push-off down the other side). Rutherford showed, using the method outlined below, that the size of the nucleus was less than about (how much less than this size, Rutherford could not tell from this experiment alone; see more below on this problem of lowest possible size). It was determined that the atom's positive charge was concentrated in a small area in its center, making the positive charge dense enough to deflect any positively charged alpha particles that came close to what was later termed the nucleus. The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. For the case of light alpha particles scattering off heavy nuclei, as in the experiment performed by Rutherford, the reduced mass, essentially the mass of the alpha particle and the nucleus off of which it scatters, is essentially stationary in the lab frame. It was found that some of the alpha particles were deflected at much larger angles than expected (at a suggestion by Rutherford to check it) and some even bounced almost directly back. Because the mass of an alpha particle is about 8000 times that of an electron, it became apparent that a very strong force must be present if it could deflect the massive and fast moving alpha particles. With a typical kinetic energy of 5 MeV; the speed of emitted alpha particles is 15,000 km/s, which is 5% of the speed of light. For the more extreme case of an electron scattering off a proton, s \approx 1/1836 and F \approx 0.00218. == See also == *Rutherford backscattering spectrometry ==References== == Textbooks == * == External links == * E. Rutherford, The Scattering of α and β Particles by Matter and the Structure of the Atom, Philosophical Magazine. Rutherford hypothesized that, assuming the ""plum pudding"" model of the atom was correct, the positively charged alpha particles would be only slightly deflected, if at all, by the dispersed positive charge predicted. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. ",+4.1,4.4,0.405,269,3.2,C
-"When an electron in a certain excited energy level in a one-dimensional box of length $2.00  Å$ makes a transition to the ground state, a photon of wavelength $8.79 \mathrm{~nm}$ is emitted. Find the quantum number of the initial state.","In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. The excited state originates from the interaction between a photon and the quantum system. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. These energy levels are described by the principal quantum number n = 1, 2, 3, ... . After excitation the atom may return to the ground state or a lower excited state, by emitting a photon with a characteristic energy. Photoexcitation is the production of an excited state of a quantum system by photon absorption. In modern physics, the concept of a quantum jump is rarely used; as a rule scientists speak of transitions between quantum states or energy levels. == Atomic electron transition == thumb|Grotrian diagram of a quantum 3-level system with characteristic transition frequencies, \omega12 and \omega13, and excited state lifetimes \Gamma2 and \Gamma3 Atomic electron transitions cause the emission or absorption of photons. The ground state of the hydrogen atom has the atom's single electron in the lowest possible orbital (that is, the spherically symmetric ""1s"" wave function, which, so far, has been demonstrated to have the lowest possible quantum numbers). By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). The larger the energy separation of the states between which the electron jumps, the shorter the wavelength of the photon emitted. thumb|EMCCD camera and photomultiplier tube signals while driving quantum jumps on the 674 nm transition of 88Sr+ In an ion trap, quantum jumps can be directly observed by addressing a trapped ion with radiation at two different frequencies to drive electron transitions. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state. == Hydrogenic potential == thumb|right|Figure 3. In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). State |3\rangle has a relatively long lifetime \Gamma3 which causes an interruption of the photon emission as the electron gets shelved in state through application of light with frequency \omega13. Quantum- mechanically, a state with abnormally high n refers to an atom in which the valence electron(s) have been excited into a formerly unpopulated electron orbital with higher energy and lower binding energy. A quantum jump is the abrupt transition of a quantum system (atom, molecule, atomic nucleus) from one quantum state to another, from one energy level to another. In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations. ",4,0.22222222,152.67,1838.50666349,3.2,A
-"For a macroscopic object of mass $1.0 \mathrm{~g}$ moving with speed $1.0 \mathrm{~cm} / \mathrm{s}$ in a one-dimensional box of length $1.0 \mathrm{~cm}$, find the quantum number $n$.","The magnitude of the momentum is given by :p=\frac{h}{2L}\sqrt{n_x^2+n_y^2+n_z^2} \qquad \qquad n_x,n_y,n_z=1,2,3,\ldots where h is Planck's constant and L is the length of a side of the box. The distance from the origin to any point will be :n=\sqrt{n_x^2+n_y^2+n_z^2}=\frac{2Lp}{h} Suppose each set of quantum numbers specify f states where f is the number of internal degrees of freedom of the particle that can be altered by collision. The principal quantum number is related to the radial quantum number, nr, by: n = n_r + \ell + 1 where ℓ is the azimuthal quantum number and nr is equal to the number of nodes in the radial wavefunction. Using a continuum approximation, the number of states with magnitude of momentum between p and p+dp is therefore :dg = \frac{\pi}{2}~f n^2\,dn = \frac{4\pi fV}{h^3}~ p^2\,dp where V=L3 is the volume of the box. In quantum mechanics, the principal quantum number (symbolized n) is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. These integers are the magnetic quantum numbers. In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions. In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The principal quantum number n represents the relative overall energy of each orbital. More complete calculations will be left to separate articles, but some simple examples will be given in this article. ==Thomas–Fermi approximation for the degeneracy of states== For both massive and massless particles in a box, the states of a particle are enumerated by a set of quantum numbers . Integrating the energy distribution function and solving for N gives the particle number :N = \left(\frac{Vf}{\Lambda^3}\right)\textrm{Li}_{3/2}(z) where Lis(z) is the polylogarithm function. The orbital magnetic quantum number takes integer values in the range from -\ell to +\ell, including zero. The spin magnetic quantum number specifies the z-axis component of the spin angular momentum for a particle having spin quantum number . For large values of n, the number of states with magnitude of momentum less than or equal to p from the above equation is approximately : g = \left(\frac{f}{8}\right) \frac{4}{3}\pi n^3 = \frac{4\pi f}{3} \left(\frac{Lp}{h}\right)^3 which is just f times the volume of a sphere of radius n divided by eight since only the octant with positive ni is considered. In particle physics, G-parity is a multiplicative quantum number that results from the generalization of C-parity to multiplets of particles. The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the principal quantum number n, the azimuthal (orbital) quantum number \ell, and the magnetic quantum numbers and . Using the Thomas−Fermi approximation, the number of particles dNE with energy between E and E+dE is: :dN_E= \frac{dg_E}{\Phi(E)} where dg_E is the number of states with energy between E and E+dE. ==Energy distribution== Using the results derived from the previous sections of this article, some distributions for the gas in a box can now be determined. The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field (the Zeeman effect) -- hence the name magnetic quantum number. The quantum number m_l refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the z-direction or quantization axis. The principal quantum number arose in the solution of the radial part of the wave equation as shown below. Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as or for the total z-axis orbital angular momentum of all the electrons in an atom. ==Derivation== thumb|These orbitals have magnetic quantum numbers m_l=-\ell, \ldots,\ell from left to right in ascending order. It turns out, however, that the above equation for particle number expresses the number of bosons in excited states rather well, and thus: : N=\frac{g_0 z}{1-z}+\left(\frac{Vf}{\Lambda^3}\right)\operatorname{Li}_{3/2}(z) where the added term is the number of particles in the ground state. ",8.44,3,2.3613,0.16,3.23,B
-"For the $\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \mathrm{eV}$. Find $\Delta H_0^{\circ}$ for $\mathrm{H}_2(g) \rightarrow 2 \mathrm{H}(g)$ in $\mathrm{kJ} / \mathrm{mol}$","The delta function model becomes particularly useful with the double-well Dirac Delta function model which represents a one-dimensional version of the Hydrogen molecule ion, as shown in the following section. == Double delta potential == thumb|300px|right| The symmetric and anti-symmetric wavefunctions for the double-well Dirac delta function model with ""internuclear"" distance The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation: -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x), where the potential is now V(x) = -q \left[ \delta \left(x + \frac{R}{2}\right) + \lambda\delta \left(x - \frac{R}{2} \right) \right], where 0 < R < \infty is the ""internuclear"" distance with Dirac delta-function (negative) peaks located at (shown in brown in the diagram). Similarly to the single band case, we can write for U^{A}_{jj'} : D_{jj'} \equiv U^{A}_{jj'} = E_{j}(0)\delta_{jj'} + \sum_{\alpha\beta} D^{\alpha\beta}_{jj'}k_{\alpha}k_{\beta}, : D^{\alpha\beta}_{jj'} = \frac{\hbar^2}{2 m_0} \left [ \delta_{jj'}\delta_{\alpha\beta} + \sum^{B}_{\gamma} \frac{ p^{\alpha}_{j\gamma}p^{\beta}_{\gamma j'} + p^{\beta}_{j\gamma}p^{\alpha}_{\gamma j'} }{ m_0 (E_0-E_{\gamma}) } \right ]. The third parameter \gamma_3 relates to the anisotropy of the energy band structure around the \Gamma point when \gamma_2 eq \gamma_3 . == Explicit Hamiltonian matrix == The Luttinger-Kohn Hamiltonian \mathbf{D_{jj'}} can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off) : \mathbf{H} = \left( \begin{array}{cccccccc} E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ P_z^{\dagger} & P+\Delta & \sqrt{2}Q^{\dagger} & -S^{\dagger}/\sqrt{2} & -\sqrt{2}P_{+}^{\dagger} & 0 & -\sqrt{3/2}S & -\sqrt{2}R \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ \end{array} \right) == Summary == == References == 2\. A flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors. The International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS), is a biannual event, where over hundred scientists meet for the presentation of new developments on the special field of two-dimensional electron systems in semiconductors. thumb|right|200px|Formation of a δ bond by the overlap of two d orbitals thumb|right|200px|3D model of a boundary surface of a δ bond in Mo2 In chemistry, delta bonds (δ bonds) are covalent chemical bonds, where four lobes of one involved atomic orbital overlap four lobes of the other involved atomic orbital. Substituting into the Schrödinger equation in Bloch approximation we obtain : H u_{n\mathbf{k}}(\mathbf{r}) = \left( H_0 + \frac{\hbar}{m_{0}}\mathbf{k}\cdot\mathbf{\Pi} + \frac{\hbar^2 k^2}{4m_{0}^{2}c^{2}} abla V \times \mathbf{p} \cdot \bar{\sigma} \right) u_{n\mathbf{k}}(\mathbf{r}) = E_{n}(\mathbf{k}) u_{n\mathbf{k}}(\mathbf{r}) , where : \mathbf{\Pi} = \mathbf{p} + \frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma} \times abla V and the perturbation Hamiltonian can be defined as : H' = \frac{\hbar}{m_0}\mathbf{k}\cdot\mathbf{\Pi}. The energy of the bound state is then E = -\frac{\hbar^2\kappa^2}{2m} = -\frac{m\lambda^2}{2\hbar^2}. === Scattering (E > 0) === right|thumb|350px|Transmission (T) and reflection (R) probability of a delta potential well. The boundary conditions thus give the following restrictions on the coefficients \begin{cases} A_r + A_l - B_r - B_l &= 0,\\\ -A_r + A_l + B_r - B_l &= \frac{2m\lambda}{ik\hbar^2} (A_r + A_l). \end{cases} === Bound state (E < 0) === right|thumb|350px|The graph of the bound state wavefunction solution to the delta function potential is continuous everywhere, but its derivative is not defined at . The delta function model is actually a one-dimensional version of the Hydrogen atom according to the dimensional scaling method developed by the group of Dudley R. HerschbachD.R. Herschbach, J.S. Avery, and O. Goscinski (eds.), Dimensional Scaling in Chemical Physics, Springer, (1992). In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Kevin Hwang (Hangul: 케빈황; born May 1, 1992), also known by his Korean name Hwang Ji-tu (Hangul: 황지투) and better known by his stage name G2 (Hangul: 지투), is a Korean-American rapper and singer. Matching of the wavefunction at the Dirac delta-function peaks yields the determinant \begin{vmatrix} q - d & q e^{-d R} \\\ q \lambda e^{-d R} & q \lambda - d \end{vmatrix} = 0, \quad \text{where } E = -\frac{d^2}{2}. Substituting the definition of into this expression yields -\frac{\hbar^2}{2m} ik (-A_r + A_l + B_r - B_l) + \lambda(A_r + A_l) = 0. They represent an approximation of the two lowest discrete energy states of the three-dimensional H2^+ and are useful in its analysis. Using Löwdin's method, only the following eigenvalue problem needs to be solved : \sum^{A}_{j'} (U^{A}_{jj'}-E\delta_{jj'})a_{j'}(\mathbf{k}) = 0, where : U^{A}_{jj'} = H_{jj'} + \sum^{B}_{\gamma eq j,j'} \frac{H_{j\gamma}H_{\gamma j'}}{E_0-E_{\gamma}} = H_{jj'} + \sum^{B}_{\gamma eq j,j'} \frac{H^{'}_{j\gamma}H^{'}_{\gamma j'}}{E_0-E_{\gamma}} , : H^{'}_{j\gamma} = \left \langle u_{j0} \right | \frac{\hbar}{m_0} \mathbf{k} \cdot \left ( \mathbf{p} + \frac{\hbar}{4 m_0 c^2} \bar{\sigma} \times abla V \right ) \left | u_{\gamma 0} \right \rangle \approx \sum_{\alpha} \frac{\hbar k_{\alpha}}{m_0}p^{\alpha}_{j \gamma}. Note that the lowest energy corresponds to the symmetric solution d_+. Ytterbium hydride is the hydride of ytterbium with the chemical formula YbH2. We now define the following parameters : A_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{x}_{\gamma x} }{ E_0-E_{\gamma} }, : B_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{y}_{x\gamma}p^{y}_{\gamma x} }{ E_0-E_{\gamma} }, : C_0 = \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{y}_{\gamma y} + p^{y}_{x\gamma}p^{x}_{\gamma y} }{ E_0-E_{\gamma} }, and the band structure parameters (or the Luttinger parameters) can be defined to be : \gamma_1 = - \frac{1}{3} \frac{2 m_0}{\hbar^2} (A_0 + 2B_0), : \gamma_2 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} (A_0 - B_0), : \gamma_3 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} C_0, These parameters are very closely related to the effective masses of the holes in various valence bands. \gamma_1 and \gamma_2 describe the coupling of the |X \rangle , |Y \rangle and |Z \rangle states to the other states. Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set \hbar = m = 1. In this compound, the ytterbium atom has an oxidation state of +2 and the hydrogen atoms have an oxidation state of -1. In this model the influence of all other bands is taken into account by using Löwdin's perturbation method. == Background == All bands can be subdivided into two classes: * Class A: six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands. ",6.0,14.5115,1.2,7.97,432.07,E
-"The contribution of molecular vibrations to the molar internal energy $U_{\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\mathrm{m}, \mathrm{vib}}=R \sum_{s=1}^{3 N-6} \theta_s /\left(e^{\theta_s / T}-1\right)$, where $\theta_s \equiv h \nu_s / k$ and $\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\mathrm{m}, \text { vib }}$ at $25^{\circ} \mathrm{C}$ of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$.","In terms of the vibrational wavenumbers we can write the partition function as Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{ u}_j}{k_\text{B} T}} } It is convenient to define a characteristic vibrational temperature \Theta_{i,\text{vib}} = \frac{h u_i}{k_\text{B}} where u is experimentally determined for each vibrational mode by taking a spectrum or by calculation. Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } where E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j is total vibrational zero point energy of the system. Using this approximation we can derive a closed form expression for the vibrational partition function. For an isolated molecule of n atoms, the number of vibrational modes (i.e. values of j) is 3n − 5 for linear molecules and 3n − 6 for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons. ==Approximations== ===Quantum harmonic oscillator=== The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. ==Definition== For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_{j,n} is the energy of j-th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . The second formula is adequate for small values of the vibrational quantum number. Number of degrees of vibrational freedom for nonlinear molecules: 3N-6 Number of degrees of vibrational freedom for linear molecules: 3N-5Housecroft, Catherine E., and A. G. Sharpe. It has also been applied to the study of unstable molecules such as dicarbon, C2, in discharges, flames and astronomical objects.Hollas, p. 211. == Principles == Electronic transitions are typically observed in the visible and ultraviolet regions, in the wavelength range approximately 200–700 nm (50,000–14,000 cm−1), whereas fundamental vibrations are observed below about 4000 cm−1.Energy is related to wavenumber by E=hc \bar u, where h=Planck's constant and c is the velocity of light When the electronic and vibrational energy changes are so different, vibronic coupling (mixing of electronic and vibrational wave functions) can be neglected and the energy of a vibronic level can be taken as the sum of the electronic and vibrational (and rotational) energies; that is, the Born–Oppenheimer approximation applies.Banwell and McCash, p. 162. Vibronic spectra of diatomic molecules in the gas phase have been analyzed in detail.Hollas, pp. 210–228 Vibrational coarse structure can sometimes be observed in the spectra of molecules in liquid or solid phases and of molecules in solution. The molecule is excited to another electronic state and to many possible vibrational states v'=0, 1, 2, 3, ... . The vibrational temperature is used commonly when finding the vibrational partition function. The overall molecular energy depends not only on the electronic state but also on vibrational and rotational quantum numbers, denoted v and J respectively for diatomic molecules. The transition energies, expressed in wavenumbers, of the lines for a particular vibronic transition are given, in the rigid rotor approximation, that is, ignoring centrifugal distortion, byBanwell and McCash, p. 171 :G(J^\prime, J^{\prime \prime}) = \bar u _{v^\prime-v^{\prime\prime}}+B^\prime J^\prime (J^\prime +1)-B^{\prime\prime} J^{\prime\prime}(J^{\prime\prime} +1) Here B are rotational constants and J are rotational quantum numbers. In the next approximation the term values are given by : G(v) = \bar u _{electronic} + \omega_e (v+{1 \over 2}) - \omega_e\chi_e (v+{1 \over 2})^2\, where χe is an anharmonicity constant. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. The Renner–Teller effect is observed in the spectra of molecules having electronic states that allow vibration through a linear configuration. Later studies on the same anion were also able to account for vibronic transitions involving low-frequency lattice vibrations. == Notes == == References == == Bibliography == * Chapter: Molecular Spectroscopy 2. : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum. == Degrees of freedom == The location of a molecule in a 3-dimensional space can be described by the total number of coordinates. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . The potential at infinite internuclear distance is the dissociation energy for pure vibrational spectra. ",0.75,16,6.0,0.14,10,D
-Calculate the magnitude of the spin magnetic moment of an electron.,"In turn, calculation of the magnitude of the total spin magnetic moment requires that () be replaced by: Thus, for a single electron, with spin quantum number the component of the magnetic moment along the field direction is, from (), while the (magnitude of the) total spin magnetic moment is, from (), or approximately 1.73 μ. The component of the orbital magnetic dipole moment for an electron with a magnetic quantum number ℓ is given by :(\boldsymbol{\mu}_\text{L})_z = -\mu_\text{B} m_\ell. ==History== The electron magnetic moment is intrinsically connected to electron spin and was first hypothesized during the early models of the atom in the early twentieth century. Again it is important to notice that is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin. : Number of unpaired electrons Spin-only moment () 1 1.73 2 2.83 3 3.87 4 4.90 5 5.92 === Elementary particles === In atomic and nuclear physics, the Greek symbol represents the magnitude of the magnetic moment, often measured in Bohr magnetons or nuclear magnetons, associated with the intrinsic spin of the particle and/or with the orbital motion of the particle in a system. Note that is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum. In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The analysis is readily extended to the spin-only magnetic moment of an atom. The spin magnetic dipole moment is approximately one B because g_{\rm s} \approx 2 and the electron is a spin- particle (): The component of the electron magnetic moment is (\boldsymbol{\mu}_\text{s})_z = -g_\text{s}\,\mu_\text{B}\,m_\text{s}\,, where s is the spin quantum number. The spin frequency of the electron is determined by the -factor. : u_s = \frac{g}{2} u_c : \frac{g}{2} = \frac{\bar{ u}_c + \bar{ u}_a}{\bar{ u}_c} ==See also== * Spin (physics) * Electron precipitation * Bohr magneton * Nuclear magnetic moment * Nucleon magnetic moment * Anomalous magnetic dipole moment * Electron electric dipole moment * Fine structure * Hyperfine structure ==References== ==Bibliography== * * Category:Atomic physics Category:Electric dipole moment Category:Magnetic moment Category:Particle physics Category:Physical constants Values of the intrinsic magnetic moments of some particles are given in the table below: : Intrinsic magnetic moments and spins of some elementary particles Particle name (symbol) Magnetic dipole moment (10 J⋅T) Spin quantum number (dimensionless) electron (e−) proton (H+) neutron (n) muon (μ) deuteron (H) 1 triton (H) helion (He) alpha particle (He) 0 0 For the relation between the notions of magnetic moment and magnetization see magnetization. == See also == * Moment (physics) * Electric dipole moment * Toroidal dipole moment * Magnetic susceptibility * Orbital magnetization * Magnetic dipole–dipole interaction == References and notes == ==External links== * Category:Magnetostatics Category:Magnetism Category:Electric and magnetic fields in matter Category:Physical quantities Category:Moment (physics) Category:Magnetic moment Here is the electron spin angular momentum. The magnetic moment of such a particle is parallel to its spin. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum , its magnetic dipole moment is given by: \boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,, where e is the electron rest mass. In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. The CODATA value for the electron magnetic moment is : ===Orbital magnetic dipole moment=== The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. The magnetic moment of the electron is : \mathbf{m}_\text{S} = -\frac{g_\text{S} \mu_\text{B} \mathbf{S}}{\hbar}, where is the Bohr magneton, is electron spin, and the g-factor is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: . However, in order to obtain the magnitude of the total spin angular momentum, be replaced by its eigenvalue, where s is the spin quantum number. The magnetic moment of the electron has been measured using a one-electron quantum cyclotron and quantum nondemolition spectroscopy. The value of the electron magnetic moment (symbol μe) is In units of the Bohr magneton (μB), it is , a value that was measured with a relative accuracy of . ==Magnetic moment of an electron== The electron is a charged particle with charge −, where is the unit of elementary charge. The sum of the proton and neutron magnetic moments gives 0.879 µN, which is within 3% of the measured value 0.857 µN. The ratio between the true spin magnetic moment and that predicted by this model is a dimensionless factor , known as the electron -factor: \boldsymbol{\mu} = g_\text{e}\,\frac{(-e)}{~2m_\text{e}~}\,\mathbf{L}\,. The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μ because and the electron's spin is also : \frac{\hbar}{2} = - \mu_\text{B}|}} Equation () is therefore normally written as: Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. ",0.0245,1.61,-0.16,5040,4,B
-"A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \mathrm{eV}$, calculate $\langle V\rangle$ for the ground state.","The average energy in this state would be \langle\psi|H|\psi\rangle = \int dx\, \left(-\frac{\hbar^2}{2m} \psi^* \frac{d^2\psi}{dx^2} + V(x)|\psi(x)|^2\right), where is the potential. The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. Formulae are given in SI units and Gaussian-cgs units. == Definition == The electromagnetic four- potential can be defined as: : SI units Gaussian units A^\alpha = \left( \frac{1}{c}\phi, \mathbf{A} \right)\,\\! For hydrogen (H), an electron in the ground state has energy , relative to the ionization threshold. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). An excited state is any state with energy greater than the ground state. An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. In quantum field theory, the ground state is usually called the vacuum state or the vacuum. The solutions are outlined in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. === Vacuum case states === Let us now consider V(r) = 0. Therefore, the potential energy is unchanged up to order \varepsilon^2, if we deform the state \psi with a node into a state without a node, and the change can be ignored. Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. Now, consider the potential energy. The correction to the potential V(r) is called the centrifugal barrier term. An object may be said to have electric potential energy by virtue of either its own electric charge or its relative position to other electrically charged objects. We can therefore remove all nodes and reduce the energy by O(\varepsilon), which implies that cannot be the ground state. However, the contribution to the potential energy from this region for the state with a node is V^\varepsilon_\text{avg} = \int_{-\varepsilon}^\varepsilon dx\, V(x)|\psi|^2 = |c|^2\int_{-\varepsilon}^\varepsilon dx\, x^2V(x) \simeq \frac{2}{3}\varepsilon^3|c|^2 V(0) + \cdots, lower, but still of the same lower order O(\varepsilon^3) as for the deformed state , and subdominant to the lowering of the average kinetic energy. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant. When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb's law can be used. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom. (The average energy may then be further lowered by eliminating undulations, to the variational absolute minimum.) === Implication === As the ground state has no nodes it is spatially non- degenerate, i.e. there are no two stationary quantum states with the energy eigenvalue of the ground state (let's name it E_g) and the same spin state and therefore would only differ in their position-space wave functions. The term ""electric potential energy"" is used to describe the potential energy in systems with time-variant electric fields, while the term ""electrostatic potential energy"" is used to describe the potential energy in systems with time-invariant electric fields. ==Definition== The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. The electrostatic potential energy can also be defined from the electric potential as follows: ==Units== The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule). ",0.14,3.333333333,17.7,460.5,0.0625,B
-"For an electron in a certain rectangular well with a depth of $20.0 \mathrm{eV}$, the lowest energy level lies $3.00 \mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\tan \theta=\sin \theta / \cos \theta$","The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. A potential well is the region surrounding a local minimum of potential energy. Depth in a well is not necessarily measured vertically or along a straight line. In the oil and gas industry, depth in a well is the measurement, for any point in that well, of the distance between a reference point or elevation, and that point. Specification of a differential depth or a thickness: in Figure 2 above, the thickness of the reservoir penetrated by the well might be 57 mMD or 42 mTVD, even though the reservoir true stratigraphic thickness in that area (or isopach) might be only 10 m, and its true vertical thickness (isochore), 14 m. ==See also== *Measured depth *True vertical depth ==References== ==External links== * Determining Lowest Astronomical Tide (LAT) * Seas and Submerged Lands Act 1973 (Australia) * Log Data Acquisition and Quality Control, Ph. A potential hill is the opposite of a potential well, and is the region surrounding a local maximum. ==Quantum confinement== thumb|500 px|Quantum confinement is responsible for the increase of energy difference between energy states and band gap, a phenomenon tightly related to the optical and electronic properties of the materials. But there, this additional width is interpreted as energy dispersion, which is, to the first order, |\Delta r_\pi|_\varepsilon = 2R_P\,\Delta E/E_P. Because wells are not always drilled vertically, there may be two ""depths"" for every given point in a wellbore: the measured depth (MD) measured along the path of the borehole, and the true vertical depth (TVD), the absolute vertical distance between the datum and the point in the wellbore. Well depth may refer to: *Depth in a well, a measurement of location in oil and gas drilling and production *The charge capacity of each pixel in a charge-coupled device 350px|right|thumb In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called ""quantum tunneling"") and wave-mechanical reflection. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge. Configurations reproduced in * * * This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Category:Electrostatics Category:Electron Category:Circle packing Category:Unsolved problems in mathematics A well may reach to many kilometers.Sakhalin-1 sets new extended reach drilling record, Rosneft says, 2015 ==Figures== thumb|left|Fig. 1: The specification of depthsthumb|right|Fig. 2: Differential depths: reservoir thickness, isochor, isopach Specification of an absolute depth: in Figure 1 above, point P1 might be at 3207 mMDRT and 2370 mTVDMSL, while point P2 might be at 2530 mMDRT and 2502 mTVDLAT. The problem consists of solving the one- dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. For example, a spherical shell of N=1 represents the uniform distribution of a single electron's charge, -e across the entire shell. ===Randomly distributed point charges=== The global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by :U_{\text{rand}}(N)=\frac{N(N-1)}{2} and is, in general, greater than the energy of every Thomson problem solution. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The angular spread, while also worsening the energy resolution, shows some focusing as the equal negative and positive deviations map to the same final spot. center|thumb|upright=3|Distance from the central trajectory at the exit of a hemispherical electron energy analyzer depending on the electron's kinetic energy, initial position within the 1 mm slit, and the angle at which it enters the radial field after passing through the slit. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: * constrained global optimization (Altschuler et al. 1994), * steepest descent (Claxton and Benson 1966, Erber and Hockney 1991), * random walk (Weinrach et al. 1990), * genetic algorithm (Morris et al. 1996) While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest. ===Continuous spherical shell charge=== thumb|The extreme upper energy limit of the Thomson Problem is given by N^2/2 for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Initial angular spread, dependent on the chosen slit and aperture width, worsens the energy resolution.|alt= When two voltages, V_{1} and V_{2}, are applied to the inner and outer hemispheres, respectively, the electric potential in the region between the two electrodes follows from the Laplace equation: : V(r)= - \left[\frac{V_{2}-V_{1}}{R_{2}-R_{1}}\right]\cdot\frac{R_{1}R_{2}}{r} + const. In these cases the measured depth will continue to increase while true vertical depth will decrease toward the toe of the wellbore. ==Depth in practice== * Unit: the usual unit of depth is the metre (m). alt=|thumb|upright=1.2|Hemispherical electron energy analyzer. thumb|400px|right|A generic potential energy well. ",0.264,9,3.0,-2,2,A
-Calculate the uncertainty $\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$.,"H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. H_p = -\sum_{j=-\infty}^\infty \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53 The entropic uncertainty is indeed larger than the limiting value. Entropic uncertainty of the normal distribution We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. Under the above definition, the entropic uncertainty relation is H_x + H_p > \ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h} \right). Its value is The number in parenthesis denotes the uncertainty of the last digits. ==Definition and value== The Bohr radius is defined asDavid J. Griffiths, Introduction to Quantum Mechanics, Prentice- Hall, 1995, p. 137. a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{e^2 m_{\text{e}}} = \frac{\varepsilon_0 h^2}{\pi e^2 m_{\text{e}}} = \frac{\hbar}{m_{\text{e}} c \alpha} , where * \varepsilon_0 is the permittivity of free space, * \hbar is the reduced Planck constant, * m_{\text{e}} is the mass of an electron, * e is the elementary charge, * c is the speed of light in vacuum, and * \alpha is the fine-structure constant. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. Since |\psi(x)|^2 is a probability density function for position, we calculate its standard deviation. Normal distribution example We demonstrate this method first on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. \psi(x)=\left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \omega x^2}{2\hbar}\right)} The probability of lying within one of these bins can be expressed in terms of the error function. \begin{align} \operatorname P[x_j] &= \sqrt{\frac{m \omega}{\pi \hbar}} \int_{(j-1/2)\delta x}^{(j+1/2)\delta x} \exp\left( -\frac{m \omega x^2}{\hbar}\right) \, dx \\\ &= \sqrt{\frac{1}{\pi}} \int_{(j-1/2)\delta x\sqrt{m \omega / \hbar}}^{(j+1/2)\delta x\sqrt{m \omega / \hbar}} e^{u^2} \, du \\\ &= \frac{1}{2} \left[ \operatorname{erf} \left( \left(j+\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right) \right] \end{align} The momentum probabilities are completely analogous. \operatorname P[p_j] = \frac{1}{2} \left[ \operatorname{erf} \left( \left(j+\frac{1}{2}\right)\delta p \cdot \frac{1}{\sqrt{\hbar m \omega}}\right)- \operatorname{erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \frac{1}{\sqrt{\hbar m \omega}}\right) \right] For simplicity, we will set the resolutions to \delta x = \sqrt{\frac{h}{m \omega}} \delta p = \sqrt{h m \omega} so that the probabilities reduce to \operatorname P[x_j] = \operatorname P[p_j] = \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] The Shannon entropy can be evaluated numerically. \begin{align} H_x = H_p &= -\sum_{j=-\infty}^\infty \operatorname P[x_j] \ln \operatorname P[x_j] \\\ &= -\sum_{j=-\infty}^\infty \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \ln \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \\\ &\approx 0.3226 \end{align} The entropic uncertainty is indeed larger than the limiting value. 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. This precision may be quantified by the standard deviations, \sigma_x=\sqrt{\langle \hat{x}^2 \rangle-\langle \hat{x}\rangle^2} \sigma_p=\sqrt{\langle \hat{p}^2 \rangle-\langle \hat{p}\rangle^2}. In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. When applying the BO approximation, two smaller, consecutive steps can be used: For a given position of the nuclei, the electronic Schrödinger equation is solved, while treating the nuclei as stationary (not ""coupled"" with the dynamics of the electrons). In the published 1927 paper, Heisenberg originally concluded that the uncertainty principle was ΔpΔq ≈ h using the full Planck constant.Werner Heisenberg, Encounters with Einstein and Other Essays on People, Places and Particles, Published October 21st 1989 by Princeton University Press, p.53.Kumar, Manjit. ""On the energy- time uncertainty relation. Orbital uncertainty is related to several parameters used in the orbit determination process including the number of observations (measurements), the time spanned by those observations (observation arc), the quality of the observations (e.g. radar vs. optical), and the geometry of the observations. The matrix element in the numerator is : \langle\chi_{k'}| [P_{A\alpha}, H_\mathrm{e}] |\chi_k\rangle_{(\mathbf{r})} = iZ_A\sum_i \left\langle\chi_{k'}\left|\frac{(\mathbf{r}_{iA})_\alpha}{r_{iA}^3}\right|\chi_k\right\rangle_{(\mathbf{r})} \quad\text{with}\quad \mathbf{r}_{iA} \equiv \mathbf{r}_i - \mathbf{R}_A. The variances of x and p can be calculated explicitly: \sigma_x^2=\frac{L^2}{12}\left(1-\frac{6}{n^2\pi^2}\right) \sigma_p^2=\left(\frac{\hbar n\pi}{L}\right)^2. On the other hand, the standard deviation of the position is \sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2} such that the uncertainty product can only increase with time as \sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2} ==Additional uncertainty relations== ===Systematic and statistical errors=== The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation \sigma. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). Everett's Dissertation proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. The second stronger uncertainty relation is given by \sigma_A^2 + \sigma_B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} \mid(A + B)\mid \Psi \rangle|^2 where | {\bar \Psi}_{A+B} \rangle is a state orthogonal to |\Psi \rangle . The product of the standard deviations is therefore \sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}. ", 9.73,0.064,0.0,8,61,C
+","Another recent paper has pointed out how a simple, yet theory motivated change to previous fits will also give the smaller radius. === 2019 measurements === In September 2019, Bezginov et al. reported the remeasurement of the proton's charge radius for electronic hydrogen and found a result consistent with Pohl's value for muonic hydrogen. Measurements of hydrogen's energy levels are now so precise that the accuracy of the proton radius is the limiting factor when comparing experimental results to theoretical calculations. His personal assumption is that past measurements have misgauged the Rydberg constant and that the current official proton size is inaccurate. ===Quantum chromodynamic calculation=== In a paper by Belushkin et al. (2007), including different constraints and perturbative quantum chromodynamics, a smaller proton radius than the then-accepted 0.877 femtometres was predicted. ===Proton radius extrapolation=== Papers from 2016 suggested that the problem was with the extrapolations that had typically been used to extract the proton radius from the electron scattering data though these explanation would require that there was also a problem with the atomic Lamb shift measurements. ===Data analysis method=== In one of the attempts to resolve the puzzle without new physics, Alarcón et al. (2018) of Jefferson Lab have proposed that a different technique to fit the experimental scattering data, in a theoretically as well as analytically justified manner, produces a proton charge radius from the existing electron scattering data that is consistent with the muonic hydrogen measurement. The result is again ~5% smaller than the previously-accepted proton radius. By measuring the energy required to excite hydrogen atoms from the 2S to the 2P state, the Rydberg constant could be calculated, and from this the proton radius inferred. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. The form of the potential, in terms of the distance r from the center of nucleus, is: V(r) = -\frac{V_0}{1+\exp({r-R\over a})} where V0 (having dimension of energy) represents the potential well depth, a is a length representing the ""surface thickness"" of the nucleus, and R = r_0 A^{1/3} is the nuclear radius where and A is the mass number. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. Effectively, this approach attributes the cause of the proton radius puzzle to a failure to use a theoretically motivated function for the extraction of the proton charge radius from the experimental data. Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). He also noted that in the asymptotic limit (far away from the nucleus), his approximate form coincides with the exact hydrogen-like wave function in the presence of a nuclear charge of Z-s and in the state with a principal quantum number n equal to his effective quantum number n*. this opinion is not yet universally held. ==Problem== Prior to 2010, the proton charge radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering. ===Spectroscopy method=== The spectroscopy method uses the energy levels of electrons orbiting the nucleus. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the ""reduced"" Bohr radius is slightly larger than the Bohr radius (a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11} meters). It is likely future experiments will be able to both explain and settle the proton radius puzzle. === 2022 analysis === A re-analysis of experimental data, published in February 2022, found a result consistent with the smaller value of approximately 0.84 fm. ==Footnotes== ==References== Category:2010 in science Category:2019 in science Category:Proton Historically the proton charge radius was measured by two independent methods, which converged to a value of about 0.877 femtometres (1 fm = 10−15 m). A hydrogen-like atom will have a Bohr radius which primarily scales as r_{Z}=a_0/Z, with Z the number of protons in the nucleus. Since it takes nearly as much energy to excite the hydrogen atom's electron from n = 1 to n = 3 (12.1 eV, via the Rydberg formula) as it does to ionize the hydrogen atom (13.6 eV), ionization is far more probable than excitation to the n = 3 level. thumb|right|300px|Woods–Saxon potential for , relative to V0 with a and R=4.6 fm The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in the nuclear shell model for the structure of the nucleus. thumb|300px|Comparison of a wavefunction in the Coulomb potential of the nucleus (blue) to the one in the pseudopotential (red). This method produces a proton radius of about , with approximately 1% relative uncertainty. ===Nuclear scattering=== The nuclear method is similar to Rutherford's scattering experiments that established the existence of the nucleus. The Bohr radius (a0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. ",6,7.136,"""-0.28""",1.2,-131.1,D
+"An electron in a three-dimensional rectangular box with dimensions of $5.00 Å, 3.00 Å$, and $6.00 Å$ makes a radiative transition from the lowest-lying excited state to the ground state. Calculate the frequency of the photon emitted.","We may derive the two- photon Rabi frequency by returning to the equations \begin{align} i \dot{c}_1(t) &= \frac{\Omega_{1i} c_2}{2} e^{i\Delta t}\\\ i \dot{c}_i(t) &= \frac{\Omega^*_{1i} c_1}{2} e^{-i\Delta t} \end{align} which now describe excitation between the ground and intermediate states. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. Photoexcitation is the production of an excited state of a quantum system by photon absorption. Transition frequency may refer to: *A measure of the high-frequency operating characteristics of a transistor, usually symbolized as *A characteristic of spectral lines *The frequency of the radiation associated with a transition between hyperfine structure energy states of an atom *Turnover frequency in enzymology In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. Also note that as the incident light frequency shifts further from the transition frequency, the amplitude of the Rabi oscillation decreases, as is illustrated by the dashed envelope in the above plot. == Two-Photon Rabi Frequency == Coherent Rabi oscillations may also be driven by two-photon transitions. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. After excitation the atom may return to the ground state or a lower excited state, by emitting a photon with a characteristic energy. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. The excited state originates from the interaction between a photon and the quantum system. A two-photon transition is not the same as excitation from the ground to intermediate state, and then out of the intermediate state to the excited state. In this case we consider a system with three atomic energy levels, |1\rangle , |i\rangle , and |2\rangle , where |i\rangle represents a so-called intermediate state with corresponding frequency \omega_i , and an electromagnetic field with two frequency components: \hat{V}(t) = e\mathbf{r} \cdot \mathbf{E}_{L1} \cos(\omega_{L1} t) + e\mathbf{r} \cdot \mathbf{E}_{L2} \cos(\omega_{L2} t) Now, \omega_i may be much greater than both \omega_1 and \omega_2 , or \omega_2 > \omega_i > \omega_1 , as illustrated in the figure on the right. thumb|Two photon excitation schema. \omega_i >> \omega_2 > \omega_1 is shown on the left, while \omega_2 > \omega_i > \omega_1 is shown on the right. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). Emission of photons from atoms in various excited states leads to an electromagnetic spectrum showing a series of characteristic emission lines (including, in the case of the hydrogen atom, the Lyman, Balmer, Paschen and Brackett series). The Rabi frequency is a semiclassical concept since it treats the atom as an object with quantized energy levels and the electromagnetic field as a continuous wave. A plot as a function of detuning and ramping the time from 0 to t = \frac{\pi}{\Omega} gives: Animation of optical resonance, frequency domain We see that for \delta = 0 the population will oscillate between the two states at the Rabi frequency. == Generalized Rabi frequency == The quantity \sqrt{\Omega^2 + \delta^2} is commonly referred to as the ""generalized Rabi frequency."" The energy released is equal to the difference in energy levels between the electron energy states. The absorption of the photon takes place in accordance with Planck's quantum theory. The next rule follows from the Frank-Condon Principle, which states that the absorption of a photon by an electron and the subsequent jump in energy levels is near-instantaneous. ",7.58,0.4772,"""24.0""",655,0.11,A
+Do $\mathrm{HF} / 6-31 \mathrm{G}^*$ geometry optimizations on one conformers of $\mathrm{HCOOH}$ with $\mathrm{OCOH}$ dihedral angle of $0^{\circ}$. Calculate the dipole moment.,"Order-6 hexagonal tiling honeycomb Order-6 hexagonal tiling honeycomb 320px Perspective projection view from center of Poincaré disk model 320px Perspective projection view from center of Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {6,3,6} {6,3[3]} Coxeter diagram ↔ ↔ Cells {6,3} 40px Faces hexagon {6} Edge figure hexagon {6} Vertex figure {3,6} or {3[3]} 40px 40px Dual Self-dual Coxeter group \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Regular, quasiregular In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. Order-6 dodecahedral honeycomb Order-6 dodecahedral honeycomb 320px Perspective projection view within Poincaré disk model 320px Perspective projection view within Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {5,3,6} {5,3[3]} Coxeter diagram ↔ Cells {5,3} 40px Faces pentagon {5} Edge figure hexagon {6} Vertex figure 80px 80px triangular tiling Dual Order-5 hexagonal tiling honeycomb Coxeter group \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Regular, quasiregular The order-6 dodecahedral honeycomb is one of 11 paracompact regular honeycombs in hyperbolic 3-space. Order-5 hexagonal tiling honeycomb Order-5 hexagonal tiling honeycomb 320px Perspective projection view from center of Poincaré disk model 320px Perspective projection view from center of Poincaré disk model Type Hyperbolic regular honeycomb Paracompact uniform honeycomb Schläfli symbol {6,3,5} Coxeter-Dynkin diagrams 80px ↔ Cells {6,3} 40px Faces hexagon {6} Edge figure pentagon {5} Vertex figure icosahedron Dual Order-6 dodecahedral honeycomb Coxeter group \overline{HV}_3, [5,3,6] Properties Regular In the field of hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). alt=|thumb|300x300px|The flow structure of the Lamb-Chaplygin dipole The Lamb–Chaplygin dipole model is a mathematical description for a particular inviscid and steady dipolar vortex flow. This dipole is the two-dimensional analogue of Hill's spherical vortex. The Lamb–Chaplygin model follows from demanding the following characteristics: * The dipole has a circular atmosphere/separatrix with radius R: \psi\left(r = R\right) = 0. It is a quasiregular honeycomb. === Cantic order-5 hexagonal tiling honeycomb === Cantic order-5 hexagonal tiling honeycomb Cantic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h2{6,3,5} Coxeter diagram ↔ Cells h2{6,3} 40px t{3,5} 40px r{5,3} 40px Faces triangle {3} pentagon {5} hexagon {6} Vertex figure 80px triangular prism Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex- transitive The cantic order-5 hexagonal tiling honeycomb, h2{6,3,5}, ↔ , has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure. === Runcic order-5 hexagonal tiling honeycomb === Runcic order-5 hexagonal tiling honeycomb Runcic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h3{6,3,5} Coxeter diagram ↔ Cells {3[3]} 40px rr{5,3} 40px {5,3} 40px {}x{3} 40px Faces triangle {3} square {4} pentagon {5} Vertex figure 80px triangular cupola Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The runcic order-5 hexagonal tiling honeycomb, h3{6,3,5}, ↔ , has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure. === Runcicantic order-5 hexagonal tiling honeycomb === Runcicantic order-5 hexagonal tiling honeycomb Runcicantic order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol h2,3{6,3,5} Coxeter diagram ↔ Cells h2{6,3} 40px tr{5,3} 40px t{5,3} 40px {}x{3} 40px Faces triangle {3} square {4} hexagon {6} decagon {10} Vertex figure 80px rectangular pyramid Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The runcicantic order-5 hexagonal tiling honeycomb, h2,3{6,3,5}, ↔ , has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, and triangular prism facets, with a rectangular pyramid vertex figure. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . All vertices are on the ideal surface. : 180px === Truncated order-5 hexagonal tiling honeycomb === Truncated order-5 hexagonal tiling honeycomb Truncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t{6,3,5} or t0,1{6,3,5} Coxeter diagram Cells {3,5} 40px t{6,3} 40px Faces triangle {3} dodecagon {12} Vertex figure 80px pentagonal pyramid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The truncated order-5 hexagonal tiling honeycomb, t0,1{6,3,5}, has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure. 480px === Bitruncated order-5 hexagonal tiling honeycomb === Bitruncated order-5 hexagonal tiling honeycomb Bitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol 2t{6,3,5} or t1,2{6,3,5} Coxeter diagram ↔ Cells t{3,6} 40px t{3,5} 40px Faces pentagon {5} hexagon {6} Vertex figure 80px digonal disphenoid Coxeter groups {\overline{HV}}_3, [5,3,6] {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive The bitruncated order-5 hexagonal tiling honeycomb, t1,2{6,3,5}, has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure. 480px === Cantellated order-5 hexagonal tiling honeycomb === Cantellated order-5 hexagonal tiling honeycomb Cantellated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol rr{6,3,5} or t0,2{6,3,5} Coxeter diagram Cells r{3,5} 40px rr{6,3} 40px {}x{5} 40px Faces triangle {3} square {4} pentagon {5} hexagon {6} Vertex figure 80px wedge Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The cantellated order-5 hexagonal tiling honeycomb, t0,2{6,3,5}, has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure. 480px === Cantitruncated order-5 hexagonal tiling honeycomb === Cantitruncated order-5 hexagonal tiling honeycomb Cantitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol tr{6,3,5} or t0,1,2{6,3,5} Coxeter diagram Cells t{3,5} 40px tr{6,3} 40px {}x{5} 40px Faces square {4} pentagon {5} hexagon {6} dodecagon {12} Vertex figure 80px mirrored sphenoid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex- transitive The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2{6,3,5}, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure. 480px === Runcinated order-5 hexagonal tiling honeycomb === Runcinated order-5 hexagonal tiling honeycomb Runcinated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,3{6,3,5} Coxeter diagram Cells {6,3} 40px {5,3} 40px {}x{6} 40px {}x{5} 40px Faces square {4} pentagon {5} hexagon {6} Vertex figure 80px irregular triangular antiprism Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The runcinated order-5 hexagonal tiling honeycomb, t0,3{6,3,5}, has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure. 480px === Runcitruncated order-5 hexagonal tiling honeycomb === Runcitruncated order-5 hexagonal tiling honeycomb Runcitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,3{6,3,5} Coxeter diagram Cells t{6,3} 40px rr{5,3} 40px {}x{5} 40px {}x{12} 40px Faces triangle {3} square {4} pentagon {5} dodecagon {12} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The runcitruncated order-5 hexagonal tiling honeycomb, t0,1,3{6,3,5}, has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure. 480px === Runcicantellated order-5 hexagonal tiling honeycomb === The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb. === Omnitruncated order-5 hexagonal tiling honeycomb === Omnitruncated order-5 hexagonal tiling honeycomb Omnitruncated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,2,3{6,3,5} Coxeter diagram Cells tr{6,3} 40px tr{5,3} 40px {}x{10} 40px {}x{12} 40px Faces square {4} hexagon {6} decagon {10} dodecagon {12} Vertex figure 80px irregular tetrahedron Coxeter groups {\overline{HV}}_3, [5,3,6] Properties Vertex-transitive The omnitruncated order-5 hexagonal tiling honeycomb, t0,1,2,3{6,3,5}, has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure. 480px === Alternated order-5 hexagonal tiling honeycomb === Alternated order-5 hexagonal tiling honeycomb Alternated order-5 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Semiregular honeycomb Schläfli symbol h{6,3,5} Coxeter diagram ↔ Cells {3[3]} 40px {3,5} 40px Faces triangle {3} Vertex figure 40px truncated icosahedron Coxeter groups {\overline{HP}}_3, [5,3[3]] Properties Vertex-transitive, edge-transitive, quasiregular The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, ↔ , has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by: :T_{abcd}=C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} + \frac{1}{4}\epsilon_{ae}{}^{hi} \epsilon_{b}{}^{ej}{}_{k} C_{hicf} C_{j}{}^{k}{}_{d}{}^{f} Alternatively, :T_{abcd} = C_{aecf}C_{b} {}^{e} {}_{d} {}^{f} - \frac{3}{2} g_{a[b} C_{jk]cf} C^{jk}{}_{d}{}^{f} where C_{abcd} is the Weyl tensor. It contains triangular tiling facets in a hexagonal tiling vertex figure. === Cantic order-6 hexagonal tiling honeycomb === Cantic order-6 hexagonal tiling honeycomb Cantic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h2{6,3,6} Coxeter diagrams ↔ Cells t{3,6} 40px r{6,3} 40px h2{6,3} 40px Faces triangle {3} hexagon {6} Vertex figure 80px triangular prism Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex- transitive, edge-transitive The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, ↔ , with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure. ===Runcic order-6 hexagonal tiling honeycomb=== Runcic order-6 hexagonal tiling honeycomb Runcic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h3{6,3,6} Coxeter diagrams ↔ Cells rr{3,6} 40px {6,3} 40px {3[3]} 40px {3}x{} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px triangular cupola Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex- transitive The runcic hexagonal tiling honeycomb, h3{6,3,6}, , or , has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure. ===Runicantic order-6 hexagonal tiling honeycomb=== Runcicantic order-6 hexagonal tiling honeycomb Runcicantic order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h2,3{6,3,6} Coxeter diagrams ↔ Cells tr{6,3} 40px t{6,3} 40px h2{6,3} 40px {}x{3} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px rectangular pyramid Coxeter groups \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The runcicantic order-6 hexagonal tiling honeycomb, h2,3{6,3,6}, , or , contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . The order-6 dodecahedral honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polytopes and honeycombs with dodecahedral cells: === Rectified order-6 dodecahedral honeycomb === Rectified order-6 dodecahedral honeycomb \--> Type Paracompact uniform honeycomb Schläfli symbols r{5,3,6} t1{5,3,6} Coxeter diagrams ↔ Cells r{5,3} 40px {3,6} 40px Faces triangle {3} pentagon {5} Vertex figure 80px hexagonal prism Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive, edge-transitive The rectified order-6 dodecahedral honeycomb, t1{5,3,6} has icosidodecahedron and triangular tiling cells connected in a hexagonal prism vertex figure. : 480px Perspective projection view within Poincaré disk model It is similar to the 2D hyperbolic pentaapeirogonal tiling, r{5,∞} with pentagon and apeirogonal faces. : 180px === Truncated order-6 dodecahedral honeycomb === Truncated order-6 dodecahedral honeycomb \--> Type Paracompact uniform honeycomb Schläfli symbols t{5,3,6} t0,1{5,3,6} Coxeter diagrams ↔ Cells t{5,3} 40px {3,6} 40px Faces triangle {3} decagon {10} Vertex figure 80px hexagonal pyramid Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The truncated order-6 dodecahedral honeycomb, t0,1{5,3,6} has truncated dodecahedron and triangular tiling cells connected in a hexagonal pyramid vertex figure. 480px === Bitruncated order-6 dodecahedral honeycomb === The bitruncated order-6 dodecahedral honeycomb is the same as the bitruncated order-5 hexagonal tiling honeycomb. === Cantellated order-6 dodecahedral honeycomb === Cantellated order-6 dodecahedral honeycomb Cantellated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols rr{5,3,6} t0,2{5,3,6} Coxeter diagrams ↔ Cells rr{5,3} 40px rr{6,3} 40px {}x{6} 40px Faces triangle {3} square {4} pentagon {5} hexagon {6} Vertex figure 80px wedge Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The cantellated order-6 dodecahedral honeycomb, t0,2{5,3,6}, has rhombicosidodecahedron, trihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure. 480px === Cantitruncated order-6 dodecahedral honeycomb === Cantitruncated order-6 dodecahedral honeycomb Cantitruncated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols tr{5,3,6} t0,1,2{5,3,6} Coxeter diagrams ↔ Cells tr{5,3} 40px t{3,6} 40px {}x{6} 40px Faces square {4} hexagon {6} decagon {10} Vertex figure 80px mirrored sphenoid Coxeter groups \overline{HV}_3, [5,3,6] \overline{HP}_3, [5,3[3]] Properties Vertex-transitive The cantitruncated order-6 dodecahedral honeycomb, t0,1,2{5,3,6} has truncated icosidodecahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure. 480px === Runcinated order-6 dodecahedral honeycomb === The runcinated order-6 dodecahedral honeycomb is the same as the runcinated order-5 hexagonal tiling honeycomb. === Runcitruncated order-6 dodecahedral honeycomb === Runcitruncated order-6 dodecahedral honeycomb Runcitruncated order-6 dodecahedral honeycomb Type Paracompact uniform honeycomb Schläfli symbols t0,1,3{5,3,6} Coxeter diagrams Cells t{5,3} 40px rr{6,3} 40px {}x{10} 40px {}x{6} 40px Faces square {4} hexagon {6} decagon {10} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups \overline{HV}_3, [5,3,6] Properties Vertex-transitive The runcitruncated order-6 dodecahedral honeycomb, t0,1,3{5,3,6} has truncated dodecahedron, rhombitrihexagonal tiling, decagonal prism, and hexagonal prism facets, with an isosceles- trapezoidal pyramid vertex figure. 480px === Runcicantellated order-6 dodecahedral honeycomb === The runcicantellated order-6 dodecahedral honeycomb is the same as the runcitruncated order-5 hexagonal tiling honeycomb. === Omnitruncated order-6 dodecahedral honeycomb === The omnitruncated order-6 dodecahedral honeycomb is the same as the omnitruncated order-5 hexagonal tiling honeycomb. == See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs == References == *Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. . Thus, infinitely many hexagonal tilings meet at each vertex of this honeycomb.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III == Related tilings == The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px It contains and that tile 2-hypercycle surfaces, which are similar to the paracompact tilings and (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively): : 120px 120px == Symmetry == 120px|thumb|left|Subgroup relations: ↔ The order-6 hexagonal tiling honeycomb has a half-symmetry construction: . The order-6 dodecahedral honeycomb is similar to the 2D hyperbolic infinite-order pentagonal tiling, {5,∞}, with pentagonal faces, and with vertices on the ideal surface. : 240px == Related polytopes and honeycombs == The order-6 dodecahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. The polar circles of the triangles of a complete quadrilateral form a coaxal system. The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures: It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells: It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures: === Rectified order-6 hexagonal tiling honeycomb === Rectified order-6 hexagonal tiling honeycomb Rectified order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols r{6,3,6} or t1{6,3,6} Coxeter diagrams ↔ ↔ ↔ ↔ Cells {3,6} 40px r{6,3} 40px Faces triangle {3} hexagon {6} Vertex figure 80px hexagonal prism Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] \overline{PP}_3, [3[3,3]] Properties Vertex-transitive, edge-transitive The rectified order-6 hexagonal tiling honeycomb, t1{6,3,6}, has triangular tiling and trihexagonal tiling facets, with a hexagonal prism vertex figure. it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, ↔ . 480px It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px ==== Related honeycombs==== The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures: It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q} === Truncated order-6 hexagonal tiling honeycomb === Truncated order-6 hexagonal tiling honeycomb Truncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t{6,3,6} or t0,1{6,3,6} Coxeter diagram ↔ Cells {3,6} 40px t{6,3} 40px Faces triangle {3} dodecagon {12} Vertex figure 80px hexagonal pyramid Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The truncated order-6 hexagonal tiling honeycomb, t0,1{6,3,6}, has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.Twitter Rotation around 3 fold axis 480px === Bitruncated order-6 hexagonal tiling honeycomb === Bitruncated order-6 hexagonal tiling honeycomb Bitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol bt{6,3,6} or t1,2{6,3,6} Coxeter diagram ↔ Cells t{3,6} 40px Faces hexagon {6} Vertex figure 80px tetrahedron Coxeter groups 2\times\overline{Z}_3, 6,3,6 \overline{VP}_3, [6,3[3]] \overline{V}_3, [3,3,6] Properties Regular The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, ↔ . Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III == Symmetry== A lower-symmetry construction of index 120, [6,(3,5)*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches. == Images== The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex. : 180px == Related polytopes and honeycombs == The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at each edge. It contains hexagonal tiling facets, with a tetrahedron vertex figure. 480px === Cantellated order-6 hexagonal tiling honeycomb === Cantellated order-6 hexagonal tiling honeycomb Cantellated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol rr{6,3,6} or t0,2{6,3,6} Coxeter diagram ↔ Cells r{3,6} 40px rr{6,3} 40px {}x{6} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px wedge Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The cantellated order-6 hexagonal tiling honeycomb, t0,2{6,3,6}, has trihexagonal tiling, rhombitrihexagonal tiling, and hexagonal prism cells, with a wedge vertex figure. 480px === Cantitruncated order-6 hexagonal tiling honeycomb === Cantitruncated order-6 hexagonal tiling honeycomb Cantitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol tr{6,3,6} or t0,1,2{6,3,6} Coxeter diagram ↔ Cells tr{3,6} 40px t{3,6} 40px {}x{6} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px mirrored sphenoid Coxeter groups \overline{Z}_3, [6,3,6] \overline{VP}_3, [6,3[3]] Properties Vertex-transitive The cantitruncated order-6 hexagonal tiling honeycomb, t0,1,2{6,3,6}, has hexagonal tiling, truncated trihexagonal tiling, and hexagonal prism cells, with a mirrored sphenoid vertex figure. 480px === Runcinated order-6 hexagonal tiling honeycomb === Runcinated order-6 hexagonal tiling honeycomb Runcinated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,3{6,3,6} Coxeter diagram ↔ Cells {6,3} 40px40px {}×{6} 40px Faces triangle {3} square {4} hexagon {6} Vertex figure 80px triangular antiprism Coxeter groups 2\times\overline{Z}_3, 6,3,6 Properties Vertex-transitive, edge-transitive The runcinated order-6 hexagonal tiling honeycomb, t0,3{6,3,6}, has hexagonal tiling and hexagonal prism cells, with a triangular antiprism vertex figure. 480px It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, with square and hexagonal faces: : 240px === Runcitruncated order-6 hexagonal tiling honeycomb === Runcitruncated order-6 hexagonal tiling honeycomb Runcitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,3{6,3,6} Coxeter diagram Cells t{6,3} 40px rr{6,3} 40px {}x{6}40px {}x{12} 40px Faces triangle {3} square {4} hexagon {6} dodecagon {12} Vertex figure 80px isosceles-trapezoidal pyramid Coxeter groups \overline{Z}_3, [6,3,6] Properties Vertex-transitive The runcitruncated order-6 hexagonal tiling honeycomb, t0,1,3{6,3,6}, has truncated hexagonal tiling, rhombitrihexagonal tiling, hexagonal prism, and dodecagonal prism cells, with an isosceles- trapezoidal pyramid vertex figure. 480px === Omnitruncated order-6 hexagonal tiling honeycomb === Omnitruncated order-6 hexagonal tiling honeycomb Omnitruncated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbol t0,1,2,3{6,3,6} Coxeter diagram Cells tr{6,3} 40px {}x{12} 40px Faces square {4} hexagon {6} dodecagon {12} Vertex figure 80px phyllic disphenoid Coxeter groups 2\times\overline{Z}_3, 6,3,6 Properties Vertex-transitive The omnitruncated order-6 hexagonal tiling honeycomb, t0,1,2,3{6,3,6}, has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure. 480px ===Alternated order-6 hexagonal tiling honeycomb=== Alternated order-6 hexagonal tiling honeycomb Alternated order-6 hexagonal tiling honeycomb Type Paracompact uniform honeycomb Schläfli symbols h{6,3,6} Coxeter diagrams ↔ Cells {3,6} 40px {3[3]} 40px Faces triangle {3} Vertex figure 80px hexagonal tiling Coxeter groups \overline{VP}_3, [6,3[3]] Properties Regular, quasiregular The alternated order-6 hexagonal tiling honeycomb is a lower- symmetry construction of the regular triangular tiling honeycomb, ↔ . A less favorable property is that the second derivative of the flow field at the dipole's edge is not continuous. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: . == Related polytopes and honeycombs == The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space. ",1.41,4.76,"""0.87""",-1270,-0.55,A
+Frozen-core $\mathrm{SCF} / \mathrm{DZP}$ and CI-SD/DZP calculations on $\mathrm{H}_2 \mathrm{O}$ at its equilibrium geometry gave energies of -76.040542 and -76.243772 hartrees. Application of the Davidson correction brought the energy to -76.254549 hartrees. Find the coefficient of $\Phi_0$ in the normalized CI-SD wave function.,"As with other perturbative approaches, the Davidson correction is not reliable when the electronic structure of CISD and the reference Hartree–Fock wave functions are significantly different (i.e. when a_0^2 is not close to 1). A solution of these equations yields the Hartree–Fock wave function and energy of the system. The Davidson correction is an energy correction often applied in calculations using the method of truncated configuration interaction, which is one of several post-Hartree-Fock ab initio quantum chemistry methods in the field of computational chemistry. It uses the formula :\Delta E_Q = (1 - a_0^2)(E_{\rm CISD} - E_{\rm HF}), \ :E_{\rm CISDTQ} \approx E_{\rm CISD} + \Delta E_Q, \ where a0 is the coefficient of the Hartree-Fock wavefunction in the CISD expansion, ECISD and EHF are the energies of the CISD and Hartree-Fock wavefunctions respectively, and ΔEQ is the correction to estimate ECISDTQ, the energy of the CISDTQ wavefunction. Developing post-Hartree–Fock methods based on a ROHF wave function is inherently more difficult than using a UHF wave function, due to the lack of a unique set of molecular orbitals. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. In atomic structure theory, calculations may be for a spectrum with many excited energy levels and consequently the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not necessarily the ground state. In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Davidson correction does not give information about the wave function. An alternative to Hartree–Fock calculations used in some cases is density functional theory, which treats both exchange and correlation energies, albeit approximately. Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation : \hat F(1)\phi_i(1) = \epsilon_i \phi_i(1), where \phi_i(1) are a set of one-electron wave functions, called the Hartree–Fock molecular orbitals. === Linear combination of atomic orbitals === Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a linear combination of atomic orbitals. They add electron correlation which is a more accurate way of including the repulsions between electrons than in the Hartree–Fock method where repulsions are only averaged. == Details == In general, the SCF procedure makes several assumptions about the nature of the multi-body Schrödinger equation and its set of solutions: * For molecules, the Born–Oppenheimer approximation is inherently assumed. Charlotte Froese Fischer (born 1929) is a Canadian-American applied mathematician and computer scientist noted for the development and implementation of the Multi-Configurational Hartree–Fock (MCHF) approach to atomic-structure calculations and its application to the description of atomic structure and spectra. In computational chemistry, post–Hartree–Fock (post-HF) methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as spin-orbitals. The Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules, nanostructures and solids but it has also found widespread use in nuclear physics. However, different choices of reference orbitals have shown to provide similar results, and thus many different post-Hartree–Fock methods have been implemented in a variety of electronic structure packages. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting exchange. (See Hartree–Fock–Bogoliubov method for a discussion of its application in nuclear structure theory). Another option is to use modern valence bond methods. == Software packages == For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the list of quantum chemistry and solid state physics software. == See also == Related fields * Quantum chemistry * Molecular physics * Quantum chemistry computer programs * Fock symmetry Concepts * Roothaan equations * Koopmans' theorem * Post-Hartree–Fock * Direct inversion of iterative subspace People * Vladimir Aleksandrovich Fock * Clemens Roothaan * George G. Hall * John Pople * Reinhart Ahlrichs == References == == Sources == * * * == External links == * The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. However, neither Davidson correction itself nor the corrected energies are size-consistent or size-extensive. ",0.2244,30,"""-2.0""",0.9731,35.2,D
+Let $w$ be the variable defined as the number of heads that show when two coins are tossed simultaneously. Find $\langle w\rangle$.,"thumb|upright=1.5|Coin of Epander. thumb|upright=1.5|Coin of Bhradrayasha. thumb|upright=1.35|Coin of Tennes. thumb|upright=1.2|Coin of Sabaces. It was designed by Nico de Haas, a Dutch national-socialist, and struck in 1941 and 1942. ==Mintage== Year Mintage Notes 1941 27,600,000 1942 ==References== Category:Netherlands in World War II Category:Coins of the Netherlands Category:Modern obsolete currencies Category:Currencies of Europe Category:Zinc and aluminum coins minted in Germany and occupied territories during World War II Japanese coins from this period are read clockwise from right to left: :""Year"" ← ""Number representing year of reign"" ← ""Emperors name"" (Ex: 年 ← 五 ← 和昭) Year of reign Japanese date Gregorian date Mintage 5th 五 1930 6th 六 1931 7th 七 1932 Unknown ==Collecting== The value of any given coin is determined by survivability rate and condition as collectors in general prefer uncleaned appealing coins. The -cent coin minted in the Netherlands during World War II was made of zinc, and worth , or .025, of the Dutch guilder. Japanese coins from this period are read clockwise from right to left: :""Year"" ← ""Number representing year of reign"" ← ""Emperors name"" (Ex: 年 ← 六 ← 正大) Year of reign Japanese date Gregorian date Mintage 1st 元 1912 2nd 二 1913 3rd 三 1914 4th 四 1915 5th 五 1916 6th 六 1917 7th 七 1918 8th 八 1919 9th 九 1920 === Shōwa === The following are mintage figures for coins minted between the 5th and the 7th year of Emperor Shōwa's reign. This system was officially put into place on May 10, 1871 setting standards for the 20 yen coin. Japanese coins from this period are read clockwise from right to left :""Year"" ← ""Number representing year of reign"" ← ""Emperors name"" (Ex: 年 ← 七十三 ← 治明) thumb|right|20 yen coin from 1870 (year 3) Design 1 - (1870 - 1892) thumb|right|20 yen coin from 1897 (year 30) Design 2 - (1897 - 1912) Year of reign Japanese date Gregorian date Mintage 3rd 三 1870 5th 五 1872 6th 六 1873 9th 九 1876 10th 十 1877 13th 三十 1880 25th 五十二 1892 Not circulated 30th 十三 1897 37th 七十三 1904 38th 八十三 1905 39th 九十三 1906 40th 十四 1907 41st 一十四 1908 42nd 二十四 1909 43rd 三十四 1910 44th 四十四 1911 45th 五十四 1912 ===Taishō=== The following are mintage figures for the coins that were minted from the 1st to the 9th year of Taishō's reign. Many of these coins were then melted or destroyed as a result of the wars between 1931 and 1945. An auction held in 2011 featuring one of these coins sold it for $230,000 (USD). Some of these coins were kept away in bank vaults for decades before being released as part of a hoard in the mid 2000s. ==Weight and size== Image Minted Size Weight 150px 1870–1880 35.06mm 33.33g 150px 1897–1932 28.78mm 16.66g ==Circulation figures== ===Meiji=== The following are mintage figures for the coins that were minted between the 3rd and 45th (last) year of Meiji's reign. Gold coins of the 20 yen denomination were last minted in 1932, it is unknown how many Shōwa era coins were later melted. Twenty yen coins dated 1877 (year 10) have an extremely low mintage of just 29 coins struck. These coins which are dated from 1870 to 1876 (year 3 to 9) are all priced in five digit dollar amounts (USD) in average condition. These new standards lowered both the size and weight of the coin, the new diameter was set at 28.78mm (previously 35.06mm), and the weight was lowered from 33.3g down to 16.6g. Coinage of the 20 yen piece had all but stopped by 1877, and those struck in 1880 were only done so as part of presentation sets for visiting dignitaries and heads of state. For this denomination all 20 yen coins are scarce as the amount remaining today are dependent on how many were saved or kept away. Twenty Yen coins spanned three different Imperial eras before mintage was halted in 1932. The was a denomination of Japanese yen. These coins were minted in gold, and during their lifespan were the highest denomination of coin that circulated in the country. ",-233,-2,"""8.3147""",0.4207,1,E
+Calculate the force on an alpha particle passing a gold atomic nucleus at a distance of $0.00300 Å$.,"Initially the alpha particles are at a very large distance from the nucleus. :\frac{1}{2} mv^2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 q_2}{r_\text{min}} Rearranging: :r_\text{min} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2 q_1 q_2}{mv^2} For an alpha particle: * (mass) = = * (for helium) = 2 × = * (for gold) = 79 × = * (initial velocity) = (for this example) Substituting these in gives the value of about , or 27 fm. The true radius of the nucleus is not recovered in these experiments because the alphas do not have enough energy to penetrate to more than 27 fm of the nuclear center, as noted, when the actual radius of gold is 7.3 fm. Rutherford realized this, and also realized that actual impact of the alphas on gold causing any force-deviation from that of the coulomb potential would change the form of his scattering curve at high scattering angles (the smallest impact parameters) from a hyperbola to something else. This was not seen, indicating that the surface of the gold nucleus had not been ""touched"" so that Rutherford also knew the gold nucleus (or the sum of the gold and alpha radii) was smaller than 27 fm. == Extension to situations with relativistic particles and target recoil == The extension of low-energy Rutherford-type scattering to relativistic energies and particles that have intrinsic spin is beyond the scope of this article. In Rutherford's gold foil experiment conducted by his students Hans Geiger and Ernest Marsden, a narrow beam of alpha particles was established, passing through very thin (a few hundred atoms thick) gold foil. Applying the inverse-square law between the charges on the alpha particle and nucleus, one can write: Assumptions: 1\. The distance from the center of the alpha particle to the center of the nucleus () at this point is an upper limit for the nuclear radius, if it is evident from the experiment that the scattering process obeys the cross section formula given above. The classical Rutherford scattering process of alpha particles against gold nuclei is an example of ""elastic scattering"" because neither the alpha particles nor the gold nuclei are internally excited. For any central potential, the differential cross- section in the lab frame is related to that in the center-of-mass frame by \frac{d\sigma}{d\Omega}_L=\frac{(1+2s\cos\Theta+s^2)^{3/2}}{1+s\cos\Theta} \frac{d\sigma}{d\Omega} To give a sense of the importance of recoil, we evaluate the head-on energy ratio F for an incident alpha particle (mass number \approx 4) scattering off a gold nucleus (mass number \approx 197): F \approx 0.0780. This same result can be expressed alternatively as : \frac{d\sigma}{d\Omega} = \left( \frac{ Z_1 Z_2 \alpha (\hbar c)} {4 E_{\mathrm{K}10} \sin^2 \frac{\Theta}{2} } \right)^2, where is the dimensionless fine structure constant, is the initial non-relativistic kinetic energy of particle 1 in MeV, and . ==Details of calculating maximal nuclear size== For head-on collisions between alpha particles and the nucleus (with zero impact parameter), all the kinetic energy of the alpha particle is turned into potential energy and the particle is at rest. The Rutherford formula (see below) further neglects the recoil kinetic energy of the massive target nucleus. When a (positive) alpha particle approached sufficiently close to the nucleus, it was repelled strongly enough to rebound at high angles. In classical physics, alpha particles do not have enough energy to escape the potential well from the strong force inside the nucleus (this well involves escaping the strong force to go up one side of the well, which is followed by the electromagnetic force causing a repulsive push-off down the other side). Rutherford showed, using the method outlined below, that the size of the nucleus was less than about (how much less than this size, Rutherford could not tell from this experiment alone; see more below on this problem of lowest possible size). It was determined that the atom's positive charge was concentrated in a small area in its center, making the positive charge dense enough to deflect any positively charged alpha particles that came close to what was later termed the nucleus. The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. For the case of light alpha particles scattering off heavy nuclei, as in the experiment performed by Rutherford, the reduced mass, essentially the mass of the alpha particle and the nucleus off of which it scatters, is essentially stationary in the lab frame. It was found that some of the alpha particles were deflected at much larger angles than expected (at a suggestion by Rutherford to check it) and some even bounced almost directly back. Because the mass of an alpha particle is about 8000 times that of an electron, it became apparent that a very strong force must be present if it could deflect the massive and fast moving alpha particles. With a typical kinetic energy of 5 MeV; the speed of emitted alpha particles is 15,000 km/s, which is 5% of the speed of light. For the more extreme case of an electron scattering off a proton, s \approx 1/1836 and F \approx 0.00218. == See also == *Rutherford backscattering spectrometry ==References== == Textbooks == * == External links == * E. Rutherford, The Scattering of α and β Particles by Matter and the Structure of the Atom, Philosophical Magazine. Rutherford hypothesized that, assuming the ""plum pudding"" model of the atom was correct, the positively charged alpha particles would be only slightly deflected, if at all, by the dispersed positive charge predicted. The diameter of the nucleus is in the range of () for hydrogen (the diameter of a single proton) to about for uranium. ",+4.1,4.4,"""0.405""",269,3.2,C
+"When an electron in a certain excited energy level in a one-dimensional box of length $2.00  Å$ makes a transition to the ground state, a photon of wavelength $8.79 \mathrm{~nm}$ is emitted. Find the quantum number of the initial state.","In the new atom, the electron may begin in any energy level, and subsequently cascades to the ground state (n = 1), emitting photons with each transition. The excited state originates from the interaction between a photon and the quantum system. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. Excited-state absorption is possible only when an electron has been already excited from the ground state to a lower excited state. In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. These energy levels are described by the principal quantum number n = 1, 2, 3, ... . After excitation the atom may return to the ground state or a lower excited state, by emitting a photon with a characteristic energy. Photoexcitation is the production of an excited state of a quantum system by photon absorption. In modern physics, the concept of a quantum jump is rarely used; as a rule scientists speak of transitions between quantum states or energy levels. == Atomic electron transition == thumb|Grotrian diagram of a quantum 3-level system with characteristic transition frequencies, \omega12 and \omega13, and excited state lifetimes \Gamma2 and \Gamma3 Atomic electron transitions cause the emission or absorption of photons. The ground state of the hydrogen atom has the atom's single electron in the lowest possible orbital (that is, the spherically symmetric ""1s"" wave function, which, so far, has been demonstrated to have the lowest possible quantum numbers). By giving the atom additional energy (for example, by absorption of a photon of an appropriate energy), the electron moves into an excited state (one with one or more quantum numbers greater than the minimum possible). The larger the energy separation of the states between which the electron jumps, the shorter the wavelength of the photon emitted. thumb|EMCCD camera and photomultiplier tube signals while driving quantum jumps on the 674 nm transition of 88Sr+ In an ion trap, quantum jumps can be directly observed by addressing a trapped ion with radiation at two different frequencies to drive electron transitions. However, it is not easy to measure them compared to ground-state absorption, and in some cases complete bleaching of the ground state is required to measure excited-state absorption. ==Reaction== A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. == See also == * Rydberg formula * Stationary state * Repulsive state == References == == External links == * NASA background information on ground and excited states Category:Quantum mechanics Although the initial step excites to a broad range of intermediate states, the precision inherent in the optical excitation process means that the laser light only interacts with a specific subset of atoms in a particular state, exciting to the chosen final state. == Hydrogenic potential == thumb|right|Figure 3. In optical excitation, the incident photon is absorbed by the target atom, resulting in a precise final state energy. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). State |3\rangle has a relatively long lifetime \Gamma3 which causes an interruption of the photon emission as the electron gets shelved in state through application of light with frequency \omega13. Quantum- mechanically, a state with abnormally high n refers to an atom in which the valence electron(s) have been excited into a formerly unpopulated electron orbital with higher energy and lower binding energy. A quantum jump is the abrupt transition of a quantum system (atom, molecule, atomic nucleus) from one quantum state to another, from one energy level to another. In quantum many-body systems, ground states of gapped Hamiltonians have exponential decay of correlations. ",4,0.22222222,"""152.67""",1838.50666349,3.2,A
+"For a macroscopic object of mass $1.0 \mathrm{~g}$ moving with speed $1.0 \mathrm{~cm} / \mathrm{s}$ in a one-dimensional box of length $1.0 \mathrm{~cm}$, find the quantum number $n$.","The magnitude of the momentum is given by :p=\frac{h}{2L}\sqrt{n_x^2+n_y^2+n_z^2} \qquad \qquad n_x,n_y,n_z=1,2,3,\ldots where h is Planck's constant and L is the length of a side of the box. The distance from the origin to any point will be :n=\sqrt{n_x^2+n_y^2+n_z^2}=\frac{2Lp}{h} Suppose each set of quantum numbers specify f states where f is the number of internal degrees of freedom of the particle that can be altered by collision. The principal quantum number is related to the radial quantum number, nr, by: n = n_r + \ell + 1 where ℓ is the azimuthal quantum number and nr is equal to the number of nodes in the radial wavefunction. Using a continuum approximation, the number of states with magnitude of momentum between p and p+dp is therefore :dg = \frac{\pi}{2}~f n^2\,dn = \frac{4\pi fV}{h^3}~ p^2\,dp where V=L3 is the volume of the box. In quantum mechanics, the principal quantum number (symbolized n) is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. These integers are the magnetic quantum numbers. In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions. In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The principal quantum number n represents the relative overall energy of each orbital. More complete calculations will be left to separate articles, but some simple examples will be given in this article. ==Thomas–Fermi approximation for the degeneracy of states== For both massive and massless particles in a box, the states of a particle are enumerated by a set of quantum numbers . Integrating the energy distribution function and solving for N gives the particle number :N = \left(\frac{Vf}{\Lambda^3}\right)\textrm{Li}_{3/2}(z) where Lis(z) is the polylogarithm function. The orbital magnetic quantum number takes integer values in the range from -\ell to +\ell, including zero. The spin magnetic quantum number specifies the z-axis component of the spin angular momentum for a particle having spin quantum number . For large values of n, the number of states with magnitude of momentum less than or equal to p from the above equation is approximately : g = \left(\frac{f}{8}\right) \frac{4}{3}\pi n^3 = \frac{4\pi f}{3} \left(\frac{Lp}{h}\right)^3 which is just f times the volume of a sphere of radius n divided by eight since only the octant with positive ni is considered. In particle physics, G-parity is a multiplicative quantum number that results from the generalization of C-parity to multiplets of particles. The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the principal quantum number n, the azimuthal (orbital) quantum number \ell, and the magnetic quantum numbers and . Using the Thomas−Fermi approximation, the number of particles dNE with energy between E and E+dE is: :dN_E= \frac{dg_E}{\Phi(E)} where dg_E is the number of states with energy between E and E+dE. ==Energy distribution== Using the results derived from the previous sections of this article, some distributions for the gas in a box can now be determined. The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field (the Zeeman effect) -- hence the name magnetic quantum number. The quantum number m_l refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the z-direction or quantization axis. The principal quantum number arose in the solution of the radial part of the wave equation as shown below. Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as or for the total z-axis orbital angular momentum of all the electrons in an atom. ==Derivation== thumb|These orbitals have magnetic quantum numbers m_l=-\ell, \ldots,\ell from left to right in ascending order. It turns out, however, that the above equation for particle number expresses the number of bosons in excited states rather well, and thus: : N=\frac{g_0 z}{1-z}+\left(\frac{Vf}{\Lambda^3}\right)\operatorname{Li}_{3/2}(z) where the added term is the number of particles in the ground state. ",8.44,3,"""2.3613""",0.16,3.23,B
+"For the $\mathrm{H}_2$ ground electronic state, $D_0=4.4781 \mathrm{eV}$. Find $\Delta H_0^{\circ}$ for $\mathrm{H}_2(g) \rightarrow 2 \mathrm{H}(g)$ in $\mathrm{kJ} / \mathrm{mol}$","The delta function model becomes particularly useful with the double-well Dirac Delta function model which represents a one-dimensional version of the Hydrogen molecule ion, as shown in the following section. == Double delta potential == thumb|300px|right| The symmetric and anti-symmetric wavefunctions for the double-well Dirac delta function model with ""internuclear"" distance The double-well Dirac delta function models a diatomic hydrogen molecule by the corresponding Schrödinger equation: -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x) \psi(x) = E \psi(x), where the potential is now V(x) = -q \left[ \delta \left(x + \frac{R}{2}\right) + \lambda\delta \left(x - \frac{R}{2} \right) \right], where 0 < R < \infty is the ""internuclear"" distance with Dirac delta-function (negative) peaks located at (shown in brown in the diagram). Similarly to the single band case, we can write for U^{A}_{jj'} : D_{jj'} \equiv U^{A}_{jj'} = E_{j}(0)\delta_{jj'} + \sum_{\alpha\beta} D^{\alpha\beta}_{jj'}k_{\alpha}k_{\beta}, : D^{\alpha\beta}_{jj'} = \frac{\hbar^2}{2 m_0} \left [ \delta_{jj'}\delta_{\alpha\beta} + \sum^{B}_{\gamma} \frac{ p^{\alpha}_{j\gamma}p^{\beta}_{\gamma j'} + p^{\beta}_{j\gamma}p^{\alpha}_{\gamma j'} }{ m_0 (E_0-E_{\gamma}) } \right ]. The third parameter \gamma_3 relates to the anisotropy of the energy band structure around the \Gamma point when \gamma_2 eq \gamma_3 . == Explicit Hamiltonian matrix == The Luttinger-Kohn Hamiltonian \mathbf{D_{jj'}} can be written explicitly as a 8X8 matrix (taking into account 8 bands - 2 conduction, 2 heavy-holes, 2 light-holes and 2 split-off) : \mathbf{H} = \left( \begin{array}{cccccccc} E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ P_z^{\dagger} & P+\Delta & \sqrt{2}Q^{\dagger} & -S^{\dagger}/\sqrt{2} & -\sqrt{2}P_{+}^{\dagger} & 0 & -\sqrt{3/2}S & -\sqrt{2}R \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\\ \end{array} \right) == Summary == == References == 2\. A flavor of the k·p perturbation theory used for calculating the structure of multiple, degenerate electronic bands in bulk and quantum well semiconductors. The International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS), is a biannual event, where over hundred scientists meet for the presentation of new developments on the special field of two-dimensional electron systems in semiconductors. thumb|right|200px|Formation of a δ bond by the overlap of two d orbitals thumb|right|200px|3D model of a boundary surface of a δ bond in Mo2 In chemistry, delta bonds (δ bonds) are covalent chemical bonds, where four lobes of one involved atomic orbital overlap four lobes of the other involved atomic orbital. Substituting into the Schrödinger equation in Bloch approximation we obtain : H u_{n\mathbf{k}}(\mathbf{r}) = \left( H_0 + \frac{\hbar}{m_{0}}\mathbf{k}\cdot\mathbf{\Pi} + \frac{\hbar^2 k^2}{4m_{0}^{2}c^{2}} abla V \times \mathbf{p} \cdot \bar{\sigma} \right) u_{n\mathbf{k}}(\mathbf{r}) = E_{n}(\mathbf{k}) u_{n\mathbf{k}}(\mathbf{r}) , where : \mathbf{\Pi} = \mathbf{p} + \frac{\hbar}{4m_{0}^{2}c^{2}}\bar{\sigma} \times abla V and the perturbation Hamiltonian can be defined as : H' = \frac{\hbar}{m_0}\mathbf{k}\cdot\mathbf{\Pi}. The energy of the bound state is then E = -\frac{\hbar^2\kappa^2}{2m} = -\frac{m\lambda^2}{2\hbar^2}. === Scattering (E > 0) === right|thumb|350px|Transmission (T) and reflection (R) probability of a delta potential well. The boundary conditions thus give the following restrictions on the coefficients \begin{cases} A_r + A_l - B_r - B_l &= 0,\\\ -A_r + A_l + B_r - B_l &= \frac{2m\lambda}{ik\hbar^2} (A_r + A_l). \end{cases} === Bound state (E < 0) === right|thumb|350px|The graph of the bound state wavefunction solution to the delta function potential is continuous everywhere, but its derivative is not defined at . The delta function model is actually a one-dimensional version of the Hydrogen atom according to the dimensional scaling method developed by the group of Dudley R. HerschbachD.R. Herschbach, J.S. Avery, and O. Goscinski (eds.), Dimensional Scaling in Chemical Physics, Springer, (1992). In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Kevin Hwang (Hangul: 케빈황; born May 1, 1992), also known by his Korean name Hwang Ji-tu (Hangul: 황지투) and better known by his stage name G2 (Hangul: 지투), is a Korean-American rapper and singer. Matching of the wavefunction at the Dirac delta-function peaks yields the determinant \begin{vmatrix} q - d & q e^{-d R} \\\ q \lambda e^{-d R} & q \lambda - d \end{vmatrix} = 0, \quad \text{where } E = -\frac{d^2}{2}. Substituting the definition of into this expression yields -\frac{\hbar^2}{2m} ik (-A_r + A_l + B_r - B_l) + \lambda(A_r + A_l) = 0. They represent an approximation of the two lowest discrete energy states of the three-dimensional H2^+ and are useful in its analysis. Using Löwdin's method, only the following eigenvalue problem needs to be solved : \sum^{A}_{j'} (U^{A}_{jj'}-E\delta_{jj'})a_{j'}(\mathbf{k}) = 0, where : U^{A}_{jj'} = H_{jj'} + \sum^{B}_{\gamma eq j,j'} \frac{H_{j\gamma}H_{\gamma j'}}{E_0-E_{\gamma}} = H_{jj'} + \sum^{B}_{\gamma eq j,j'} \frac{H^{'}_{j\gamma}H^{'}_{\gamma j'}}{E_0-E_{\gamma}} , : H^{'}_{j\gamma} = \left \langle u_{j0} \right | \frac{\hbar}{m_0} \mathbf{k} \cdot \left ( \mathbf{p} + \frac{\hbar}{4 m_0 c^2} \bar{\sigma} \times abla V \right ) \left | u_{\gamma 0} \right \rangle \approx \sum_{\alpha} \frac{\hbar k_{\alpha}}{m_0}p^{\alpha}_{j \gamma}. Note that the lowest energy corresponds to the symmetric solution d_+. Ytterbium hydride is the hydride of ytterbium with the chemical formula YbH2. We now define the following parameters : A_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{x}_{\gamma x} }{ E_0-E_{\gamma} }, : B_0 = \frac{\hbar^2}{2 m_0} + \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{y}_{x\gamma}p^{y}_{\gamma x} }{ E_0-E_{\gamma} }, : C_0 = \frac{\hbar^2}{m_0^2} \sum^{B}_{\gamma} \frac{ p^{x}_{x\gamma}p^{y}_{\gamma y} + p^{y}_{x\gamma}p^{x}_{\gamma y} }{ E_0-E_{\gamma} }, and the band structure parameters (or the Luttinger parameters) can be defined to be : \gamma_1 = - \frac{1}{3} \frac{2 m_0}{\hbar^2} (A_0 + 2B_0), : \gamma_2 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} (A_0 - B_0), : \gamma_3 = - \frac{1}{6} \frac{2 m_0}{\hbar^2} C_0, These parameters are very closely related to the effective masses of the holes in various valence bands. \gamma_1 and \gamma_2 describe the coupling of the |X \rangle , |Y \rangle and |Z \rangle states to the other states. Keeping in mind the relationship of this model with its three-dimensional molecular counterpart, we use atomic units and set \hbar = m = 1. In this compound, the ytterbium atom has an oxidation state of +2 and the hydrogen atoms have an oxidation state of -1. In this model the influence of all other bands is taken into account by using Löwdin's perturbation method. == Background == All bands can be subdivided into two classes: * Class A: six valence bands (heavy hole, light hole, split off band and their spin counterparts) and two conduction bands. ",6.0,14.5115,"""1.2""",7.97,432.07,E
+"The contribution of molecular vibrations to the molar internal energy $U_{\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\mathrm{m}, \mathrm{vib}}=R \sum_{s=1}^{3 N-6} \theta_s /\left(e^{\theta_s / T}-1\right)$, where $\theta_s \equiv h \nu_s / k$ and $\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\mathrm{m}, \text { vib }}$ at $25^{\circ} \mathrm{C}$ of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$.","In terms of the vibrational wavenumbers we can write the partition function as Q_\text{vib}(T) = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{ h c \tilde{ u}_j}{k_\text{B} T}} } It is convenient to define a characteristic vibrational temperature \Theta_{i,\text{vib}} = \frac{h u_i}{k_\text{B}} where u is experimentally determined for each vibrational mode by taking a spectrum or by calculation. Q_\text{vib}(T) =\prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} = \prod_j e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}} \sum_n \left( e^{-\frac{\hbar \omega_j}{k_\text{B} T}} \right)^n = \prod_j \frac{e^{-\frac{\hbar \omega_j}{2 k_\text{B} T}}}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } = e^{- \frac{E_\text{ZP}}{k_\text{B} T}} \prod_j \frac{1}{ 1 - e^{-\frac{\hbar \omega_j}{k_\text{B} T}} } where E_\text{ZP} = \frac{1}{2} \sum_j \hbar \omega_j is total vibrational zero point energy of the system. Using this approximation we can derive a closed form expression for the vibrational partition function. For an isolated molecule of n atoms, the number of vibrational modes (i.e. values of j) is 3n − 5 for linear molecules and 3n − 6 for non-linear ones.G. Herzberg, Infrared and Raman Spectra, Van Nostrand Reinhold, 1945 In crystals, the vibrational normal modes are commonly known as phonons. ==Approximations== ===Quantum harmonic oscillator=== The most common approximation to the vibrational partition function uses a model in which the vibrational eigenmodes or normal modes of the system are considered to be a set of uncoupled quantum harmonic oscillators. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics The vibrational partition function is only well-defined in model systems where the vibrational motion is relatively uncoupled with the system's other degrees of freedom. ==Definition== For a system (such as a molecule or solid) with uncoupled vibrational modes the vibrational partition function is defined by Q_\text{vib}(T) = \prod_j{\sum_n{e^{-\frac{E_{j,n}}{k_\text{B} T}}}} where T is the absolute temperature of the system, k_B is the Boltzmann constant, and E_{j,n} is the energy of j-th mode when it has vibrational quantum number n = 0, 1, 2, \ldots . The second formula is adequate for small values of the vibrational quantum number. Number of degrees of vibrational freedom for nonlinear molecules: 3N-6 Number of degrees of vibrational freedom for linear molecules: 3N-5Housecroft, Catherine E., and A. G. Sharpe. It has also been applied to the study of unstable molecules such as dicarbon, C2, in discharges, flames and astronomical objects.Hollas, p. 211. == Principles == Electronic transitions are typically observed in the visible and ultraviolet regions, in the wavelength range approximately 200–700 nm (50,000–14,000 cm−1), whereas fundamental vibrations are observed below about 4000 cm−1.Energy is related to wavenumber by E=hc \bar u, where h=Planck's constant and c is the velocity of light When the electronic and vibrational energy changes are so different, vibronic coupling (mixing of electronic and vibrational wave functions) can be neglected and the energy of a vibronic level can be taken as the sum of the electronic and vibrational (and rotational) energies; that is, the Born–Oppenheimer approximation applies.Banwell and McCash, p. 162. Vibronic spectra of diatomic molecules in the gas phase have been analyzed in detail.Hollas, pp. 210–228 Vibrational coarse structure can sometimes be observed in the spectra of molecules in liquid or solid phases and of molecules in solution. The molecule is excited to another electronic state and to many possible vibrational states v'=0, 1, 2, 3, ... . The vibrational temperature is used commonly when finding the vibrational partition function. The overall molecular energy depends not only on the electronic state but also on vibrational and rotational quantum numbers, denoted v and J respectively for diatomic molecules. The transition energies, expressed in wavenumbers, of the lines for a particular vibronic transition are given, in the rigid rotor approximation, that is, ignoring centrifugal distortion, byBanwell and McCash, p. 171 :G(J^\prime, J^{\prime \prime}) = \bar u _{v^\prime-v^{\prime\prime}}+B^\prime J^\prime (J^\prime +1)-B^{\prime\prime} J^{\prime\prime}(J^{\prime\prime} +1) Here B are rotational constants and J are rotational quantum numbers. In the next approximation the term values are given by : G(v) = \bar u _{electronic} + \omega_e (v+{1 \over 2}) - \omega_e\chi_e (v+{1 \over 2})^2\, where χe is an anharmonicity constant. It is a first order approximation to the partition function which allows one to calculate the contribution of the vibrational degrees of freedom of molecules towards its thermodynamic variables. The Renner–Teller effect is observed in the spectra of molecules having electronic states that allow vibration through a linear configuration. Later studies on the same anion were also able to account for vibronic transitions involving low-frequency lattice vibrations. == Notes == == References == == Bibliography == * Chapter: Molecular Spectroscopy 2. : R_{\rm specific} = \frac{R}{M} Just as the molar gas constant can be related to the Boltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by the molecular mass of the gas. To determine the vibrational spectroscopy of linear molecules, the rotation and vibration of linear molecules are taken into account to predict which vibrational (normal) modes are active in the infrared spectrum and the Raman spectrum. == Degrees of freedom == The location of a molecule in a 3-dimensional space can be described by the total number of coordinates. Value of R Unit SI units J⋅K−1⋅mol−1 m3⋅Pa⋅K−1⋅mol−1 kg⋅m2⋅s−2⋅K−1⋅mol−1 Other common units L⋅Pa⋅K−1⋅mol−1 L⋅kPa⋅K−1⋅mol−1 L⋅bar⋅K−1⋅mol−1 erg⋅K−1⋅mol−1 atm⋅ft3⋅lbmol−1⋅°R−1 psi⋅ft3⋅lbmol−1⋅°R−1 BTU⋅lbmol−1⋅°R−1 inH2O⋅ft3⋅lbmol−1⋅°R−1 torr⋅ft3⋅lbmol−1⋅°R−1 L⋅atm⋅K−1⋅mol−1 L⋅Torr⋅K−1⋅mol−1 cal⋅K−1⋅mol−1 m3⋅atm⋅K−1⋅mol−1 The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . The potential at infinite internuclear distance is the dissociation energy for pure vibrational spectra. ",0.75,16,"""6.0""",0.14,10,D
+Calculate the magnitude of the spin magnetic moment of an electron.,"In turn, calculation of the magnitude of the total spin magnetic moment requires that () be replaced by: Thus, for a single electron, with spin quantum number the component of the magnetic moment along the field direction is, from (), while the (magnitude of the) total spin magnetic moment is, from (), or approximately 1.73 μ. The component of the orbital magnetic dipole moment for an electron with a magnetic quantum number ℓ is given by :(\boldsymbol{\mu}_\text{L})_z = -\mu_\text{B} m_\ell. ==History== The electron magnetic moment is intrinsically connected to electron spin and was first hypothesized during the early models of the atom in the early twentieth century. Again it is important to notice that is a negative constant multiplied by the spin, so the magnetic moment of the electron is antiparallel to the spin. : Number of unpaired electrons Spin-only moment () 1 1.73 2 2.83 3 3.87 4 4.90 5 5.92 === Elementary particles === In atomic and nuclear physics, the Greek symbol represents the magnitude of the magnetic moment, often measured in Bohr magnetons or nuclear magnetons, associated with the intrinsic spin of the particle and/or with the orbital motion of the particle in a system. Note that is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum. In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The analysis is readily extended to the spin-only magnetic moment of an atom. The spin magnetic dipole moment is approximately one B because g_{\rm s} \approx 2 and the electron is a spin- particle (): The component of the electron magnetic moment is (\boldsymbol{\mu}_\text{s})_z = -g_\text{s}\,\mu_\text{B}\,m_\text{s}\,, where s is the spin quantum number. The spin frequency of the electron is determined by the -factor. : u_s = \frac{g}{2} u_c : \frac{g}{2} = \frac{\bar{ u}_c + \bar{ u}_a}{\bar{ u}_c} ==See also== * Spin (physics) * Electron precipitation * Bohr magneton * Nuclear magnetic moment * Nucleon magnetic moment * Anomalous magnetic dipole moment * Electron electric dipole moment * Fine structure * Hyperfine structure ==References== ==Bibliography== * * Category:Atomic physics Category:Electric dipole moment Category:Magnetic moment Category:Particle physics Category:Physical constants Values of the intrinsic magnetic moments of some particles are given in the table below: : Intrinsic magnetic moments and spins of some elementary particles Particle name (symbol) Magnetic dipole moment (10 J⋅T) Spin quantum number (dimensionless) electron (e−) proton (H+) neutron (n) muon (μ) deuteron (H) 1 triton (H) helion (He) alpha particle (He) 0 0 For the relation between the notions of magnetic moment and magnetization see magnetization. == See also == * Moment (physics) * Electric dipole moment * Toroidal dipole moment * Magnetic susceptibility * Orbital magnetization * Magnetic dipole–dipole interaction == References and notes == ==External links== * Category:Magnetostatics Category:Magnetism Category:Electric and magnetic fields in matter Category:Physical quantities Category:Moment (physics) Category:Magnetic moment Here is the electron spin angular momentum. The magnetic moment of such a particle is parallel to its spin. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with angular momentum , its magnetic dipole moment is given by: \boldsymbol{\mu} = \frac{-e}{2m_\text{e}}\,\mathbf{L}\,, where e is the electron rest mass. In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. The CODATA value for the electron magnetic moment is : ===Orbital magnetic dipole moment=== The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. The magnetic moment of the electron is : \mathbf{m}_\text{S} = -\frac{g_\text{S} \mu_\text{B} \mathbf{S}}{\hbar}, where is the Bohr magneton, is electron spin, and the g-factor is 2 according to Dirac's theory, but due to quantum electrodynamic effects it is slightly larger in reality: . However, in order to obtain the magnitude of the total spin angular momentum, be replaced by its eigenvalue, where s is the spin quantum number. The magnetic moment of the electron has been measured using a one-electron quantum cyclotron and quantum nondemolition spectroscopy. The value of the electron magnetic moment (symbol μe) is In units of the Bohr magneton (μB), it is , a value that was measured with a relative accuracy of . ==Magnetic moment of an electron== The electron is a charged particle with charge −, where is the unit of elementary charge. The sum of the proton and neutron magnetic moments gives 0.879 µN, which is within 3% of the measured value 0.857 µN. The ratio between the true spin magnetic moment and that predicted by this model is a dimensionless factor , known as the electron -factor: \boldsymbol{\mu} = g_\text{e}\,\frac{(-e)}{~2m_\text{e}~}\,\mathbf{L}\,. The intrinsic electron magnetic dipole moment is approximately equal to the Bohr magneton μ because and the electron's spin is also : \frac{\hbar}{2} = - \mu_\text{B}|}} Equation () is therefore normally written as: Just like the total spin angular momentum cannot be measured, neither can the total spin magnetic moment be measured. ",0.0245,1.61,"""-0.16""",5040,4,B
+"A particle is subject to the potential energy $V=a x^4+b y^4+c z^4$. If its ground-state energy is $10 \mathrm{eV}$, calculate $\langle V\rangle$ for the ground state.","The average energy in this state would be \langle\psi|H|\psi\rangle = \int dx\, \left(-\frac{\hbar^2}{2m} \psi^* \frac{d^2\psi}{dx^2} + V(x)|\psi(x)|^2\right), where is the potential. The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. Formulae are given in SI units and Gaussian-cgs units. == Definition == The electromagnetic four- potential can be defined as: : SI units Gaussian units A^\alpha = \left( \frac{1}{c}\phi, \mathbf{A} \right)\,\\! For hydrogen (H), an electron in the ground state has energy , relative to the ionization threshold. Also electronvolts may be used, 1 eV = 1.602×10−19 Joules. ==Electrostatic potential energy of one point charge== ===One point charge q in the presence of another point charge Q=== right|A point charge q in the electric field of another charge Q.|thumb|434px The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: where k_\text{e} = \frac{1}{4\pi\varepsilon_0} is the Coulomb constant, r is the distance between the point charges q and Q, and q and Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). An excited state is any state with energy greater than the ground state. An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. In quantum field theory, the ground state is usually called the vacuum state or the vacuum. The solutions are outlined in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. === Vacuum case states === Let us now consider V(r) = 0. Therefore, the potential energy is unchanged up to order \varepsilon^2, if we deform the state \psi with a node into a state without a node, and the change can be ignored. Electric potential energy is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. Now, consider the potential energy. The correction to the potential V(r) is called the centrifugal barrier term. An object may be said to have electric potential energy by virtue of either its own electric charge or its relative position to other electrically charged objects. We can therefore remove all nodes and reduce the energy by O(\varepsilon), which implies that cannot be the ground state. However, the contribution to the potential energy from this region for the state with a node is V^\varepsilon_\text{avg} = \int_{-\varepsilon}^\varepsilon dx\, V(x)|\psi|^2 = |c|^2\int_{-\varepsilon}^\varepsilon dx\, x^2V(x) \simeq \frac{2}{3}\varepsilon^3|c|^2 V(0) + \cdots, lower, but still of the same lower order O(\varepsilon^3) as for the deformed state , and subdominant to the lowering of the average kinetic energy. While both the scalar and vector potential depend upon the frame, the electromagnetic four-potential is Lorentz covariant. When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb's law can be used. In other words, 13.6 eV is the energy input required for the electron to no longer be bound to the atom. (The average energy may then be further lowered by eliminating undulations, to the variational absolute minimum.) === Implication === As the ground state has no nodes it is spatially non- degenerate, i.e. there are no two stationary quantum states with the energy eigenvalue of the ground state (let's name it E_g) and the same spin state and therefore would only differ in their position-space wave functions. The term ""electric potential energy"" is used to describe the potential energy in systems with time-variant electric fields, while the term ""electrostatic potential energy"" is used to describe the potential energy in systems with time-invariant electric fields. ==Definition== The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. The electrostatic potential energy can also be defined from the electric potential as follows: ==Units== The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule). ",0.14,3.333333333,"""17.7""",460.5,0.0625,B
+"For an electron in a certain rectangular well with a depth of $20.0 \mathrm{eV}$, the lowest energy level lies $3.00 \mathrm{eV}$ above the bottom of the well. Find the width of this well. Hint: Use $\tan \theta=\sin \theta / \cos \theta$","The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. A potential well is the region surrounding a local minimum of potential energy. Depth in a well is not necessarily measured vertically or along a straight line. In the oil and gas industry, depth in a well is the measurement, for any point in that well, of the distance between a reference point or elevation, and that point. Specification of a differential depth or a thickness: in Figure 2 above, the thickness of the reservoir penetrated by the well might be 57 mMD or 42 mTVD, even though the reservoir true stratigraphic thickness in that area (or isopach) might be only 10 m, and its true vertical thickness (isochore), 14 m. ==See also== *Measured depth *True vertical depth ==References== ==External links== * Determining Lowest Astronomical Tide (LAT) * Seas and Submerged Lands Act 1973 (Australia) * Log Data Acquisition and Quality Control, Ph. A potential hill is the opposite of a potential well, and is the region surrounding a local maximum. ==Quantum confinement== thumb|500 px|Quantum confinement is responsible for the increase of energy difference between energy states and band gap, a phenomenon tightly related to the optical and electronic properties of the materials. But there, this additional width is interpreted as energy dispersion, which is, to the first order, |\Delta r_\pi|_\varepsilon = 2R_P\,\Delta E/E_P. Because wells are not always drilled vertically, there may be two ""depths"" for every given point in a wellbore: the measured depth (MD) measured along the path of the borehole, and the true vertical depth (TVD), the absolute vertical distance between the datum and the point in the wellbore. Well depth may refer to: *Depth in a well, a measurement of location in oil and gas drilling and production *The charge capacity of each pixel in a charge-coupled device 350px|right|thumb In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called ""quantum tunneling"") and wave-mechanical reflection. The total energy of the configuration is defined as zero because the charge of the electron is subject to no electric field due to other sources of charge. Configurations reproduced in * * * This webpage contains many more electron configurations with the lowest known energy: https://www.hars.us. Category:Electrostatics Category:Electron Category:Circle packing Category:Unsolved problems in mathematics A well may reach to many kilometers.Sakhalin-1 sets new extended reach drilling record, Rosneft says, 2015 ==Figures== thumb|left|Fig. 1: The specification of depthsthumb|right|Fig. 2: Differential depths: reservoir thickness, isochor, isopach Specification of an absolute depth: in Figure 1 above, point P1 might be at 3207 mMDRT and 2370 mTVDMSL, while point P2 might be at 2530 mMDRT and 2502 mTVDLAT. The problem consists of solving the one- dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. For example, a spherical shell of N=1 represents the uniform distribution of a single electron's charge, -e across the entire shell. ===Randomly distributed point charges=== The global energy of a system of electrons distributed in a purely random manner across the surface of the sphere is given by :U_{\text{rand}}(N)=\frac{N(N-1)}{2} and is, in general, greater than the energy of every Thomson problem solution. The energy of a continuous spherical shell of charge distributed across its surface is given by :U_{\text{shell}}(N)=\frac{N^2}{2} and is, in general, greater than the energy of every Thomson problem solution. The angular spread, while also worsening the energy resolution, shows some focusing as the equal negative and positive deviations map to the same final spot. center|thumb|upright=3|Distance from the central trajectory at the exit of a hemispherical electron energy analyzer depending on the electron's kinetic energy, initial position within the 1 mm slit, and the angle at which it enters the radial field after passing through the slit. The focus since the millennium has been on local optimization methods applied to the energy function, although random walks have made their appearance: * constrained global optimization (Altschuler et al. 1994), * steepest descent (Claxton and Benson 1966, Erber and Hockney 1991), * random walk (Weinrach et al. 1990), * genetic algorithm (Morris et al. 1996) While the objective is to minimize the global electrostatic potential energy of each N-electron case, several algorithmic starting cases are of interest. ===Continuous spherical shell charge=== thumb|The extreme upper energy limit of the Thomson Problem is given by N^2/2 for a continuous shell charge followed by N(N − 1)/2, the energy associated with a random distribution of N electrons. Initial angular spread, dependent on the chosen slit and aperture width, worsens the energy resolution.|alt= When two voltages, V_{1} and V_{2}, are applied to the inner and outer hemispheres, respectively, the electric potential in the region between the two electrodes follows from the Laplace equation: : V(r)= - \left[\frac{V_{2}-V_{1}}{R_{2}-R_{1}}\right]\cdot\frac{R_{1}R_{2}}{r} + const. In these cases the measured depth will continue to increase while true vertical depth will decrease toward the toe of the wellbore. ==Depth in practice== * Unit: the usual unit of depth is the metre (m). alt=|thumb|upright=1.2|Hemispherical electron energy analyzer. thumb|400px|right|A generic potential energy well. ",0.264,9,"""3.0""",-2,2,A
+Calculate the uncertainty $\Delta L_z$ for the hydrogen-atom stationary state: $2 p_z$.,"H_x+H_p \approx 0.69 + 0.53 = 1.22 >\ln\left(\frac{e}{2}\right)-\ln 1 \approx 0.31 ==Uncertainty relation with three angular momentum components== For a particle of spin-j the following uncertainty relation holds \sigma_{J_x}^2+\sigma_{J_y}^2+\sigma_{J_z}^2\ge j, where J_l are angular momentum components. H_p = -\sum_{j=-\infty}^\infty \operatorname P[p_j] \ln \operatorname P[p_j] = -\operatorname P[p_0] \ln \operatorname P[p_0]-2 \cdot \sum_{j=1}^{\infty} \operatorname P[p_j] \ln \operatorname P[p_j] \approx 0.53 The entropic uncertainty is indeed larger than the limiting value. Entropic uncertainty of the normal distribution We demonstrate this method on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. Under the above definition, the entropic uncertainty relation is H_x + H_p > \ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x \delta p}{h} \right). Its value is The number in parenthesis denotes the uncertainty of the last digits. ==Definition and value== The Bohr radius is defined asDavid J. Griffiths, Introduction to Quantum Mechanics, Prentice- Hall, 1995, p. 137. a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{e^2 m_{\text{e}}} = \frac{\varepsilon_0 h^2}{\pi e^2 m_{\text{e}}} = \frac{\hbar}{m_{\text{e}} c \alpha} , where * \varepsilon_0 is the permittivity of free space, * \hbar is the reduced Planck constant, * m_{\text{e}} is the mass of an electron, * e is the elementary charge, * c is the speed of light in vacuum, and * \alpha is the fine-structure constant. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. Since |\psi(x)|^2 is a probability density function for position, we calculate its standard deviation. Normal distribution example We demonstrate this method first on the ground state of the QHO, which as discussed above saturates the usual uncertainty based on standard deviations. \psi(x)=\left(\frac{m \omega}{\pi \hbar}\right)^{1/4} \exp{\left( -\frac{m \omega x^2}{2\hbar}\right)} The probability of lying within one of these bins can be expressed in terms of the error function. \begin{align} \operatorname P[x_j] &= \sqrt{\frac{m \omega}{\pi \hbar}} \int_{(j-1/2)\delta x}^{(j+1/2)\delta x} \exp\left( -\frac{m \omega x^2}{\hbar}\right) \, dx \\\ &= \sqrt{\frac{1}{\pi}} \int_{(j-1/2)\delta x\sqrt{m \omega / \hbar}}^{(j+1/2)\delta x\sqrt{m \omega / \hbar}} e^{u^2} \, du \\\ &= \frac{1}{2} \left[ \operatorname{erf} \left( \left(j+\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \sqrt{\frac{m \omega}{\hbar}}\right) \right] \end{align} The momentum probabilities are completely analogous. \operatorname P[p_j] = \frac{1}{2} \left[ \operatorname{erf} \left( \left(j+\frac{1}{2}\right)\delta p \cdot \frac{1}{\sqrt{\hbar m \omega}}\right)- \operatorname{erf} \left( \left(j-\frac{1}{2}\right)\delta x \cdot \frac{1}{\sqrt{\hbar m \omega}}\right) \right] For simplicity, we will set the resolutions to \delta x = \sqrt{\frac{h}{m \omega}} \delta p = \sqrt{h m \omega} so that the probabilities reduce to \operatorname P[x_j] = \operatorname P[p_j] = \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] The Shannon entropy can be evaluated numerically. \begin{align} H_x = H_p &= -\sum_{j=-\infty}^\infty \operatorname P[x_j] \ln \operatorname P[x_j] \\\ &= -\sum_{j=-\infty}^\infty \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \ln \frac{1}{2} \left[ \operatorname {erf} \left( \left(j+\frac{1}{2}\right) \sqrt{2\pi} \right)- \operatorname {erf} \left( \left(j-\frac{1}{2}\right) \sqrt{2\pi} \right) \right] \\\ &\approx 0.3226 \end{align} The entropic uncertainty is indeed larger than the limiting value. 180px|thumb|right|Diagram of a helium atom, showing the electron probability density as shades of gray. This precision may be quantified by the standard deviations, \sigma_x=\sqrt{\langle \hat{x}^2 \rangle-\langle \hat{x}\rangle^2} \sigma_p=\sqrt{\langle \hat{p}^2 \rangle-\langle \hat{p}\rangle^2}. In his celebrated 1927 paper, ""Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik"" (""On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics""), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement, but he did not give a precise definition for the uncertainties Δx and Δp. When applying the BO approximation, two smaller, consecutive steps can be used: For a given position of the nuclei, the electronic Schrödinger equation is solved, while treating the nuclei as stationary (not ""coupled"" with the dynamics of the electrons). In the published 1927 paper, Heisenberg originally concluded that the uncertainty principle was ΔpΔq ≈ h using the full Planck constant.Werner Heisenberg, Encounters with Einstein and Other Essays on People, Places and Particles, Published October 21st 1989 by Princeton University Press, p.53.Kumar, Manjit. ""On the energy- time uncertainty relation. Orbital uncertainty is related to several parameters used in the orbit determination process including the number of observations (measurements), the time spanned by those observations (observation arc), the quality of the observations (e.g. radar vs. optical), and the geometry of the observations. The matrix element in the numerator is : \langle\chi_{k'}| [P_{A\alpha}, H_\mathrm{e}] |\chi_k\rangle_{(\mathbf{r})} = iZ_A\sum_i \left\langle\chi_{k'}\left|\frac{(\mathbf{r}_{iA})_\alpha}{r_{iA}^3}\right|\chi_k\right\rangle_{(\mathbf{r})} \quad\text{with}\quad \mathbf{r}_{iA} \equiv \mathbf{r}_i - \mathbf{R}_A. The variances of x and p can be calculated explicitly: \sigma_x^2=\frac{L^2}{12}\left(1-\frac{6}{n^2\pi^2}\right) \sigma_p^2=\left(\frac{\hbar n\pi}{L}\right)^2. On the other hand, the standard deviation of the position is \sigma_x = \frac{x_0}{\sqrt{2}} \sqrt{1+\omega_0^2 t^2} such that the uncertainty product can only increase with time as \sigma_x(t) \sigma_p(t) = \frac{\hbar}{2} \sqrt{1+\omega_0^2 t^2} ==Additional uncertainty relations== ===Systematic and statistical errors=== The inequalities above focus on the statistical imprecision of observables as quantified by the standard deviation \sigma. Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). Everett's Dissertation proven in 1975 by W. Beckner and in the same year interpreted as a generalized quantum mechanical uncertainty principle by Białynicki-Birula and Mycielski. The second stronger uncertainty relation is given by \sigma_A^2 + \sigma_B^2 \ge \frac{1}{2}| \langle {\bar \Psi}_{A+B} \mid(A + B)\mid \Psi \rangle|^2 where | {\bar \Psi}_{A+B} \rangle is a state orthogonal to |\Psi \rangle . The product of the standard deviations is therefore \sigma_x \sigma_p = \frac{\hbar}{2} \sqrt{\frac{n^2\pi^2}{3}-2}. ", 9.73,0.064,"""0.0""",8,61,C
 "5.8-5. If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when
 (a) $n=100$.
-","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. The lower bound is expressed in terms of the probabilities for pairs of events. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... thumb|Plot of S_n/n (red), its standard deviation 1/\sqrt{n} (blue) and its bound \sqrt{2\log\log n/n} given by LIL (green). right|thumb|300px| Probability mass function for Fisher's noncentral hypergeometric distribution for different values of the odds ratio ω. m1 = 80, m2 = 60, n = 100, ω = 0.01, ..., 1000 In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The probability function and a simple approximation to the mean are given to the right. Bound the desired probability using the Chebyshev inequality: :\operatorname{P}\left(|T- n H_n| \geq cn\right) \le \frac{\pi^2}{6c^2}. ===Tail estimates=== A stronger tail estimate for the upper tail be obtained as follows. Probability. Then : \begin{align} P\left [ {Z}_i^r \right ] = \left(1-\frac{1}{n}\right)^r \le e^{-r / n}. \end{align} Thus, for r = \beta n \log n, we have P\left [ {Z}_i^r \right ] \le e^{(-\beta n \log n ) / n} = n^{-\beta}. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Their odds ratio is given as : \omega = \frac{\omega_X}{\omega_Y} = \frac{\pi_X/(1-\pi_X)}{\pi_Y/(1-\pi_Y)} . It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). The commonly used approximate value of 1.96 is therefore accurate to better than one part in 50,000, which is more than adequate for applied work. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons. ==Solution== ===Calculating the expectation=== Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. The trial can be summarized and analyzed in terms of the following contingency table. responder non-responder Total X x . mX Y y . mY Total n . ",0.2,6.3,0.25,0.8185,0.5,C
+","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. The lower bound is expressed in terms of the probabilities for pairs of events. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... thumb|Plot of S_n/n (red), its standard deviation 1/\sqrt{n} (blue) and its bound \sqrt{2\log\log n/n} given by LIL (green). right|thumb|300px| Probability mass function for Fisher's noncentral hypergeometric distribution for different values of the odds ratio ω. m1 = 80, m2 = 60, n = 100, ω = 0.01, ..., 1000 In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The probability function and a simple approximation to the mean are given to the right. Bound the desired probability using the Chebyshev inequality: :\operatorname{P}\left(|T- n H_n| \geq cn\right) \le \frac{\pi^2}{6c^2}. ===Tail estimates=== A stronger tail estimate for the upper tail be obtained as follows. Probability. Then : \begin{align} P\left [ {Z}_i^r \right ] = \left(1-\frac{1}{n}\right)^r \le e^{-r / n}. \end{align} Thus, for r = \beta n \log n, we have P\left [ {Z}_i^r \right ] \le e^{(-\beta n \log n ) / n} = n^{-\beta}. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Their odds ratio is given as : \omega = \frac{\omega_X}{\omega_Y} = \frac{\pi_X/(1-\pi_X)}{\pi_Y/(1-\pi_Y)} . It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). The commonly used approximate value of 1.96 is therefore accurate to better than one part in 50,000, which is more than adequate for applied work. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons. ==Solution== ===Calculating the expectation=== Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. The trial can be summarized and analyzed in terms of the following contingency table. responder non-responder Total X x . mX Y y . mY Total n . ",0.2,6.3,"""0.25""",0.8185,0.5,C
 "5.3-13. A device contains three components, each of which has a lifetime in hours with the pdf
 $$
 f(x)=\frac{2 x}{10^2} e^{-(x / 10)^2}, \quad 0 < x < \infty .
 $$
-The device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).","Note that this is a conditional probability, where the condition is that no failure has occurred before time t. Although the failure rate, \lambda (t), is often thought of as the probability that a failure occurs in a specified interval given no failure before time t, it is not actually a probability because it can exceed 1. It is based on an exponential failure distribution (see failure rate for a full derivation). A continuous failure rate depends on the existence of a failure distribution, F(t), which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t, :\operatorname{Pr}(T\le t)=F(t)=1-R(t),\quad t\ge 0. \\! where {T} is the failure time. It can be defined with the aid of the reliability function, also called the survival function, R(t)=1-F(t), the probability of no failure before time t. ::\lambda(t) = \frac{f(t)}{R(t)}, where f(t) is the time to (first) failure distribution (i.e. the failure density function). ::\lambda(t) = \frac{R(t_1)-R(t_2)}{(t_2-t_1) \cdot R(t_1)} = \frac{R(t)-R(t+\Delta t)}{\Delta t \cdot R(t)} \\! over a time interval \Delta t = (t_2-t_1) from t_1 (or t) to t_2. Solving the differential equation :h(t)=\frac{f(t)}{1-F(t)}=\frac{F'(t)}{1-F(t)} for F(t), it can be shown that :F(t) = 1 - \exp{\left(-\int_0^t h(t) dt \right)}. ==Decreasing failure rate== A decreasing failure rate (DFR) describes a phenomenon where the probability of an event in a fixed time interval in the future decreases over time. The Failures In Time (FIT) rate of a device is the number of failures that can be expected in one billion (109) device-hours of operation. The failure rate of a system usually depends on time, with the rate varying over the life cycle of the system. The pdf for the standard fatigue life distribution reduces to : f(x) = \frac{\sqrt{x}+\sqrt{\frac{1}{x}}}{2\gamma x}\phi\left(\frac{\sqrt{x}-\sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma >0 Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function. ==Cumulative distribution function== The formula for the cumulative distribution function is : F(x) = \Phi\left(\frac{\sqrt{x} - \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma > 0 where Φ is the cumulative distribution function of the standard normal distribution. ==Quantile function== The formula for the quantile function is : G(p) = \frac{1}{4}\left[\gamma\Phi^{-1}(p) + \sqrt{4+\left(\gamma\Phi^{-1}(p)\right)^2}\right]^2 where Φ −1 is the quantile function of the standard normal distribution. ==References== * * * * * * * ==External links== *Fatigue life distribution Category:Continuous distributions The results are as follows: Estimated failure rate is : \frac{6\text{ failures}}{7502\text{ hours}} = 0.0007998\, \frac{\text{failures}}{\text{hour}} = 799.8 \times 10^{-6}\, \frac{\text{failures}}{\text{hour}}, or 799.8 failures for every million hours of operation. ==See also== *Annualized failure rate *Burn-in *Failure *Failure mode *Failure modes, effects, and diagnostic analysis *Force of mortality *Frequency of exceedance *Reliability engineering *Reliability theory *Reliability theory of aging and longevity *Survival analysis *Weibull distribution ==References== ==Further reading== * * * *Federal Standard 1037C * * * * * * * *U.S. Department of Defense, (1991) Military Handbook, “Reliability Prediction of Electronic Equipment, MIL-HDBK-217F, 2 ==External links== *Bathtub curve issues , ASQC *Fault Tolerant Computing in Industrial Automation by Hubert Kirrmann, ABB Research Center, Switzerland Category:Actuarial science Category:Engineering failures Category:Reliability engineering Category:Survival analysis Category:Maintenance Category:Statistical ratios Category:Error measures Category:Rates Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is ""memory-less""). The failure distribution function is the integral of the failure density function, f(t), :F(t)=\int_{0}^{t} f(\tau)\, d\tau. Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. Many probability distributions can be used to model the failure distribution (see List of important probability distributions). The Birnbaum-Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. Failures most commonly occur near the beginning and near the ending of the lifetime of the parts, resulting in the bathtub curve graph of failure rates. # Reliability of semiconductor devices may depend on assembly, use, environmental, and cooling conditions. X is then distributed normally with a mean of zero and a variance of α2 / 4. ==Probability density function== The general formula for the probability density function (pdf) is : f(x) = \frac{\sqrt{\frac{x-\mu}{\beta}}+\sqrt{\frac{\beta}{x-\mu}}}{2\gamma\left(x-\mu\right)}\phi\left(\frac{\sqrt{\frac{x-\mu}{\beta}}-\sqrt{\frac{\beta}{x-\mu}}}{\gamma}\right)\quad x > \mu; \gamma,\beta>0 where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and \phi is the probability density function of the standard normal distribution. ==Standard fatigue life distribution== The case where μ = 0 and β = 1 is called the standard fatigue life distribution. Failure rates are often expressed in engineering notation as failures per million, or 10−6, especially for individual components, since their failure rates are often very low. Reliability of semiconductor devices can be summarized as follows: # Semiconductor devices are very sensitive to impurities and particles. For many devices, the wear-out failure point is measured by the number of cycles performed before the device fails, and can be discovered by cycle testing. The failure can occur invisibly inside the packaging and is measurable. ",0.1800,41.40,0.03,109,5.4,C
-"5.6-13. The tensile strength $X$ of paper, in pounds per square inch, has $\mu=30$ and $\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\bar{X}$ is greater than 29.5 pounds per square inch.","thumb|300px|Probability density of stress S (red, top) and resistance R (blue, top), and of equality (m = R - S = 0, black, bottom). thumb|300px|Distribution of stress S and strength R: all the (R, S) situations have a probability density (grey level surface). The ""ISO 534:2011, Paper and board — Determination of thickness, density and specific volume"" indicates that the paper density is expressed in grams per cubic centimeter (g/cm3). ==See also== * Grammage * Density ** Area density ** Linear density * ==References== ==External links== * Paper Weight – Conversion Chart * Understanding Paper Weights * Understanding paper weight (Staples, Inc.) * M-weight Calculator * Paper Weight Calculator Category:Paper Category:Printing Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. thumb|Weighing scale to determine paper weight Paper density is a paper product's mass per unit volume. thumb|upright=1.3|Each row of points is a sample from the same normal distribution. :Human hair strength varies by ethnicity and chemical treatments. == Typical properties of annealed elements == Typical properties for annealed elementsA.M. Howatson, P. G. Lund, and J. D. Todd, Engineering Tables and Data, p. 41 Element Young's modulus (GPa) Yield strength (MPa) Ultimate strength (MPa) Silicon 107 5000–9000 Tungsten 411 550 550–620 Iron 211 80–100 350 Titanium 120 100–225 246–370 Copper 130 117 210 Tantalum 186 180 200 Tin 47 9–14 15–200 Zinc 85–105 200–400 200–400 Nickel 170 140–350 140–195 Silver 83 170 Gold 79 100 Aluminium 70 15–20 40–50 Lead 16 12 ==See also== *Flexural strength *Strength of materials *Tensile structure *Toughness *Failure *Tension (physics) *Young's modulus ==References== ==Further reading== *Giancoli, Douglas, Physics for Scientists & Engineers Third Edition (2000). Bond paper is a high-quality durable writing paper similar to bank paper but having a weight greater than 50 g/m2. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. The ultimate tensile strength is a common engineering parameter to design members made of brittle material because such materials have no yield point. ==Testing== thumb|Round bar specimen after tensile stress testing Typically, the testing involves taking a small sample with a fixed cross-sectional area, and then pulling it with a tensometer at a constant strain (change in gauge length divided by initial gauge length) rate until the sample breaks. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. Suppose we wanted to calculate a 95% confidence interval for μ. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... They are tabulated for common materials such as alloys, composite materials, ceramics, plastics, and wood. == Definition == The ultimate tensile strength of a material is an intensive property; therefore its value does not depend on the size of the test specimen. Tensile strength is defined as a stress, which is measured as force per unit area. Environmental stresses have a distribution with a mean \left(\mu_x\right) and a standard deviation \left(s_x\right) and component strengths have a distribution with a mean \left(\mu_y\right) and a standard deviation \left(s_y\right). This important relation permits economically important nondestructive testing of bulk metal deliveries with lightweight, even portable equipment, such as hand-held Rockwell hardness testers.E.J. Pavlina and C.J. Van Tyne, ""Correlation of Yield Strength and Tensile Strength with Hardness for Steels"", Journal of Materials Engineering and Performance, 17:6 (December 2008) This practical correlation helps quality assurance in metalworking industries to extend well beyond the laboratory and universal testing machines. ==Typical tensile strengths== Typical tensile strengths of some materials Material Yield strength (MPa) Ultimate tensile strength (MPa) Density (g/cm3) Steel, structural ASTM A36 steel 250 400–550 7.8 Steel, 1090 mild 247 841 7.58 Chromium-vanadium steel AISI 6150 620 940 7.8 Steel, 2800 Maraging steel 2617 2693 8.00 Steel, AerMet 340 2160 2430 7.86 Steel, Sandvik Sanicro 36Mo logging cable precision wire 1758 2070 8.00 Steel, AISI 4130, water quenched 855 °C (1570 °F), 480 °C (900 °F) temper 951 1110 7.85 Steel, API 5L X65 448 531 7.8 Steel, high strength alloy ASTM A514 690 760 7.8 Acrylic, clear cast sheet (PMMA) IAPD Typical Properties of Acrylics 72 87strictly speaking this figure is the flexural strength (or modulus of rupture), which is a more appropriate measure for brittle materials than ""ultimate strength."" The ultimate tensile strength is usually found by performing a tensile test and recording the engineering stress versus strain. The density depends on the manufacturing method, and the lowest value is 0.037 or 0.55 (solid). The density can be calculated by dividing the grammage of paper (in grams per square metre or ""gsm"") by its caliper (usually in micrometres, occasionally in mils). ",0.6247,0.166666666,-1.78,0.9522,6.3,D
+The device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).","Note that this is a conditional probability, where the condition is that no failure has occurred before time t. Although the failure rate, \lambda (t), is often thought of as the probability that a failure occurs in a specified interval given no failure before time t, it is not actually a probability because it can exceed 1. It is based on an exponential failure distribution (see failure rate for a full derivation). A continuous failure rate depends on the existence of a failure distribution, F(t), which is a cumulative distribution function that describes the probability of failure (at least) up to and including time t, :\operatorname{Pr}(T\le t)=F(t)=1-R(t),\quad t\ge 0. \\! where {T} is the failure time. It can be defined with the aid of the reliability function, also called the survival function, R(t)=1-F(t), the probability of no failure before time t. ::\lambda(t) = \frac{f(t)}{R(t)}, where f(t) is the time to (first) failure distribution (i.e. the failure density function). ::\lambda(t) = \frac{R(t_1)-R(t_2)}{(t_2-t_1) \cdot R(t_1)} = \frac{R(t)-R(t+\Delta t)}{\Delta t \cdot R(t)} \\! over a time interval \Delta t = (t_2-t_1) from t_1 (or t) to t_2. Solving the differential equation :h(t)=\frac{f(t)}{1-F(t)}=\frac{F'(t)}{1-F(t)} for F(t), it can be shown that :F(t) = 1 - \exp{\left(-\int_0^t h(t) dt \right)}. ==Decreasing failure rate== A decreasing failure rate (DFR) describes a phenomenon where the probability of an event in a fixed time interval in the future decreases over time. The Failures In Time (FIT) rate of a device is the number of failures that can be expected in one billion (109) device-hours of operation. The failure rate of a system usually depends on time, with the rate varying over the life cycle of the system. The pdf for the standard fatigue life distribution reduces to : f(x) = \frac{\sqrt{x}+\sqrt{\frac{1}{x}}}{2\gamma x}\phi\left(\frac{\sqrt{x}-\sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma >0 Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function. ==Cumulative distribution function== The formula for the cumulative distribution function is : F(x) = \Phi\left(\frac{\sqrt{x} - \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma > 0 where Φ is the cumulative distribution function of the standard normal distribution. ==Quantile function== The formula for the quantile function is : G(p) = \frac{1}{4}\left[\gamma\Phi^{-1}(p) + \sqrt{4+\left(\gamma\Phi^{-1}(p)\right)^2}\right]^2 where Φ −1 is the quantile function of the standard normal distribution. ==References== * * * * * * * ==External links== *Fatigue life distribution Category:Continuous distributions The results are as follows: Estimated failure rate is : \frac{6\text{ failures}}{7502\text{ hours}} = 0.0007998\, \frac{\text{failures}}{\text{hour}} = 799.8 \times 10^{-6}\, \frac{\text{failures}}{\text{hour}}, or 799.8 failures for every million hours of operation. ==See also== *Annualized failure rate *Burn-in *Failure *Failure mode *Failure modes, effects, and diagnostic analysis *Force of mortality *Frequency of exceedance *Reliability engineering *Reliability theory *Reliability theory of aging and longevity *Survival analysis *Weibull distribution ==References== ==Further reading== * * * *Federal Standard 1037C * * * * * * * *U.S. Department of Defense, (1991) Military Handbook, “Reliability Prediction of Electronic Equipment, MIL-HDBK-217F, 2 ==External links== *Bathtub curve issues , ASQC *Fault Tolerant Computing in Industrial Automation by Hubert Kirrmann, ABB Research Center, Switzerland Category:Actuarial science Category:Engineering failures Category:Reliability engineering Category:Survival analysis Category:Maintenance Category:Statistical ratios Category:Error measures Category:Rates Thus, for an exponential failure distribution, the hazard rate is a constant with respect to time (that is, the distribution is ""memory-less""). The failure distribution function is the integral of the failure density function, f(t), :F(t)=\int_{0}^{t} f(\tau)\, d\tau. Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. Many probability distributions can be used to model the failure distribution (see List of important probability distributions). The Birnbaum-Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. Failures most commonly occur near the beginning and near the ending of the lifetime of the parts, resulting in the bathtub curve graph of failure rates. # Reliability of semiconductor devices may depend on assembly, use, environmental, and cooling conditions. X is then distributed normally with a mean of zero and a variance of α2 / 4. ==Probability density function== The general formula for the probability density function (pdf) is : f(x) = \frac{\sqrt{\frac{x-\mu}{\beta}}+\sqrt{\frac{\beta}{x-\mu}}}{2\gamma\left(x-\mu\right)}\phi\left(\frac{\sqrt{\frac{x-\mu}{\beta}}-\sqrt{\frac{\beta}{x-\mu}}}{\gamma}\right)\quad x > \mu; \gamma,\beta>0 where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and \phi is the probability density function of the standard normal distribution. ==Standard fatigue life distribution== The case where μ = 0 and β = 1 is called the standard fatigue life distribution. Failure rates are often expressed in engineering notation as failures per million, or 10−6, especially for individual components, since their failure rates are often very low. Reliability of semiconductor devices can be summarized as follows: # Semiconductor devices are very sensitive to impurities and particles. For many devices, the wear-out failure point is measured by the number of cycles performed before the device fails, and can be discovered by cycle testing. The failure can occur invisibly inside the packaging and is measurable. ",0.1800,41.40,"""0.03""",109,5.4,C
+"5.6-13. The tensile strength $X$ of paper, in pounds per square inch, has $\mu=30$ and $\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\bar{X}$ is greater than 29.5 pounds per square inch.","thumb|300px|Probability density of stress S (red, top) and resistance R (blue, top), and of equality (m = R - S = 0, black, bottom). thumb|300px|Distribution of stress S and strength R: all the (R, S) situations have a probability density (grey level surface). The ""ISO 534:2011, Paper and board — Determination of thickness, density and specific volume"" indicates that the paper density is expressed in grams per cubic centimeter (g/cm3). ==See also== * Grammage * Density ** Area density ** Linear density * ==References== ==External links== * Paper Weight – Conversion Chart * Understanding Paper Weights * Understanding paper weight (Staples, Inc.) * M-weight Calculator * Paper Weight Calculator Category:Paper Category:Printing Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. thumb|Weighing scale to determine paper weight Paper density is a paper product's mass per unit volume. thumb|upright=1.3|Each row of points is a sample from the same normal distribution. :Human hair strength varies by ethnicity and chemical treatments. == Typical properties of annealed elements == Typical properties for annealed elementsA.M. Howatson, P. G. Lund, and J. D. Todd, Engineering Tables and Data, p. 41 Element Young's modulus (GPa) Yield strength (MPa) Ultimate strength (MPa) Silicon 107 5000–9000 Tungsten 411 550 550–620 Iron 211 80–100 350 Titanium 120 100–225 246–370 Copper 130 117 210 Tantalum 186 180 200 Tin 47 9–14 15–200 Zinc 85–105 200–400 200–400 Nickel 170 140–350 140–195 Silver 83 170 Gold 79 100 Aluminium 70 15–20 40–50 Lead 16 12 ==See also== *Flexural strength *Strength of materials *Tensile structure *Toughness *Failure *Tension (physics) *Young's modulus ==References== ==Further reading== *Giancoli, Douglas, Physics for Scientists & Engineers Third Edition (2000). Bond paper is a high-quality durable writing paper similar to bank paper but having a weight greater than 50 g/m2. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. The ultimate tensile strength is a common engineering parameter to design members made of brittle material because such materials have no yield point. ==Testing== thumb|Round bar specimen after tensile stress testing Typically, the testing involves taking a small sample with a fixed cross-sectional area, and then pulling it with a tensometer at a constant strain (change in gauge length divided by initial gauge length) rate until the sample breaks. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. Suppose we wanted to calculate a 95% confidence interval for μ. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... They are tabulated for common materials such as alloys, composite materials, ceramics, plastics, and wood. == Definition == The ultimate tensile strength of a material is an intensive property; therefore its value does not depend on the size of the test specimen. Tensile strength is defined as a stress, which is measured as force per unit area. Environmental stresses have a distribution with a mean \left(\mu_x\right) and a standard deviation \left(s_x\right) and component strengths have a distribution with a mean \left(\mu_y\right) and a standard deviation \left(s_y\right). This important relation permits economically important nondestructive testing of bulk metal deliveries with lightweight, even portable equipment, such as hand-held Rockwell hardness testers.E.J. Pavlina and C.J. Van Tyne, ""Correlation of Yield Strength and Tensile Strength with Hardness for Steels"", Journal of Materials Engineering and Performance, 17:6 (December 2008) This practical correlation helps quality assurance in metalworking industries to extend well beyond the laboratory and universal testing machines. ==Typical tensile strengths== Typical tensile strengths of some materials Material Yield strength (MPa) Ultimate tensile strength (MPa) Density (g/cm3) Steel, structural ASTM A36 steel 250 400–550 7.8 Steel, 1090 mild 247 841 7.58 Chromium-vanadium steel AISI 6150 620 940 7.8 Steel, 2800 Maraging steel 2617 2693 8.00 Steel, AerMet 340 2160 2430 7.86 Steel, Sandvik Sanicro 36Mo logging cable precision wire 1758 2070 8.00 Steel, AISI 4130, water quenched 855 °C (1570 °F), 480 °C (900 °F) temper 951 1110 7.85 Steel, API 5L X65 448 531 7.8 Steel, high strength alloy ASTM A514 690 760 7.8 Acrylic, clear cast sheet (PMMA) IAPD Typical Properties of Acrylics 72 87strictly speaking this figure is the flexural strength (or modulus of rupture), which is a more appropriate measure for brittle materials than ""ultimate strength."" The ultimate tensile strength is usually found by performing a tensile test and recording the engineering stress versus strain. The density depends on the manufacturing method, and the lowest value is 0.037 or 0.55 (solid). The density can be calculated by dividing the grammage of paper (in grams per square metre or ""gsm"") by its caliper (usually in micrometres, occasionally in mils). ",0.6247,0.166666666,"""-1.78""",0.9522,6.3,D
 "5.6-3. Let $\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \leq \bar{X} \leq 4)$
-","The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin{align} \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end{align} The usefulness of this heuristic especially depends on the question under consideration. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545 This is related to confidence interval as used in statistics: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size n. ==Normality tests== The ""68–95–99.7 rule"" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. The assumed mean is the centre of the range from 174 to 177 which is 175.5. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... There are other rapid calculation methods which are more suited for computers which also ensure more accurate results than the obvious methods. ==Example== First: The mean of the following numbers is sought: : 219, 223, 226, 228, 231, 234, 235, 236, 240, 241, 244, 247, 249, 255, 262 Suppose we start with a plausible initial guess that the mean is about 240. Therefore, that is what we need to add to the assumed mean to get the correct mean: : correct mean = 240 − 2 = 238. ==Method== The method depends on estimating the mean and rounding to an easy value to calculate with. The average of these 15 deviations from the assumed mean is therefore −30/15 = −2\. We only need to calculate each integral for the cases n = 1,2,3. \begin{align} &\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{u^2}{2}}du \approx 0.6827 \\\ &\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) =\frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{u^2}{2}}du \approx 0.9545 \\\ &\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) = \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{u^2}{2}}du \approx 0.9973. \end{align} ==Cumulative distribution function== These numerical values ""68%, 95%, 99.7%"" come from the cumulative distribution function of the normal distribution. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. In the empirical sciences, the so-called three-sigma rule of thumb (or 3 rule) expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty.This usage of ""three-sigma rule"" entered common usage in the 2000s, e.g. cited in * * In the social sciences, a result may be considered ""significant"" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a discovery. In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. For a data set with assumed mean x0 suppose: :d_i=x_i-x_0 \, :A = \sum_{i=1}^N d_i \, :B = \sum_{i=1}^N d_i^2 \, :D = \frac{A}{N} \, Then :\overline{x} = x_0 + D \, :\sigma = \sqrt{\frac{B - N D^2}{N}} \, or for a sample standard deviation using Bessel's correction: :\sigma = \sqrt{\frac{ B - N D^2}{N-1}} \, ==Example using class ranges== Where there are a large number of samples a quick reasonable estimate of the mean and standard deviation can be got by grouping the samples into classes using equal size ranges. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. In statistical hypothesis testing, the error exponent of a hypothesis testing procedure is the rate at which the probabilities of Type I and Type II decay exponentially with the size of the sample used in the test. Then the deviations from this ""assumed"" mean are the following: :−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22 In adding these up, one finds that: : 22 and −21 almost cancel, leaving +1, : 15 and −17 almost cancel, leaving −2, : 9 and −9 cancel, : 7 + 4 cancels −6 − 5, and so on. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means In statistics the assumed mean is a method for calculating the arithmetic mean and standard deviation of a data set. ",0.70710678,0.8185,1.4,0.9974,1.61,B
+","The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin{align} \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end{align} The usefulness of this heuristic especially depends on the question under consideration. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545 This is related to confidence interval as used in statistics: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size n. ==Normality tests== The ""68–95–99.7 rule"" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. The assumed mean is the centre of the range from 174 to 177 which is 175.5. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... There are other rapid calculation methods which are more suited for computers which also ensure more accurate results than the obvious methods. ==Example== First: The mean of the following numbers is sought: : 219, 223, 226, 228, 231, 234, 235, 236, 240, 241, 244, 247, 249, 255, 262 Suppose we start with a plausible initial guess that the mean is about 240. Therefore, that is what we need to add to the assumed mean to get the correct mean: : correct mean = 240 − 2 = 238. ==Method== The method depends on estimating the mean and rounding to an easy value to calculate with. The average of these 15 deviations from the assumed mean is therefore −30/15 = −2\. We only need to calculate each integral for the cases n = 1,2,3. \begin{align} &\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{u^2}{2}}du \approx 0.6827 \\\ &\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) =\frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{u^2}{2}}du \approx 0.9545 \\\ &\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) = \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{u^2}{2}}du \approx 0.9973. \end{align} ==Cumulative distribution function== These numerical values ""68%, 95%, 99.7%"" come from the cumulative distribution function of the normal distribution. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. In the empirical sciences, the so-called three-sigma rule of thumb (or 3 rule) expresses a conventional heuristic that nearly all values are taken to lie within three standard deviations of the mean, and thus it is empirically useful to treat 99.7% probability as near certainty.This usage of ""three-sigma rule"" entered common usage in the 2000s, e.g. cited in * * In the social sciences, a result may be considered ""significant"" if its confidence level is of the order of a two-sigma effect (95%), while in particle physics, there is a convention of a five-sigma effect (99.99994% confidence) being required to qualify as a discovery. In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. For a data set with assumed mean x0 suppose: :d_i=x_i-x_0 \, :A = \sum_{i=1}^N d_i \, :B = \sum_{i=1}^N d_i^2 \, :D = \frac{A}{N} \, Then :\overline{x} = x_0 + D \, :\sigma = \sqrt{\frac{B - N D^2}{N}} \, or for a sample standard deviation using Bessel's correction: :\sigma = \sqrt{\frac{ B - N D^2}{N-1}} \, ==Example using class ranges== Where there are a large number of samples a quick reasonable estimate of the mean and standard deviation can be got by grouping the samples into classes using equal size ranges. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. In statistical hypothesis testing, the error exponent of a hypothesis testing procedure is the rate at which the probabilities of Type I and Type II decay exponentially with the size of the sample used in the test. Then the deviations from this ""assumed"" mean are the following: :−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22 In adding these up, one finds that: : 22 and −21 almost cancel, leaving +1, : 15 and −17 almost cancel, leaving −2, : 9 and −9 cancel, : 7 + 4 cancels −6 − 5, and so on. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. A weaker three-sigma rule can be derived from Chebyshev's inequality, stating that even for non-normally distributed variables, at least 88.8% of cases should fall within properly calculated three-sigma intervals. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means In statistics the assumed mean is a method for calculating the arithmetic mean and standard deviation of a data set. ",0.70710678,0.8185,"""1.4""",0.9974,1.61,B
 "5.3-9. Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine
-(a) $P\left(\max X_i < 3 / 4\right)=P\left(X_1<3 / 4, X_2<3 / 4\right)$","Thus if one has a sample \\{X_1,\dots,X_n\\}, and one picks another observation X_{n+1}, then this has 1/(n+1) probability of being the largest value seen so far, 1/(n+1) probability of being the smallest value seen so far, and thus the other (n-1)/(n+1) of the time, X_{n+1} falls between the sample maximum and sample minimum of \\{X_1,\dots,X_n\\}. * Generalized extreme value distribution, possible limit distributions of sample maximum (opposite question). Thus the sampling distribution of the quantile of the sample maximum is the graph x1/k from 0 to 1: the p-th to q-th quantile of the sample maximum m are the interval [p1/kN, q1/kN]. The minimum and the maximum value are the first and last order statistics (often denoted X(1) and X(n) respectively, for a sample size of n). In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. A related bound is Edelman's : P\left( \left| \sum_{ i = 1 }^n a_i X_i \right| \ge k \right) \le 2 \left( 1 - \Phi\left[ k - \frac{ 1.5 }{ k } \right] \right) = 2 B_{ Ed }( k ) , where Φ(x) is cumulative distribution function of the standard normal distribution. A derivation of the expected value and the variance of the sample maximum are shown in the page of the discrete uniform distribution. Ann Probab 22(4):1679–1706 : P( S_n \ge x ) \le \frac{ 2e^3 }{ 9 } P( Z \ge x ) The constant in the last inequality is approximately 4.4634. Eaton showed that : P\left( \left| \sum_{ i = 1 }^n a_i X_i \right| \ge k \right) \le 2 \inf_{ 0 \le c \le k } \int_c^\infty \left( \frac{ z - c }{ k - c } \right)^3 \phi( z ) \, dz = 2 B_E( k ) , where φ(x) is the probability density function of the standard normal distribution. A smooth maximum, for example, : g(x1, x2, …, xn) = log( exp(x1) + exp(x2) + … + exp(xn) ) is a good approximation of the sample maximum. ===Summary statistics=== The sample maximum and minimum are basic summary statistics, showing the most extreme observations, and are used in the five-number summary and a version of the seven-number summary and the associated box plot. ===Prediction interval=== The sample maximum and minimum provide a non- parametric prediction interval: in a sample from a population, or more generally an exchangeable sequence of random variables, each observation is equally likely to be the maximum or minimum. In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables. * The range shrinks rapidly, reflecting the exponentially decaying probability that all observations in the sample will be significantly below the maximum. If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. Likewise, n = 39 gives a 95% prediction interval, and n = 199 gives a 99% prediction interval. ===Estimation=== Due to their sensitivity to outliers, the sample extrema cannot reliably be used as estimators unless data is clean – robust alternatives include the first and last deciles. This did not count as an official maximum, however, as the break was made on a non- templated table used during the event. Annals of Statistics 2(3) 609–614 ==Statement of the inequality== Let {Xi} be a set of real independent random variables, each with an expected value of zero and bounded above by 1 ( |Xi | ≤ 1, for 1 ≤ i ≤ n). * The confidence interval exhibits positive skew, as N can never be below the sample maximum, but can potentially be arbitrarily high above it. Inverting this yields the corresponding confidence interval for the population maximum of [m/q1/k, m/p1/k]. J Amer Statist Assoc 58: 13–30 MR144363 Let : S_n = a_i b_i + \cdots + a_n b_n ThenPinelis I (1994) Optimum bounds for the distributions of martingales in Banach spaces. If only the top endpoint is unknown, the sample maximum is a biased estimator for the population maximum, but the unbiased estimator \frac{k+1}{k}m - 1 (where m is the sample maximum and k is the sample size) is the UMVU estimator; see German tank problem for details. The largest sample serial number is m. In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. ",234.4,-17,0.178,399,15,C
+(a) $P\left(\max X_i < 3 / 4\right)=P\left(X_1<3 / 4, X_2<3 / 4\right)$","Thus if one has a sample \\{X_1,\dots,X_n\\}, and one picks another observation X_{n+1}, then this has 1/(n+1) probability of being the largest value seen so far, 1/(n+1) probability of being the smallest value seen so far, and thus the other (n-1)/(n+1) of the time, X_{n+1} falls between the sample maximum and sample minimum of \\{X_1,\dots,X_n\\}. * Generalized extreme value distribution, possible limit distributions of sample maximum (opposite question). Thus the sampling distribution of the quantile of the sample maximum is the graph x1/k from 0 to 1: the p-th to q-th quantile of the sample maximum m are the interval [p1/kN, q1/kN]. The minimum and the maximum value are the first and last order statistics (often denoted X(1) and X(n) respectively, for a sample size of n). In statistics, the sample maximum and sample minimum, also called the largest observation and smallest observation, are the values of the greatest and least elements of a sample. A related bound is Edelman's : P\left( \left| \sum_{ i = 1 }^n a_i X_i \right| \ge k \right) \le 2 \left( 1 - \Phi\left[ k - \frac{ 1.5 }{ k } \right] \right) = 2 B_{ Ed }( k ) , where Φ(x) is cumulative distribution function of the standard normal distribution. A derivation of the expected value and the variance of the sample maximum are shown in the page of the discrete uniform distribution. Ann Probab 22(4):1679–1706 : P( S_n \ge x ) \le \frac{ 2e^3 }{ 9 } P( Z \ge x ) The constant in the last inequality is approximately 4.4634. Eaton showed that : P\left( \left| \sum_{ i = 1 }^n a_i X_i \right| \ge k \right) \le 2 \inf_{ 0 \le c \le k } \int_c^\infty \left( \frac{ z - c }{ k - c } \right)^3 \phi( z ) \, dz = 2 B_E( k ) , where φ(x) is the probability density function of the standard normal distribution. A smooth maximum, for example, : g(x1, x2, …, xn) = log( exp(x1) + exp(x2) + … + exp(xn) ) is a good approximation of the sample maximum. ===Summary statistics=== The sample maximum and minimum are basic summary statistics, showing the most extreme observations, and are used in the five-number summary and a version of the seven-number summary and the associated box plot. ===Prediction interval=== The sample maximum and minimum provide a non- parametric prediction interval: in a sample from a population, or more generally an exchangeable sequence of random variables, each observation is equally likely to be the maximum or minimum. In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables. * The range shrinks rapidly, reflecting the exponentially decaying probability that all observations in the sample will be significantly below the maximum. If the sample has outliers, they necessarily include the sample maximum or sample minimum, or both, depending on whether they are extremely high or low. Likewise, n = 39 gives a 95% prediction interval, and n = 199 gives a 99% prediction interval. ===Estimation=== Due to their sensitivity to outliers, the sample extrema cannot reliably be used as estimators unless data is clean – robust alternatives include the first and last deciles. This did not count as an official maximum, however, as the break was made on a non- templated table used during the event. Annals of Statistics 2(3) 609–614 ==Statement of the inequality== Let {Xi} be a set of real independent random variables, each with an expected value of zero and bounded above by 1 ( |Xi | ≤ 1, for 1 ≤ i ≤ n). * The confidence interval exhibits positive skew, as N can never be below the sample maximum, but can potentially be arbitrarily high above it. Inverting this yields the corresponding confidence interval for the population maximum of [m/q1/k, m/p1/k]. J Amer Statist Assoc 58: 13–30 MR144363 Let : S_n = a_i b_i + \cdots + a_n b_n ThenPinelis I (1994) Optimum bounds for the distributions of martingales in Banach spaces. If only the top endpoint is unknown, the sample maximum is a biased estimator for the population maximum, but the unbiased estimator \frac{k+1}{k}m - 1 (where m is the sample maximum and k is the sample size) is the UMVU estimator; see German tank problem for details. The largest sample serial number is m. In the statistical theory of estimation, the German tank problem consists of estimating the maximum of a discrete uniform distribution from sampling without replacement. ",234.4,-17,"""0.178""",399,15,C
 "7.4-1. Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \%$ confident that the maximum error of the estimate of $\mu$ is 0.4 .
-","Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. ===Bootstrapping and Standard Error=== As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation. The likely size of the sampling error can generally be reduced by taking a larger sample. ===Sample Size Determination=== The cost of increasing a sample size may be prohibitive in reality. thumb|350px|Estimation of distribution algorithm. thumb|right|A specimen sheet for the regular weight of Normal-Grotesk. Since sampling is almost always done to estimate population parameters that are unknown, by definition exact measurement of the sampling errors will not be possible; however they can often be estimated, either by general methods such as bootstrapping, or by specific methods incorporating some assumptions (or guesses) regarding the true population distribution and parameters thereof. ==Description== ===Sampling Error=== The sampling error is the error caused by observing a sample instead of the whole population. Franklin's gull (Leucophaeus pipixcan) is a small (length 12.6–14.2 in, 32–36 cm) gull. This is a source of genetic drift, as certain alleles become more or less common), and has been referred to as ""sampling error"", despite not being an ""error"" in the statistical sense. ==See also== * Margin of error * Propagation of uncertainty * Ratio estimator * Sampling (statistics) ==References== Category:Sampling (statistics) Category:Errors and residuals Category:Auditing terms By comparing many samples, or splitting a larger sample up into smaller ones (potentially with overlap), the spread of the resulting sample statistics can be used to estimate the standard error on the sample. ==In Genetics== The term ""sampling error"" has also been used in a related but fundamentally different sense in the field of genetics; for example in the bottleneck effect or founder effect, when natural disasters or migrations dramatically reduce the size of a population, resulting in a smaller population that may or may not fairly represent the original one. The difference between the sample statistic and population parameter is considered the sampling error.Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, For example, if one measures the height of a thousand individuals from a population of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. The sampling error is the difference between a sample statistic used to estimate a population parameter and the actual but unknown value of the parameter. ===Effective Sampling=== In statistics, a truly random sample means selecting individuals from a population with an equivalent probability; in other words, picking individuals from a group without bias. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle. - P. 288, nota 35. The sampling (following a normal distribution N) concentrates around the optimum as one goes along unwinding algorithm. Even in a perfectly non-biased sample, the sample error will still exist due to the remaining statistical component; consider that measuring only two or three individuals and taking the average would produce a wildly varying result each time. At each generation, \mu individuals are sampled and \lambda\leq \mu are selected. Measurements: * Length: 12.6-14.2 in (32-36 cm) * Weight: 8.1-10.6 oz (230-300 g) * Wingspan: 33.5-37.4 in (85-95 cm) Although the bird is uncommon on the coasts of North America, it occurs as a rare vagrant to northwest Europe, south and west Africa, Australia and Japan, with a single record from Eilat, Israel, in 2011 (Smith 2011), and a single record from Larnaca, Cyprus, July 2006. Without assuming the Riemann hypothesis, the best known upper bound is :V(r)=\frac{6}{\pi}r^2+O(r\exp(-c(\log r)^{3/5}(\log\log r^2)^{-1/5})) for a positive constant c. At the beginning of 2017 has been observed also in Southern Romania, southeast Europe. ===Behaviour=== They are omnivores like most gulls, and they will scavenge as well as seeking suitable small prey. ",+11,117,-22.1,-501,260,B
+","Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. ===Bootstrapping and Standard Error=== As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation. The likely size of the sampling error can generally be reduced by taking a larger sample. ===Sample Size Determination=== The cost of increasing a sample size may be prohibitive in reality. thumb|350px|Estimation of distribution algorithm. thumb|right|A specimen sheet for the regular weight of Normal-Grotesk. Since sampling is almost always done to estimate population parameters that are unknown, by definition exact measurement of the sampling errors will not be possible; however they can often be estimated, either by general methods such as bootstrapping, or by specific methods incorporating some assumptions (or guesses) regarding the true population distribution and parameters thereof. ==Description== ===Sampling Error=== The sampling error is the error caused by observing a sample instead of the whole population. Franklin's gull (Leucophaeus pipixcan) is a small (length 12.6–14.2 in, 32–36 cm) gull. This is a source of genetic drift, as certain alleles become more or less common), and has been referred to as ""sampling error"", despite not being an ""error"" in the statistical sense. ==See also== * Margin of error * Propagation of uncertainty * Ratio estimator * Sampling (statistics) ==References== Category:Sampling (statistics) Category:Errors and residuals Category:Auditing terms By comparing many samples, or splitting a larger sample up into smaller ones (potentially with overlap), the spread of the resulting sample statistics can be used to estimate the standard error on the sample. ==In Genetics== The term ""sampling error"" has also been used in a related but fundamentally different sense in the field of genetics; for example in the bottleneck effect or founder effect, when natural disasters or migrations dramatically reduce the size of a population, resulting in a smaller population that may or may not fairly represent the original one. The difference between the sample statistic and population parameter is considered the sampling error.Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, For example, if one measures the height of a thousand individuals from a population of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. The sampling error is the difference between a sample statistic used to estimate a population parameter and the actual but unknown value of the parameter. ===Effective Sampling=== In statistics, a truly random sample means selecting individuals from a population with an equivalent probability; in other words, picking individuals from a group without bias. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle. - P. 288, nota 35. The sampling (following a normal distribution N) concentrates around the optimum as one goes along unwinding algorithm. Even in a perfectly non-biased sample, the sample error will still exist due to the remaining statistical component; consider that measuring only two or three individuals and taking the average would produce a wildly varying result each time. At each generation, \mu individuals are sampled and \lambda\leq \mu are selected. Measurements: * Length: 12.6-14.2 in (32-36 cm) * Weight: 8.1-10.6 oz (230-300 g) * Wingspan: 33.5-37.4 in (85-95 cm) Although the bird is uncommon on the coasts of North America, it occurs as a rare vagrant to northwest Europe, south and west Africa, Australia and Japan, with a single record from Eilat, Israel, in 2011 (Smith 2011), and a single record from Larnaca, Cyprus, July 2006. Without assuming the Riemann hypothesis, the best known upper bound is :V(r)=\frac{6}{\pi}r^2+O(r\exp(-c(\log r)^{3/5}(\log\log r^2)^{-1/5})) for a positive constant c. At the beginning of 2017 has been observed also in Southern Romania, southeast Europe. ===Behaviour=== They are omnivores like most gulls, and they will scavenge as well as seeking suitable small prey. ",+11,117,"""-22.1""",-501,260,B
 "5.4-17. In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \geq X+2$. HINT: What is the distribution of $Y-X+25$ ?
-","* The event dropout rate estimates the percentage of high school students who left high school between the beginning of one school year and the beginning of the next without earning a high school diploma or its equivalent (e.g., a GED). * The status dropout rate reports the percentage of individuals in a given age range who are not in school and have not earned a high school diploma or equivalent credential. It is estimated 1.2 million students annually drop out of high school in the United States, where high school graduation rates rank 19th in the world.High School Dropouts - Do Something. In 2010 the dropout rates of 16- through 24-year- olds who are not enrolled in school and have not earned a high school credential were: 5.1% for white students, 8% for black students, 15.1% for Hispanic students, and 4.2% for Asian students. ===Academic risk factors=== Academic risk factors refer to the students' performance in school and are highly related to school level problems. ""The Influence of Selected Academic, Demographic and Instructional Program Related Factors on High School Student Dropout Rates"". Large schools, enrolling between 1,500 and 2,500 students, were found to have the largest proportion of students who dropped out, 12%. Event rates can be used to track annual changes in the dropout behavior of students in the U.S. school system. This percentage jumps to 38% in adolescents aged 15 to 17 years who also provided this reason for their disengagement with the education system. ==Dropout recovery== A ""dropout recovery"" initiative is any community, government, non-profit or business program in which students who have previously left school are sought out for the purpose of re-enrollment. Using this tool, assessing educational attainment and school attendance can calculate a dropout rate (Gilmore, 2010). A study by Battin- Pearson found that these two factors did not contribute significantly to dropout beyond what was explained by poor academic achievement. ==Motivation for dropping out== While the above factors certainly place a student at risk for dropout, they are not always the reason the student identifies as their motivation for dropping out. Although since 1990 dropout rates have gone down from 20% to a low of 9% in 2010, the rate does not seem to be dropping since this time (2010). A dropout is a momentary loss of signal in a communications system, usually caused by noise, propagation anomalies, or system malfunctions. There has been contention over the influence of ethnicity on dropout rates. Because of these factors, an average high school dropout will cost the government over $292,000. ==Measurement of the dropout rate== The U.S. Department of Education identifies four different rates to measure high school dropout and completion in the United States. Grade retention can increase the odds of dropping out by as much as 250 percent above those of similar students who were not retained.McNeil 2008 Students who drop out typically have a history of absenteeism, grade retention and academic trouble and are more disengaged from school life. The study also found that men still have higher dropout rates than women, and that students outside of major cities and in the northern territories also have a higher risk of dropping out. Allowing students to interact with support dogs and their owners allowed students to feel connected to their peers, school and school community (Binfet et al., 2016., & Binfet et al., 2018). ==United Kingdom== In the United Kingdom, a dropout is anyone who leaves school, college or university without either completing their course of study or transferring to another educational institution. The United States Department of Education's measurement of the status dropout rate is the percentage of 16-24-year-olds who are not enrolled in school and have not earned a high school credential.NCES 2011 This rate is different from the event dropout rate and related measures of the status completion and average freshman completion rates.NCES 2009 The status high school dropout rate in 2009 was 8.1%. The average Canadian dropout earns $70 less per week than their peers with a high school diploma. Graduates (without post-secondary) earned an average of $621 per week, whereas dropout students earned an average of $551 (Gilmore, 2010). The United States Department of Education's measurement of the status dropout rate is the percentage of 16 to 24-year-olds who are not enrolled in school and have not earned a high school credential.NCES 2011 This rate is different from the event dropout rate and related measures of the status completion and average freshman completion rates.NCES 2009 The status high school dropout rate in 2009 was 8.1%. As such, this theory examines the relationship between family background and dropout rates. ",4.8,0.3359,54.394,1590,0.118,B
-"9.6-111. Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\rho$. To test $H_0: \rho=0$ against $H_1: \rho \neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.","For example, if we were expecting a population correlation between intelligence and job performance of around 0.50, a sample size of 20 will give us approximately 80% power ( = 0.05, two-tail) to reject the null hypothesis of zero correlation. Furthermore, assume that the null hypothesis will be rejected at the significance level of \alpha = 0.05\,. If rrb is calculated as above then the smaller of : (1+r_{rb})\frac{n_1n_0}{2} and : (1-r_{rb})\frac{n_1n_0}{2} is distributed as Mann–Whitney U with sample sizes n1 and n0 when the null hypothesis is true. == Notes == * MacCallum Robert C. et all Psychological Methods. 2002, Vol. 7, N°1, 49-40 ==References== ==External links== *Point Biserial Coefficient (Keith Calkins, 2005) Category:Correlation indicators It remains the case that very small values are relatively unlikely if the null-hypothesis is true, and that a significance test at level \alpha is obtained by rejecting the null-hypothesis if the significance level is less than or equal to \alpha. For a simple hypothesis, :\alpha = P(\text{test rejects } H_0 \mid H_0). The minimum (infimum) value of the power is equal to the confidence level of the test, \alpha, in this example 0.05. In the case of a composite null hypothesis, the size is the supremum over all data generating processes that satisfy the null hypotheses. :\alpha = \sup_{h\in H_0} P(\text{test rejects } H_0 \mid h). If the criterion is 0.05, the probability of the data implying an effect at least as large as the observed effect when the null hypothesis is true must be less than 0.05, for the null hypothesis of no effect to be rejected. If it is desirable to have enough power, say at least 0.90, to detect values of \theta > 1, the required sample size can be calculated approximately: B(1) \approx 1 - \Phi \left (1.64-\frac{\sqrt{n}}{\hat{\sigma}_D}\right) >0.90, from which it follows that \Phi \left( 1.64 - \frac{\sqrt{n}}{\hat{\sigma}_D} \right) < 0.10\,. It is possible to use this to test the null hypothesis of zero correlation in the population from which the sample was drawn. A little algebra shows that the usual formula for assessing the significance of a correlation coefficient, when applied to rpb, is the same as the formula for an unpaired t-test and so : r_{pb} \sqrt{ \frac{n_1+n_0-2}{1-r_{pb}^2}} follows Student's t-distribution with (n1+n0 − 2) degrees of freedom when the null hypothesis is true. However, in doing this study we are probably more interested in knowing whether the correlation is 0.30 or 0.60 or 0.50. If we set the significance level alpha to 0.05, and only reject the null hypothesis if the p-value is less than or equal to 0.05, then our hypothesis test will indeed have significance level (maximal type 1 error rate) 0.05. A test is said to have significance level \alpha if its size is less than or equal to \alpha . In the above example: * Null hypothesis (H0): The coin is fair, with Pr(heads) = 0.5 * Test statistic: Number of heads * Alpha level (designated threshold of significance): 0.05 * Observation O: 14 heads out of 20 flips; and * Two-tailed p-value of observation O given H0 = 2 × min(Pr(no. of heads ≥ 14 heads), Pr(no. of heads ≤ 14 heads)) = 2 × min(0.058, 0.978) = 2*0.058 = 0.115. Correlation (iii) is : r_{upb}=\frac{M_1-M_0-1}{\sqrt{\frac{n^2s_n^2}{n_1n_0}-2(M_1-M_0)+1}}. If the distribution of T is symmetric about zero, then p =\Pr(|T| \geq |t| \mid H_0) === Interpretations === ==== p-value as the statistic for performing significance tests ==== In a significance test, the null hypothesis H_0 is rejected if the p-value is less than or equal to a predefined threshold value \alpha, which is referred to as the alpha level or significance level. \alpha is not derived from the data, but rather is set by the researcher before examining the data. \alpha is commonly set to 0.05, though lower alpha levels are sometimes used. We can test the null hypothesis that the correlation is zero in the population. Some factors may be particular to a specific testing situation, but at a minimum, power nearly always depends on the following three factors: * the statistical significance criterion used in the test * the magnitude of the effect of interest in the population * the sample size used to detect the effect A significance criterion is a statement of how unlikely a positive result must be, if the null hypothesis of no effect is true, for the null hypothesis to be rejected. Hence, the null hypothesis is not rejected at the .05 level. The lower the p-value is, the lower the probability of getting that result if the null hypothesis were true. Thus, \text{power} = \Pr \big( \text{reject } H_0 \mid H_1 \text{ is true} \big). ",9,5300,2.19,0.59,-11.2,A
+","* The event dropout rate estimates the percentage of high school students who left high school between the beginning of one school year and the beginning of the next without earning a high school diploma or its equivalent (e.g., a GED). * The status dropout rate reports the percentage of individuals in a given age range who are not in school and have not earned a high school diploma or equivalent credential. It is estimated 1.2 million students annually drop out of high school in the United States, where high school graduation rates rank 19th in the world.High School Dropouts - Do Something. In 2010 the dropout rates of 16- through 24-year- olds who are not enrolled in school and have not earned a high school credential were: 5.1% for white students, 8% for black students, 15.1% for Hispanic students, and 4.2% for Asian students. ===Academic risk factors=== Academic risk factors refer to the students' performance in school and are highly related to school level problems. ""The Influence of Selected Academic, Demographic and Instructional Program Related Factors on High School Student Dropout Rates"". Large schools, enrolling between 1,500 and 2,500 students, were found to have the largest proportion of students who dropped out, 12%. Event rates can be used to track annual changes in the dropout behavior of students in the U.S. school system. This percentage jumps to 38% in adolescents aged 15 to 17 years who also provided this reason for their disengagement with the education system. ==Dropout recovery== A ""dropout recovery"" initiative is any community, government, non-profit or business program in which students who have previously left school are sought out for the purpose of re-enrollment. Using this tool, assessing educational attainment and school attendance can calculate a dropout rate (Gilmore, 2010). A study by Battin- Pearson found that these two factors did not contribute significantly to dropout beyond what was explained by poor academic achievement. ==Motivation for dropping out== While the above factors certainly place a student at risk for dropout, they are not always the reason the student identifies as their motivation for dropping out. Although since 1990 dropout rates have gone down from 20% to a low of 9% in 2010, the rate does not seem to be dropping since this time (2010). A dropout is a momentary loss of signal in a communications system, usually caused by noise, propagation anomalies, or system malfunctions. There has been contention over the influence of ethnicity on dropout rates. Because of these factors, an average high school dropout will cost the government over $292,000. ==Measurement of the dropout rate== The U.S. Department of Education identifies four different rates to measure high school dropout and completion in the United States. Grade retention can increase the odds of dropping out by as much as 250 percent above those of similar students who were not retained.McNeil 2008 Students who drop out typically have a history of absenteeism, grade retention and academic trouble and are more disengaged from school life. The study also found that men still have higher dropout rates than women, and that students outside of major cities and in the northern territories also have a higher risk of dropping out. Allowing students to interact with support dogs and their owners allowed students to feel connected to their peers, school and school community (Binfet et al., 2016., & Binfet et al., 2018). ==United Kingdom== In the United Kingdom, a dropout is anyone who leaves school, college or university without either completing their course of study or transferring to another educational institution. The United States Department of Education's measurement of the status dropout rate is the percentage of 16-24-year-olds who are not enrolled in school and have not earned a high school credential.NCES 2011 This rate is different from the event dropout rate and related measures of the status completion and average freshman completion rates.NCES 2009 The status high school dropout rate in 2009 was 8.1%. The average Canadian dropout earns $70 less per week than their peers with a high school diploma. Graduates (without post-secondary) earned an average of $621 per week, whereas dropout students earned an average of $551 (Gilmore, 2010). The United States Department of Education's measurement of the status dropout rate is the percentage of 16 to 24-year-olds who are not enrolled in school and have not earned a high school credential.NCES 2011 This rate is different from the event dropout rate and related measures of the status completion and average freshman completion rates.NCES 2009 The status high school dropout rate in 2009 was 8.1%. As such, this theory examines the relationship between family background and dropout rates. ",4.8,0.3359,"""54.394""",1590,0.118,B
+"9.6-111. Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\rho$. To test $H_0: \rho=0$ against $H_1: \rho \neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.","For example, if we were expecting a population correlation between intelligence and job performance of around 0.50, a sample size of 20 will give us approximately 80% power ( = 0.05, two-tail) to reject the null hypothesis of zero correlation. Furthermore, assume that the null hypothesis will be rejected at the significance level of \alpha = 0.05\,. If rrb is calculated as above then the smaller of : (1+r_{rb})\frac{n_1n_0}{2} and : (1-r_{rb})\frac{n_1n_0}{2} is distributed as Mann–Whitney U with sample sizes n1 and n0 when the null hypothesis is true. == Notes == * MacCallum Robert C. et all Psychological Methods. 2002, Vol. 7, N°1, 49-40 ==References== ==External links== *Point Biserial Coefficient (Keith Calkins, 2005) Category:Correlation indicators It remains the case that very small values are relatively unlikely if the null-hypothesis is true, and that a significance test at level \alpha is obtained by rejecting the null-hypothesis if the significance level is less than or equal to \alpha. For a simple hypothesis, :\alpha = P(\text{test rejects } H_0 \mid H_0). The minimum (infimum) value of the power is equal to the confidence level of the test, \alpha, in this example 0.05. In the case of a composite null hypothesis, the size is the supremum over all data generating processes that satisfy the null hypotheses. :\alpha = \sup_{h\in H_0} P(\text{test rejects } H_0 \mid h). If the criterion is 0.05, the probability of the data implying an effect at least as large as the observed effect when the null hypothesis is true must be less than 0.05, for the null hypothesis of no effect to be rejected. If it is desirable to have enough power, say at least 0.90, to detect values of \theta > 1, the required sample size can be calculated approximately: B(1) \approx 1 - \Phi \left (1.64-\frac{\sqrt{n}}{\hat{\sigma}_D}\right) >0.90, from which it follows that \Phi \left( 1.64 - \frac{\sqrt{n}}{\hat{\sigma}_D} \right) < 0.10\,. It is possible to use this to test the null hypothesis of zero correlation in the population from which the sample was drawn. A little algebra shows that the usual formula for assessing the significance of a correlation coefficient, when applied to rpb, is the same as the formula for an unpaired t-test and so : r_{pb} \sqrt{ \frac{n_1+n_0-2}{1-r_{pb}^2}} follows Student's t-distribution with (n1+n0 − 2) degrees of freedom when the null hypothesis is true. However, in doing this study we are probably more interested in knowing whether the correlation is 0.30 or 0.60 or 0.50. If we set the significance level alpha to 0.05, and only reject the null hypothesis if the p-value is less than or equal to 0.05, then our hypothesis test will indeed have significance level (maximal type 1 error rate) 0.05. A test is said to have significance level \alpha if its size is less than or equal to \alpha . In the above example: * Null hypothesis (H0): The coin is fair, with Pr(heads) = 0.5 * Test statistic: Number of heads * Alpha level (designated threshold of significance): 0.05 * Observation O: 14 heads out of 20 flips; and * Two-tailed p-value of observation O given H0 = 2 × min(Pr(no. of heads ≥ 14 heads), Pr(no. of heads ≤ 14 heads)) = 2 × min(0.058, 0.978) = 2*0.058 = 0.115. Correlation (iii) is : r_{upb}=\frac{M_1-M_0-1}{\sqrt{\frac{n^2s_n^2}{n_1n_0}-2(M_1-M_0)+1}}. If the distribution of T is symmetric about zero, then p =\Pr(|T| \geq |t| \mid H_0) === Interpretations === ==== p-value as the statistic for performing significance tests ==== In a significance test, the null hypothesis H_0 is rejected if the p-value is less than or equal to a predefined threshold value \alpha, which is referred to as the alpha level or significance level. \alpha is not derived from the data, but rather is set by the researcher before examining the data. \alpha is commonly set to 0.05, though lower alpha levels are sometimes used. We can test the null hypothesis that the correlation is zero in the population. Some factors may be particular to a specific testing situation, but at a minimum, power nearly always depends on the following three factors: * the statistical significance criterion used in the test * the magnitude of the effect of interest in the population * the sample size used to detect the effect A significance criterion is a statement of how unlikely a positive result must be, if the null hypothesis of no effect is true, for the null hypothesis to be rejected. Hence, the null hypothesis is not rejected at the .05 level. The lower the p-value is, the lower the probability of getting that result if the null hypothesis were true. Thus, \text{power} = \Pr \big( \text{reject } H_0 \mid H_1 \text{ is true} \big). ",9,5300,"""2.19""",0.59,-11.2,A
 "7.3-5. In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval.
 (a) Give a point estimate of $p$.
-","* The averaged freshman graduation rate estimates the proportion of public high school freshmen who graduate with a regular diploma four years after starting ninth grade. Large schools, enrolling between 1,500 and 2,500 students, were found to have the largest proportion of students who dropped out, 12%. The rate focuses on public high school students as opposed to all high school students or the general population and is designed to provide an estimate of on-time graduation from high school. ""The Influence of Selected Academic, Demographic and Instructional Program Related Factors on High School Student Dropout Rates"". ""The Impact of High School Size on Math Achievement and Dropout Rate"". More precisely, the tertiary enrollment rate is the percentage of total enrollment, regardless of age, in post-secondary institutions to the population of people within five years of the age at which students normally graduate high school. ==Rankings== 1 United States: 72.6% 2 Finland: 70.4% 3 Norway: 70% 3 Sweden: 70% 5 New Zealand: 69.2% 6 Russia: 64.1% 7 Australia: 63.3% 8 Latvia: 63.1% 9 Slovenia: 60.5% 10 Canada: 60% ==References== Category:Higher education Tertiary enrollment rates are an expression of the percentage of high school graduates that successfully enroll into university. This is a list of States and Union Territories of India ranked according to Gross Enrollment Ratio (GER) of students in Classes I to VIII (6–13 yrs). * The event dropout rate estimates the percentage of high school students who left high school between the beginning of one school year and the beginning of the next without earning a high school diploma or its equivalent (e.g., a GED). In 2010 the dropout rates of 16- through 24-year- olds who are not enrolled in school and have not earned a high school credential were: 5.1% for white students, 8% for black students, 15.1% for Hispanic students, and 4.2% for Asian students. ===Academic risk factors=== Academic risk factors refer to the students' performance in school and are highly related to school level problems. While in the U.S. highly competitive students have A grades, in Chile these same students tend to average 6,8 , 6,9 or 7,0, all of which are considered near perfect grades. Graduating students from high school who are not prepared for college, however, also generates problems, as the college dropout rate exceeds the high school rate. The list is compiled from the Statistics of School Education- 2010–11 Report by Ministry of HRD, Government of India. == List == Gross enrolment ratio (GER) is a statistical measure used in the education sector and by the UN in its Education Index to determine the number of students enrolled in school at several different grade levels (like elementary, middle school and high school), and examine it to analyze the ratio of the number of students who live in that country to those who qualify for the particular grade level. The grade point average (GPA) in Chile ranges from 1.0 up to 7.0 (with one decimal place). ""Predictors of Early High School Dropout: A Test of Five Theories"". “Exceptions to High School Dropout Predictions in a Low-Income Sample: Do Adults Make a Difference?” Thus, it provides a measure of the extent to which public high schools are graduating students within the expected period of four years. ==Notable dropouts== * Don Adams (1923–2005), actor; dropped out of DeWitt Clinton High SchoolSmith, Austin. Table of Chilean GPA GPA % Achievement Meaning Honours 6.0 - 7.0 83% - 100% Outstanding (7.0) Highest Honours 5.0 - 5.9 66% - 82% Good Honours 4.0 - 4.9 50% - 65% Sufficient Passed 3.0 - 3.9 33% - 49% Less than Sufficient Failed 2.0 - 2.9 16% - 32% Deficient Failed 1.0 - 1.9 0% - 15% Very Deficient Failed ==References== Chile Grading Grading An overall GPA in university degrees that ranges from 5.5 to 5.9 is uncommon and is considered a ""very good"" academic standing. Students who attended schools that offered Calculus or fewer courses below the level of Algebra 1 had a reduced risk of dropping out of school by 56%. The United States Department of Education's measurement of the status dropout rate is the percentage of 16 to 24-year-olds who are not enrolled in school and have not earned a high school credential.NCES 2011 This rate is different from the event dropout rate and related measures of the status completion and average freshman completion rates.NCES 2009 The status high school dropout rate in 2009 was 8.1%. This rate focuses on an overall age group as opposed to individuals in the U.S. school system, so it can be used to study general population issues. ",0.011,0.1800,6.0,2.89,-4564.7,B
-"7.4-15. If $\bar{X}$ and $\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\bar{x}-\bar{y} \pm 4$ to be a $90 \%$ confidence interval for $\mu_X-\mu_Y$. Assume that the standard deviations are known to be $\sigma_X=15$ and $\sigma_Y=25$.","The mean value calculated from the sample, \bar{x}, will have an associated standard error on the mean, {\sigma}_\bar{x}, given by: :{\sigma}_\bar{x}\ = \frac{\sigma}{\sqrt{n}}. The following expressions can be used to calculate the upper and lower 95% confidence limits, where \bar{x} is equal to the sample mean, \operatorname{SE} is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution: :Upper 95% limit = \bar{x} + (\operatorname{SE}\times 1.96) , and :Lower 95% limit = \bar{x} - (\operatorname{SE}\times 1.96) . Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. Therefore, the standard error of the mean is usually estimated by replacing \sigma with the sample standard deviation \sigma_{x} instead: :{\sigma}_\bar{x}\ \approx \frac{\sigma_{x}}{\sqrt{n}}. The variance of the mean is then :\operatorname{Var}(\bar{x}) = \operatorname{Var}\left(\frac{T}{n}\right) = \frac{1}{n^2}\operatorname{Var}(T) = \frac{1}{n^2}n\sigma^2 = \frac{\sigma^2}{n}. The standard error is, by definition, the standard deviation of \bar{x} which is simply the square root of the variance: :\sigma_{\bar{x}} = \sqrt{\frac{\sigma^2}{n}} = \frac{\sigma}{\sqrt{n}} . Suppose we wanted to calculate a 95% confidence interval for μ. The mean of these measurements \bar{x} is simply given by :\bar{x} = T/n . After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. Practically this tells us that when trying to estimate the value of a population mean, due to the factor 1/\sqrt{n}, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations. === Estimate === The standard deviation \sigma of the population being sampled is seldom known. Hence the estimator of \operatorname{Var}(T) becomes nS^2_X + n\bar{X}^2, leading the following formula for standard error: :\operatorname{Standard~Error}(\bar{X})= \sqrt{\frac{S^2_X + \bar{X}^2}{n}} (since the standard deviation is the square root of the variance) ==Student approximation when σ value is unknown== In many practical applications, the true value of σ is unknown. As this is only an estimator for the true ""standard error"", it is common to see other notations here such as: :\widehat{\sigma}_{\bar{x}} \approx \frac{\sigma_{x}}{\sqrt{n}} or alternately {s}_\bar{x}\ \approx \frac{s}{\sqrt{n}}. Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation ""s"" instead of σ, and we could use this value to calculate confidence intervals. Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. * The confidence interval can be expressed in terms of statistical significance, e.g.: ",0.321,2,4.16,435,144,E
+","* The averaged freshman graduation rate estimates the proportion of public high school freshmen who graduate with a regular diploma four years after starting ninth grade. Large schools, enrolling between 1,500 and 2,500 students, were found to have the largest proportion of students who dropped out, 12%. The rate focuses on public high school students as opposed to all high school students or the general population and is designed to provide an estimate of on-time graduation from high school. ""The Influence of Selected Academic, Demographic and Instructional Program Related Factors on High School Student Dropout Rates"". ""The Impact of High School Size on Math Achievement and Dropout Rate"". More precisely, the tertiary enrollment rate is the percentage of total enrollment, regardless of age, in post-secondary institutions to the population of people within five years of the age at which students normally graduate high school. ==Rankings== 1 United States: 72.6% 2 Finland: 70.4% 3 Norway: 70% 3 Sweden: 70% 5 New Zealand: 69.2% 6 Russia: 64.1% 7 Australia: 63.3% 8 Latvia: 63.1% 9 Slovenia: 60.5% 10 Canada: 60% ==References== Category:Higher education Tertiary enrollment rates are an expression of the percentage of high school graduates that successfully enroll into university. This is a list of States and Union Territories of India ranked according to Gross Enrollment Ratio (GER) of students in Classes I to VIII (6–13 yrs). * The event dropout rate estimates the percentage of high school students who left high school between the beginning of one school year and the beginning of the next without earning a high school diploma or its equivalent (e.g., a GED). In 2010 the dropout rates of 16- through 24-year- olds who are not enrolled in school and have not earned a high school credential were: 5.1% for white students, 8% for black students, 15.1% for Hispanic students, and 4.2% for Asian students. ===Academic risk factors=== Academic risk factors refer to the students' performance in school and are highly related to school level problems. While in the U.S. highly competitive students have A grades, in Chile these same students tend to average 6,8 , 6,9 or 7,0, all of which are considered near perfect grades. Graduating students from high school who are not prepared for college, however, also generates problems, as the college dropout rate exceeds the high school rate. The list is compiled from the Statistics of School Education- 2010–11 Report by Ministry of HRD, Government of India. == List == Gross enrolment ratio (GER) is a statistical measure used in the education sector and by the UN in its Education Index to determine the number of students enrolled in school at several different grade levels (like elementary, middle school and high school), and examine it to analyze the ratio of the number of students who live in that country to those who qualify for the particular grade level. The grade point average (GPA) in Chile ranges from 1.0 up to 7.0 (with one decimal place). ""Predictors of Early High School Dropout: A Test of Five Theories"". “Exceptions to High School Dropout Predictions in a Low-Income Sample: Do Adults Make a Difference?” Thus, it provides a measure of the extent to which public high schools are graduating students within the expected period of four years. ==Notable dropouts== * Don Adams (1923–2005), actor; dropped out of DeWitt Clinton High SchoolSmith, Austin. Table of Chilean GPA GPA % Achievement Meaning Honours 6.0 - 7.0 83% - 100% Outstanding (7.0) Highest Honours 5.0 - 5.9 66% - 82% Good Honours 4.0 - 4.9 50% - 65% Sufficient Passed 3.0 - 3.9 33% - 49% Less than Sufficient Failed 2.0 - 2.9 16% - 32% Deficient Failed 1.0 - 1.9 0% - 15% Very Deficient Failed ==References== Chile Grading Grading An overall GPA in university degrees that ranges from 5.5 to 5.9 is uncommon and is considered a ""very good"" academic standing. Students who attended schools that offered Calculus or fewer courses below the level of Algebra 1 had a reduced risk of dropping out of school by 56%. The United States Department of Education's measurement of the status dropout rate is the percentage of 16 to 24-year-olds who are not enrolled in school and have not earned a high school credential.NCES 2011 This rate is different from the event dropout rate and related measures of the status completion and average freshman completion rates.NCES 2009 The status high school dropout rate in 2009 was 8.1%. This rate focuses on an overall age group as opposed to individuals in the U.S. school system, so it can be used to study general population issues. ",0.011,0.1800,"""6.0""",2.89,-4564.7,B
+"7.4-15. If $\bar{X}$ and $\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\bar{x}-\bar{y} \pm 4$ to be a $90 \%$ confidence interval for $\mu_X-\mu_Y$. Assume that the standard deviations are known to be $\sigma_X=15$ and $\sigma_Y=25$.","The mean value calculated from the sample, \bar{x}, will have an associated standard error on the mean, {\sigma}_\bar{x}, given by: :{\sigma}_\bar{x}\ = \frac{\sigma}{\sqrt{n}}. The following expressions can be used to calculate the upper and lower 95% confidence limits, where \bar{x} is equal to the sample mean, \operatorname{SE} is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution: :Upper 95% limit = \bar{x} + (\operatorname{SE}\times 1.96) , and :Lower 95% limit = \bar{x} - (\operatorname{SE}\times 1.96) . Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. Therefore, the standard error of the mean is usually estimated by replacing \sigma with the sample standard deviation \sigma_{x} instead: :{\sigma}_\bar{x}\ \approx \frac{\sigma_{x}}{\sqrt{n}}. The variance of the mean is then :\operatorname{Var}(\bar{x}) = \operatorname{Var}\left(\frac{T}{n}\right) = \frac{1}{n^2}\operatorname{Var}(T) = \frac{1}{n^2}n\sigma^2 = \frac{\sigma^2}{n}. The standard error is, by definition, the standard deviation of \bar{x} which is simply the square root of the variance: :\sigma_{\bar{x}} = \sqrt{\frac{\sigma^2}{n}} = \frac{\sigma}{\sqrt{n}} . Suppose we wanted to calculate a 95% confidence interval for μ. The mean of these measurements \bar{x} is simply given by :\bar{x} = T/n . After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. Practically this tells us that when trying to estimate the value of a population mean, due to the factor 1/\sqrt{n}, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations. === Estimate === The standard deviation \sigma of the population being sampled is seldom known. Hence the estimator of \operatorname{Var}(T) becomes nS^2_X + n\bar{X}^2, leading the following formula for standard error: :\operatorname{Standard~Error}(\bar{X})= \sqrt{\frac{S^2_X + \bar{X}^2}{n}} (since the standard deviation is the square root of the variance) ==Student approximation when σ value is unknown== In many practical applications, the true value of σ is unknown. As this is only an estimator for the true ""standard error"", it is common to see other notations here such as: :\widehat{\sigma}_{\bar{x}} \approx \frac{\sigma_{x}}{\sqrt{n}} or alternately {s}_\bar{x}\ \approx \frac{s}{\sqrt{n}}. Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation ""s"" instead of σ, and we could use this value to calculate confidence intervals. Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. * The confidence interval can be expressed in terms of statistical significance, e.g.: ",0.321,2,"""4.16""",435,144,E
 "7.4-7. For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to
-(a) 0.03 with $95 \%$ confidence?","The values for \hat{p} = 0.68, n = 400, z^* = 1.96 can now be substituted into the formula for one- sample proportion in the Z-interval: \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \Rightarrow (0.68) \pm (1.96) \sqrt{\frac{(0.68)(1-0.68)}{(400)}} \Rightarrow 0.68 \pm 1.96 \sqrt{0.000544} \Rightarrow \bigl(0.63429,0.72571\bigr) Based on the conditions of inference and the formula for the one-sample proportion in the Z-interval, it can be concluded with a 95% confidence level that the percentage of the voter population in this democracy supporting candidate B is between 63.429% and 72.571%. === Value of the parameter in the confidence interval range === A commonly asked question in inferential statistics is whether the parameter is included within a confidence interval. To answer the political scientist's question, a one- sample proportion in the Z-interval with a confidence level of 95% can be constructed in order to determine the population proportion of eligible voters in this democracy that support candidate B. ==== Solution ==== It is known from the random sample that \hat{p} = \frac{272}{400} = 0.68 with sample size n = 400. Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. ===Bootstrapping and Standard Error=== As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation. :N \geq 10(400) \Rightarrow N \geq 4000 :The population size N for this democracy's voters can be assumed to be at least 4,000. Thus, p_{\min} < p < p_{\max}, where: :\frac{\Gamma(n+1)}{\Gamma(x )\Gamma(n-x+1)}\int_0^{ p_{\min}} t^{x-1}(1-t)^{n-x}dt = \frac{\alpha}{2} :\frac{\Gamma(n+1)}{\Gamma(x+1)\Gamma(n-x)}\int_0^{ p_{\max}} t^{x}(1-t)^{n-x-1}dt = 1-\frac{\alpha}{2} The binomial proportion confidence interval is then ( p_{\min}, p_{\max}), as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. A population proportion can be estimated through the usage of a confidence interval known as a one-sample proportion in the Z-interval whose formula is given below: :\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} where \hat{p} is the sample proportion, n is the sample size, and z^* is the upper \frac{1-C}{2} critical value of the standard normal distribution for a level of confidence C. === Proof === In order to derive the formula for the one-sample proportion in the Z-interval, a sampling distribution of sample proportions needs to be taken into consideration. This can be verified mathematically with the following definition: #* Let n be the sample size of a given random sample and let \hat{p} be its sample proportion. * Since a random sample of 400 voters was obtained from the voting population, the condition for a simple random sample has been met. In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes nS are known. This approximation is based on the central limit theorem and is unreliable when the sample size is small or the success probability is close to 0 or 1. The value of 72% is a sample proportion. Suppose the following probability is calculated: :P(-z^*<\frac{\hat{p}-P}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}, where 0 and \pm z^* are the standard critical values. thumb|The sampling distribution of sample proportions is approximately normal when it satisfies the requirements of the Central Limit Theorem. A political scientist wants to determine what percentage of the voter population support candidate B. A random sample of 400 eligible voters in the democracy's voter population shows that 272 voters support candidate B. The likely size of the sampling error can generally be reduced by taking a larger sample. ===Sample Size Determination=== The cost of increasing a sample size may be prohibitive in reality. The mean of the sampling distribution of sample proportions is usually denoted as \mu_\hat{p} = P and its standard deviation is denoted as: :\sigma_\hat{p} = \sqrt{\frac{P(1-P)}{n}} Since the value of P is unknown, an unbiased statistic \hat{p} will be used for P. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed. ==Normal approximation interval or Wald interval == A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, \hat p, with a normal distribution. Hence, the values of C fall between 0 and 1, exclusively. === Estimation of P using ranked set sampling === A more precise estimate of P can be obtained by choosing ranked set sampling instead of simple random sampling ==See also== *Binomial proportion confidence interval *Confidence interval *Prevalence *Statistical hypothesis testing *Statistical inference *Statistical parameter *Tolerance interval == References == Category:Ratios In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). The solution for estimates the upper and lower limits of the confidence interval for . Since the X_i are independent and each one has variance \text{Var}(X_i) = p(1-p), the sampling variance of the proportion therefore is:How to calculate the standard error of a proportion using weighted data? :\text{Var}(\hat p) = \sum_{i=1}^n \text{Var}(w_i X_i) = p(1-p)\sum_{i=1}^n w_i^2. ",1068,-167,199.4,0.3359,0.082,A
+(a) 0.03 with $95 \%$ confidence?","The values for \hat{p} = 0.68, n = 400, z^* = 1.96 can now be substituted into the formula for one- sample proportion in the Z-interval: \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \Rightarrow (0.68) \pm (1.96) \sqrt{\frac{(0.68)(1-0.68)}{(400)}} \Rightarrow 0.68 \pm 1.96 \sqrt{0.000544} \Rightarrow \bigl(0.63429,0.72571\bigr) Based on the conditions of inference and the formula for the one-sample proportion in the Z-interval, it can be concluded with a 95% confidence level that the percentage of the voter population in this democracy supporting candidate B is between 63.429% and 72.571%. === Value of the parameter in the confidence interval range === A commonly asked question in inferential statistics is whether the parameter is included within a confidence interval. To answer the political scientist's question, a one- sample proportion in the Z-interval with a confidence level of 95% can be constructed in order to determine the population proportion of eligible voters in this democracy that support candidate B. ==== Solution ==== It is known from the random sample that \hat{p} = \frac{272}{400} = 0.68 with sample size n = 400. Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. ===Bootstrapping and Standard Error=== As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation. :N \geq 10(400) \Rightarrow N \geq 4000 :The population size N for this democracy's voters can be assumed to be at least 4,000. Thus, p_{\min} < p < p_{\max}, where: :\frac{\Gamma(n+1)}{\Gamma(x )\Gamma(n-x+1)}\int_0^{ p_{\min}} t^{x-1}(1-t)^{n-x}dt = \frac{\alpha}{2} :\frac{\Gamma(n+1)}{\Gamma(x+1)\Gamma(n-x)}\int_0^{ p_{\max}} t^{x}(1-t)^{n-x-1}dt = 1-\frac{\alpha}{2} The binomial proportion confidence interval is then ( p_{\min}, p_{\max}), as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. A population proportion can be estimated through the usage of a confidence interval known as a one-sample proportion in the Z-interval whose formula is given below: :\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} where \hat{p} is the sample proportion, n is the sample size, and z^* is the upper \frac{1-C}{2} critical value of the standard normal distribution for a level of confidence C. === Proof === In order to derive the formula for the one-sample proportion in the Z-interval, a sampling distribution of sample proportions needs to be taken into consideration. This can be verified mathematically with the following definition: #* Let n be the sample size of a given random sample and let \hat{p} be its sample proportion. * Since a random sample of 400 voters was obtained from the voting population, the condition for a simple random sample has been met. In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes nS are known. This approximation is based on the central limit theorem and is unreliable when the sample size is small or the success probability is close to 0 or 1. The value of 72% is a sample proportion. Suppose the following probability is calculated: :P(-z^*<\frac{\hat{p}-P}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}, where 0 and \pm z^* are the standard critical values. thumb|The sampling distribution of sample proportions is approximately normal when it satisfies the requirements of the Central Limit Theorem. A political scientist wants to determine what percentage of the voter population support candidate B. A random sample of 400 eligible voters in the democracy's voter population shows that 272 voters support candidate B. The likely size of the sampling error can generally be reduced by taking a larger sample. ===Sample Size Determination=== The cost of increasing a sample size may be prohibitive in reality. The mean of the sampling distribution of sample proportions is usually denoted as \mu_\hat{p} = P and its standard deviation is denoted as: :\sigma_\hat{p} = \sqrt{\frac{P(1-P)}{n}} Since the value of P is unknown, an unbiased statistic \hat{p} will be used for P. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed. ==Normal approximation interval or Wald interval == A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, \hat p, with a normal distribution. Hence, the values of C fall between 0 and 1, exclusively. === Estimation of P using ranked set sampling === A more precise estimate of P can be obtained by choosing ranked set sampling instead of simple random sampling ==See also== *Binomial proportion confidence interval *Confidence interval *Prevalence *Statistical hypothesis testing *Statistical inference *Statistical parameter *Tolerance interval == References == Category:Ratios In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). The solution for estimates the upper and lower limits of the confidence interval for . Since the X_i are independent and each one has variance \text{Var}(X_i) = p(1-p), the sampling variance of the proportion therefore is:How to calculate the standard error of a proportion using weighted data? :\text{Var}(\hat p) = \sum_{i=1}^n \text{Var}(w_i X_i) = p(1-p)\sum_{i=1}^n w_i^2. ",1068,-167,"""199.4""",0.3359,0.082,A
 "5.5-15. Let the distribution of $T$ be $t(17)$. Find
 (a) $t_{0.01}(17)$.
-","""Noncentral Student's t-Distribution."" The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's t distribution. Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution. However, in practice the distribution is rarely used, since tabulated values for T are hard to find. The t statistic is calculated as :t = \frac{\bar{X}_D - \mu_0}{s_D/\sqrt n} where \bar{X}_D and s_D are the average and standard deviation of the differences between all pairs. However, the central t-distribution can be used as an approximation to the noncentral t-distribution. The location/scale generalization of the central t-distribution is a different distribution from the noncentral t-distribution discussed in this article. In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. Another is Hotelling's T statistic follows a T distribution. The test statistic is approximately equal to 1.959, which gives a two-tailed p-value of 0.07857. ==Related statistical tests== ===Alternatives to the t-test for location problems=== The t-test provides an exact test for the equality of the means of two i.i.d. normal populations with unknown, but equal, variances. T helper 17 cells (Th17) are a subset of pro-inflammatory T helper cells defined by their production of interleukin 17 (IL-17). * The t-test p-value for the difference in means, and the regression p-value for the slope, are both 0.00805. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given. The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. Usually, T is converted instead to an F statistic. Galaxy 17 is a communications satellite owned by Intelsat to be located at 91° West longitude, serving the North American market. The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. If the test procedure rejects the null hypothesis whenever |T|>t_{1-\alpha/2}\,\\!, where t_{1-\alpha/2}\,\\! is the upper α/2 quantile of the (central) Student's t-distribution for a pre-specified α ∈ (0, 1), then the power of this test is given by :1-F_{n-1,\sqrt{n}\theta/\sigma}(t_{1-\alpha/2})+F_{n-1,\sqrt{n}\theta/\sigma}(-t_{1-\alpha/2}) . ",+2.9,1.1,2.567,1.88,0.139,C
+","""Noncentral Student's t-Distribution."" The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. When the scaling term is estimated based on the data, the test statistic—under certain conditions—follows a Student's t distribution. Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution. However, in practice the distribution is rarely used, since tabulated values for T are hard to find. The t statistic is calculated as :t = \frac{\bar{X}_D - \mu_0}{s_D/\sqrt n} where \bar{X}_D and s_D are the average and standard deviation of the differences between all pairs. However, the central t-distribution can be used as an approximation to the noncentral t-distribution. The location/scale generalization of the central t-distribution is a different distribution from the noncentral t-distribution discussed in this article. In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. Another is Hotelling's T statistic follows a T distribution. The test statistic is approximately equal to 1.959, which gives a two-tailed p-value of 0.07857. ==Related statistical tests== ===Alternatives to the t-test for location problems=== The t-test provides an exact test for the equality of the means of two i.i.d. normal populations with unknown, but equal, variances. T helper 17 cells (Th17) are a subset of pro-inflammatory T helper cells defined by their production of interleukin 17 (IL-17). * The t-test p-value for the difference in means, and the regression p-value for the slope, are both 0.00805. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given. The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. Usually, T is converted instead to an F statistic. Galaxy 17 is a communications satellite owned by Intelsat to be located at 91° West longitude, serving the North American market. The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. If the test procedure rejects the null hypothesis whenever |T|>t_{1-\alpha/2}\,\\!, where t_{1-\alpha/2}\,\\! is the upper α/2 quantile of the (central) Student's t-distribution for a pre-specified α ∈ (0, 1), then the power of this test is given by :1-F_{n-1,\sqrt{n}\theta/\sigma}(t_{1-\alpha/2})+F_{n-1,\sqrt{n}\theta/\sigma}(-t_{1-\alpha/2}) . ",+2.9,1.1,"""2.567""",1.88,0.139,C
 "5.5-1. Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute
-(a) $P(77<\bar{X}<79.5)$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545 This is related to confidence interval as used in statistics: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size n. ==Normality tests== The ""68–95–99.7 rule"" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. Let X_1, X_2, \dots, X_n denote a random sample of n independent observations from a population with overall expected value (average) \mu and finite variance, and denote the sample mean of that sample – itself a random variable – by \bar{X}_n. In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin{align} \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end{align} The usefulness of this heuristic especially depends on the question under consideration. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Suppose we are interested in the sample average \bar{X}_n \equiv \frac{X_1 + \cdots + X_n}{n}. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. For large enough n, the distribution of \bar{X}_n gets arbitrarily close to the normal distribution with mean \mu and variance \sigma^2/n. After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). We only need to calculate each integral for the cases n = 1,2,3. \begin{align} &\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{u^2}{2}}du \approx 0.6827 \\\ &\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) =\frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{u^2}{2}}du \approx 0.9545 \\\ &\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) = \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{u^2}{2}}du \approx 0.9973. \end{align} ==Cumulative distribution function== These numerical values ""68%, 95%, 99.7%"" come from the cumulative distribution function of the normal distribution. 77 (seventy-seven) is the natural number following 76 and preceding 78. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution. In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. Then the limit as n\to\infty of the distribution of \frac{\bar{X}_n-\mu}{\sigma_{\bar{X}_n}}, where \sigma_{\bar{X}_n}=\frac{\sigma}{\sqrt{n}}, is the standard normal distribution. The number , whose typical value is close to but not greater than 1, is sometimes given in the form \ 1 - \alpha\ (or as a percentage \ 100%\cdot( 1 - \alpha )\ ), where \ \alpha\ is a small positive number, often 0.05 . thumb|upright=1.3|Each row of points is a sample from the same normal distribution. There is no single accepted name for this number; it is also commonly referred to as the ""standard normal deviate"", ""normal score"" or ""Z score"" for the 97.5 percentile point, the .975 point, or just its approximate value, 1.96. Suppose we wanted to calculate a 95% confidence interval for μ. ",0.0526315789,0.66,0.1353,0.4772,-1.00,D
+(a) $P(77<\bar{X}<79.5)$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545 This is related to confidence interval as used in statistics: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size n. ==Normality tests== The ""68–95–99.7 rule"" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. Let X_1, X_2, \dots, X_n denote a random sample of n independent observations from a population with overall expected value (average) \mu and finite variance, and denote the sample mean of that sample – itself a random variable – by \bar{X}_n. In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin{align} \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end{align} The usefulness of this heuristic especially depends on the question under consideration. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Suppose we are interested in the sample average \bar{X}_n \equiv \frac{X_1 + \cdots + X_n}{n}. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. For large enough n, the distribution of \bar{X}_n gets arbitrarily close to the normal distribution with mean \mu and variance \sigma^2/n. After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). We only need to calculate each integral for the cases n = 1,2,3. \begin{align} &\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{u^2}{2}}du \approx 0.6827 \\\ &\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) =\frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{u^2}{2}}du \approx 0.9545 \\\ &\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) = \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{u^2}{2}}du \approx 0.9973. \end{align} ==Cumulative distribution function== These numerical values ""68%, 95%, 99.7%"" come from the cumulative distribution function of the normal distribution. 77 (seventy-seven) is the natural number following 76 and preceding 78. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution. In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. Then the limit as n\to\infty of the distribution of \frac{\bar{X}_n-\mu}{\sigma_{\bar{X}_n}}, where \sigma_{\bar{X}_n}=\frac{\sigma}{\sqrt{n}}, is the standard normal distribution. The number , whose typical value is close to but not greater than 1, is sometimes given in the form \ 1 - \alpha\ (or as a percentage \ 100%\cdot( 1 - \alpha )\ ), where \ \alpha\ is a small positive number, often 0.05 . thumb|upright=1.3|Each row of points is a sample from the same normal distribution. There is no single accepted name for this number; it is also commonly referred to as the ""standard normal deviate"", ""normal score"" or ""Z score"" for the 97.5 percentile point, the .975 point, or just its approximate value, 1.96. Suppose we wanted to calculate a 95% confidence interval for μ. ",0.0526315789,0.66,"""0.1353""",0.4772,-1.00,D
 "5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?
-","* Exponential distribution is a special case of type 3 Pearson distribution. In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. Reliability theory and reliability engineering also make extensive use of the exponential distribution. The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a probability of occurring. And for the group of 23 people, the probability of sharing is :p(23) \approx 1 - \left(\frac{364}{365}\right)^\binom{23}{2} = 1 - \left(\frac{364}{365}\right)^{253} \approx 0.500477 . ===Poisson approximation=== Applying the Poisson approximation for the binomial on the group of 23 people, :\operatorname{Poi}\left(\frac{\binom{23}{2}}{365}\right) =\operatorname{Poi}\left(\frac{253}{365}\right) \approx \operatorname{Poi}(0.6932) so :\Pr(X>0)=1-\Pr(X=0) \approx 1-e^{-0.6932} \approx 1-0.499998=0.500002. In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. X =\sum_{i=1}^k \sum_{j=i+1}^k X_{ij} \begin{alignat}{3} E[X] & = \sum_{i=1}^k \sum_{j=i+1}^k E[X_{ij}]\\\ & = \binom{k}{2} \frac{1}{n}\\\ & = \frac{k(k-1)}{2n}\\\ \end{alignat} For , if , the expected number of people with the same birthday is ≈ 1.0356. (The arrival of customers for instance is also modeled by the Poisson distribution if the arrivals are independent and distributed identically.) This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. ==Definitions== ===Probability density function=== The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin{cases} \lambda e^{ - \lambda x} & x \ge 0, \\\ 0 & x < 0\. \end{cases} Here λ > 0 is the parameter of the distribution, often called the rate parameter. This can be seen by considering the complementary cumulative distribution function: \begin{align} \Pr\left(T > s + t \mid T > s\right) &= \frac{\Pr\left(T > s + t \cap T > s\right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{\Pr\left(T > s + t \right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\\\[4pt] &= e^{-\lambda t} \\\\[4pt] &= \Pr(T > t). \end{align} When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the ""failure rates"" here are not constant: more failures occur for very young and for very old systems. Similar caveats apply to the following examples which yield approximately exponentially distributed variables: * The time until a radioactive particle decays, or the time between clicks of a Geiger counter * The time it takes before your next telephone call * The time until default (on payment to company debt holders) in reduced-form credit risk modeling Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand, or between roadkills on a given road. The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people . ===Probability of a shared birthday (collision)=== The birthday problem can be generalized as follows: :Given random integers drawn from a discrete uniform distribution with range , what is the probability that at least two numbers are the same? ( gives the usual birthday problem.) But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. \\\ V_{t} &= n^{k} = 365^{23} \\\ P(A) &= \frac{V_{nr}}{V_{t}} \approx 0.492703 \\\ P(B) &= 1 - P(A) \approx 1 - 0.492703 \approx 0.507297 (50.7297%)\end{align} Another way the birthday problem can be solved is by asking for an approximate probability that in a group of people at least two have the same birthday. The exponential distribution is not the same as the class of exponential families of distributions. This implies that the expected number of people with a non-shared (unique) birthday is: : n \left( \frac{d-1}{d} \right)^{n-1} Similar formulas can be derived for the expected number of people who share with three, four, etc. other people. === Number of people until every birthday is achieved === The expected number of people needed until every birthday is achieved is called the Coupon collector's problem. Note that the vertical scale is logarithmic (each step down is 1020 times less likely). : 1 0.0% 5 2.7% 10 11.7% 20 41.1% 23 50.7% 30 70.6% 40 89.1% 50 97.0% 60 99.4% 70 99.9% 75 99.97% 100 % 200 % 300 (100 − )% 350 (100 − )% 365 (100 − )% ≥ 366 100% ==Approximations== thumb|right|upright=1.4|Graphs showing the approximate probabilities of at least two people sharing a birthday () and its complementary event () thumb|right|upright=1.4|A graph showing the accuracy of the approximation () The Taylor series expansion of the exponential function (the constant ) : e^x = 1 + x + \frac{x^2}{2!}+\cdots provides a first-order approximation for for |x| \ll 1: : e^x \approx 1 + x. Consequently, the desired probability is . In operating-rooms management, the distribution of surgery duration for a category of surgeries with no typical work-content (like in an emergency room, encompassing all types of surgeries). ===Prediction=== Having observed a sample of n data points from an unknown exponential distribution a common task is to use these samples to make predictions about future data from the same source. In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. ",1.60,1.88,1.86,2.25,0.5768,E
+","* Exponential distribution is a special case of type 3 Pearson distribution. In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. Reliability theory and reliability engineering also make extensive use of the exponential distribution. The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a probability of occurring. And for the group of 23 people, the probability of sharing is :p(23) \approx 1 - \left(\frac{364}{365}\right)^\binom{23}{2} = 1 - \left(\frac{364}{365}\right)^{253} \approx 0.500477 . ===Poisson approximation=== Applying the Poisson approximation for the binomial on the group of 23 people, :\operatorname{Poi}\left(\frac{\binom{23}{2}}{365}\right) =\operatorname{Poi}\left(\frac{253}{365}\right) \approx \operatorname{Poi}(0.6932) so :\Pr(X>0)=1-\Pr(X=0) \approx 1-e^{-0.6932} \approx 1-0.499998=0.500002. In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. X =\sum_{i=1}^k \sum_{j=i+1}^k X_{ij} \begin{alignat}{3} E[X] & = \sum_{i=1}^k \sum_{j=i+1}^k E[X_{ij}]\\\ & = \binom{k}{2} \frac{1}{n}\\\ & = \frac{k(k-1)}{2n}\\\ \end{alignat} For , if , the expected number of people with the same birthday is ≈ 1.0356. (The arrival of customers for instance is also modeled by the Poisson distribution if the arrivals are independent and distributed identically.) This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. ==Definitions== ===Probability density function=== The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin{cases} \lambda e^{ - \lambda x} & x \ge 0, \\\ 0 & x < 0\. \end{cases} Here λ > 0 is the parameter of the distribution, often called the rate parameter. This can be seen by considering the complementary cumulative distribution function: \begin{align} \Pr\left(T > s + t \mid T > s\right) &= \frac{\Pr\left(T > s + t \cap T > s\right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{\Pr\left(T > s + t \right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\\\[4pt] &= e^{-\lambda t} \\\\[4pt] &= \Pr(T > t). \end{align} When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the ""failure rates"" here are not constant: more failures occur for very young and for very old systems. Similar caveats apply to the following examples which yield approximately exponentially distributed variables: * The time until a radioactive particle decays, or the time between clicks of a Geiger counter * The time it takes before your next telephone call * The time until default (on payment to company debt holders) in reduced-form credit risk modeling Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand, or between roadkills on a given road. The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people . ===Probability of a shared birthday (collision)=== The birthday problem can be generalized as follows: :Given random integers drawn from a discrete uniform distribution with range , what is the probability that at least two numbers are the same? ( gives the usual birthday problem.) But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. \\\ V_{t} &= n^{k} = 365^{23} \\\ P(A) &= \frac{V_{nr}}{V_{t}} \approx 0.492703 \\\ P(B) &= 1 - P(A) \approx 1 - 0.492703 \approx 0.507297 (50.7297%)\end{align} Another way the birthday problem can be solved is by asking for an approximate probability that in a group of people at least two have the same birthday. The exponential distribution is not the same as the class of exponential families of distributions. This implies that the expected number of people with a non-shared (unique) birthday is: : n \left( \frac{d-1}{d} \right)^{n-1} Similar formulas can be derived for the expected number of people who share with three, four, etc. other people. === Number of people until every birthday is achieved === The expected number of people needed until every birthday is achieved is called the Coupon collector's problem. Note that the vertical scale is logarithmic (each step down is 1020 times less likely). : 1 0.0% 5 2.7% 10 11.7% 20 41.1% 23 50.7% 30 70.6% 40 89.1% 50 97.0% 60 99.4% 70 99.9% 75 99.97% 100 % 200 % 300 (100 − )% 350 (100 − )% 365 (100 − )% ≥ 366 100% ==Approximations== thumb|right|upright=1.4|Graphs showing the approximate probabilities of at least two people sharing a birthday () and its complementary event () thumb|right|upright=1.4|A graph showing the accuracy of the approximation () The Taylor series expansion of the exponential function (the constant ) : e^x = 1 + x + \frac{x^2}{2!}+\cdots provides a first-order approximation for for |x| \ll 1: : e^x \approx 1 + x. Consequently, the desired probability is . In operating-rooms management, the distribution of surgery duration for a category of surgeries with no typical work-content (like in an emergency room, encompassing all types of surgeries). ===Prediction=== Having observed a sample of n data points from an unknown exponential distribution a common task is to use these samples to make predictions about future data from the same source. In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. ",1.60,1.88,"""1.86""",2.25,0.5768,E
 "7.3-9. Consider the following two groups of women: Group 1 consists of women who spend less than $\$ 500$ annually on clothes; Group 2 comprises women who spend over $\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive,
 (a) Give a point estimate of $p_1-p_2$.
-","The joint hypothesis problem is the problem that testing for market efficiency is difficult, or even impossible. The following estimate only replaces the population variances by the sample variances: :\hat u \approx \frac{(g_1 + g_2)^2}{g_1^2/(n_1-1) + g_2^2/(n_2-1)} \quad \text{ where } g_i = s_i^2/n_i. Many other methods of treating the problem have been proposed since, and the effect on the resulting confidence intervals have been investigated. ===Welch's approximate t solution=== A widely used method is that of B. L. Welch,Welch (1938, 1947) who, like Fisher, was at University College London. This p_i is proportional to some known quantity x_i so that p_i = \frac{x_i}{\sum_{i=1}^N x_i}.Skinner, Chris J. ""Probability proportional to size (PPS) sampling."" thumb|250px|Budget constraint, where A=\frac{m}{P_y} and B=\frac{m}{P_x} In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. ===Bootstrapping and Standard Error=== As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation. * Estimator of true probability (Frequentist approach). ""On the theory of sampling from finite populations."" *In statistics, the estimate of a proportion of a sample (denoted by p) has a standard error given by: :s_p = \sqrt{ \frac {p \, (1-p) } {n} } where n is the number of trials (which was denoted by N in the previous section). The likely size of the sampling error can generally be reduced by taking a larger sample. ===Sample Size Determination=== The cost of increasing a sample size may be prohibitive in reality. If consideration is restricted to classical statistical inference only, it is possible to seek solutions to the inference problem that are simple to apply in a practical sense, giving preference to this simplicity over any inaccuracy in the corresponding probability statements. In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Sampling Techniques (3rd ed.). :E = \frac {Z}{ 2 \, \sqrt{n} } :E = \frac {Z}{ 2 \, \sqrt{ 10000 } } = \frac {Z}{ 200 } :E = 0.0050\, at 68.27% level of confidence (Z=1) :E = 0.0100\, at 95.45% level of confidence (Z=2) :E = 0.0165\, at 99.90% level of confidence (Z=3.3) 3\. The difference between the sample statistic and population parameter is considered the sampling error.Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, For example, if one measures the height of a thousand individuals from a population of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. For example, attempting to measure the average height of the entire human population of the Earth, but measuring a sample only from one country, could result in a large over- or under-estimation. (The TOTs are given by the price ratio Px/Py, where x is the exportable commodity and y is the importable). == Many goods == While low-level demonstrations of budget constraints are often limited to less than two good situations which provide easy graphical representation, it is possible to demonstrate the relationship between multiple goods through a budget constraint. In such a case, assuming there are n\, goods, called x_i\, for i=1,\dots,n\,, that the price of good x_i\, is denoted by p_i\,, and if \,W\, is the total amount that may be spent, then the budget constraint is: :\sum_{i=1}^np_ix_i\leq W. The pps sampling results in a fixed sample size n (as opposed to Poisson sampling which is similar but results in a random sample size with expectancy of n). In survey methodology, probability-proportional-to-size (pps) sampling is a sampling process where each element of the population (of size N) has some (independent) chance p_i to be selected to the sample when performing one draw. The sampling error is the difference between a sample statistic used to estimate a population parameter and the actual but unknown value of the parameter. ===Effective Sampling=== In statistics, a truly random sample means selecting individuals from a population with an equivalent probability; in other words, picking individuals from a group without bias. ",4152,0.2115,-24.0,1.45,-1,B
+","The joint hypothesis problem is the problem that testing for market efficiency is difficult, or even impossible. The following estimate only replaces the population variances by the sample variances: :\hat u \approx \frac{(g_1 + g_2)^2}{g_1^2/(n_1-1) + g_2^2/(n_2-1)} \quad \text{ where } g_i = s_i^2/n_i. Many other methods of treating the problem have been proposed since, and the effect on the resulting confidence intervals have been investigated. ===Welch's approximate t solution=== A widely used method is that of B. L. Welch,Welch (1938, 1947) who, like Fisher, was at University College London. This p_i is proportional to some known quantity x_i so that p_i = \frac{x_i}{\sum_{i=1}^N x_i}.Skinner, Chris J. ""Probability proportional to size (PPS) sampling."" thumb|250px|Budget constraint, where A=\frac{m}{P_y} and B=\frac{m}{P_x} In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Since the sample error can often be estimated beforehand as a function of the sample size, various methods of sample size determination are used to weigh the predicted accuracy of an estimator against the predicted cost of taking a larger sample. ===Bootstrapping and Standard Error=== As discussed, a sample statistic, such as an average or percentage, will generally be subject to sample-to-sample variation. * Estimator of true probability (Frequentist approach). ""On the theory of sampling from finite populations."" *In statistics, the estimate of a proportion of a sample (denoted by p) has a standard error given by: :s_p = \sqrt{ \frac {p \, (1-p) } {n} } where n is the number of trials (which was denoted by N in the previous section). The likely size of the sampling error can generally be reduced by taking a larger sample. ===Sample Size Determination=== The cost of increasing a sample size may be prohibitive in reality. If consideration is restricted to classical statistical inference only, it is possible to seek solutions to the inference problem that are simple to apply in a practical sense, giving preference to this simplicity over any inaccuracy in the corresponding probability statements. In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Sampling Techniques (3rd ed.). :E = \frac {Z}{ 2 \, \sqrt{n} } :E = \frac {Z}{ 2 \, \sqrt{ 10000 } } = \frac {Z}{ 200 } :E = 0.0050\, at 68.27% level of confidence (Z=1) :E = 0.0100\, at 95.45% level of confidence (Z=2) :E = 0.0165\, at 99.90% level of confidence (Z=3.3) 3\. The difference between the sample statistic and population parameter is considered the sampling error.Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, For example, if one measures the height of a thousand individuals from a population of one million, the average height of the thousand is typically not the same as the average height of all one million people in the country. For example, attempting to measure the average height of the entire human population of the Earth, but measuring a sample only from one country, could result in a large over- or under-estimation. (The TOTs are given by the price ratio Px/Py, where x is the exportable commodity and y is the importable). == Many goods == While low-level demonstrations of budget constraints are often limited to less than two good situations which provide easy graphical representation, it is possible to demonstrate the relationship between multiple goods through a budget constraint. In such a case, assuming there are n\, goods, called x_i\, for i=1,\dots,n\,, that the price of good x_i\, is denoted by p_i\,, and if \,W\, is the total amount that may be spent, then the budget constraint is: :\sum_{i=1}^np_ix_i\leq W. The pps sampling results in a fixed sample size n (as opposed to Poisson sampling which is similar but results in a random sample size with expectancy of n). In survey methodology, probability-proportional-to-size (pps) sampling is a sampling process where each element of the population (of size N) has some (independent) chance p_i to be selected to the sample when performing one draw. The sampling error is the difference between a sample statistic used to estimate a population parameter and the actual but unknown value of the parameter. ===Effective Sampling=== In statistics, a truly random sample means selecting individuals from a population with an equivalent probability; in other words, picking individuals from a group without bias. ",4152,0.2115,"""-24.0""",1.45,-1,B
 "5.6-9. In Example 5.6-4, compute $P(1.7 \leq Y \leq 3.2)$ with $n=4$ and compare your answer with the normal approximation of this probability.
-","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. Inverse Probability. There is no single accepted name for this number; it is also commonly referred to as the ""standard normal deviate"", ""normal score"" or ""Z score"" for the 97.5 percentile point, the .975 point, or just its approximate value, 1.96. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. right|thumb|Lines 10580–10594, columns 21–40, from A Million Random Digits with 100,000 Normal Deviates A Million Random Digits with 100,000 Normal Deviates is a random number book by the RAND Corporation, originally published in 1955. Probability. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. thumb|upright=1.3|Each row of points is a sample from the same normal distribution. The upper bounds for C0 were subsequently lowered from the original estimate 7.59 due to to (considering recent results only) 0.9051 due to , 0.7975 due to , 0.7915 due to , 0.6379 and 0.5606 due to and . the best estimate is 0.5600 obtained by . ===Multidimensional version=== As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.Bentkus, Vidmantas. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval. Thus, the probability that T will be between -c and +c is 95%. Approximation Theorems of Mathematical Statistics. * A particular confidence level of 95% calculated from an experiment does not mean that there is a 95% probability of a sample parameter from a repeat of the experiment falling within this interval. == Counterexamples == Since confidence interval theory was proposed, a number of counter-examples to the theory have been developed to show how the interpretation of confidence intervals can be problematic, at least if one interprets them naïvely. === Confidence procedure for uniform location === Welch presented an example which clearly shows the difference between the theory of confidence intervals and other theories of interval estimation (including Fisher's fiducial intervals and objective Bayesian intervals). * Category:Probabilistic inequalities Category:Theorems in statistics Category:Central limit theorem Welch showed that the first confidence procedure dominates the second, according to desiderata from confidence interval theory; for every \theta_1 eq\theta, the probability that the first procedure contains \theta_1 is less than or equal to the probability that the second procedure contains \theta_1. ",0.6749,435,1.2,4.5,15.1,A
+","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. Inverse Probability. There is no single accepted name for this number; it is also commonly referred to as the ""standard normal deviate"", ""normal score"" or ""Z score"" for the 97.5 percentile point, the .975 point, or just its approximate value, 1.96. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. right|thumb|Lines 10580–10594, columns 21–40, from A Million Random Digits with 100,000 Normal Deviates A Million Random Digits with 100,000 Normal Deviates is a random number book by the RAND Corporation, originally published in 1955. Probability. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. thumb|upright=1.3|Each row of points is a sample from the same normal distribution. The upper bounds for C0 were subsequently lowered from the original estimate 7.59 due to to (considering recent results only) 0.9051 due to , 0.7975 due to , 0.7915 due to , 0.6379 and 0.5606 due to and . the best estimate is 0.5600 obtained by . ===Multidimensional version=== As with the multidimensional central limit theorem, there is a multidimensional version of the Berry–Esseen theorem.Bentkus, Vidmantas. The 95% probability relates to the reliability of the estimation procedure, not to a specific calculated interval. Thus, the probability that T will be between -c and +c is 95%. Approximation Theorems of Mathematical Statistics. * A particular confidence level of 95% calculated from an experiment does not mean that there is a 95% probability of a sample parameter from a repeat of the experiment falling within this interval. == Counterexamples == Since confidence interval theory was proposed, a number of counter-examples to the theory have been developed to show how the interpretation of confidence intervals can be problematic, at least if one interprets them naïvely. === Confidence procedure for uniform location === Welch presented an example which clearly shows the difference between the theory of confidence intervals and other theories of interval estimation (including Fisher's fiducial intervals and objective Bayesian intervals). * Category:Probabilistic inequalities Category:Theorems in statistics Category:Central limit theorem Welch showed that the first confidence procedure dominates the second, according to desiderata from confidence interval theory; for every \theta_1 eq\theta, the probability that the first procedure contains \theta_1 is less than or equal to the probability that the second procedure contains \theta_1. ",0.6749,435,"""1.2""",4.5,15.1,A
 "5.8-5. If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when
-(c) $n=1000$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The lower bound is expressed in terms of the probabilities for pairs of events. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. thumb|Plot of S_n/n (red), its standard deviation 1/\sqrt{n} (blue) and its bound \sqrt{2\log\log n/n} given by LIL (green). From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. Then : \begin{align} P\left [ {Z}_i^r \right ] = \left(1-\frac{1}{n}\right)^r \le e^{-r / n}. \end{align} Thus, for r = \beta n \log n, we have P\left [ {Z}_i^r \right ] \le e^{(-\beta n \log n ) / n} = n^{-\beta}. right|thumb|300px| Probability mass function for Fisher's noncentral hypergeometric distribution for different values of the odds ratio ω. m1 = 80, m2 = 60, n = 100, ω = 0.01, ..., 1000 In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. Bound the desired probability using the Chebyshev inequality: :\operatorname{P}\left(|T- n H_n| \geq cn\right) \le \frac{\pi^2}{6c^2}. ===Tail estimates=== A stronger tail estimate for the upper tail be obtained as follows. The probability function and a simple approximation to the mean are given to the right. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). Probability. In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. The calculation time for the probability function can be high when the sum in P0 has many terms. Using the Markov inequality to bound the desired probability: :\operatorname{P}(T \geq cn H_n) \le \frac{1}{c}. Their odds ratio is given as : \omega = \frac{\omega_X}{\omega_Y} = \frac{\pi_X/(1-\pi_X)}{\pi_Y/(1-\pi_Y)} . Then : \Pr\left( \limsup_n \frac{S_n}{\sqrt{n}} \geq M \right) \geqslant \limsup_n \Pr\left( \frac{S_n}{\sqrt{n}} \geq M \right) = \Pr\left( \mathcal{N}(0, 1) \geq M \right) > 0 so :\limsup_n \frac{S_n}{\sqrt{n}}=\infty \qquad \text{with probability 1.} ",0.925,2.3,1.5377,-1.00,71,A
+(c) $n=1000$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The lower bound is expressed in terms of the probabilities for pairs of events. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. thumb|Plot of S_n/n (red), its standard deviation 1/\sqrt{n} (blue) and its bound \sqrt{2\log\log n/n} given by LIL (green). From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. Then : \begin{align} P\left [ {Z}_i^r \right ] = \left(1-\frac{1}{n}\right)^r \le e^{-r / n}. \end{align} Thus, for r = \beta n \log n, we have P\left [ {Z}_i^r \right ] \le e^{(-\beta n \log n ) / n} = n^{-\beta}. right|thumb|300px| Probability mass function for Fisher's noncentral hypergeometric distribution for different values of the odds ratio ω. m1 = 80, m2 = 60, n = 100, ω = 0.01, ..., 1000 In probability theory and statistics, Fisher's noncentral hypergeometric distribution is a generalization of the hypergeometric distribution where sampling probabilities are modified by weight factors. Bound the desired probability using the Chebyshev inequality: :\operatorname{P}\left(|T- n H_n| \geq cn\right) \le \frac{\pi^2}{6c^2}. ===Tail estimates=== A stronger tail estimate for the upper tail be obtained as follows. The probability function and a simple approximation to the mean are given to the right. It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). Probability. In probability theory, the Chung–Erdős inequality provides a lower bound on the probability that one out of many (possibly dependent) events occurs. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. The calculation time for the probability function can be high when the sum in P0 has many terms. Using the Markov inequality to bound the desired probability: :\operatorname{P}(T \geq cn H_n) \le \frac{1}{c}. Their odds ratio is given as : \omega = \frac{\omega_X}{\omega_Y} = \frac{\pi_X/(1-\pi_X)}{\pi_Y/(1-\pi_Y)} . Then : \Pr\left( \limsup_n \frac{S_n}{\sqrt{n}} \geq M \right) \geqslant \limsup_n \Pr\left( \frac{S_n}{\sqrt{n}} \geq M \right) = \Pr\left( \mathcal{N}(0, 1) \geq M \right) > 0 so :\limsup_n \frac{S_n}{\sqrt{n}}=\infty \qquad \text{with probability 1.} ",0.925,2.3,"""1.5377""",-1.00,71,A
 "7.5-1. Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\pi_p$. Compute
-(a) $P\left(Y_2 < \pi_{0.5} < Y_5\right)$.","If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sample. Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median. === Large sample sizes === For the uniform distribution, as n tends to infinity, the pth sample quantile is asymptotically normally distributed, since it is approximated by : U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right). In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n). == See also == * Rankit * Box plot * BRS-inequality * Concomitant (statistics) * Fisher–Tippett distribution * Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables * Bernstein polynomial * L-estimator – linear combinations of order statistics * Rank-size distribution * Selection algorithm === Examples of order statistics === * Sample maximum and minimum * Quantile * Percentile * Decile * Quartile * Median == References == == External links == * Retrieved Feb 02,2005 * Retrieved Feb 02,2005 * C++ source Dynamic Order Statistics Category:Nonparametric statistics Category:Summary statistics Category:Permutations Similar remarks apply to all sample quantiles. == Probabilistic analysis == Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order. We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the cdf. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. == Notation and examples == For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. In this particular case, a better confidence interval for the median is the one delimited by the 2nd and 5th order statistics, which contains the population median with probability :\left[{6\choose 2}+{6\choose 3}+{6\choose 4}\right](1/2)^{6} = {25\over 32} \approx 78\%. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. To find the probabilities of the k^\text{th} order statistics, three values are first needed, namely :p_1=P(Xx)=1-F(x). In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of U_{(n)}-U_{(1)}, i.e. maximum minus the minimum. The cumulative distribution function of the k^\text{th} order statistic can be computed by noting that : \begin{align} P(X_{(k)}\leq x)& =P(\text{there are at least }k\text{ observations less than or equal to }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}p_3^j(p_1+p_2)^{n-j} . \end{align} Similarly, P(X_{(k)} is given by : \begin{align} P(X_{(k)}< x)& =P(\text{there are at least }k\text{ observations less than }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than or equal to }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}(p_2+p_3)^j(p_1)^{n-j} . \end{align} Note that the probability mass function of X_{(k)} is just the difference of these values, that is to say : \begin{align} P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}< x) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^j(p_1)^{n-j}\right) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left((1-F(x))^j(F(x))^{n-j}-(1-F(x)+f(x))^j(F(x)-f(x))^{n-j}\right). \end{align} == Computing order statistics == The problem of computing the kth smallest (or largest) element of a list is called the selection problem and is solved by a selection algorithm. In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.Roscoe, J. T. (1975). With such a small sample size, if one wants at least 95% confidence, one is reduced to saying that the median is between the minimum and the maximum of the 6 observations with probability 31/32 or approximately 97%. If the distribution is normally distributed, the percentile rank can be inferred from the standard score. ==See also== * Quantile * Percentile ==References== Category:Summary statistics The mean of this distribution is k / (n + 1). ==== The joint distribution of the order statistics of the uniform distribution ==== Similarly, for i < j, the joint probability density function of the two order statistics U(i) < U(j) can be shown to be :f_{U_{(i)},U_{(j)}}(u,v) = n!{u^{i-1}\over (i-1)!}{(v-u)^{j-i-1}\over(j-i-1)!}{(1-v)^{n-j}\over (n-j)!} which is (up to terms of higher order than O(du\,dv)) the probability that i − 1, 1, j − 1 − i, 1 and n − j sample elements fall in the intervals (0,u), (u,u+du), (u+du,v), (v,v+dv), (v+dv,1) respectively. However, we know from the preceding discussion that the probability that this interval actually contains the population median is :{6\choose 3}(1/2)^{6} = {5\over 16} \approx 31\%. One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. In other words, all n order statistics are needed from the n observations in a sample. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant: :f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!. The sample median may or may not be an order statistic, since there is a single middle value only when the number of observations is odd. ",3.54,-11.2,240.0,0.7812,1.27,D
+(a) $P\left(Y_2 < \pi_{0.5} < Y_5\right)$.","If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sample. Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median. === Large sample sizes === For the uniform distribution, as n tends to infinity, the pth sample quantile is asymptotically normally distributed, since it is approximated by : U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right). In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n). == See also == * Rankit * Box plot * BRS-inequality * Concomitant (statistics) * Fisher–Tippett distribution * Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables * Bernstein polynomial * L-estimator – linear combinations of order statistics * Rank-size distribution * Selection algorithm === Examples of order statistics === * Sample maximum and minimum * Quantile * Percentile * Decile * Quartile * Median == References == == External links == * Retrieved Feb 02,2005 * Retrieved Feb 02,2005 * C++ source Dynamic Order Statistics Category:Nonparametric statistics Category:Summary statistics Category:Permutations Similar remarks apply to all sample quantiles. == Probabilistic analysis == Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order. We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the cdf. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. == Notation and examples == For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. In this particular case, a better confidence interval for the median is the one delimited by the 2nd and 5th order statistics, which contains the population median with probability :\left[{6\choose 2}+{6\choose 3}+{6\choose 4}\right](1/2)^{6} = {25\over 32} \approx 78\%. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. To find the probabilities of the k^\text{th} order statistics, three values are first needed, namely :p_1=P(Xx)=1-F(x). In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of U_{(n)}-U_{(1)}, i.e. maximum minus the minimum. The cumulative distribution function of the k^\text{th} order statistic can be computed by noting that : \begin{align} P(X_{(k)}\leq x)& =P(\text{there are at least }k\text{ observations less than or equal to }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}p_3^j(p_1+p_2)^{n-j} . \end{align} Similarly, P(X_{(k)} is given by : \begin{align} P(X_{(k)}< x)& =P(\text{there are at least }k\text{ observations less than }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than or equal to }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}(p_2+p_3)^j(p_1)^{n-j} . \end{align} Note that the probability mass function of X_{(k)} is just the difference of these values, that is to say : \begin{align} P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}< x) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^j(p_1)^{n-j}\right) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left((1-F(x))^j(F(x))^{n-j}-(1-F(x)+f(x))^j(F(x)-f(x))^{n-j}\right). \end{align} == Computing order statistics == The problem of computing the kth smallest (or largest) element of a list is called the selection problem and is solved by a selection algorithm. In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.Roscoe, J. T. (1975). With such a small sample size, if one wants at least 95% confidence, one is reduced to saying that the median is between the minimum and the maximum of the 6 observations with probability 31/32 or approximately 97%. If the distribution is normally distributed, the percentile rank can be inferred from the standard score. ==See also== * Quantile * Percentile ==References== Category:Summary statistics The mean of this distribution is k / (n + 1). ==== The joint distribution of the order statistics of the uniform distribution ==== Similarly, for i < j, the joint probability density function of the two order statistics U(i) < U(j) can be shown to be :f_{U_{(i)},U_{(j)}}(u,v) = n!{u^{i-1}\over (i-1)!}{(v-u)^{j-i-1}\over(j-i-1)!}{(1-v)^{n-j}\over (n-j)!} which is (up to terms of higher order than O(du\,dv)) the probability that i − 1, 1, j − 1 − i, 1 and n − j sample elements fall in the intervals (0,u), (u,u+du), (u+du,v), (v,v+dv), (v+dv,1) respectively. However, we know from the preceding discussion that the probability that this interval actually contains the population median is :{6\choose 3}(1/2)^{6} = {5\over 16} \approx 31\%. One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. In other words, all n order statistics are needed from the n observations in a sample. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant: :f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!. The sample median may or may not be an order statistic, since there is a single middle value only when the number of observations is odd. ",3.54,-11.2,"""240.0""",0.7812,1.27,D
 "5.2-13. Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a
 device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \infty $
 (a) Find $G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right)$.
-","The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008).Tahmasbi, R., Rezaei, S., (2008), ""A two-parameter lifetime distribution with decreasing failure rate"", Computational Statistics and Data Analysis, 52 (8), 3889-3901. The pdf for the standard fatigue life distribution reduces to : f(x) = \frac{\sqrt{x}+\sqrt{\frac{1}{x}}}{2\gamma x}\phi\left(\frac{\sqrt{x}-\sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma >0 Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function. ==Cumulative distribution function== The formula for the cumulative distribution function is : F(x) = \Phi\left(\frac{\sqrt{x} - \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma > 0 where Φ is the cumulative distribution function of the standard normal distribution. ==Quantile function== The formula for the quantile function is : G(p) = \frac{1}{4}\left[\gamma\Phi^{-1}(p) + \sqrt{4+\left(\gamma\Phi^{-1}(p)\right)^2}\right]^2 where Φ −1 is the quantile function of the standard normal distribution. ==References== * * * * * * * ==External links== *Fatigue life distribution Category:Continuous distributions Then there are seven nonempty subsets of { 1, ..., b } = { 1, 2, 3 }; hence seven different exponential random variables: : E_{\\{1\\}}, E_{\\{2\\}}, E_{\\{3\\}}, E_{\\{1,2\\}}, E_{\\{1,3\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} Then we have: : \begin{align} T_1 & = \min\\{ E_{\\{1\\}}, E_{\\{1,2\\}}, E_{\\{1,3\\}}, E_{\\{1,2,3\\}} \\} \\\ T_2 & = \min\\{ E_{\\{2\\}}, E_{\\{1,2\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} \\} \\\ T_3 & = \min\\{ E_{\\{3\\}}, E_{\\{1,3\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} \\} \\\ \end{align} ==References== * Xu M, Xu S. The joint distribution of T=(T_1,\ldots,T_b) is called the Marshall–Olkin exponential distribution with parameters \\{\lambda _B,B\subset \\{1,2,\ldots,b\\}\\}. === Concrete example === Suppose b = 3\. In this situation, the energy distance is zero if and only if X and Y are identically distributed. Vilnius, 2009 If X is defined to be the random variable which is the minimum of N independent realisations from an exponential distribution with rate parameter β, and if N is a realisation from a logarithmic distribution (where the parameter p in the usual parameterisation is replaced by ), then X has the exponential- logarithmic distribution in the parameterisation used above. ==References== Category:Continuous distributions Category:Survival analysis X is then distributed normally with a mean of zero and a variance of α2 / 4. ==Probability density function== The general formula for the probability density function (pdf) is : f(x) = \frac{\sqrt{\frac{x-\mu}{\beta}}+\sqrt{\frac{\beta}{x-\mu}}}{2\gamma\left(x-\mu\right)}\phi\left(\frac{\sqrt{\frac{x-\mu}{\beta}}-\sqrt{\frac{\beta}{x-\mu}}}{\gamma}\right)\quad x > \mu; \gamma,\beta>0 where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and \phi is the probability density function of the standard normal distribution. ==Standard fatigue life distribution== The case where μ = 0 and β = 1 is called the standard fatigue life distribution. thumb|Diagram showing queueing system equivalent of a hyperexponential distribution In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable X is given by : f_X(x) = \sum_{i=1}^n f_{Y_i}(x)\;p_i, where each Yi is an exponentially distributed random variable with rate parameter λi, and pi is the probability that X will take on the form of the exponential distribution with rate λi. {1-(1-p) e^{-\beta x}} | cdf = 1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p} | mean = -\frac{\text{polylog}(2,1-p)}{\beta\ln p} | median = \frac{\ln(1+\sqrt{p})}{\beta} | mode = 0 | variance = -\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p} -\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p} | skewness = | kurtosis = | entropy = | mgf = -\frac{\beta(1-p)}{\ln p (\beta-t)} \text{hypergeom}_{2,1} ([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p) | cf = | pgf = | fisher = }} In probability theory and statistics, the Exponential-Logarithmic (EL) distribution is a family of lifetime distributions with decreasing failure rate, defined on the interval [0, ∞). If T is the number of cycles to failure then the cumulative distribution function (cdf) of T is : P( T \le t ) = 1 - \Phi\left( \frac{ \omega - t \mu }{ \sigma \sqrt{ t } } \right) = \Phi\left( \frac{ t \mu - \omega }{ \sigma \sqrt{ t } } \right) = \Phi\left( \frac{ \mu \sqrt{ t } }{ \sigma } - \frac{ \omega }{ \sigma \sqrt{t} } \right) = \Phi\left( \frac{ \sqrt{ \mu \omega } }{ \sigma } \left[ \left( \frac{ t }{ \omega / \mu } \right)^{ 0.5 } - \left( \frac{ \omega / \mu }{ t } \right)^{ 0.5 } \right] \right) The more usual form of this distribution is: : F( x; \alpha, \beta ) = \Phi\left( \frac{ 1 }{ \alpha } \left[ \left( \frac{ x }{ \beta } \right)^{0.5} - \left( \frac{ \beta }{ x } \right)^{0.5} \right] \right) Here α is the shape parameter and β is the scale parameter. ==Properties== The Birnbaum–Saunders distribution is unimodal with a median of β. In applied statistics, the Marshall–Olkin exponential distribution is any member of a certain family of continuous multivariate probability distributions with positive-valued components. The Birnbaum-Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. Category:Statistics articles needing expert attention Category:Continuous distributions Category:Exponentials Category:Exponential family distributions In general, the lifetime of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms). If X and Y are independent random vectors in Rd with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of : D^2(F, G) = 2\operatorname E\|X - Y\| - \operatorname E\|X - X'\| - \operatorname E\|Y - Y'\| \geq 0, where (X, X', Y, Y') are independent, the cdf of X and X' is F, the cdf of Y and Y' is G, \operatorname E is the expected value, and || . || denotes the length of a vector. Hence the mean and variance of the EL distribution are given, respectively, by :E(X)=-\frac{\operatorname{Li}_2(1-p)}{\beta\ln p}, :\operatorname{Var}(X)=-\frac{2 \operatorname{Li}_3(1-p)}{\beta^2\ln p}-\left(\frac{ \operatorname{Li}_2(1-p)}{\beta\ln p}\right)^2. === The survival, hazard and mean residual life functions === thumb|300px|Hazard function The survival function (also known as the reliability function) and hazard function (also known as the failure rate function) of the EL distribution are given, respectively, by : s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p}, : h(x)=\frac{-\beta(1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}. The mean residual lifetime of the EL distribution is given by : m(x_0;p,\beta)=E(X-x_0|X\geq x_0;\beta,p)=-\frac{\operatorname{Li}_2(1-(1-p)e^{-\beta x_0})}{\beta \ln(1-(1-p)e^{-\beta x_0})} where \operatorname{Li}_2 is the dilogarithm function === Random number generation === Let U be a random variate from the standard uniform distribution. The EM iteration is given by : \beta^{(h+1)} = n \left( \sum_{i=1}^n\frac{x_i}{1-(1-p^{(h)})e^{-\beta^{(h)}x_i}} \right)^{-1}, : p^{(h+1)}=\frac{-n(1-p^{(h+1)})} { \ln( p^{(h+1)}) \sum_{i=1}^n \\{1-(1-p^{(h)})e^{-\beta^{(h)} x_i}\\}^{-1}}. ==Related distributions== The EL distribution has been generalized to form the Weibull-logarithmic distribution.Ciumara, Roxana; Preda, Vasile (2009) ""The Weibull-logarithmic distribution in lifetime analysis and its properties"". The power system reliability is the probability of a normal operation of the electrical grid at a given time. Energy distance and E-statistic were considered as N-distances and N-statistic in Zinger A.A., Kakosyan A.V., Klebanov L.B. Characterization of distributions by means of mean values of some statistics in connection with some probability metrics, Stability Problems for Stochastic Models. A class of Probability Metrics and its Statistical Applications, Statistics in Industry and Technology: Statistical Data Analysis, Yadolah Dodge, Ed. Energy distance is a statistical distance between probability distributions. ",362880,0.5117,0.8,15,+10,B
+","The exponential-logarithmic model, together with its various properties, are studied by Tahmasbi and Rezaei (2008).Tahmasbi, R., Rezaei, S., (2008), ""A two-parameter lifetime distribution with decreasing failure rate"", Computational Statistics and Data Analysis, 52 (8), 3889-3901. The pdf for the standard fatigue life distribution reduces to : f(x) = \frac{\sqrt{x}+\sqrt{\frac{1}{x}}}{2\gamma x}\phi\left(\frac{\sqrt{x}-\sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma >0 Since the general form of probability functions can be expressed in terms of the standard distribution, all of the subsequent formulas are given for the standard form of the function. ==Cumulative distribution function== The formula for the cumulative distribution function is : F(x) = \Phi\left(\frac{\sqrt{x} - \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x > 0; \gamma > 0 where Φ is the cumulative distribution function of the standard normal distribution. ==Quantile function== The formula for the quantile function is : G(p) = \frac{1}{4}\left[\gamma\Phi^{-1}(p) + \sqrt{4+\left(\gamma\Phi^{-1}(p)\right)^2}\right]^2 where Φ −1 is the quantile function of the standard normal distribution. ==References== * * * * * * * ==External links== *Fatigue life distribution Category:Continuous distributions Then there are seven nonempty subsets of { 1, ..., b } = { 1, 2, 3 }; hence seven different exponential random variables: : E_{\\{1\\}}, E_{\\{2\\}}, E_{\\{3\\}}, E_{\\{1,2\\}}, E_{\\{1,3\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} Then we have: : \begin{align} T_1 & = \min\\{ E_{\\{1\\}}, E_{\\{1,2\\}}, E_{\\{1,3\\}}, E_{\\{1,2,3\\}} \\} \\\ T_2 & = \min\\{ E_{\\{2\\}}, E_{\\{1,2\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} \\} \\\ T_3 & = \min\\{ E_{\\{3\\}}, E_{\\{1,3\\}}, E_{\\{2,3\\}}, E_{\\{1,2,3\\}} \\} \\\ \end{align} ==References== * Xu M, Xu S. The joint distribution of T=(T_1,\ldots,T_b) is called the Marshall–Olkin exponential distribution with parameters \\{\lambda _B,B\subset \\{1,2,\ldots,b\\}\\}. === Concrete example === Suppose b = 3\. In this situation, the energy distance is zero if and only if X and Y are identically distributed. Vilnius, 2009 If X is defined to be the random variable which is the minimum of N independent realisations from an exponential distribution with rate parameter β, and if N is a realisation from a logarithmic distribution (where the parameter p in the usual parameterisation is replaced by ), then X has the exponential- logarithmic distribution in the parameterisation used above. ==References== Category:Continuous distributions Category:Survival analysis X is then distributed normally with a mean of zero and a variance of α2 / 4. ==Probability density function== The general formula for the probability density function (pdf) is : f(x) = \frac{\sqrt{\frac{x-\mu}{\beta}}+\sqrt{\frac{\beta}{x-\mu}}}{2\gamma\left(x-\mu\right)}\phi\left(\frac{\sqrt{\frac{x-\mu}{\beta}}-\sqrt{\frac{\beta}{x-\mu}}}{\gamma}\right)\quad x > \mu; \gamma,\beta>0 where γ is the shape parameter, μ is the location parameter, β is the scale parameter, and \phi is the probability density function of the standard normal distribution. ==Standard fatigue life distribution== The case where μ = 0 and β = 1 is called the standard fatigue life distribution. thumb|Diagram showing queueing system equivalent of a hyperexponential distribution In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable X is given by : f_X(x) = \sum_{i=1}^n f_{Y_i}(x)\;p_i, where each Yi is an exponentially distributed random variable with rate parameter λi, and pi is the probability that X will take on the form of the exponential distribution with rate λi. {1-(1-p) e^{-\beta x}} | cdf = 1-\frac{\ln(1-(1-p) e^{-\beta x})}{\ln p} | mean = -\frac{\text{polylog}(2,1-p)}{\beta\ln p} | median = \frac{\ln(1+\sqrt{p})}{\beta} | mode = 0 | variance = -\frac{2 \text{polylog}(3,1-p)}{\beta^2\ln p} -\frac{ \text{polylog}^2(2,1-p)}{\beta^2\ln^2 p} | skewness = | kurtosis = | entropy = | mgf = -\frac{\beta(1-p)}{\ln p (\beta-t)} \text{hypergeom}_{2,1} ([1,\frac{\beta-t}{\beta}],[\frac{2\beta-t}{\beta}],1-p) | cf = | pgf = | fisher = }} In probability theory and statistics, the Exponential-Logarithmic (EL) distribution is a family of lifetime distributions with decreasing failure rate, defined on the interval [0, ∞). If T is the number of cycles to failure then the cumulative distribution function (cdf) of T is : P( T \le t ) = 1 - \Phi\left( \frac{ \omega - t \mu }{ \sigma \sqrt{ t } } \right) = \Phi\left( \frac{ t \mu - \omega }{ \sigma \sqrt{ t } } \right) = \Phi\left( \frac{ \mu \sqrt{ t } }{ \sigma } - \frac{ \omega }{ \sigma \sqrt{t} } \right) = \Phi\left( \frac{ \sqrt{ \mu \omega } }{ \sigma } \left[ \left( \frac{ t }{ \omega / \mu } \right)^{ 0.5 } - \left( \frac{ \omega / \mu }{ t } \right)^{ 0.5 } \right] \right) The more usual form of this distribution is: : F( x; \alpha, \beta ) = \Phi\left( \frac{ 1 }{ \alpha } \left[ \left( \frac{ x }{ \beta } \right)^{0.5} - \left( \frac{ \beta }{ x } \right)^{0.5} \right] \right) Here α is the shape parameter and β is the scale parameter. ==Properties== The Birnbaum–Saunders distribution is unimodal with a median of β. In applied statistics, the Marshall–Olkin exponential distribution is any member of a certain family of continuous multivariate probability distributions with positive-valued components. The Birnbaum-Saunders distribution, also known as the fatigue life distribution, is a probability distribution used extensively in reliability applications to model failure times. Category:Statistics articles needing expert attention Category:Continuous distributions Category:Exponentials Category:Exponential family distributions In general, the lifetime of a device is expected to exhibit decreasing failure rate (DFR) when its behavior over time is characterized by 'work-hardening' (in engineering terms) or 'immunity' (in biological terms). If X and Y are independent random vectors in Rd with cumulative distribution functions (cdf) F and G respectively, then the energy distance between the distributions F and G is defined to be the square root of : D^2(F, G) = 2\operatorname E\|X - Y\| - \operatorname E\|X - X'\| - \operatorname E\|Y - Y'\| \geq 0, where (X, X', Y, Y') are independent, the cdf of X and X' is F, the cdf of Y and Y' is G, \operatorname E is the expected value, and || . || denotes the length of a vector. Hence the mean and variance of the EL distribution are given, respectively, by :E(X)=-\frac{\operatorname{Li}_2(1-p)}{\beta\ln p}, :\operatorname{Var}(X)=-\frac{2 \operatorname{Li}_3(1-p)}{\beta^2\ln p}-\left(\frac{ \operatorname{Li}_2(1-p)}{\beta\ln p}\right)^2. === The survival, hazard and mean residual life functions === thumb|300px|Hazard function The survival function (also known as the reliability function) and hazard function (also known as the failure rate function) of the EL distribution are given, respectively, by : s(x)=\frac{\ln(1-(1-p)e^{-\beta x})}{\ln p}, : h(x)=\frac{-\beta(1-p)e^{-\beta x}}{(1-(1-p)e^{-\beta x})\ln(1-(1-p)e^{-\beta x})}. The mean residual lifetime of the EL distribution is given by : m(x_0;p,\beta)=E(X-x_0|X\geq x_0;\beta,p)=-\frac{\operatorname{Li}_2(1-(1-p)e^{-\beta x_0})}{\beta \ln(1-(1-p)e^{-\beta x_0})} where \operatorname{Li}_2 is the dilogarithm function === Random number generation === Let U be a random variate from the standard uniform distribution. The EM iteration is given by : \beta^{(h+1)} = n \left( \sum_{i=1}^n\frac{x_i}{1-(1-p^{(h)})e^{-\beta^{(h)}x_i}} \right)^{-1}, : p^{(h+1)}=\frac{-n(1-p^{(h+1)})} { \ln( p^{(h+1)}) \sum_{i=1}^n \\{1-(1-p^{(h)})e^{-\beta^{(h)} x_i}\\}^{-1}}. ==Related distributions== The EL distribution has been generalized to form the Weibull-logarithmic distribution.Ciumara, Roxana; Preda, Vasile (2009) ""The Weibull-logarithmic distribution in lifetime analysis and its properties"". The power system reliability is the probability of a normal operation of the electrical grid at a given time. Energy distance and E-statistic were considered as N-distances and N-statistic in Zinger A.A., Kakosyan A.V., Klebanov L.B. Characterization of distributions by means of mean values of some statistics in connection with some probability metrics, Stability Problems for Stochastic Models. A class of Probability Metrics and its Statistical Applications, Statistics in Industry and Technology: Statistical Data Analysis, Yadolah Dodge, Ed. Energy distance is a statistical distance between probability distributions. ",362880,0.5117,"""0.8""",15,+10,B
 "5.4-5. Let $Z_1, Z_2, \ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$
-","To find a negative value such as -0.83, one could use a cumulative table for negative z-values which yield a probability of 0.20327. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... The probability distribution fZ(z) is given in this case by :f_Z(z)=\frac{1}{\sqrt{2 \pi}\sigma_+ }\exp\left(-\frac{z^2}{2\sigma_+^2}\right) where :\sigma_+ = \sqrt{\sigma_x^2+\sigma_y^2+2\rho\sigma_x \sigma_y}. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities. ==Normal and standard normal distribution== Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. * What is the probability that a student scores an 82 or less? \begin{align} P(X \le 82) &= P \\!\\! \left(Z \le \frac{82 - 80}{5}\right) \\\ &= P(Z \le 0.40) \\\\[2pt] &= 0.15542 + 0.5 \\\\[2pt] &= 0.65542 \end{align} * What is the probability that a student scores a 90 or more? \begin{align} P(X \ge 90) &= P \\!\\! \left(Z \ge \frac{90 - 80}{5}\right) \\\ &= P(Z \ge 2.00) \\\\[2pt] &= 1 - P(Z \le 2.00) \\\\[2pt] &= 1 - (0.47725 + 0.5) \\\\[2pt] &= 0.02275 \end{align} * What is the probability that a student scores a 74 or less? \begin{align} P(X \le 74) &= P \\!\\! \left(Z \le \frac{74 - 80}{5}\right) \\\ &= P(Z \le - 1.20) \end{align} Since this table does not include negatives, the process involves the following additional step: \begin{align} \qquad \qquad \quad ={} & P(Z \ge 1.20) \\\\[2pt] ={} & 1 - (0.38493 + 0.5) \\\\[2pt] ={} & 0.11507 \end{align} * What is the probability that a student scores between 74 and 82? \begin{align} P(74 \le X \le 82) &= P(X \le 82) - P(X \le 74) \\\\[2pt] &= 0.65542 - 0.11507 \\\\[2pt] &= 0.54035 \end{align} * What is the probability that an average of three scores is 82 or less? \begin{align} P(X \le 82) &= P\left(Z \le \frac{82 - 80}{5/\sqrt{3}}\right) \\\ &= P(Z \le 0.69) \\\\[2pt] &= 0.2549 + 0.5 \\\\[2pt] &= 0.7549 \end{align} ==See also== * 68–95–99.7 rule * t-distribution table ==References== Category:Normal distribution Category:Mathematical tables In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships. This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. The standard normal distribution, represented by , is the normal distribution having a mean of 0 and a standard deviation of 1. ===Conversion=== If is a random variable from a normal distribution with mean and standard deviation , its Z-score may be calculated from by subtracting and dividing by the standard deviation: : Z = \frac{X - \mu}{\sigma } If \overline{X} is the mean of a sample of size from some population in which the mean is and the standard deviation is , the standard error is : Z = \frac{\overline{X} - \mu}{\sigma / \sqrt n} If \sum X is the total of a sample of size from some population in which the mean is and the standard deviation is , the expected total is and the standard error is : Z = \frac{\sum{X} - n\mu}{\sigma \sqrt{n}} ==Reading a Z table== ===Formatting / layout=== tables are typically composed as follows: * The label for rows contains the integer part and the first decimal place of . The 7TP (siedmiotonowy polski - 7-tonne Polish) was a Polish light tank of the Second World War. In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. This is not to be confused with the sum of normal distributions which forms a mixture distribution. ==Independent random variables== Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if :X \sim N(\mu_X, \sigma_X^2) :Y \sim N(\mu_Y, \sigma_Y^2) :Z=X+Y, then :Z \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2). At the same time, one 7TP was captured by the Soviets during their invasion of Poland. This equates to the area of the distribution below . But since the normal distribution curve is symmetrical, probabilities for only positive values of are typically given. These probabilities are calculations of the area under the normal curve from the starting point (0 for cumulative from mean, negative infinity for cumulative and positive infinity for complementary cumulative) to . Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table. * The values within the table are the probabilities corresponding to the table type. : \Phi(z) = \frac12\left[1 + \operatorname{erf}\left( \frac z {\sqrt 2} \right) \right] Note that for , one obtains (after multiplying by 2 to account for the interval) the results , characteristic of the 68–95–99.7 rule. ===Cumulative (less than Z)=== This table gives a probability that a statistic is less than (i.e. between negative infinity and ). z −0.00 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 -4.0 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 -3.9 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003 0.00003 -3.8 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005 0.00005 -3.7 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008 -3.6 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012 0.00011 -3.5 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017 0.00017 −3.4 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025 0.00024 −3.3 0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036 0.00035 −3.2 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052 0.00050 −3.1 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074 0.00071 −3.0 0.00135 0.00131 0.00126 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104 0.00100 −2.9 0.00187 0.00181 0.00175 0.00169 0.00164 0.00159 0.00154 0.00149 0.00144 0.00139 −2.8 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 −2.7 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272 0.00264 −2.6 0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368 0.00357 −2.5 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00508 0.00494 0.00480 −2.4 0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657 0.00639 −2.3 0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866 0.00842 −2.2 0.01390 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.01160 0.01130 0.01101 −2.1 0.01786 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463 0.01426 −2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876 0.01831 −1.9 0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385 0.02330 −1.8 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938 −1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754 0.03673 −1.6 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648 0.04551 −1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592 −1.4 0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944 0.06811 −1.3 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08692 0.08534 0.08379 0.08226 −1.2 0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10383 0.10204 0.10027 0.09853 −1.1 0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900 0.11702 −1.0 0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007 0.13786 −0.9 0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354 0.16109 −0.8 0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19489 0.19215 0.18943 0.18673 −0.7 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770 0.21476 −0.6 0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825 0.24510 −0.5 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096 0.27760 −0.4 0.34458 0.34090 0.33724 0.33360 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 −0.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 −0.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 −0.1 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.43644 0.43251 0.42858 0.42465 −0.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414 z −0.00 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 z + 0.00 + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 + 0.08 + 0.09 0.0 0.50000 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.52790 0.53188 0.53586 0.1 0.53983 0.54380 0.54776 0.55172 0.55567 0.55962 0.56360 0.56749 0.57142 0.57535 0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409 0.3 0.61791 0.62172 0.62552 0.62930 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173 0.4 0.65542 0.65910 0.66276 0.66640 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793 0.5 0.69146 0.69497 0.69847 0.70194 0.70540 0.70884 0.71226 0.71566 0.71904 0.72240 0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.75490 0.7 0.75804 0.76115 0.76424 0.76730 0.77035 0.77337 0.77637 0.77935 0.78230 0.78524 0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327 0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891 1.0 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214 1.1 0.86433 0.86650 0.86864 0.87076 0.87286 0.87493 0.87698 0.87900 0.88100 0.88298 1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147 1.3 0.90320 0.90490 0.90658 0.90824 0.90988 0.91149 0.91308 0.91466 0.91621 0.91774 1.4 0.91924 0.92073 0.92220 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189 1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408 1.6 0.94520 0.94630 0.94738 0.94845 0.94950 0.95053 0.95154 0.95254 0.95352 0.95449 1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.96080 0.96164 0.96246 0.96327 1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062 1.9 0.97128 0.97193 0.97257 0.97320 0.97381 0.97441 0.97500 0.97558 0.97615 0.97670 2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169 2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98461 0.98500 0.98537 0.98574 2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.98840 0.98870 0.98899 2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158 2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361 2.5 0.99379 0.99396 0.99413 0.99430 0.99446 0.99461 0.99477 0.99492 0.99506 0.99520 2.6 0.99534 0.99547 0.99560 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643 2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.99720 0.99728 0.99736 2.8 0.99744 0.99752 0.99760 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807 2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861 3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900 3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929 3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950 3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965 3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976 3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983 3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989 3.7 0.99989 0.99990 0.99990 0.99990 0.99991 0.99991 0.99992 0.99992 0.99992 0.99992 3.8 0.99993 0.99993 0.99993 0.99994 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995 3.9 0.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997 4.0 0.99997 0.99997 0.99997 0.99997 0.99997 0.99997 0.99998 0.99998 0.99998 0.99998 z +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 0.5 + each value in Cumulative from mean table ===Complementary cumulative=== This table gives a probability that a statistic is greater than . :f(z) = 1 - \Phi(z) z +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 0.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414 0.1 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.43640 0.43251 0.42858 0.42465 0.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 0.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 0.4 0.34458 0.34090 0.33724 0.33360 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 0.5 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096 0.27760 0.6 0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825 0.24510 0.7 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770 0.21476 0.8 0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19489 0.19215 0.18943 0.18673 0.9 0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354 0.16109 1.0 0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007 0.13786 1.1 0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900 0.11702 1.2 0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10383 0.10204 0.10027 0.09853 1.3 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08692 0.08534 0.08379 0.08226 1.4 0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944 0.06811 1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592 1.6 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648 0.04551 1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754 0.03673 1.8 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938 1.9 0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385 0.02330 2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876 0.01831 2.1 0.01786 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463 0.01426 2.2 0.01390 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.01160 0.01130 0.01101 2.3 0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866 0.00842 2.4 0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657 0.00639 2.5 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00508 0.00494 0.00480 2.6 0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368 0.00357 2.7 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272 0.00264 2.8 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 2.9 0.00187 0.00181 0.00175 0.00169 0.00164 0.00159 0.00154 0.00149 0.00144 0.00139 3.0 0.00135 0.00131 0.00126 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104 0.00100 3.1 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074 0.00071 3.2 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052 0.00050 3.3 0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036 0.00035 3.4 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025 0.00024 3.5 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017 0.00017 3.6 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012 0.00011 3.7 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008 3.8 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005 0.00005 3.9 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003 0.00003 4.0 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 0.5 − each value in Cumulative from mean (0 to Z) table This table gives a probability that a statistic is greater than Z, for large integer Z values. z +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 0 5.00000 E −1 1.58655 E −1 2.27501 E −2 1.34990 E −3 3.16712 E −5 2.86652 E −7 9.86588 E −10 1.27981 E −12 6.22096 E −16 1.12859 E −19 10 7.61985 E −24 1.91066 E −28 1.77648 E −33 6.11716 E −39 7.79354 E −45 3.67097 E −51 6.38875 E −58 4.10600 E −65 9.74095 E −73 8.52722 E −81 20 2.75362 E -89 3.27928 E -98 1.43989 E -107 2.33064 E -117 1.39039 E -127 3.05670 E -138 2.47606 E -149 7.38948 E -161 8.12387 E -173 3.28979 E -185 30 4.90671 E -198 2.69525 E -211 5.45208 E -225 4.06119 E -239 1.11390 E -253 1.12491 E -268 4.18262 E -284 5.72557 E -300 2.88543 E -316 5.35312 E -333 40 3.65589 E -350 9.19086 E -368 8.50515 E -386 2.89707 E -404 3.63224 E -423 1.67618 E -442 2.84699 E -462 1.77976 E -482 4.09484 E -503 3.46743 E -524 50 1.08060 E -545 1.23937 E -567 5.23127 E -590 8.12606 E -613 4.64529 E -636 9.77237 E -660 7.56547 E -684 2.15534 E -708 2.25962 E -733 8.71741 E -759 60 1.23757 E -784 6.46517 E -811 1.24283 E -837 8.79146 E -865 2.28836 E -892 2.19180 E -920 7.72476 E -949 1.00178 E -977 4.78041 E -1007 8.39374 E -1037 70 5.42304 E -1067 1.28921 E -1097 1.12771 E -1128 3.62960 E -1160 4.29841 E -1192 1.87302 E -1224 3.00302 E -1257 1.77155 E -1290 3.84530 E -1324 3.07102 E -1358 ==Examples of use== A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. The 7.39 is a British drama television film that was broadcast in two parts on BBC One on 6 January and 7 January 2014. ",4.09, 0.0024,0.33333333,0.24995,0.925,E
-"5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?","* Exponential distribution is a special case of type 3 Pearson distribution. In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. Reliability theory and reliability engineering also make extensive use of the exponential distribution. The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a probability of occurring. And for the group of 23 people, the probability of sharing is :p(23) \approx 1 - \left(\frac{364}{365}\right)^\binom{23}{2} = 1 - \left(\frac{364}{365}\right)^{253} \approx 0.500477 . ===Poisson approximation=== Applying the Poisson approximation for the binomial on the group of 23 people, :\operatorname{Poi}\left(\frac{\binom{23}{2}}{365}\right) =\operatorname{Poi}\left(\frac{253}{365}\right) \approx \operatorname{Poi}(0.6932) so :\Pr(X>0)=1-\Pr(X=0) \approx 1-e^{-0.6932} \approx 1-0.499998=0.500002. In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. X =\sum_{i=1}^k \sum_{j=i+1}^k X_{ij} \begin{alignat}{3} E[X] & = \sum_{i=1}^k \sum_{j=i+1}^k E[X_{ij}]\\\ & = \binom{k}{2} \frac{1}{n}\\\ & = \frac{k(k-1)}{2n}\\\ \end{alignat} For , if , the expected number of people with the same birthday is ≈ 1.0356. Similar caveats apply to the following examples which yield approximately exponentially distributed variables: * The time until a radioactive particle decays, or the time between clicks of a Geiger counter * The time it takes before your next telephone call * The time until default (on payment to company debt holders) in reduced-form credit risk modeling Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand, or between roadkills on a given road. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. ==Definitions== ===Probability density function=== The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin{cases} \lambda e^{ - \lambda x} & x \ge 0, \\\ 0 & x < 0\. \end{cases} Here λ > 0 is the parameter of the distribution, often called the rate parameter. (The arrival of customers for instance is also modeled by the Poisson distribution if the arrivals are independent and distributed identically.) The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the ""failure rates"" here are not constant: more failures occur for very young and for very old systems. This can be seen by considering the complementary cumulative distribution function: \begin{align} \Pr\left(T > s + t \mid T > s\right) &= \frac{\Pr\left(T > s + t \cap T > s\right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{\Pr\left(T > s + t \right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\\\[4pt] &= e^{-\lambda t} \\\\[4pt] &= \Pr(T > t). \end{align} When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. In operating-rooms management, the distribution of surgery duration for a category of surgeries with no typical work-content (like in an emergency room, encompassing all types of surgeries). ===Prediction=== Having observed a sample of n data points from an unknown exponential distribution a common task is to use these samples to make predictions about future data from the same source. The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people . ===Probability of a shared birthday (collision)=== The birthday problem can be generalized as follows: :Given random integers drawn from a discrete uniform distribution with range , what is the probability that at least two numbers are the same? ( gives the usual birthday problem.) Note that the vertical scale is logarithmic (each step down is 1020 times less likely). : 1 0.0% 5 2.7% 10 11.7% 20 41.1% 23 50.7% 30 70.6% 40 89.1% 50 97.0% 60 99.4% 70 99.9% 75 99.97% 100 % 200 % 300 (100 − )% 350 (100 − )% 365 (100 − )% ≥ 366 100% ==Approximations== thumb|right|upright=1.4|Graphs showing the approximate probabilities of at least two people sharing a birthday () and its complementary event () thumb|right|upright=1.4|A graph showing the accuracy of the approximation () The Taylor series expansion of the exponential function (the constant ) : e^x = 1 + x + \frac{x^2}{2!}+\cdots provides a first-order approximation for for |x| \ll 1: : e^x \approx 1 + x. Consequently, the desired probability is . \\\ V_{t} &= n^{k} = 365^{23} \\\ P(A) &= \frac{V_{nr}}{V_{t}} \approx 0.492703 \\\ P(B) &= 1 - P(A) \approx 1 - 0.492703 \approx 0.507297 (50.7297%)\end{align} Another way the birthday problem can be solved is by asking for an approximate probability that in a group of people at least two have the same birthday. The exponential distribution is not the same as the class of exponential families of distributions. This implies that the expected number of people with a non-shared (unique) birthday is: : n \left( \frac{d-1}{d} \right)^{n-1} Similar formulas can be derived for the expected number of people who share with three, four, etc. other people. === Number of people until every birthday is achieved === The expected number of people needed until every birthday is achieved is called the Coupon collector's problem. In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. ",4.8,-0.041,167.0,5,0.15,D
+","To find a negative value such as -0.83, one could use a cumulative table for negative z-values which yield a probability of 0.20327. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... The probability distribution fZ(z) is given in this case by :f_Z(z)=\frac{1}{\sqrt{2 \pi}\sigma_+ }\exp\left(-\frac{z^2}{2\sigma_+^2}\right) where :\sigma_+ = \sqrt{\sigma_x^2+\sigma_y^2+2\rho\sigma_x \sigma_y}. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities. ==Normal and standard normal distribution== Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. * What is the probability that a student scores an 82 or less? \begin{align} P(X \le 82) &= P \\!\\! \left(Z \le \frac{82 - 80}{5}\right) \\\ &= P(Z \le 0.40) \\\\[2pt] &= 0.15542 + 0.5 \\\\[2pt] &= 0.65542 \end{align} * What is the probability that a student scores a 90 or more? \begin{align} P(X \ge 90) &= P \\!\\! \left(Z \ge \frac{90 - 80}{5}\right) \\\ &= P(Z \ge 2.00) \\\\[2pt] &= 1 - P(Z \le 2.00) \\\\[2pt] &= 1 - (0.47725 + 0.5) \\\\[2pt] &= 0.02275 \end{align} * What is the probability that a student scores a 74 or less? \begin{align} P(X \le 74) &= P \\!\\! \left(Z \le \frac{74 - 80}{5}\right) \\\ &= P(Z \le - 1.20) \end{align} Since this table does not include negatives, the process involves the following additional step: \begin{align} \qquad \qquad \quad ={} & P(Z \ge 1.20) \\\\[2pt] ={} & 1 - (0.38493 + 0.5) \\\\[2pt] ={} & 0.11507 \end{align} * What is the probability that a student scores between 74 and 82? \begin{align} P(74 \le X \le 82) &= P(X \le 82) - P(X \le 74) \\\\[2pt] &= 0.65542 - 0.11507 \\\\[2pt] &= 0.54035 \end{align} * What is the probability that an average of three scores is 82 or less? \begin{align} P(X \le 82) &= P\left(Z \le \frac{82 - 80}{5/\sqrt{3}}\right) \\\ &= P(Z \le 0.69) \\\\[2pt] &= 0.2549 + 0.5 \\\\[2pt] &= 0.7549 \end{align} ==See also== * 68–95–99.7 rule * t-distribution table ==References== Category:Normal distribution Category:Mathematical tables In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. In probability theory, calculation of the sum of normally distributed random variables is an instance of the arithmetic of random variables, which can be quite complex based on the probability distributions of the random variables involved and their relationships. This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. The standard normal distribution, represented by , is the normal distribution having a mean of 0 and a standard deviation of 1. ===Conversion=== If is a random variable from a normal distribution with mean and standard deviation , its Z-score may be calculated from by subtracting and dividing by the standard deviation: : Z = \frac{X - \mu}{\sigma } If \overline{X} is the mean of a sample of size from some population in which the mean is and the standard deviation is , the standard error is : Z = \frac{\overline{X} - \mu}{\sigma / \sqrt n} If \sum X is the total of a sample of size from some population in which the mean is and the standard deviation is , the expected total is and the standard error is : Z = \frac{\sum{X} - n\mu}{\sigma \sqrt{n}} ==Reading a Z table== ===Formatting / layout=== tables are typically composed as follows: * The label for rows contains the integer part and the first decimal place of . The 7TP (siedmiotonowy polski - 7-tonne Polish) was a Polish light tank of the Second World War. In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. This is not to be confused with the sum of normal distributions which forms a mixture distribution. ==Independent random variables== Let X and Y be independent random variables that are normally distributed (and therefore also jointly so), then their sum is also normally distributed. i.e., if :X \sim N(\mu_X, \sigma_X^2) :Y \sim N(\mu_Y, \sigma_Y^2) :Z=X+Y, then :Z \sim N(\mu_X + \mu_Y, \sigma_X^2 + \sigma_Y^2). At the same time, one 7TP was captured by the Soviets during their invasion of Poland. This equates to the area of the distribution below . But since the normal distribution curve is symmetrical, probabilities for only positive values of are typically given. These probabilities are calculations of the area under the normal curve from the starting point (0 for cumulative from mean, negative infinity for cumulative and positive infinity for complementary cumulative) to . Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table. * The values within the table are the probabilities corresponding to the table type. : \Phi(z) = \frac12\left[1 + \operatorname{erf}\left( \frac z {\sqrt 2} \right) \right] Note that for , one obtains (after multiplying by 2 to account for the interval) the results , characteristic of the 68–95–99.7 rule. ===Cumulative (less than Z)=== This table gives a probability that a statistic is less than (i.e. between negative infinity and ). z −0.00 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 -4.0 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 -3.9 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003 0.00003 -3.8 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005 0.00005 -3.7 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008 -3.6 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012 0.00011 -3.5 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017 0.00017 −3.4 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025 0.00024 −3.3 0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036 0.00035 −3.2 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052 0.00050 −3.1 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074 0.00071 −3.0 0.00135 0.00131 0.00126 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104 0.00100 −2.9 0.00187 0.00181 0.00175 0.00169 0.00164 0.00159 0.00154 0.00149 0.00144 0.00139 −2.8 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 −2.7 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272 0.00264 −2.6 0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368 0.00357 −2.5 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00508 0.00494 0.00480 −2.4 0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657 0.00639 −2.3 0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866 0.00842 −2.2 0.01390 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.01160 0.01130 0.01101 −2.1 0.01786 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463 0.01426 −2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876 0.01831 −1.9 0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385 0.02330 −1.8 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938 −1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754 0.03673 −1.6 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648 0.04551 −1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592 −1.4 0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944 0.06811 −1.3 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08692 0.08534 0.08379 0.08226 −1.2 0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10383 0.10204 0.10027 0.09853 −1.1 0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900 0.11702 −1.0 0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007 0.13786 −0.9 0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354 0.16109 −0.8 0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19489 0.19215 0.18943 0.18673 −0.7 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770 0.21476 −0.6 0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825 0.24510 −0.5 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096 0.27760 −0.4 0.34458 0.34090 0.33724 0.33360 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 −0.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 −0.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 −0.1 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.43644 0.43251 0.42858 0.42465 −0.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414 z −0.00 −0.01 −0.02 −0.03 −0.04 −0.05 −0.06 −0.07 −0.08 −0.09 z + 0.00 + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 + 0.08 + 0.09 0.0 0.50000 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.52790 0.53188 0.53586 0.1 0.53983 0.54380 0.54776 0.55172 0.55567 0.55962 0.56360 0.56749 0.57142 0.57535 0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409 0.3 0.61791 0.62172 0.62552 0.62930 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173 0.4 0.65542 0.65910 0.66276 0.66640 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793 0.5 0.69146 0.69497 0.69847 0.70194 0.70540 0.70884 0.71226 0.71566 0.71904 0.72240 0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.75490 0.7 0.75804 0.76115 0.76424 0.76730 0.77035 0.77337 0.77637 0.77935 0.78230 0.78524 0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327 0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891 1.0 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214 1.1 0.86433 0.86650 0.86864 0.87076 0.87286 0.87493 0.87698 0.87900 0.88100 0.88298 1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147 1.3 0.90320 0.90490 0.90658 0.90824 0.90988 0.91149 0.91308 0.91466 0.91621 0.91774 1.4 0.91924 0.92073 0.92220 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189 1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408 1.6 0.94520 0.94630 0.94738 0.94845 0.94950 0.95053 0.95154 0.95254 0.95352 0.95449 1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.96080 0.96164 0.96246 0.96327 1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062 1.9 0.97128 0.97193 0.97257 0.97320 0.97381 0.97441 0.97500 0.97558 0.97615 0.97670 2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169 2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98461 0.98500 0.98537 0.98574 2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.98840 0.98870 0.98899 2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158 2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361 2.5 0.99379 0.99396 0.99413 0.99430 0.99446 0.99461 0.99477 0.99492 0.99506 0.99520 2.6 0.99534 0.99547 0.99560 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643 2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.99720 0.99728 0.99736 2.8 0.99744 0.99752 0.99760 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807 2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861 3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900 3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929 3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950 3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965 3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976 3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983 3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989 3.7 0.99989 0.99990 0.99990 0.99990 0.99991 0.99991 0.99992 0.99992 0.99992 0.99992 3.8 0.99993 0.99993 0.99993 0.99994 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995 3.9 0.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997 4.0 0.99997 0.99997 0.99997 0.99997 0.99997 0.99997 0.99998 0.99998 0.99998 0.99998 z +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 0.5 + each value in Cumulative from mean table ===Complementary cumulative=== This table gives a probability that a statistic is greater than . :f(z) = 1 - \Phi(z) z +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 0.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414 0.1 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.43640 0.43251 0.42858 0.42465 0.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 0.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 0.4 0.34458 0.34090 0.33724 0.33360 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 0.5 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096 0.27760 0.6 0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825 0.24510 0.7 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770 0.21476 0.8 0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19489 0.19215 0.18943 0.18673 0.9 0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354 0.16109 1.0 0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007 0.13786 1.1 0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900 0.11702 1.2 0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10383 0.10204 0.10027 0.09853 1.3 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08692 0.08534 0.08379 0.08226 1.4 0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944 0.06811 1.5 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592 1.6 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648 0.04551 1.7 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754 0.03673 1.8 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938 1.9 0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385 0.02330 2.0 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876 0.01831 2.1 0.01786 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463 0.01426 2.2 0.01390 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.01160 0.01130 0.01101 2.3 0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866 0.00842 2.4 0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657 0.00639 2.5 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00508 0.00494 0.00480 2.6 0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368 0.00357 2.7 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272 0.00264 2.8 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 2.9 0.00187 0.00181 0.00175 0.00169 0.00164 0.00159 0.00154 0.00149 0.00144 0.00139 3.0 0.00135 0.00131 0.00126 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104 0.00100 3.1 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074 0.00071 3.2 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052 0.00050 3.3 0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036 0.00035 3.4 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025 0.00024 3.5 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017 0.00017 3.6 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012 0.00011 3.7 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008 3.8 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005 0.00005 3.9 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003 0.00003 4.0 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 0.5 − each value in Cumulative from mean (0 to Z) table This table gives a probability that a statistic is greater than Z, for large integer Z values. z +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 0 5.00000 E −1 1.58655 E −1 2.27501 E −2 1.34990 E −3 3.16712 E −5 2.86652 E −7 9.86588 E −10 1.27981 E −12 6.22096 E −16 1.12859 E −19 10 7.61985 E −24 1.91066 E −28 1.77648 E −33 6.11716 E −39 7.79354 E −45 3.67097 E −51 6.38875 E −58 4.10600 E −65 9.74095 E −73 8.52722 E −81 20 2.75362 E -89 3.27928 E -98 1.43989 E -107 2.33064 E -117 1.39039 E -127 3.05670 E -138 2.47606 E -149 7.38948 E -161 8.12387 E -173 3.28979 E -185 30 4.90671 E -198 2.69525 E -211 5.45208 E -225 4.06119 E -239 1.11390 E -253 1.12491 E -268 4.18262 E -284 5.72557 E -300 2.88543 E -316 5.35312 E -333 40 3.65589 E -350 9.19086 E -368 8.50515 E -386 2.89707 E -404 3.63224 E -423 1.67618 E -442 2.84699 E -462 1.77976 E -482 4.09484 E -503 3.46743 E -524 50 1.08060 E -545 1.23937 E -567 5.23127 E -590 8.12606 E -613 4.64529 E -636 9.77237 E -660 7.56547 E -684 2.15534 E -708 2.25962 E -733 8.71741 E -759 60 1.23757 E -784 6.46517 E -811 1.24283 E -837 8.79146 E -865 2.28836 E -892 2.19180 E -920 7.72476 E -949 1.00178 E -977 4.78041 E -1007 8.39374 E -1037 70 5.42304 E -1067 1.28921 E -1097 1.12771 E -1128 3.62960 E -1160 4.29841 E -1192 1.87302 E -1224 3.00302 E -1257 1.77155 E -1290 3.84530 E -1324 3.07102 E -1358 ==Examples of use== A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. The 7.39 is a British drama television film that was broadcast in two parts on BBC One on 6 January and 7 January 2014. ",4.09, 0.0024,"""0.33333333""",0.24995,0.925,E
+"5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?","* Exponential distribution is a special case of type 3 Pearson distribution. In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. Reliability theory and reliability engineering also make extensive use of the exponential distribution. The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a probability of occurring. And for the group of 23 people, the probability of sharing is :p(23) \approx 1 - \left(\frac{364}{365}\right)^\binom{23}{2} = 1 - \left(\frac{364}{365}\right)^{253} \approx 0.500477 . ===Poisson approximation=== Applying the Poisson approximation for the binomial on the group of 23 people, :\operatorname{Poi}\left(\frac{\binom{23}{2}}{365}\right) =\operatorname{Poi}\left(\frac{253}{365}\right) \approx \operatorname{Poi}(0.6932) so :\Pr(X>0)=1-\Pr(X=0) \approx 1-e^{-0.6932} \approx 1-0.499998=0.500002. In queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. X =\sum_{i=1}^k \sum_{j=i+1}^k X_{ij} \begin{alignat}{3} E[X] & = \sum_{i=1}^k \sum_{j=i+1}^k E[X_{ij}]\\\ & = \binom{k}{2} \frac{1}{n}\\\ & = \frac{k(k-1)}{2n}\\\ \end{alignat} For , if , the expected number of people with the same birthday is ≈ 1.0356. Similar caveats apply to the following examples which yield approximately exponentially distributed variables: * The time until a radioactive particle decays, or the time between clicks of a Geiger counter * The time it takes before your next telephone call * The time until default (on payment to company debt holders) in reduced-form credit risk modeling Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between mutations on a DNA strand, or between roadkills on a given road. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the normal, binomial, gamma, and Poisson distributions. ==Definitions== ===Probability density function=== The probability density function (pdf) of an exponential distribution is : f(x;\lambda) = \begin{cases} \lambda e^{ - \lambda x} & x \ge 0, \\\ 0 & x < 0\. \end{cases} Here λ > 0 is the parameter of the distribution, often called the rate parameter. (The arrival of customers for instance is also modeled by the Poisson distribution if the arrivals are independent and distributed identically.) The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the ""failure rates"" here are not constant: more failures occur for very young and for very old systems. This can be seen by considering the complementary cumulative distribution function: \begin{align} \Pr\left(T > s + t \mid T > s\right) &= \frac{\Pr\left(T > s + t \cap T > s\right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{\Pr\left(T > s + t \right)}{\Pr\left(T > s\right)} \\\\[4pt] &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\\\[4pt] &= e^{-\lambda t} \\\\[4pt] &= \Pr(T > t). \end{align} When T is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s, the distribution of the remaining waiting time is the same as the original unconditional distribution. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. In operating-rooms management, the distribution of surgery duration for a category of surgeries with no typical work-content (like in an emergency room, encompassing all types of surgeries). ===Prediction=== Having observed a sample of n data points from an unknown exponential distribution a common task is to use these samples to make predictions about future data from the same source. The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people . ===Probability of a shared birthday (collision)=== The birthday problem can be generalized as follows: :Given random integers drawn from a discrete uniform distribution with range , what is the probability that at least two numbers are the same? ( gives the usual birthday problem.) Note that the vertical scale is logarithmic (each step down is 1020 times less likely). : 1 0.0% 5 2.7% 10 11.7% 20 41.1% 23 50.7% 30 70.6% 40 89.1% 50 97.0% 60 99.4% 70 99.9% 75 99.97% 100 % 200 % 300 (100 − )% 350 (100 − )% 365 (100 − )% ≥ 366 100% ==Approximations== thumb|right|upright=1.4|Graphs showing the approximate probabilities of at least two people sharing a birthday () and its complementary event () thumb|right|upright=1.4|A graph showing the accuracy of the approximation () The Taylor series expansion of the exponential function (the constant ) : e^x = 1 + x + \frac{x^2}{2!}+\cdots provides a first-order approximation for for |x| \ll 1: : e^x \approx 1 + x. Consequently, the desired probability is . \\\ V_{t} &= n^{k} = 365^{23} \\\ P(A) &= \frac{V_{nr}}{V_{t}} \approx 0.492703 \\\ P(B) &= 1 - P(A) \approx 1 - 0.492703 \approx 0.507297 (50.7297%)\end{align} Another way the birthday problem can be solved is by asking for an approximate probability that in a group of people at least two have the same birthday. The exponential distribution is not the same as the class of exponential families of distributions. This implies that the expected number of people with a non-shared (unique) birthday is: : n \left( \frac{d-1}{d} \right)^{n-1} Similar formulas can be derived for the expected number of people who share with three, four, etc. other people. === Number of people until every birthday is achieved === The expected number of people needed until every birthday is achieved is called the Coupon collector's problem. In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. ",4.8,-0.041,"""167.0""",5,0.15,D
 "5.3-1. Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\lambda_1=2$ and $\lambda_2=3$. Find
 (a) $P\left(X_1=3, X_2=5\right)$.
-HINT. Note that this event can occur if and only if $\left\{X_1=1, X_2=0\right\}$ or $\left\{X_1=0, X_2=1\right\}$.","The multiple Poisson distribution, its characteristics and a variety of forms. Mixed poisson processes, volume 77. In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson- distributed variable. It should not be confused with compound Poisson distribution or compound Poisson process. == Definition == A random variable X satisfies the mixed Poisson distribution with density (λ) if it has the probability distribution : \operatorname{P}(X=k) = \int_0^\infty \frac{\lambda^k}{k!}e^{-\lambda} \,\,\pi(\lambda)\,\mathrm d\lambda. \, Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X1. In this situation, the number of points at \textstyle x is a Poisson random variable with mean \textstyle \Lambda({x}). If the points belong to a homogeneous Poisson process with parameter \textstyle \lambda>0, then the probability of \textstyle n points existing in \textstyle B is given by: : \Pr \\{N(B)=n\\}=\frac{(\lambda|B|)^n}{n!} e^{-\lambda|B|} where \textstyle |B| denotes the area of \textstyle B. The two separate Poisson point processes formed respectively from the removed and kept points are stochastically independent of each other. It follows that \lambda is the expected number of arrivals that occur per unit of time. ====Key properties==== The previous definition has two important features shared by Poisson point processes in general: * the number of arrivals in each finite interval has a Poisson distribution; * the number of arrivals in disjoint intervals are independent random variables. In other words, for each point of the original Poisson process, there is an independent and identically distributed non-negative random variable, and then the compound Poisson process is formed from the sum of all the random variables corresponding to points of the Poisson process located in some region of the underlying mathematical space. The two properties are not logically independent; indeed, independence implies the Poisson distribution of point counts, but not the converse. ===Poisson distribution of point counts=== A Poisson point process is characterized via the Poisson distribution. If a point x is sampled from a countable n union of Poisson processes, then the probability that the point \textstyle x belongs to the jth Poisson process N_j is given by: : \Pr \\{x\in N_j\\}=\frac{\Lambda_j}{\sum_{i=1}^n\Lambda_i}. The result can be either a continuous or a discrete distribution. ==Definition== Suppose that :N\sim\operatorname{Poisson}(\lambda), i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that :X_1, X_2, X_3, \dots are identically distributed random variables that are mutually independent and also independent of N. For two real numbers \textstyle a and \textstyle b, where \textstyle a\leq b, denote by \textstyle N(a,b] the number points of an inhomogeneous Poisson process with intensity function \textstyle \lambda(t) occurring in the interval \textstyle (a,b]. The Poisson distribution is the probability distribution of a random variable N (called a Poisson random variable) such that the probability that \textstyle N equals \textstyle n is given by: : \Pr \\{N=n\\}=\frac{\Lambda^n}{n!} e^{-\Lambda} where n! denotes factorial and the parameter \Lambda determines the shape of the distribution. A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The probability of \textstyle n points existing in the above interval \textstyle (a,b] is given by: : \Pr \\{N(a,b]=n\\}=\frac{[\Lambda(a,b)]^n}{n!} e^{-\Lambda(a,b)}. where the mean or intensity measure is: : \Lambda(a,b)=\int_a^b \lambda (t)\,\mathrm dt, which means that the random variable \textstyle N(a,b] is a Poisson random variable with mean \textstyle \operatorname E[N(a,b]] = \Lambda(a,b). When r = 1,2, DCP becomes Poisson distribution and Hermite distribution, respectively. If we denote the probabilities of the Poisson distribution by qλ(k), then : \operatorname{P}(X=k) = \int_0^\infty q_\lambda(k) \,\,\pi(\lambda)\,\mathrm d\lambda. == Properties == * The variance is always bigger than the expected value. In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. Then the probability distribution of the sum of N i.i.d. random variables :Y = \sum_{n=1}^N X_n is a compound Poisson distribution. ",76,35.2,0.0182,0.38,0.064,C
+HINT. Note that this event can occur if and only if $\left\{X_1=1, X_2=0\right\}$ or $\left\{X_1=0, X_2=1\right\}$.","The multiple Poisson distribution, its characteristics and a variety of forms. Mixed poisson processes, volume 77. In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson- distributed variable. It should not be confused with compound Poisson distribution or compound Poisson process. == Definition == A random variable X satisfies the mixed Poisson distribution with density (λ) if it has the probability distribution : \operatorname{P}(X=k) = \int_0^\infty \frac{\lambda^k}{k!}e^{-\lambda} \,\,\pi(\lambda)\,\mathrm d\lambda. \, Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X1. In this situation, the number of points at \textstyle x is a Poisson random variable with mean \textstyle \Lambda({x}). If the points belong to a homogeneous Poisson process with parameter \textstyle \lambda>0, then the probability of \textstyle n points existing in \textstyle B is given by: : \Pr \\{N(B)=n\\}=\frac{(\lambda|B|)^n}{n!} e^{-\lambda|B|} where \textstyle |B| denotes the area of \textstyle B. The two separate Poisson point processes formed respectively from the removed and kept points are stochastically independent of each other. It follows that \lambda is the expected number of arrivals that occur per unit of time. ====Key properties==== The previous definition has two important features shared by Poisson point processes in general: * the number of arrivals in each finite interval has a Poisson distribution; * the number of arrivals in disjoint intervals are independent random variables. In other words, for each point of the original Poisson process, there is an independent and identically distributed non-negative random variable, and then the compound Poisson process is formed from the sum of all the random variables corresponding to points of the Poisson process located in some region of the underlying mathematical space. The two properties are not logically independent; indeed, independence implies the Poisson distribution of point counts, but not the converse. ===Poisson distribution of point counts=== A Poisson point process is characterized via the Poisson distribution. If a point x is sampled from a countable n union of Poisson processes, then the probability that the point \textstyle x belongs to the jth Poisson process N_j is given by: : \Pr \\{x\in N_j\\}=\frac{\Lambda_j}{\sum_{i=1}^n\Lambda_i}. The result can be either a continuous or a discrete distribution. ==Definition== Suppose that :N\sim\operatorname{Poisson}(\lambda), i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that :X_1, X_2, X_3, \dots are identically distributed random variables that are mutually independent and also independent of N. For two real numbers \textstyle a and \textstyle b, where \textstyle a\leq b, denote by \textstyle N(a,b] the number points of an inhomogeneous Poisson process with intensity function \textstyle \lambda(t) occurring in the interval \textstyle (a,b]. The Poisson distribution is the probability distribution of a random variable N (called a Poisson random variable) such that the probability that \textstyle N equals \textstyle n is given by: : \Pr \\{N=n\\}=\frac{\Lambda^n}{n!} e^{-\Lambda} where n! denotes factorial and the parameter \Lambda determines the shape of the distribution. A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. In probability theory and statistics, the Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed. The probability of \textstyle n points existing in the above interval \textstyle (a,b] is given by: : \Pr \\{N(a,b]=n\\}=\frac{[\Lambda(a,b)]^n}{n!} e^{-\Lambda(a,b)}. where the mean or intensity measure is: : \Lambda(a,b)=\int_a^b \lambda (t)\,\mathrm dt, which means that the random variable \textstyle N(a,b] is a Poisson random variable with mean \textstyle \operatorname E[N(a,b]] = \Lambda(a,b). When r = 1,2, DCP becomes Poisson distribution and Hermite distribution, respectively. If we denote the probabilities of the Poisson distribution by qλ(k), then : \operatorname{P}(X=k) = \int_0^\infty q_\lambda(k) \,\,\pi(\lambda)\,\mathrm d\lambda. == Properties == * The variance is always bigger than the expected value. In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. Then the probability distribution of the sum of N i.i.d. random variables :Y = \sum_{n=1}^N X_n is a compound Poisson distribution. ",76,35.2,"""0.0182""",0.38,0.064,C
 "5.9-1. Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3
 (a) By using the binomial distribution.
-","In such a case, the probability distribution of the number of failures that appear will be a negative binomial distribution. Thus, the probability of failure, q, is given by :q = 1 - p = 1 - \tfrac{1}{2} = \tfrac{1}{2}. That number of successes is a negative-binomially distributed random variable. When counting the number of successes before the r-th failure, as in alternative formulation (3) above, the variance is rp/(1 − p)2. ===Relation to the binomial theorem=== Suppose Y is a random variable with a binomial distribution with parameters n and p. Let p be the probability of success in a Bernoulli trial, and q be the probability of failure. The following table describes four distributions related to the number of successes in a sequence of draws: With replacements No replacements Given number of draws binomial distribution hypergeometric distribution Given number of failures negative binomial distribution negative hypergeometric distribution ===(a,b,0) class of distributions=== The negative binomial, along with the Poisson and binomial distributions, is a member of the (a,b,0) class of distributions. * Beta-binomial distribution. In each trial the probability of success is p and of failure is 1-p. The number of successes before the third failure belongs to the infinite set { 0, 1, 2, 3, ... In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted r) occurs. A random variable corresponding to a binomial experiment is denoted by B(n,p), and is said to have a binomial distribution. In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is the number of successes in a series of independent Bernoulli trials, where each trial has probability of success . In other words, the negative binomial distribution is the probability distribution of the number of successes before the rth failure in a Bernoulli process, with probability p of successes on each trial. If r is a counting number, the coin tosses show that the count of successes before the rth failure follows a negative binomial distribution with parameters r and p. Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, ""success"" and ""failure"", in which the probability of success is the same every time the experiment is conducted. Probability in the Engineering and Informational Sciences is an international journal published by Cambridge University Press. Then the random number of observed failures, X, follows the negative binomial (or Pascal) distribution: : X\sim\operatorname{NB}(r, p) ===Probability mass function=== The probability mass function of the negative binomial distribution is : f(k; r, p) \equiv \Pr(X = k) = \binom{k+r-1}{k} (1-p)^k p^r where r is the number of successes, k is the number of failures, and p is the probability of success on each trial. * When k = 2, the multinomial distribution is the binomial distribution. Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. Thus, the expected number of failures would be this value, minus the successes: : E[\operatorname{NB}(r, p)] = \frac{r}{p} - r = \frac{r(1-p)}{p} ===Expectation of successes=== The expected total number of failures in a negative binomial distribution with parameters is r(1 − p)/p. In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables. ",-32,0.9984,1.07,0.648004372,0.3359,B
+","In such a case, the probability distribution of the number of failures that appear will be a negative binomial distribution. Thus, the probability of failure, q, is given by :q = 1 - p = 1 - \tfrac{1}{2} = \tfrac{1}{2}. That number of successes is a negative-binomially distributed random variable. When counting the number of successes before the r-th failure, as in alternative formulation (3) above, the variance is rp/(1 − p)2. ===Relation to the binomial theorem=== Suppose Y is a random variable with a binomial distribution with parameters n and p. Let p be the probability of success in a Bernoulli trial, and q be the probability of failure. The following table describes four distributions related to the number of successes in a sequence of draws: With replacements No replacements Given number of draws binomial distribution hypergeometric distribution Given number of failures negative binomial distribution negative hypergeometric distribution ===(a,b,0) class of distributions=== The negative binomial, along with the Poisson and binomial distributions, is a member of the (a,b,0) class of distributions. * Beta-binomial distribution. In each trial the probability of success is p and of failure is 1-p. The number of successes before the third failure belongs to the infinite set { 0, 1, 2, 3, ... In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted r) occurs. A random variable corresponding to a binomial experiment is denoted by B(n,p), and is said to have a binomial distribution. In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is the number of successes in a series of independent Bernoulli trials, where each trial has probability of success . In other words, the negative binomial distribution is the probability distribution of the number of successes before the rth failure in a Bernoulli process, with probability p of successes on each trial. If r is a counting number, the coin tosses show that the count of successes before the rth failure follows a negative binomial distribution with parameters r and p. Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, ""success"" and ""failure"", in which the probability of success is the same every time the experiment is conducted. Probability in the Engineering and Informational Sciences is an international journal published by Cambridge University Press. Then the random number of observed failures, X, follows the negative binomial (or Pascal) distribution: : X\sim\operatorname{NB}(r, p) ===Probability mass function=== The probability mass function of the negative binomial distribution is : f(k; r, p) \equiv \Pr(X = k) = \binom{k+r-1}{k} (1-p)^k p^r where r is the number of successes, k is the number of failures, and p is the probability of success on each trial. * When k = 2, the multinomial distribution is the binomial distribution. Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. Thus, the expected number of failures would be this value, minus the successes: : E[\operatorname{NB}(r, p)] = \frac{r}{p} - r = \frac{r(1-p)}{p} ===Expectation of successes=== The expected total number of failures in a negative binomial distribution with parameters is r(1 − p)/p. In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables. ",-32,0.9984,"""1.07""",0.648004372,0.3359,B
 "7.4-11. Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking.
-(a) How large a sample $n$ of rat embryos is needed for $y / n \pm 0.10$ to be a $95 \%$ confidence interval for $p$ ?","Given n_S successes in n trials, define :\tilde{n} = n + z^2 and :\tilde{p} = \frac{1}{\tilde{n}}\left(n_S + \frac{z^2}{2}\right) Then, a confidence interval for p is given by : \tilde{p} \pm z \sqrt{\frac{\tilde{p}}{\tilde{n}}\left(1 - \tilde{p} \right)} where z = \Phi^{-1}\\!\left(1 - \frac{\alpha}{2}\\!\right) is the quantile of a standard normal distribution, as before (for example, a 95% confidence interval requires \alpha = 0.05, thereby producing z = 1.96). In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes nS are known. By symmetry, one could expect for only successes (\hat p = 1), the interval is . ==Comparison and discussion== There are several research papers that compare these and other confidence intervals for the binomial proportion. Follow the examples below for guidance: Fusion range: * PFR cc (6m) 8Δ BI → 20Δ BO * PFR sc (1/3m) 16Δ BI → 45Δ BO c diplopia Break + recovery: * PFR sc (6m) -8/6Δ → +20/15Δ c diplopia * PFR cc (1/3m) -16/14Δ → +45/40Δ c diplopia Patient results should be compared to the normal values for prism fusional amplitudes to determine if the patient has any anomalies. The parameter a has to be estimated for the data set. ==Rule of three — for when no successes are observed== The rule of three is used to provide a simple way of stating an approximate 95% confidence interval for p, in the special case that no successes (\hat p = 0) have been observed.Steve Simon (2010) ""Confidence interval with zero events"", The Children's Mercy Hospital, Kansas City, Mo. (website: ""Ask Professor Mean at Stats topics or Medical Research ) The interval is . This method may be used to estimate the variance of p but its use is problematic when p is close to 0 or 1\. ==ta transform== Let p be the proportion of successes. For a 95% confidence level, the error \alpha=1-0.95=0.05, so 1 - \tfrac \alpha 2=0.975 and z=1.96. Combining the two, and squaring out the radical, gives an equation that is quadratic in : : \left(\, \hat{p} - p \,\right)^{2} = z^{2}\cdot\frac{\,p\left(1-p\right)\,}{n} Transforming the relation into a standard-form quadratic equation for , treating \hat p and as known values from the sample (see prior section), and using the value of that corresponds to the desired confidence for the estimate of gives this: \left( 1 + \frac{\,z^2\,}{n} \right) p^2 + \left( - 2 {\hat p} - \frac{\,z^2\,}{n} \right) p + \biggl( {\hat p}^2 \biggr) = 0 ~, where all of the values in parentheses are known quantities. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. Tooth fusion arises through union of two normally separated tooth germs, and depending upon the stage of development of the teeth at the time of union, it may be either complete or incomplete. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed. ==Normal approximation interval or Wald interval == A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, \hat p, with a normal distribution. The solution for estimates the upper and lower limits of the confidence interval for . However, fusion can also be the union of a normal tooth bud to a supernumerary tooth germ. thumb|400px|Lawson criterion of important magnetic confinement fusion experiments The Lawson criterion is a figure of merit used in nuclear fusion research. In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. Because we do not know p(1-p), we have to estimate it. The Clopper–Pearson interval can be written as : S_{\le} \cap S_{\ge} or equivalently, : \left( \inf S_{\ge}\,,\, \sup S_{\le} \right) with : S_{\le} := \left\\{ p \,\,\Big|\,\, P \left[ \operatorname{Bin}\left( n; p \right) \le x \right] > \frac{\alpha}{2} \right\\} \text{ and } S_{\ge} := \left\\{ p \,\,\Big|\,\, P \left[ \operatorname{Bin}\left( n; p \right) \ge x \right] > \frac{\alpha}{2} \right\\}, where 0 ≤ x ≤ n is the number of successes observed in the sample and Bin(n; p) is a binomial random variable with n trials and probability of success p. Thus, p_{\min} < p < p_{\max}, where: :\frac{\Gamma(n+1)}{\Gamma(x )\Gamma(n-x+1)}\int_0^{ p_{\min}} t^{x-1}(1-t)^{n-x}dt = \frac{\alpha}{2} :\frac{\Gamma(n+1)}{\Gamma(x+1)\Gamma(n-x)}\int_0^{ p_{\max}} t^{x}(1-t)^{n-x-1}dt = 1-\frac{\alpha}{2} The binomial proportion confidence interval is then ( p_{\min}, p_{\max}), as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function. Using the normal approximation, the success probability p is estimated as : \hat p \pm z \sqrt{\frac{\hat p \left(1 - \hat p\right)}{n}}, or the equivalent : \frac{n_S}{n} \pm \frac{z}{n \sqrt{n}} \sqrt{n_S n_F}, where \hat p = n_S / n is the proportion of successes in a Bernoulli trial process, measured with n trials yielding n_S successes and n_F = n - n_S failures, and z is the 1 - \tfrac{\alpha}{2} quantile of a standard normal distribution (i.e., the probit) corresponding to the target error rate \alpha. When such is the case, the fusion power density is proportional to p2<σv>/T 2. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approximations are used to calculate this confidence interval, all with their own tradeoffs in accuracy and computational intensity. ",1.41,2.74,12.0,38,9.8,D
+(a) How large a sample $n$ of rat embryos is needed for $y / n \pm 0.10$ to be a $95 \%$ confidence interval for $p$ ?","Given n_S successes in n trials, define :\tilde{n} = n + z^2 and :\tilde{p} = \frac{1}{\tilde{n}}\left(n_S + \frac{z^2}{2}\right) Then, a confidence interval for p is given by : \tilde{p} \pm z \sqrt{\frac{\tilde{p}}{\tilde{n}}\left(1 - \tilde{p} \right)} where z = \Phi^{-1}\\!\left(1 - \frac{\alpha}{2}\\!\right) is the quantile of a standard normal distribution, as before (for example, a 95% confidence interval requires \alpha = 0.05, thereby producing z = 1.96). In other words, a binomial proportion confidence interval is an interval estimate of a success probability p when only the number of experiments n and the number of successes nS are known. By symmetry, one could expect for only successes (\hat p = 1), the interval is . ==Comparison and discussion== There are several research papers that compare these and other confidence intervals for the binomial proportion. Follow the examples below for guidance: Fusion range: * PFR cc (6m) 8Δ BI → 20Δ BO * PFR sc (1/3m) 16Δ BI → 45Δ BO c diplopia Break + recovery: * PFR sc (6m) -8/6Δ → +20/15Δ c diplopia * PFR cc (1/3m) -16/14Δ → +45/40Δ c diplopia Patient results should be compared to the normal values for prism fusional amplitudes to determine if the patient has any anomalies. The parameter a has to be estimated for the data set. ==Rule of three — for when no successes are observed== The rule of three is used to provide a simple way of stating an approximate 95% confidence interval for p, in the special case that no successes (\hat p = 0) have been observed.Steve Simon (2010) ""Confidence interval with zero events"", The Children's Mercy Hospital, Kansas City, Mo. (website: ""Ask Professor Mean at Stats topics or Medical Research ) The interval is . This method may be used to estimate the variance of p but its use is problematic when p is close to 0 or 1\. ==ta transform== Let p be the proportion of successes. For a 95% confidence level, the error \alpha=1-0.95=0.05, so 1 - \tfrac \alpha 2=0.975 and z=1.96. Combining the two, and squaring out the radical, gives an equation that is quadratic in : : \left(\, \hat{p} - p \,\right)^{2} = z^{2}\cdot\frac{\,p\left(1-p\right)\,}{n} Transforming the relation into a standard-form quadratic equation for , treating \hat p and as known values from the sample (see prior section), and using the value of that corresponds to the desired confidence for the estimate of gives this: \left( 1 + \frac{\,z^2\,}{n} \right) p^2 + \left( - 2 {\hat p} - \frac{\,z^2\,}{n} \right) p + \biggl( {\hat p}^2 \biggr) = 0 ~, where all of the values in parentheses are known quantities. Under this formulation, the confidence interval represents those values of the population parameter that would have large p-values if they were tested as a hypothesized population proportion. Tooth fusion arises through union of two normally separated tooth germs, and depending upon the stage of development of the teeth at the time of union, it may be either complete or incomplete. A 95% confidence interval for the proportion, for instance, will contain the true proportion 95% of the times that the procedure for constructing the confidence interval is employed. ==Normal approximation interval or Wald interval == A commonly used formula for a binomial confidence interval relies on approximating the distribution of error about a binomially-distributed observation, \hat p, with a normal distribution. The solution for estimates the upper and lower limits of the confidence interval for . However, fusion can also be the union of a normal tooth bud to a supernumerary tooth germ. thumb|400px|Lawson criterion of important magnetic confinement fusion experiments The Lawson criterion is a figure of merit used in nuclear fusion research. In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. Because we do not know p(1-p), we have to estimate it. The Clopper–Pearson interval can be written as : S_{\le} \cap S_{\ge} or equivalently, : \left( \inf S_{\ge}\,,\, \sup S_{\le} \right) with : S_{\le} := \left\\{ p \,\,\Big|\,\, P \left[ \operatorname{Bin}\left( n; p \right) \le x \right] > \frac{\alpha}{2} \right\\} \text{ and } S_{\ge} := \left\\{ p \,\,\Big|\,\, P \left[ \operatorname{Bin}\left( n; p \right) \ge x \right] > \frac{\alpha}{2} \right\\}, where 0 ≤ x ≤ n is the number of successes observed in the sample and Bin(n; p) is a binomial random variable with n trials and probability of success p. Thus, p_{\min} < p < p_{\max}, where: :\frac{\Gamma(n+1)}{\Gamma(x )\Gamma(n-x+1)}\int_0^{ p_{\min}} t^{x-1}(1-t)^{n-x}dt = \frac{\alpha}{2} :\frac{\Gamma(n+1)}{\Gamma(x+1)\Gamma(n-x)}\int_0^{ p_{\max}} t^{x}(1-t)^{n-x-1}dt = 1-\frac{\alpha}{2} The binomial proportion confidence interval is then ( p_{\min}, p_{\max}), as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function. Using the normal approximation, the success probability p is estimated as : \hat p \pm z \sqrt{\frac{\hat p \left(1 - \hat p\right)}{n}}, or the equivalent : \frac{n_S}{n} \pm \frac{z}{n \sqrt{n}} \sqrt{n_S n_F}, where \hat p = n_S / n is the proportion of successes in a Bernoulli trial process, measured with n trials yielding n_S successes and n_F = n - n_S failures, and z is the 1 - \tfrac{\alpha}{2} quantile of a standard normal distribution (i.e., the probit) corresponding to the target error rate \alpha. When such is the case, the fusion power density is proportional to p2<σv>/T 2. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approximations are used to calculate this confidence interval, all with their own tradeoffs in accuracy and computational intensity. ",1.41,2.74,"""12.0""",38,9.8,D
 "7.1-3. To determine the effect of $100 \%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \%$ nitrate every day. At the end of
 two weeks, the plants were measured. Here are data on seven of them:
 $$
@@ -781,86 +781,86 @@ $$
 $$
 Assume that these data are a random sample from a normal distribution $N\left(\mu, \sigma^2\right)$.
 (a) Find the value of a point estimate of $\mu$.
-","The average of these 15 deviations from the assumed mean is therefore −30/15 = −2\. Suppose we wanted to calculate a 95% confidence interval for μ. The assumed mean is the centre of the range from 174 to 177 which is 175.5. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means thumb|upright=1.3|Each row of points is a sample from the same normal distribution. thumb|Plot of the standard deviation line (SD line), dashed, and the regression line, solid, for a scatter diagram of 20 points. thumb|upright=1.3|right|Relationship of phosphate to nitrate uptake for photosynthesis in various regions of the ocean. This value is then subtracted from all the sample values. Therefore, that is what we need to add to the assumed mean to get the correct mean: : correct mean = 240 − 2 = 238. ==Method== The method depends on estimating the mean and rounding to an easy value to calculate with. The colored lines are 50% confidence intervals for the mean, μ. Observed numbers in ranges Range tally-count frequency class diff freq×diff freq×diff2 159—161 / 1 −5 −5 25 162—164 ~~////~~ / 6 −4 −24 96 165—167 ~~////~~ ~~////~~ 10 −3 −30 90 168—170 ~~////~~ ~~////~~ /// 13 −2 −26 52 171—173 ~~////~~ ~~////~~ ~~////~~ / 16 −1 −16 16 174—176 ~~////~~ ~~////~~ ~~////~~ ~~////~~ ~~////~~ 25 0 0 0 177—179 ~~////~~ ~~////~~ ~~////~~ / 16 1 16 16 180—182 ~~////~~ ~~////~~ / 11 2 22 44 183—185 0 3 0 0 186—188 // 2 4 8 32 Sum N = 100 A = −55 B = 371 The mean is then estimated to be :x_0 + CS \times \frac{A}{N} = 175.5+3\times -55 / 100 = 173.85 which is very close to the actual mean of 173.846. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Then the deviations from this ""assumed"" mean are the following: :−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22 In adding these up, one finds that: : 22 and −21 almost cancel, leaving +1, : 15 and −17 almost cancel, leaving −2, : 9 and −9 cancel, : 7 + 4 cancels −6 − 5, and so on. After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). (Section 9.5) Note that the distribution of T does not depend on the values of the unobservable parameters μ and σ2; i.e., it is a pivotal quantity. A confidence interval for the true mean can be constructed centered on the sample mean with a width which is a multiple of the square root of the sample variance. ====Likelihood theory==== Estimates can be constructed using the maximum likelihood principle, the likelihood theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates. ====Estimating equations==== The estimation approach here can be considered as both a generalization of the method of moments and a generalization of the maximum likelihood approach. * The confidence interval can be expressed in terms of statistical significance, e.g.: If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. At the center of each interval is the sample mean, marked with a diamond. ",20.2,15.757,8.8,420,4.0,B
+","The average of these 15 deviations from the assumed mean is therefore −30/15 = −2\. Suppose we wanted to calculate a 95% confidence interval for μ. The assumed mean is the centre of the range from 174 to 177 which is 175.5. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means thumb|upright=1.3|Each row of points is a sample from the same normal distribution. thumb|Plot of the standard deviation line (SD line), dashed, and the regression line, solid, for a scatter diagram of 20 points. thumb|upright=1.3|right|Relationship of phosphate to nitrate uptake for photosynthesis in various regions of the ocean. This value is then subtracted from all the sample values. Therefore, that is what we need to add to the assumed mean to get the correct mean: : correct mean = 240 − 2 = 238. ==Method== The method depends on estimating the mean and rounding to an easy value to calculate with. The colored lines are 50% confidence intervals for the mean, μ. Observed numbers in ranges Range tally-count frequency class diff freq×diff freq×diff2 159—161 / 1 −5 −5 25 162—164 ~~////~~ / 6 −4 −24 96 165—167 ~~////~~ ~~////~~ 10 −3 −30 90 168—170 ~~////~~ ~~////~~ /// 13 −2 −26 52 171—173 ~~////~~ ~~////~~ ~~////~~ / 16 −1 −16 16 174—176 ~~////~~ ~~////~~ ~~////~~ ~~////~~ ~~////~~ 25 0 0 0 177—179 ~~////~~ ~~////~~ ~~////~~ / 16 1 16 16 180—182 ~~////~~ ~~////~~ / 11 2 22 44 183—185 0 3 0 0 186—188 // 2 4 8 32 Sum N = 100 A = −55 B = 371 The mean is then estimated to be :x_0 + CS \times \frac{A}{N} = 175.5+3\times -55 / 100 = 173.85 which is very close to the actual mean of 173.846. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Then the deviations from this ""assumed"" mean are the following: :−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22 In adding these up, one finds that: : 22 and −21 almost cancel, leaving +1, : 15 and −17 almost cancel, leaving −2, : 9 and −9 cancel, : 7 + 4 cancels −6 − 5, and so on. After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). (Section 9.5) Note that the distribution of T does not depend on the values of the unobservable parameters μ and σ2; i.e., it is a pivotal quantity. A confidence interval for the true mean can be constructed centered on the sample mean with a width which is a multiple of the square root of the sample variance. ====Likelihood theory==== Estimates can be constructed using the maximum likelihood principle, the likelihood theory for this provides two ways of constructing confidence intervals or confidence regions for the estimates. ====Estimating equations==== The estimation approach here can be considered as both a generalization of the method of moments and a generalization of the maximum likelihood approach. * The confidence interval can be expressed in terms of statistical significance, e.g.: If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. At the center of each interval is the sample mean, marked with a diamond. ",20.2,15.757,"""8.8""",420,4.0,B
 "5.5-7. Suppose that the distribution of the weight of a prepackaged ""1-pound bag"" of carrots is $N\left(1.18,0.07^2\right)$ and the distribution of the weight of a prepackaged ""3-pound bag"" of carrots is $N\left(3.22,0.09^2\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag.
-","Even if P≠NP, the O(nW) complexity does not contradict the fact that the knapsack problem is NP-complete, since W, unlike n, is not polynomial in the length of the input to the problem. Baggett v. During the process of the running of this method, how do we get the weight w? Nevertheless, a simple modification allows us to solve this case: Assume for simplicity that all items individually fit in the sack (w_i \le W for all i). The target is to maximize the sum of the values of the items in the knapsack so that the sum of weights in each dimension d does not exceed W_d. Define value[n, W] Initialize all value[i, j] = -1 Define m:=(i,j) // Define function m so that it represents the maximum value we can get under the condition: use first i items, total weight limit is j { if i == 0 or j <= 0 then: value[i, j] = 0 return if (value[i-1, j] == -1) then: // m[i-1, j] has not been calculated, we have to call function m m(i-1, j) if w[i] > j then: // item cannot fit in the bag value[i, j] = value[i-1, j] else: if (value[i-1, j-w[i]] == -1) then: // m[i-1,j-w[i]] has not been calculated, we have to call function m m(i-1, j-w[i]) value[i, j] = max(value[i-1,j], value[i-1, j-w[i]] + v[i]) } Run m(n, W) For example, there are 10 different items and the weight limit is 67. For a given item i, suppose we could find a set of items J such that their total weight is less than the weight of i, and their total value is greater than the value of i. His version sorts the items in decreasing order of value per unit of weight, v_1/w_1\ge\cdots\ge v_n/w_n. Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity. The knapsack problem is the following problem in combinatorial optimization: :Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. Given a set of n items numbered from 1 up to n, each with a weight w_i and a value v_i, along with a maximum weight capacity W, : maximize \sum_{i=1}^n v_i x_i : subject to \sum_{i=1}^n w_i x_i \leq W and x_i \in \\{0,1\\}. There are only i ways and the previous weights are w-w_1, w-w_2,..., w-w_i where there are total i kinds of different item (by saying different, we mean that the weight and the value are not completely the same). Here x_i represents the number of instances of item i to include in the knapsack. In the Bag is a 1956 American animated short comedy film produced by Walt Disney Productions, directed by Jack Hannah,“Bearly” a Star: A Tribute To Disney’s Humphrey the Bear-Cartoon Research and featuring park ranger J. Audubon Woodlore and his comedic foil Humphrey the Bear.BCDB.com This was the last Disney theatrical cartoon short subject distributed by RKO Radio Pictures.Amazon.com ==Plot== Tourists have departed Brownstone National Park where Humphrey lives, leaving trash everywhere, despite signs asking tourists not to litter the park. Where are the hard knapsack problems? Three Bags Full: A Sheep Detective Story (original German title: Glennkill: Ein Schafskrimi) is 2005 novel by Leonie Swann. It can be shown that the average performance converges to the optimal solution in distribution at the error rate n^{-1/2} ==== Fully polynomial time approximation scheme ==== The fully polynomial time approximation scheme (FPTAS) for the knapsack problem takes advantage of the fact that the reason the problem has no known polynomial time solutions is because the profits associated with the items are not restricted. Baggett is a surname. The ranger prepares to reward Humphrey with a dish full of cacciatore, but before Humphrey can take it, the geyser suddenly erupts, spouting the garbage everywhere, resulting in Humphrey having to start all over again at cleaning up the park.Internet Archive ==In the Bag song== The song featured in In The Bag was so popular that Disney released a version of it (with similar instrumentation and different vocals) as a single, ""The Humphrey Hop"". Weight distribution is the apportioning of weight within a vehicle, especially cars, airplanes, and trains. The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number x_i of copies of each kind of item to a maximum non-negative integer value c: : maximize \sum_{i=1}^n v_i x_i : subject to \sum_{i=1}^n w_i x_i \leq W and x_i \in \\{0,1,2,\dots,c\\}. For this reason weight distribution varies with the vehicle's intended usage. ", 0.01961,210,22.0,0.9830,2500,D
+","Even if P≠NP, the O(nW) complexity does not contradict the fact that the knapsack problem is NP-complete, since W, unlike n, is not polynomial in the length of the input to the problem. Baggett v. During the process of the running of this method, how do we get the weight w? Nevertheless, a simple modification allows us to solve this case: Assume for simplicity that all items individually fit in the sack (w_i \le W for all i). The target is to maximize the sum of the values of the items in the knapsack so that the sum of weights in each dimension d does not exceed W_d. Define value[n, W] Initialize all value[i, j] = -1 Define m:=(i,j) // Define function m so that it represents the maximum value we can get under the condition: use first i items, total weight limit is j { if i == 0 or j <= 0 then: value[i, j] = 0 return if (value[i-1, j] == -1) then: // m[i-1, j] has not been calculated, we have to call function m m(i-1, j) if w[i] > j then: // item cannot fit in the bag value[i, j] = value[i-1, j] else: if (value[i-1, j-w[i]] == -1) then: // m[i-1,j-w[i]] has not been calculated, we have to call function m m(i-1, j-w[i]) value[i, j] = max(value[i-1,j], value[i-1, j-w[i]] + v[i]) } Run m(n, W) For example, there are 10 different items and the weight limit is 67. For a given item i, suppose we could find a set of items J such that their total weight is less than the weight of i, and their total value is greater than the value of i. His version sorts the items in decreasing order of value per unit of weight, v_1/w_1\ge\cdots\ge v_n/w_n. Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity. The knapsack problem is the following problem in combinatorial optimization: :Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. Given a set of n items numbered from 1 up to n, each with a weight w_i and a value v_i, along with a maximum weight capacity W, : maximize \sum_{i=1}^n v_i x_i : subject to \sum_{i=1}^n w_i x_i \leq W and x_i \in \\{0,1\\}. There are only i ways and the previous weights are w-w_1, w-w_2,..., w-w_i where there are total i kinds of different item (by saying different, we mean that the weight and the value are not completely the same). Here x_i represents the number of instances of item i to include in the knapsack. In the Bag is a 1956 American animated short comedy film produced by Walt Disney Productions, directed by Jack Hannah,“Bearly” a Star: A Tribute To Disney’s Humphrey the Bear-Cartoon Research and featuring park ranger J. Audubon Woodlore and his comedic foil Humphrey the Bear.BCDB.com This was the last Disney theatrical cartoon short subject distributed by RKO Radio Pictures.Amazon.com ==Plot== Tourists have departed Brownstone National Park where Humphrey lives, leaving trash everywhere, despite signs asking tourists not to litter the park. Where are the hard knapsack problems? Three Bags Full: A Sheep Detective Story (original German title: Glennkill: Ein Schafskrimi) is 2005 novel by Leonie Swann. It can be shown that the average performance converges to the optimal solution in distribution at the error rate n^{-1/2} ==== Fully polynomial time approximation scheme ==== The fully polynomial time approximation scheme (FPTAS) for the knapsack problem takes advantage of the fact that the reason the problem has no known polynomial time solutions is because the profits associated with the items are not restricted. Baggett is a surname. The ranger prepares to reward Humphrey with a dish full of cacciatore, but before Humphrey can take it, the geyser suddenly erupts, spouting the garbage everywhere, resulting in Humphrey having to start all over again at cleaning up the park.Internet Archive ==In the Bag song== The song featured in In The Bag was so popular that Disney released a version of it (with similar instrumentation and different vocals) as a single, ""The Humphrey Hop"". Weight distribution is the apportioning of weight within a vehicle, especially cars, airplanes, and trains. The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number x_i of copies of each kind of item to a maximum non-negative integer value c: : maximize \sum_{i=1}^n v_i x_i : subject to \sum_{i=1}^n w_i x_i \leq W and x_i \in \\{0,1,2,\dots,c\\}. For this reason weight distribution varies with the vehicle's intended usage. ", 0.01961,210,"""22.0""",0.9830,2500,D
 "5.3-7. The distributions of incomes in two cities follow the two Pareto-type pdfs
 $$
 f(x)=\frac{2}{x^3}, 1 < x < \infty , \text { and } g(y)= \frac{3}{y^4} ,  \quad 1 < y < \infty,
 $$
-respectively. Here one unit represents $\$ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.","As applied to distribution of incomes, this means that the larger the value of the Pareto index θ the smaller the proportion of incomes many times as big as the smallest incomes. The family of Pareto distributions is parameterized by * a positive number κ that is the smallest value that a random variable with a Pareto distribution can take. For example, it may be observed that 45% of individuals in the sample have incomes below a = $35,000 per year, and 55% have incomes below b = $40,000 per year. Multivariate Pareto distributions have been defined for many of these types. ==Bivariate Pareto distributions== ===Bivariate Pareto distribution of the first kind=== Mardia (1962) defined a bivariate distribution with cumulative distribution function (CDF) given by : F(x_1, x_2) = 1 -\sum_{i=1}^2\left(\frac{x_i}{\theta_i}\right)^{-a}+ \left(\sum_{i=1}^2 \frac{x_i}{\theta_i} - 1\right)^{-a}, \qquad x_i > \theta_i > 0, i=1,2; a>0, and joint density function : f(x_1, x_2) = (a+1)a(\theta_1 \theta_2)^{a+1}(\theta_2x_1 + \theta_1x_2 - \theta_1 \theta_2)^{-(a+2)}, \qquad x_i \geq \theta_i>0, i=1,2; a>0. This distribution is called a multivariate Pareto distribution of type II by Arnold. This distribution is called a multivariate Pareto distribution of type II by Arnold. In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution. There are several different types of univariate Pareto distributions including Pareto Types I−IV and Feller−Pareto. (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.) There were huge differences between white and the other people, not only in wages, but also in the place they can enter and so on. == Development of income distribution as a stochastic process == It is difficult to create a realistic and not complicated theoretical model, because the forces determining the distribution of income (DoI) are varied and complex and they continuously interact and fluctuate. The marginal distributions are Pareto Type 1 with density functions : f(x_i)=a\theta_i^a x_i^{-(a+1)}, \qquad x_i \geq \theta_i>0, i=1,2. Pareto interpolation is a method of estimating the median and other properties of a population that follows a Pareto distribution. Income inequality is amount to which income is distributed unequally in a population., additional text. If the location and scale parameter are allowed to differ, the complementary CDF is : \overline{F}(x_1,x_2) = \left(1 + \sum_{i=1}^2 \frac{x_i-\mu_i}{\sigma_i} \right)^{-a}, \qquad x_i > \mu_i, i=1,2, which has Pareto Type II univariate marginal distributions. In a model by Champernowne, the author assumes that the income scale is divided into an enumerable infinity of income ranges, which have uniform proportionate distribution. Modern economists have also addressed issues of income distribution, but have focused more on the distribution of income across individuals and households. As applied to distribution of incomes, κ is the lowest income of any person in the population; and * a positive number θ the ""Pareto index""; as this increases, the tail of the distribution gets thinner. The top 1% had a 71.9% of the overall shared income. == Household inequality == Household inequality is the extent to which income is distributed unequally among people living in a houses collectively in a population. If the location and scale parameter are allowed to differ, the complementary CDF is : \overline{F}(x_1,\dots,x_k) = \left(1 + \sum_{i=1}^k \frac{x_i-\mu_i}{\sigma_i} \right)^{-a}, \qquad x_i > \mu_i, \quad i=1,\dots,k, \qquad (3) which has marginal distributions of the same type (3) and Pareto Type II univariate marginal distributions. Category:Estimation methods Category:Income inequality metrics Category:Theory of probability distributions Category:Parametric statistics Category:Vilfredo Pareto While it is common to refer to pareto as ""80/20"" rule, under the assumption that, in all situations, 20% of causes determine 80% of problems, this ratio is merely a convenient rule of thumb and is not, nor should it be considered, an immutable law of nature. The marginal distributions and conditional distributions are of the same type (5); that is, they are multivariate Feller–Pareto distributions. ",5.85,0.4,93.4,1.07,0.66666666666,B
+respectively. Here one unit represents $\$ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.","As applied to distribution of incomes, this means that the larger the value of the Pareto index θ the smaller the proportion of incomes many times as big as the smallest incomes. The family of Pareto distributions is parameterized by * a positive number κ that is the smallest value that a random variable with a Pareto distribution can take. For example, it may be observed that 45% of individuals in the sample have incomes below a = $35,000 per year, and 55% have incomes below b = $40,000 per year. Multivariate Pareto distributions have been defined for many of these types. ==Bivariate Pareto distributions== ===Bivariate Pareto distribution of the first kind=== Mardia (1962) defined a bivariate distribution with cumulative distribution function (CDF) given by : F(x_1, x_2) = 1 -\sum_{i=1}^2\left(\frac{x_i}{\theta_i}\right)^{-a}+ \left(\sum_{i=1}^2 \frac{x_i}{\theta_i} - 1\right)^{-a}, \qquad x_i > \theta_i > 0, i=1,2; a>0, and joint density function : f(x_1, x_2) = (a+1)a(\theta_1 \theta_2)^{a+1}(\theta_2x_1 + \theta_1x_2 - \theta_1 \theta_2)^{-(a+2)}, \qquad x_i \geq \theta_i>0, i=1,2; a>0. This distribution is called a multivariate Pareto distribution of type II by Arnold. This distribution is called a multivariate Pareto distribution of type II by Arnold. In statistics, a multivariate Pareto distribution is a multivariate extension of a univariate Pareto distribution. There are several different types of univariate Pareto distributions including Pareto Types I−IV and Feller−Pareto. (This definition is not equivalent to Mardia's bivariate Pareto distribution of the second kind.) There were huge differences between white and the other people, not only in wages, but also in the place they can enter and so on. == Development of income distribution as a stochastic process == It is difficult to create a realistic and not complicated theoretical model, because the forces determining the distribution of income (DoI) are varied and complex and they continuously interact and fluctuate. The marginal distributions are Pareto Type 1 with density functions : f(x_i)=a\theta_i^a x_i^{-(a+1)}, \qquad x_i \geq \theta_i>0, i=1,2. Pareto interpolation is a method of estimating the median and other properties of a population that follows a Pareto distribution. Income inequality is amount to which income is distributed unequally in a population., additional text. If the location and scale parameter are allowed to differ, the complementary CDF is : \overline{F}(x_1,x_2) = \left(1 + \sum_{i=1}^2 \frac{x_i-\mu_i}{\sigma_i} \right)^{-a}, \qquad x_i > \mu_i, i=1,2, which has Pareto Type II univariate marginal distributions. In a model by Champernowne, the author assumes that the income scale is divided into an enumerable infinity of income ranges, which have uniform proportionate distribution. Modern economists have also addressed issues of income distribution, but have focused more on the distribution of income across individuals and households. As applied to distribution of incomes, κ is the lowest income of any person in the population; and * a positive number θ the ""Pareto index""; as this increases, the tail of the distribution gets thinner. The top 1% had a 71.9% of the overall shared income. == Household inequality == Household inequality is the extent to which income is distributed unequally among people living in a houses collectively in a population. If the location and scale parameter are allowed to differ, the complementary CDF is : \overline{F}(x_1,\dots,x_k) = \left(1 + \sum_{i=1}^k \frac{x_i-\mu_i}{\sigma_i} \right)^{-a}, \qquad x_i > \mu_i, \quad i=1,\dots,k, \qquad (3) which has marginal distributions of the same type (3) and Pareto Type II univariate marginal distributions. Category:Estimation methods Category:Income inequality metrics Category:Theory of probability distributions Category:Parametric statistics Category:Vilfredo Pareto While it is common to refer to pareto as ""80/20"" rule, under the assumption that, in all situations, 20% of causes determine 80% of problems, this ratio is merely a convenient rule of thumb and is not, nor should it be considered, an immutable law of nature. The marginal distributions and conditional distributions are of the same type (5); that is, they are multivariate Feller–Pareto distributions. ",5.85,0.4,"""93.4""",1.07,0.66666666666,B
 "7.3-3. Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year.
-(a) Use these data to give a point estimate of $p$.","A prospective cohort study of 76 runners followed for one year showed that 51 percent reported an injury. ""A prospective cohort study of 300 runners followed for two years showed that 73 percent of women and 62 percent of men sustained an injury, with 56 percent of the injured runners sustaining more than one injury during the study period."" Many of the common injuries that affect runners are chronic, developing over longer periods as the result of overuse. Because of this mechanism, stress fractures are common overuse injuries in athletes. ""Over 60% of male injured runners and over 50% of female injured runners had increased their weekly running distance by >30% between consecutive weeks at least once in the 4 weeks prior to injury."" However, this has not been proven and is still debated. == Overview == > ""The causes of running injuries are so multifactorial and diverse, and > apparently vary greatly from individual to individual, that any preventive > measure proposed would probably be of help to only a small minority. Common overuse injuries include shin splints, stress fractures, Achilles tendinitis, Iliotibial band syndrome, Patellofemoral pain (runner's knee), and plantar fasciitis. In general, overuse injuries are the result of repetitive impact between the foot and the ground. The 2013 European Triathlon Championships was held in Alanya, Turkey from 14 June to 16 June 2013. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:42:09 1:42:16 1:42:22 Women 1:55:43 1:55:45 1:55:53 Mixed Relay 1:32:05 1:32:25 1:32:29 Junior Junior Junior Junior Junior Junior Junior Men 0:52:40 0:52:59 0:53:05 Women 0:58:46 0:59:03 0:59:11 Mixed Relay 1:34:18 1:34:38 1:34:46 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 1 16:46 0:53:18 30:34 1:42:09 — 2 16:42 0:53:23 30:43 1:42:16 +00:07 3 17:04 0:54:46 29:05 1:42:22 +00:13 4 46 16:39 0:53:23 31:07 1:42:42 +00:33 5 7 16:37 0:54:46 31:16 1:42:48 +00:39 6 8 16:41 0:53:28 31:26 1:42:57 +00:48 7 4 16:44 0:53:27 30:07 1:43:29 +01:20 8 9 16:51 0:53:22 30:14 1:43:31 +01:22 9 22 16:46 0:55:11 32:05 1:43:43 +01:34 10 11 17:06 0:55:02 30:31 1:43:50 +01:41 11 5 17:09 0:53:19 30:34 1:43:54 +01:45 12 55 17:08 0:54:44 30:34 1:44:01 +01:52 13 29 17:14 0:54:40 30:50 1:44:16 +02:07 14 19 17:06 0:54:49 31:08 1:44:25 +02:16 15 12 17:02 0:54:40 31:18 1:44:34 +02:25 16 10 16:49 0:54:46 31:29 1:44:48 +02:39 17 23 17:08 0:54:47 31:38 1:44:59 +02:50 18 21 16:44 0:55:01 33:32 1:45:10 +03:01 19 31 16:50 0:54:41 32:06 1:45:27 +03:18 20 24 17:10 0:53:20 32:36 1:45:53 +03:44 21 41 17:05 0:54:59 32:43 1:46:03 +03:54 22 33 17:30 0:54:42 30:48 1:46:13 +04:04 23 14 17:42 0:54:46 31:00 1:46:24 +04:15 24 42 17:07 0:56:31 33:29 1:46:54 +04:45 25 26 16:44 0:56:19 35:44 1:47:17 +05:08 Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 9 18:09 1:02:11 33:53 1:55:43 — 22 18:17 1:02:07 33:55 1:55:45 +00:02 4 18:21 1:02:03 34:06 1:55:53 +00:10 4 11 18:27 1:02:07 34:17 1:56:03 +00:20 5 37 18:25 1:02:03 34:24 1:56:15 +00:32 6 10 18:16 1:01:56 34:55 1:56:39 +00:56 7 23 18:19 1:02:03 34:49 1:56:42 +00:59 8 20 18:22 1:02:03 35:05 1:56:55 +01:12 9 12 18:12 1:02:01 35:15 1:57:07 +01:24 10 14 19:03 1:02:03 33:56 1:57:09 +01:26 11 3 18:20 1:02:11 35:33 1:57:18 +01:35 12 6 18:20 1:02:47 35:37 1:57:26 +01:43 13 28 18:22 1:02:02 35:49 1:57:40 +01:57 14 17 18:13 1:02:02 36:01 1:57:50 +02:07 15 5 18:39 1:01:56 34:38 1:57:52 +02:09 16 21 18:19 1:02:10 36:07 1:57:59 +02:16 17 26 19:03 1:03:11 34:50 1:58:08 +02:25 18 24 18:07 1:02:04 36:28 1:58:19 +02:36 19 33 18:19 1:02:44 36:32 1:58:24 +02:41 20 1 19:07 1:02:14 35:24 1:58:40 +02:57 21 16 18:24 1:02:04 37:07 1:59:02 +03:19 22 15 18:48 1:02:39 35:59 1:59:19 +03:36 23 18 18:59 1:02:01 36:29 1:59:42 +03:59 24 19 19:06 1:02:55 34:37 1:59:47 +04:04 25 2 19:00 1:02:47 36:43 1:59:57 +04:14 Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Turkey Category:International sports competitions hosted by Turkey Category:Alanya The 2015 European Triathlon Championships was held in Geneva, Switzerland from 9 July to 12 July 2015. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:52:55 1:53:13 1:53:16 Women 2:07:15 2:08:14 2:09:16 Mixed Relay Jeanne Lehair David Hauss Emmie Charayron Simon Viain 1:25:21 Jolanda Annen Andrea Salvisberg Nicola Spirig Sven Riederer 1:25:30 Jodie Stimpson Lucy Hall Thomas Bishop Matthew Sharp 1:25:31 Junior Junior Junior Junior Junior Junior Junior Men 0:57:41 0:57:42 0:57:42 Women 1:04:06 1:04:42 1:04:50 Mixed Relay Margot Garabedian Maxime Hueber-Moosbrugger Emilie Morier Léo Bergere 1:28:37 Lena Meißner Linus Stimmel Lisa Tertsch Lasse Lührs 1:28:59 Alberte Kjær Pedersen Daniel Bækkegård Anne Holm Emil Deleuran Hansen 1:29:14 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course 0 Rank # Triathlete Swimming Cycling Running Total time Difference 16 17:54 1:02:39 31:19 1:52:55 — 7 17:55 1:02:38 31:42 1:53:13 +00:18 24 18:06 1:02:25 31:43 1:53:16 +00:21 4 9 17:41 1:02:38 31:46 1:53:20 +00:25 5 18 18:08 1:02:25 31:42 1:53:22 +00:27 6 1 17:41 1:02:52 32:09 1:53:46 +00:51 7 23 17:54 1:02:26 32:29 1:54:00 +01:05 8 3 17:50 1:02:51 32:30 1:54:03 +01:08 9 5 28:08 1:02:38 00:00 1:54:07 +01:12 10 32 17:56 1:02:40 33:20 1:54:59 +02:04 11 41 18:10 0:00:00 33:30 1:55:08 +02:13 12 8 17:55 1:02:36 33:39 1:55:09 +02:14 13 50 17:49 1:02:24 33:55 1:55:27 +02:32 14 31 17:55 1:02:40 34:07 1:55:43 +02:48 15 15 17:53 1:02:42 34:15 1:55:49 +02:54 16 61 17:44 1:02:38 34:13 1:55:52 +02:57 17 21 18:46 1:02:40 32:23 1:56:42 +03:47 18 2 17:55 1:02:45 33:13 1:56:47 +03:52 19 36 17:56 1:04:35 35:11 1:56:52 +03:57 20 19 18:50 1:04:36 32:47 1:57:16 +04:21 21 14 18:58 1:02:34 33:03 1:57:27 +04:32 22 26 17:51 1:04:34 33:20 1:57:40 +04:45 23 35 17:59 1:04:23 36:16 1:57:53 +04:58 24 34 19:01 1:05:13 33:40 1:58:05 +05:10 25 25 18:47 1:02:40 33:44 1:58:12 +05:17 Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course 0 Rank # Triathlete Swimming Cycling Running Total time Difference 1 19:48 1:10:49 35:33 2:07:15 — 2 19:34 1:11:03 36:31 2:08:14 +00:59 7 19:46 1:10:51 37:31 2:09:16 +02:01 4 5 19:50 1:11:03 35:44 2:09:45 +02:30 5 9 18:48 1:10:51 38:14 2:09:59 +02:44 6 22 20:39 1:13:07 36:13 2:10:20 +03:05 7 30 19:42 1:11:48 38:40 2:10:26 +03:11 8 20 20:33 1:12:21 36:25 2:10:33 +03:18 9 6 19:34 1:10:54 37:54 2:10:50 +03:35 10 11 20:23 1:12:27 36:52 2:10:56 +03:41 11 37 18:50 1:12:18 39:11 2:11:00 +03:45 12 24 20:32 1:12:36 00:00 2:11:11 +03:56 13 17 20:28 1:11:45 37:04 2:11:14 +03:59 14 16 19:33 0:00:00 38:25 2:11:23 +04:08 15 8 18:55 1:12:33 39:35 2:11:28 +04:13 16 25 19:56 1:12:17 37:49 2:11:57 +04:42 17 35 20:27 1:11:42 38:20 2:12:24 +05:09 18 23 20:34 1:12:58 00:00 2:12:38 +05:23 19 36 20:44 1:12:30 35:40 2:12:49 +05:34 20 21 20:30 0:00:00 38:51 2:13:02 +05:47 21 12 19:44 1:15:07 40:01 2:13:04 +05:49 22 44 19:36 1:12:26 39:11 2:13:19 +06:04 23 39 19:52 1:12:09 39:15 2:13:24 +06:09 24 18 20:27 1:13:20 39:19 2:13:27 +06:12 25 15 19:53 1:13:06 40:13 2:14:24 +07:09 Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Switzerland Category:International sports competitions hosted by Switzerland Category:Geneva The 2017 European Triathlon Championships was held in Kitzbühel, Austria from 16 June to 18 June 2017. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:45:31 1:45:32 1:45:35 Women 1:57:50 1:58:05 1:58:31 Mixed Relay Anne Holm Andreas Schilling Sif Bendix Madsen Emil Deleuran Hansen 1:15:17 Cassandre Beaugrand Simon Viain Emilie Morier Raphael Montoya 1:15:24 Anastasia Gorbunova Dmitry Polyanskiy Anastasia Abrosimova Vladimir Turbaevskiy 1:15:32 Junior Junior Junior Junior Junior Junior Junior Men 53:39 53:40 53:40 Women 59:20 59:23 59:34 Mixed Relay Lili Mátyus Gergő Soós Dorka Putnóczki Csongor Lehmann 1:18:31 Daria Lushnikova Mikhail Antipov Ekaterina Matiukh Grigory Antipov 1:19:01 Bianca Bogen Moritz Horn Nina Eim Tim Siepmann 1:19:15 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 4 18:52 53:34 31:54 1:45:31 — 21 18:49 53:36 31:52 1:45:32 +0:01 5 18:57 53:29 31:58 1:45:35 +0:04 4 9 18:49 53:35 32:02 1:45:40 +0:09 5 10 18:44 53:42 32:12 1:45:47 +0:16 6 7 19:25 53:47 31:29 1:45:51 +0:20 7 6 19:11 54:03 31:32 1:45:54 +0:23 8 17 19:00 53:29 32:32 1:46:12 +0:81 9 16 19:21 53:46 32:01 1:46:16 +0:85 10 1 18:38 53:49 32:45 1:46:21 +0:90 11 11 19:23 53:47 32:12 1:46:33 +1:02 12 20 18:46 54:22 32:24 1:46:47 +1:16 13 18 19:00 53:26 33:18 1:46:50 +1:19 14 8 19:04 53:21 33:16 1:46:51 +1:20 15 37 19:02 53:20 33:12 1:46:51 +1:20 16 28 18:41 53:44 33:18 1:46:52 +1:21 17 3 18:39 53:44 33:16 1:46:52 +1:21 18 39 19:13 53:39 32:27 1:46:54 +1:23 19 14 19:23 53:48 33:04 1:47:20 +1:89 20 38 19:00 53:25 33:49 1:47:25 +1:94 21 40 19:16 47:00 33:01 1:47:27 +1:96 22 49 19:15 53:55 33:06 1:47:30 +1:99 23 15 19:10 54:00 33:34 1:47:50 +2:19 24 2 18:43 53:41 34:31 1:48:03 +2:72 25 25 18:42 53:44 34:43 1:48:15 +2:84 26 29 19:25 53:48 33:53 1:48:21 +2:90 27 33 19:20 53:49 34:07 1:48:23 +2:92 28 27 19:20 53:51 34:07 1:48:30 +2:99 29 43 19:14 53:54 34:15 1:48:35 +3:04 30 45 18:41 53:42 35:08 1:48:44 +3:13 31 32 19:17 53:56 34:40 1:48:58 +3:27 32 26 19:21 53:50 34:53 1:49:09 +3:78 33 23 18:45 53:41 36:04 1:49:44 +4:13 34 53 19:18 53:52 36:12 1:50:37 +5:06 35 50 18:54 53:31 37:26 1:51:07 +5:76 36 31 18:59 54:11 36:58 1:51:18 +5:87 37 22 19:32 57:38 33:10 1:51:30 +5:99 38 36 19:32 56:31 34:38 1:51:53 +6:22 39 48 19:08 54:02 37:41 1:52:03 +6:72 40 47 19:17 57:51 35:10 1:53:33 +8:02 41 41 19:30 57:39 36:05 1:54:22 +8:91 42 34 19:06 58:02 36:11 1:54:31 +9:00 43 19 20:55 56:11 36:58 1:55:14 +9:83 44 55 19:27 59:18 35:20 1:55:24 +9:93 45 51 20:55 56:11 37:26 1:55:44 +10:13 46 52 19:22 57:47 38:25 1:56:47 +11:16 47 54 19:31 59:18 37:55 1:57:54 +12:23 — 35 20:13 56:55 did not finish did not finish did not finish — 12 18:36 53:47 did not finish did not finish did not finish — 24 19:23 did not finish did not finish did not finish did not finish — 42 19:31 did not finish did not finish did not finish did not finish — 44 18:57 did not advance did not advance did not advance did not advance — 46 19:08 Lapped Lapped Lapped Lapped Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 8 19:09 1:00:11 37:14 1:57:50 15 19:49 59:34 37:27 1:58:05 +0:15 14 19:45 59:36 37:53 1:58:31 +0:41 4 3 20:53 1:00:44 35:51 1:58:41 +0:51 5 9 20:56 1:00:45 35:44 1:58:41 +0:51 6 6 20:18 1:01:21 35:55 1:58:47 +0:57 7 7 20:51 1:00:46 36:08 1:59:00 +1:10 8 10 19:50 1:01:51 36:29 1:59:24 +1:34 9 2 21:00 1:00:43 36:30 1:59:28 +1:38 10 23 20:05 1:03:13 34:59 1:59:37 +1:47 11 19 19:27 59:52 39:06 1:59:46 +1:56 12 11 21:00 1:00:39 37:15 2:00:13 +2:23 13 12 20:59 1:00:41 37:29 2:00:24 +2:34 14 22 20:57 1:00:43 37:42 2:00:37 +2:47 15 1 20:16 1:01:22 38:27 2:01:24 +3:34 16 17 Michelle Flipo 20:56 1:02:16 37:38 2:02:14 +4:24 17 27 20:48 1:00:47 39:34 2:02:28 +4:38 18 26 21:13 1:02:02 38:01 2:02:41 +4:51 19 18 21:02 1:02:13 38:37 2:03:09 +5:19 20 4 21:14 1:02:00 38:46 2:03:19 +5:29 21 24 21:01 1:00:39 42:08 2:05:07 +7:17 22 20 20:17 1:01:20 44:13 2:07:13 +9:23 16 20:17 1:01:26 did not finish did not finish did not finish did not finish 28 22:20 did not finish did not finish did not finish did not finish 21 21:13 did not finish did not finish did not finish did not finish 29 23:34 did not finish did not finish did not finish did not finish 25 21:12 Lapped Lapped Lapped Lapped 30 22:50 Lapped Lapped Lapped Lapped Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Austria Category:2017 in Austrian sport Category:June 2017 sports events in Europe Category:International sports competitions hosted by Austria Category:Kitzbühel Running injuries (or running-related injuries, RRI) affect about half of runners annually. The 2016 European Triathlon Championships was held in Lisbon, Portugal from 26 May to 29 May 2016. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:49:30 1:50:09 1:50:32 Women 2:04:03 2:04:19 2:04:24 Mixed Relay Lucy Hall Thomas Bishop India Lee Grant Sheldon 1:07:03 Mariya Shorets Igor Polyanskiy Alexandra Razarenova Dmitry Polyanskiy 1:07:08 Zsófia Kovács Tamás Tóth Margit Vanek Ákos Vanek 1:07:19 Junior Junior Junior Junior Junior Junior Junior Men 0:58:03 0:58:04 0:58:08 Women 1:02:42 1:02:54 1:03:14 Mixed Relay Sian Rainsley Samuel Dickinson Kate Waugh Alex Yee 1:07:48 Ines Santiago Moron Alberto Gonzalez Garcia Cecilia Santamaria Surroca Javier Lluch Perez 1:07:59 Lena Meißner Paul Weindl Lisa Tertsch Moritz Horn 1:08:00 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 9 16:55 0:59:45 31:25 1:49:30 — 6 16:51 0:59:51 32:02 1:50:09 +00:39 19 16:49 0:59:47 32:31 1:50:32 +01:02 4 8 16:58 0:59:51 32:32 1:50:37 +01:07 5 15 16:46 0:59:47 32:30 1:50:38 +01:08 6 10 17:13 0:59:39 31:00 1:50:39 +01:09 7 16 17:07 0:59:51 30:57 1:50:40 +01:10 8 7 16:45 1:01:03 32:40 1:50:48 +01:18 9 14 17:14 1:01:09 31:12 1:51:01 +01:31 10 52 16:58 0:59:57 31:44 1:51:05 +01:35 11 38 16:52 1:01:07 33:01 1:51:09 +01:39 12 4 16:44 1:00:55 33:07 1:51:12 +01:42 13 12 17:10 0:59:48 31:44 1:51:31 +02:01 14 24 17:14 0:59:53 00:00 1:51:46 +02:16 15 48 16:54 1:01:09 34:14 1:51:49 +02:19 16 45 17:18 0:00:00 32:00 1:51:50 +02:20 17 30 17:06 0:59:45 32:23 1:52:03 +02:33 18 26 17:05 1:01:04 32:19 1:52:05 +02:35 19 5 16:50 1:01:10 34:00 1:52:09 +02:39 20 42 17:19 1:01:09 32:32 1:52:14 +02:44 21 36 17:06 0:59:46 33:07 1:52:22 +02:52 22 29 17:09 1:00:55 32:46 1:52:24 +02:54 23 23 17:13 1:01:08 32:50 1:52:29 +02:59 24 33 17:07 1:01:05 33:40 1:52:31 +03:01 25 18 17:12 1:01:00 32:45 1:52:34 +03:04 Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course 0 Rank # Triathlete Swimming Cycling Running Total time Difference 20 18:41 1:06:05 36:57 2:04:03 — 6 19:11 1:07:09 35:36 2:04:19 +00:16 9 18:53 1:07:43 35:38 2:04:24 +00:21 4 1 18:04 1:07:09 36:03 2:04:40 +00:37 5 14 19:26 1:07:43 36:03 2:04:45 +00:42 6 7 19:05 1:08:27 36:03 2:04:51 +00:48 7 25 18:28 1:07:02 36:13 2:05:04 +01:01 8 11 18:29 1:07:21 36:28 2:05:09 +01:06 9 3 19:24 1:08:03 36:41 2:05:23 +01:20 10 2 17:55 1:07:49 38:20 2:05:29 +01:26 11 16 18:28 1:07:05 36:55 2:05:43 +01:40 12 26 18:09 1:06:58 37:07 2:05:48 +01:45 13 17 19:08 1:07:57 37:07 2:05:53 +01:50 14 19 18:48 1:08:21 37:09 2:06:00 +01:57 15 8 18:03 1:07:21 37:12 2:06:02 +01:59 16 24 18:06 1:07:42 37:17 2:06:03 +02:00 17 30 18:05 1:08:27 38:05 2:06:49 +02:46 18 22 18:31 1:08:25 38:42 2:07:23 +03:20 19 15 18:07 1:08:19 39:17 2:08:05 +04:02 20 29 18:08 1:07:52 39:56 2:08:36 +04:33 21 27 18:40 1:08:24 40:35 2:09:11 +05:08 22 32 18:03 1:08:07 40:57 2:09:39 +05:36 23 21 18:44 1:07:46 42:42 2:11:22 +07:19 24 34 18:32 1:08:21 43:10 2:12:01 +07:58 25 28 19:27 1:07:30 42:23 2:13:02 +08:59 Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Portugal Category:International sports competitions hosted by Portugal Category:Lisbon Injured runners were heavier. Some injuries are acute, caused by sudden overstress, such as side stitch, strains, and sprains. Instead of resulting from a single severe impact, stress fractures are the result of accumulated injury from repeated submaximal loading, such as running or jumping. In the 1984 Bern 16 km race questionnaire, runners who had no shoe brand preference and presumably changed brands frequently had significantly fewer running injuries. These findings suggest that focusing on proper running form, particularly when fatigued, could reduce the risk of running-related injuries. Pete Jacobs File:silver medal icon.svg 2011 File:gold medal icon.svg 2012 Sebastian Kienle File:bronze medal icon.svg 2013 File:gold medal icon.svg 2014 File:silver medal icon.svg 2016 Patrick Lange File:bronze medal icon.svg 2016 File:gold medal icon.svg 2017 He is the record holder for the Ironman World Championship James Lawrence Holds record for most triathlons completed in a single year Chris Lieto File:silver medal icon.svg 2009 Eneko Llanos (23) 2000 (20) 2004 File:silver medal icon.svg 2008 Chris McCormack File:gold medal icon.svg 2010 File:gold medal icon.svg 2007 File:silver medal icon.svg 2006 File:gold medal icon.svg 1997 File:gold medal icon.svg 1997 Javier Gomez File:silver medal icon.svg 2012 File:silver medal icon.svg 2007 File:gold medal icon.svg 2008 File:silver medal icon.svg 2009 File:gold medal icon.svg 2010 File:bronze medal icon.svg 2011 File:silver medal icon.svg 2012 File:gold medal icon.svg 2013 File:gold medal icon.svg 2014 File:gold medal icon.svg 2006 File:gold medal icon.svg 2007 File:gold medal icon.svg 2008 Andreas Raelert (12) 2000 (6) 2004 File:bronze medal icon.svg 2009 File:silver medal icon.svg 2010 File:bronze medal icon.svg 2011 File:silver medal icon.svg 2012 File:silver medal icon.svg 2015 Jan Rehula File:bronze medal icon.svg 2000 Sven Riederer File:bronze medal icon.svg 2004 (23) 2008 Marino Vanhoenacker File:bronze medal icon.svg 2010 Stephan Vuckovic File:silver medal icon.svg 2000 Simon Whitfield File:gold medal icon.svg 2000 (11) 2004 File:silver medal icon.svg 2008 (DNF) 2012 First man to win a gold medal at the Olympics ==Women== Name Country Olympics Ironman WTS WC Other Ref Kate Allen File:gold medal icon.svg 2004 (14) 2008 Erin Densham (22) 2008 File:bronze medal icon.svg 2012 Vanessa Fernandes (8) 2004 File:silver medal icon.svg 2008 Jude Flannery From 1991-96, won six US age group national championship and four world age-group triathlon championships. Injuries are more likely to occur in novice barefoot runners. Rutger Beke File:silver medal icon.svg 2008 Alistair Brownlee (12) 2008 File:gold medal icon.svg 2012 File:gold medal icon.svg 2016 File:gold medal icon.svg 2009 File:gold medal icon.svg 2011 Jonathan Brownlee File:bronze medal icon.svg 2012 File:silver medal icon.svg 2016 File:gold medal icon.svg 2012 File:silver medal icon.svg 2013 File:bronze medal icon.svg 2014 Hamish Carter (26) 2000 File:gold medal icon.svg 2004 Bevan Docherty File:silver medal icon.svg 2004 File:bronze medal icon.svg 2008 (12) 2012 Jan Frodeno File:gold medal icon.svg 2008 (6) 2012 File:bronze medal icon.svg 2014 File:gold medal icon.svg 2015 File:gold medal icon.svg 2016 Arthur Gilbert At 90 years of age, confirmed as the world's oldest competing triathlete in 2011. Extrinsic risk factors include deconditioning, hard surfaces, inadequate stretching and poor footwear. == Footwear == === Traditional running shoes === Study participants wearing running shoes with moderate lateral torsional stiffness ""were 49% less likely to incur any type of lower extremity injury and 52% less likely to incur an overuse lower extremity injury than"" participants wearing running shoes with minimal lateral torsional stiffness, both of which were statistically significant observations."" ",129,3.8,0.08,0.5061,-20,D
+(a) Use these data to give a point estimate of $p$.","A prospective cohort study of 76 runners followed for one year showed that 51 percent reported an injury. ""A prospective cohort study of 300 runners followed for two years showed that 73 percent of women and 62 percent of men sustained an injury, with 56 percent of the injured runners sustaining more than one injury during the study period."" Many of the common injuries that affect runners are chronic, developing over longer periods as the result of overuse. Because of this mechanism, stress fractures are common overuse injuries in athletes. ""Over 60% of male injured runners and over 50% of female injured runners had increased their weekly running distance by >30% between consecutive weeks at least once in the 4 weeks prior to injury."" However, this has not been proven and is still debated. == Overview == > ""The causes of running injuries are so multifactorial and diverse, and > apparently vary greatly from individual to individual, that any preventive > measure proposed would probably be of help to only a small minority. Common overuse injuries include shin splints, stress fractures, Achilles tendinitis, Iliotibial band syndrome, Patellofemoral pain (runner's knee), and plantar fasciitis. In general, overuse injuries are the result of repetitive impact between the foot and the ground. The 2013 European Triathlon Championships was held in Alanya, Turkey from 14 June to 16 June 2013. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:42:09 1:42:16 1:42:22 Women 1:55:43 1:55:45 1:55:53 Mixed Relay 1:32:05 1:32:25 1:32:29 Junior Junior Junior Junior Junior Junior Junior Men 0:52:40 0:52:59 0:53:05 Women 0:58:46 0:59:03 0:59:11 Mixed Relay 1:34:18 1:34:38 1:34:46 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 1 16:46 0:53:18 30:34 1:42:09 — 2 16:42 0:53:23 30:43 1:42:16 +00:07 3 17:04 0:54:46 29:05 1:42:22 +00:13 4 46 16:39 0:53:23 31:07 1:42:42 +00:33 5 7 16:37 0:54:46 31:16 1:42:48 +00:39 6 8 16:41 0:53:28 31:26 1:42:57 +00:48 7 4 16:44 0:53:27 30:07 1:43:29 +01:20 8 9 16:51 0:53:22 30:14 1:43:31 +01:22 9 22 16:46 0:55:11 32:05 1:43:43 +01:34 10 11 17:06 0:55:02 30:31 1:43:50 +01:41 11 5 17:09 0:53:19 30:34 1:43:54 +01:45 12 55 17:08 0:54:44 30:34 1:44:01 +01:52 13 29 17:14 0:54:40 30:50 1:44:16 +02:07 14 19 17:06 0:54:49 31:08 1:44:25 +02:16 15 12 17:02 0:54:40 31:18 1:44:34 +02:25 16 10 16:49 0:54:46 31:29 1:44:48 +02:39 17 23 17:08 0:54:47 31:38 1:44:59 +02:50 18 21 16:44 0:55:01 33:32 1:45:10 +03:01 19 31 16:50 0:54:41 32:06 1:45:27 +03:18 20 24 17:10 0:53:20 32:36 1:45:53 +03:44 21 41 17:05 0:54:59 32:43 1:46:03 +03:54 22 33 17:30 0:54:42 30:48 1:46:13 +04:04 23 14 17:42 0:54:46 31:00 1:46:24 +04:15 24 42 17:07 0:56:31 33:29 1:46:54 +04:45 25 26 16:44 0:56:19 35:44 1:47:17 +05:08 Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 9 18:09 1:02:11 33:53 1:55:43 — 22 18:17 1:02:07 33:55 1:55:45 +00:02 4 18:21 1:02:03 34:06 1:55:53 +00:10 4 11 18:27 1:02:07 34:17 1:56:03 +00:20 5 37 18:25 1:02:03 34:24 1:56:15 +00:32 6 10 18:16 1:01:56 34:55 1:56:39 +00:56 7 23 18:19 1:02:03 34:49 1:56:42 +00:59 8 20 18:22 1:02:03 35:05 1:56:55 +01:12 9 12 18:12 1:02:01 35:15 1:57:07 +01:24 10 14 19:03 1:02:03 33:56 1:57:09 +01:26 11 3 18:20 1:02:11 35:33 1:57:18 +01:35 12 6 18:20 1:02:47 35:37 1:57:26 +01:43 13 28 18:22 1:02:02 35:49 1:57:40 +01:57 14 17 18:13 1:02:02 36:01 1:57:50 +02:07 15 5 18:39 1:01:56 34:38 1:57:52 +02:09 16 21 18:19 1:02:10 36:07 1:57:59 +02:16 17 26 19:03 1:03:11 34:50 1:58:08 +02:25 18 24 18:07 1:02:04 36:28 1:58:19 +02:36 19 33 18:19 1:02:44 36:32 1:58:24 +02:41 20 1 19:07 1:02:14 35:24 1:58:40 +02:57 21 16 18:24 1:02:04 37:07 1:59:02 +03:19 22 15 18:48 1:02:39 35:59 1:59:19 +03:36 23 18 18:59 1:02:01 36:29 1:59:42 +03:59 24 19 19:06 1:02:55 34:37 1:59:47 +04:04 25 2 19:00 1:02:47 36:43 1:59:57 +04:14 Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Turkey Category:International sports competitions hosted by Turkey Category:Alanya The 2015 European Triathlon Championships was held in Geneva, Switzerland from 9 July to 12 July 2015. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:52:55 1:53:13 1:53:16 Women 2:07:15 2:08:14 2:09:16 Mixed Relay Jeanne Lehair David Hauss Emmie Charayron Simon Viain 1:25:21 Jolanda Annen Andrea Salvisberg Nicola Spirig Sven Riederer 1:25:30 Jodie Stimpson Lucy Hall Thomas Bishop Matthew Sharp 1:25:31 Junior Junior Junior Junior Junior Junior Junior Men 0:57:41 0:57:42 0:57:42 Women 1:04:06 1:04:42 1:04:50 Mixed Relay Margot Garabedian Maxime Hueber-Moosbrugger Emilie Morier Léo Bergere 1:28:37 Lena Meißner Linus Stimmel Lisa Tertsch Lasse Lührs 1:28:59 Alberte Kjær Pedersen Daniel Bækkegård Anne Holm Emil Deleuran Hansen 1:29:14 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course 0 Rank # Triathlete Swimming Cycling Running Total time Difference 16 17:54 1:02:39 31:19 1:52:55 — 7 17:55 1:02:38 31:42 1:53:13 +00:18 24 18:06 1:02:25 31:43 1:53:16 +00:21 4 9 17:41 1:02:38 31:46 1:53:20 +00:25 5 18 18:08 1:02:25 31:42 1:53:22 +00:27 6 1 17:41 1:02:52 32:09 1:53:46 +00:51 7 23 17:54 1:02:26 32:29 1:54:00 +01:05 8 3 17:50 1:02:51 32:30 1:54:03 +01:08 9 5 28:08 1:02:38 00:00 1:54:07 +01:12 10 32 17:56 1:02:40 33:20 1:54:59 +02:04 11 41 18:10 0:00:00 33:30 1:55:08 +02:13 12 8 17:55 1:02:36 33:39 1:55:09 +02:14 13 50 17:49 1:02:24 33:55 1:55:27 +02:32 14 31 17:55 1:02:40 34:07 1:55:43 +02:48 15 15 17:53 1:02:42 34:15 1:55:49 +02:54 16 61 17:44 1:02:38 34:13 1:55:52 +02:57 17 21 18:46 1:02:40 32:23 1:56:42 +03:47 18 2 17:55 1:02:45 33:13 1:56:47 +03:52 19 36 17:56 1:04:35 35:11 1:56:52 +03:57 20 19 18:50 1:04:36 32:47 1:57:16 +04:21 21 14 18:58 1:02:34 33:03 1:57:27 +04:32 22 26 17:51 1:04:34 33:20 1:57:40 +04:45 23 35 17:59 1:04:23 36:16 1:57:53 +04:58 24 34 19:01 1:05:13 33:40 1:58:05 +05:10 25 25 18:47 1:02:40 33:44 1:58:12 +05:17 Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course 0 Rank # Triathlete Swimming Cycling Running Total time Difference 1 19:48 1:10:49 35:33 2:07:15 — 2 19:34 1:11:03 36:31 2:08:14 +00:59 7 19:46 1:10:51 37:31 2:09:16 +02:01 4 5 19:50 1:11:03 35:44 2:09:45 +02:30 5 9 18:48 1:10:51 38:14 2:09:59 +02:44 6 22 20:39 1:13:07 36:13 2:10:20 +03:05 7 30 19:42 1:11:48 38:40 2:10:26 +03:11 8 20 20:33 1:12:21 36:25 2:10:33 +03:18 9 6 19:34 1:10:54 37:54 2:10:50 +03:35 10 11 20:23 1:12:27 36:52 2:10:56 +03:41 11 37 18:50 1:12:18 39:11 2:11:00 +03:45 12 24 20:32 1:12:36 00:00 2:11:11 +03:56 13 17 20:28 1:11:45 37:04 2:11:14 +03:59 14 16 19:33 0:00:00 38:25 2:11:23 +04:08 15 8 18:55 1:12:33 39:35 2:11:28 +04:13 16 25 19:56 1:12:17 37:49 2:11:57 +04:42 17 35 20:27 1:11:42 38:20 2:12:24 +05:09 18 23 20:34 1:12:58 00:00 2:12:38 +05:23 19 36 20:44 1:12:30 35:40 2:12:49 +05:34 20 21 20:30 0:00:00 38:51 2:13:02 +05:47 21 12 19:44 1:15:07 40:01 2:13:04 +05:49 22 44 19:36 1:12:26 39:11 2:13:19 +06:04 23 39 19:52 1:12:09 39:15 2:13:24 +06:09 24 18 20:27 1:13:20 39:19 2:13:27 +06:12 25 15 19:53 1:13:06 40:13 2:14:24 +07:09 Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Switzerland Category:International sports competitions hosted by Switzerland Category:Geneva The 2017 European Triathlon Championships was held in Kitzbühel, Austria from 16 June to 18 June 2017. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:45:31 1:45:32 1:45:35 Women 1:57:50 1:58:05 1:58:31 Mixed Relay Anne Holm Andreas Schilling Sif Bendix Madsen Emil Deleuran Hansen 1:15:17 Cassandre Beaugrand Simon Viain Emilie Morier Raphael Montoya 1:15:24 Anastasia Gorbunova Dmitry Polyanskiy Anastasia Abrosimova Vladimir Turbaevskiy 1:15:32 Junior Junior Junior Junior Junior Junior Junior Men 53:39 53:40 53:40 Women 59:20 59:23 59:34 Mixed Relay Lili Mátyus Gergő Soós Dorka Putnóczki Csongor Lehmann 1:18:31 Daria Lushnikova Mikhail Antipov Ekaterina Matiukh Grigory Antipov 1:19:01 Bianca Bogen Moritz Horn Nina Eim Tim Siepmann 1:19:15 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 4 18:52 53:34 31:54 1:45:31 — 21 18:49 53:36 31:52 1:45:32 +0:01 5 18:57 53:29 31:58 1:45:35 +0:04 4 9 18:49 53:35 32:02 1:45:40 +0:09 5 10 18:44 53:42 32:12 1:45:47 +0:16 6 7 19:25 53:47 31:29 1:45:51 +0:20 7 6 19:11 54:03 31:32 1:45:54 +0:23 8 17 19:00 53:29 32:32 1:46:12 +0:81 9 16 19:21 53:46 32:01 1:46:16 +0:85 10 1 18:38 53:49 32:45 1:46:21 +0:90 11 11 19:23 53:47 32:12 1:46:33 +1:02 12 20 18:46 54:22 32:24 1:46:47 +1:16 13 18 19:00 53:26 33:18 1:46:50 +1:19 14 8 19:04 53:21 33:16 1:46:51 +1:20 15 37 19:02 53:20 33:12 1:46:51 +1:20 16 28 18:41 53:44 33:18 1:46:52 +1:21 17 3 18:39 53:44 33:16 1:46:52 +1:21 18 39 19:13 53:39 32:27 1:46:54 +1:23 19 14 19:23 53:48 33:04 1:47:20 +1:89 20 38 19:00 53:25 33:49 1:47:25 +1:94 21 40 19:16 47:00 33:01 1:47:27 +1:96 22 49 19:15 53:55 33:06 1:47:30 +1:99 23 15 19:10 54:00 33:34 1:47:50 +2:19 24 2 18:43 53:41 34:31 1:48:03 +2:72 25 25 18:42 53:44 34:43 1:48:15 +2:84 26 29 19:25 53:48 33:53 1:48:21 +2:90 27 33 19:20 53:49 34:07 1:48:23 +2:92 28 27 19:20 53:51 34:07 1:48:30 +2:99 29 43 19:14 53:54 34:15 1:48:35 +3:04 30 45 18:41 53:42 35:08 1:48:44 +3:13 31 32 19:17 53:56 34:40 1:48:58 +3:27 32 26 19:21 53:50 34:53 1:49:09 +3:78 33 23 18:45 53:41 36:04 1:49:44 +4:13 34 53 19:18 53:52 36:12 1:50:37 +5:06 35 50 18:54 53:31 37:26 1:51:07 +5:76 36 31 18:59 54:11 36:58 1:51:18 +5:87 37 22 19:32 57:38 33:10 1:51:30 +5:99 38 36 19:32 56:31 34:38 1:51:53 +6:22 39 48 19:08 54:02 37:41 1:52:03 +6:72 40 47 19:17 57:51 35:10 1:53:33 +8:02 41 41 19:30 57:39 36:05 1:54:22 +8:91 42 34 19:06 58:02 36:11 1:54:31 +9:00 43 19 20:55 56:11 36:58 1:55:14 +9:83 44 55 19:27 59:18 35:20 1:55:24 +9:93 45 51 20:55 56:11 37:26 1:55:44 +10:13 46 52 19:22 57:47 38:25 1:56:47 +11:16 47 54 19:31 59:18 37:55 1:57:54 +12:23 — 35 20:13 56:55 did not finish did not finish did not finish — 12 18:36 53:47 did not finish did not finish did not finish — 24 19:23 did not finish did not finish did not finish did not finish — 42 19:31 did not finish did not finish did not finish did not finish — 44 18:57 did not advance did not advance did not advance did not advance — 46 19:08 Lapped Lapped Lapped Lapped Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 8 19:09 1:00:11 37:14 1:57:50 15 19:49 59:34 37:27 1:58:05 +0:15 14 19:45 59:36 37:53 1:58:31 +0:41 4 3 20:53 1:00:44 35:51 1:58:41 +0:51 5 9 20:56 1:00:45 35:44 1:58:41 +0:51 6 6 20:18 1:01:21 35:55 1:58:47 +0:57 7 7 20:51 1:00:46 36:08 1:59:00 +1:10 8 10 19:50 1:01:51 36:29 1:59:24 +1:34 9 2 21:00 1:00:43 36:30 1:59:28 +1:38 10 23 20:05 1:03:13 34:59 1:59:37 +1:47 11 19 19:27 59:52 39:06 1:59:46 +1:56 12 11 21:00 1:00:39 37:15 2:00:13 +2:23 13 12 20:59 1:00:41 37:29 2:00:24 +2:34 14 22 20:57 1:00:43 37:42 2:00:37 +2:47 15 1 20:16 1:01:22 38:27 2:01:24 +3:34 16 17 Michelle Flipo 20:56 1:02:16 37:38 2:02:14 +4:24 17 27 20:48 1:00:47 39:34 2:02:28 +4:38 18 26 21:13 1:02:02 38:01 2:02:41 +4:51 19 18 21:02 1:02:13 38:37 2:03:09 +5:19 20 4 21:14 1:02:00 38:46 2:03:19 +5:29 21 24 21:01 1:00:39 42:08 2:05:07 +7:17 22 20 20:17 1:01:20 44:13 2:07:13 +9:23 16 20:17 1:01:26 did not finish did not finish did not finish did not finish 28 22:20 did not finish did not finish did not finish did not finish 21 21:13 did not finish did not finish did not finish did not finish 29 23:34 did not finish did not finish did not finish did not finish 25 21:12 Lapped Lapped Lapped Lapped 30 22:50 Lapped Lapped Lapped Lapped Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Austria Category:2017 in Austrian sport Category:June 2017 sports events in Europe Category:International sports competitions hosted by Austria Category:Kitzbühel Running injuries (or running-related injuries, RRI) affect about half of runners annually. The 2016 European Triathlon Championships was held in Lisbon, Portugal from 26 May to 29 May 2016. ==Medallists== Elite Elite Elite Elite Elite Elite Elite Men 1:49:30 1:50:09 1:50:32 Women 2:04:03 2:04:19 2:04:24 Mixed Relay Lucy Hall Thomas Bishop India Lee Grant Sheldon 1:07:03 Mariya Shorets Igor Polyanskiy Alexandra Razarenova Dmitry Polyanskiy 1:07:08 Zsófia Kovács Tamás Tóth Margit Vanek Ákos Vanek 1:07:19 Junior Junior Junior Junior Junior Junior Junior Men 0:58:03 0:58:04 0:58:08 Women 1:02:42 1:02:54 1:03:14 Mixed Relay Sian Rainsley Samuel Dickinson Kate Waugh Alex Yee 1:07:48 Ines Santiago Moron Alberto Gonzalez Garcia Cecilia Santamaria Surroca Javier Lluch Perez 1:07:59 Lena Meißner Paul Weindl Lisa Tertsch Moritz Horn 1:08:00 == Results == === Men's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course Rank # Triathlete Swimming Cycling Running Total time Difference 9 16:55 0:59:45 31:25 1:49:30 — 6 16:51 0:59:51 32:02 1:50:09 +00:39 19 16:49 0:59:47 32:31 1:50:32 +01:02 4 8 16:58 0:59:51 32:32 1:50:37 +01:07 5 15 16:46 0:59:47 32:30 1:50:38 +01:08 6 10 17:13 0:59:39 31:00 1:50:39 +01:09 7 16 17:07 0:59:51 30:57 1:50:40 +01:10 8 7 16:45 1:01:03 32:40 1:50:48 +01:18 9 14 17:14 1:01:09 31:12 1:51:01 +01:31 10 52 16:58 0:59:57 31:44 1:51:05 +01:35 11 38 16:52 1:01:07 33:01 1:51:09 +01:39 12 4 16:44 1:00:55 33:07 1:51:12 +01:42 13 12 17:10 0:59:48 31:44 1:51:31 +02:01 14 24 17:14 0:59:53 00:00 1:51:46 +02:16 15 48 16:54 1:01:09 34:14 1:51:49 +02:19 16 45 17:18 0:00:00 32:00 1:51:50 +02:20 17 30 17:06 0:59:45 32:23 1:52:03 +02:33 18 26 17:05 1:01:04 32:19 1:52:05 +02:35 19 5 16:50 1:01:10 34:00 1:52:09 +02:39 20 42 17:19 1:01:09 32:32 1:52:14 +02:44 21 36 17:06 0:59:46 33:07 1:52:22 +02:52 22 29 17:09 1:00:55 32:46 1:52:24 +02:54 23 23 17:13 1:01:08 32:50 1:52:29 +02:59 24 33 17:07 1:01:05 33:40 1:52:31 +03:01 25 18 17:12 1:01:00 32:45 1:52:34 +03:04 Source: Official results === Women's === ;Key * # denotes the athlete's bib number for the event * Swimming denotes the time it took the athlete to complete the swimming leg * Cycling denotes the time it took the athlete to complete the cycling leg * Running denotes the time it took the athlete to complete the running leg * Difference denotes the time difference between the athlete and the event winner * Lapped denotes that the athlete was lapped and removed from the course 0 Rank # Triathlete Swimming Cycling Running Total time Difference 20 18:41 1:06:05 36:57 2:04:03 — 6 19:11 1:07:09 35:36 2:04:19 +00:16 9 18:53 1:07:43 35:38 2:04:24 +00:21 4 1 18:04 1:07:09 36:03 2:04:40 +00:37 5 14 19:26 1:07:43 36:03 2:04:45 +00:42 6 7 19:05 1:08:27 36:03 2:04:51 +00:48 7 25 18:28 1:07:02 36:13 2:05:04 +01:01 8 11 18:29 1:07:21 36:28 2:05:09 +01:06 9 3 19:24 1:08:03 36:41 2:05:23 +01:20 10 2 17:55 1:07:49 38:20 2:05:29 +01:26 11 16 18:28 1:07:05 36:55 2:05:43 +01:40 12 26 18:09 1:06:58 37:07 2:05:48 +01:45 13 17 19:08 1:07:57 37:07 2:05:53 +01:50 14 19 18:48 1:08:21 37:09 2:06:00 +01:57 15 8 18:03 1:07:21 37:12 2:06:02 +01:59 16 24 18:06 1:07:42 37:17 2:06:03 +02:00 17 30 18:05 1:08:27 38:05 2:06:49 +02:46 18 22 18:31 1:08:25 38:42 2:07:23 +03:20 19 15 18:07 1:08:19 39:17 2:08:05 +04:02 20 29 18:08 1:07:52 39:56 2:08:36 +04:33 21 27 18:40 1:08:24 40:35 2:09:11 +05:08 22 32 18:03 1:08:07 40:57 2:09:39 +05:36 23 21 18:44 1:07:46 42:42 2:11:22 +07:19 24 34 18:32 1:08:21 43:10 2:12:01 +07:58 25 28 19:27 1:07:30 42:23 2:13:02 +08:59 Source: Official results == References == == External links == * Official page Category:European Triathlon Championships Category:Triathlon in Portugal Category:International sports competitions hosted by Portugal Category:Lisbon Injured runners were heavier. Some injuries are acute, caused by sudden overstress, such as side stitch, strains, and sprains. Instead of resulting from a single severe impact, stress fractures are the result of accumulated injury from repeated submaximal loading, such as running or jumping. In the 1984 Bern 16 km race questionnaire, runners who had no shoe brand preference and presumably changed brands frequently had significantly fewer running injuries. These findings suggest that focusing on proper running form, particularly when fatigued, could reduce the risk of running-related injuries. Pete Jacobs File:silver medal icon.svg 2011 File:gold medal icon.svg 2012 Sebastian Kienle File:bronze medal icon.svg 2013 File:gold medal icon.svg 2014 File:silver medal icon.svg 2016 Patrick Lange File:bronze medal icon.svg 2016 File:gold medal icon.svg 2017 He is the record holder for the Ironman World Championship James Lawrence Holds record for most triathlons completed in a single year Chris Lieto File:silver medal icon.svg 2009 Eneko Llanos (23) 2000 (20) 2004 File:silver medal icon.svg 2008 Chris McCormack File:gold medal icon.svg 2010 File:gold medal icon.svg 2007 File:silver medal icon.svg 2006 File:gold medal icon.svg 1997 File:gold medal icon.svg 1997 Javier Gomez File:silver medal icon.svg 2012 File:silver medal icon.svg 2007 File:gold medal icon.svg 2008 File:silver medal icon.svg 2009 File:gold medal icon.svg 2010 File:bronze medal icon.svg 2011 File:silver medal icon.svg 2012 File:gold medal icon.svg 2013 File:gold medal icon.svg 2014 File:gold medal icon.svg 2006 File:gold medal icon.svg 2007 File:gold medal icon.svg 2008 Andreas Raelert (12) 2000 (6) 2004 File:bronze medal icon.svg 2009 File:silver medal icon.svg 2010 File:bronze medal icon.svg 2011 File:silver medal icon.svg 2012 File:silver medal icon.svg 2015 Jan Rehula File:bronze medal icon.svg 2000 Sven Riederer File:bronze medal icon.svg 2004 (23) 2008 Marino Vanhoenacker File:bronze medal icon.svg 2010 Stephan Vuckovic File:silver medal icon.svg 2000 Simon Whitfield File:gold medal icon.svg 2000 (11) 2004 File:silver medal icon.svg 2008 (DNF) 2012 First man to win a gold medal at the Olympics ==Women== Name Country Olympics Ironman WTS WC Other Ref Kate Allen File:gold medal icon.svg 2004 (14) 2008 Erin Densham (22) 2008 File:bronze medal icon.svg 2012 Vanessa Fernandes (8) 2004 File:silver medal icon.svg 2008 Jude Flannery From 1991-96, won six US age group national championship and four world age-group triathlon championships. Injuries are more likely to occur in novice barefoot runners. Rutger Beke File:silver medal icon.svg 2008 Alistair Brownlee (12) 2008 File:gold medal icon.svg 2012 File:gold medal icon.svg 2016 File:gold medal icon.svg 2009 File:gold medal icon.svg 2011 Jonathan Brownlee File:bronze medal icon.svg 2012 File:silver medal icon.svg 2016 File:gold medal icon.svg 2012 File:silver medal icon.svg 2013 File:bronze medal icon.svg 2014 Hamish Carter (26) 2000 File:gold medal icon.svg 2004 Bevan Docherty File:silver medal icon.svg 2004 File:bronze medal icon.svg 2008 (12) 2012 Jan Frodeno File:gold medal icon.svg 2008 (6) 2012 File:bronze medal icon.svg 2014 File:gold medal icon.svg 2015 File:gold medal icon.svg 2016 Arthur Gilbert At 90 years of age, confirmed as the world's oldest competing triathlete in 2011. Extrinsic risk factors include deconditioning, hard surfaces, inadequate stretching and poor footwear. == Footwear == === Traditional running shoes === Study participants wearing running shoes with moderate lateral torsional stiffness ""were 49% less likely to incur any type of lower extremity injury and 52% less likely to incur an overuse lower extremity injury than"" participants wearing running shoes with minimal lateral torsional stiffness, both of which were statistically significant observations."" ",129,3.8,"""0.08""",0.5061,-20,D
 "6.I-I. One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:
 $\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\end{array}$
-(a) Find the sample mean, $\bar{x}$.","Where \bar{x_i} and w_i are the mean and size of sample i respectively. The mean of is the residence time, : \bar{\tau}(y_0) = E[\tau(y_0)\mid y_0]. Similarly, the mean of a sample x_1,x_2,\ldots,x_n, usually denoted by \bar{x}, is the sum of the sampled values divided by the number of items in the sample. : \bar{x} = \frac{1}{n}\left (\sum_{i=1}^n{x_i}\right ) = \frac{x_1+x_2+\cdots +x_n}{n} For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: :\frac{4+36+45+50+75}{5} = \frac{210}{5} = 42. ==== Geometric mean (GM) ==== The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean): :\bar{x} = \left( \prod_{i=1}^n{x_i} \right )^\frac{1}{n} = \left(x_1 x_2 \cdots x_n \right)^\frac{1}{n} For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: :(4 \times 36 \times 45 \times 50 \times 75)^\frac{1}{5} = \sqrt[5]{24\;300\;000} = 30. ==== Harmonic mean (HM) ==== The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, as in the case of speed (i.e., distance per unit of time): : \bar{x} = n \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1} For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is :\frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75}} = \frac{5}{\;\tfrac{1}{3}\;} = 15. ==== Relationship between AM, GM, and HM ==== AM, GM, and HM satisfy these inequalities: : \mathrm{AM} \ge \mathrm{GM} \ge \mathrm{HM} \, Equality holds if all the elements of the given sample are equal. ===Statistical location=== thumb|100px|Geometric visualization of the mode, median and mean of an arbitrary probability density function. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (\bar{x}) to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) Introstat, Juta and Company Ltd. p. 181 Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below. ==Types of means== ===Pythagorean means=== ==== Arithmetic mean (AM) ==== The arithmetic mean (or simply mean) of a list of numbers, is the sum of all of the numbers divided by the number of numbers. The mean of a probability distribution is the long-run average value of a random variable with that distribution. The mean of this time is the residence time, : \bar{\tau}(y_0) = \operatorname{E}[\tau(y_0)\mid y_0]. ===Logarithmic residence time=== The logarithmic residence time is a dimensionless variation of the residence time. It is simply the arithmetic mean after removing the lowest and the highest quarter of values. : \bar{x} = \frac{2}{n} \;\sum_{i = \frac{n}{4} + 1}^{\frac{3}{4}n}\\!\\! x_i assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights. ===Mean of a function=== In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. For a discrete probability distribution, the mean is given by \textstyle \sum xP(x), where the sum is taken over all possible values of the random variable and P(x) is the probability mass function. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, \bar{x}. The mean sojourn time (or sometimes mean waiting time) for an object in a system is the amount of time an object is expected to spend in a system before leaving the system for good. == Calculation == Imagine you are standing in line to buy a ticket at the counter. For a continuous distribution, the mean is \textstyle \int_{-\infty}^{\infty} xf(x)\,dx, where f(x) is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons. ==Solution== ===Calculating the expectation=== Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. thumb|400px|Graph of number of coupons, n vs the expected number of trials (i.e., time) needed to collect them all, E (T ) In probability theory, the coupon collector's problem describes ""collect all coupons and win"" contests. In statistics, the residence time is the average amount of time it takes for a random process to reach a certain boundary value, usually a boundary far from the mean. ==Definition== Suppose is a real, scalar stochastic process with initial value , mean and two critical values }, where and . The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability. A typical estimate for the sample variance from a set of sample values x_i uses a divisor of the number of values minus one, n-1, rather than n as in a simple quadratic mean, and this is still called the ""mean square"" (e.g. in analysis of variance): :s^2=\textstyle\frac{1}{n-1}\sum(x_i-\bar{x})^2 The second moment of a random variable, E(X^{2}) is also called the mean square. There are several kinds of mean in mathematics, especially in statistics. That is, the mean sojourn time of a subsystem is the total time a particle is expected to spend in the subsystem s before leaving the system S for good. In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may incorrectly be called an ""average"" (more formally, a measure of central tendency). ",24,0.1792,1.1,3.2,2,C
+(a) Find the sample mean, $\bar{x}$.","Where \bar{x_i} and w_i are the mean and size of sample i respectively. The mean of is the residence time, : \bar{\tau}(y_0) = E[\tau(y_0)\mid y_0]. Similarly, the mean of a sample x_1,x_2,\ldots,x_n, usually denoted by \bar{x}, is the sum of the sampled values divided by the number of items in the sample. : \bar{x} = \frac{1}{n}\left (\sum_{i=1}^n{x_i}\right ) = \frac{x_1+x_2+\cdots +x_n}{n} For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is: :\frac{4+36+45+50+75}{5} = \frac{210}{5} = 42. ==== Geometric mean (GM) ==== The geometric mean is an average that is useful for sets of positive numbers, that are interpreted according to their product (as is the case with rates of growth) and not their sum (as is the case with the arithmetic mean): :\bar{x} = \left( \prod_{i=1}^n{x_i} \right )^\frac{1}{n} = \left(x_1 x_2 \cdots x_n \right)^\frac{1}{n} For example, the geometric mean of five values: 4, 36, 45, 50, 75 is: :(4 \times 36 \times 45 \times 50 \times 75)^\frac{1}{5} = \sqrt[5]{24\;300\;000} = 30. ==== Harmonic mean (HM) ==== The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, as in the case of speed (i.e., distance per unit of time): : \bar{x} = n \left ( \sum_{i=1}^n \frac{1}{x_i} \right ) ^{-1} For example, the harmonic mean of the five values: 4, 36, 45, 50, 75 is :\frac{5}{\tfrac{1}{4}+\tfrac{1}{36}+\tfrac{1}{45} + \tfrac{1}{50} + \tfrac{1}{75}} = \frac{5}{\;\tfrac{1}{3}\;} = 15. ==== Relationship between AM, GM, and HM ==== AM, GM, and HM satisfy these inequalities: : \mathrm{AM} \ge \mathrm{GM} \ge \mathrm{HM} \, Equality holds if all the elements of the given sample are equal. ===Statistical location=== thumb|100px|Geometric visualization of the mode, median and mean of an arbitrary probability density function. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (\bar{x}) to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) Introstat, Juta and Company Ltd. p. 181 Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below. ==Types of means== ===Pythagorean means=== ==== Arithmetic mean (AM) ==== The arithmetic mean (or simply mean) of a list of numbers, is the sum of all of the numbers divided by the number of numbers. The mean of a probability distribution is the long-run average value of a random variable with that distribution. The mean of this time is the residence time, : \bar{\tau}(y_0) = \operatorname{E}[\tau(y_0)\mid y_0]. ===Logarithmic residence time=== The logarithmic residence time is a dimensionless variation of the residence time. It is simply the arithmetic mean after removing the lowest and the highest quarter of values. : \bar{x} = \frac{2}{n} \;\sum_{i = \frac{n}{4} + 1}^{\frac{3}{4}n}\\!\\! x_i assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights. ===Mean of a function=== In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. For a discrete probability distribution, the mean is given by \textstyle \sum xP(x), where the sum is taken over all possible values of the random variable and P(x) is the probability mass function. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted using an overhead bar, \bar{x}. The mean sojourn time (or sometimes mean waiting time) for an object in a system is the amount of time an object is expected to spend in a system before leaving the system for good. == Calculation == Imagine you are standing in line to buy a ticket at the counter. For a continuous distribution, the mean is \textstyle \int_{-\infty}^{\infty} xf(x)\,dx, where f(x) is the probability density function. In all cases, including those in which the distribution is neither discrete nor continuous, the mean is the Lebesgue integral of the random variable with respect to its probability measure. For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons. ==Solution== ===Calculating the expectation=== Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. thumb|400px|Graph of number of coupons, n vs the expected number of trials (i.e., time) needed to collect them all, E (T ) In probability theory, the coupon collector's problem describes ""collect all coupons and win"" contests. In statistics, the residence time is the average amount of time it takes for a random process to reach a certain boundary value, usually a boundary far from the mean. ==Definition== Suppose is a real, scalar stochastic process with initial value , mean and two critical values }, where and . The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). The law of averages is the commonly held belief that a particular outcome or event will, over certain periods of time, occur at a frequency that is similar to its probability. A typical estimate for the sample variance from a set of sample values x_i uses a divisor of the number of values minus one, n-1, rather than n as in a simple quadratic mean, and this is still called the ""mean square"" (e.g. in analysis of variance): :s^2=\textstyle\frac{1}{n-1}\sum(x_i-\bar{x})^2 The second moment of a random variable, E(X^{2}) is also called the mean square. There are several kinds of mean in mathematics, especially in statistics. That is, the mean sojourn time of a subsystem is the total time a particle is expected to spend in the subsystem s before leaving the system S for good. In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may incorrectly be called an ""average"" (more formally, a measure of central tendency). ",24,0.1792,"""1.1""",3.2,2,C
 "5.5-1. Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute
-(b) $P(74.2<\bar{X}<78.4)$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545 This is related to confidence interval as used in statistics: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size n. ==Normality tests== The ""68–95–99.7 rule"" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin{align} \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end{align} The usefulness of this heuristic especially depends on the question under consideration. We only need to calculate each integral for the cases n = 1,2,3. \begin{align} &\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{u^2}{2}}du \approx 0.6827 \\\ &\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) =\frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{u^2}{2}}du \approx 0.9545 \\\ &\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) = \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{u^2}{2}}du \approx 0.9973. \end{align} ==Cumulative distribution function== These numerical values ""68%, 95%, 99.7%"" come from the cumulative distribution function of the normal distribution. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). thumb|upright=1.3|Each row of points is a sample from the same normal distribution. 77 (seventy-seven) is the natural number following 76 and preceding 78. Then the optimal 50% confidence procedure for \theta is : \bar{X} \pm \begin{cases} \dfrac{|X_1-X_2|}{2} & \text{if } |X_1-X_2| < 1/2 \\\\[8pt] \dfrac{1-|X_1-X_2|}{2} &\text{if } |X_1-X_2| \geq 1/2 . \end{cases} A fiducial or objective Bayesian argument can be used to derive the interval estimate : \bar{X} \pm \frac{1-|X_1-X_2|}{4}, which is also a 50% confidence procedure. In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Thus, the probability that T will be between -c and +c is 95%. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. Suppose we wanted to calculate a 95% confidence interval for μ. * The confidence interval can be expressed in terms of statistical significance, e.g.: For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. 74 (seventy-four) is the natural number following 73 and preceding 75. ==In mathematics== 74 is: * the twenty-first distinct semiprime and the eleventh of the form 2×q. * a palindromic number in bases 6 (2026) and 36 (2236). * a nontotient. * the number of collections of subsets of {1, 2, 3} that are closed under union and intersection. * φ(74) = φ(σ(74)). ",13.2,0.8561,226.0,432,-7.5,B
+(b) $P(74.2<\bar{X}<78.4)$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. To compute the probability that an observation is within two standard deviations of the mean (small differences due to rounding): \Pr(\mu-2\sigma \le X \le \mu+2\sigma) = \Phi(2) - \Phi(-2) \approx 0.9772 - (1 - 0.9772) \approx 0.9545 This is related to confidence interval as used in statistics: \bar{X} \pm 2\frac{\sigma}{\sqrt{n}} is approximately a 95% confidence interval when \bar{X} is the average of a sample of size n. ==Normality tests== The ""68–95–99.7 rule"" is often used to quickly get a rough probability estimate of something, given its standard deviation, if the population is assumed to be normal. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. Then, denoting c as the 97.5th percentile of this distribution, : \Pr(-c\le T \le c)=0.95 Note that ""97.5th"" and ""0.95"" are correct in the preceding expressions. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... In mathematical notation, these facts can be expressed as follows, where is the probability function, is an observation from a normally distributed random variable, (mu) is the mean of the distribution, and (sigma) is its standard deviation: \begin{align} \Pr(\mu-1\sigma \le X \le \mu+1\sigma) & \approx 68.27\% \\\ \Pr(\mu-2\sigma \le X \le \mu+2\sigma) & \approx 95.45\% \\\ \Pr(\mu-3\sigma \le X \le \mu+3\sigma) & \approx 99.73\% \end{align} The usefulness of this heuristic especially depends on the question under consideration. We only need to calculate each integral for the cases n = 1,2,3. \begin{align} &\Pr(\mu -1\sigma \leq X \leq \mu + 1\sigma) = \frac{1}{\sqrt{2\pi}} \int_{-1}^{1} e^{-\frac{u^2}{2}}du \approx 0.6827 \\\ &\Pr(\mu -2\sigma \leq X \leq \mu + 2\sigma) =\frac{1}{\sqrt{2\pi}}\int_{-2}^{2} e^{-\frac{u^2}{2}}du \approx 0.9545 \\\ &\Pr(\mu -3\sigma \leq X \leq \mu + 3\sigma) = \frac{1}{\sqrt{2\pi}}\int_{-3}^{3} e^{-\frac{u^2}{2}}du \approx 0.9973. \end{align} ==Cumulative distribution function== These numerical values ""68%, 95%, 99.7%"" come from the cumulative distribution function of the normal distribution. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. After observing the sample we find values for and s for S, from which we compute the confidence interval : \left[ \bar{x} - \frac{cs}{\sqrt{n}}, \bar{x} + \frac{cs}{\sqrt{n}} \right]. == Interpretation == Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following). thumb|upright=1.3|Each row of points is a sample from the same normal distribution. 77 (seventy-seven) is the natural number following 76 and preceding 78. Then the optimal 50% confidence procedure for \theta is : \bar{X} \pm \begin{cases} \dfrac{|X_1-X_2|}{2} & \text{if } |X_1-X_2| < 1/2 \\\\[8pt] \dfrac{1-|X_1-X_2|}{2} &\text{if } |X_1-X_2| \geq 1/2 . \end{cases} A fiducial or objective Bayesian argument can be used to derive the interval estimate : \bar{X} \pm \frac{1-|X_1-X_2|}{4}, which is also a 50% confidence procedure. In statistics, the 68–95–99.7 rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively. The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Thus, the probability that T will be between -c and +c is 95%. Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. Suppose we wanted to calculate a 95% confidence interval for μ. * The confidence interval can be expressed in terms of statistical significance, e.g.: For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. 74 (seventy-four) is the natural number following 73 and preceding 75. ==In mathematics== 74 is: * the twenty-first distinct semiprime and the eleventh of the form 2×q. * a palindromic number in bases 6 (2026) and 36 (2236). * a nontotient. * the number of collections of subsets of {1, 2, 3} that are closed under union and intersection. * φ(74) = φ(σ(74)). ",13.2,0.8561,"""226.0""",432,-7.5,B
 "5.3-3. Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\left(x_1\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \left(x_2\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute
-(a) $P \left(0.5 < X_1 < 1\right.$ and $\left.0.4 < X_2 < 0.8\right)$.","That means, If X1,X2,…,Xn are discrete random variables, then the marginal probability mass function should be p_{X_i}(k)=\sum p(x_1,x_2,\dots,x_{i-1},k,x_{i+1},\dots,x_n); if X1,X2,…,Xn are continuous random variables, then the marginal probability density function should be f_{X_i}(x_i)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2,\dots,x_n) dx_1 dx_2 \cdots dx_{i-1} dx_{i+1} \cdots dx_n . ==See also== * Compound probability distribution * Joint probability distribution * Marginal likelihood * Wasserstein metric * Conditional distribution ==References== ==Bibliography== * * Category:Theory of probability distributions Assuming that X and Y are discrete random variables, the joint distribution of X and Y can be described by listing all the possible values of p(xi,yj), as shown in Table.3. That is, for any two random variables X1, X2, both have the same probability distribution if and only if \varphi_{X_1}=\varphi_{X_2}. The Pearson type III distribution is a gamma distribution or chi-squared distribution. ==== The Pearson type V distribution ==== Defining new parameters: :\begin{align} C_1 &= \frac{b_1}{2 b_2}, \\\ \lambda &= \mu_1-\frac{a-C_1} {1-2 b_2}, \end{align} x-\lambda follows an \operatorname{InverseGamma}(\frac{1}{b_2}-1,\frac{a-C_1}{b_2}). Apply the substitution :x = a_1 + y (a_2 - a_1), where 0, which yields a solution in terms of y that is supported on the interval (0, 1): :p(y) \propto \left(\frac{a_1-a_2}{a_1}y\right)^{(-a_1+a) u} \left(\frac{a_2-a_1}{a_2}(1-y)\right)^{(a_2-a) u}. The diagram on the right shows which Pearson type a given concrete distribution (identified by a point (β1, β2)) belongs to. Since this density is only known up to a hidden constant of proportionality, that constant can be changed and the density written as follows: :p(x) \propto \left(1-\frac{x}{a_1}\right)^{- u (a_1-a)} \left(1-\frac{x}{a_2}\right)^{ u (a_2-a)}. ==== The Pearson type I distribution ==== The Pearson type I distribution (a generalization of the beta distribution) arises when the roots of the quadratic equation (2) are of opposite sign, that is, a_1 < 0 < a_2. thumb|280px|right|The characteristic function of a uniform U(–1,1) random variable. Recall that: * For discrete random variables, F(x,y) = P(X\leq x, Y\leq y) * For continuous random variables, F(x,y) = \int_{a}^{x} \int_{c}^{y} f(x',y') \, dy' dx' If X and Y jointly take values on [a, b] × [c, d] then :F_X(x)=F(x,d) and F_Y(y)=F(b,y) If d is ∞, then this becomes a limit F_X(x) = \lim_{y \to \infty} F(x,y). The marginal distributions are shown in red and blue. 300px|thumb|Diagram of the Pearson system, showing distributions of types I, III, VI, V, and IV in terms of β1 (squared skewness) and β2 (traditional kurtosis) The Pearson distribution is a family of continuous probability distributions. Conditional distribution: P(H\mid L) Red Yellow Green Not Hit 0.99 0.9 0.2 Hit 0.01 0.1 0.8 To find the joint probability distribution, more data is required. Several different analyses may be done, each treating a different subset of variables as the marginal distribution. == Definition == === Marginal probability mass function === Given a known joint distribution of two discrete random variables, say, and , the marginal distribution of either variable – for example – is the probability distribution of when the values of are not taken into consideration. (Modern treatments define kurtosis γ2 in terms of cumulants instead of moments, so that for a normal distribution we have γ2 = 0 and β2 = 3. One can often start with the next step below, if bounds of the form (5.1) are already available (which is the case for many distributions). ===An abstract approximation theorem=== We are now in a position to bound the left hand side of (3.1). Multiplying each column in the conditional distribution by the probability of that column occurring results in the joint probability distribution of H and L, given in the central 2×3 block of entries. Joint distribution: Red Yellow Green Marginal probability P(H) Not Hit 0.198 0.09 0.14 0.428 Hit 0.002 0.01 0.56 0.572 Total 0.2 0.1 0.7 1 The marginal probability P(H = Hit) is the sum 0.572 along the H = Hit row of this joint distribution table, as this is the probability of being hit when the lights are red OR yellow OR green. That is :f_X(x) = \int_{c}^{d} f(x,y) \, dy :f_Y(y) = \int_{a}^{b} f(x,y) \, dx where x\in[a,b], and y\in[c,d]. === Marginal cumulative distribution function === Finding the marginal cumulative distribution function from the joint cumulative distribution function is easy. One specific case is the sum of two independent random variables X1 and X2 in which case one has \varphi_{X_1+X_2}(t) = \varphi_{X_1}(t)\cdot\varphi_{X_2}(t). * Let X and Y be two random variables with characteristic functions \varphi_{X} and \varphi_{Y}. Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. From Theorem A we obtain that : (7.1)\quad d_W(\mathcal{L}(W),N(0,1)) \leq \frac{5 E|X_1|^3}{n^{1/2}}. ",3.7,1.44,210.0,7, 0.01961,B
+(a) $P \left(0.5 < X_1 < 1\right.$ and $\left.0.4 < X_2 < 0.8\right)$.","That means, If X1,X2,…,Xn are discrete random variables, then the marginal probability mass function should be p_{X_i}(k)=\sum p(x_1,x_2,\dots,x_{i-1},k,x_{i+1},\dots,x_n); if X1,X2,…,Xn are continuous random variables, then the marginal probability density function should be f_{X_i}(x_i)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} f(x_1,x_2,\dots,x_n) dx_1 dx_2 \cdots dx_{i-1} dx_{i+1} \cdots dx_n . ==See also== * Compound probability distribution * Joint probability distribution * Marginal likelihood * Wasserstein metric * Conditional distribution ==References== ==Bibliography== * * Category:Theory of probability distributions Assuming that X and Y are discrete random variables, the joint distribution of X and Y can be described by listing all the possible values of p(xi,yj), as shown in Table.3. That is, for any two random variables X1, X2, both have the same probability distribution if and only if \varphi_{X_1}=\varphi_{X_2}. The Pearson type III distribution is a gamma distribution or chi-squared distribution. ==== The Pearson type V distribution ==== Defining new parameters: :\begin{align} C_1 &= \frac{b_1}{2 b_2}, \\\ \lambda &= \mu_1-\frac{a-C_1} {1-2 b_2}, \end{align} x-\lambda follows an \operatorname{InverseGamma}(\frac{1}{b_2}-1,\frac{a-C_1}{b_2}). Apply the substitution :x = a_1 + y (a_2 - a_1), where 0, which yields a solution in terms of y that is supported on the interval (0, 1): :p(y) \propto \left(\frac{a_1-a_2}{a_1}y\right)^{(-a_1+a) u} \left(\frac{a_2-a_1}{a_2}(1-y)\right)^{(a_2-a) u}. The diagram on the right shows which Pearson type a given concrete distribution (identified by a point (β1, β2)) belongs to. Since this density is only known up to a hidden constant of proportionality, that constant can be changed and the density written as follows: :p(x) \propto \left(1-\frac{x}{a_1}\right)^{- u (a_1-a)} \left(1-\frac{x}{a_2}\right)^{ u (a_2-a)}. ==== The Pearson type I distribution ==== The Pearson type I distribution (a generalization of the beta distribution) arises when the roots of the quadratic equation (2) are of opposite sign, that is, a_1 < 0 < a_2. thumb|280px|right|The characteristic function of a uniform U(–1,1) random variable. Recall that: * For discrete random variables, F(x,y) = P(X\leq x, Y\leq y) * For continuous random variables, F(x,y) = \int_{a}^{x} \int_{c}^{y} f(x',y') \, dy' dx' If X and Y jointly take values on [a, b] × [c, d] then :F_X(x)=F(x,d) and F_Y(y)=F(b,y) If d is ∞, then this becomes a limit F_X(x) = \lim_{y \to \infty} F(x,y). The marginal distributions are shown in red and blue. 300px|thumb|Diagram of the Pearson system, showing distributions of types I, III, VI, V, and IV in terms of β1 (squared skewness) and β2 (traditional kurtosis) The Pearson distribution is a family of continuous probability distributions. Conditional distribution: P(H\mid L) Red Yellow Green Not Hit 0.99 0.9 0.2 Hit 0.01 0.1 0.8 To find the joint probability distribution, more data is required. Several different analyses may be done, each treating a different subset of variables as the marginal distribution. == Definition == === Marginal probability mass function === Given a known joint distribution of two discrete random variables, say, and , the marginal distribution of either variable – for example – is the probability distribution of when the values of are not taken into consideration. (Modern treatments define kurtosis γ2 in terms of cumulants instead of moments, so that for a normal distribution we have γ2 = 0 and β2 = 3. One can often start with the next step below, if bounds of the form (5.1) are already available (which is the case for many distributions). ===An abstract approximation theorem=== We are now in a position to bound the left hand side of (3.1). Multiplying each column in the conditional distribution by the probability of that column occurring results in the joint probability distribution of H and L, given in the central 2×3 block of entries. Joint distribution: Red Yellow Green Marginal probability P(H) Not Hit 0.198 0.09 0.14 0.428 Hit 0.002 0.01 0.56 0.572 Total 0.2 0.1 0.7 1 The marginal probability P(H = Hit) is the sum 0.572 along the H = Hit row of this joint distribution table, as this is the probability of being hit when the lights are red OR yellow OR green. That is :f_X(x) = \int_{c}^{d} f(x,y) \, dy :f_Y(y) = \int_{a}^{b} f(x,y) \, dx where x\in[a,b], and y\in[c,d]. === Marginal cumulative distribution function === Finding the marginal cumulative distribution function from the joint cumulative distribution function is easy. One specific case is the sum of two independent random variables X1 and X2 in which case one has \varphi_{X_1+X_2}(t) = \varphi_{X_1}(t)\cdot\varphi_{X_2}(t). * Let X and Y be two random variables with characteristic functions \varphi_{X} and \varphi_{Y}. Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. From Theorem A we obtain that : (7.1)\quad d_W(\mathcal{L}(W),N(0,1)) \leq \frac{5 E|X_1|^3}{n^{1/2}}. ",3.7,1.44,"""210.0""",7, 0.01961,B
 "5.8-1. If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find
-(a) A lower bound for $P(23 < X < 43)$.","Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.89% within three standard deviations for a broad range of different probability distributions. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean. The additional fraction of 4/9 present in these tail bounds lead to better confidence intervals than Chebyshev's inequality. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg.Savage, I. Richard. The second of these inequalities with is the Chebyshev bound. The Chebyshev inequality for the distribution gives 95% and 99% confidence intervals of approximately ±4.472 standard deviations and ±10 standard deviations respectively. ====Samuelson's inequality==== Although Chebyshev's inequality is the best possible bound for an arbitrary distribution, this is not necessarily true for finite samples. However, the benefit of Chebyshev's inequality is that it can be applied more generally to get confidence bounds for ranges of standard deviations that do not depend on the number of samples. ====Semivariances==== An alternative method of obtaining sharper bounds is through the use of semivariances (partial variances). By comparison, Chebyshev's inequality states that all but a 1/N fraction of the sample will lie within standard deviations of the mean. *Grechuk et al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any deviation risk measure in place of standard deviation. This inequality is related to Jensen's inequality, Kantorovich's inequality, the Hermite–Hadamard inequality and Walter's conjecture. ===Other inequalities=== There are also a number of other inequalities associated with Chebyshev: *Chebyshev's sum inequality *Chebyshev–Markov–Stieltjes inequalities ==Notes== The Environmental Protection Agency has suggested best practices for the use of Chebyshev's inequality for estimating confidence intervals. ==See also== *Multidimensional Chebyshev's inequality *Concentration inequality – a summary of tail-bounds on random variables. In this setting we can state the following: :General version of Chebyshev's inequality. \forall k > 0: \quad \Pr\left( \|X - \mu\|_\alpha \ge k \sigma_\alpha \right) \le \frac{1}{ k^2 }. One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable with a = (kσ)2: : \Pr(|X - \mu| \geq k\sigma) = \Pr((X - \mu)^2 \geq k^2\sigma^2) \leq \frac{\mathbb{E}[(X - \mu)^2]}{k^2\sigma^2} = \frac{\sigma^2}{k^2\sigma^2} = \frac{1}{k^2}. The American Statistician, 56(3), pp.186-190 When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. The first provides a lower bound for the value of P(x). ==Finite samples== === Univariate case === Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known and may not exist, but the sample mean and sample standard deviation from N samples are to be employed to bound the expected value of a new drawing from the same distribution. Chebyshev Inequalities with Law Invariant Deviation Measures, Probability in the Engineering and Informational Sciences, 24(1), 145-170. ==Related inequalities== Several other related inequalities are also known. ===Paley–Zygmund inequality=== The Paley–Zygmund inequality gives a lower bound on tail probabilities, as opposed to Chebyshev's inequality which gives an upper bound.Godwin H. J. (1964) Inequalities on distribution functions. For k ≥ 1, n > 4 and assuming that the nth moment exists, this bound is tighter than Chebyshev's inequality. Chebyshev's inequality can now be written : \Pr(x \le m - k \sigma) \le \frac { 1 } { k^2 } \frac { \sigma_-^2 } { \sigma^2 }. On the other hand, for two-sided tail bounds, Cantelli's inequality gives : \Pr(|X-\mathbb{E}[X]|\ge\lambda) = \Pr(X-\mathbb{E}[X]\ge\lambda) + \Pr(X-\mathbb{E}[X]\le-\lambda) \le \frac{2\sigma^2}{\sigma^2 + \lambda^2}, which is always worse than Chebyshev's inequality (when \lambda \geq \sigma; otherwise, both inequalities bound a probability by a value greater than one, and so are trivial). ==Proof== Let X be a real-valued random variable with finite variance \sigma^2 and expectation \mu, and define Y = X - \mathbb{E}[X] (so that \mathbb{E}[Y] = 0 and \operatorname{Var}(Y) = \sigma^2). In terms of the lower semivariance Chebyshev's inequality can be written : \Pr(x \le m - a \sigma_-) \le \frac { 1 } { a^2 }. The Chebyshev inequality has ""higher moments versions"" and ""vector versions"", and so does the Cantelli inequality. == Comparison to Chebyshev's inequality == For one-sided tail bounds, Cantelli's inequality is better, since Chebyshev's inequality can only get : \Pr(X - \mathbb{E}[X] \geq \lambda) \leq \Pr(|X-\mathbb{E}[X]|\ge\lambda) \le \frac{\sigma^2}{\lambda^2}. For n = 2 we obtain Chebyshev's inequality. ",0.84, 1.16,1.2,257,0.33333333,A
+(a) A lower bound for $P(23 < X < 43)$.","Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.89% within three standard deviations for a broad range of different probability distributions. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean. The additional fraction of 4/9 present in these tail bounds lead to better confidence intervals than Chebyshev's inequality. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg.Savage, I. Richard. The second of these inequalities with is the Chebyshev bound. The Chebyshev inequality for the distribution gives 95% and 99% confidence intervals of approximately ±4.472 standard deviations and ±10 standard deviations respectively. ====Samuelson's inequality==== Although Chebyshev's inequality is the best possible bound for an arbitrary distribution, this is not necessarily true for finite samples. However, the benefit of Chebyshev's inequality is that it can be applied more generally to get confidence bounds for ranges of standard deviations that do not depend on the number of samples. ====Semivariances==== An alternative method of obtaining sharper bounds is through the use of semivariances (partial variances). By comparison, Chebyshev's inequality states that all but a 1/N fraction of the sample will lie within standard deviations of the mean. *Grechuk et al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any deviation risk measure in place of standard deviation. This inequality is related to Jensen's inequality, Kantorovich's inequality, the Hermite–Hadamard inequality and Walter's conjecture. ===Other inequalities=== There are also a number of other inequalities associated with Chebyshev: *Chebyshev's sum inequality *Chebyshev–Markov–Stieltjes inequalities ==Notes== The Environmental Protection Agency has suggested best practices for the use of Chebyshev's inequality for estimating confidence intervals. ==See also== *Multidimensional Chebyshev's inequality *Concentration inequality – a summary of tail-bounds on random variables. In this setting we can state the following: :General version of Chebyshev's inequality. \forall k > 0: \quad \Pr\left( \|X - \mu\|_\alpha \ge k \sigma_\alpha \right) \le \frac{1}{ k^2 }. One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable with a = (kσ)2: : \Pr(|X - \mu| \geq k\sigma) = \Pr((X - \mu)^2 \geq k^2\sigma^2) \leq \frac{\mathbb{E}[(X - \mu)^2]}{k^2\sigma^2} = \frac{\sigma^2}{k^2\sigma^2} = \frac{1}{k^2}. The American Statistician, 56(3), pp.186-190 When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. The first provides a lower bound for the value of P(x). ==Finite samples== === Univariate case === Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known and may not exist, but the sample mean and sample standard deviation from N samples are to be employed to bound the expected value of a new drawing from the same distribution. Chebyshev Inequalities with Law Invariant Deviation Measures, Probability in the Engineering and Informational Sciences, 24(1), 145-170. ==Related inequalities== Several other related inequalities are also known. ===Paley–Zygmund inequality=== The Paley–Zygmund inequality gives a lower bound on tail probabilities, as opposed to Chebyshev's inequality which gives an upper bound.Godwin H. J. (1964) Inequalities on distribution functions. For k ≥ 1, n > 4 and assuming that the nth moment exists, this bound is tighter than Chebyshev's inequality. Chebyshev's inequality can now be written : \Pr(x \le m - k \sigma) \le \frac { 1 } { k^2 } \frac { \sigma_-^2 } { \sigma^2 }. On the other hand, for two-sided tail bounds, Cantelli's inequality gives : \Pr(|X-\mathbb{E}[X]|\ge\lambda) = \Pr(X-\mathbb{E}[X]\ge\lambda) + \Pr(X-\mathbb{E}[X]\le-\lambda) \le \frac{2\sigma^2}{\sigma^2 + \lambda^2}, which is always worse than Chebyshev's inequality (when \lambda \geq \sigma; otherwise, both inequalities bound a probability by a value greater than one, and so are trivial). ==Proof== Let X be a real-valued random variable with finite variance \sigma^2 and expectation \mu, and define Y = X - \mathbb{E}[X] (so that \mathbb{E}[Y] = 0 and \operatorname{Var}(Y) = \sigma^2). In terms of the lower semivariance Chebyshev's inequality can be written : \Pr(x \le m - a \sigma_-) \le \frac { 1 } { a^2 }. The Chebyshev inequality has ""higher moments versions"" and ""vector versions"", and so does the Cantelli inequality. == Comparison to Chebyshev's inequality == For one-sided tail bounds, Cantelli's inequality is better, since Chebyshev's inequality can only get : \Pr(X - \mathbb{E}[X] \geq \lambda) \leq \Pr(|X-\mathbb{E}[X]|\ge\lambda) \le \frac{\sigma^2}{\lambda^2}. For n = 2 we obtain Chebyshev's inequality. ",0.84, 1.16,"""1.2""",257,0.33333333,A
 "6.3-5. Let $Y_1 < Y_2 < \cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\pi_{0.7}=27.3$.
 (a) Determine $P\left(Y_7<27.3\right)$.
-","If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sample. Similar remarks apply to all sample quantiles. == Probabilistic analysis == Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order. We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the cdf. To find the probabilities of the k^\text{th} order statistics, three values are first needed, namely :p_1=P(Xx)=1-F(x). In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n). == See also == * Rankit * Box plot * BRS-inequality * Concomitant (statistics) * Fisher–Tippett distribution * Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables * Bernstein polynomial * L-estimator – linear combinations of order statistics * Rank-size distribution * Selection algorithm === Examples of order statistics === * Sample maximum and minimum * Quantile * Percentile * Decile * Quartile * Median == References == == External links == * Retrieved Feb 02,2005 * Retrieved Feb 02,2005 * C++ source Dynamic Order Statistics Category:Nonparametric statistics Category:Summary statistics Category:Permutations When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. == Notation and examples == For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. In other words, all n order statistics are needed from the n observations in a sample. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. thumb|right|Major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. Note that the order statistics also satisfy U_{(i)}=F_X(X_{(i)}). Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median. === Large sample sizes === For the uniform distribution, as n tends to infinity, the pth sample quantile is asymptotically normally distributed, since it is approximated by : U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right). One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. The 7.39 is a British drama television film that was broadcast in two parts on BBC One on 6 January and 7 January 2014. In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of U_{(n)}-U_{(1)}, i.e. maximum minus the minimum. In statistics, some Monte Carlo methods require independent observations in a sample to be drawn from a one-dimensional distribution in sorted order. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant: :f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!. The small major seventh is a ratio of 9:5,Royal Society (Great Britain) (1880, digitized Feb 26, 2008). The sample median may or may not be an order statistic, since there is a single middle value only when the number of observations is odd. On the other hand, when is even, and there are two middle values, X_{(m)} and X_{(m+1)}, and the sample median is some function of the two (usually the average) and hence not an order statistic. The probability density function of the order statistic U_{(k)} is equal to. :f_{U_{(k)}}(u)={n!\over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k} that is, the kth order statistic of the uniform distribution is a beta-distributed random variable. ",92,257,27.0,0.2553,2.84367,D
+","If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sample. Similar remarks apply to all sample quantiles. == Probabilistic analysis == Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order. We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the cdf. To find the probabilities of the k^\text{th} order statistics, three values are first needed, namely :p_1=P(Xx)=1-F(x). In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n). == See also == * Rankit * Box plot * BRS-inequality * Concomitant (statistics) * Fisher–Tippett distribution * Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables * Bernstein polynomial * L-estimator – linear combinations of order statistics * Rank-size distribution * Selection algorithm === Examples of order statistics === * Sample maximum and minimum * Quantile * Percentile * Decile * Quartile * Median == References == == External links == * Retrieved Feb 02,2005 * Retrieved Feb 02,2005 * C++ source Dynamic Order Statistics Category:Nonparametric statistics Category:Summary statistics Category:Permutations When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. == Notation and examples == For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. In other words, all n order statistics are needed from the n observations in a sample. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. thumb|right|Major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. Note that the order statistics also satisfy U_{(i)}=F_X(X_{(i)}). Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median. === Large sample sizes === For the uniform distribution, as n tends to infinity, the pth sample quantile is asymptotically normally distributed, since it is approximated by : U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right). One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. The 7.39 is a British drama television film that was broadcast in two parts on BBC One on 6 January and 7 January 2014. In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of U_{(n)}-U_{(1)}, i.e. maximum minus the minimum. In statistics, some Monte Carlo methods require independent observations in a sample to be drawn from a one-dimensional distribution in sorted order. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant: :f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!. The small major seventh is a ratio of 9:5,Royal Society (Great Britain) (1880, digitized Feb 26, 2008). The sample median may or may not be an order statistic, since there is a single middle value only when the number of observations is odd. On the other hand, when is even, and there are two middle values, X_{(m)} and X_{(m+1)}, and the sample median is some function of the two (usually the average) and hence not an order statistic. The probability density function of the order statistic U_{(k)} is equal to. :f_{U_{(k)}}(u)={n!\over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k} that is, the kth order statistic of the uniform distribution is a beta-distributed random variable. ",92,257,"""27.0""",0.2553,2.84367,D
 "6.3-5. Let $Y_1 < Y_2 < \cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\pi_{0.7}=27.3$.
 (a) Determine $P\left(Y_7<27.3\right)$.
-","If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sample. Similar remarks apply to all sample quantiles. == Probabilistic analysis == Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order. To find the probabilities of the k^\text{th} order statistics, three values are first needed, namely :p_1=P(Xx)=1-F(x). We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the cdf. In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n). == See also == * Rankit * Box plot * BRS-inequality * Concomitant (statistics) * Fisher–Tippett distribution * Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables * Bernstein polynomial * L-estimator – linear combinations of order statistics * Rank-size distribution * Selection algorithm === Examples of order statistics === * Sample maximum and minimum * Quantile * Percentile * Decile * Quartile * Median == References == == External links == * Retrieved Feb 02,2005 * Retrieved Feb 02,2005 * C++ source Dynamic Order Statistics Category:Nonparametric statistics Category:Summary statistics Category:Permutations When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. == Notation and examples == For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. In other words, all n order statistics are needed from the n observations in a sample. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. thumb|right|Major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median. === Large sample sizes === For the uniform distribution, as n tends to infinity, the pth sample quantile is asymptotically normally distributed, since it is approximated by : U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right). Note that the order statistics also satisfy U_{(i)}=F_X(X_{(i)}). In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. The 7.39 is a British drama television film that was broadcast in two parts on BBC One on 6 January and 7 January 2014. One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of U_{(n)}-U_{(1)}, i.e. maximum minus the minimum. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant: :f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!. In statistics, some Monte Carlo methods require independent observations in a sample to be drawn from a one-dimensional distribution in sorted order. The sample median may or may not be an order statistic, since there is a single middle value only when the number of observations is odd. The small major seventh is a ratio of 9:5,Royal Society (Great Britain) (1880, digitized Feb 26, 2008). On the other hand, when is even, and there are two middle values, X_{(m)} and X_{(m+1)}, and the sample median is some function of the two (usually the average) and hence not an order statistic. The cumulative distribution function of the k^\text{th} order statistic can be computed by noting that : \begin{align} P(X_{(k)}\leq x)& =P(\text{there are at least }k\text{ observations less than or equal to }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}p_3^j(p_1+p_2)^{n-j} . \end{align} Similarly, P(X_{(k)} is given by : \begin{align} P(X_{(k)}< x)& =P(\text{there are at least }k\text{ observations less than }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than or equal to }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}(p_2+p_3)^j(p_1)^{n-j} . \end{align} Note that the probability mass function of X_{(k)} is just the difference of these values, that is to say : \begin{align} P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}< x) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^j(p_1)^{n-j}\right) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left((1-F(x))^j(F(x))^{n-j}-(1-F(x)+f(x))^j(F(x)-f(x))^{n-j}\right). \end{align} == Computing order statistics == The problem of computing the kth smallest (or largest) element of a list is called the selection problem and is solved by a selection algorithm. ",72,0.318,0.9731,0.2553,7,D
+","If the sample values are :6, 9, 3, 8, the order statistics would be denoted :x_{(1)}=3,\ \ x_{(2)}=6,\ \ x_{(3)}=8,\ \ x_{(4)}=9,\, where the subscript enclosed in parentheses indicates the th order statistic of the sample. Similar remarks apply to all sample quantiles. == Probabilistic analysis == Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order. To find the probabilities of the k^\text{th} order statistics, three values are first needed, namely :p_1=P(Xx)=1-F(x). We also give a simple method to derive the joint distribution of any number of order statistics, and finally translate these results to arbitrary continuous distributions using the cdf. In many applications all order statistics are required, in which case a sorting algorithm can be used and the time taken is O(n log n). == See also == * Rankit * Box plot * BRS-inequality * Concomitant (statistics) * Fisher–Tippett distribution * Bapat–Beg theorem for the order statistics of independent but not necessarily identically distributed random variables * Bernstein polynomial * L-estimator – linear combinations of order statistics * Rank-size distribution * Selection algorithm === Examples of order statistics === * Sample maximum and minimum * Quantile * Percentile * Decile * Quartile * Median == References == == External links == * Retrieved Feb 02,2005 * Retrieved Feb 02,2005 * C++ source Dynamic Order Statistics Category:Nonparametric statistics Category:Summary statistics Category:Permutations When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution. == Notation and examples == For example, suppose that four numbers are observed or recorded, resulting in a sample of size 4. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. In other words, all n order statistics are needed from the n observations in a sample. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. thumb|right|Major seventh In music from Western culture, a seventh is a musical interval encompassing seven staff positions (see Interval number for more details), and the major seventh is one of two commonly occurring sevenths. Size 6 is, in fact, the smallest sample size such that the interval determined by the minimum and the maximum is at least a 95% confidence interval for the population median. === Large sample sizes === For the uniform distribution, as n tends to infinity, the pth sample quantile is asymptotically normally distributed, since it is approximated by : U_{(\lceil np \rceil)} \sim AN\left(p,\frac{p(1-p)}{n}\right). Note that the order statistics also satisfy U_{(i)}=F_X(X_{(i)}). In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. The 7.39 is a British drama television film that was broadcast in two parts on BBC One on 6 January and 7 January 2014. One way to understand this is that the unordered sample does have constant density equal to 1, and that there are n! different permutations of the sample corresponding to the same sequence of order statistics. Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of U_{(n)}-U_{(1)}, i.e. maximum minus the minimum. Perhaps surprisingly, the joint density of the n order statistics turns out to be constant: :f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!. In statistics, some Monte Carlo methods require independent observations in a sample to be drawn from a one-dimensional distribution in sorted order. The sample median may or may not be an order statistic, since there is a single middle value only when the number of observations is odd. The small major seventh is a ratio of 9:5,Royal Society (Great Britain) (1880, digitized Feb 26, 2008). On the other hand, when is even, and there are two middle values, X_{(m)} and X_{(m+1)}, and the sample median is some function of the two (usually the average) and hence not an order statistic. The cumulative distribution function of the k^\text{th} order statistic can be computed by noting that : \begin{align} P(X_{(k)}\leq x)& =P(\text{there are at least }k\text{ observations less than or equal to }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}p_3^j(p_1+p_2)^{n-j} . \end{align} Similarly, P(X_{(k)} is given by : \begin{align} P(X_{(k)}< x)& =P(\text{there are at least }k\text{ observations less than }x) ,\\\ & =P(\text{there are at most }n-k\text{ observations greater than or equal to }x) ,\\\ & =\sum_{j=0}^{n-k}{n\choose j}(p_2+p_3)^j(p_1)^{n-j} . \end{align} Note that the probability mass function of X_{(k)} is just the difference of these values, that is to say : \begin{align} P(X_{(k)}=x)&=P(X_{(k)}\leq x)-P(X_{(k)}< x) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left(p_3^j(p_1+p_2)^{n-j}-(p_2+p_3)^j(p_1)^{n-j}\right) ,\\\ &=\sum_{j=0}^{n-k}{n\choose j}\left((1-F(x))^j(F(x))^{n-j}-(1-F(x)+f(x))^j(F(x)-f(x))^{n-j}\right). \end{align} == Computing order statistics == The problem of computing the kth smallest (or largest) element of a list is called the selection problem and is solved by a selection algorithm. ",72,0.318,"""0.9731""",0.2553,7,D
 "5.4-21. Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$.
-","Y = 2 Y = 4 Y = 6 Y = 8 X = 1 0 0.1 0 0.1 X = 3 0 0 0.2 0 X = 5 0.3 0 0 0.15 X = 7 0 0 0.15 0 Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form: Y < 2 2 ≤ Y < 4 4 ≤ Y < 6 6 ≤ Y < 8 Y ≥ 8 X < 1 0 0 0 0 0 1 ≤ X < 3 0 0 0.1 0.1 0.2 3 ≤ X < 5 0 0 0.1 0.3 0.4 5 ≤ X < 7 0 0.3 0.4 0.6 0.85 X ≥ 7 0 0.3 0.4 0.75 1 ===Definition for more than two random variables=== For N random variables X_1,\ldots,X_N, the joint CDF F_{X_1,\ldots,X_N} is given by Interpreting the N random variables as a random vector \mathbf{X} = (X_1, \ldots, X_N)^T yields a shorter notation: F_{\mathbf{X}}(\mathbf{x}) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_N) ===Properties=== Every multivariate CDF is: # Monotonically non-decreasing for each of its variables, # Right-continuous in each of its variables, # 0\leq F_{X_1 \ldots X_n}(x_1,\ldots,x_n)\leq 1, # \lim_{x_1,\ldots,x_n \rightarrow+\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=1 \text{ and } \lim_{x_i\rightarrow-\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=0, \text{for all } i. Example of joint cumulative distribution function: For two continuous variables X and Y: \Pr(a < X < b \text{ and } c < Y < d) = \int_a^b \int_c^d f(x,y) \, dy \, dx; For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example: given the joint probability mass function in tabular form, determine the joint cumulative distribution function. To see the distribution of Y conditional on X=70, one can first visualize the line X=70 in the X,Y plane, and then visualize the plane containing that line and perpendicular to the X,Y plane. The probability of success on each trial is 5/6. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρXY is near +1 (or −1). Sum those probabilities: : f(5) = 0.07776 \, : f(6) = 0.15552 \, : f(7) = 0.18662 \, : f(8) = 0.17418 \, :\sum_{j=5}^8 f(j) = 0.59408. That number of successes is a negative-binomially distributed random variable. For discrete random variables this means P(Y=y|X=x) = P(Y=y) for all possible y and x with P(X=x)>0. The relation with the probability distribution of X given Y is given by: :f_{Y\mid X}(y \mid x)f_X(x) = f_{X,Y}(x, y) = f_{X|Y}(x \mid y)f_Y(y). The intersection of that plane with the joint normal density, once rescaled to give unit area under the intersection, is the relevant conditional density of Y. Y\mid X=70 \ \sim\ \mathcal{N}\left(\mu_1+\frac{\sigma_1}{\sigma_2}\rho( 70 - \mu_2),\, (1-\rho^2)\sigma_1^2\right). ==Relation to independence== Random variables X, Y are independent if and only if the conditional distribution of Y given X is, for all possible realizations of X, equal to the unconditional distribution of Y. The joint probability distribution is presented in the following table: A=Red A=Blue P(B) B=Red (2/3)(2/3)=4/9 (1/3)(2/3)=2/9 4/9+2/9=2/3 B=Blue (2/3)(1/3)=2/9 (1/3)(1/3)=1/9 2/9+1/9=1/3 P(A) 4/9+2/9=2/3 2/9+1/9=1/3 Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are the joint distribution. The relation with the probability distribution of X given Y is: :P(Y=y \mid X=x) P(X=x) = P(\\{X=x\\} \cap \\{Y=y\\}) = P(X=x \mid Y=y)P(Y=y). ===Example=== Consider the roll of a fair and let X=1 if the number is even (i.e., 2, 4, or 6) and X=0 otherwise. If the joint probability density function of random variable X and Y is f_{X,Y}(x,y) , the marginal probability density function of X and Y, which defines the marginal distribution, is given by: f_{X}(x)= \int f_{X,Y}(x,y) \; dy f_{Y}(y)= \int f_{X,Y}(x,y) \; dx where the first integral is over all points in the range of (X,Y) for which X=x and the second integral is over all points in the range of (X,Y) for which Y=y. ==Joint cumulative distribution function== For a pair of random variables X,Y, the joint cumulative distribution function (CDF) F_{XY} is given by where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x and that Y takes on a value less than or equal to y. Furthermore, if Bs+r is a random variable following the binomial distribution with parameters s + r and p, then : \begin{align} \Pr(Y_r \leq s) & {} = 1 - I_p(s+1, r) \\\\[5pt] & {} = 1 - I_{p}((s+r)-(r-1), (r-1)+1) \\\\[5pt] & {} = 1 - \Pr(B_{s+r} \leq r-1) \\\\[5pt] & {} = \Pr(B_{s+r} \geq r) \\\\[5pt] & {} = \Pr(\text{after } s+r \text{ trials, there are at least } r \text{ successes}). \end{align} In this sense, the negative binomial distribution is the ""inverse"" of the binomial distribution. The probability that X lies in the semi-closed interval (a,b], where a < b, is therefore In the definition above, the ""less than or equal to"" sign, ""≤"", is a convention, not a universally used one (e.g. Hungarian literature uses ""<""), but the distinction is important for discrete distributions. At each house, there is a 0.6 probability of selling one candy bar and a 0.4 probability of selling nothing. What's the probability of selling the last candy bar at the nth house? One must use the ""mixed"" joint density when finding the cumulative distribution of this binary outcome because the input variables (X,Y) were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. The joint probability mass function of A and B defines probabilities for each pair of outcomes. In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. Formally, f_{X,Y}(x,y) is the probability density function of (X,Y) with respect to the product measure on the respective supports of X and Y. Either of these two decompositions can then be used to recover the joint cumulative distribution function: : \begin{align} F_{X,Y}(x,y)&=\sum\limits_{t\le y}\int_{s=-\infty}^x f_{X,Y}(s,t)\;ds. \end{align} The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables. ==Additional properties== ===Joint distribution for independent variables=== In general two random variables X and Y are independent if and only if the joint cumulative distribution function satisfies : F_{X,Y}(x,y) = F_X(x) \cdot F_Y(y) Two discrete random variables X and Y are independent if and only if the joint probability mass function satisfies : P(X = x \ \mbox{and} \ Y = y ) = P( X = x) \cdot P( Y = y) for all x and y. Then the CDF of X is given by F(k;n,p) = \Pr(X\leq k) = \sum _{i=0}^{\lfloor k\rfloor }{n \choose i} p^{i} (1-p)^{n-i} Here p is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of n independent experiments, and \lfloor k\rfloor is the ""floor"" under k, i.e. the greatest integer less than or equal to k. ==Derived functions== ===Complementary cumulative distribution function (tail distribution)=== Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. ",0.24995,-45,15.757,0.4207,0.9984,D
+","Y = 2 Y = 4 Y = 6 Y = 8 X = 1 0 0.1 0 0.1 X = 3 0 0 0.2 0 X = 5 0.3 0 0 0.15 X = 7 0 0 0.15 0 Solution: using the given table of probabilities for each potential range of X and Y, the joint cumulative distribution function may be constructed in tabular form: Y < 2 2 ≤ Y < 4 4 ≤ Y < 6 6 ≤ Y < 8 Y ≥ 8 X < 1 0 0 0 0 0 1 ≤ X < 3 0 0 0.1 0.1 0.2 3 ≤ X < 5 0 0 0.1 0.3 0.4 5 ≤ X < 7 0 0.3 0.4 0.6 0.85 X ≥ 7 0 0.3 0.4 0.75 1 ===Definition for more than two random variables=== For N random variables X_1,\ldots,X_N, the joint CDF F_{X_1,\ldots,X_N} is given by Interpreting the N random variables as a random vector \mathbf{X} = (X_1, \ldots, X_N)^T yields a shorter notation: F_{\mathbf{X}}(\mathbf{x}) = \operatorname{P}(X_1 \leq x_1,\ldots,X_N \leq x_N) ===Properties=== Every multivariate CDF is: # Monotonically non-decreasing for each of its variables, # Right-continuous in each of its variables, # 0\leq F_{X_1 \ldots X_n}(x_1,\ldots,x_n)\leq 1, # \lim_{x_1,\ldots,x_n \rightarrow+\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=1 \text{ and } \lim_{x_i\rightarrow-\infty}F_{X_1 \ldots X_n}(x_1,\ldots,x_n)=0, \text{for all } i. Example of joint cumulative distribution function: For two continuous variables X and Y: \Pr(a < X < b \text{ and } c < Y < d) = \int_a^b \int_c^d f(x,y) \, dy \, dx; For two discrete random variables, it is beneficial to generate a table of probabilities and address the cumulative probability for each potential range of X and Y, and here is the example: given the joint probability mass function in tabular form, determine the joint cumulative distribution function. To see the distribution of Y conditional on X=70, one can first visualize the line X=70 in the X,Y plane, and then visualize the plane containing that line and perpendicular to the X,Y plane. The probability of success on each trial is 5/6. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρXY is near +1 (or −1). Sum those probabilities: : f(5) = 0.07776 \, : f(6) = 0.15552 \, : f(7) = 0.18662 \, : f(8) = 0.17418 \, :\sum_{j=5}^8 f(j) = 0.59408. That number of successes is a negative-binomially distributed random variable. For discrete random variables this means P(Y=y|X=x) = P(Y=y) for all possible y and x with P(X=x)>0. The relation with the probability distribution of X given Y is given by: :f_{Y\mid X}(y \mid x)f_X(x) = f_{X,Y}(x, y) = f_{X|Y}(x \mid y)f_Y(y). The intersection of that plane with the joint normal density, once rescaled to give unit area under the intersection, is the relevant conditional density of Y. Y\mid X=70 \ \sim\ \mathcal{N}\left(\mu_1+\frac{\sigma_1}{\sigma_2}\rho( 70 - \mu_2),\, (1-\rho^2)\sigma_1^2\right). ==Relation to independence== Random variables X, Y are independent if and only if the conditional distribution of Y given X is, for all possible realizations of X, equal to the unconditional distribution of Y. The joint probability distribution is presented in the following table: A=Red A=Blue P(B) B=Red (2/3)(2/3)=4/9 (1/3)(2/3)=2/9 4/9+2/9=2/3 B=Blue (2/3)(1/3)=2/9 (1/3)(1/3)=1/9 2/9+1/9=1/3 P(A) 4/9+2/9=2/3 2/9+1/9=1/3 Each of the four inner cells shows the probability of a particular combination of results from the two draws; these probabilities are the joint distribution. The relation with the probability distribution of X given Y is: :P(Y=y \mid X=x) P(X=x) = P(\\{X=x\\} \cap \\{Y=y\\}) = P(X=x \mid Y=y)P(Y=y). ===Example=== Consider the roll of a fair and let X=1 if the number is even (i.e., 2, 4, or 6) and X=0 otherwise. If the joint probability density function of random variable X and Y is f_{X,Y}(x,y) , the marginal probability density function of X and Y, which defines the marginal distribution, is given by: f_{X}(x)= \int f_{X,Y}(x,y) \; dy f_{Y}(y)= \int f_{X,Y}(x,y) \; dx where the first integral is over all points in the range of (X,Y) for which X=x and the second integral is over all points in the range of (X,Y) for which Y=y. ==Joint cumulative distribution function== For a pair of random variables X,Y, the joint cumulative distribution function (CDF) F_{XY} is given by where the right-hand side represents the probability that the random variable X takes on a value less than or equal to x and that Y takes on a value less than or equal to y. Furthermore, if Bs+r is a random variable following the binomial distribution with parameters s + r and p, then : \begin{align} \Pr(Y_r \leq s) & {} = 1 - I_p(s+1, r) \\\\[5pt] & {} = 1 - I_{p}((s+r)-(r-1), (r-1)+1) \\\\[5pt] & {} = 1 - \Pr(B_{s+r} \leq r-1) \\\\[5pt] & {} = \Pr(B_{s+r} \geq r) \\\\[5pt] & {} = \Pr(\text{after } s+r \text{ trials, there are at least } r \text{ successes}). \end{align} In this sense, the negative binomial distribution is the ""inverse"" of the binomial distribution. The probability that X lies in the semi-closed interval (a,b], where a < b, is therefore In the definition above, the ""less than or equal to"" sign, ""≤"", is a convention, not a universally used one (e.g. Hungarian literature uses ""<""), but the distinction is important for discrete distributions. At each house, there is a 0.6 probability of selling one candy bar and a 0.4 probability of selling nothing. What's the probability of selling the last candy bar at the nth house? One must use the ""mixed"" joint density when finding the cumulative distribution of this binary outcome because the input variables (X,Y) were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. The joint probability mass function of A and B defines probabilities for each pair of outcomes. In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. Formally, f_{X,Y}(x,y) is the probability density function of (X,Y) with respect to the product measure on the respective supports of X and Y. Either of these two decompositions can then be used to recover the joint cumulative distribution function: : \begin{align} F_{X,Y}(x,y)&=\sum\limits_{t\le y}\int_{s=-\infty}^x f_{X,Y}(s,t)\;ds. \end{align} The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables. ==Additional properties== ===Joint distribution for independent variables=== In general two random variables X and Y are independent if and only if the joint cumulative distribution function satisfies : F_{X,Y}(x,y) = F_X(x) \cdot F_Y(y) Two discrete random variables X and Y are independent if and only if the joint probability mass function satisfies : P(X = x \ \mbox{and} \ Y = y ) = P( X = x) \cdot P( Y = y) for all x and y. Then the CDF of X is given by F(k;n,p) = \Pr(X\leq k) = \sum _{i=0}^{\lfloor k\rfloor }{n \choose i} p^{i} (1-p)^{n-i} Here p is the probability of success and the function denotes the discrete probability distribution of the number of successes in a sequence of n independent experiments, and \lfloor k\rfloor is the ""floor"" under k, i.e. the greatest integer less than or equal to k. ==Derived functions== ===Complementary cumulative distribution function (tail distribution)=== Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. ",0.24995,-45,"""15.757""",0.4207,0.9984,D
 "7.4-5. A quality engineer wanted to be $98 \%$ confident that the maximum error of the estimate of the mean strength, $\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\bar{x}=35.68$ and a standard deviation of $s=1.723$.
-(a) How large a sample is required?","For example, if we are interested in estimating the amount by which a drug lowers a subject's blood pressure with a 95% confidence interval that is six units wide, and we know that the standard deviation of blood pressure in the population is 15, then the required sample size is \frac{4\times1.96^2\times15^2}{6^2} = 96.04, which would be rounded up to 97, because the obtained value is the minimum sample size, and sample sizes must be integers and must lie on or above the calculated minimum. ==Required sample sizes for hypothesis tests == A common problem faced by statisticians is calculating the sample size required to yield a certain power for a test, given a predetermined Type I error rate α. The mean value calculated from the sample, \bar{x}, will have an associated standard error on the mean, {\sigma}_\bar{x}, given by: :{\sigma}_\bar{x}\ = \frac{\sigma}{\sqrt{n}}. Generally, at a confidence level \gamma, a sample sized n of a population having expected standard deviation \sigma has a margin of error :MOE_\gamma = z_\gamma \times \sqrt{\frac{\sigma^2}{n}} where z_\gamma denotes the quantile (also, commonly, a z-score), and \sqrt{\frac{\sigma^2}{n}} is the standard error. == Standard deviation and standard error == We would expect the average of normally distributed values p_1,p_2,\ldots to have a standard deviation which somehow varies with n. Therefore, the standard error of the mean is usually estimated by replacing \sigma with the sample standard deviation \sigma_{x} instead: :{\sigma}_\bar{x}\ \approx \frac{\sigma_{x}}{\sqrt{n}}. The standard error is, by definition, the standard deviation of \bar{x} which is simply the square root of the variance: :\sigma_{\bar{x}} = \sqrt{\frac{\sigma^2}{n}} = \frac{\sigma}{\sqrt{n}} . Practically this tells us that when trying to estimate the value of a population mean, due to the factor 1/\sqrt{n}, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations. === Estimate === The standard deviation \sigma of the population being sampled is seldom known. Since \max \sigma_P^2 = \max P(1-P) = 0.25 at p = 0.5, we can arbitrarily set p=\overline{p} = 0.5, calculate \sigma_{P}, \sigma_\overline{p}, and z_\gamma\sigma_\overline{p} to obtain the maximum margin of error for P at a given confidence level \gamma and sample size n, even before having actual results. Knowing that the value of the n is the minimum number of sample points needed to acquire the desired result, the number of respondents then must lie on or above the minimum. ===Estimation of a mean=== When estimating the population mean using an independent and identically distributed (iid) sample of size n, where each data value has variance σ2, the standard error of the sample mean is: :\frac{\sigma}{\sqrt{n}}. Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. The arrows show that the maximum margin error for a sample size of 1000 is ±3.1% at 95% confidence level, and ±4.1% at 99%. Using the central limit theorem to justify approximating the sample mean with a normal distribution yields a confidence interval of the form : \left(\bar x - \frac{Z\sigma}{\sqrt{n}}, \quad \bar x + \frac{Z\sigma}{\sqrt{n}} \right ) , :where Z is a standard Z-score for the desired level of confidence (1.96 for a 95% confidence interval). Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. The following expressions can be used to calculate the upper and lower 95% confidence limits, where \bar{x} is equal to the sample mean, \operatorname{SE} is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution: :Upper 95% limit = \bar{x} + (\operatorname{SE}\times 1.96) , and :Lower 95% limit = \bar{x} - (\operatorname{SE}\times 1.96) . The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem. Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means For the single result from our survey, we assume that p = \overline{p}, and that all subsequent results p_1,p_2,\ldots together would have a variance \sigma_{P}^2=P(1-P). : \text{Standard error} = \sigma_\overline{p} \approx \sqrt{\frac{\sigma_{P}^2}{n}} \approx \sqrt{\frac{p(1-p)}{n}} Note that p(1-p) corresponds to the variance of a Bernoulli distribution. == Maximum margin of error at different confidence levels == thumb|350pxFor a confidence level \gamma, there is a corresponding confidence interval about the mean \mu\plusmn z_\gamma\sigma, that is, the interval [\mu-z_\gamma\sigma,\mu+z_\gamma\sigma] within which values of P should fall with probability \gamma. If we wish to have a confidence interval that is W units total in width (W/2 being the margin of error on each side of the sample mean), we would solve : \frac{Z\sigma}{\sqrt{n}} = W/2 for n, yielding the sample size n = \frac{4Z^2\sigma^2}{W^2}. For example, if we are interested in estimating the proportion of the US population who supports a particular presidential candidate, and we want the width of 95% confidence interval to be at most 2 percentage points (0.02), then we would need a sample size of (1.96)2/ (0.022) = 9604. In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. Standard errors provide simple measures of uncertainty in a value and are often used because: *in many cases, if the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated; *when the probability distribution of the value is known, it can be used to calculate an exact confidence interval; *when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and * as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal. ===Standard error of mean versus standard deviation=== In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. According to the 68-95-99.7 rule, we would expect that 95% of the results p_1,p_2,\ldots will fall within about two standard deviations (\plusmn2\sigma_{P}) either side of the true mean \overline{p}. ",0.15,0.6296296296,257.0,0.3359,-1.49,C
+(a) How large a sample is required?","For example, if we are interested in estimating the amount by which a drug lowers a subject's blood pressure with a 95% confidence interval that is six units wide, and we know that the standard deviation of blood pressure in the population is 15, then the required sample size is \frac{4\times1.96^2\times15^2}{6^2} = 96.04, which would be rounded up to 97, because the obtained value is the minimum sample size, and sample sizes must be integers and must lie on or above the calculated minimum. ==Required sample sizes for hypothesis tests == A common problem faced by statisticians is calculating the sample size required to yield a certain power for a test, given a predetermined Type I error rate α. The mean value calculated from the sample, \bar{x}, will have an associated standard error on the mean, {\sigma}_\bar{x}, given by: :{\sigma}_\bar{x}\ = \frac{\sigma}{\sqrt{n}}. Generally, at a confidence level \gamma, a sample sized n of a population having expected standard deviation \sigma has a margin of error :MOE_\gamma = z_\gamma \times \sqrt{\frac{\sigma^2}{n}} where z_\gamma denotes the quantile (also, commonly, a z-score), and \sqrt{\frac{\sigma^2}{n}} is the standard error. == Standard deviation and standard error == We would expect the average of normally distributed values p_1,p_2,\ldots to have a standard deviation which somehow varies with n. Therefore, the standard error of the mean is usually estimated by replacing \sigma with the sample standard deviation \sigma_{x} instead: :{\sigma}_\bar{x}\ \approx \frac{\sigma_{x}}{\sqrt{n}}. The standard error is, by definition, the standard deviation of \bar{x} which is simply the square root of the variance: :\sigma_{\bar{x}} = \sqrt{\frac{\sigma^2}{n}} = \frac{\sigma}{\sqrt{n}} . Practically this tells us that when trying to estimate the value of a population mean, due to the factor 1/\sqrt{n}, reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations. === Estimate === The standard deviation \sigma of the population being sampled is seldom known. Since \max \sigma_P^2 = \max P(1-P) = 0.25 at p = 0.5, we can arbitrarily set p=\overline{p} = 0.5, calculate \sigma_{P}, \sigma_\overline{p}, and z_\gamma\sigma_\overline{p} to obtain the maximum margin of error for P at a given confidence level \gamma and sample size n, even before having actual results. Knowing that the value of the n is the minimum number of sample points needed to acquire the desired result, the number of respondents then must lie on or above the minimum. ===Estimation of a mean=== When estimating the population mean using an independent and identically distributed (iid) sample of size n, where each data value has variance σ2, the standard error of the sample mean is: :\frac{\sigma}{\sqrt{n}}. Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean. The arrows show that the maximum margin error for a sample size of 1000 is ±3.1% at 95% confidence level, and ±4.1% at 99%. Using the central limit theorem to justify approximating the sample mean with a normal distribution yields a confidence interval of the form : \left(\bar x - \frac{Z\sigma}{\sqrt{n}}, \quad \bar x + \frac{Z\sigma}{\sqrt{n}} \right ) , :where Z is a standard Z-score for the desired level of confidence (1.96 for a 95% confidence interval). Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. The following expressions can be used to calculate the upper and lower 95% confidence limits, where \bar{x} is equal to the sample mean, \operatorname{SE} is equal to the standard error for the sample mean, and 1.96 is the approximate value of the 97.5 percentile point of the normal distribution: :Upper 95% limit = \bar{x} + (\operatorname{SE}\times 1.96) , and :Lower 95% limit = \bar{x} - (\operatorname{SE}\times 1.96) . The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem. Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means For the single result from our survey, we assume that p = \overline{p}, and that all subsequent results p_1,p_2,\ldots together would have a variance \sigma_{P}^2=P(1-P). : \text{Standard error} = \sigma_\overline{p} \approx \sqrt{\frac{\sigma_{P}^2}{n}} \approx \sqrt{\frac{p(1-p)}{n}} Note that p(1-p) corresponds to the variance of a Bernoulli distribution. == Maximum margin of error at different confidence levels == thumb|350pxFor a confidence level \gamma, there is a corresponding confidence interval about the mean \mu\plusmn z_\gamma\sigma, that is, the interval [\mu-z_\gamma\sigma,\mu+z_\gamma\sigma] within which values of P should fall with probability \gamma. If we wish to have a confidence interval that is W units total in width (W/2 being the margin of error on each side of the sample mean), we would solve : \frac{Z\sigma}{\sqrt{n}} = W/2 for n, yielding the sample size n = \frac{4Z^2\sigma^2}{W^2}. For example, if we are interested in estimating the proportion of the US population who supports a particular presidential candidate, and we want the width of 95% confidence interval to be at most 2 percentage points (0.02), then we would need a sample size of (1.96)2/ (0.022) = 9604. In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the sample mean in the process by which it was generated. Standard errors provide simple measures of uncertainty in a value and are often used because: *in many cases, if the standard error of several individual quantities is known then the standard error of some function of the quantities can be easily calculated; *when the probability distribution of the value is known, it can be used to calculate an exact confidence interval; *when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and * as the sample size tends to infinity the central limit theorem guarantees that the sampling distribution of the mean is asymptotically normal. ===Standard error of mean versus standard deviation=== In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. According to the 68-95-99.7 rule, we would expect that 95% of the results p_1,p_2,\ldots will fall within about two standard deviations (\plusmn2\sigma_{P}) either side of the true mean \overline{p}. ",0.15,0.6296296296,"""257.0""",0.3359,-1.49,C
 "5.2-5. Let the distribution of $W$ be $F(8,4)$. Find the following:
 (a) $F_{0.01}(8,4)$.
-","When filling out a Form W-4 an employee calculates the number of Form W-4 allowances to claim based on their expected tax filing situation for the year. There are typically three Tracy–Widom distributions, F_\beta, with \beta \in \\{1, 2, 4\\}. These distributions have been tabulated in to four significant digits for values of the argument in increments of 0.01; a statistical table for p-values was also given in this work. gave accurate and fast algorithms for the numerical evaluation of F_\beta and the density functions f_\beta(s)=dF_\beta/ds for \beta=1,2,4. :Note: the Wigner distribution function is abbreviated here as WD rather than WDF as used at Wigner distribution function A Modified Wigner distribution function is a variation of the Wigner distribution function (WD) with reduced or removed cross-terms. (I.e., withholding is calculated as if the employee earned this amount every payday on an annual basis.) == Filing == The W-4 Form includes a series of worksheets for calculating the number of allowances to claim. thumb|right|300px|Densities of Tracy–Widom distributions for β = 1, 2, 4 The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by . The W-4 Form is usually not sent to the IRS; rather, the employer uses the form in order to calculate how much of an employee's salary is withheld. An alternative to the above method is to define the PDF parametrically as (W(p),1/w(p)), \ 0\le p \le 1. The W-4 form tells the employer the correct amount of federal tax to withhold from an employee's paycheck. == Motivation == The W-4 is based on the idea of ""allowances""; the more allowances claimed, the less money the employer withholds for tax purposes. thumb|Form W-4, 2012 Form W-4 (officially, the ""Employee's Withholding Allowance Certificate"") is an Internal Revenue Service (IRS) tax form completed by an employee in the United States to indicate his or her tax situation (exemptions, status, etc.) to the employer. thumb|250px|right|Full width at half maximum In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. The distribution F_1 is of particular interest in multivariate statistics.. : W_x(t,f)=\int_{-\infty}^{\infty} C_x\left(t + \frac{\tau}{2}, t - \frac{\tau}{2}\right) \, e^{-2\pi i\tau f} \, d\tau . The definition of the Tracy–Widom distributions F_\beta may be extended to all \beta >0 (Slide 56 in , ). By WDF :\begin{align} W_x(t,f) &= \int_{-\infty}^{\infty}\delta\left(t + \frac{\tau}{2}\right)\delta\left(t - \frac{\tau}{2}\right) e^{-i2\pi\tau\,f}\,d\tau \\\ &= 4\int_{-\infty}^{\infty}\delta(2t + \tau)\delta(2t - \tau)e^{-i2\pi\tau f}\,d\tau \\\ &= 4\delta(4t)e^{i4\pi tf} \\\ &= \delta(t)e^{i4\pi tf} \\\ &= \delta(t). \end{align} The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower. In the latter case, this creates an oddity in that the employee will have one more exemption on the W-4 than on the 1040 tax return. There are specialized versions of the W-4 Form for other types of payment; for example, W-4P for pensions, and the voluntary W-4V for certain government payments such as unemployment compensation. == See also == * Form W-2 * Form W-9 * Form 1040 * Personal exemption * Tax withholding in the United States == References == ==External links== * IRS Form W-4 W-4 W-4 The corresponding area within this FWHM accounts to approximately 76%. For example, if x(t) = 1, then :W_x(t,f)=\int_{-\infty}^\infty e^{-i2\pi\tau\,f}\,d\tau=\delta(f). ===Sinusoidal input signal=== When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the time axis, displaced from it by the sinusoidal signal's frequency. This can be set up as a probability density function, f(x), by solving for the unique p in the equation W(p)=x and returning 1/w(p). == See also == * Generalized Pareto distribution == References == == External links == * Discussion of the naming of the distribution on Stack Exchange :Note: this work is based on a NIST document that is in the public domain as a work of the U.S. federal government Category:Continuous distributions For those signals, WDF can exactly generate the time frequency distribution of the input signal. ===Boxcar function=== :x(t) = \begin{cases} 1 & |t|<1/2 \\\ 0 & \text{otherwise} \end{cases} \qquad , the rectangular function ⇒ : W_x(t,f) = \begin{cases} \frac{1}{\pi f}\sin (2\pi f\\{1 - 2|t|\\}) &|t|<1/2 \\\ 0 & \mbox{otherwise} \end{cases} ==Cross term property== The Wigner distribution function is not a linear transform. See and for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution F_2 (or F_1) as predicted by . ",2.84367,0.042,0.0,14.80,3.51,D
+","When filling out a Form W-4 an employee calculates the number of Form W-4 allowances to claim based on their expected tax filing situation for the year. There are typically three Tracy–Widom distributions, F_\beta, with \beta \in \\{1, 2, 4\\}. These distributions have been tabulated in to four significant digits for values of the argument in increments of 0.01; a statistical table for p-values was also given in this work. gave accurate and fast algorithms for the numerical evaluation of F_\beta and the density functions f_\beta(s)=dF_\beta/ds for \beta=1,2,4. :Note: the Wigner distribution function is abbreviated here as WD rather than WDF as used at Wigner distribution function A Modified Wigner distribution function is a variation of the Wigner distribution function (WD) with reduced or removed cross-terms. (I.e., withholding is calculated as if the employee earned this amount every payday on an annual basis.) == Filing == The W-4 Form includes a series of worksheets for calculating the number of allowances to claim. thumb|right|300px|Densities of Tracy–Widom distributions for β = 1, 2, 4 The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by . The W-4 Form is usually not sent to the IRS; rather, the employer uses the form in order to calculate how much of an employee's salary is withheld. An alternative to the above method is to define the PDF parametrically as (W(p),1/w(p)), \ 0\le p \le 1. The W-4 form tells the employer the correct amount of federal tax to withhold from an employee's paycheck. == Motivation == The W-4 is based on the idea of ""allowances""; the more allowances claimed, the less money the employer withholds for tax purposes. thumb|Form W-4, 2012 Form W-4 (officially, the ""Employee's Withholding Allowance Certificate"") is an Internal Revenue Service (IRS) tax form completed by an employee in the United States to indicate his or her tax situation (exemptions, status, etc.) to the employer. thumb|250px|right|Full width at half maximum In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. The distribution F_1 is of particular interest in multivariate statistics.. : W_x(t,f)=\int_{-\infty}^{\infty} C_x\left(t + \frac{\tau}{2}, t - \frac{\tau}{2}\right) \, e^{-2\pi i\tau f} \, d\tau . The definition of the Tracy–Widom distributions F_\beta may be extended to all \beta >0 (Slide 56 in , ). By WDF :\begin{align} W_x(t,f) &= \int_{-\infty}^{\infty}\delta\left(t + \frac{\tau}{2}\right)\delta\left(t - \frac{\tau}{2}\right) e^{-i2\pi\tau\,f}\,d\tau \\\ &= 4\int_{-\infty}^{\infty}\delta(2t + \tau)\delta(2t - \tau)e^{-i2\pi\tau f}\,d\tau \\\ &= 4\delta(4t)e^{i4\pi tf} \\\ &= \delta(t)e^{i4\pi tf} \\\ &= \delta(t). \end{align} The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower. In the latter case, this creates an oddity in that the employee will have one more exemption on the W-4 than on the 1040 tax return. There are specialized versions of the W-4 Form for other types of payment; for example, W-4P for pensions, and the voluntary W-4V for certain government payments such as unemployment compensation. == See also == * Form W-2 * Form W-9 * Form 1040 * Personal exemption * Tax withholding in the United States == References == ==External links== * IRS Form W-4 W-4 W-4 The corresponding area within this FWHM accounts to approximately 76%. For example, if x(t) = 1, then :W_x(t,f)=\int_{-\infty}^\infty e^{-i2\pi\tau\,f}\,d\tau=\delta(f). ===Sinusoidal input signal=== When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the time axis, displaced from it by the sinusoidal signal's frequency. This can be set up as a probability density function, f(x), by solving for the unique p in the equation W(p)=x and returning 1/w(p). == See also == * Generalized Pareto distribution == References == == External links == * Discussion of the naming of the distribution on Stack Exchange :Note: this work is based on a NIST document that is in the public domain as a work of the U.S. federal government Category:Continuous distributions For those signals, WDF can exactly generate the time frequency distribution of the input signal. ===Boxcar function=== :x(t) = \begin{cases} 1 & |t|<1/2 \\\ 0 & \text{otherwise} \end{cases} \qquad , the rectangular function ⇒ : W_x(t,f) = \begin{cases} \frac{1}{\pi f}\sin (2\pi f\\{1 - 2|t|\\}) &|t|<1/2 \\\ 0 & \mbox{otherwise} \end{cases} ==Cross term property== The Wigner distribution function is not a linear transform. See and for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution F_2 (or F_1) as predicted by . ",2.84367,0.042,"""0.0""",14.80,3.51,D
 "5.8-5. If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when
-(b) $n=500$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. thumb|Plot of S_n/n (red), its standard deviation 1/\sqrt{n} (blue) and its bound \sqrt{2\log\log n/n} given by LIL (green). From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. Bound the desired probability using the Chebyshev inequality: :\operatorname{P}\left(|T- n H_n| \geq cn\right) \le \frac{\pi^2}{6c^2}. ===Tail estimates=== A stronger tail estimate for the upper tail be obtained as follows. Then : \begin{align} P\left [ {Z}_i^r \right ] = \left(1-\frac{1}{n}\right)^r \le e^{-r / n}. \end{align} Thus, for r = \beta n \log n, we have P\left [ {Z}_i^r \right ] \le e^{(-\beta n \log n ) / n} = n^{-\beta}. For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Probability. The commonly used approximate value of 1.96 is therefore accurate to better than one part in 50,000, which is more than adequate for applied work. It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions. ==See also== *Vysochanskiï–Petunin inequality, a similar result for the distance from the mean rather than the mode *Chebyshev's inequality, concerns distance from the mean without requiring unimodality * Concentration inequality – a summary of tail-bounds on random variables. ==References== * * * * Category:Probabilistic inequalities It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? thumb|The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle. Then : \Pr\left( \limsup_n \frac{S_n}{\sqrt{n}} \geq M \right) \geqslant \limsup_n \Pr\left( \frac{S_n}{\sqrt{n}} \geq M \right) = \Pr\left( \mathcal{N}(0, 1) \geq M \right) > 0 so :\limsup_n \frac{S_n}{\sqrt{n}}=\infty \qquad \text{with probability 1.} In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons. ==Solution== ===Calculating the expectation=== Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. By Kolmogorov's zero–one law, for any fixed M, the probability that the event \limsup_n \frac{S_n}{\sqrt{n}} \geq M occurs is 0 or 1. thumb|400px|Graph of number of coupons, n vs the expected number of trials (i.e., time) needed to collect them all, E (T ) In probability theory, the coupon collector's problem describes ""collect all coupons and win"" contests. ",1.07,96.4365076099,17.4,58.2,0.85,E
+(b) $n=500$.","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. thumb|Plot of S_n/n (red), its standard deviation 1/\sqrt{n} (blue) and its bound \sqrt{2\log\log n/n} given by LIL (green). From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. In statistics, probable error defines the half-range of an interval about a central point for the distribution, such that half of the values from the distribution will lie within the interval and half outside.Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. Bound the desired probability using the Chebyshev inequality: :\operatorname{P}\left(|T- n H_n| \geq cn\right) \le \frac{\pi^2}{6c^2}. ===Tail estimates=== A stronger tail estimate for the upper tail be obtained as follows. Then : \begin{align} P\left [ {Z}_i^r \right ] = \left(1-\frac{1}{n}\right)^r \le e^{-r / n}. \end{align} Thus, for r = \beta n \log n, we have P\left [ {Z}_i^r \right ] \le e^{(-\beta n \log n ) / n} = n^{-\beta}. For example, when n = 50 it takes about 225E(50) = 50(1 + 1/2 + 1/3 + ... + 1/50) = 224.9603, the expected number of trials to collect all 50 coupons. The mathematical analysis of the problem reveals that the expected number of trials needed grows as \Theta(n\log(n)). The probable error can also be expressed as a multiple of the standard deviation σ,Zwillinger, D.; Kokosa, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, Chapman & Hall/CRC. Probability. The commonly used approximate value of 1.96 is therefore accurate to better than one part in 50,000, which is more than adequate for applied work. It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions. ==See also== *Vysochanskiï–Petunin inequality, a similar result for the distance from the mean rather than the mode *Chebyshev's inequality, concerns distance from the mean without requiring unimodality * Concentration inequality – a summary of tail-bounds on random variables. ==References== * * * * Category:Probabilistic inequalities It asks the following question: If each box of a brand of cereals contains a coupon, and there are n different types of coupons, what is the probability that more than t boxes need to be bought to collect all n coupons? thumb|The gaussian correlation inequality states that probability of hitting both circle and rectangle with a dart is greater than or equal to the product of the individual probabilities of hitting the circle or the rectangle. Then : \Pr\left( \limsup_n \frac{S_n}{\sqrt{n}} \geq M \right) \geqslant \limsup_n \Pr\left( \frac{S_n}{\sqrt{n}} \geq M \right) = \Pr\left( \mathcal{N}(0, 1) \geq M \right) > 0 so :\limsup_n \frac{S_n}{\sqrt{n}}=\infty \qquad \text{with probability 1.} In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode. The approximation n\log n+\gamma n+1/2 for this expected number gives in this case 50\log 50+50\gamma+1/2 \approx 195.6011+28.8608+0.5\approx 224.9619. trials on average to collect all 50 coupons. ==Solution== ===Calculating the expectation=== Let time T be the number of draws needed to collect all n coupons, and let ti be the time to collect the i-th coupon after i − 1 coupons have been collected. By Kolmogorov's zero–one law, for any fixed M, the probability that the event \limsup_n \frac{S_n}{\sqrt{n}} \geq M occurs is 0 or 1. thumb|400px|Graph of number of coupons, n vs the expected number of trials (i.e., time) needed to collect them all, E (T ) In probability theory, the coupon collector's problem describes ""collect all coupons and win"" contests. ",1.07,96.4365076099,"""17.4""",58.2,0.85,E
 "5.6-1. Let $\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \leq \bar{X} \leq 2 / 3)$.
-","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. The lower bound is very close to m, thus more informative is the asymmetric confidence interval from p = 5% to 100%; for k = 5 this yields 0.051/5 ≈ 0.55 and the interval [m, 1.82m]. For example, taking the symmetric 95% interval p = 2.5% and q = 97.5% for k = 5 yields 0.0251/5 ≈ 0.48, 0.9751/5 ≈ 0.995, so the confidence interval is approximately [1.005m, 2.08m]. An approximation can be given by replacing with , yielding: \hat\sigma = \sqrt{\frac{1}{N - 1.5} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, The error in this approximation decays quadratically (as ), and it is suited for all but the smallest samples or highest precision: for the bias is equal to 1.3%, and for the bias is already less than 0.1%. For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation: \hat\sigma = \sqrt{\frac{1}{N - 1.5 - \frac{1}{4}\gamma_2} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, where denotes the population excess kurtosis. This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. The median in this example is 74.5, in close agreement with the frequentist formula. For various values of , the percentage of values expected to lie in and outside the symmetric interval, , are as follows: thumb|Percentage within(z) thumb|z(Percentage within) Confidence interval Proportion within Proportion without Proportion without Confidence interval Percentage Percentage Fraction 25% 75% 3 / 4 % % 1 / 66.6667% 33.3333% 1 / 3 68% 32% 1 / 3.125 1 % % 1 / 80% 20% 1 / 5 90% 10% 1 / 10 95% 5% 1 / 20 2 % % 1 / 99% 1% 1 / 100 3 % % 1 / 370.398 99.9% 0.1% 1 / 99.99% 0.01% 1 / 4 % % 1 / 99.999% 0.001% 1 / 1 / 6.8 / % % 1 / 5 % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / 7 % 1 / ==Relationship between standard deviation and mean== The mean and the standard deviation of a set of data are descriptive statistics usually reported together. If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, , where is the arithmetic mean), about 95 percent are within two standard deviations (), and about 99.7 percent lie within three standard deviations (). The Pareto distribution with parameter \alpha \in (1,2] has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The ratio of uniforms is a method initially proposed by Kinderman and Monahan in 1977 for pseudo-random number sampling, that is, for drawing random samples from a statistical distribution. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by: \sigma_\text{mean} = \frac{1}{\sqrt{N}}\sigma where is the number of observations in the sample used to estimate the mean. For a sample population , this is down to 0.88 × SD to 1.16 × SD. Distance from mean Minimum population \sqrt{2}\,\sigma 50% 2\sigma 75% 3\sigma 89% 4\sigma 94% 5\sigma 96% 6\sigma 97% k\sigma 1 - \frac{1}{k^2} \frac{1}{\sqrt{1 - \ell}}\, \sigma \ell ===Rules for normally distributed data=== The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of f\left(x, \mu, \sigma^2\right) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2} where is the expected value of the random variables, equals their distribution's standard deviation divided by , and is the number of random variables. This has a variance : \operatorname{var}\left(\widehat{N}\right) = \frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N, so the standard deviation is approximately N/k, the expected size of the gap between sorted observations in the sample. These are easily computed, based on the observation that the probability that k observations in the sample will fall in an interval covering p of the range (0 ≤ p ≤ 1) is pk (assuming in this section that draws are with replacement, to simplify computations; if draws are without replacement, this overstates the likelihood, and intervals will be overly conservative). An estimate of the standard deviation for data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values represents four standard deviations so that . From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Thus the sampling distribution of the quantile of the sample maximum is the graph x1/k from 0 to 1: the p-th to q-th quantile of the sample maximum m are the interval [p1/kN, q1/kN]. More generally, the (downward biased) 95% confidence interval is [m, m/0.051/k] = [m, m·201/k]. ", 135.36,1.56,3.23,-1270,0.4772,E
-"5.2-9. Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.","thumb|300px|As the degree of the Taylor polynomial rises, it approaches the correct function. C63 or C-63 may refer to: * Caldwell 63, a planetary nebula * Convention concerning Statistics of Wages and Hours of Work, 1938 of the International Labour Organization * JNR Class C63, a proposed Japanese steam locomotive * Lockheed C-63 Hudson, an American military transport aircraft * Mercedes-AMG C 63, a German automobile * Ruy Lopez, a chess opening This image shows and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at . The molecular formula C6O6 (molar mass: 168.06 g/mol, exact mass: 167.9695 u) may refer to: * Cyclohexanehexone, also known as hexaketocyclohexane or triquinoyl * Ethylenetetracarboxylic dianhydride The molecular formula C6H6O3 may refer to: * Cyclohexanetriones * Hydroxymethylfurfural * Hydroxyquinol * Isomaltol * Levoglucosenone * Maltol * Phloroglucinol * Pyrogallol * Triacetic acid lactone The molecular formula C3F6O (molar mass: 166.02 g/mol, exact mass: 165.9853 u) may refer to: * Hexafluoroacetone (HFA) * Hexafluoropropylene oxide (HFPO) The molecular formula C3H2F6O (molar mass: 168.038 g/mol, exact mass: 168.0010 u) may refer to: * Desflurane * Hexafluoro-2-propanol (HFIP) (In addition, the series for converges for , and the series for converges for .) === Geometric series === The geometric series and its derivatives have Maclaurin series :\begin{align} \frac{1}{1-x} &= \sum^\infty_{n=0} x^n \\\ \frac{1}{(1-x)^2} &= \sum^\infty_{n=1} nx^{n-1}\\\ \frac{1}{(1-x)^3} &= \sum^\infty_{n=2} \frac{(n-1)n}{2} x^{n-2}. \end{align} All are convergent for |x| < 1. In order to compute a second-order Taylor series expansion around point of the function :f(x,y)=e^x\ln(1+y), one first computes all the necessary partial derivatives: :\begin{align} f_x &= e^x\ln(1+y) \\\\[6pt] f_y &= \frac{e^x}{1+y} \\\\[6pt] f_{xx} &= e^x\ln(1+y) \\\\[6pt] f_{yy} &= - \frac{e^x}{(1+y)^2} \\\\[6pt] f_{xy} &=f_{yx} = \frac{e^x}{1+y} . \end{align} Evaluating these derivatives at the origin gives the Taylor coefficients :\begin{align} f_x(0,0) &= 0 \\\ f_y(0,0) &=1 \\\ f_{xx}(0,0) &=0 \\\ f_{yy}(0,0) &=-1 \\\ f_{xy}(0,0) &=f_{yx}(0,0)=1. \end{align} Substituting these values in to the general formula :\begin{align} T(x,y) = &f;(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) \\\ &{}+\frac{1}{2!}\left( (x-a)^2f_{xx}(a,b) \+ 2(x-a)(y-b)f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \right)+ \cdots \end{align} produces :\begin{align} T(x,y) &= 0 + 0(x-0) + 1(y-0) + \frac{1}{2}\big( 0(x-0)^2 + 2(x-0)(y-0) + (-1)(y-0)^2 \big) + \cdots \\\ &= y + xy - \tfrac12 y^2 + \cdots \end{align} Since is analytic in , we have :e^x\ln(1+y)= y + xy - \tfrac12 y^2 + \cdots, \qquad |y| < 1. == Comparison with Fourier series == The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval ) as an infinite sum of trigonometric functions (sines and cosines). In particular, for , the error is less than 0.000003. So, by substituting for , the Taylor series of at is :1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots. In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum. For , Taylor polynomials of higher degree provide worse approximations. 300px|thumb|right|The Taylor approximations for (black). For most common functions, the function and the sum of its Taylor series are equal near this point. The error in this approximation is no more than . Collecting the terms up to fourth order yields : e^x =c_0 + c_1x + \left(c_2 - \frac{c_0}{2}\right)x^2 + \left(c_3 - \frac{c_1}{2}\right)x^3+\left(c_4-\frac{c_2}{2}+\frac{c_0}{4!}\right)x^4 + \cdots\\! However, is not the zero function, so does not equal its Taylor series around the origin. Taylor polynomials are approximations of a function, which become generally better as increases. By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the natural logarithm: :-x - \tfrac{1}{2}x^2 - \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 - \cdots. For these functions the Taylor series do not converge if is far from . This method uses the known Taylor expansion of the exponential function. #The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). ", -1,14.44,16.0,840,1.7,D
+","If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. The lower bound is very close to m, thus more informative is the asymmetric confidence interval from p = 5% to 100%; for k = 5 this yields 0.051/5 ≈ 0.55 and the interval [m, 1.82m]. For example, taking the symmetric 95% interval p = 2.5% and q = 97.5% for k = 5 yields 0.0251/5 ≈ 0.48, 0.9751/5 ≈ 0.995, so the confidence interval is approximately [1.005m, 2.08m]. An approximation can be given by replacing with , yielding: \hat\sigma = \sqrt{\frac{1}{N - 1.5} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, The error in this approximation decays quadratically (as ), and it is suited for all but the smallest samples or highest precision: for the bias is equal to 1.3%, and for the bias is already less than 0.1%. For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation: \hat\sigma = \sqrt{\frac{1}{N - 1.5 - \frac{1}{4}\gamma_2} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, where denotes the population excess kurtosis. This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. The median in this example is 74.5, in close agreement with the frequentist formula. For various values of , the percentage of values expected to lie in and outside the symmetric interval, , are as follows: thumb|Percentage within(z) thumb|z(Percentage within) Confidence interval Proportion within Proportion without Proportion without Confidence interval Percentage Percentage Fraction 25% 75% 3 / 4 % % 1 / 66.6667% 33.3333% 1 / 3 68% 32% 1 / 3.125 1 % % 1 / 80% 20% 1 / 5 90% 10% 1 / 10 95% 5% 1 / 20 2 % % 1 / 99% 1% 1 / 100 3 % % 1 / 370.398 99.9% 0.1% 1 / 99.99% 0.01% 1 / 4 % % 1 / 99.999% 0.001% 1 / 1 / 6.8 / % % 1 / 5 % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / 7 % 1 / ==Relationship between standard deviation and mean== The mean and the standard deviation of a set of data are descriptive statistics usually reported together. If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, , where is the arithmetic mean), about 95 percent are within two standard deviations (), and about 99.7 percent lie within three standard deviations (). The Pareto distribution with parameter \alpha \in (1,2] has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The ratio of uniforms is a method initially proposed by Kinderman and Monahan in 1977 for pseudo-random number sampling, that is, for drawing random samples from a statistical distribution. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by: \sigma_\text{mean} = \frac{1}{\sqrt{N}}\sigma where is the number of observations in the sample used to estimate the mean. For a sample population , this is down to 0.88 × SD to 1.16 × SD. Distance from mean Minimum population \sqrt{2}\,\sigma 50% 2\sigma 75% 3\sigma 89% 4\sigma 94% 5\sigma 96% 6\sigma 97% k\sigma 1 - \frac{1}{k^2} \frac{1}{\sqrt{1 - \ell}}\, \sigma \ell ===Rules for normally distributed data=== The central limit theorem states that the distribution of an average of many independent, identically distributed random variables tends toward the famous bell-shaped normal distribution with a probability density function of f\left(x, \mu, \sigma^2\right) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2} where is the expected value of the random variables, equals their distribution's standard deviation divided by , and is the number of random variables. This has a variance : \operatorname{var}\left(\widehat{N}\right) = \frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N, so the standard deviation is approximately N/k, the expected size of the gap between sorted observations in the sample. These are easily computed, based on the observation that the probability that k observations in the sample will fall in an interval covering p of the range (0 ≤ p ≤ 1) is pk (assuming in this section that draws are with replacement, to simplify computations; if draws are without replacement, this overstates the likelihood, and intervals will be overly conservative). An estimate of the standard deviation for data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values represents four standard deviations so that . From the probability density function of the standard normal distribution, the exact value of z.975 is determined by : \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z_{.975}} e^{-x^2/2} \, \mathrm{d}x = 0.975. == History == thumb|right|200px|Ronald Fisher The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925: In Table 1 of the same work, he gave the more precise value 1.959964. , Table 1 In 1970, the value truncated to 20 decimal places was calculated to be :1.95996 39845 40054 23552... Thus the sampling distribution of the quantile of the sample maximum is the graph x1/k from 0 to 1: the p-th to q-th quantile of the sample maximum m are the interval [p1/kN, q1/kN]. More generally, the (downward biased) 95% confidence interval is [m, m/0.051/k] = [m, m·201/k]. ", 135.36,1.56,"""3.23""",-1270,0.4772,E
+"5.2-9. Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.","thumb|300px|As the degree of the Taylor polynomial rises, it approaches the correct function. C63 or C-63 may refer to: * Caldwell 63, a planetary nebula * Convention concerning Statistics of Wages and Hours of Work, 1938 of the International Labour Organization * JNR Class C63, a proposed Japanese steam locomotive * Lockheed C-63 Hudson, an American military transport aircraft * Mercedes-AMG C 63, a German automobile * Ruy Lopez, a chess opening This image shows and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at . The molecular formula C6O6 (molar mass: 168.06 g/mol, exact mass: 167.9695 u) may refer to: * Cyclohexanehexone, also known as hexaketocyclohexane or triquinoyl * Ethylenetetracarboxylic dianhydride The molecular formula C6H6O3 may refer to: * Cyclohexanetriones * Hydroxymethylfurfural * Hydroxyquinol * Isomaltol * Levoglucosenone * Maltol * Phloroglucinol * Pyrogallol * Triacetic acid lactone The molecular formula C3F6O (molar mass: 166.02 g/mol, exact mass: 165.9853 u) may refer to: * Hexafluoroacetone (HFA) * Hexafluoropropylene oxide (HFPO) The molecular formula C3H2F6O (molar mass: 168.038 g/mol, exact mass: 168.0010 u) may refer to: * Desflurane * Hexafluoro-2-propanol (HFIP) (In addition, the series for converges for , and the series for converges for .) === Geometric series === The geometric series and its derivatives have Maclaurin series :\begin{align} \frac{1}{1-x} &= \sum^\infty_{n=0} x^n \\\ \frac{1}{(1-x)^2} &= \sum^\infty_{n=1} nx^{n-1}\\\ \frac{1}{(1-x)^3} &= \sum^\infty_{n=2} \frac{(n-1)n}{2} x^{n-2}. \end{align} All are convergent for |x| < 1. In order to compute a second-order Taylor series expansion around point of the function :f(x,y)=e^x\ln(1+y), one first computes all the necessary partial derivatives: :\begin{align} f_x &= e^x\ln(1+y) \\\\[6pt] f_y &= \frac{e^x}{1+y} \\\\[6pt] f_{xx} &= e^x\ln(1+y) \\\\[6pt] f_{yy} &= - \frac{e^x}{(1+y)^2} \\\\[6pt] f_{xy} &=f_{yx} = \frac{e^x}{1+y} . \end{align} Evaluating these derivatives at the origin gives the Taylor coefficients :\begin{align} f_x(0,0) &= 0 \\\ f_y(0,0) &=1 \\\ f_{xx}(0,0) &=0 \\\ f_{yy}(0,0) &=-1 \\\ f_{xy}(0,0) &=f_{yx}(0,0)=1. \end{align} Substituting these values in to the general formula :\begin{align} T(x,y) = &f;(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) \\\ &{}+\frac{1}{2!}\left( (x-a)^2f_{xx}(a,b) \+ 2(x-a)(y-b)f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \right)+ \cdots \end{align} produces :\begin{align} T(x,y) &= 0 + 0(x-0) + 1(y-0) + \frac{1}{2}\big( 0(x-0)^2 + 2(x-0)(y-0) + (-1)(y-0)^2 \big) + \cdots \\\ &= y + xy - \tfrac12 y^2 + \cdots \end{align} Since is analytic in , we have :e^x\ln(1+y)= y + xy - \tfrac12 y^2 + \cdots, \qquad |y| < 1. == Comparison with Fourier series == The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval ) as an infinite sum of trigonometric functions (sines and cosines). In particular, for , the error is less than 0.000003. So, by substituting for , the Taylor series of at is :1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots. In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum. For , Taylor polynomials of higher degree provide worse approximations. 300px|thumb|right|The Taylor approximations for (black). For most common functions, the function and the sum of its Taylor series are equal near this point. The error in this approximation is no more than . Collecting the terms up to fourth order yields : e^x =c_0 + c_1x + \left(c_2 - \frac{c_0}{2}\right)x^2 + \left(c_3 - \frac{c_1}{2}\right)x^3+\left(c_4-\frac{c_2}{2}+\frac{c_0}{4!}\right)x^4 + \cdots\\! However, is not the zero function, so does not equal its Taylor series around the origin. Taylor polynomials are approximations of a function, which become generally better as increases. By integrating the above Maclaurin series, we find the Maclaurin series of , where denotes the natural logarithm: :-x - \tfrac{1}{2}x^2 - \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 - \cdots. For these functions the Taylor series do not converge if is far from . This method uses the known Taylor expansion of the exponential function. #The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). ", -1,14.44,"""16.0""",840,1.7,D
 "5.3-15. Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ ""triples"" and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$
-","Let p be the probability of success in a Bernoulli trial, and q be the probability of failure. The problem was: Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability. ==Solution== The probabilities of outcomes A, B and C are: :P(A)=1-\left(\frac{5}{6}\right)^{6} = \frac{31031}{46656} \approx 0.6651\, , :P(B)=1-\sum_{x=0}^1\binom{12}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{12-x} = \frac{1346704211}{2176782336} \approx 0.6187\, , :P(C)=1-\sum_{x=0}^2\binom{18}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{18-x} = \frac{15166600495229}{25389989167104} \approx 0.5973\, . When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials.Rajeev Motwani and P. Raghavan. Then the probability of success and the probability of failure sum to one, since these are complementary events: ""success"" and ""failure"" are mutually exclusive and exhaustive. Alternatively, these can be stated in terms of odds: given probability p of success and q of failure, the odds for are p:q and the odds against are q:p. The probability of success (POS) is a statistics concept commonly used in the pharmaceutical industry including by health authorities to support decision making. Thus, the probability of failure, q, is given by :q = 1 - p = 1 - \tfrac{1}{2} = \tfrac{1}{2}. Triple therapy may refer to : * a first line therapy in Helicobacter pylori eradication protocols * any of the three drug treatments used in Management of HIV/AIDS * the combination of methotrexate, sulfasalazine, and hydroxychloroquine used to treat rheumatoid arthritis The probability of exactly k successes in the experiment B(n,p) is given by: :P(k)={n \choose k} p^k q^{n-k} where {n \choose k} is a binomial coefficient. Statistics & Probability Letters 83 (5), 1472-1478. If u_1, u_2 , n, k are positive natural numbers, and u_1 \le u_2, k \le n, p \in [0, 1] then P(r = u_1 k ; u_1 n, p) \ge P(r = u_2 k ; u_2 n, p). ==References== Category:Factorial and binomial topics Category:Probability problems Category:Isaac Newton Category:Mathematical problems The first criterion ensures that the probability of success is large. Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, ""success"" and ""failure"", in which the probability of success is the same every time the experiment is conducted. However, it is a very important method for counts when the appropriate order of magnitude is unknown a priori and sampling is necessarily destructive. ==See also== *Dilution assay == External links == *A downloadable MPN calculator to take your data and get estimates *A five-replicate MPN table *Details of practical implementation, but not theory *US FDA article on MPN method *Information on the MPN method and ballast water treatment *Downloadable EXCEL program for the determination of the Most Probable Numbers (MPN), their standard deviations, confidence bounds and rarity values according to Jarvis, B., Wilrich, C., and P.-T. Wilrich: Reconsideration of the derivation of Most Probable Numbers, their standard deviations, confidence bounds and rarity values. Finding the optimal design is equivalent to finding the solution to the following equations: # mCPOS=c1 # lCPOS=c2 == See also == * Credible interval * Posterior probability * Interim analysis == References == Category:Pharmaceutical statistics He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by: :\begin{align} P(2) &= {4 \choose 2} p^{2} q^{4-2} \\\ &= 6 \times \left(\tfrac{1}{2}\right)^2 \times \left(\tfrac{1}{2}\right)^2 \\\ &= \dfrac {3}{8}. \end{align} ==See also== *Bernoulli scheme *Bernoulli sampling *Bernoulli distribution *Binomial distribution *Binomial coefficient *Binomial proportion confidence interval *Poisson sampling *Sampling design *Coin flipping *Jacob Bernoulli *Fisher's exact test *Boschloo's test ==References== ==External links== * * Category:Discrete distributions Category:Coin flipping Category:Experiment (probability theory) Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. If k, n_1, n_2 are positive natural numbers, and n_1 < n_2, then P(r \ge k ; k n_1, \frac{1}{n_1}) > P(r \ge k ; k n_2, \frac{1}{n_2}). (from Varagnolo, Pillonetto and Schenato (2013)):D. Varagnolo, L. Schenato, G. Pillonetto, 2013. As n grows, P(n) decreases monotonically towards an asymptotic limit of 1/2. ==Example in R == The solution outlined above can be implemented in R as follows: for (s in 1:3) { # looking for s = 1, 2 or 3 sixes n = 6*s # ... in n = 6, 12 or 18 dice q = pbinom(s-1, n, 1/6) # q = Prob( ~~==Newton's explanation== Although Newton correctly calculated the odds of each bet, he provided a separate intuitive explanation to Pepys. ",0.082,+93.4,4.86,0.0384,30,D
+","Let p be the probability of success in a Bernoulli trial, and q be the probability of failure. The problem was: Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability. ==Solution== The probabilities of outcomes A, B and C are: :P(A)=1-\left(\frac{5}{6}\right)^{6} = \frac{31031}{46656} \approx 0.6651\, , :P(B)=1-\sum_{x=0}^1\binom{12}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{12-x} = \frac{1346704211}{2176782336} \approx 0.6187\, , :P(C)=1-\sum_{x=0}^2\binom{18}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{18-x} = \frac{15166600495229}{25389989167104} \approx 0.5973\, . When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials.Rajeev Motwani and P. Raghavan. Then the probability of success and the probability of failure sum to one, since these are complementary events: ""success"" and ""failure"" are mutually exclusive and exhaustive. Alternatively, these can be stated in terms of odds: given probability p of success and q of failure, the odds for are p:q and the odds against are q:p. The probability of success (POS) is a statistics concept commonly used in the pharmaceutical industry including by health authorities to support decision making. Thus, the probability of failure, q, is given by :q = 1 - p = 1 - \tfrac{1}{2} = \tfrac{1}{2}. Triple therapy may refer to : * a first line therapy in Helicobacter pylori eradication protocols * any of the three drug treatments used in Management of HIV/AIDS * the combination of methotrexate, sulfasalazine, and hydroxychloroquine used to treat rheumatoid arthritis The probability of exactly k successes in the experiment B(n,p) is given by: :P(k)={n \choose k} p^k q^{n-k} where {n \choose k} is a binomial coefficient. Statistics & Probability Letters 83 (5), 1472-1478. If u_1, u_2 , n, k are positive natural numbers, and u_1 \le u_2, k \le n, p \in [0, 1] then P(r = u_1 k ; u_1 n, p) \ge P(r = u_2 k ; u_2 n, p). ==References== Category:Factorial and binomial topics Category:Probability problems Category:Isaac Newton Category:Mathematical problems The first criterion ensures that the probability of success is large. Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, ""success"" and ""failure"", in which the probability of success is the same every time the experiment is conducted. However, it is a very important method for counts when the appropriate order of magnitude is unknown a priori and sampling is necessarily destructive. ==See also== *Dilution assay == External links == *A downloadable MPN calculator to take your data and get estimates *A five-replicate MPN table *Details of practical implementation, but not theory *US FDA article on MPN method *Information on the MPN method and ballast water treatment *Downloadable EXCEL program for the determination of the Most Probable Numbers (MPN), their standard deviations, confidence bounds and rarity values according to Jarvis, B., Wilrich, C., and P.-T. Wilrich: Reconsideration of the derivation of Most Probable Numbers, their standard deviations, confidence bounds and rarity values. Finding the optimal design is equivalent to finding the solution to the following equations: # mCPOS=c1 # lCPOS=c2 == See also == * Credible interval * Posterior probability * Interim analysis == References == Category:Pharmaceutical statistics He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by: :\begin{align} P(2) &= {4 \choose 2} p^{2} q^{4-2} \\\ &= 6 \times \left(\tfrac{1}{2}\right)^2 \times \left(\tfrac{1}{2}\right)^2 \\\ &= \dfrac {3}{8}. \end{align} ==See also== *Bernoulli scheme *Bernoulli sampling *Bernoulli distribution *Binomial distribution *Binomial coefficient *Binomial proportion confidence interval *Poisson sampling *Sampling design *Coin flipping *Jacob Bernoulli *Fisher's exact test *Boschloo's test ==References== ==External links== * * Category:Discrete distributions Category:Coin flipping Category:Experiment (probability theory) Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. If k, n_1, n_2 are positive natural numbers, and n_1 < n_2, then P(r \ge k ; k n_1, \frac{1}{n_1}) > P(r \ge k ; k n_2, \frac{1}{n_2}). (from Varagnolo, Pillonetto and Schenato (2013)):D. Varagnolo, L. Schenato, G. Pillonetto, 2013. As n grows, P(n) decreases monotonically towards an asymptotic limit of 1/2. ==Example in R == The solution outlined above can be implemented in R as follows: for (s in 1:3) { # looking for s = 1, 2 or 3 sixes n = 6*s # ... in n = 6, 12 or 18 dice q = pbinom(s-1, n, 1/6) # q = Prob( ~~==Newton's explanation== Although Newton correctly calculated the odds of each bet, he provided a separate intuitive explanation to Pepys. ",0.082,+93.4,"""4.86""",0.0384,30,D
 "5.2- II. Evaluate
 $$
 \int_0^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^3(1-y)^2 d y
 $$
-(a) Using integration.","Thus * \Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2) * \Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1) and so on. Note that \Gamma_1(a) reduces to the ordinary gamma function. Introduction to the Gamma Function * S. Finch. The other one, more useful to obtain a numerical result is: : \Gamma_p(a)= \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2). Numerically, :\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337 :\Gamma\left(\tfrac14\right) \approx 3.625\,609\,908\,221\,908\,3119 :\Gamma\left(\tfrac15\right) \approx 4.590\,843\,711\,998\,803\,0532 :\Gamma\left(\tfrac16\right) \approx 5.566\,316\,001\,780\,235\,2043 :\Gamma\left(\tfrac17\right) \approx 6.548\,062\,940\,247\,824\,4377 :\Gamma\left(\tfrac18\right) \approx 7.533\,941\,598\,797\,611\,9047 . The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the right of all poles of factors of the form Γ(a + s) and to the left of all poles of factors of the form Γ(a − s). ==Hypergeometric series== The hypergeometric function is given as a Barnes integral by :{}_2F_1(a,b;c;z) =\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^s\,ds, see also . Integrating the reciprocal gamma function along the positive real axis also gives the Fransén–Robinson constant. The second Barnes lemma states :\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}ds :=\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1-d+a)\Gamma(1-d+b)\Gamma(1-d+c)}{\Gamma(e-a)\Gamma(e-b)\Gamma(e-c)} where e = a + b + c − d + 1\. The gamma function is an important special function in mathematics. The following two representations for were given by I. Mező :\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma^2\left(\frac34\right)}=i\sum_{k=-\infty}^\infty e^{\pi(k-2k^2)}\theta_1\left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right), and :\sqrt{\frac{\pi}{2}}\frac{1}{\Gamma^2\left(\frac34\right)}=\sum_{k=-\infty}^\infty\frac{\theta_4(ik\pi,e^{-\pi})}{e^{2\pi k^2}}, where and are two of the Jacobi theta functions. == Products == Some product identities include: : \prod_{r=1}^2 \Gamma\left(\tfrac{r}{3}\right) = \frac{2\pi}{\sqrt 3} \approx 3.627\,598\,728\,468\,435\,7012 : \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) = \sqrt{2\pi^3} \approx 7.874\,804\,972\,861\,209\,8721 : \prod_{r=1}^4 \Gamma\left(\tfrac{r}{5}\right) = \frac{4\pi^2}{\sqrt 5} \approx 17.655\,285\,081\,493\,524\,2483 : \prod_{r=1}^5 \Gamma\left(\tfrac{r}{6}\right) = 4\sqrt{\frac{\pi^5}3} \approx 40.399\,319\,122\,003\,790\,0785 : \prod_{r=1}^6 \Gamma\left(\tfrac{r}{7}\right) = \frac{8\pi^3}{\sqrt 7} \approx 93.754\,168\,203\,582\,503\,7970 : \prod_{r=1}^7 \Gamma\left(\tfrac{r}{8}\right) = 4\sqrt{\pi^7} \approx 219.828\,778\,016\,957\,263\,6207 In general: : \prod_{r=1}^n \Gamma\left(\tfrac{r}{n+1}\right) = \sqrt{\frac{(2\pi)^n}{n+1}} From those products can be deduced other values, for example, from the former equations for \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) , \Gamma\left(\tfrac{1}{4}\right) and \Gamma\left(\tfrac{2}{4}\right) , can be deduced: \Gamma\left(\tfrac{3}{4}\right) =\left(\tfrac{\pi} {2}\right) ^{\tfrac{1}{4}} {\operatorname{AGM}\left(\sqrt 2, 1\right)}^{\tfrac{1}{2}} Other rational relations include :\frac{\Gamma\left(\tfrac15\right)\Gamma\left(\tfrac{4}{15}\right)}{\Gamma\left(\tfrac13\right)\Gamma\left(\tfrac{2}{15}\right)} = \frac{\sqrt{2}\,\sqrt[20]{3}}{\sqrt[6]{5}\,\sqrt[4]{5-\frac{7}{\sqrt 5}+\sqrt{6-\frac{6}{\sqrt 5}}}} :\frac{\Gamma\left(\tfrac{1}{20}\right)\Gamma\left(\tfrac{9}{20}\right)}{\Gamma\left(\tfrac{3}{20}\right)\Gamma\left(\tfrac{7}{20}\right)} = \frac{\sqrt[4]{5}\left(1+\sqrt{5}\right)}{2} :\frac{\Gamma\left(\frac{1}{5}\right)^2}{\Gamma\left(\frac{1}{10}\right)\Gamma\left(\frac{3}{10}\right)} = \frac{\sqrt{1+\sqrt{5}}}{2^{\tfrac{7}{10}}\sqrt[4]{5}} and many more relations for where the denominator d divides 24 or 60.Raimundas Vidūnas, Expressions for Values of the Gamma Function Gamma quotients with algebraic values must be ""poised"" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator. Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions pFq in a similar way . ==Barnes lemmas== The first Barnes lemma states :\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds =\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)}. Euler Gamma Function Constants * * * * * Category:Gamma and related functions Category:Mathematical constants In mathematics and mathematical physics, Slater integrals are certain integrals of products of three spherical harmonics. In particular, where AGM() is the arithmetic–geometric mean, we have :\Gamma\left(\tfrac13\right) = \frac{2^\frac{7}{9}\cdot \pi^\frac23}{3^\frac{1}{12}\cdot \operatorname{AGM}\left(2,\sqrt{2+\sqrt{3}}\right)^\frac13} :\Gamma\left(\tfrac14\right) = \sqrt \frac{(2 \pi)^\frac32}{\operatorname{AGM}\left(\sqrt 2, 1\right)} :\Gamma\left(\tfrac16\right) = \frac{2^\frac{14}{9}\cdot 3^\frac13\cdot \pi^\frac56}{\operatorname{AGM}\left(1+\sqrt{3},\sqrt{8}\right)^\frac23}. For non-positive integers, the gamma function is not defined. Beta integral may refer to: *beta function *Barnes beta integral It may also be given in terms of the Barnes -function: :\Gamma(i) = \frac{G(1+i)}{G(i)} = e^{-\log G(i)+ \log G(1+i)}. Category:Gamma and related functions In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. The gamma function with other complex arguments returns :\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i :\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i :\Gamma(\tfrac12 + \tfrac12 i) \approx 0.818\,163\,9995 - 0.763\,313\,8287\, i :\Gamma(\tfrac12 - \tfrac12 i) \approx 0.818\,163\,9995 + 0.763\,313\,8287\, i :\Gamma(5 + 3i) \approx 0.016\,041\,8827 - 9.433\,293\,2898\, i :\Gamma(5 - 3i) \approx 0.016\,041\,8827 + 9.433\,293\,2898\, i. ==Other constants== The gamma function has a local minimum on the positive real axis :x_{\min} = 1.461\,632\,144\,968\,362\,341\,262\ldots\, with the value :\Gamma\left(x_{\min}\right) = 0.885\,603\,194\,410\,888\ldots\, . In mathematics, a Barnes integral or Mellin-Barnes integral is a contour integral involving a product of gamma functions. Curiously enough, \Gamma(i) appears in the below integral evaluation:The webpage of István Mező :\int_0^{\pi/2}\\{\cot(x)\\}\,dx=1-\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\pi}{\sinh(\pi)\Gamma(i)^2}\right). ",3857,0.36,-0.029,3.2,0.1792,E
+(a) Using integration.","Thus * \Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2) * \Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1) and so on. Note that \Gamma_1(a) reduces to the ordinary gamma function. Introduction to the Gamma Function * S. Finch. The other one, more useful to obtain a numerical result is: : \Gamma_p(a)= \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2). Numerically, :\Gamma\left(\tfrac13\right) \approx 2.678\,938\,534\,707\,747\,6337 :\Gamma\left(\tfrac14\right) \approx 3.625\,609\,908\,221\,908\,3119 :\Gamma\left(\tfrac15\right) \approx 4.590\,843\,711\,998\,803\,0532 :\Gamma\left(\tfrac16\right) \approx 5.566\,316\,001\,780\,235\,2043 :\Gamma\left(\tfrac17\right) \approx 6.548\,062\,940\,247\,824\,4377 :\Gamma\left(\tfrac18\right) \approx 7.533\,941\,598\,797\,611\,9047 . The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the right of all poles of factors of the form Γ(a + s) and to the left of all poles of factors of the form Γ(a − s). ==Hypergeometric series== The hypergeometric function is given as a Barnes integral by :{}_2F_1(a,b;c;z) =\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^s\,ds, see also . Integrating the reciprocal gamma function along the positive real axis also gives the Fransén–Robinson constant. The second Barnes lemma states :\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}ds :=\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1-d+a)\Gamma(1-d+b)\Gamma(1-d+c)}{\Gamma(e-a)\Gamma(e-b)\Gamma(e-c)} where e = a + b + c − d + 1\. The gamma function is an important special function in mathematics. The following two representations for were given by I. Mező :\sqrt{\frac{\pi\sqrt{e^\pi}}{2}}\frac{1}{\Gamma^2\left(\frac34\right)}=i\sum_{k=-\infty}^\infty e^{\pi(k-2k^2)}\theta_1\left(\frac{i\pi}{2}(2k-1),e^{-\pi}\right), and :\sqrt{\frac{\pi}{2}}\frac{1}{\Gamma^2\left(\frac34\right)}=\sum_{k=-\infty}^\infty\frac{\theta_4(ik\pi,e^{-\pi})}{e^{2\pi k^2}}, where and are two of the Jacobi theta functions. == Products == Some product identities include: : \prod_{r=1}^2 \Gamma\left(\tfrac{r}{3}\right) = \frac{2\pi}{\sqrt 3} \approx 3.627\,598\,728\,468\,435\,7012 : \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) = \sqrt{2\pi^3} \approx 7.874\,804\,972\,861\,209\,8721 : \prod_{r=1}^4 \Gamma\left(\tfrac{r}{5}\right) = \frac{4\pi^2}{\sqrt 5} \approx 17.655\,285\,081\,493\,524\,2483 : \prod_{r=1}^5 \Gamma\left(\tfrac{r}{6}\right) = 4\sqrt{\frac{\pi^5}3} \approx 40.399\,319\,122\,003\,790\,0785 : \prod_{r=1}^6 \Gamma\left(\tfrac{r}{7}\right) = \frac{8\pi^3}{\sqrt 7} \approx 93.754\,168\,203\,582\,503\,7970 : \prod_{r=1}^7 \Gamma\left(\tfrac{r}{8}\right) = 4\sqrt{\pi^7} \approx 219.828\,778\,016\,957\,263\,6207 In general: : \prod_{r=1}^n \Gamma\left(\tfrac{r}{n+1}\right) = \sqrt{\frac{(2\pi)^n}{n+1}} From those products can be deduced other values, for example, from the former equations for \prod_{r=1}^3 \Gamma\left(\tfrac{r}{4}\right) , \Gamma\left(\tfrac{1}{4}\right) and \Gamma\left(\tfrac{2}{4}\right) , can be deduced: \Gamma\left(\tfrac{3}{4}\right) =\left(\tfrac{\pi} {2}\right) ^{\tfrac{1}{4}} {\operatorname{AGM}\left(\sqrt 2, 1\right)}^{\tfrac{1}{2}} Other rational relations include :\frac{\Gamma\left(\tfrac15\right)\Gamma\left(\tfrac{4}{15}\right)}{\Gamma\left(\tfrac13\right)\Gamma\left(\tfrac{2}{15}\right)} = \frac{\sqrt{2}\,\sqrt[20]{3}}{\sqrt[6]{5}\,\sqrt[4]{5-\frac{7}{\sqrt 5}+\sqrt{6-\frac{6}{\sqrt 5}}}} :\frac{\Gamma\left(\tfrac{1}{20}\right)\Gamma\left(\tfrac{9}{20}\right)}{\Gamma\left(\tfrac{3}{20}\right)\Gamma\left(\tfrac{7}{20}\right)} = \frac{\sqrt[4]{5}\left(1+\sqrt{5}\right)}{2} :\frac{\Gamma\left(\frac{1}{5}\right)^2}{\Gamma\left(\frac{1}{10}\right)\Gamma\left(\frac{3}{10}\right)} = \frac{\sqrt{1+\sqrt{5}}}{2^{\tfrac{7}{10}}\sqrt[4]{5}} and many more relations for where the denominator d divides 24 or 60.Raimundas Vidūnas, Expressions for Values of the Gamma Function Gamma quotients with algebraic values must be ""poised"" in the sense that the sum of arguments is the same (modulo 1) for the denominator and the numerator. Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions pFq in a similar way . ==Barnes lemmas== The first Barnes lemma states :\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds =\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)}. Euler Gamma Function Constants * * * * * Category:Gamma and related functions Category:Mathematical constants In mathematics and mathematical physics, Slater integrals are certain integrals of products of three spherical harmonics. In particular, where AGM() is the arithmetic–geometric mean, we have :\Gamma\left(\tfrac13\right) = \frac{2^\frac{7}{9}\cdot \pi^\frac23}{3^\frac{1}{12}\cdot \operatorname{AGM}\left(2,\sqrt{2+\sqrt{3}}\right)^\frac13} :\Gamma\left(\tfrac14\right) = \sqrt \frac{(2 \pi)^\frac32}{\operatorname{AGM}\left(\sqrt 2, 1\right)} :\Gamma\left(\tfrac16\right) = \frac{2^\frac{14}{9}\cdot 3^\frac13\cdot \pi^\frac56}{\operatorname{AGM}\left(1+\sqrt{3},\sqrt{8}\right)^\frac23}. For non-positive integers, the gamma function is not defined. Beta integral may refer to: *beta function *Barnes beta integral It may also be given in terms of the Barnes -function: :\Gamma(i) = \frac{G(1+i)}{G(i)} = e^{-\log G(i)+ \log G(1+i)}. Category:Gamma and related functions In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. The gamma function with other complex arguments returns :\Gamma(1 + i) = i\Gamma(i) \approx 0.498 - 0.155i :\Gamma(1 - i) = -i\Gamma(-i) \approx 0.498 + 0.155i :\Gamma(\tfrac12 + \tfrac12 i) \approx 0.818\,163\,9995 - 0.763\,313\,8287\, i :\Gamma(\tfrac12 - \tfrac12 i) \approx 0.818\,163\,9995 + 0.763\,313\,8287\, i :\Gamma(5 + 3i) \approx 0.016\,041\,8827 - 9.433\,293\,2898\, i :\Gamma(5 - 3i) \approx 0.016\,041\,8827 + 9.433\,293\,2898\, i. ==Other constants== The gamma function has a local minimum on the positive real axis :x_{\min} = 1.461\,632\,144\,968\,362\,341\,262\ldots\, with the value :\Gamma\left(x_{\min}\right) = 0.885\,603\,194\,410\,888\ldots\, . In mathematics, a Barnes integral or Mellin-Barnes integral is a contour integral involving a product of gamma functions. Curiously enough, \Gamma(i) appears in the below integral evaluation:The webpage of István Mező :\int_0^{\pi/2}\\{\cot(x)\\}\,dx=1-\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\pi}{\sinh(\pi)\Gamma(i)^2}\right). ",3857,0.36,"""-0.029""",3.2,0.1792,E
 "5.6-7. Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\mu=$ 54.030 and the standard deviation is $\sigma=5.8$. Let $\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \leq \bar{X} \leq 54.453)$, approximately.
-","Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. For a finite population with equal probabilities at all points, we have \sqrt{\frac{1}{N}\sum_{i=1}^N\left(x_i - \bar{x}\right)^2} = \sqrt{\frac{1}{N}\left(\sum_{i=1}^N x_i^2\right) - {\bar{x}}^2} = \sqrt{\left(\frac{1}{N}\sum_{i=1}^N x_i^2\right) - \left(\frac{1}{N} \sum_{i=1}^{N} x_i\right)^2}, which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by: \sigma_\text{mean} = \frac{1}{\sqrt{N}}\sigma where is the number of observations in the sample used to estimate the mean. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68–95–99.7 rule, or the empirical rule, for more information). ==Definition of population values== Let μ be the expected value (the average) of random variable with density : \mu \equiv \operatorname{E}[X] = \int_{-\infty}^{+\infty} x f(x) \, \mathrm dx The standard deviation of is defined as \sigma \equiv \sqrt{\operatorname E\left[(X - \mu)^2\right]} = \sqrt{ \int_{-\infty}^{+\infty} (x-\mu)^2 f(x) \, \mathrm dx }, which can be shown to equal \sqrt{\operatorname E\left[X^2\right] - (\operatorname E[X])^2}. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. An estimate of the standard deviation for data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values represents four standard deviations so that . This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. The sample standard deviation can be computed as: s(X) = \sqrt{\frac{N}{N-1}} \sqrt{\operatorname E\left[(X - \operatorname E[X])^2\right]}. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means The average of these 15 deviations from the assumed mean is therefore −30/15 = −2\. If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, , where is the arithmetic mean), about 95 percent are within two standard deviations (), and about 99.7 percent lie within three standard deviations (). For various values of , the percentage of values expected to lie in and outside the symmetric interval, , are as follows: thumb|Percentage within(z) thumb|z(Percentage within) Confidence interval Proportion within Proportion without Proportion without Confidence interval Percentage Percentage Fraction 25% 75% 3 / 4 % % 1 / 66.6667% 33.3333% 1 / 3 68% 32% 1 / 3.125 1 % % 1 / 80% 20% 1 / 5 90% 10% 1 / 10 95% 5% 1 / 20 2 % % 1 / 99% 1% 1 / 100 3 % % 1 / 370.398 99.9% 0.1% 1 / 99.99% 0.01% 1 / 4 % % 1 / 99.999% 0.001% 1 / 1 / 6.8 / % % 1 / 5 % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / 7 % 1 / ==Relationship between standard deviation and mean== The mean and the standard deviation of a set of data are descriptive statistics usually reported together. Suppose we wanted to calculate a 95% confidence interval for μ. In this case, the standard deviation will be \sigma = \sqrt{\sum_{i=1}^N p_i(x_i - \mu)^2},\text{ where } \mu = \sum_{i=1}^N p_i x_i. ===Continuous random variable=== The standard deviation of a continuous real-valued random variable with probability density function is \sigma = \sqrt{\int_\mathbf{X} (x - \mu)^2\, p(x)\, \mathrm dx},\text{ where } \mu = \int_\mathbf{X} x\, p(x)\, \mathrm dx, and where the integrals are definite integrals taken for ranging over the set of possible values of the random variable . It has a mean of 1007 meters, and a standard deviation of 5 meters. For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation: \hat\sigma = \sqrt{\frac{1}{N - 1.5 - \frac{1}{4}\gamma_2} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, where denotes the population excess kurtosis. An approximation can be given by replacing with , yielding: \hat\sigma = \sqrt{\frac{1}{N - 1.5} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, The error in this approximation decays quadratically (as ), and it is suited for all but the smallest samples or highest precision: for the bias is equal to 1.3%, and for the bias is already less than 0.1%. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows: \Pr\left(q_\frac{\alpha}{2} < k \frac{s^2}{\sigma^2} < q_{1 - \frac{\alpha}{2}}\right) = 1 - \alpha, where q_p is the -th quantile of the chi-square distribution with degrees of freedom, and is the confidence level. Similarly for sample standard deviation, s = \sqrt{\frac{Ns_2 - s_1^2}{N(N - 1)}}. The proportion that is less than or equal to a number, , is given by the cumulative distribution function: \text{Proportion} \le x = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{x - \mu}{\sigma\sqrt{2}}\right)\right] = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right]. ",0.6247,34,-131.1,1.51, 13.45,A
+","Consequently, : \Pr\left(\bar{X} - \frac{cS}{\sqrt{n}} \le \mu \le \bar{X} + \frac{cS}{\sqrt{n}} \right)=0.95\, and we have a theoretical (stochastic) 95% confidence interval for μ. If X has a standard normal distribution, i.e. X ~ N(0,1), : \mathrm{P}(X > 1.96) \approx 0.025, \, : \mathrm{P}(X < 1.96) \approx 0.975, \, and as the normal distribution is symmetric, : \mathrm{P}(-1.96 < X < 1.96) \approx 0.95. For a finite population with equal probabilities at all points, we have \sqrt{\frac{1}{N}\sum_{i=1}^N\left(x_i - \bar{x}\right)^2} = \sqrt{\frac{1}{N}\left(\sum_{i=1}^N x_i^2\right) - {\bar{x}}^2} = \sqrt{\left(\frac{1}{N}\sum_{i=1}^N x_i^2\right) - \left(\frac{1}{N} \sum_{i=1}^{N} x_i\right)^2}, which means that the standard deviation is equal to the square root of the difference between the average of the squares of the values and the square of the average value. For a large number of independent identically distributed random variables \ X_1, ..., X_n\ , with finite variance, the average \ \overline{X}_n\ approximately has a normal distribution, no matter what the distribution of the \ X_i\ is, with the approximation roughly improving in proportion to \ \sqrt{n\ }. == Example == Suppose {X1, …, Xn} is an independent sample from a normally distributed population with unknown parameters mean μ and variance σ2. Assuming statistical independence of the values in the sample, the standard deviation of the mean is related to the standard deviation of the distribution by: \sigma_\text{mean} = \frac{1}{\sqrt{N}}\sigma where is the number of observations in the sample used to estimate the mean. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68–95–99.7 rule, or the empirical rule, for more information). ==Definition of population values== Let μ be the expected value (the average) of random variable with density : \mu \equiv \operatorname{E}[X] = \int_{-\infty}^{+\infty} x f(x) \, \mathrm dx The standard deviation of is defined as \sigma \equiv \sqrt{\operatorname E\left[(X - \mu)^2\right]} = \sqrt{ \int_{-\infty}^{+\infty} (x-\mu)^2 f(x) \, \mathrm dx }, which can be shown to equal \sqrt{\operatorname E\left[X^2\right] - (\operatorname E[X])^2}. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. An estimate of the standard deviation for data taken to be approximately normal follows from the heuristic that 95% of the area under the normal curve lies roughly two standard deviations to either side of the mean, so that, with 95% probability the total range of values represents four standard deviations so that . This means that most men (about 68%, assuming a normal distribution) have a height within 3 inches of the mean (67–73 inches)one standard deviationand almost all men (about 95%) have a height within 6 inches of the mean (64–76 inches)two standard deviations. The sample standard deviation can be computed as: s(X) = \sqrt{\frac{N}{N-1}} \sqrt{\operatorname E\left[(X - \operatorname E[X])^2\right]}. The standard deviation is estimated as :CS \sqrt{\frac{B-\frac{A^2}{N}}{N-1}}=5.57 ==References== Category:Means The average of these 15 deviations from the assumed mean is therefore −30/15 = −2\. If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, , where is the arithmetic mean), about 95 percent are within two standard deviations (), and about 99.7 percent lie within three standard deviations (). For various values of , the percentage of values expected to lie in and outside the symmetric interval, , are as follows: thumb|Percentage within(z) thumb|z(Percentage within) Confidence interval Proportion within Proportion without Proportion without Confidence interval Percentage Percentage Fraction 25% 75% 3 / 4 % % 1 / 66.6667% 33.3333% 1 / 3 68% 32% 1 / 3.125 1 % % 1 / 80% 20% 1 / 5 90% 10% 1 / 10 95% 5% 1 / 20 2 % % 1 / 99% 1% 1 / 100 3 % % 1 / 370.398 99.9% 0.1% 1 / 99.99% 0.01% 1 / 4 % % 1 / 99.999% 0.001% 1 / 1 / 6.8 / % % 1 / 5 % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / % % 1 / 7 % 1 / ==Relationship between standard deviation and mean== The mean and the standard deviation of a set of data are descriptive statistics usually reported together. Suppose we wanted to calculate a 95% confidence interval for μ. In this case, the standard deviation will be \sigma = \sqrt{\sum_{i=1}^N p_i(x_i - \mu)^2},\text{ where } \mu = \sum_{i=1}^N p_i x_i. ===Continuous random variable=== The standard deviation of a continuous real-valued random variable with probability density function is \sigma = \sqrt{\int_\mathbf{X} (x - \mu)^2\, p(x)\, \mathrm dx},\text{ where } \mu = \int_\mathbf{X} x\, p(x)\, \mathrm dx, and where the integrals are definite integrals taken for ranging over the set of possible values of the random variable . It has a mean of 1007 meters, and a standard deviation of 5 meters. For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of the approximation: \hat\sigma = \sqrt{\frac{1}{N - 1.5 - \frac{1}{4}\gamma_2} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, where denotes the population excess kurtosis. An approximation can be given by replacing with , yielding: \hat\sigma = \sqrt{\frac{1}{N - 1.5} \sum_{i=1}^N \left(x_i - \bar{x}\right)^2}, The error in this approximation decays quadratically (as ), and it is suited for all but the smallest samples or highest precision: for the bias is equal to 1.3%, and for the bias is already less than 0.1%. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows: \Pr\left(q_\frac{\alpha}{2} < k \frac{s^2}{\sigma^2} < q_{1 - \frac{\alpha}{2}}\right) = 1 - \alpha, where q_p is the -th quantile of the chi-square distribution with degrees of freedom, and is the confidence level. Similarly for sample standard deviation, s = \sqrt{\frac{Ns_2 - s_1^2}{N(N - 1)}}. The proportion that is less than or equal to a number, , is given by the cumulative distribution function: \text{Proportion} \le x = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{x - \mu}{\sigma\sqrt{2}}\right)\right] = \frac{1}{2}\left[1 + \operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right]. ",0.6247,34,"""-131.1""",1.51, 13.45,A
 "5.3-19. Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$. Compute $E(Z)$.
-","It is based on an exponential failure distribution (see failure rate for a full derivation). Inputs to this process include unit and system failure rates. This failure rate changes throughout the life of the product. :* Provide necessary input to unit and system-level life cycle cost analyses. In semiconductor devices, problems in the device package may cause failures due to contamination, mechanical stress of the device, or open or short circuits. This degradation drastically limits the overall operating life of a relay or contactor to a range of perhaps 100,000 operations, a level representing 1% or less than the mechanical life expectancy of the same device. ==Semiconductor failures== Many failures result in generation of hot electrons. Life cycle cost studies determine the cost of a product over its entire life. Another important factor in estimating a NPPs lifetime cost derives from its capacity factor. A CMU 2007 study showed an estimated 3% mean AFR over 1–5 years based on replacement logs for a large sample of drives.. ==See also== * Failure rate * Frequency of exceedance == References == Category:Engineering failures Category:Rates Electronic components have a wide range of failure modes. Member of Optical Society of America, IEEE, ""Automated Reliability Prediction, SR-332, Issue 3"", January 2011; ""Automated Reliability Prediction (ARPP), FD-ARPP-01, Issue 11"", January 2011 Every product has a failure rate, λ which is the number of units failing per unit time. It is necessary to know how often different parts of the system are going to fail even for redundant components. If this part of the sample is the only option and is weaker than the bond itself, the sample will fail before the bond. ==See also== * Reliability (semiconductor) ==References== ==Further reading== *Herfst, R.W., Steeneken, P.G., Schmitz, J., Time and voltage dependence of dielectric charging in RF MEMS capacitive switches, (2007) Annual Proceedings – Reliability Physics (Symposium), art. no. 4227667, pp. 417–421. ==External links== * http://www.esda.org - ESD Association Category:Semiconductor device defects Category:Engineering failures This leaves a product with a useful life period during which failures occur randomly i.e., λ is constant, and finally a wear-out period, usually beyond the products useful life, where λ is increasing. == Definition of reliability == A practical definition of reliability is “the probability that a piece of equipment operating under specified conditions shall perform satisfactorily for a given period of time”. Gallium arsenide monolithic microwave integrated circuits can have these failures:Chapter 4. Gaudenzio Meneghesso from the University of Padova, Padova, Italy was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2013 for contributions to the reliability physics of compound semiconductors devices. ==References== Category:Fellow Members of the IEEE Category:Living people Category:Year of birth missing (living people) Category:Place of birth missing (living people) Analysis of the statistical properties of failures can give guidance in designs to establish a given level of reliability. Annualized failure rate (AFR) gives the estimated probability that a device or component will fail during a full year of use. In semiconductor devices, parasitic structures, irrelevant for normal operation, become important in the context of failures; they can be both a source and protection against failure. [[File:Flamanville-3 2010-07-15.jpg|thumb|upright=1.35|right|EDF has said its third-generation Flamanville 3 project (seen here in 2010) will be delayed until 2018, due to ""both structural and economic reasons,"" and the project's total cost had climbed to EUR 11 billion by 2012.EDF raises French EPR reactor cost to over $11 billion, Reuters, Dec 3, 2012. During the ‘useful life period’ assuming a constant failure rate, MTBF is the inverse of the failure rate and the terms can be used interchangeably. == Importance of reliability prediction == Reliability predictions: :* Help assess the effect of product reliability on the maintenance activity and on the quantity of spare units required for acceptable field performance of any particular system. Parametric failures occur at intermediate discharge voltages and occur more often, with latent failures the most common. ",0.000226,1.7,5275.0,+3.03,5,E
+","It is based on an exponential failure distribution (see failure rate for a full derivation). Inputs to this process include unit and system failure rates. This failure rate changes throughout the life of the product. :* Provide necessary input to unit and system-level life cycle cost analyses. In semiconductor devices, problems in the device package may cause failures due to contamination, mechanical stress of the device, or open or short circuits. This degradation drastically limits the overall operating life of a relay or contactor to a range of perhaps 100,000 operations, a level representing 1% or less than the mechanical life expectancy of the same device. ==Semiconductor failures== Many failures result in generation of hot electrons. Life cycle cost studies determine the cost of a product over its entire life. Another important factor in estimating a NPPs lifetime cost derives from its capacity factor. A CMU 2007 study showed an estimated 3% mean AFR over 1–5 years based on replacement logs for a large sample of drives.. ==See also== * Failure rate * Frequency of exceedance == References == Category:Engineering failures Category:Rates Electronic components have a wide range of failure modes. Member of Optical Society of America, IEEE, ""Automated Reliability Prediction, SR-332, Issue 3"", January 2011; ""Automated Reliability Prediction (ARPP), FD-ARPP-01, Issue 11"", January 2011 Every product has a failure rate, λ which is the number of units failing per unit time. It is necessary to know how often different parts of the system are going to fail even for redundant components. If this part of the sample is the only option and is weaker than the bond itself, the sample will fail before the bond. ==See also== * Reliability (semiconductor) ==References== ==Further reading== *Herfst, R.W., Steeneken, P.G., Schmitz, J., Time and voltage dependence of dielectric charging in RF MEMS capacitive switches, (2007) Annual Proceedings – Reliability Physics (Symposium), art. no. 4227667, pp. 417–421. ==External links== * http://www.esda.org - ESD Association Category:Semiconductor device defects Category:Engineering failures This leaves a product with a useful life period during which failures occur randomly i.e., λ is constant, and finally a wear-out period, usually beyond the products useful life, where λ is increasing. == Definition of reliability == A practical definition of reliability is “the probability that a piece of equipment operating under specified conditions shall perform satisfactorily for a given period of time”. Gallium arsenide monolithic microwave integrated circuits can have these failures:Chapter 4. Gaudenzio Meneghesso from the University of Padova, Padova, Italy was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2013 for contributions to the reliability physics of compound semiconductors devices. ==References== Category:Fellow Members of the IEEE Category:Living people Category:Year of birth missing (living people) Category:Place of birth missing (living people) Analysis of the statistical properties of failures can give guidance in designs to establish a given level of reliability. Annualized failure rate (AFR) gives the estimated probability that a device or component will fail during a full year of use. In semiconductor devices, parasitic structures, irrelevant for normal operation, become important in the context of failures; they can be both a source and protection against failure. [[File:Flamanville-3 2010-07-15.jpg|thumb|upright=1.35|right|EDF has said its third-generation Flamanville 3 project (seen here in 2010) will be delayed until 2018, due to ""both structural and economic reasons,"" and the project's total cost had climbed to EUR 11 billion by 2012.EDF raises French EPR reactor cost to over $11 billion, Reuters, Dec 3, 2012. During the ‘useful life period’ assuming a constant failure rate, MTBF is the inverse of the failure rate and the terms can be used interchangeably. == Importance of reliability prediction == Reliability predictions: :* Help assess the effect of product reliability on the maintenance activity and on the quantity of spare units required for acceptable field performance of any particular system. Parametric failures occur at intermediate discharge voltages and occur more often, with latent failures the most common. ",0.000226,1.7,"""5275.0""",+3.03,5,E
 "5.8-1. If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find
 (b) An upper bound for $P(|X-33| \geq 14)$.
-","Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean. Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.89% within three standard deviations for a broad range of different probability distributions. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg.Savage, I. Richard. The second of these inequalities with is the Chebyshev bound. In this setting we can state the following: :General version of Chebyshev's inequality. \forall k > 0: \quad \Pr\left( \|X - \mu\|_\alpha \ge k \sigma_\alpha \right) \le \frac{1}{ k^2 }. However, the benefit of Chebyshev's inequality is that it can be applied more generally to get confidence bounds for ranges of standard deviations that do not depend on the number of samples. ====Semivariances==== An alternative method of obtaining sharper bounds is through the use of semivariances (partial variances). *Grechuk et al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any deviation risk measure in place of standard deviation. : P( | X - \mu | \ge k \sigma ) \le \frac{ 4 }{ 3k^2 } - \frac13 \quad \text{if} \quad k \le \sqrt{8/3}. The bounds are sharp for the following example: for any k ≥ 1, : X = \begin{cases} -1, & \text{with probability }\frac{1}{2k^2} \\\ 0, & \text{with probability }1 - \frac{1}{k^2} \\\ 1, & \text{with probability }\frac{1}{2k^2} \end{cases} For this distribution, the mean μ = 0 and the standard deviation σ = , so : \Pr(|X-\mu| \ge k\sigma) = \Pr(|X| \ge 1) = \frac{1}{k^2}. One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable with a = (kσ)2: : \Pr(|X - \mu| \geq k\sigma) = \Pr((X - \mu)^2 \geq k^2\sigma^2) \leq \frac{\mathbb{E}[(X - \mu)^2]}{k^2\sigma^2} = \frac{\sigma^2}{k^2\sigma^2} = \frac{1}{k^2}. By comparison, Chebyshev's inequality states that all but a 1/N fraction of the sample will lie within standard deviations of the mean. For k ≥ 1, n > 4 and assuming that the nth moment exists, this bound is tighter than Chebyshev's inequality. Chebyshev's inequality can now be written : \Pr(x \le m - k \sigma) \le \frac { 1 } { k^2 } \frac { \sigma_-^2 } { \sigma^2 }. The additional fraction of 4/9 present in these tail bounds lead to better confidence intervals than Chebyshev's inequality. The Chebyshev inequality for the distribution gives 95% and 99% confidence intervals of approximately ±4.472 standard deviations and ±10 standard deviations respectively. ====Samuelson's inequality==== Although Chebyshev's inequality is the best possible bound for an arbitrary distribution, this is not necessarily true for finite samples. * If 1\le r\le \sqrt{8/3}, the bound is tight when X=r with probability \frac{4}{3r^2}-\frac{1}{3} and is otherwise distributed uniformly in the interval \left[-\frac{r}{2},r\right]. === Specialization to mean and variance === If X has mean \mu and finite, non-zero variance \sigma^2, then taking \alpha=\mu and r=\lambda \sigma gives that for any \lambda > \sqrt{\frac{8}{3}} = 1.63299..., :\operatorname{Pr}(\left|X-\mu\right|\geq \lambda\sigma)\leq\frac{4}{9\lambda^2}. === Proof Sketch === For a relatively elementary proof see.Pukelsheim, F., 1994. The first provides a lower bound for the value of P(x). ==Finite samples== === Univariate case === Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known and may not exist, but the sample mean and sample standard deviation from N samples are to be employed to bound the expected value of a new drawing from the same distribution. If we put : \sigma_u^2 = \max(\sigma_-^2, \sigma_+^2) , Chebyshev's inequality can be written : \Pr(| x \le m - k \sigma |) \le \frac 1 {k^2} \frac { \sigma_u^2 } { \sigma^2 } . If X is a unimodal distribution with mean μ and variance σ2, then the inequality states that : P( | X - \mu | \ge k \sigma ) \le \frac{ 4 }{ 9k^2 } \quad \text{if} \quad k \ge \sqrt{8/3} = 1.633. In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. In terms of the lower semivariance Chebyshev's inequality can be written : \Pr(x \le m - a \sigma_-) \le \frac { 1 } { a^2 }. The American Statistician, 56(3), pp.186-190 When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. ",0.444444444444444 , 135.36,-0.1,0.082,-45,D
+","Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean. Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.89% within three standard deviations for a broad range of different probability distributions. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg.Savage, I. Richard. The second of these inequalities with is the Chebyshev bound. In this setting we can state the following: :General version of Chebyshev's inequality. \forall k > 0: \quad \Pr\left( \|X - \mu\|_\alpha \ge k \sigma_\alpha \right) \le \frac{1}{ k^2 }. However, the benefit of Chebyshev's inequality is that it can be applied more generally to get confidence bounds for ranges of standard deviations that do not depend on the number of samples. ====Semivariances==== An alternative method of obtaining sharper bounds is through the use of semivariances (partial variances). *Grechuk et al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any deviation risk measure in place of standard deviation. : P( | X - \mu | \ge k \sigma ) \le \frac{ 4 }{ 3k^2 } - \frac13 \quad \text{if} \quad k \le \sqrt{8/3}. The bounds are sharp for the following example: for any k ≥ 1, : X = \begin{cases} -1, & \text{with probability }\frac{1}{2k^2} \\\ 0, & \text{with probability }1 - \frac{1}{k^2} \\\ 1, & \text{with probability }\frac{1}{2k^2} \end{cases} For this distribution, the mean μ = 0 and the standard deviation σ = , so : \Pr(|X-\mu| \ge k\sigma) = \Pr(|X| \ge 1) = \frac{1}{k^2}. One way to prove Chebyshev's inequality is to apply Markov's inequality to the random variable with a = (kσ)2: : \Pr(|X - \mu| \geq k\sigma) = \Pr((X - \mu)^2 \geq k^2\sigma^2) \leq \frac{\mathbb{E}[(X - \mu)^2]}{k^2\sigma^2} = \frac{\sigma^2}{k^2\sigma^2} = \frac{1}{k^2}. By comparison, Chebyshev's inequality states that all but a 1/N fraction of the sample will lie within standard deviations of the mean. For k ≥ 1, n > 4 and assuming that the nth moment exists, this bound is tighter than Chebyshev's inequality. Chebyshev's inequality can now be written : \Pr(x \le m - k \sigma) \le \frac { 1 } { k^2 } \frac { \sigma_-^2 } { \sigma^2 }. The additional fraction of 4/9 present in these tail bounds lead to better confidence intervals than Chebyshev's inequality. The Chebyshev inequality for the distribution gives 95% and 99% confidence intervals of approximately ±4.472 standard deviations and ±10 standard deviations respectively. ====Samuelson's inequality==== Although Chebyshev's inequality is the best possible bound for an arbitrary distribution, this is not necessarily true for finite samples. * If 1\le r\le \sqrt{8/3}, the bound is tight when X=r with probability \frac{4}{3r^2}-\frac{1}{3} and is otherwise distributed uniformly in the interval \left[-\frac{r}{2},r\right]. === Specialization to mean and variance === If X has mean \mu and finite, non-zero variance \sigma^2, then taking \alpha=\mu and r=\lambda \sigma gives that for any \lambda > \sqrt{\frac{8}{3}} = 1.63299..., :\operatorname{Pr}(\left|X-\mu\right|\geq \lambda\sigma)\leq\frac{4}{9\lambda^2}. === Proof Sketch === For a relatively elementary proof see.Pukelsheim, F., 1994. The first provides a lower bound for the value of P(x). ==Finite samples== === Univariate case === Saw et al extended Chebyshev's inequality to cases where the population mean and variance are not known and may not exist, but the sample mean and sample standard deviation from N samples are to be employed to bound the expected value of a new drawing from the same distribution. If we put : \sigma_u^2 = \max(\sigma_-^2, \sigma_+^2) , Chebyshev's inequality can be written : \Pr(| x \le m - k \sigma |) \le \frac 1 {k^2} \frac { \sigma_u^2 } { \sigma^2 } . If X is a unimodal distribution with mean μ and variance σ2, then the inequality states that : P( | X - \mu | \ge k \sigma ) \le \frac{ 4 }{ 9k^2 } \quad \text{if} \quad k \ge \sqrt{8/3} = 1.633. In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. In terms of the lower semivariance Chebyshev's inequality can be written : \Pr(x \le m - a \sigma_-) \le \frac { 1 } { a^2 }. The American Statistician, 56(3), pp.186-190 When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. ",0.444444444444444 , 135.36,"""-0.1""",0.082,-45,D
 "5.5-9. Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until ""death.""
 (a) Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours.
-","Additionally, the consistency of the power delivery to the bulb and how little it fluctuates prevents the bulb's filaments from being damaged by dirty power (brown-outs cause damage to electrical systems). ==Other long-lasting light bulbs== ===Second=== The second-longest-lasting light bulb is in Fort Worth, Texas. The Livermore-Pleasanton Fire Department plans to house and maintain the bulb for the rest of its life, regardless of length. While it might seem astonishing that so many longest-lasting light bulbs have been so infrequently turned off, this is the precise reason for their longevity. The bulb has been on ever since, and may in fact have the longest continuous service in the world with other bulbs having interruptions in operation during their existence. === Fourth === The Fourth-longest-lasting light bulb was above the back door of Gasnick Supply, a New York City hardware store on Second Avenue, between 52nd and 53rd Streets. The Mangum Light Bulb burned out on Friday, December 13, 2019. ===Sixth=== The sixth-longest-lasting light bulb was in a washroom at the Martin & Newby Electrical Shop in Ipswich, England. The Centennial Light is the world's longest-lasting light bulb, burning since 1901, and almost never turned off. Another reason for the longevity of bulbs is the size, quality and material of the filament. This indicates that the broken bulb must be one of the last three (B). The store, as well as the entire half-block on which it stood, was razed in 2003. ===Fifth=== The fifth-longest-lasting light bulb was located in a fire house in Mangum, Oklahoma. Due to its longevity, the bulb has been noted by The Guinness Book of World Records,. This is a list of the longest-lasting incandescent light bulbs. ==Longest- lasting light bulb== The world's longest-lasting light bulb is the Centennial Light located at 4550 East Avenue, Livermore, California. Research continued with inoculated canning pack studies that were published by the NCA in 1968. ==Mathematical formulas== Thermal death time can be determined one of two ways: 1) by using graphs or 2) by using mathematical formulas. ===Graphical method=== This is usually expressed in minutes at the temperature of . It burned out in 2001.Martin & Newby Bulb ===Seventh=== The seventh-longest- lasting light bulb is located in the Cinema Napoleón in Río Chico, Miranda, Venezuela. Thermal death time is how long it takes to kill a specific bacterium at a specific temperature. It is titled ""A Million Hours of Service"". thumb|right|130px|The pendant light at Fire Station #6 in which the bulb is installed.|alt=A photo of the pendant light at Fire Station #6 in which the bulb is installed.In 1976, the fire department moved to Fire Station #6 with the bulb; the bulb socket's cord was severed for fear that unscrewing the bulb could damage it. The bulb, known as the Eternal Light, was credited as being the longest-lasting bulb in the 1970 edition of the Guinness Book of World Records, two years before the discovery of the Livermore bulb.Livermore's Centennial Light Guinness Book of World Records The bulb was originally at the Byers Opera House, and was installed by a stage-hand, Barry Burke, on , above the backstage door. thumb|400px|An illustration of the lightbulb problem, where one is searching for a broken bulb among six lightbulbs. The objective is to find the broken bulb using the smallest number of tests (where a test is when some of the bulbs are connected to a power supply). The wagon is now part of a museum, and the light bulb is in use several times per week. ===Third=== The third longest lasting light bulb began operation in 1929-30 when BC Electric's Ruskin Generating Station (British Columbia Canada) commenced service. Dunstan contacted the Guinness Book of World Records, Ripley's Believe It or Not, and General Electric, who all confirmed it as the longest-lasting bulb known in existence. The bulb is cared for by the Centennial Light Bulb Committee, a partnership of the Livermore-Pleasanton Fire Department, Livermore Heritage Guild, Lawrence Livermore National Laboratories, and Sandia National Laboratories. Bulb customer numbers Date Customers (approx.) Source January 2017 15,000 August 2017 100,000 January 2018 200,000 300,000 August 2018 670,000 January 2019 870,000 March 2019 1,130,000 November 2021 1,700,000 ==References== Category:2022 mergers and acquisitions Category:Electric power companies of the United Kingdom Category:Utilities of the United Kingdom Category:Companies based in London Category:British companies established in 2015 Category:Companies that have entered administration in the United Kingdom ",0.0547,27.211,435.0,7,0.3085,E
-"An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.","Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. The probability that the red ball is not taken in the first draw is 1000/2000 = . This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. See Pólya urn model. * binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls. One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. * Mixed replacement/non-replacement: the urn contains black and white balls. (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. We want to calculate the probability that the red ball is not taken. * Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour. In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. * Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector's problem and birthday problem. thumb|Two urns containing white and red balls. The probability that the second ball picked is red depends on whether the first ball was red or white. Here the draws are independent and the probabilities are therefore not multiplied together. * The probability of taking a particular item at a particular draw is equal to its fraction of the total ""weight"" of all items that have not yet been taken at that moment. ",0.3359,0.0625,0.0,1.39,524,B
-" If $P(A)=0.8, P(B)=0.5$, and $P(A \cup B)=0.9$. What is $P(A \cap B)$?","* This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. In this event, the event B can be analyzed by a conditional probability with respect to A. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. Moreover, this ""multiplication rule"" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} ==== As the probability of a conditional event ==== Conditional probability can be defined as the probability of a conditional event A_B. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. For a value in and an event , the conditional probability is given by P(A \mid X=x) . These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. It may be tempting to say that : Pr([""1"" on 1st trial] or [""1"" on second trial] or ... or [""1"" on 8th trial]) := Pr(""1"" on 1st trial) + Pr(""1"" on second trial) + ... + P(""1"" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... Equivalently, the probabilities of an event and its complement must always total to 1. Therefore, the probability of an event's complement must be unity minus the probability of the event. That is, for an event A, :P(A^c) = 1 - P(A). That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A after having accounted for evidence E or after having updated P(A). Similar reasoning can be used to show that P(Ā|B) = etc. ",+107,479,0.9,41.40, 10.7598,C
-"Suppose that the alleles for eye color for a certain male fruit fly are $(R, W)$ and the alleles for eye color for the mating female fruit fly are $(R, W)$, where $R$ and $W$ represent red and white, respectively. Their offspring receive one allele for eye color from each parent. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offspring's eyes look white, what is the conditional probability that it has two white alleles for eye color?","When assessing phenotype from this, ""3"" of the offspring have ""Brown"" eyes and only one offspring has ""green"" eyes. (3 are ""B?"" thumb|Punnett squares for each combination of parents' colour vision status giving probabilities of their offsprings' status, each cell having 25% probability in theory. However, when they crossed a red-eyed male with a white-eyed female, the male offspring had white eyes while the female offspring had red eyes. The probability of an individual offspring's having the genotype BB is 25%, Bb is 50%, and bb is 25%. As every individual has a 50% chance of passing on an allele to the next generation, the formula depends on 0.5 raised to the power of however many generations separate the individual from the common ancestor of its parents, on both the father's side and mother's side. These tables can be used to examine the genotypical outcome probabilities of the offspring of a single trait (allele), or when crossing multiple traits from the parents. Via principles of dominant and recessive alleles, they could then (perhaps after cross-breeding the offspring as well) make an inference as to which sex chromosome contains the gene Z, if either in fact did. ==Reciprocal cross in practice== Given that the trait of interest is either autosomal or sex-linked and follows by either complete dominance or incomplete dominance, a reciprocal cross following two generations will determine the mode of inheritance of the trait. ===White-eye mutation in Drosophila melanogaster=== Sex linkage was first reported by Doncaster and Raynor in 1906Doncaster L and Raynor GH (1906). The ratio 9:3:3:1 is the expected outcome when crossing two double-heterozygous parents with unlinked genes. Mutant Male x Wild- type Female ( X(mut)Y x X(wt)X(wt) ) X (wt) X (wt) X (mut) X (mut) X (wt) Red eye Female X (mut) X (wt) Red eye Female Y X (wt) Y Red eye Male X (wt) Y Red eye Male As shown in Table 1, the male offspring are white-eyed and the female offspring are red-eyed. The Punnett square works, however, only if the genes are independent of each other, which means that having a particular allele of gene ""A"" does not alter the probability of possessing an allele of gene ""B"". In this example, both parents have the genotype Bb. A represents the dominant allele for color (yellow), while a represents the recessive allele (green). He found that a white-eyed male crossed with a red-eyed female produced only red-eyed offspring. and 1 is ""bb"") B b B BB Bb b Bb bb The way in which the B and b alleles interact with each other to affect the appearance of the offspring depends on how the gene products (proteins) interact (see Mendelian inheritance). The reason was that the white eye allele is sex-linked (more specifically, on the X chromosome) and recessive. RA Ra rA ra RA RRAA RRAa RrAA RrAa Ra RRAa RRaa RrAa Rraa rA RrAA RrAa rrAA rrAa ra RrAa Rraa rrAa rraa Since dominant traits mask recessive traits (assuming no epistasis), there are nine combinations that have the phenotype round yellow, three that are round green, three that are wrinkled yellow, and one that is wrinkled green. The female offspring are carrying the mutant white-eye allele X(mut), but do not express it phenotypically because it is recessive. Next, they would cross an A-trait female with a Z-trait male and observe the offspring. As stated above, the phenotypic ratio is expected to be 9:3:3:1 if crossing unlinked genes from two double-heterozygotes. In genetics, a gametic phase represents the original allelic combinations that a diploid individual inherits from both parents. As shown in Table 2, all offspring are Red-eyed. The diagram is used by biologists to determine the probability of an offspring having a particular genotype. ",273,0.25,0.36,0.33333333,7,D
+","Additionally, the consistency of the power delivery to the bulb and how little it fluctuates prevents the bulb's filaments from being damaged by dirty power (brown-outs cause damage to electrical systems). ==Other long-lasting light bulbs== ===Second=== The second-longest-lasting light bulb is in Fort Worth, Texas. The Livermore-Pleasanton Fire Department plans to house and maintain the bulb for the rest of its life, regardless of length. While it might seem astonishing that so many longest-lasting light bulbs have been so infrequently turned off, this is the precise reason for their longevity. The bulb has been on ever since, and may in fact have the longest continuous service in the world with other bulbs having interruptions in operation during their existence. === Fourth === The Fourth-longest-lasting light bulb was above the back door of Gasnick Supply, a New York City hardware store on Second Avenue, between 52nd and 53rd Streets. The Mangum Light Bulb burned out on Friday, December 13, 2019. ===Sixth=== The sixth-longest-lasting light bulb was in a washroom at the Martin & Newby Electrical Shop in Ipswich, England. The Centennial Light is the world's longest-lasting light bulb, burning since 1901, and almost never turned off. Another reason for the longevity of bulbs is the size, quality and material of the filament. This indicates that the broken bulb must be one of the last three (B). The store, as well as the entire half-block on which it stood, was razed in 2003. ===Fifth=== The fifth-longest-lasting light bulb was located in a fire house in Mangum, Oklahoma. Due to its longevity, the bulb has been noted by The Guinness Book of World Records,. This is a list of the longest-lasting incandescent light bulbs. ==Longest- lasting light bulb== The world's longest-lasting light bulb is the Centennial Light located at 4550 East Avenue, Livermore, California. Research continued with inoculated canning pack studies that were published by the NCA in 1968. ==Mathematical formulas== Thermal death time can be determined one of two ways: 1) by using graphs or 2) by using mathematical formulas. ===Graphical method=== This is usually expressed in minutes at the temperature of . It burned out in 2001.Martin & Newby Bulb ===Seventh=== The seventh-longest- lasting light bulb is located in the Cinema Napoleón in Río Chico, Miranda, Venezuela. Thermal death time is how long it takes to kill a specific bacterium at a specific temperature. It is titled ""A Million Hours of Service"". thumb|right|130px|The pendant light at Fire Station #6 in which the bulb is installed.|alt=A photo of the pendant light at Fire Station #6 in which the bulb is installed.In 1976, the fire department moved to Fire Station #6 with the bulb; the bulb socket's cord was severed for fear that unscrewing the bulb could damage it. The bulb, known as the Eternal Light, was credited as being the longest-lasting bulb in the 1970 edition of the Guinness Book of World Records, two years before the discovery of the Livermore bulb.Livermore's Centennial Light Guinness Book of World Records The bulb was originally at the Byers Opera House, and was installed by a stage-hand, Barry Burke, on , above the backstage door. thumb|400px|An illustration of the lightbulb problem, where one is searching for a broken bulb among six lightbulbs. The objective is to find the broken bulb using the smallest number of tests (where a test is when some of the bulbs are connected to a power supply). The wagon is now part of a museum, and the light bulb is in use several times per week. ===Third=== The third longest lasting light bulb began operation in 1929-30 when BC Electric's Ruskin Generating Station (British Columbia Canada) commenced service. Dunstan contacted the Guinness Book of World Records, Ripley's Believe It or Not, and General Electric, who all confirmed it as the longest-lasting bulb known in existence. The bulb is cared for by the Centennial Light Bulb Committee, a partnership of the Livermore-Pleasanton Fire Department, Livermore Heritage Guild, Lawrence Livermore National Laboratories, and Sandia National Laboratories. Bulb customer numbers Date Customers (approx.) Source January 2017 15,000 August 2017 100,000 January 2018 200,000 300,000 August 2018 670,000 January 2019 870,000 March 2019 1,130,000 November 2021 1,700,000 ==References== Category:2022 mergers and acquisitions Category:Electric power companies of the United Kingdom Category:Utilities of the United Kingdom Category:Companies based in London Category:British companies established in 2015 Category:Companies that have entered administration in the United Kingdom ",0.0547,27.211,"""435.0""",7,0.3085,E
+"An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.","Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. The probability that the red ball is not taken in the first draw is 1000/2000 = . This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. See Pólya urn model. * binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls. One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. * Mixed replacement/non-replacement: the urn contains black and white balls. (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. We want to calculate the probability that the red ball is not taken. * Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour. In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. * Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector's problem and birthday problem. thumb|Two urns containing white and red balls. The probability that the second ball picked is red depends on whether the first ball was red or white. Here the draws are independent and the probabilities are therefore not multiplied together. * The probability of taking a particular item at a particular draw is equal to its fraction of the total ""weight"" of all items that have not yet been taken at that moment. ",0.3359,0.0625,"""0.0""",1.39,524,B
+" If $P(A)=0.8, P(B)=0.5$, and $P(A \cup B)=0.9$. What is $P(A \cap B)$?","* This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. In this event, the event B can be analyzed by a conditional probability with respect to A. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. Moreover, this ""multiplication rule"" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} ==== As the probability of a conditional event ==== Conditional probability can be defined as the probability of a conditional event A_B. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. For a value in and an event , the conditional probability is given by P(A \mid X=x) . These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. It may be tempting to say that : Pr([""1"" on 1st trial] or [""1"" on second trial] or ... or [""1"" on 8th trial]) := Pr(""1"" on 1st trial) + Pr(""1"" on second trial) + ... + P(""1"" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... Equivalently, the probabilities of an event and its complement must always total to 1. Therefore, the probability of an event's complement must be unity minus the probability of the event. That is, for an event A, :P(A^c) = 1 - P(A). That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A after having accounted for evidence E or after having updated P(A). Similar reasoning can be used to show that P(Ā|B) = etc. ",+107,479,"""0.9""",41.40, 10.7598,C
+"Suppose that the alleles for eye color for a certain male fruit fly are $(R, W)$ and the alleles for eye color for the mating female fruit fly are $(R, W)$, where $R$ and $W$ represent red and white, respectively. Their offspring receive one allele for eye color from each parent. Assume that each of the four possible outcomes has equal probability. If an offspring ends up with either two white alleles or one red and one white allele for eye color, its eyes will look white. Given that an offspring's eyes look white, what is the conditional probability that it has two white alleles for eye color?","When assessing phenotype from this, ""3"" of the offspring have ""Brown"" eyes and only one offspring has ""green"" eyes. (3 are ""B?"" thumb|Punnett squares for each combination of parents' colour vision status giving probabilities of their offsprings' status, each cell having 25% probability in theory. However, when they crossed a red-eyed male with a white-eyed female, the male offspring had white eyes while the female offspring had red eyes. The probability of an individual offspring's having the genotype BB is 25%, Bb is 50%, and bb is 25%. As every individual has a 50% chance of passing on an allele to the next generation, the formula depends on 0.5 raised to the power of however many generations separate the individual from the common ancestor of its parents, on both the father's side and mother's side. These tables can be used to examine the genotypical outcome probabilities of the offspring of a single trait (allele), or when crossing multiple traits from the parents. Via principles of dominant and recessive alleles, they could then (perhaps after cross-breeding the offspring as well) make an inference as to which sex chromosome contains the gene Z, if either in fact did. ==Reciprocal cross in practice== Given that the trait of interest is either autosomal or sex-linked and follows by either complete dominance or incomplete dominance, a reciprocal cross following two generations will determine the mode of inheritance of the trait. ===White-eye mutation in Drosophila melanogaster=== Sex linkage was first reported by Doncaster and Raynor in 1906Doncaster L and Raynor GH (1906). The ratio 9:3:3:1 is the expected outcome when crossing two double-heterozygous parents with unlinked genes. Mutant Male x Wild- type Female ( X(mut)Y x X(wt)X(wt) ) X (wt) X (wt) X (mut) X (mut) X (wt) Red eye Female X (mut) X (wt) Red eye Female Y X (wt) Y Red eye Male X (wt) Y Red eye Male As shown in Table 1, the male offspring are white-eyed and the female offspring are red-eyed. The Punnett square works, however, only if the genes are independent of each other, which means that having a particular allele of gene ""A"" does not alter the probability of possessing an allele of gene ""B"". In this example, both parents have the genotype Bb. A represents the dominant allele for color (yellow), while a represents the recessive allele (green). He found that a white-eyed male crossed with a red-eyed female produced only red-eyed offspring. and 1 is ""bb"") B b B BB Bb b Bb bb The way in which the B and b alleles interact with each other to affect the appearance of the offspring depends on how the gene products (proteins) interact (see Mendelian inheritance). The reason was that the white eye allele is sex-linked (more specifically, on the X chromosome) and recessive. RA Ra rA ra RA RRAA RRAa RrAA RrAa Ra RRAa RRaa RrAa Rraa rA RrAA RrAa rrAA rrAa ra RrAa Rraa rrAa rraa Since dominant traits mask recessive traits (assuming no epistasis), there are nine combinations that have the phenotype round yellow, three that are round green, three that are wrinkled yellow, and one that is wrinkled green. The female offspring are carrying the mutant white-eye allele X(mut), but do not express it phenotypically because it is recessive. Next, they would cross an A-trait female with a Z-trait male and observe the offspring. As stated above, the phenotypic ratio is expected to be 9:3:3:1 if crossing unlinked genes from two double-heterozygotes. In genetics, a gametic phase represents the original allelic combinations that a diploid individual inherits from both parents. As shown in Table 2, all offspring are Red-eyed. The diagram is used by biologists to determine the probability of an offspring having a particular genotype. ",273,0.25,"""0.36""",0.33333333,7,D
 "Consider the trial on which a 3 is first observed in successive rolls of a six-sided die. Let $A$ be the event that 3 is observed on the first trial. Let $B$ be the event that at least two trials are required to observe a 3 . Assuming that each side has probability $1 / 6$, find (a) $P(A)$.
-","So the likelihood of B beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Die C beats D two-thirds of the time but beats B only one-third of the time. So the likelihood of A beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {4 \over 9} \right) = {13 \over 27} Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. With the second set of dice, die C′ will win with the lowest probability () and dice A′ and B′ will each win with a probability of . ==Variations== ===Efron's dice=== Efron's dice are a set of four intransitive dice invented by Bradley Efron. thumb|320px|Representation of Efron's dice The four dice A, B, C, D have the following numbers on their six faces: * A: 4, 4, 4, 4, 0, 0 * B: 3, 3, 3, 3, 3, 3 * C: 6, 6, 2, 2, 2, 2 * D: 5, 5, 5, 1, 1, 1 ====Probabilities==== Each die is beaten by the previous die in the list, with a probability of : :P(A>B) = P(B>C) = P(C>D) = P(D>A) = {2 \over 3} B's value is constant; A beats it on rolls because four of its six faces are higher. P(C>D) can be calculated by summing conditional probabilities for two events: * C rolls 6 (probability ); wins regardless of D (probability 1) * C rolls 2 (probability ); wins only if D rolls 1 (probability ) The total probability of win for C is therefore :\left( {1 \over 3}\times1 \right) + \left( {2 \over 3}\times{1 \over 2} \right) = {2 \over 3} With a similar calculation, the probability of D winning over A is :\left( {1 \over 2}\times1 \right) + \left( {1 \over 2}\times{1 \over 3} \right) = {2 \over 3} ====Best overall die==== The four dice have unequal probabilities of beating a die chosen at random from the remaining three: As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. Consider a set of three dice, III, IV and V such that * die III has sides 1, 2, 5, 6, 7, 9 * die IV has sides 1, 3, 4, 5, 8, 9 * die V has sides 2, 3, 4, 6, 7, 8 Then: * the probability that III rolls a higher number than IV is * the probability that IV rolls a higher number than V is * the probability that V rolls a higher number than III is === Three-dice set with minimal alterations to standard dice === The following intransitive dice have only a few differences compared to 1 through 6 standard dice: * as with standard dice, the total number of pips is always 21 * as with standard dice, the sides only carry pip numbers between 1 and 6 * faces with the same number of pips occur a maximum of twice per dice * only two sides on each die have numbers different from standard dice: ** A: 1, 1, 3, 5, 5, 6 ** B: 2, 3, 3, 4, 4, 5 ** C: 1, 2, 2, 4, 6, 6 Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. So the likelihood of C beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {5 \over 9} \right) = {14 \over 27} Finally, die D beats A two-thirds of the time but beats C only one-third of the time. The probability of die D beating B is (only when D rolls 5). He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. The probability of die B beating D is (only when D rolls 1). Die 1 1 4 Die 2 2 3 === Three players === An optimal and permutation- fair solution for 3 six-sided dice was found by Robert Ford in 2010. Player 1 chooses die A Player 2 chooses die C Player 1 chooses die B Player 2 chooses die A Player 1 chooses die C Player 2 chooses die B 2 4 9 1 6 8 3 5 7 3 C A A 2 A B B 1 C C C 5 C C A 4 A B B 6 B B C 7 C C A 9 A A A 8 B B B == Comment regarding the equivalency of intransitive dice == Though the three intransitive dice A, B, C (first set of dice) * A: 2, 2, 6, 6, 7, 7 * B: 1, 1, 5, 5, 9, 9 * C: 3, 3, 4, 4, 8, 8 P(A > B) = P(B > C) = P(C > A) = and the three intransitive dice A′, B′, C′ (second set of dice) * A′: 2, 2, 4, 4, 9, 9 * B′: 1, 1, 6, 6, 8, 8 * C′: 3, 3, 5, 5, 7, 7 P(A′ > B′) = P(B′ > C′) = P(C′ > A′) = win against each other with equal probability they are not equivalent. ;Set 2: * A: 3, 3, 3, 6 * B: 2, 2, 5, 5 * C: 1, 4, 4, 4 P(A > B) = P(B > C) = , P(C > A) = 9/16 == Intransitive 12-sided dice == In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. The following tables show all possible outcomes for all three pairs of dice. So the likelihood of D beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Therefore, the best overall die is C with a probability of winning of 0.5185. With the first set of dice, die B will win with the highest probability () and dice A and C will each win with a probability of . The probability of die C beating A is . This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem. ==Generalizations== A natural generalization of the problem is to consider n non-necessarily fair dice, with p the probability that each die will select the 6 face when thrown (notice that actually the number of faces of the dice and which face should be selected are irrelevant). The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability that C rolls higher than A are all , so this set of dice is intransitive. With adjacent pairs, one die's probability of winning is 2/3. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. The probability of die A beating C is (A must roll 4 and C must roll 2). ", 35.91,2.19,0.166666666,2,0.4908,C
-"An urn contains four balls numbered 1 through 4 . The balls are selected one at a time without replacement. A match occurs if the ball numbered $m$ is the $m$ th ball selected. Let the event $A_i$ denote a match on the $i$ th draw, $i=1,2,3,4$. Extend this exercise so that there are $n$ balls in the urn. What is the limit of this probability as $n$ increases without bound?","The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. * Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector's problem and birthday problem. In other words, the probability of not taking a very heavy ball in n draws falls almost exponentially with n in Wallenius' model. Hence, the number of total balls in the urn grows. The probability that the red ball is not taken in the first draw is 1000/2000 = . This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). And the weight of the competing balls depends on the outcomes of all preceding draws. (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . See Pólya urn model. * binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls. In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. What is the distribution of the number of black balls drawn after m draws? * multinomial distribution: there are balls of more than two colors. Hence, the number of total marbles in the urn decreases. The probability of not taking the heavy red ball in Fisher's model is approximately 1/(n + 1). * The probability of taking a particular item at a particular draw is equal to its fraction of the total ""weight"" of all items that have not yet been taken at that moment. ",0.5,425,773.0,1,0.6321205588,E
+","So the likelihood of B beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Die C beats D two-thirds of the time but beats B only one-third of the time. So the likelihood of A beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {4 \over 9} \right) = {13 \over 27} Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. With the second set of dice, die C′ will win with the lowest probability () and dice A′ and B′ will each win with a probability of . ==Variations== ===Efron's dice=== Efron's dice are a set of four intransitive dice invented by Bradley Efron. thumb|320px|Representation of Efron's dice The four dice A, B, C, D have the following numbers on their six faces: * A: 4, 4, 4, 4, 0, 0 * B: 3, 3, 3, 3, 3, 3 * C: 6, 6, 2, 2, 2, 2 * D: 5, 5, 5, 1, 1, 1 ====Probabilities==== Each die is beaten by the previous die in the list, with a probability of : :P(A>B) = P(B>C) = P(C>D) = P(D>A) = {2 \over 3} B's value is constant; A beats it on rolls because four of its six faces are higher. P(C>D) can be calculated by summing conditional probabilities for two events: * C rolls 6 (probability ); wins regardless of D (probability 1) * C rolls 2 (probability ); wins only if D rolls 1 (probability ) The total probability of win for C is therefore :\left( {1 \over 3}\times1 \right) + \left( {2 \over 3}\times{1 \over 2} \right) = {2 \over 3} With a similar calculation, the probability of D winning over A is :\left( {1 \over 2}\times1 \right) + \left( {1 \over 2}\times{1 \over 3} \right) = {2 \over 3} ====Best overall die==== The four dice have unequal probabilities of beating a die chosen at random from the remaining three: As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. Consider a set of three dice, III, IV and V such that * die III has sides 1, 2, 5, 6, 7, 9 * die IV has sides 1, 3, 4, 5, 8, 9 * die V has sides 2, 3, 4, 6, 7, 8 Then: * the probability that III rolls a higher number than IV is * the probability that IV rolls a higher number than V is * the probability that V rolls a higher number than III is === Three-dice set with minimal alterations to standard dice === The following intransitive dice have only a few differences compared to 1 through 6 standard dice: * as with standard dice, the total number of pips is always 21 * as with standard dice, the sides only carry pip numbers between 1 and 6 * faces with the same number of pips occur a maximum of twice per dice * only two sides on each die have numbers different from standard dice: ** A: 1, 1, 3, 5, 5, 6 ** B: 2, 3, 3, 4, 4, 5 ** C: 1, 2, 2, 4, 6, 6 Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. So the likelihood of C beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {5 \over 9} \right) = {14 \over 27} Finally, die D beats A two-thirds of the time but beats C only one-third of the time. The probability of die D beating B is (only when D rolls 5). He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. The probability of die B beating D is (only when D rolls 1). Die 1 1 4 Die 2 2 3 === Three players === An optimal and permutation- fair solution for 3 six-sided dice was found by Robert Ford in 2010. Player 1 chooses die A Player 2 chooses die C Player 1 chooses die B Player 2 chooses die A Player 1 chooses die C Player 2 chooses die B 2 4 9 1 6 8 3 5 7 3 C A A 2 A B B 1 C C C 5 C C A 4 A B B 6 B B C 7 C C A 9 A A A 8 B B B == Comment regarding the equivalency of intransitive dice == Though the three intransitive dice A, B, C (first set of dice) * A: 2, 2, 6, 6, 7, 7 * B: 1, 1, 5, 5, 9, 9 * C: 3, 3, 4, 4, 8, 8 P(A > B) = P(B > C) = P(C > A) = and the three intransitive dice A′, B′, C′ (second set of dice) * A′: 2, 2, 4, 4, 9, 9 * B′: 1, 1, 6, 6, 8, 8 * C′: 3, 3, 5, 5, 7, 7 P(A′ > B′) = P(B′ > C′) = P(C′ > A′) = win against each other with equal probability they are not equivalent. ;Set 2: * A: 3, 3, 3, 6 * B: 2, 2, 5, 5 * C: 1, 4, 4, 4 P(A > B) = P(B > C) = , P(C > A) = 9/16 == Intransitive 12-sided dice == In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. The following tables show all possible outcomes for all three pairs of dice. So the likelihood of D beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Therefore, the best overall die is C with a probability of winning of 0.5185. With the first set of dice, die B will win with the highest probability () and dice A and C will each win with a probability of . The probability of die C beating A is . This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem. ==Generalizations== A natural generalization of the problem is to consider n non-necessarily fair dice, with p the probability that each die will select the 6 face when thrown (notice that actually the number of faces of the dice and which face should be selected are irrelevant). The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability that C rolls higher than A are all , so this set of dice is intransitive. With adjacent pairs, one die's probability of winning is 2/3. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. The probability of die A beating C is (A must roll 4 and C must roll 2). ", 35.91,2.19,"""0.166666666""",2,0.4908,C
+"An urn contains four balls numbered 1 through 4 . The balls are selected one at a time without replacement. A match occurs if the ball numbered $m$ is the $m$ th ball selected. Let the event $A_i$ denote a match on the $i$ th draw, $i=1,2,3,4$. Extend this exercise so that there are $n$ balls in the urn. What is the limit of this probability as $n$ increases without bound?","The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. * Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector's problem and birthday problem. In other words, the probability of not taking a very heavy ball in n draws falls almost exponentially with n in Wallenius' model. Hence, the number of total balls in the urn grows. The probability that the red ball is not taken in the first draw is 1000/2000 = . This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). And the weight of the competing balls depends on the outcomes of all preceding draws. (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . See Pólya urn model. * binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls. In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. What is the distribution of the number of black balls drawn after m draws? * multinomial distribution: there are balls of more than two colors. Hence, the number of total marbles in the urn decreases. The probability of not taking the heavy red ball in Fisher's model is approximately 1/(n + 1). * The probability of taking a particular item at a particular draw is equal to its fraction of the total ""weight"" of all items that have not yet been taken at that moment. ",0.5,425,"""773.0""",1,0.6321205588,E
 " Of a group of patients having injuries, $28 \%$ visit both a physical therapist and a chiropractor and $8 \%$ visit neither. Say that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by $16 \%$. What is the probability of a randomly selected person from this group visiting a physical therapist?
-","Finally, the principle of conditional probability implies that is equal to the product of these individual probabilities: The terms of equation () can be collected to arrive at: Evaluating equation () gives Therefore, (50.7297%). Further results showed that psychology students and women did better on the task than casino visitors/personnel or men, but were less confident about their estimates. ===Reverse problem=== The reverse problem is to find, for a fixed probability , the greatest for which the probability is smaller than the given , or the smallest for which the probability is greater than the given . Consequently, the desired probability is . The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people . ===Probability of a shared birthday (collision)=== The birthday problem can be generalized as follows: :Given random integers drawn from a discrete uniform distribution with range , what is the probability that at least two numbers are the same? ( gives the usual birthday problem.) In the standard case of , substituting gives about 6.1%, which is less than 1 chance in 16. The following table shows the probability for some other values of (for this table, the existence of leap years is ignored, and each birthday is assumed to be equally likely): thumb|right|upright=1.4|The probability that no two people share a birthday in a group of people. This is a list of people in the chiropractic profession, comprising chiropractors and other people who have been notably connected with the profession. thumb|upright=1.3|The computed probability of at least two people sharing a birthday versus the number of people In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. Then, because and are the only two possibilities and are also mutually exclusive, Here is the calculation of for 23 people. The answer is 20—if there is a prize for first match, the best position in line is 20th. ===Same birthday as you=== thumb|right|upright=1.4|Comparing = probability of a birthday match with = probability of matching your birthday In the birthday problem, neither of the two people is chosen in advance. For example, the usual 50% probability value is realized for both a 32-member group of 16 men and 16 women and a 49-member group of 43 women and 6 men. ==Other birthday problems== ===First match=== A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? Where the event is the probability of finding a group of 23 people with at least two people sharing same birthday, . is the ratio of the total number of birthdays, V_{nr}, without repetitions and order matters (e.g. for a group of 2 people, mm/dd birthday format, one possible outcome is \left \\{ \left \\{01/02,05/20\right \\},\left \\{05/20,01/02\right \\},\left \\{10/02,08/04\right\\},...\right \\} divided by the total number of birthdays with repetition and order matters, V_{t}, as it is the total space of outcomes from the experiment (e.g. 2 people, one possible outcome is \left \\{ \left \\{01/02,01/02\right \\},\left \\{10/02,08/04\right \\},...\right \\}. This number is significantly higher than : the reason is that it is likely that there are some birthday matches among the other people in the room. === Number of people with a shared birthday === For any one person in a group of n people the probability that he or she shares his birthday with someone else is q(n-1;d) , as explained above. *List Chiropractors Category:Chiropractic And for the group of 23 people, the probability of sharing is :p(23) \approx 1 - \left(\frac{364}{365}\right)^\binom{23}{2} = 1 - \left(\frac{364}{365}\right)^{253} \approx 0.500477 . ===Poisson approximation=== Applying the Poisson approximation for the binomial on the group of 23 people, :\operatorname{Poi}\left(\frac{\binom{23}{2}}{365}\right) =\operatorname{Poi}\left(\frac{253}{365}\right) \approx \operatorname{Poi}(0.6932) so :\Pr(X>0)=1-\Pr(X=0) \approx 1-e^{-0.6932} \approx 1-0.499998=0.500002. Probability in the Engineering and Informational Sciences is an international journal published by Cambridge University Press. Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. ==Scope== Much research involving probability is done under the auspices of applied probability. In short can be multiplied by itself times, which gives us :\bar p(n) \approx \left(\frac{364}{365}\right)^\binom{n}{2}. Therefore, its probability is : p(n) = 1 - \bar p(n). The formula :n(d)=\left\lceil \sqrt{2d\ln2}+\frac{3-2\ln2}{6}+\frac{9-4(\ln2)^2}{72\sqrt{2d\ln2}}-\frac{2(\ln2)^2}{135d}\right\rceil holds for all , and it is conjectured that this formula holds for all . ===More than two people sharing a birthday=== It is possible to extend the problem to ask how many people in a group are necessary for there to be a greater than 50% probability that at least 3, 4, 5, etc. of the group share the same birthday. Note that the vertical scale is logarithmic (each step down is 1020 times less likely). : 1 0.0% 5 2.7% 10 11.7% 20 41.1% 23 50.7% 30 70.6% 40 89.1% 50 97.0% 60 99.4% 70 99.9% 75 99.97% 100 % 200 % 300 (100 − )% 350 (100 − )% 365 (100 − )% ≥ 366 100% ==Approximations== thumb|right|upright=1.4|Graphs showing the approximate probabilities of at least two people sharing a birthday () and its complementary event () thumb|right|upright=1.4|A graph showing the accuracy of the approximation () The Taylor series expansion of the exponential function (the constant ) : e^x = 1 + x + \frac{x^2}{2!}+\cdots provides a first-order approximation for for |x| \ll 1: : e^x \approx 1 + x. The birthday problem has been generalized to consider an arbitrary number of types.M. C. Wendl (2003) Collision Probability Between Sets of Random Variables, Statistics and Probability Letters 64(3), 249–254. ",35.64,0.68,8.87,0.166666666,0.1353,B
-"A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or irregular and finds that 16\% have high blood pressure; (b) 19\% have low blood pressure; (c) $17 \%$ have an irregular heartbeat; (d) of those with an irregular heartbeat, $35 \%$ have high blood pressure; and (e) of those with normal blood pressure, $11 \%$ have an irregular heartbeat. What percentage of her patients have a regular heartbeat and low blood pressure?","Clinicians consider a pulse pressure of 60 mmHg to likely be associated with diseases, with a pulse pressure of 50 mmHg or more increasing the risk of cardiovascular disease. == Calculation == Pulse pressure is calculated as the difference between the systolic blood pressure and the diastolic blood pressure. It is measured by right heart catheterization or may be estimated by transthoracic echocardiography Normal pulmonary artery pressure is between 8mmHg -20 mm Hg at rest. : e.g. normal: 15mmHg - 8mmHg = 7mmHg : high: 25mmHg - 10mmHg = 15mmHg ==Values and variation== ===Low (narrow) pulse pressure === A pulse pressure is considered abnormally low if it is less than 25% of the systolic value. Many studies further indicate a J-shaped relationship between blood pressure and mortality, whereby both very high and very low levels are associated with notable increases in mortality. However, pulse pressure has usually been found to be a stronger independent predictor of cardiovascular events, especially in older populations, than has systolic, diastolic, or mean arterial pressure. The systemic pulse pressure is approximately proportional to stroke volume, or the amount of blood ejected from the left ventricle during systole (pump action) and inversely proportional to the compliance (similar to elasticity) of the aorta. * Systemic pulse pressure (usually measured at upper arm artery) = Psystolic \- Pdiastolic :e.g. normal 120mmHg - 80mmHg = 40mmHg : low: 107mmHg - 80mmHg = 27mmHg : high: 160mmHg - 80mmHg = 80mmHg * Pulmonary pulse pressure is normally much lower than systemic blood pressure due to the higher compliance of the pulmonary system compared to the arterial circulation. Pulse pressure is the difference between systolic and diastolic blood pressure. Proportion can be written as \frac{a}{b}=\frac{c}{d}, where ratios are expressed as fractions. Readings greater than or equal to 130/80 mm Hg are considered hypertension by ACC/AHA and if greater than or equal to 140/90 mm Hg by ESC/ESH. Clonidine (decrease of 6.3 mm Hg), diltiazem (decrease of 5.5 mm Hg), and prazosin (decrease of 5.0 mm Hg) were intermediate. == See also == * Mean arterial pressure * Cold pressor test * Hypertension * Prehypertension * Antihypertensive * Patent ductus arteriosus == References == Category:Medical signs Category:Cardiovascular physiology Heart is a biweekly peer-reviewed medical journal covering all areas of cardiovascular medicine and surgery. Thus, blood pressures above normal can go undiagnosed for a long period of time. ==Causes== Elevated blood pressure develops gradually over many years usually without a specific identifiable cause. Normal pulse pressure is around 40 mmHg. If the aorta becomes rigid because of disorders, such as arteriosclerosis or atherosclerosis, the pulse pressure would be high due to less compliance of the aorta. The ACC/AHA define elevated blood pressure as readings with a systolic pressure from 120 to 129 mm Hg and a diastolic pressure under 80 mm Hg, and the European Society of Cardiology and European Society of Hypertension (ESC/ESH) define ""high normal blood pressure"" as readings with a systolic pressure from 130 to 139 mm Hg and a diastolic pressure 85-89 mm Hg. On the other hand, the National Heart, Lung, and Blood Institute suggests that people with prehypertension are at a higher risk for developing hypertension, or high blood pressure, compared to people with normal blood pressure.National Heart, Lung and Blood Institute<> A 2014 meta-analysis concluded that prehypertension increases the risk of stroke, and that even low-range prehypertension significantly increases stroke risk and a 2019 meta-analysis found elevated blood pressure increases the risk of heart attack by 86% and stroke by 66%. ==Epidemiology== Data from the 1999 and 2000 National Health and Nutrition Examination Survey (NHANES III) estimated that the prevalence of prehypertension among adults in the United States was approximately 31 percent and decreased to 28 percent in the 2011–2012 National Health and Nutrition Examination Survey. If the usual resting pulse pressure is consistently greater than 100 mmHg, potential factors are stiffness of the major arteries, aortic regurgitation (a leak in the aortic valve), or arteriovenous malformation, among others. Such a proportion is known as geometrical proportion, not to be confused with arithmetical proportion and harmonic proportion. ==Properties of proportions== * Fundamental rule of proportion. The prevalence was higher among men than women. === Risk factors === A primary risk factor for prehypertension is being overweight. This suggests that interventions that lower diastolic pressure without also lowering systolic pressure (and thus lowering pulse pressure) could actually be counterproductive. The most common cause of a low (narrow) pulse pressure is a drop in left ventricular stroke volume. High blood pressure that develops over time without a specific cause is considered benign or essential hypertension. ",3.51,72,2.3,1.7,15.1,E
-"Roll a fair six-sided die three times. Let $A_1=$ $\{1$ or 2 on the first roll $\}, A_2=\{3$ or 4 on the second roll $\}$, and $A_3=\{5$ or 6 on the third roll $\}$. It is given that $P\left(A_i\right)=1 / 3, i=1,2,3 ; P\left(A_i \cap A_j\right)=(1 / 3)^2, i \neq j$; and $P\left(A_1 \cap A_2 \cap A_3\right)=(1 / 3)^3$. Use Theorem 1.1-6 to find $P\left(A_1 \cup A_2 \cup A_3\right)$.","Consider a set of three dice, III, IV and V such that * die III has sides 1, 2, 5, 6, 7, 9 * die IV has sides 1, 3, 4, 5, 8, 9 * die V has sides 2, 3, 4, 6, 7, 8 Then: * the probability that III rolls a higher number than IV is * the probability that IV rolls a higher number than V is * the probability that V rolls a higher number than III is === Three-dice set with minimal alterations to standard dice === The following intransitive dice have only a few differences compared to 1 through 6 standard dice: * as with standard dice, the total number of pips is always 21 * as with standard dice, the sides only carry pip numbers between 1 and 6 * faces with the same number of pips occur a maximum of twice per dice * only two sides on each die have numbers different from standard dice: ** A: 1, 1, 3, 5, 5, 6 ** B: 2, 3, 3, 4, 4, 5 ** C: 1, 2, 2, 4, 6, 6 Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. P(C>D) can be calculated by summing conditional probabilities for two events: * C rolls 6 (probability ); wins regardless of D (probability 1) * C rolls 2 (probability ); wins only if D rolls 1 (probability ) The total probability of win for C is therefore :\left( {1 \over 3}\times1 \right) + \left( {2 \over 3}\times{1 \over 2} \right) = {2 \over 3} With a similar calculation, the probability of D winning over A is :\left( {1 \over 2}\times1 \right) + \left( {1 \over 2}\times{1 \over 3} \right) = {2 \over 3} ====Best overall die==== The four dice have unequal probabilities of beating a die chosen at random from the remaining three: As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. The following tables show all possible outcomes for all three pairs of dice. With the second set of dice, die C′ will win with the lowest probability () and dice A′ and B′ will each win with a probability of . ==Variations== ===Efron's dice=== Efron's dice are a set of four intransitive dice invented by Bradley Efron. thumb|320px|Representation of Efron's dice The four dice A, B, C, D have the following numbers on their six faces: * A: 4, 4, 4, 4, 0, 0 * B: 3, 3, 3, 3, 3, 3 * C: 6, 6, 2, 2, 2, 2 * D: 5, 5, 5, 1, 1, 1 ====Probabilities==== Each die is beaten by the previous die in the list, with a probability of : :P(A>B) = P(B>C) = P(C>D) = P(D>A) = {2 \over 3} B's value is constant; A beats it on rolls because four of its six faces are higher. Player 1 chooses die A Player 2 chooses die C Player 1 chooses die B Player 2 chooses die A Player 1 chooses die C Player 2 chooses die B 2 4 9 1 6 8 3 5 7 3 C A A 2 A B B 1 C C C 5 C C A 4 A B B 6 B B C 7 C C A 9 A A A 8 B B B == Comment regarding the equivalency of intransitive dice == Though the three intransitive dice A, B, C (first set of dice) * A: 2, 2, 6, 6, 7, 7 * B: 1, 1, 5, 5, 9, 9 * C: 3, 3, 4, 4, 8, 8 P(A > B) = P(B > C) = P(C > A) = and the three intransitive dice A′, B′, C′ (second set of dice) * A′: 2, 2, 4, 4, 9, 9 * B′: 1, 1, 6, 6, 8, 8 * C′: 3, 3, 5, 5, 7, 7 P(A′ > B′) = P(B′ > C′) = P(C′ > A′) = win against each other with equal probability they are not equivalent. Rolling the three dice of a set and always using the highest score for evaluation will show a different winning pattern for the two sets of dice. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. The 2003 A3 Champions Cup was first edition of A3 Champions Cup. Consider the following set of dice. With adjacent pairs, one die's probability of winning is 2/3. So the likelihood of B beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Die C beats D two-thirds of the time but beats B only one-third of the time. ;Set 2: * A: 3, 3, 3, 6 * B: 2, 2, 5, 5 * C: 1, 4, 4, 4 P(A > B) = P(B > C) = , P(C > A) = 9/16 == Intransitive 12-sided dice == In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice. The 2005 A3 Champions Cup was third edition of A3 Champions Cup. The 2004 A3 Champions Cup was second edition of A3 Champions Cup. So the likelihood of A beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {4 \over 9} \right) = {13 \over 27} Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. A set of dice is intransitive (or nontransitive) if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. So the likelihood of D beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Therefore, the best overall die is C with a probability of winning of 0.5185. Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both (as long as the player is then allowed to choose between the one-die option and the two-die option): : Sets chosen by opponents Winning set of dice Type Number A B E 1 A C E 2 A D C 2 A E D 1 B C A 1 B D A 2 B E D 2 C D B 1 C E B 2 D E C 1 There are two major issues with this set, however. So the likelihood of C beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {5 \over 9} \right) = {14 \over 27} Finally, die D beats A two-thirds of the time but beats C only one-third of the time. With the first set of dice, die B will win with the highest probability () and dice A and C will each win with a probability of . Awarded by Mathematical Association of America * Timothy Gowers' project on intransitive dice * Category:Probability theory paradoxes Category:Dice The probability of die C beating A is . ",-0.40864,0.6296296296,-3.141592,2.6,3,B
-Let $A$ and $B$ be independent events with $P(A)=$ $1 / 4$ and $P(B)=2 / 3$. Compute $P(A \cap B)$,"* This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. In this event, the event B can be analyzed by a conditional probability with respect to A. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))��the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios All events that are not in B will have null probability in the new distribution. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. This particular method relies on event B occurring with some sort of relationship with another event A. It can be interpreted as ""the probability of B occurring multiplied by the probability of A occurring, provided that B has occurred, is equal to the probability of the A and B occurrences together, although not necessarily occurring at the same time"". Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. That is, for an event A, :P(A^c) = 1 - P(A). Moreover, this ""multiplication rule"" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} ==== As the probability of a conditional event ==== Conditional probability can be defined as the probability of a conditional event A_B. If statistically independent If mutually exclusive P(A\mid B)= P(A) 0 P(B\mid A)= P(B) 0 P(A \cap B)= P(A) P(B) 0 In fact, mutually exclusive events cannot be statistically independent (unless both of them are impossible), since knowing that one occurs gives information about the other (in particular, that the latter will certainly not occur). == Common fallacies == :These fallacies should not be confused with Robert K. Shope's 1978 ""conditional fallacy"", which deals with counterfactual examples that beg the question. === Assuming conditional probability is of similar size to its inverse === thumb|450x450px|A geometric visualization of Bayes' theorem. The former is required by the axioms of probability, and the latter stems from the fact that the new probability measure has to be the analog of P in which the probability of B is one - and every event that is not in B, therefore, has a null probability. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . ",0.396,0.166666666,102.0,7.00,8.3147,B
-How many four-letter code words are possible using the letters in IOWA if The letters may not be repeated?,"The state of Iowa is covered by five area codes. None of the Iowa codes are expected to need relief in the immediate future. * 319: Cedar Rapids, Waterloo, Iowa City, and Cedar Falls (original area code created in 1947) * 515: Des Moines, Ames, West Des Moines, Urbandale and Fort Dodge (original area code created in 1947) * 563: Davenport, Dubuque, Bettendorf, Clinton, Muscatine (split from 319 in 2001) * 641: Mason City, Marshalltown, Ottumwa, Tama (split from 515 in 2000) * 712: Sioux City, Council Bluffs (original area code created in 1947) ==See also== *State of Iowa Area codes Iowa thumb|right|The area codes of Kentucky: 270 and 364 in light green. The Code of Iowa contains the statutory laws of the U.S. state of Iowa. 734 is an area code in the North American Numbering Plan. ""Splitting the area code in two avoids ten digit dialing but requires changing all current area code 270 numbers within the new area code to 364. It is republished in full every odd year, and is supplemented in even years. ==External reference== *Iowa Code online at Iowa General Assembly. Such codes are half the size of two-part codes but are more vulnerable since an attacker who recovers some code word meanings can often infer the meaning of nearby code words. thumb|Page 187 of the State Department 1899 code book, a one part code with a choice of code word or numeric ciphertext. Area code 270 is a telephone area code in the North American Numbering Plan (NANP) for the Commonwealth of Kentucky's western and south central counties. On June 13, 2007, the PSC announced that the new area code will be 364, but also announced that the previously announced implementation would be delayed in favor of number conservation measures including expanded number pooling. Planning for the introduction of a second area code for the region, area code 364, was assigned in 2007. These subsequent delays of the implementation of the 270 / 364 area code split were due to further use of number conservation measures, including mandatory and expanded number pooling, as well as a weakened economy and a reduced usage of telephone numbers dedicated for use by computer and fax modems. Surprisingly, Kentucky's two most urbanized area codes, 502 (Louisville) and 859 (serving Lexington and Northern Kentucky), were not expected to exhaust until 2017 at the earliest, even though they have fewer numbers than 270. Under the NANPA proposal, existing 270 numbers would be retained by customers, but 10-digit dialing for local calls would be required across western Kentucky. Codebook come in two forms, one-part or two-part: * In one part codes, the plain text words and phrases and the corresponding code words are in the same alphabetical order. *Iowa Online Law Reference Category:Iowa statutes Iowa The distribution and physical security of codebooks presents a special difficulty in the use of codes, compared to the secret information used in ciphers, the key, which is typically much shorter. The JN-25 code used in World War II used a code book of 30,000 code groups superencrypted with 30,000 random additives. ABC Codes are five-digit alpha codes (e.g., AAAAA) used by licensed and non- licensed healthcare practitioners to supplement medical codes (e.g. CPT and HCPCS II) on standard electronic (e.g. American National Standards Institute, Accredited Standards Committee X12 N 837P healthcare claims and on standard paper claims (e.g., CMS 1500 Form) to describe services, remedies and/or supply items provided and/or used during patient visits. Numbers of the new area code were made available for assignment on March 3, 2014. This area had been historically served by area code 313, which today only applies to Detroit and its closest suburbs. ==See also== * List of Michigan area codes ==External links== * Map of Michigan area codes at North American Numbering Plan Administration's website * List of exchanges from AreaCodeDownload.com, 734 Area Code Category:Telecommunications-related introductions in 1997 734 734 ",773,0.5,24.0,0.333333333333333,269,C
-"A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, $d_1 d_2 d_3 d_4$, where $d_i, i=1,2,3,4$, is selected from $1,2,3,4,5,6,7$, and 8 . How many different lock combinations are possible with such a lock?","A combination lock is a type of locking device in which a sequence of symbols, usually numbers, is used to open the lock. The number of wheels in the mechanism determines the number of specific dial positions that must be entered to open the lock, so a three- sequence combination is required for a three-wheel lock. In 1978 a combination lock which could be set by the user to a sequence of his own choosing was invented by Andrew Elliot Rae. If the arrangement of numbers is fixed, it is easy to determine the lock sequence by viewing several successful accesses. Many combination locks have three wheels, but the lock may be equipped with additional wheels, each with a drive pin and fly, in a similar manner. The first commercially viable single-dial combination lock was patented on 1 February 1910 by John Junkunc, owner of American Lock Company. == Types == ===Multiple-dial locks=== One of the simplest types of combination lock, often seen in low-security bicycle locks and in briefcases, uses several rotating discs with notches cut into them. The other side of the lock, or the other end of the cable, has a pin with several protruding teeth. frame|When the toothed pin is inserted and the discs are rotated to an incorrect combination, the inner faces of the discs block the pin from being extracted. thumb|right|250px|A simple combination lock. ==History== The earliest known combination lock was excavated in a Roman period tomb on the Kerameikos, Athens. In this case, the combination is 9-2-4. frame|The discs are mounted on one side of the lock, which may in turn be attached to the end of a chain or cable. Unlike ordinary padlocks, combination locks do not use keys. frame|Exploded view of the rotating discs. This leads to some limitations on what combinations are possible. Types range from inexpensive three-digit luggage locks to high-security safes. Nearly all safes made after World War II have relock triggers in their combination locks. ==Manufacturers== *ABUS *Master Lock *Sargent and Greenleaf *Wordlock *Dudley *Conair *Kaba Mas *CJSJ ==See also== * Electronic lock * Password * Immobiliser * Keycard ==References== ==External links== * How Combination Locks Work HowStuffWorks.com Category:Locks (security device) Category:Locksmithing de:Schloss (Technik)#Zahlenschloss Wheels may be made of radiotransparent materials such as Nylon, Lexan, or Delrin to prevent the use of X-ray imaging to determine the wheel position and required combination. == See also == * Combination lock * Safes ==References== ==External links== * How rotary combination locks work, HowStuffWorks * * Locraker - Automatic combination lock cracker, Neil Fraser, 13 March 2002 - rotary combination lock cracking machine * - contains a detailed description, with photographs, of rotary combination locks and their security concerns Category:Locks (security device) The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a probability of occurring. This type of locking mechanism consists of a single dial which must be rotated left and right in a certain combination in order to open the lock. ==Design and operation== thumb|right|upright=1.5|Internal mechanism of a rotary combination lock with a retractable bolt. There is a variation of the traditional dial based combination lock wherein the ""secret"" is encoded in an electronic microcontroller. A rotary combination lock is a lock commonly used to secure safes and as an unkeyed padlock mechanism. When the notches in the discs align with the teeth on the pin, the lock can be opened. thumb|right|The component parts of a Stoplock combination padlock. ===Single-dial locks=== The rotary combination locks found on padlocks, lockers, or safes may use a single dial which interacts with several parallel discs or cams. The remaining numbers can be arranged in (100-l)! ways. US Patents regarding combination padlocks by J.B. Gray in 1841Permutation padlock. In the case of eight prisoners, this cycle- following strategy is successful if and only if the length of the longest cycle of the permutation is at most 4. This in turn allows the owner to set a custom combination. ===Additional security=== Some rotary combination locks include internal relockers or relocking devices that separately lock the shackle or bolt when an attack is detected, including mechanical levers that respond to attempts to dislodge the locking mechanism (""punching""), thermal (fusible) links that melt in response to a cutting attempt, or tempered glass that breaks in response to a drilling attempt. ",+93.4,1.44,4096.0,+116.0,−2,C
-"An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is $151 / 300$. How many red balls are in the second urn?","Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. The probability that the second ball picked is red depends on whether the first ball was red or white. In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? The probability that the red ball is not taken in the first draw is 1000/2000 = . When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. thumb|Two urns containing white and red balls. * Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour. We want to calculate the probability that the red ball is not taken. The probability that the first ball picked is red is equal to the weight fraction of red balls: : p_1 = \frac{m_1 \omega_1}{m_1 \omega_1 + m_2 \omega_2}. What is the distribution of the number of black balls drawn after m draws? * multinomial distribution: there are balls of more than two colors. While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. Here the draws are independent and the probabilities are therefore not multiplied together. See Pólya urn model. * binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls. * Mixed replacement/non-replacement: the urn contains black and white balls. ",11, 258.14,48.6,1.91,0.14,A
-" A typical roulette wheel used in a casino has 38 slots that are numbered $1,2,3, \ldots, 36,0,00$, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of $1 / 38$, and we are interested in the number of the slot into which the ball falls. Let $A=\{0,00\}$. Give the value of $P(A)$.","The payout given by the casino for a win is based on the roulette wheel having 36 outcomes, and the payout for a bet is given by \frac{36}{p}. It is worth noting that the odds for the player in American roulette are even worse, as the bet profitability is at worst -\frac{3}{38}r \approx -0.0789r, and never better than -\frac{r}{19} \approx -0.0526r. ===Simplified mathematical model=== For a roulette wheel with n green numbers and 36 other unique numbers, the chance of the ball landing on a given number is \frac{1}{(36+n)}. By law, the game must use cards and not slots on the roulette wheel to pick the winning number. ==Roulette wheel number sequence== The pockets of the roulette wheel are numbered from 0 to 36. In the United Kingdom, the farthest outside bets (low/high, red/black, even/odd) result in the player losing only half of their bet if a zero comes up. ==Bet odds table== The expected value of a $1 bet (except for the special case of Top line bets), for American and European roulette, can be calculated as :\mathrm{expected value} = \frac{1}{n} (36 - n)= \frac{36}{n} - 1, where n is the number of pockets in the wheel. Since this roulette has 37 cells with equal odds of hitting, this is a final model of field probability (\Omega, 2^\Omega, \mathbb{P}), where \Omega = \\{0, \ldots, 36\\}, \mathbb{P}(A) = \frac{|A|}{37} for all A \in 2^\Omega. Therefore, SD for Roulette even-money bet is equal to 2b\sqrt{npq}, where b is the flat bet per round, n is the number of rounds, p=18/38, and q=20/38. The roulette wheel. The ball eventually loses momentum, passes through an area of deflectors, and falls onto the wheel and into one of thirty-seven (single-zero, French or European style roulette) or thirty-eight (double-zero, American style roulette) or thirty-nine (triple- zero, ""Sands Roulette"") colored and numbered pockets on the wheel. The expected value is: :−1 × + 35 × = −0.0526 (5.26% house edge) For European roulette, a single number wins and loses : :−1 × + 35 × = −0.0270 (2.70% house edge) For triple-zero wheels, a single number wins and loses : :−1 × + 35 × = −0.0769 (7.69% house edge) ==Mathematical model== As an example, the European roulette model, that is, roulette with only one zero, can be examined. All roulette tables operated by a casino have the same basic mechanics: * There is a balanced mechanical wheel with colored pockets separated by identical vanes and the wheel which spins freely on a supporting post. Pocket number order on the roulette wheel adheres to the following clockwise sequence in most casinos: ;Single- zero wheel : 0-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23-10-5-24-16-33-1-20-14-31-9-22-18-29-7-28-12-35-3-26 ;Double-zero wheel : 0-28-9-26-30-11-7-20-32-17-5-22-34-15-3-24-36-13-1-00-27-10-25-29-12-8-19-31-18-6-21-33-16-4-23-35-14-2 ;Triple-zero wheel : 0-000-00-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23-10-5-24-16-33-1-20-14-31-9-22-18-29-7-28-12-35-3-26 ==Roulette table layout== thumbnail|upright|French style layout, French single zero wheel The cloth-covered betting area on a roulette table is known as the layout. The ball landed on ""Red 7"" and Revell walked away with $270,600. ==See also== *Bauernroulette *Boule *Eudaemons *Monte Carlo Paradox *Russian roulette *Straperlo *The Gambler, a novel written by Fyodor Dostoevsky inspired by his addiction to roulette *Le multicolore; a game similar to roulette ==Notes== ==External links== Category:Gambling games Category:Roulette and wheel games Category:French inventions The sum of all the numbers on the roulette wheel (from 0 to 36) is 666, which is the ""Number of the Beast"".The last term in a sequence of partial sums composed of either sequence is 666, the ""beast number"". ==Rules of play against a casino== thumb|left|220px|Roulette with red 12 as the winner Roulette players have a variety of betting options. In some casinos, a player may bet full complete for less than the table straight-up maximum, for example, ""number 17 full complete by $25"" would cost $1000, that is 40 chips each at $25 value. ==Betting strategies and tactics== Over the years, many people have tried to beat the casino, and turn roulette—a game designed to turn a profit for the house—into one on which the player expects to win. The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games. Take the coin toss for example, the chances of heads and tails are equal, 50% each, if a player bets $10 on the coin landing heads up and they win, the casino pays them $10. Players can continue to place bets as the ball spins around the wheel until the dealer announces ""no more bets"" or ""rien ne va plus"". thumb|220px|Croupier's rake pushing chips across a roulette layout When a winning number and color is determined by the roulette wheel, the dealer will place a marker, also known as a dolly, on that winning number on the roulette table layout. In the early 1990s, Gonzalo Garcia-Pelayo believed that casino roulette wheels were not perfectly random, and that by recording the results and analysing them with a computer, he could gain an edge on the house by predicting that certain numbers were more likely to occur next than the 1-in-36 odds offered by the house suggested. According to Hoyle ""the single 0, the double 0, and eagle are never bars; but when the ball falls into either of them, the banker sweeps every thing upon the table, except what may happen to be bet on either one of them, when he pays twenty-seven for one, which is the amount paid for all sums bet upon any single figure"". thumb|left|250px|1800s engraving of the French roulette In the 19th century, roulette spread all over Europe and the US, becoming one of the most famous and most popular casino games. The initial bet is returned in addition to the mentioned payout: it can be easily demonstrated that this payout formula would lead to a zero expected value of profit if there were only 36 numbers (that is, the casino would break even). Therefore, the VI for the even-money American Roulette bet is \sqrt{18/38\cdot20/38}\approx0.499. Here, the profit margin for the roulette owner is equal to approximately 2.7%. ",5.4,1.3,5.5,0.0526315789,2.3613,D
-"In the gambling game ""craps,"" a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2,3 , or 12 . If the sum is $4,5,6$, 8,9 , or 10 , that number is called the bettor's ""point."" Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7 , the bettor wins. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11 , $P(7$ or 11$)$.","Take the coin toss for example, the chances of heads and tails are equal, 50% each, if a player bets $10 on the coin landing heads up and they win, the casino pays them $10. Note: Expert bet settlers before the introduction of bet-settling software would have invariably used an algebraic- type method together with a simple calculator to determine the return on a bet (see below). ==Algebraic interpretation== If a, b, c, d... represent the decimal odds, i.e. (fractional odds + 1), then an 'odds multiplier' OM can be calculated algebraically by multiplying the expressions (a + 1), (b + 1), (c + 1), ... etc. together in the required manner and adding or subtracting additional components. If they lose, all the $10 is lost to the casino, in this case, the casino advantage is zero (the casino is certainly not stupid enough to open this game); but if they win, the casino only pays them $9, if they lose, all the $10 is lost to the casino. Of course, the casino can't win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round. The place part of each-way bets is calculated separately from the win part; the method is identical but the odds are reduced by whatever the place factor is for the particular event (see Accumulator below for detailed example). In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits. When gambles are selected through a choice process – when people indicate which gamble they prefer from a set of gambles (e.g., win/lose, over/under) – people tend to prefer to bet on the outcome that is more likely to occur. In algebraic terms the OM for the Yankee bet is given by: :OM = (a + 1)(b + 1)(c + 1)(d + 1) − 1 − (a + b + c + d) In the days before software became available for use by bookmakers and those settling bets in Licensed Betting Offices (LBOs) this method was virtually de rigueur for saving time and avoiding the multiple repetitious calculations necessary in settling bets of the full cover type. ==Settling other types of winning bets== Up and down :Returns (£20 single at 7-2 ATC £20 single at 15-8) = £20 × 7/2 + £20 × (15/8 + 1) = £127.50 :Returns (£20 single at 15-8 ATC £20 single at 7-2) = £20 × 15/8 + £20 × (7/2 + 1) = £127.50 :Total returns = £255.00 :Note: This is the same as two £20 single bets at twice the odds; i.e. £20 singles at 7-1 and 15-4 and is the preferred manual way of calculating the bet. Spread betting allows gamblers to wagering on the outcome of an event where the pay-off is based on the accuracy of the wager, rather than a simple ""win or lose"" outcome. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field. === Bingo probability === The probability of winning a game of Bingo (ignoring simultaneous winners, making wins mutually exclusive) may be calculated as: : P(Win)=1-P(Loss) since winning and losing are mutually exclusive. Gambling (also known as betting or gaming) is the wagering of something of value (""the stakes"") on a random event with the intent of winning something else of value, where instances of strategy are discounted. A Round Robin with 1 winner is calculated as two Up and Down bets with one winner in each. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6 probability of any single number appearing. The mathematics of gambling is a collection of probability applications encountered in games of chance and can get included in game theory. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38. Thus, an each-way Lucky 63 on six horses with three winners and a further two placed horses is settled as a win Patent and a place Lucky 31. ==Algebraic interpretation== Returns on any bet may be considered to be calculated as 'stake unit' × 'odds multiplier'. Gambling is not luck, but a contest of intellect, strategy, and yield. All bets are taken as 'win' bets unless 'each-way' is specifically stated. A Round Robin with 2 winners is calculated as a double plus one Up and Down bet with 2 winners plus two Up and Down bets with 1 winner in each. * Carnival Games such as The Razzle or Hanky Pank * Coin-tossing games such as Head and Tail, Two-up* * Confidence tricks such as Three-card Monte or the Shell game * Dice-based games, such as Backgammon, Liar's dice, Passe-dix, Hazard, Threes, Pig, or Mexico (or Perudo); *Although coin tossing is not usually played in a casino, it has been known to be an official gambling game in some Australian casinos ===Fixed-odds betting=== Fixed-odds betting and Parimutuel betting frequently occur at many types of sporting events, and political elections. The end-of-the-day betting effect is a cognitive bias reflected in the tendency for bettors to take gambles with higher risk and higher reward at the end of their betting session to try to make up for losses. ",0.5117,0,0.6247,11,0.22222222,E
-"Given that $P(A \cup B)=0.76$ and $P\left(A \cup B^{\prime}\right)=0.87$, find $P(A)$.","That is, for an event A, :P(A^c) = 1 - P(A). 1983 Emperor's Cup Final was the 63rd final of the Emperor's Cup competition. It may be tempting to say that : Pr([""1"" on 1st trial] or [""1"" on second trial] or ... or [""1"" on 8th trial]) := Pr(""1"" on 1st trial) + Pr(""1"" on second trial) + ... + P(""1"" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. The 1983–84 Gold Cup was the 65th edition of the Gold Cup, a cup competition in Northern Irish football. Therefore, the probability of an event's complement must be unity minus the probability of the event. The 1983–84 Ulster Cup was the 36th edition of the Ulster Cup, a cup competition in Northern Irish football. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. The 1983–84 Irish Cup was the 104th edition of the Irish Cup, Northern Ireland's premier football knock-out cup competition. The 1884–85 Belfast Charity Cup was the 2nd edition of the Belfast Charity Cup, a cup competition in Irish football. Equivalently, the probabilities of an event and its complement must always total to 1. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. This result cannot be right because a probability cannot be more than 1. One may resolve this overlap by the principle of inclusion- exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: : Pr(at least one ""1"") = 1 − Pr(no ""1""s) := 1 − Pr([no ""1"" on 1st trial] and [no ""1"" on 2nd trial] and ... and [no ""1"" on 8th trial]) := 1 − Pr(no ""1"" on 1st trial) × Pr(no ""1"" on 2nd trial) × ... × Pr(no ""1"" on 8th trial) := 1 −(5/6) × (5/6) × ... × (5/6) := 1 − (5/6)8 := 0.7674... ==See also== *Logical complement *Exclusive disjunction *Binomial probability ==References== ==External links== *Complementary events - (free) page from probability book of McGraw-Hill Category:Experiment (probability theory) The complement of an event A is usually denoted as A′, Ac, egA or . For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. Oldpark won the tournament for the 1st time, defeating Cliftonville 1–0 in the final. ==Results== ===Semi- finals=== |} ====Replay==== |} ===Final=== ==References== ==External links== * Northern Ireland - List of Belfast Charity Cup Winners Category:1884–85 in Irish association football What is the probability that one sees a ""1"" at least once? Given an event, the event and its complementary event define a Bernoulli trial: did the event occur or not? This does not, however, mean that any two events whose probabilities total to 1 are each other's complements; complementary events must also fulfill the condition of mutual exclusivity. ===Example of the utility of this concept=== Suppose one throws an ordinary six-sided die eight times. Ballymena United won their fourth Irish Cup, defeating Carrick Rangers 4–1 in the final. ==Results== ===First round=== |} ====Replays==== |} ===Second round=== |} ====Replays==== |} ====Second replay==== |} ===Quarter-finals=== |} ===Semi-finals=== |} ===Final=== ==References== 1983–84 Category:1983–84 domestic association football cups Category:1983–84 in Northern Ireland association football Category:1983 in Northern Ireland sport Category:1984 in Northern Ireland sport ",-87.8,0.63,205.0,2.84367,0.72,B
-Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?,"Table of Six () is a political conference established by the Republican People's Party, Good Party, Felicity Party, Democrat Party, Democracy and Progress Party and Future Party, with the first meeting held on 12 February 2022. In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. Table topics are topics on various subjects that are discussed by a group of people around a table. There will be a table topic master for each meeting, who will prepare questions beforehand and ask the participants questions one by one for which they are called upon to answer. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bézout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 65 = 7776, which was Steiner's incorrect solution. If the five conics have the properties that *there is no line such that every one of the 5 conics is either tangent to it or passes through one of two fixed points on it (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics) *no three of the conics pass through any point (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics passing through this triple intersection point) *no two of the conics are tangent *no three of the five conics are tangent to a line *a pair of lines each tangent to two of the conics do not intersect on the fifth conic (otherwise this pair is a degenerate conic tangent to all 5 conics) then the total number of conics C tangent to all 5 (counted with multiplicities) is 3264. The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848. ==History== claimed that the number of conics tangent to 5 given conics in general position is 7776 = 65, but later realized this was wrong. Many personality or public speaking clubs like the 'Toastmasters' have a separate session in their meetings known as a table topic session. Student Number 1 was panned by critics and flopped at the box office. ==Cast== ==Production== Shooting was commenced at Chennai, for a fifteen-days schedule, after which the unit moved to Russia to shoot two songs. The Table of Six was originally an independent entity from the Nation Alliance. In particular if C intersects each of the five conics in exactly 3 points (one double point of tangency and two others) then the multiplicity is 1, and if this condition always holds then there are exactly 3264 conics tangent to the 5 given conics. Some chapters of Toastmasters also host Table Topics contests. ==See also== * TableTopics ==References== Category:Public speaking However the number of conics is not (6H)5 but (6H−2E)5 because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E. On 21 January 2023, Table of Six defined itself as the ""Nation Alliance"" for the first time after its 11th meeting. As practiced by Toastmasters International, the topics to be discussed are written on pieces of paper which are placed in a box in the middle of a table. Five Guys Walk into a Bar... has received a largely positive response from critics since its release. The participants pick up one paper each and start talking about the topic written on the paper. Graphs and Combinatorics (ISSN 0911-0119, abbreviated Graphs Combin.) is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. In the text of the memorandum, lowering the electoral threshold to 3%, treasury aid to the parties that received at least 1% of the votes, ending the omnibus law practice, removing the veto power of the president and extending his term of office to 7 years, recognising the authority to issue a no-confidence question on the government, human rights and human rights in the education curriculum. Steiner observed that the conics tangent to a given conic form a degree 6 hypersurface in CP5. Five Guys Walk into a Bar... is a comprehensive four-disc retrospective of the British rock group Faces released in 2004, collecting sixty-seven tracks from among the group's four studio albums, assorted rare single A and B-sides, BBC sessions, rehearsal tapes and one track from a promotional flexi-disc, ""Dishevelment Blues"" - a deliberately-sloppy studio romp, captured during the sessions for their Ooh La La album, which was never actually intended for official release. On 3 March 2023, Good Party leader Meral Akşener announced that she took the decision to withdraw from the Table of Six and said her party would not support main opposition Republican People's Party leader Kemal Kılıçdaroğlu as the joint candidate in the 2023 Turkish presidential election. ",-0.55,0.00539,1.4,+116.0,-0.10,B
-How many different license plates are possible if a state uses two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)?,"In New Zealand, vehicle registration plates (usually called number plates) contain up to six alphanumeric characters, depending on the type of vehicle and the date of registration. The vehicle registration plates of Cyprus are composed of three letters and three digits (e.g. ABC 123). Greek vehicle registration plates are composed of three letters and four digits per plate (e.g. `ΑΑΑ–1000`) printed in black on a white background. From 1964 until March 2001 these number plates had two letters followed by one to four numbers (format LLnnnn), the sequence having started with AA1 and continuing through to ZZ9989 chronologically (for example, XE3782 would have been issued in 1998). Each of the 49 states of the United States of America plus several of its territories and the District of Columbia issued individual passenger license plates for the year 1959. ==Passenger baseplates== Image Region Design Slogan Serial format Serials issued Notes 150px Alabama Embossed blue lettering, heart logo and rims on white base. The letters represent the district (prefecture) that issues the plates while the numbers range from 1000 to 9999. None 123-456 1 to approximately 999-999 150px Washington Embossed white numbers on green plate; ""54 WASHINGTON"" embossed in white block letters at bottom. Number plates had to be changed before the end of January 2019. 1973 - 2018 150px 150px 2018 - today 150px ===Special plates=== Taxis Taxi plates show the prefix T, followed by two letters and three digits, formerly one letter only. Since 2013, some numbers have been reissued on white plates. 2013 - today Composed of five numerals and the prefix P, on white plates. The final one or two letters in the sequence changes in Greek alphabetical order after 8,999 issued plates. Similar plates but of square size with numbers ranging from 1 to 999 are issued for motorcycles which exceed 50 cc in engine size. If the visibility of a regular number plate is obstructed, for example by a bike rack mounted to a car's trailer hitch, a supplementary plate with the same registration number must be obtained and affixed to the obstruction (or the vehicle) such that it will be visible from the same direction as the regular number plate would have been. == Standard numbering sequences == thumb|right|A vehicle registration plate of New Zealand in the optional 'Europlate' style === Cars and heavy vehicles === * 1964–1987: AAnnnn * 1987–2001: AAnnnn * 2001–present: AAAnnn Private cars, taxis, and heavier road vehicles in New Zealand have number plates with up to six characters. Hence these so-called ""six-figure plates"" can still occasionally (as of 2018) be spotted on a few old vehicles. ===1973–1985=== In 1972, they became lettered and the system was `LL–NNNN` while trucks used `L–NNNN`. In some later instances issuers coded plates to the area of registration, such as in 1966 with the allocation of plates beginning with ""CE"" to the Manawatu-Wanganui region, in 1974–1976 with the allocation of plates beginning with ""HB"" to the Hawke's Bay region, in May 1989 with the allocation of plates beginning with ""OG"" to Wellington region, and in July 2000 with the allocation of plates beginning with ""ZI"" to Auckland region. === Motorcycles and tractors === These vehicles use one of several five-character systems. Some early numbers were printed on remaining yellow plates. === Commercial trucks === 1973 - 1990 1990 - 2003 2003 - 2013 2013 - today Composed of three letters and three numerals, on yellow plates. None 1-12345 10-1234 County-coded ==Non-passenger plates== Image (standard) Region Type Design and slogan Serial format Serials issued Notes 150px ==See also== *Antique vehicle registration *Electronic license plate *Motor vehicle registration *Vehicle license ==References== ==External links== Category:1959 in the United States 1959 Although plate character/number combinations can contain ""spaces"", they do not form part of the unique identification and are typically not stored (for example, in Police computer-systems). Cypriot National Guard plates are composed of an old version of the Greek flag followed by the letters ΕΦ and up to five digits. ==Northern Cyprus== thumb|upright|Northern Cyprus plate thumb|upright|Northern Cyprus rear plate thumb|Taxi number plate (T) thumb|upright=0.5|Trailer number plate (R) thumb|Rental car number plate thumb|Police number plate ===Style and numbering=== Northern Cyprus civilian number plates still use the old format (1973–1990) of Cyprus number plates (AB 123). White front plates were omitted after 2013. thumb|Temporary number plates === Temporary / Visitors === 1973 - 2003 Up to four numerals followed by the letter V, followed by two numerals indicating the year of registration. None 1-1234 1-A123 12-1234 12-A123 Coded by county (1 or 10 prefix). 1957 base plates revalidated for 1959 with red tabs. 150px Tennessee Embossed yellow numbers on black state- shaped plate with border line; ""TENN. 54"" embossed in yellow block letters centered at bottom. None AA1 to WW999 Letters I, Q and U not used, and X, Y and Z used only on replacement plates. 150px South Carolina Embossed white numbers on blue plate; ""SOUTH CAROLINA 59"" embossed in white block letters at top. Export plates, from 1973 until 1990, showed the letter E followed by four numerals. ",0.333333333333333,0.03,6760000.0,0.16,1.27,C
+","Finally, the principle of conditional probability implies that is equal to the product of these individual probabilities: The terms of equation () can be collected to arrive at: Evaluating equation () gives Therefore, (50.7297%). Further results showed that psychology students and women did better on the task than casino visitors/personnel or men, but were less confident about their estimates. ===Reverse problem=== The reverse problem is to find, for a fixed probability , the greatest for which the probability is smaller than the given , or the smallest for which the probability is greater than the given . Consequently, the desired probability is . The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people . ===Probability of a shared birthday (collision)=== The birthday problem can be generalized as follows: :Given random integers drawn from a discrete uniform distribution with range , what is the probability that at least two numbers are the same? ( gives the usual birthday problem.) In the standard case of , substituting gives about 6.1%, which is less than 1 chance in 16. The following table shows the probability for some other values of (for this table, the existence of leap years is ignored, and each birthday is assumed to be equally likely): thumb|right|upright=1.4|The probability that no two people share a birthday in a group of people. This is a list of people in the chiropractic profession, comprising chiropractors and other people who have been notably connected with the profession. thumb|upright=1.3|The computed probability of at least two people sharing a birthday versus the number of people In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. Then, because and are the only two possibilities and are also mutually exclusive, Here is the calculation of for 23 people. The answer is 20—if there is a prize for first match, the best position in line is 20th. ===Same birthday as you=== thumb|right|upright=1.4|Comparing = probability of a birthday match with = probability of matching your birthday In the birthday problem, neither of the two people is chosen in advance. For example, the usual 50% probability value is realized for both a 32-member group of 16 men and 16 women and a 49-member group of 43 women and 6 men. ==Other birthday problems== ===First match=== A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? Where the event is the probability of finding a group of 23 people with at least two people sharing same birthday, . is the ratio of the total number of birthdays, V_{nr}, without repetitions and order matters (e.g. for a group of 2 people, mm/dd birthday format, one possible outcome is \left \\{ \left \\{01/02,05/20\right \\},\left \\{05/20,01/02\right \\},\left \\{10/02,08/04\right\\},...\right \\} divided by the total number of birthdays with repetition and order matters, V_{t}, as it is the total space of outcomes from the experiment (e.g. 2 people, one possible outcome is \left \\{ \left \\{01/02,01/02\right \\},\left \\{10/02,08/04\right \\},...\right \\}. This number is significantly higher than : the reason is that it is likely that there are some birthday matches among the other people in the room. === Number of people with a shared birthday === For any one person in a group of n people the probability that he or she shares his birthday with someone else is q(n-1;d) , as explained above. *List Chiropractors Category:Chiropractic And for the group of 23 people, the probability of sharing is :p(23) \approx 1 - \left(\frac{364}{365}\right)^\binom{23}{2} = 1 - \left(\frac{364}{365}\right)^{253} \approx 0.500477 . ===Poisson approximation=== Applying the Poisson approximation for the binomial on the group of 23 people, :\operatorname{Poi}\left(\frac{\binom{23}{2}}{365}\right) =\operatorname{Poi}\left(\frac{253}{365}\right) \approx \operatorname{Poi}(0.6932) so :\Pr(X>0)=1-\Pr(X=0) \approx 1-e^{-0.6932} \approx 1-0.499998=0.500002. Probability in the Engineering and Informational Sciences is an international journal published by Cambridge University Press. Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. ==Scope== Much research involving probability is done under the auspices of applied probability. In short can be multiplied by itself times, which gives us :\bar p(n) \approx \left(\frac{364}{365}\right)^\binom{n}{2}. Therefore, its probability is : p(n) = 1 - \bar p(n). The formula :n(d)=\left\lceil \sqrt{2d\ln2}+\frac{3-2\ln2}{6}+\frac{9-4(\ln2)^2}{72\sqrt{2d\ln2}}-\frac{2(\ln2)^2}{135d}\right\rceil holds for all , and it is conjectured that this formula holds for all . ===More than two people sharing a birthday=== It is possible to extend the problem to ask how many people in a group are necessary for there to be a greater than 50% probability that at least 3, 4, 5, etc. of the group share the same birthday. Note that the vertical scale is logarithmic (each step down is 1020 times less likely). : 1 0.0% 5 2.7% 10 11.7% 20 41.1% 23 50.7% 30 70.6% 40 89.1% 50 97.0% 60 99.4% 70 99.9% 75 99.97% 100 % 200 % 300 (100 − )% 350 (100 − )% 365 (100 − )% ≥ 366 100% ==Approximations== thumb|right|upright=1.4|Graphs showing the approximate probabilities of at least two people sharing a birthday () and its complementary event () thumb|right|upright=1.4|A graph showing the accuracy of the approximation () The Taylor series expansion of the exponential function (the constant ) : e^x = 1 + x + \frac{x^2}{2!}+\cdots provides a first-order approximation for for |x| \ll 1: : e^x \approx 1 + x. The birthday problem has been generalized to consider an arbitrary number of types.M. C. Wendl (2003) Collision Probability Between Sets of Random Variables, Statistics and Probability Letters 64(3), 249–254. ",35.64,0.68,"""8.87""",0.166666666,0.1353,B
+"A doctor is concerned about the relationship between blood pressure and irregular heartbeats. Among her patients, she classifies blood pressures as high, normal, or low and heartbeats as regular or irregular and finds that 16\% have high blood pressure; (b) 19\% have low blood pressure; (c) $17 \%$ have an irregular heartbeat; (d) of those with an irregular heartbeat, $35 \%$ have high blood pressure; and (e) of those with normal blood pressure, $11 \%$ have an irregular heartbeat. What percentage of her patients have a regular heartbeat and low blood pressure?","Clinicians consider a pulse pressure of 60 mmHg to likely be associated with diseases, with a pulse pressure of 50 mmHg or more increasing the risk of cardiovascular disease. == Calculation == Pulse pressure is calculated as the difference between the systolic blood pressure and the diastolic blood pressure. It is measured by right heart catheterization or may be estimated by transthoracic echocardiography Normal pulmonary artery pressure is between 8mmHg -20 mm Hg at rest. : e.g. normal: 15mmHg - 8mmHg = 7mmHg : high: 25mmHg - 10mmHg = 15mmHg ==Values and variation== ===Low (narrow) pulse pressure === A pulse pressure is considered abnormally low if it is less than 25% of the systolic value. Many studies further indicate a J-shaped relationship between blood pressure and mortality, whereby both very high and very low levels are associated with notable increases in mortality. However, pulse pressure has usually been found to be a stronger independent predictor of cardiovascular events, especially in older populations, than has systolic, diastolic, or mean arterial pressure. The systemic pulse pressure is approximately proportional to stroke volume, or the amount of blood ejected from the left ventricle during systole (pump action) and inversely proportional to the compliance (similar to elasticity) of the aorta. * Systemic pulse pressure (usually measured at upper arm artery) = Psystolic \- Pdiastolic :e.g. normal 120mmHg - 80mmHg = 40mmHg : low: 107mmHg - 80mmHg = 27mmHg : high: 160mmHg - 80mmHg = 80mmHg * Pulmonary pulse pressure is normally much lower than systemic blood pressure due to the higher compliance of the pulmonary system compared to the arterial circulation. Pulse pressure is the difference between systolic and diastolic blood pressure. Proportion can be written as \frac{a}{b}=\frac{c}{d}, where ratios are expressed as fractions. Readings greater than or equal to 130/80 mm Hg are considered hypertension by ACC/AHA and if greater than or equal to 140/90 mm Hg by ESC/ESH. Clonidine (decrease of 6.3 mm Hg), diltiazem (decrease of 5.5 mm Hg), and prazosin (decrease of 5.0 mm Hg) were intermediate. == See also == * Mean arterial pressure * Cold pressor test * Hypertension * Prehypertension * Antihypertensive * Patent ductus arteriosus == References == Category:Medical signs Category:Cardiovascular physiology Heart is a biweekly peer-reviewed medical journal covering all areas of cardiovascular medicine and surgery. Thus, blood pressures above normal can go undiagnosed for a long period of time. ==Causes== Elevated blood pressure develops gradually over many years usually without a specific identifiable cause. Normal pulse pressure is around 40 mmHg. If the aorta becomes rigid because of disorders, such as arteriosclerosis or atherosclerosis, the pulse pressure would be high due to less compliance of the aorta. The ACC/AHA define elevated blood pressure as readings with a systolic pressure from 120 to 129 mm Hg and a diastolic pressure under 80 mm Hg, and the European Society of Cardiology and European Society of Hypertension (ESC/ESH) define ""high normal blood pressure"" as readings with a systolic pressure from 130 to 139 mm Hg and a diastolic pressure 85-89 mm Hg. On the other hand, the National Heart, Lung, and Blood Institute suggests that people with prehypertension are at a higher risk for developing hypertension, or high blood pressure, compared to people with normal blood pressure.National Heart, Lung and Blood Institute<> A 2014 meta-analysis concluded that prehypertension increases the risk of stroke, and that even low-range prehypertension significantly increases stroke risk and a 2019 meta-analysis found elevated blood pressure increases the risk of heart attack by 86% and stroke by 66%. ==Epidemiology== Data from the 1999 and 2000 National Health and Nutrition Examination Survey (NHANES III) estimated that the prevalence of prehypertension among adults in the United States was approximately 31 percent and decreased to 28 percent in the 2011–2012 National Health and Nutrition Examination Survey. If the usual resting pulse pressure is consistently greater than 100 mmHg, potential factors are stiffness of the major arteries, aortic regurgitation (a leak in the aortic valve), or arteriovenous malformation, among others. Such a proportion is known as geometrical proportion, not to be confused with arithmetical proportion and harmonic proportion. ==Properties of proportions== * Fundamental rule of proportion. The prevalence was higher among men than women. === Risk factors === A primary risk factor for prehypertension is being overweight. This suggests that interventions that lower diastolic pressure without also lowering systolic pressure (and thus lowering pulse pressure) could actually be counterproductive. The most common cause of a low (narrow) pulse pressure is a drop in left ventricular stroke volume. High blood pressure that develops over time without a specific cause is considered benign or essential hypertension. ",3.51,72,"""2.3""",1.7,15.1,E
+"Roll a fair six-sided die three times. Let $A_1=$ $\{1$ or 2 on the first roll $\}, A_2=\{3$ or 4 on the second roll $\}$, and $A_3=\{5$ or 6 on the third roll $\}$. It is given that $P\left(A_i\right)=1 / 3, i=1,2,3 ; P\left(A_i \cap A_j\right)=(1 / 3)^2, i \neq j$; and $P\left(A_1 \cap A_2 \cap A_3\right)=(1 / 3)^3$. Use Theorem 1.1-6 to find $P\left(A_1 \cup A_2 \cup A_3\right)$.","Consider a set of three dice, III, IV and V such that * die III has sides 1, 2, 5, 6, 7, 9 * die IV has sides 1, 3, 4, 5, 8, 9 * die V has sides 2, 3, 4, 6, 7, 8 Then: * the probability that III rolls a higher number than IV is * the probability that IV rolls a higher number than V is * the probability that V rolls a higher number than III is === Three-dice set with minimal alterations to standard dice === The following intransitive dice have only a few differences compared to 1 through 6 standard dice: * as with standard dice, the total number of pips is always 21 * as with standard dice, the sides only carry pip numbers between 1 and 6 * faces with the same number of pips occur a maximum of twice per dice * only two sides on each die have numbers different from standard dice: ** A: 1, 1, 3, 5, 5, 6 ** B: 2, 3, 3, 4, 4, 5 ** C: 1, 2, 2, 4, 6, 6 Like Miwin’s set, the probability of A winning versus B (or B vs. C, C vs. P(C>D) can be calculated by summing conditional probabilities for two events: * C rolls 6 (probability ); wins regardless of D (probability 1) * C rolls 2 (probability ); wins only if D rolls 1 (probability ) The total probability of win for C is therefore :\left( {1 \over 3}\times1 \right) + \left( {2 \over 3}\times{1 \over 2} \right) = {2 \over 3} With a similar calculation, the probability of D winning over A is :\left( {1 \over 2}\times1 \right) + \left( {1 \over 2}\times{1 \over 3} \right) = {2 \over 3} ====Best overall die==== The four dice have unequal probabilities of beating a die chosen at random from the remaining three: As proven above, die A beats B two-thirds of the time but beats D only one-third of the time. The following tables show all possible outcomes for all three pairs of dice. With the second set of dice, die C′ will win with the lowest probability () and dice A′ and B′ will each win with a probability of . ==Variations== ===Efron's dice=== Efron's dice are a set of four intransitive dice invented by Bradley Efron. thumb|320px|Representation of Efron's dice The four dice A, B, C, D have the following numbers on their six faces: * A: 4, 4, 4, 4, 0, 0 * B: 3, 3, 3, 3, 3, 3 * C: 6, 6, 2, 2, 2, 2 * D: 5, 5, 5, 1, 1, 1 ====Probabilities==== Each die is beaten by the previous die in the list, with a probability of : :P(A>B) = P(B>C) = P(C>D) = P(D>A) = {2 \over 3} B's value is constant; A beats it on rolls because four of its six faces are higher. Player 1 chooses die A Player 2 chooses die C Player 1 chooses die B Player 2 chooses die A Player 1 chooses die C Player 2 chooses die B 2 4 9 1 6 8 3 5 7 3 C A A 2 A B B 1 C C C 5 C C A 4 A B B 6 B B C 7 C C A 9 A A A 8 B B B == Comment regarding the equivalency of intransitive dice == Though the three intransitive dice A, B, C (first set of dice) * A: 2, 2, 6, 6, 7, 7 * B: 1, 1, 5, 5, 9, 9 * C: 3, 3, 4, 4, 8, 8 P(A > B) = P(B > C) = P(C > A) = and the three intransitive dice A′, B′, C′ (second set of dice) * A′: 2, 2, 4, 4, 9, 9 * B′: 1, 1, 6, 6, 8, 8 * C′: 3, 3, 5, 5, 7, 7 P(A′ > B′) = P(B′ > C′) = P(C′ > A′) = win against each other with equal probability they are not equivalent. Rolling the three dice of a set and always using the highest score for evaluation will show a different winning pattern for the two sets of dice. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. The 2003 A3 Champions Cup was first edition of A3 Champions Cup. Consider the following set of dice. With adjacent pairs, one die's probability of winning is 2/3. So the likelihood of B beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Die C beats D two-thirds of the time but beats B only one-third of the time. ;Set 2: * A: 3, 3, 3, 6 * B: 2, 2, 5, 5 * C: 1, 4, 4, 4 P(A > B) = P(B > C) = , P(C > A) = 9/16 == Intransitive 12-sided dice == In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive twelve-sided dice. The 2005 A3 Champions Cup was third edition of A3 Champions Cup. The 2004 A3 Champions Cup was second edition of A3 Champions Cup. So the likelihood of A beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {4 \over 9} \right) = {13 \over 27} Similarly, die B beats C two-thirds of the time but beats A only one-third of the time. A set of dice is intransitive (or nontransitive) if it contains three dice, A, B, and C, with the property that A rolls higher than B more than half the time, and B rolls higher than C more than half the time, but it is not true that A rolls higher than C more than half the time. So the likelihood of D beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {1 \over 2} \right) = {1 \over 2} Therefore, the best overall die is C with a probability of winning of 0.5185. Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both (as long as the player is then allowed to choose between the one-die option and the two-die option): : Sets chosen by opponents Winning set of dice Type Number A B E 1 A C E 2 A D C 2 A E D 1 B C A 1 B D A 2 B E D 2 C D B 1 C E B 2 D E C 1 There are two major issues with this set, however. So the likelihood of C beating any other randomly selected die is: :{1 \over 3}\times \left( {2 \over 3} + {1 \over 3} + {5 \over 9} \right) = {14 \over 27} Finally, die D beats A two-thirds of the time but beats C only one-third of the time. With the first set of dice, die B will win with the highest probability () and dice A and C will each win with a probability of . Awarded by Mathematical Association of America * Timothy Gowers' project on intransitive dice * Category:Probability theory paradoxes Category:Dice The probability of die C beating A is . ",-0.40864,0.6296296296,"""-3.141592""",2.6,3,B
+Let $A$ and $B$ be independent events with $P(A)=$ $1 / 4$ and $P(B)=2 / 3$. Compute $P(A \cap B)$,"* This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. In this event, the event B can be analyzed by a conditional probability with respect to A. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios All events that are not in B will have null probability in the new distribution. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. This particular method relies on event B occurring with some sort of relationship with another event A. It can be interpreted as ""the probability of B occurring multiplied by the probability of A occurring, provided that B has occurred, is equal to the probability of the A and B occurrences together, although not necessarily occurring at the same time"". Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. That is, for an event A, :P(A^c) = 1 - P(A). Moreover, this ""multiplication rule"" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} ==== As the probability of a conditional event ==== Conditional probability can be defined as the probability of a conditional event A_B. If statistically independent If mutually exclusive P(A\mid B)= P(A) 0 P(B\mid A)= P(B) 0 P(A \cap B)= P(A) P(B) 0 In fact, mutually exclusive events cannot be statistically independent (unless both of them are impossible), since knowing that one occurs gives information about the other (in particular, that the latter will certainly not occur). == Common fallacies == :These fallacies should not be confused with Robert K. Shope's 1978 ""conditional fallacy"", which deals with counterfactual examples that beg the question. === Assuming conditional probability is of similar size to its inverse === thumb|450x450px|A geometric visualization of Bayes' theorem. The former is required by the axioms of probability, and the latter stems from the fact that the new probability measure has to be the analog of P in which the probability of B is one - and every event that is not in B, therefore, has a null probability. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . ",0.396,0.166666666,"""102.0""",7.00,8.3147,B
+How many four-letter code words are possible using the letters in IOWA if The letters may not be repeated?,"The state of Iowa is covered by five area codes. None of the Iowa codes are expected to need relief in the immediate future. * 319: Cedar Rapids, Waterloo, Iowa City, and Cedar Falls (original area code created in 1947) * 515: Des Moines, Ames, West Des Moines, Urbandale and Fort Dodge (original area code created in 1947) * 563: Davenport, Dubuque, Bettendorf, Clinton, Muscatine (split from 319 in 2001) * 641: Mason City, Marshalltown, Ottumwa, Tama (split from 515 in 2000) * 712: Sioux City, Council Bluffs (original area code created in 1947) ==See also== *State of Iowa Area codes Iowa thumb|right|The area codes of Kentucky: 270 and 364 in light green. The Code of Iowa contains the statutory laws of the U.S. state of Iowa. 734 is an area code in the North American Numbering Plan. ""Splitting the area code in two avoids ten digit dialing but requires changing all current area code 270 numbers within the new area code to 364. It is republished in full every odd year, and is supplemented in even years. ==External reference== *Iowa Code online at Iowa General Assembly. Such codes are half the size of two-part codes but are more vulnerable since an attacker who recovers some code word meanings can often infer the meaning of nearby code words. thumb|Page 187 of the State Department 1899 code book, a one part code with a choice of code word or numeric ciphertext. Area code 270 is a telephone area code in the North American Numbering Plan (NANP) for the Commonwealth of Kentucky's western and south central counties. On June 13, 2007, the PSC announced that the new area code will be 364, but also announced that the previously announced implementation would be delayed in favor of number conservation measures including expanded number pooling. Planning for the introduction of a second area code for the region, area code 364, was assigned in 2007. These subsequent delays of the implementation of the 270 / 364 area code split were due to further use of number conservation measures, including mandatory and expanded number pooling, as well as a weakened economy and a reduced usage of telephone numbers dedicated for use by computer and fax modems. Surprisingly, Kentucky's two most urbanized area codes, 502 (Louisville) and 859 (serving Lexington and Northern Kentucky), were not expected to exhaust until 2017 at the earliest, even though they have fewer numbers than 270. Under the NANPA proposal, existing 270 numbers would be retained by customers, but 10-digit dialing for local calls would be required across western Kentucky. Codebook come in two forms, one-part or two-part: * In one part codes, the plain text words and phrases and the corresponding code words are in the same alphabetical order. *Iowa Online Law Reference Category:Iowa statutes Iowa The distribution and physical security of codebooks presents a special difficulty in the use of codes, compared to the secret information used in ciphers, the key, which is typically much shorter. The JN-25 code used in World War II used a code book of 30,000 code groups superencrypted with 30,000 random additives. ABC Codes are five-digit alpha codes (e.g., AAAAA) used by licensed and non- licensed healthcare practitioners to supplement medical codes (e.g. CPT and HCPCS II) on standard electronic (e.g. American National Standards Institute, Accredited Standards Committee X12 N 837P healthcare claims and on standard paper claims (e.g., CMS 1500 Form) to describe services, remedies and/or supply items provided and/or used during patient visits. Numbers of the new area code were made available for assignment on March 3, 2014. This area had been historically served by area code 313, which today only applies to Detroit and its closest suburbs. ==See also== * List of Michigan area codes ==External links== * Map of Michigan area codes at North American Numbering Plan Administration's website * List of exchanges from AreaCodeDownload.com, 734 Area Code Category:Telecommunications-related introductions in 1997 734 734 ",773,0.5,"""24.0""",0.333333333333333,269,C
+"A boy found a bicycle lock for which the combination was unknown. The correct combination is a four-digit number, $d_1 d_2 d_3 d_4$, where $d_i, i=1,2,3,4$, is selected from $1,2,3,4,5,6,7$, and 8 . How many different lock combinations are possible with such a lock?","A combination lock is a type of locking device in which a sequence of symbols, usually numbers, is used to open the lock. The number of wheels in the mechanism determines the number of specific dial positions that must be entered to open the lock, so a three- sequence combination is required for a three-wheel lock. In 1978 a combination lock which could be set by the user to a sequence of his own choosing was invented by Andrew Elliot Rae. If the arrangement of numbers is fixed, it is easy to determine the lock sequence by viewing several successful accesses. Many combination locks have three wheels, but the lock may be equipped with additional wheels, each with a drive pin and fly, in a similar manner. The first commercially viable single-dial combination lock was patented on 1 February 1910 by John Junkunc, owner of American Lock Company. == Types == ===Multiple-dial locks=== One of the simplest types of combination lock, often seen in low-security bicycle locks and in briefcases, uses several rotating discs with notches cut into them. The other side of the lock, or the other end of the cable, has a pin with several protruding teeth. frame|When the toothed pin is inserted and the discs are rotated to an incorrect combination, the inner faces of the discs block the pin from being extracted. thumb|right|250px|A simple combination lock. ==History== The earliest known combination lock was excavated in a Roman period tomb on the Kerameikos, Athens. In this case, the combination is 9-2-4. frame|The discs are mounted on one side of the lock, which may in turn be attached to the end of a chain or cable. Unlike ordinary padlocks, combination locks do not use keys. frame|Exploded view of the rotating discs. This leads to some limitations on what combinations are possible. Types range from inexpensive three-digit luggage locks to high-security safes. Nearly all safes made after World War II have relock triggers in their combination locks. ==Manufacturers== *ABUS *Master Lock *Sargent and Greenleaf *Wordlock *Dudley *Conair *Kaba Mas *CJSJ ==See also== * Electronic lock * Password * Immobiliser * Keycard ==References== ==External links== * How Combination Locks Work HowStuffWorks.com Category:Locks (security device) Category:Locksmithing de:Schloss (Technik)#Zahlenschloss Wheels may be made of radiotransparent materials such as Nylon, Lexan, or Delrin to prevent the use of X-ray imaging to determine the wheel position and required combination. == See also == * Combination lock * Safes ==References== ==External links== * How rotary combination locks work, HowStuffWorks * * Locraker - Automatic combination lock cracker, Neil Fraser, 13 March 2002 - rotary combination lock cracking machine * - contains a detailed description, with photographs, of rotary combination locks and their security concerns Category:Locks (security device) The original Three Prisoners problem can be seen in this light: The warden in that problem still has these six cases, each with a probability of occurring. This type of locking mechanism consists of a single dial which must be rotated left and right in a certain combination in order to open the lock. ==Design and operation== thumb|right|upright=1.5|Internal mechanism of a rotary combination lock with a retractable bolt. There is a variation of the traditional dial based combination lock wherein the ""secret"" is encoded in an electronic microcontroller. A rotary combination lock is a lock commonly used to secure safes and as an unkeyed padlock mechanism. When the notches in the discs align with the teeth on the pin, the lock can be opened. thumb|right|The component parts of a Stoplock combination padlock. ===Single-dial locks=== The rotary combination locks found on padlocks, lockers, or safes may use a single dial which interacts with several parallel discs or cams. The remaining numbers can be arranged in (100-l)! ways. US Patents regarding combination padlocks by J.B. Gray in 1841Permutation padlock. In the case of eight prisoners, this cycle- following strategy is successful if and only if the length of the longest cycle of the permutation is at most 4. This in turn allows the owner to set a custom combination. ===Additional security=== Some rotary combination locks include internal relockers or relocking devices that separately lock the shackle or bolt when an attack is detected, including mechanical levers that respond to attempts to dislodge the locking mechanism (""punching""), thermal (fusible) links that melt in response to a cutting attempt, or tempered glass that breaks in response to a drilling attempt. ",+93.4,1.44,"""4096.0""",+116.0,−2,C
+"An urn contains eight red and seven blue balls. A second urn contains an unknown number of red balls and nine blue balls. A ball is drawn from each urn at random, and the probability of getting two balls of the same color is $151 / 300$. How many red balls are in the second urn?","Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. The probability that the second ball picked is red depends on whether the first ball was red or white. In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? The probability that the red ball is not taken in the first draw is 1000/2000 = . When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. thumb|Two urns containing white and red balls. * Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour. We want to calculate the probability that the red ball is not taken. The probability that the first ball picked is red is equal to the weight fraction of red balls: : p_1 = \frac{m_1 \omega_1}{m_1 \omega_1 + m_2 \omega_2}. What is the distribution of the number of black balls drawn after m draws? * multinomial distribution: there are balls of more than two colors. While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. Here the draws are independent and the probabilities are therefore not multiplied together. See Pólya urn model. * binomial distribution: the distribution of the number of successful draws (trials), i.e. extraction of white balls, given n draws with replacement in an urn with black and white balls. * Mixed replacement/non-replacement: the urn contains black and white balls. ",11, 258.14,"""48.6""",1.91,0.14,A
+" A typical roulette wheel used in a casino has 38 slots that are numbered $1,2,3, \ldots, 36,0,00$, respectively. The 0 and 00 slots are colored green. Half of the remaining slots are red and half are black. Also, half of the integers between 1 and 36 inclusive are odd, half are even, and 0 and 00 are defined to be neither odd nor even. A ball is rolled around the wheel and ends up in one of the slots; we assume that each slot has equal probability of $1 / 38$, and we are interested in the number of the slot into which the ball falls. Let $A=\{0,00\}$. Give the value of $P(A)$.","The payout given by the casino for a win is based on the roulette wheel having 36 outcomes, and the payout for a bet is given by \frac{36}{p}. It is worth noting that the odds for the player in American roulette are even worse, as the bet profitability is at worst -\frac{3}{38}r \approx -0.0789r, and never better than -\frac{r}{19} \approx -0.0526r. ===Simplified mathematical model=== For a roulette wheel with n green numbers and 36 other unique numbers, the chance of the ball landing on a given number is \frac{1}{(36+n)}. By law, the game must use cards and not slots on the roulette wheel to pick the winning number. ==Roulette wheel number sequence== The pockets of the roulette wheel are numbered from 0 to 36. In the United Kingdom, the farthest outside bets (low/high, red/black, even/odd) result in the player losing only half of their bet if a zero comes up. ==Bet odds table== The expected value of a $1 bet (except for the special case of Top line bets), for American and European roulette, can be calculated as :\mathrm{expected value} = \frac{1}{n} (36 - n)= \frac{36}{n} - 1, where n is the number of pockets in the wheel. Since this roulette has 37 cells with equal odds of hitting, this is a final model of field probability (\Omega, 2^\Omega, \mathbb{P}), where \Omega = \\{0, \ldots, 36\\}, \mathbb{P}(A) = \frac{|A|}{37} for all A \in 2^\Omega. Therefore, SD for Roulette even-money bet is equal to 2b\sqrt{npq}, where b is the flat bet per round, n is the number of rounds, p=18/38, and q=20/38. The roulette wheel. The ball eventually loses momentum, passes through an area of deflectors, and falls onto the wheel and into one of thirty-seven (single-zero, French or European style roulette) or thirty-eight (double-zero, American style roulette) or thirty-nine (triple- zero, ""Sands Roulette"") colored and numbered pockets on the wheel. The expected value is: :−1 × + 35 × = −0.0526 (5.26% house edge) For European roulette, a single number wins and loses : :−1 × + 35 × = −0.0270 (2.70% house edge) For triple-zero wheels, a single number wins and loses : :−1 × + 35 × = −0.0769 (7.69% house edge) ==Mathematical model== As an example, the European roulette model, that is, roulette with only one zero, can be examined. All roulette tables operated by a casino have the same basic mechanics: * There is a balanced mechanical wheel with colored pockets separated by identical vanes and the wheel which spins freely on a supporting post. Pocket number order on the roulette wheel adheres to the following clockwise sequence in most casinos: ;Single- zero wheel : 0-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23-10-5-24-16-33-1-20-14-31-9-22-18-29-7-28-12-35-3-26 ;Double-zero wheel : 0-28-9-26-30-11-7-20-32-17-5-22-34-15-3-24-36-13-1-00-27-10-25-29-12-8-19-31-18-6-21-33-16-4-23-35-14-2 ;Triple-zero wheel : 0-000-00-32-15-19-4-21-2-25-17-34-6-27-13-36-11-30-8-23-10-5-24-16-33-1-20-14-31-9-22-18-29-7-28-12-35-3-26 ==Roulette table layout== thumbnail|upright|French style layout, French single zero wheel The cloth-covered betting area on a roulette table is known as the layout. The ball landed on ""Red 7"" and Revell walked away with $270,600. ==See also== *Bauernroulette *Boule *Eudaemons *Monte Carlo Paradox *Russian roulette *Straperlo *The Gambler, a novel written by Fyodor Dostoevsky inspired by his addiction to roulette *Le multicolore; a game similar to roulette ==Notes== ==External links== Category:Gambling games Category:Roulette and wheel games Category:French inventions The sum of all the numbers on the roulette wheel (from 0 to 36) is 666, which is the ""Number of the Beast"".The last term in a sequence of partial sums composed of either sequence is 666, the ""beast number"". ==Rules of play against a casino== thumb|left|220px|Roulette with red 12 as the winner Roulette players have a variety of betting options. In some casinos, a player may bet full complete for less than the table straight-up maximum, for example, ""number 17 full complete by $25"" would cost $1000, that is 40 chips each at $25 value. ==Betting strategies and tactics== Over the years, many people have tried to beat the casino, and turn roulette—a game designed to turn a profit for the house—into one on which the player expects to win. The standard deviation for the even-money Roulette bet is one of the lowest out of all casinos games. Take the coin toss for example, the chances of heads and tails are equal, 50% each, if a player bets $10 on the coin landing heads up and they win, the casino pays them $10. Players can continue to place bets as the ball spins around the wheel until the dealer announces ""no more bets"" or ""rien ne va plus"". thumb|220px|Croupier's rake pushing chips across a roulette layout When a winning number and color is determined by the roulette wheel, the dealer will place a marker, also known as a dolly, on that winning number on the roulette table layout. In the early 1990s, Gonzalo Garcia-Pelayo believed that casino roulette wheels were not perfectly random, and that by recording the results and analysing them with a computer, he could gain an edge on the house by predicting that certain numbers were more likely to occur next than the 1-in-36 odds offered by the house suggested. According to Hoyle ""the single 0, the double 0, and eagle are never bars; but when the ball falls into either of them, the banker sweeps every thing upon the table, except what may happen to be bet on either one of them, when he pays twenty-seven for one, which is the amount paid for all sums bet upon any single figure"". thumb|left|250px|1800s engraving of the French roulette In the 19th century, roulette spread all over Europe and the US, becoming one of the most famous and most popular casino games. The initial bet is returned in addition to the mentioned payout: it can be easily demonstrated that this payout formula would lead to a zero expected value of profit if there were only 36 numbers (that is, the casino would break even). Therefore, the VI for the even-money American Roulette bet is \sqrt{18/38\cdot20/38}\approx0.499. Here, the profit margin for the roulette owner is equal to approximately 2.7%. ",5.4,1.3,"""5.5""",0.0526315789,2.3613,D
+"In the gambling game ""craps,"" a pair of dice is rolled and the outcome of the experiment is the sum of the points on the up sides of the six-sided dice. The bettor wins on the first roll if the sum is 7 or 11. The bettor loses on the first roll if the sum is 2,3 , or 12 . If the sum is $4,5,6$, 8,9 , or 10 , that number is called the bettor's ""point."" Once the point is established, the rule is as follows: If the bettor rolls a 7 before the point, the bettor loses; but if the point is rolled before a 7 , the bettor wins. Find the probability that the bettor wins on the first roll. That is, find the probability of rolling a 7 or 11 , $P(7$ or 11$)$.","Take the coin toss for example, the chances of heads and tails are equal, 50% each, if a player bets $10 on the coin landing heads up and they win, the casino pays them $10. Note: Expert bet settlers before the introduction of bet-settling software would have invariably used an algebraic- type method together with a simple calculator to determine the return on a bet (see below). ==Algebraic interpretation== If a, b, c, d... represent the decimal odds, i.e. (fractional odds + 1), then an 'odds multiplier' OM can be calculated algebraically by multiplying the expressions (a + 1), (b + 1), (c + 1), ... etc. together in the required manner and adding or subtracting additional components. If they lose, all the $10 is lost to the casino, in this case, the casino advantage is zero (the casino is certainly not stupid enough to open this game); but if they win, the casino only pays them $9, if they lose, all the $10 is lost to the casino. Of course, the casino can't win exactly 53 cents; this figure is the average casino profit from each player if it had millions of players each betting 10 rounds at $1 per round. The place part of each-way bets is calculated separately from the win part; the method is identical but the odds are reduced by whatever the place factor is for the particular event (see Accumulator below for detailed example). In games such as Blackjack or Spanish 21, the final bet may be several times the original bet, if the player doubles or splits. When gambles are selected through a choice process – when people indicate which gamble they prefer from a set of gambles (e.g., win/lose, over/under) – people tend to prefer to bet on the outcome that is more likely to occur. In algebraic terms the OM for the Yankee bet is given by: :OM = (a + 1)(b + 1)(c + 1)(d + 1) − 1 − (a + b + c + d) In the days before software became available for use by bookmakers and those settling bets in Licensed Betting Offices (LBOs) this method was virtually de rigueur for saving time and avoiding the multiple repetitious calculations necessary in settling bets of the full cover type. ==Settling other types of winning bets== Up and down :Returns (£20 single at 7-2 ATC £20 single at 15-8) = £20 × 7/2 + £20 × (15/8 + 1) = £127.50 :Returns (£20 single at 15-8 ATC £20 single at 7-2) = £20 × 15/8 + £20 × (7/2 + 1) = £127.50 :Total returns = £255.00 :Note: This is the same as two £20 single bets at twice the odds; i.e. £20 singles at 7-1 and 15-4 and is the preferred manual way of calculating the bet. Spread betting allows gamblers to wagering on the outcome of an event where the pay-off is based on the accuracy of the wager, rather than a simple ""win or lose"" outcome. Casinos do not have in-house expertise in this field, so they outsource their requirements to experts in the gaming analysis field. === Bingo probability === The probability of winning a game of Bingo (ignoring simultaneous winners, making wins mutually exclusive) may be calculated as: : P(Win)=1-P(Loss) since winning and losing are mutually exclusive. Gambling (also known as betting or gaming) is the wagering of something of value (""the stakes"") on a random event with the intent of winning something else of value, where instances of strategy are discounted. A Round Robin with 1 winner is calculated as two Up and Down bets with one winner in each. For example, if a game is played by wagering on the number that would result from the roll of one die, true odds would be 5 times the amount wagered since there is a 1/6 probability of any single number appearing. The mathematics of gambling is a collection of probability applications encountered in games of chance and can get included in game theory. Furthermore, if we flat bet at 10 units per round instead of 1 unit, the range of possible outcomes increases 10 fold. If a player bets $1 on red, his chance of winning $1 is therefore 18/38 and his chance of losing $1 (or winning -$1) is 20/38. Thus, an each-way Lucky 63 on six horses with three winners and a further two placed horses is settled as a win Patent and a place Lucky 31. ==Algebraic interpretation== Returns on any bet may be considered to be calculated as 'stake unit' × 'odds multiplier'. Gambling is not luck, but a contest of intellect, strategy, and yield. All bets are taken as 'win' bets unless 'each-way' is specifically stated. A Round Robin with 2 winners is calculated as a double plus one Up and Down bet with 2 winners plus two Up and Down bets with 1 winner in each. * Carnival Games such as The Razzle or Hanky Pank * Coin-tossing games such as Head and Tail, Two-up* * Confidence tricks such as Three-card Monte or the Shell game * Dice-based games, such as Backgammon, Liar's dice, Passe-dix, Hazard, Threes, Pig, or Mexico (or Perudo); *Although coin tossing is not usually played in a casino, it has been known to be an official gambling game in some Australian casinos ===Fixed-odds betting=== Fixed-odds betting and Parimutuel betting frequently occur at many types of sporting events, and political elections. The end-of-the-day betting effect is a cognitive bias reflected in the tendency for bettors to take gambles with higher risk and higher reward at the end of their betting session to try to make up for losses. ",0.5117,0,"""0.6247""",11,0.22222222,E
+"Given that $P(A \cup B)=0.76$ and $P\left(A \cup B^{\prime}\right)=0.87$, find $P(A)$.","That is, for an event A, :P(A^c) = 1 - P(A). 1983 Emperor's Cup Final was the 63rd final of the Emperor's Cup competition. It may be tempting to say that : Pr([""1"" on 1st trial] or [""1"" on second trial] or ... or [""1"" on 8th trial]) := Pr(""1"" on 1st trial) + Pr(""1"" on second trial) + ... + P(""1"" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. The 1983–84 Gold Cup was the 65th edition of the Gold Cup, a cup competition in Northern Irish football. Therefore, the probability of an event's complement must be unity minus the probability of the event. The 1983–84 Ulster Cup was the 36th edition of the Ulster Cup, a cup competition in Northern Irish football. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. The 1983–84 Irish Cup was the 104th edition of the Irish Cup, Northern Ireland's premier football knock-out cup competition. The 1884–85 Belfast Charity Cup was the 2nd edition of the Belfast Charity Cup, a cup competition in Irish football. Equivalently, the probabilities of an event and its complement must always total to 1. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. This result cannot be right because a probability cannot be more than 1. One may resolve this overlap by the principle of inclusion- exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: : Pr(at least one ""1"") = 1 − Pr(no ""1""s) := 1 − Pr([no ""1"" on 1st trial] and [no ""1"" on 2nd trial] and ... and [no ""1"" on 8th trial]) := 1 − Pr(no ""1"" on 1st trial) × Pr(no ""1"" on 2nd trial) × ... × Pr(no ""1"" on 8th trial) := 1 −(5/6) �� (5/6) × ... × (5/6) := 1 − (5/6)8 := 0.7674... ==See also== *Logical complement *Exclusive disjunction *Binomial probability ==References== ==External links== *Complementary events - (free) page from probability book of McGraw-Hill Category:Experiment (probability theory) The complement of an event A is usually denoted as A′, Ac, egA or . For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. Oldpark won the tournament for the 1st time, defeating Cliftonville 1–0 in the final. ==Results== ===Semi- finals=== |} ====Replay==== |} ===Final=== ==References== ==External links== * Northern Ireland - List of Belfast Charity Cup Winners Category:1884–85 in Irish association football What is the probability that one sees a ""1"" at least once? Given an event, the event and its complementary event define a Bernoulli trial: did the event occur or not? This does not, however, mean that any two events whose probabilities total to 1 are each other's complements; complementary events must also fulfill the condition of mutual exclusivity. ===Example of the utility of this concept=== Suppose one throws an ordinary six-sided die eight times. Ballymena United won their fourth Irish Cup, defeating Carrick Rangers 4–1 in the final. ==Results== ===First round=== |} ====Replays==== |} ===Second round=== |} ====Replays==== |} ====Second replay==== |} ===Quarter-finals=== |} ===Semi-finals=== |} ===Final=== ==References== 1983–84 Category:1983–84 domestic association football cups Category:1983–84 in Northern Ireland association football Category:1983 in Northern Ireland sport Category:1984 in Northern Ireland sport ",-87.8,0.63,"""205.0""",2.84367,0.72,B
+Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?,"Table of Six () is a political conference established by the Republican People's Party, Good Party, Felicity Party, Democrat Party, Democracy and Progress Party and Future Party, with the first meeting held on 12 February 2022. In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. Table topics are topics on various subjects that are discussed by a group of people around a table. There will be a table topic master for each meeting, who will prepare questions beforehand and ask the participants questions one by one for which they are called upon to answer. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bézout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 65 = 7776, which was Steiner's incorrect solution. If the five conics have the properties that *there is no line such that every one of the 5 conics is either tangent to it or passes through one of two fixed points on it (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics) *no three of the conics pass through any point (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics passing through this triple intersection point) *no two of the conics are tangent *no three of the five conics are tangent to a line *a pair of lines each tangent to two of the conics do not intersect on the fifth conic (otherwise this pair is a degenerate conic tangent to all 5 conics) then the total number of conics C tangent to all 5 (counted with multiplicities) is 3264. The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848. ==History== claimed that the number of conics tangent to 5 given conics in general position is 7776 = 65, but later realized this was wrong. Many personality or public speaking clubs like the 'Toastmasters' have a separate session in their meetings known as a table topic session. Student Number 1 was panned by critics and flopped at the box office. ==Cast== ==Production== Shooting was commenced at Chennai, for a fifteen-days schedule, after which the unit moved to Russia to shoot two songs. The Table of Six was originally an independent entity from the Nation Alliance. In particular if C intersects each of the five conics in exactly 3 points (one double point of tangency and two others) then the multiplicity is 1, and if this condition always holds then there are exactly 3264 conics tangent to the 5 given conics. Some chapters of Toastmasters also host Table Topics contests. ==See also== * TableTopics ==References== Category:Public speaking However the number of conics is not (6H)5 but (6H−2E)5 because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E. On 21 January 2023, Table of Six defined itself as the ""Nation Alliance"" for the first time after its 11th meeting. As practiced by Toastmasters International, the topics to be discussed are written on pieces of paper which are placed in a box in the middle of a table. Five Guys Walk into a Bar... has received a largely positive response from critics since its release. The participants pick up one paper each and start talking about the topic written on the paper. Graphs and Combinatorics (ISSN 0911-0119, abbreviated Graphs Combin.) is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. In the text of the memorandum, lowering the electoral threshold to 3%, treasury aid to the parties that received at least 1% of the votes, ending the omnibus law practice, removing the veto power of the president and extending his term of office to 7 years, recognising the authority to issue a no-confidence question on the government, human rights and human rights in the education curriculum. Steiner observed that the conics tangent to a given conic form a degree 6 hypersurface in CP5. Five Guys Walk into a Bar... is a comprehensive four-disc retrospective of the British rock group Faces released in 2004, collecting sixty-seven tracks from among the group's four studio albums, assorted rare single A and B-sides, BBC sessions, rehearsal tapes and one track from a promotional flexi-disc, ""Dishevelment Blues"" - a deliberately-sloppy studio romp, captured during the sessions for their Ooh La La album, which was never actually intended for official release. On 3 March 2023, Good Party leader Meral Akşener announced that she took the decision to withdraw from the Table of Six and said her party would not support main opposition Republican People's Party leader Kemal Kılıçdaroğlu as the joint candidate in the 2023 Turkish presidential election. ",-0.55,0.00539,"""1.4""",+116.0,-0.10,B
+How many different license plates are possible if a state uses two letters followed by a four-digit integer (leading zeros are permissible and the letters and digits can be repeated)?,"In New Zealand, vehicle registration plates (usually called number plates) contain up to six alphanumeric characters, depending on the type of vehicle and the date of registration. The vehicle registration plates of Cyprus are composed of three letters and three digits (e.g. ABC 123). Greek vehicle registration plates are composed of three letters and four digits per plate (e.g. `ΑΑΑ–1000`) printed in black on a white background. From 1964 until March 2001 these number plates had two letters followed by one to four numbers (format LLnnnn), the sequence having started with AA1 and continuing through to ZZ9989 chronologically (for example, XE3782 would have been issued in 1998). Each of the 49 states of the United States of America plus several of its territories and the District of Columbia issued individual passenger license plates for the year 1959. ==Passenger baseplates== Image Region Design Slogan Serial format Serials issued Notes 150px Alabama Embossed blue lettering, heart logo and rims on white base. The letters represent the district (prefecture) that issues the plates while the numbers range from 1000 to 9999. None 123-456 1 to approximately 999-999 150px Washington Embossed white numbers on green plate; ""54 WASHINGTON"" embossed in white block letters at bottom. Number plates had to be changed before the end of January 2019. 1973 - 2018 150px 150px 2018 - today 150px ===Special plates=== Taxis Taxi plates show the prefix T, followed by two letters and three digits, formerly one letter only. Since 2013, some numbers have been reissued on white plates. 2013 - today Composed of five numerals and the prefix P, on white plates. The final one or two letters in the sequence changes in Greek alphabetical order after 8,999 issued plates. Similar plates but of square size with numbers ranging from 1 to 999 are issued for motorcycles which exceed 50 cc in engine size. If the visibility of a regular number plate is obstructed, for example by a bike rack mounted to a car's trailer hitch, a supplementary plate with the same registration number must be obtained and affixed to the obstruction (or the vehicle) such that it will be visible from the same direction as the regular number plate would have been. == Standard numbering sequences == thumb|right|A vehicle registration plate of New Zealand in the optional 'Europlate' style === Cars and heavy vehicles === * 1964–1987: AAnnnn * 1987–2001: AAnnnn * 2001–present: AAAnnn Private cars, taxis, and heavier road vehicles in New Zealand have number plates with up to six characters. Hence these so-called ""six-figure plates"" can still occasionally (as of 2018) be spotted on a few old vehicles. ===1973–1985=== In 1972, they became lettered and the system was `LL–NNNN` while trucks used `L–NNNN`. In some later instances issuers coded plates to the area of registration, such as in 1966 with the allocation of plates beginning with ""CE"" to the Manawatu-Wanganui region, in 1974–1976 with the allocation of plates beginning with ""HB"" to the Hawke's Bay region, in May 1989 with the allocation of plates beginning with ""OG"" to Wellington region, and in July 2000 with the allocation of plates beginning with ""ZI"" to Auckland region. === Motorcycles and tractors === These vehicles use one of several five-character systems. Some early numbers were printed on remaining yellow plates. === Commercial trucks === 1973 - 1990 1990 - 2003 2003 - 2013 2013 - today Composed of three letters and three numerals, on yellow plates. None 1-12345 10-1234 County-coded ==Non-passenger plates== Image (standard) Region Type Design and slogan Serial format Serials issued Notes 150px ==See also== *Antique vehicle registration *Electronic license plate *Motor vehicle registration *Vehicle license ==References== ==External links== Category:1959 in the United States 1959 Although plate character/number combinations can contain ""spaces"", they do not form part of the unique identification and are typically not stored (for example, in Police computer-systems). Cypriot National Guard plates are composed of an old version of the Greek flag followed by the letters ΕΦ and up to five digits. ==Northern Cyprus== thumb|upright|Northern Cyprus plate thumb|upright|Northern Cyprus rear plate thumb|Taxi number plate (T) thumb|upright=0.5|Trailer number plate (R) thumb|Rental car number plate thumb|Police number plate ===Style and numbering=== Northern Cyprus civilian number plates still use the old format (1973–1990) of Cyprus number plates (AB 123). White front plates were omitted after 2013. thumb|Temporary number plates === Temporary / Visitors === 1973 - 2003 Up to four numerals followed by the letter V, followed by two numerals indicating the year of registration. None 1-1234 1-A123 12-1234 12-A123 Coded by county (1 or 10 prefix). 1957 base plates revalidated for 1959 with red tabs. 150px Tennessee Embossed yellow numbers on black state- shaped plate with border line; ""TENN. 54"" embossed in yellow block letters centered at bottom. None AA1 to WW999 Letters I, Q and U not used, and X, Y and Z used only on replacement plates. 150px South Carolina Embossed white numbers on blue plate; ""SOUTH CAROLINA 59"" embossed in white block letters at top. Export plates, from 1973 until 1990, showed the letter E followed by four numerals. ",0.333333333333333,0.03,"""6760000.0""",0.16,1.27,C
 "Let $A$ and $B$ be independent events with $P(A)=$ 0.7 and $P(B)=0.2$. Compute $P(A \cap B)$.
-","In this event, the event B can be analyzed by a conditional probability with respect to A. * This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" All events that are not in B will have null probability in the new distribution. These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. This particular method relies on event B occurring with some sort of relationship with another event A. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. It can be interpreted as ""the probability of B occurring multiplied by the probability of A occurring, provided that B has occurred, is equal to the probability of the A and B occurrences together, although not necessarily occurring at the same time"". P(\text{dot received}) = P(\text{dot received } \cap \text{ dot sent}) + P(\text{dot received } \cap \text{ dash sent}) P(\text{dot received}) = P(\text{dot received } \mid \text{ dot sent})P(\text{dot sent}) + P(\text{dot received } \mid \text{ dash sent})P(\text{dash sent}) P(\text{dot received}) = \frac{9}{10}\times\frac{3}{7} + \frac{1}{10}\times\frac{4}{7} = \frac{31}{70} Now, P(\text{dot sent } \mid \text{ dot received}) can be calculated: P(\text{dot sent } \mid \text{ dot received}) = P(\text{dot received } \mid \text{ dot sent}) \frac{P(\text{dot sent})}{P(\text{dot received})} = \frac{9}{10}\times \frac{\frac{3}{7}}{\frac{31}{70}} = \frac{27}{31} == Statistical independence == Events A and B are defined to be statistically independent if the probability of the intersection of A and B is equal to the product of the probabilities of A and B: :P(A \cap B) = P(A) P(B). The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . That is, for an event A, :P(A^c) = 1 - P(A). In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. Moreover, this ""multiplication rule"" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} ==== As the probability of a conditional event ==== Conditional probability can be defined as the probability of a conditional event A_B. ",0.323,0.14,11.0,322,1.44,B
-"Suppose that $A, B$, and $C$ are mutually independent events and that $P(A)=0.5, P(B)=0.8$, and $P(C)=$ 0.9 . Find the probabilities that all three events occur?","Event \operatorname{P}(s_1)=1/4 \operatorname{P}(s_2)=1/4 \operatorname{P}(s_3)=1/4 \operatorname{P}(s_4)=1/4 Probability of event A 0 1 0 1 \tfrac{1}{2} B 0 0 1 1 \tfrac{1}{2} C 0 1 1 1 \tfrac{3}{4} and so Event s_1 s_2 s_3 s_4 Probability of event A \cap B 0 0 0 1 \tfrac{1}{4} A \cap C 0 1 0 1 \tfrac{1}{2} B \cap C 0 0 1 1 \tfrac{1}{2} A \cap B \cap C 0 0 0 1 \tfrac{1}{4} In this example, C occurs if and only if at least one of A, B occurs. Unconditionally (that is, without reference to C), A and B are independent of each other because \operatorname{P}(A)—the sum of the probabilities associated with a 1 in row A—is \tfrac{1}{2}, while \operatorname{P}(A\mid B) = \operatorname{P}(A \text{ and } B) / \operatorname{P}(B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} = \operatorname{P}(A). Since in the presence of C the probability of A is affected by the presence or absence of B, A and B are mutually dependent conditional on C. == See also == * * * == References == Category:Independence (probability theory) When A and B are mutually exclusive, .Stats: Probability Rules. But conditional on C having occurred (the last three columns in the table), we have \operatorname{P}(A \mid C) = \operatorname{P}(A \text{ and } C) / \operatorname{P}(C) = \tfrac{1/2}{3/4} = \tfrac{2}{3} while \operatorname{P}(A \mid C \text{ and } B) = \operatorname{P}(A \text{ and } C \text{ and } B) / \operatorname{P}(C \text{ and } B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} < \operatorname{P}(A \mid C). The probability of one or both events occurring is denoted P(A ∪ B) and in general, it equals P(A) + P(B) – P(A ∩ B). In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 ""Unit 3: Conditional Dependence""Introduction to learning Bayesian Networks from Data by Dirk Husmeier ""Introduction to Learning Bayesian Networks from Data -Dirk Husmeier"" For example, if A and B are two events that individually increase the probability of a third event C, and do not directly affect each other, then initially (when it has not been observed whether or not the event C occurs)Conditional Independence in Statistical theory ""Conditional Independence in Statistical Theory"", A. P. Dawid"" Probabilistic independence on Britannica ""Probability->Applications of conditional probability->independence (equation 7) "" \operatorname{P}(A \mid B) = \operatorname{P}(A) \quad \text{ and } \quad \operatorname{P}(B \mid A) = \operatorname{P}(B) (A \text{ and } B are independent). If event B occurs then the probability of occurrence of the event A will decrease because its positive relation to C is less necessary as an explanation for the occurrence of C (similarly, event A occurring will decrease the probability of occurrence of B). The probability that at least one of the events will occur is equal to one.Scott Bierman. Conditional dependence of A and B given C is the logical negation of conditional independence ((A \perp\\!\\!\\!\perp B) \mid C). In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.intmath.com; Mutually Exclusive Events. In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. The probabilities of the individual events (red, and club) are multiplied rather than added. Hence, now the two events A and B are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. Obviously, we get the following probabilities :\mathbb P(A_1) = \frac 4{52}, \qquad \mathbb P(A_2 \mid A_1) = \frac 3{51}, \qquad \mathbb P(A_3 \mid A_1 \cap A_2) = \frac 2{50}, \qquad \mathbb P(A_4 \mid A_1 \cap A_2 \cap A_3) = \frac 1{49}. We haveIntroduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 ""Unit 3: Explaining Away"" \operatorname{P}(A \mid C \text{ and } B) < \operatorname{P}(A \mid C). When heads occurs, tails can't occur, or p (heads and tails) = 0, so the outcomes are also mutually exclusive. In the case of flipping a coin, flipping a head and flipping a tail are also mutually exclusive events. Another example of events being collectively exhaustive and mutually exclusive at same time are, event ""even"" (2,4 or 6) and event ""odd"" (1,3 or 5) in a random experiment of rolling a six-sided die. The events 1 and 6 are mutually exclusive but not collectively exhaustive. ",0.6749,10.4,0.36,4,5275,C
-A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of four of a kind (four cards of equal face value and one card of a different value).,"*The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. There are 7,462 distinct poker hands. ===7-card poker hands=== In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall,https://www.casinodaniabeach.com/most-popular-types-of-poker/ a player uses the best five-card poker hand out of seven cards. Hand The five cards (or less) dealt on the screen are known as a hand. ==See also== *Casino comps *Draw poker *Gambling *Gambling mathematics *Problem gambling *Video blackjack *Video Lottery Terminal ==References== ==External links== * Category:Arcade video games Various payout variations are possible, depending on the casino, resulting in a house edge ranging from 1.98% to 6.15%.Wizard of Odds: Four Card Poker ==Rank of hands== The possible four-card hands are (from best to worst): *Four of a kind *Straight flush *Three of a kind *Flush *Straight *Two pair *One pair *High card ==References== ==External links== *ShuffleMaster flash demo of the game Category:Gambling games Category:Poker variants The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. The Total line also needs adjusting. ==See also== * Binomial coefficient * Combination * Combinatorial game theory * Effective hand strength algorithm * Event (probability theory) * Game complexity * Gaming mathematics * Odds * Permutation * Probability * Sample space * Set theory ==References== ==External links== * Brian Alspach's mathematics and poker page * MathWorld: Poker * Poker probabilities including conditional calculations * Numerous poker probability tables * 5, 6, and 7 card poker probabilities * Hold'em poker probabilities All the other hand combinations in video poker are the same as in table poker, including such hands as two pair, three of a kind, straight (a sequence of 5 cards of consecutive value), flush (any 5 cards of the same suit), full house (a pair and a three of a kind), four of a kind (four cards of the same value), straight flush (5 consecutive cards of the same suit) and royal flush (a Ten, a Jack, a Queen, a King and an Ace of the same suit). frameless|right Four Card Poker is a casino card game similar to Three Card Poker, invented by Roger Snow and owned by Shuffle Master.ShuffleMaster: Four Card Poker ==Description of play== The player can place an ante bet or an ""Aces Up"" bet or both. A four flush (also flush draw) is a poker draw or non-standard poker hand that is one card short of being a full flush. Video poker is a casino game based on five-card draw poker. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. Four of a kind may refer to: *Four of a kind (poker), a type of poker hand *Four of a Kind (card game), a patience or solitaire *Four of a Kind (TV series), an American reality series about quadruplets *Four of a Kind (film), an Australian feature film *4 of a Kind, the fourth album by American thrash band D.R.I. For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. The number of distinct poker hands is even smaller. This pejorative term originated in the 19th century when bluffing poker players misrepresented that they had a flush—a poker hand with five cards all of one suit—when they only had four cards of one suit. Flush A five-card hand that contains cards of the same suit. ",0.00024,-1.49,0.00131,1.51,-167,A
-Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?,"Table of Six () is a political conference established by the Republican People's Party, Good Party, Felicity Party, Democrat Party, Democracy and Progress Party and Future Party, with the first meeting held on 12 February 2022. thumb|Table seating arrangement A seating plan is a diagram or a set of written or spoken instructions that determines where people should take their seats. Table topics are topics on various subjects that are discussed by a group of people around a table. In this case, it is customary to arrange the host and hostess at the opposite sides of the table, and alternate male and female guests throughout. In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. There will be a table topic master for each meeting, who will prepare questions beforehand and ask the participants questions one by one for which they are called upon to answer. A seating plan is of crucial importance for musical ensembles or orchestras, where every type of instrument is allocated a specific section. == See also == * Seating assignment * Seating capacity * Table setting * Kids' table == References == ==External links == * Category:Etiquette Category:Diagrams Plan If the five conics have the properties that *there is no line such that every one of the 5 conics is either tangent to it or passes through one of two fixed points on it (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics) *no three of the conics pass through any point (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics passing through this triple intersection point) *no two of the conics are tangent *no three of the five conics are tangent to a line *a pair of lines each tangent to two of the conics do not intersect on the fifth conic (otherwise this pair is a degenerate conic tangent to all 5 conics) then the total number of conics C tangent to all 5 (counted with multiplicities) is 3264. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bézout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 65 = 7776, which was Steiner's incorrect solution. Many personality or public speaking clubs like the 'Toastmasters' have a separate session in their meetings known as a table topic session. The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848. ==History== claimed that the number of conics tangent to 5 given conics in general position is 7776 = 65, but later realized this was wrong. Honored guests (moms, dads, and in-laws) are placed to the host's and hostess's right and then left."" Student Number 1 was panned by critics and flopped at the box office. ==Cast== ==Production== Shooting was commenced at Chennai, for a fifteen-days schedule, after which the unit moved to Russia to shoot two songs. The Table of Six was originally an independent entity from the Nation Alliance. Some chapters of Toastmasters also host Table Topics contests. ==See also== * TableTopics ==References== Category:Public speaking In particular if C intersects each of the five conics in exactly 3 points (one double point of tangency and two others) then the multiplicity is 1, and if this condition always holds then there are exactly 3264 conics tangent to the 5 given conics. However the number of conics is not (6H)5 but (6H−2E)5 because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E. In the United States according to Peggy Post, ""tradition dictates that when everyone is seated together, the host and hostess sit at either end of the table. On 21 January 2023, Table of Six defined itself as the ""Nation Alliance"" for the first time after its 11th meeting. As practiced by Toastmasters International, the topics to be discussed are written on pieces of paper which are placed in a box in the middle of a table. In the text of the memorandum, lowering the electoral threshold to 3%, treasury aid to the parties that received at least 1% of the votes, ending the omnibus law practice, removing the veto power of the president and extending his term of office to 7 years, recognising the authority to issue a no-confidence question on the government, human rights and human rights in the education curriculum. The participants pick up one paper each and start talking about the topic written on the paper. ",22,432.07,0.00539,1.01,362880,E
-"Each of the 12 students in a class is given a fair 12 -sided die. In addition, each student is numbered from 1 to 12 . If the students roll their dice, what is the probability that there is at least one ""match"" (e.g., student 4 rolls a 4)?","Numbers on each die Die 1 1 5 10 11 13 17 Die 2 3 4 7 12 15 16 Die 3 2 6 8 9 14 18 === Four players === An optimal and permutation-fair solution for 4 twelve-sided dice was found by Robert Ford in 2010. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Die 1 1 4 Die 2 2 3 === Three players === An optimal and permutation- fair solution for 3 six-sided dice was found by Robert Ford in 2010. If r is the total number of dice selecting the 6 face, then P(r \ge k ; n, p) is the probability of having at least k correct selections when throwing exactly n dice. Many players collect or acquire a large number of mixed and unmatching dice. The d20 System includes a four-sided tetrahedral die among other dice with 6, 8, 10, 12 and 20 faces. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline. ====Common variations==== Dice are often sold in sets, matching in color, of six different shapes. Numbers on each die Die 1 1 8 11 14 19 22 27 30 35 38 41 48 Die 2 2 7 10 15 18 23 26 31 34 39 42 47 Die 3 3 6 12 13 17 24 25 32 36 37 43 46 Die 4 4 5 9 16 20 21 28 29 33 40 44 45 === Five players === Several candidates exist for a set of 5 dice, but none is known to be optimal. == See also == * Intransitive dice ==References== ==External links== * Go First Dice - Numberphile Category:Dice Some dice, such as those with 10 sides, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces. Note the older hand-inked green 12-sided die (showing an 11), manufactured before pre-inked dice were common. Normally, the faces on a die will be placed so opposite faces will add up to one more than the number of faces. These are six-sided dice with sides numbered `2, 3, 3, 4, 4, 5`, which have the same arithmetic mean as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for a pair). Optimal results have been proven by exhaustion for up to 4 dice. ==Configurations== === Two players === The two player case is somewhat trivial. On some four- sided dice, each face features multiple numbers, with the same number printed near each vertex on all sides. The sum of the numbers on opposite faces is 21 if numbered 1–20. ====Rarer variations==== upright=2.9|thumb|Dice collection: D2–D22, D24, D26, D28, D30, D36, D48, D60 and D100. If several dice of the same type are to be rolled, this is indicated by a leading number specifying the number of dice. Due to circumstances or character skill, the initial roll may have a number added to or subtracted from the final result, or have the player roll extra or fewer dice. This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem. ==Generalizations== A natural generalization of the problem is to consider n non-necessarily fair dice, with p the probability that each die will select the 6 face when thrown (notice that actually the number of faces of the dice and which face should be selected are irrelevant). Another configuration places only one number on each face, and the rolled number is taken from the downward face. ==References== Category:Dice In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: :P(n)=1-\sum_{x=0}^{n-1}\binom{6n}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{6n-x}\, . ""Uniform fair dice"" are dice where all faces have equal probability of outcome due to the symmetry of the die as it is face-transitive. Configurations where all die have the same number of sides are presented here, but alternative configurations might instead choose mismatched dice to minimize the number of sides, or minimize the largest number of sides on a single die. ",273,0.648004372,0.00024,+11,0.178,B
-"The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes four games?","In Major League Baseball (MLB), a game seven can occur in the World Series or in a League Championship Series (LCS), which are contested as best-of-seven series. From 2003 to 2010, the AL and NL had each won the World Series four times, but none of them had gone the full seven games. In baseball, a series refers to two or more consecutive games played between the same two teams. During the Major League Baseball Postseason, there are four wild card series (two in each League), each of which are a best-of-3 series. The World Series has been contested 118 times as of 2022, with the AL winning 67 and the NL winning 51. == Precursors to the modern World Series (1857–1902) == === The original World Series === Until the formation of the American Association in 1882 as a second major league, the National Association of Professional Base Ball Players (1871–1875) and then the National League (founded 1876) represented the top level of organized baseball in the United States. These 16 franchises, all of which are still in existence, have each won at least two World Series titles. The National League Championship Series (NLCS) and American League Championship Series (ALCS), since the expansion to best-of-seven, are always played in a 2–3–2 format: Games 1, 2, 6, and 7 are played in the stadium of the team that has home-field advantage, and Games 3, 4, and 5 are played in the stadium of the team that does not. === 1970s === ==== 1971: World Series at night ==== Night games were played in the major leagues beginning with the Cincinnati Reds in 1935, but the World Series remained a strictly daytime event for years thereafter. The two division winners within each league played each other in a best-of-five League Championship Series to determine who would advance to the World Series. Since then, the 2011, 2014, 2016, 2017, and 2019 World Series have gone the full seven games. When the first modern World Series was played in 1903, there were eight teams in each league. This is known in baseball as a road trip, and a team can be on the road for up to 20 games, or 4-5 series. This is the only time in World Series history in which three teams have won consecutive series in succession. A game seven cannot occur in earlier rounds of the MLB postseason, as Division Series and Wild Card rounds use shorter series. ==Key== (#) Extra innings (the number indicates the number of extra innings played) † Indicates the team that won a game seven after coming back from an 0–3 series deficit § Indicates the team that lost a game seven after coming back from an 0–3 series deficit ∞ Indicates a game seven that was played at a neutral site Road* Indicates a game seven that was won by the road team Year (X) Indicates the number of games seven played in that year's postseason (from 1985 on) Each year is linked to an article about that particular Major League Baseball season Team (#) Indicates team and the number of game sevens played by that team at that point ==All-time game sevens== Year Playoff round Date Venue Winner Result Loser Ref. World Series Bennett Park Pittsburgh Pirates (1) 8–0 Detroit Tigers (1) World Series Fenway Park Boston Red Sox (1) 3–2 (10) New York Giants (1) World Series Griffith Stadium Washington Senators (1) 4–3 (12) New York Giants (2) World Series Forbes Field Pittsburgh Pirates (2) 9��7 Washington Senators (2) World Series Yankee Stadium (I) St. Louis Cardinals (1) 3–2 New York Yankees (1) World Series Sportsman's Park (III) St. Louis Cardinals (2) 4–2 Philadelphia Athletics (1) World Series Navin Field St. Louis Cardinals (3) 11–0 Detroit Tigers (2) World Series Crosley Field Cincinnati Reds (1) 2–1 Detroit Tigers (3) World Series Wrigley Field Detroit Tigers (4) 9–3 Chicago Cubs (1) World Series Sportsman's Park (III) St. Louis Cardinals (4) 4–3 Boston Red Sox (2) World Series Yankee Stadium (I) New York Yankees (2) 5–2 Brooklyn Dodgers (1) World Series Ebbets Field New York Yankees (3) 4–2 Brooklyn Dodgers (2) World Series Yankee Stadium (I) Brooklyn Dodgers (3) 2–0 New York Yankees (4) World Series Ebbets Field New York Yankees (5) 9–0 Brooklyn Dodgers (4) World Series Yankee Stadium (I) Milwaukee Braves (1) 5–0 New York Yankees (6) World Series Milwaukee County Stadium New York Yankees (7) 6–2 Milwaukee Braves (2) World Series Forbes Field Pittsburgh Pirates (3) 10–9 New York Yankees (8) World Series Candlestick Park New York Yankees (9) 1–0 San Francisco Giants (3) World Series Busch Stadium (I) St. Louis Cardinals (5) 7–5 New York Yankees (10) World Series Metropolitan Stadium Los Angeles Dodgers (5) 2–0 Minnesota Twins (3) World Series Fenway Park St. Louis Cardinals (6) 7–2 Boston Red Sox (3) World Series Busch Stadium (II) Detroit Tigers (5) 4–1 St. Louis Cardinals (7) World Series Memorial Stadium Pittsburgh Pirates (4) 2–1 Baltimore Orioles (1) World Series Riverfront Stadium Oakland Athletics (2) 3–2^ Cincinnati Reds (2) World Series Oakland–Alameda County Coliseum Oakland Athletics (3) 5–2 New York Mets (1) World Series Fenway Park Cincinnati Reds (3) 4–3 Boston Red Sox (4) World Series Memorial Stadium Pittsburgh Pirates (5) 4–1 Baltimore Orioles (2) World Series Busch Stadium (II) St. Louis Cardinals (8) 6–3 Milwaukee Brewers (1) ALCS Exhibition Stadium Kansas City Royals (1) 6–2 Toronto Blue Jays (1) World Series Royals Stadium Kansas City Royals (2) 11–0 St. Louis Cardinals (9) ALCS Fenway Park Boston Red Sox (5) 8–1 California Angels (1) World Series Shea Stadium New York Mets (2) 8–5 Boston Red Sox (6) NLCS Busch Stadium (II) St. Louis Cardinals (10) 6–0 San Francisco Giants (4) World Series Hubert H. Humphrey Metrodome Minnesota Twins (4) 4–2 St. Louis Cardinals (11) NLCS Dodger Stadium Los Angeles Dodgers (6) 6–0 New York Mets (3) NLCS Three Rivers Stadium Atlanta Braves (3) 4–0 Pittsburgh Pirates (6) World Series Hubert H. Humphrey Metrodome Minnesota Twins (5) 1–0 (10) Atlanta Braves (4) NLCS Atlanta–Fulton County Stadium Atlanta Braves (5) 3–2 Pittsburgh Pirates (7) NLCS Atlanta–Fulton County Stadium Atlanta Braves (6) 15–0 St. Louis Cardinals (12) World Series Pro Player Stadium Florida Marlins (1) 3–2 (11) Cleveland Indians (1) World Series Bank One Ballpark Arizona Diamondbacks (1) 3–2 New York Yankees (11) World Series Edison International Field Anaheim Angels (2) 4–1 San Francisco Giants (5) NLCS Wrigley Field Florida Marlins (2) 9–6 Chicago Cubs (2) ALCS Yankee Stadium (I) New York Yankees (12) 6–5 (11) Boston Red Sox (7) ALCS Yankee Stadium (I) Boston Red Sox (8)† 10–3 New York Yankees (13) NLCS Busch Stadium (II) St. Louis Cardinals (13) 5–2 Houston Astros (1) NLCS Shea Stadium St. Louis Cardinals (14) 3–1 New York Mets (4) ALCS Fenway Park Boston Red Sox (9) 11–2 Cleveland Indians (2) ALCS Tropicana Field Tampa Bay Rays (1) 3–1 Boston Red Sox (10) World Series Busch Stadium St. Louis Cardinals (15) 6–2 Texas Rangers (1) NLCS AT&T; Park San Francisco Giants (6) 9–0 St. Louis Cardinals (16) World Series Kaufmann Stadium San Francisco Giants (7) 3–2 Kansas City Royals (2) World Series Progressive Field Chicago Cubs (3) 8–7 (10) Cleveland Indians (3) ALCS Minute Maid Park Houston Astros (2) 4–0 New York Yankees (14) World Series Dodger Stadium Houston Astros (3) 5–1 Los Angeles Dodgers (7) NLCS Miller Park Los Angeles Dodgers (8) 5–1 Milwaukee Brewers (2) World Series Minute Maid Park Washington Nationals (1) 6–2 Houston Astros (4) ALCS Petco Park∞ Tampa Bay Rays (2) 4–2 Houston Astros (5)§ NLCS Globe Life Field∞ Los Angeles Dodgers (9) 4–3 Atlanta Braves (7) ==All-time standings== Team Games played Wins Losses Win–loss % St. Louis Cardinals 16 11 5 New York Yankees 14 6 8 Boston Red Sox 10 4 6 Brooklyn / Los Angeles Dodgers 9 5 4 Pittsburgh Pirates 7 5 2 Milwaukee / Atlanta Braves 7 4 3 New York / San Francisco Giants 7 2 5 Washington Senators / Minnesota Twins 5 3 2 Detroit Tigers 5 2 3 Houston Astros 5 2 3 New York Mets 4 1 3 Cincinnati Reds 3 2 1 Kansas City Royals 3 2 1 Chicago Cubs 3 1 2 Philadelphia / Oakland Athletics 3 1 2 Cleveland Indians / Guardians 3 0 3 Florida Marlins 2 2 0 Tampa Bay Rays 2 2 0 California / Anaheim Angels 2 1 1 Milwaukee Brewers 2 0 2 Arizona Diamondbacks 1 1 0 Washington Nationals 1 1 0 Texas Rangers 1 0 1 Toronto Blue Jays 1 0 1 Note: Five teams have never played a game seven: Philadelphia Phillies, Chicago White Sox, San Diego Padres, Seattle Mariners, Colorado Rockies. ESPN selected it as the ""Greatest of All Time"" in their ""World Series 100th Anniversary"" countdown, with five of its games being decided by a single run, four games decided in the final at-bat and three games going into extra innings. The remainder of the Postseason consists of the League Division Series, which is a best-of-5 series, and the League Championship Series, which is a best-of-7 series, followed by the World Series, a best-of-7 series to determine the Major League Baseball Champion. There are only two other occasions when a team has won at least three consecutive World Series: 1972 to 1974 by the Oakland Athletics, and 1998 to 2000 by the Yankees. ==== 1947–1964: New York City teams dominate World Series play ==== In an 18-year span from 1947 to 1964, except for 1948 and 1959, the World Series was played in New York City, featuring at least one of the three teams located in New York at the time. Source: MLB.com ;Notes American League (AL) teams have won 67 of the 118 World Series played (56.8%). Starting in 1976, the DH rule was used in the World Series held in even-numbered years. The World Series is the annual championship series of Major League Baseball (MLB) in the United States and Canada. Since then each league has conducted a League Championship Series (ALCS and NLCS) preceding the World Series to determine which teams will advance, while those series have been preceded in turn by Division Series (ALDS and NLDS) since 1995, and Wild Card games or series in each league since 2012. Historically and currently, professional baseball season revolves around a schedule of series, each typically lasting three or four games. The ""series"" schedule gives its name to the MLB championship series, the World Series. ",2,16,26.9,1.6,4.8,A
+","In this event, the event B can be analyzed by a conditional probability with respect to A. * This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" All events that are not in B will have null probability in the new distribution. These probabilities are linked through the law of total probability: :P(A) = \sum_n P(A \cap B_n) = \sum_n P(A\mid B_n)P(B_n). where the events (B_n) form a countable partition of \Omega. This particular method relies on event B occurring with some sort of relationship with another event A. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. It can be interpreted as ""the probability of B occurring multiplied by the probability of A occurring, provided that B has occurred, is equal to the probability of the A and B occurrences together, although not necessarily occurring at the same time"". P(\text{dot received}) = P(\text{dot received } \cap \text{ dot sent}) + P(\text{dot received } \cap \text{ dash sent}) P(\text{dot received}) = P(\text{dot received } \mid \text{ dot sent})P(\text{dot sent}) + P(\text{dot received } \mid \text{ dash sent})P(\text{dash sent}) P(\text{dot received}) = \frac{9}{10}\times\frac{3}{7} + \frac{1}{10}\times\frac{4}{7} = \frac{31}{70} Now, P(\text{dot sent } \mid \text{ dot received}) can be calculated: P(\text{dot sent } \mid \text{ dot received}) = P(\text{dot received } \mid \text{ dot sent}) \frac{P(\text{dot sent})}{P(\text{dot received})} = \frac{9}{10}\times \frac{\frac{3}{7}}{\frac{31}{70}} = \frac{27}{31} == Statistical independence == Events A and B are defined to be statistically independent if the probability of the intersection of A and B is equal to the product of the probabilities of A and B: :P(A \cap B) = P(A) P(B). The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . That is, for an event A, :P(A^c) = 1 - P(A). In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. Moreover, this ""multiplication rule"" can be practically useful in computing the probability of A \cap B and introduces a symmetry with the summation axiom for Poincaré Formula: :P(A \cup B) = P(A) + P(B) - P(A \cap B) :Thus the equations can be combined to find a new representation of the : : P(A \cap B)= P(A) + P(B) - P(A \cup B) = P(A \mid B)P(B) : P(A \cup B)= {P(A) + P(B) - P(A \mid B){P(B)}} ==== As the probability of a conditional event ==== Conditional probability can be defined as the probability of a conditional event A_B. ",0.323,0.14,"""11.0""",322,1.44,B
+"Suppose that $A, B$, and $C$ are mutually independent events and that $P(A)=0.5, P(B)=0.8$, and $P(C)=$ 0.9 . Find the probabilities that all three events occur?","Event \operatorname{P}(s_1)=1/4 \operatorname{P}(s_2)=1/4 \operatorname{P}(s_3)=1/4 \operatorname{P}(s_4)=1/4 Probability of event A 0 1 0 1 \tfrac{1}{2} B 0 0 1 1 \tfrac{1}{2} C 0 1 1 1 \tfrac{3}{4} and so Event s_1 s_2 s_3 s_4 Probability of event A \cap B 0 0 0 1 \tfrac{1}{4} A \cap C 0 1 0 1 \tfrac{1}{2} B \cap C 0 0 1 1 \tfrac{1}{2} A \cap B \cap C 0 0 0 1 \tfrac{1}{4} In this example, C occurs if and only if at least one of A, B occurs. Unconditionally (that is, without reference to C), A and B are independent of each other because \operatorname{P}(A)—the sum of the probabilities associated with a 1 in row A—is \tfrac{1}{2}, while \operatorname{P}(A\mid B) = \operatorname{P}(A \text{ and } B) / \operatorname{P}(B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} = \operatorname{P}(A). Since in the presence of C the probability of A is affected by the presence or absence of B, A and B are mutually dependent conditional on C. == See also == * * * == References == Category:Independence (probability theory) When A and B are mutually exclusive, .Stats: Probability Rules. But conditional on C having occurred (the last three columns in the table), we have \operatorname{P}(A \mid C) = \operatorname{P}(A \text{ and } C) / \operatorname{P}(C) = \tfrac{1/2}{3/4} = \tfrac{2}{3} while \operatorname{P}(A \mid C \text{ and } B) = \operatorname{P}(A \text{ and } C \text{ and } B) / \operatorname{P}(C \text{ and } B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} < \operatorname{P}(A \mid C). The probability of one or both events occurring is denoted P(A ∪ B) and in general, it equals P(A) + P(B) – P(A ∩ B). In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.Introduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 ""Unit 3: Conditional Dependence""Introduction to learning Bayesian Networks from Data by Dirk Husmeier ""Introduction to Learning Bayesian Networks from Data -Dirk Husmeier"" For example, if A and B are two events that individually increase the probability of a third event C, and do not directly affect each other, then initially (when it has not been observed whether or not the event C occurs)Conditional Independence in Statistical theory ""Conditional Independence in Statistical Theory"", A. P. Dawid"" Probabilistic independence on Britannica ""Probability->Applications of conditional probability->independence (equation 7) "" \operatorname{P}(A \mid B) = \operatorname{P}(A) \quad \text{ and } \quad \operatorname{P}(B \mid A) = \operatorname{P}(B) (A \text{ and } B are independent). If event B occurs then the probability of occurrence of the event A will decrease because its positive relation to C is less necessary as an explanation for the occurrence of C (similarly, event A occurring will decrease the probability of occurrence of B). The probability that at least one of the events will occur is equal to one.Scott Bierman. Conditional dependence of A and B given C is the logical negation of conditional independence ((A \perp\\!\\!\\!\perp B) \mid C). In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.intmath.com; Mutually Exclusive Events. In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. The probabilities of the individual events (red, and club) are multiplied rather than added. Hence, now the two events A and B are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. Obviously, we get the following probabilities :\mathbb P(A_1) = \frac 4{52}, \qquad \mathbb P(A_2 \mid A_1) = \frac 3{51}, \qquad \mathbb P(A_3 \mid A_1 \cap A_2) = \frac 2{50}, \qquad \mathbb P(A_4 \mid A_1 \cap A_2 \cap A_3) = \frac 1{49}. We haveIntroduction to Artificial Intelligence by Sebastian Thrun and Peter Norvig, 2011 ""Unit 3: Explaining Away"" \operatorname{P}(A \mid C \text{ and } B) < \operatorname{P}(A \mid C). When heads occurs, tails can't occur, or p (heads and tails) = 0, so the outcomes are also mutually exclusive. In the case of flipping a coin, flipping a head and flipping a tail are also mutually exclusive events. Another example of events being collectively exhaustive and mutually exclusive at same time are, event ""even"" (2,4 or 6) and event ""odd"" (1,3 or 5) in a random experiment of rolling a six-sided die. The events 1 and 6 are mutually exclusive but not collectively exhaustive. ",0.6749,10.4,"""0.36""",4,5275,C
+A poker hand is defined as drawing 5 cards at random without replacement from a deck of 52 playing cards. Find the probability of four of a kind (four cards of equal face value and one card of a different value).,"*The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. There are 7,462 distinct poker hands. ===7-card poker hands=== In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall,https://www.casinodaniabeach.com/most-popular-types-of-poker/ a player uses the best five-card poker hand out of seven cards. Hand The five cards (or less) dealt on the screen are known as a hand. ==See also== *Casino comps *Draw poker *Gambling *Gambling mathematics *Problem gambling *Video blackjack *Video Lottery Terminal ==References== ==External links== * Category:Arcade video games Various payout variations are possible, depending on the casino, resulting in a house edge ranging from 1.98% to 6.15%.Wizard of Odds: Four Card Poker ==Rank of hands== The possible four-card hands are (from best to worst): *Four of a kind *Straight flush *Three of a kind *Flush *Straight *Two pair *One pair *High card ==References== ==External links== *ShuffleMaster flash demo of the game Category:Gambling games Category:Poker variants The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. The Total line also needs adjusting. ==See also== * Binomial coefficient * Combination * Combinatorial game theory * Effective hand strength algorithm * Event (probability theory) * Game complexity * Gaming mathematics * Odds * Permutation * Probability * Sample space * Set theory ==References== ==External links== * Brian Alspach's mathematics and poker page * MathWorld: Poker * Poker probabilities including conditional calculations * Numerous poker probability tables * 5, 6, and 7 card poker probabilities * Hold'em poker probabilities All the other hand combinations in video poker are the same as in table poker, including such hands as two pair, three of a kind, straight (a sequence of 5 cards of consecutive value), flush (any 5 cards of the same suit), full house (a pair and a three of a kind), four of a kind (four cards of the same value), straight flush (5 consecutive cards of the same suit) and royal flush (a Ten, a Jack, a Queen, a King and an Ace of the same suit). frameless|right Four Card Poker is a casino card game similar to Three Card Poker, invented by Roger Snow and owned by Shuffle Master.ShuffleMaster: Four Card Poker ==Description of play== The player can place an ante bet or an ""Aces Up"" bet or both. A four flush (also flush draw) is a poker draw or non-standard poker hand that is one card short of being a full flush. Video poker is a casino game based on five-card draw poker. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. Four of a kind may refer to: *Four of a kind (poker), a type of poker hand *Four of a Kind (card game), a patience or solitaire *Four of a Kind (TV series), an American reality series about quadruplets *Four of a Kind (film), an Australian feature film *4 of a Kind, the fourth album by American thrash band D.R.I. For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. The number of distinct poker hands is even smaller. This pejorative term originated in the 19th century when bluffing poker players misrepresented that they had a flush—a poker hand with five cards all of one suit—when they only had four cards of one suit. Flush A five-card hand that contains cards of the same suit. ",0.00024,-1.49,"""0.00131""",1.51,-167,A
+Three students $(S)$ and six faculty members $(F)$ are on a panel discussing a new college policy. In how many different ways can the nine participants be lined up at a table in the front of the auditorium?,"Table of Six () is a political conference established by the Republican People's Party, Good Party, Felicity Party, Democrat Party, Democracy and Progress Party and Future Party, with the first meeting held on 12 February 2022. thumb|Table seating arrangement A seating plan is a diagram or a set of written or spoken instructions that determines where people should take their seats. Table topics are topics on various subjects that are discussed by a group of people around a table. In this case, it is customary to arrange the host and hostess at the opposite sides of the table, and alternate male and female guests throughout. In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. There will be a table topic master for each meeting, who will prepare questions beforehand and ask the participants questions one by one for which they are called upon to answer. A seating plan is of crucial importance for musical ensembles or orchestras, where every type of instrument is allocated a specific section. == See also == * Seating assignment * Seating capacity * Table setting * Kids' table == References == ==External links == * Category:Etiquette Category:Diagrams Plan If the five conics have the properties that *there is no line such that every one of the 5 conics is either tangent to it or passes through one of two fixed points on it (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics) *no three of the conics pass through any point (otherwise there is a ""double line with 2 marked points"" tangent to all 5 conics passing through this triple intersection point) *no two of the conics are tangent *no three of the five conics are tangent to a line *a pair of lines each tangent to two of the conics do not intersect on the fifth conic (otherwise this pair is a degenerate conic tangent to all 5 conics) then the total number of conics C tangent to all 5 (counted with multiplicities) is 3264. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bézout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 65 = 7776, which was Steiner's incorrect solution. Many personality or public speaking clubs like the 'Toastmasters' have a separate session in their meetings known as a table topic session. The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848. ==History== claimed that the number of conics tangent to 5 given conics in general position is 7776 = 65, but later realized this was wrong. Honored guests (moms, dads, and in-laws) are placed to the host's and hostess's right and then left."" Student Number 1 was panned by critics and flopped at the box office. ==Cast== ==Production== Shooting was commenced at Chennai, for a fifteen-days schedule, after which the unit moved to Russia to shoot two songs. The Table of Six was originally an independent entity from the Nation Alliance. Some chapters of Toastmasters also host Table Topics contests. ==See also== * TableTopics ==References== Category:Public speaking In particular if C intersects each of the five conics in exactly 3 points (one double point of tangency and two others) then the multiplicity is 1, and if this condition always holds then there are exactly 3264 conics tangent to the 5 given conics. However the number of conics is not (6H)5 but (6H−2E)5 because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E. In the United States according to Peggy Post, ""tradition dictates that when everyone is seated together, the host and hostess sit at either end of the table. On 21 January 2023, Table of Six defined itself as the ""Nation Alliance"" for the first time after its 11th meeting. As practiced by Toastmasters International, the topics to be discussed are written on pieces of paper which are placed in a box in the middle of a table. In the text of the memorandum, lowering the electoral threshold to 3%, treasury aid to the parties that received at least 1% of the votes, ending the omnibus law practice, removing the veto power of the president and extending his term of office to 7 years, recognising the authority to issue a no-confidence question on the government, human rights and human rights in the education curriculum. The participants pick up one paper each and start talking about the topic written on the paper. ",22,432.07,"""0.00539""",1.01,362880,E
+"Each of the 12 students in a class is given a fair 12 -sided die. In addition, each student is numbered from 1 to 12 . If the students roll their dice, what is the probability that there is at least one ""match"" (e.g., student 4 rolls a 4)?","Numbers on each die Die 1 1 5 10 11 13 17 Die 2 3 4 7 12 15 16 Die 3 2 6 8 9 14 18 === Four players === An optimal and permutation-fair solution for 4 twelve-sided dice was found by Robert Ford in 2010. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Die 1 1 4 Die 2 2 3 === Three players === An optimal and permutation- fair solution for 3 six-sided dice was found by Robert Ford in 2010. If r is the total number of dice selecting the 6 face, then P(r \ge k ; n, p) is the probability of having at least k correct selections when throwing exactly n dice. Many players collect or acquire a large number of mixed and unmatching dice. The d20 System includes a four-sided tetrahedral die among other dice with 6, 8, 10, 12 and 20 faces. Dice using both the numerals 6 and 9, which are reciprocally symmetric through rotation, typically distinguish them with a dot or underline. ====Common variations==== Dice are often sold in sets, matching in color, of six different shapes. Numbers on each die Die 1 1 8 11 14 19 22 27 30 35 38 41 48 Die 2 2 7 10 15 18 23 26 31 34 39 42 47 Die 3 3 6 12 13 17 24 25 32 36 37 43 46 Die 4 4 5 9 16 20 21 28 29 33 40 44 45 === Five players === Several candidates exist for a set of 5 dice, but none is known to be optimal. == See also == * Intransitive dice ==References== ==External links== * Go First Dice - Numberphile Category:Dice Some dice, such as those with 10 sides, are usually numbered sequentially beginning with 0, in which case the opposite faces will add to one less than the number of faces. Note the older hand-inked green 12-sided die (showing an 11), manufactured before pre-inked dice were common. Normally, the faces on a die will be placed so opposite faces will add up to one more than the number of faces. These are six-sided dice with sides numbered `2, 3, 3, 4, 4, 5`, which have the same arithmetic mean as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for a pair). Optimal results have been proven by exhaustion for up to 4 dice. ==Configurations== === Two players === The two player case is somewhat trivial. On some four- sided dice, each face features multiple numbers, with the same number printed near each vertex on all sides. The sum of the numbers on opposite faces is 21 if numbered 1–20. ====Rarer variations==== upright=2.9|thumb|Dice collection: D2–D22, D24, D26, D28, D30, D36, D48, D60 and D100. If several dice of the same type are to be rolled, this is indicated by a leading number specifying the number of dice. Due to circumstances or character skill, the initial roll may have a number added to or subtracted from the final result, or have the player roll extra or fewer dice. This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem. ==Generalizations== A natural generalization of the problem is to consider n non-necessarily fair dice, with p the probability that each die will select the 6 face when thrown (notice that actually the number of faces of the dice and which face should be selected are irrelevant). Another configuration places only one number on each face, and the rolled number is taken from the downward face. ==References== Category:Dice In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: :P(n)=1-\sum_{x=0}^{n-1}\binom{6n}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{6n-x}\, . ""Uniform fair dice"" are dice where all faces have equal probability of outcome due to the symmetry of the die as it is face-transitive. Configurations where all die have the same number of sides are presented here, but alternative configurations might instead choose mismatched dice to minimize the number of sides, or minimize the largest number of sides on a single die. ",273,0.648004372,"""0.00024""",+11,0.178,B
+"The World Series in baseball continues until either the American League team or the National League team wins four games. How many different orders are possible (e.g., ANNAAA means the American League team wins in six games) if the series goes four games?","In Major League Baseball (MLB), a game seven can occur in the World Series or in a League Championship Series (LCS), which are contested as best-of-seven series. From 2003 to 2010, the AL and NL had each won the World Series four times, but none of them had gone the full seven games. In baseball, a series refers to two or more consecutive games played between the same two teams. During the Major League Baseball Postseason, there are four wild card series (two in each League), each of which are a best-of-3 series. The World Series has been contested 118 times as of 2022, with the AL winning 67 and the NL winning 51. == Precursors to the modern World Series (1857–1902) == === The original World Series === Until the formation of the American Association in 1882 as a second major league, the National Association of Professional Base Ball Players (1871–1875) and then the National League (founded 1876) represented the top level of organized baseball in the United States. These 16 franchises, all of which are still in existence, have each won at least two World Series titles. The National League Championship Series (NLCS) and American League Championship Series (ALCS), since the expansion to best-of-seven, are always played in a 2–3–2 format: Games 1, 2, 6, and 7 are played in the stadium of the team that has home-field advantage, and Games 3, 4, and 5 are played in the stadium of the team that does not. === 1970s === ==== 1971: World Series at night ==== Night games were played in the major leagues beginning with the Cincinnati Reds in 1935, but the World Series remained a strictly daytime event for years thereafter. The two division winners within each league played each other in a best-of-five League Championship Series to determine who would advance to the World Series. Since then, the 2011, 2014, 2016, 2017, and 2019 World Series have gone the full seven games. When the first modern World Series was played in 1903, there were eight teams in each league. This is known in baseball as a road trip, and a team can be on the road for up to 20 games, or 4-5 series. This is the only time in World Series history in which three teams have won consecutive series in succession. A game seven cannot occur in earlier rounds of the MLB postseason, as Division Series and Wild Card rounds use shorter series. ==Key== (#) Extra innings (the number indicates the number of extra innings played) † Indicates the team that won a game seven after coming back from an 0–3 series deficit § Indicates the team that lost a game seven after coming back from an 0–3 series deficit ∞ Indicates a game seven that was played at a neutral site Road* Indicates a game seven that was won by the road team Year (X) Indicates the number of games seven played in that year's postseason (from 1985 on) Each year is linked to an article about that particular Major League Baseball season Team (#) Indicates team and the number of game sevens played by that team at that point ==All-time game sevens== Year Playoff round Date Venue Winner Result Loser Ref. World Series Bennett Park Pittsburgh Pirates (1) 8–0 Detroit Tigers (1) World Series Fenway Park Boston Red Sox (1) 3–2 (10) New York Giants (1) World Series Griffith Stadium Washington Senators (1) 4–3 (12) New York Giants (2) World Series Forbes Field Pittsburgh Pirates (2) 9–7 Washington Senators (2) World Series Yankee Stadium (I) St. Louis Cardinals (1) 3–2 New York Yankees (1) World Series Sportsman's Park (III) St. Louis Cardinals (2) 4–2 Philadelphia Athletics (1) World Series Navin Field St. Louis Cardinals (3) 11–0 Detroit Tigers (2) World Series Crosley Field Cincinnati Reds (1) 2–1 Detroit Tigers (3) World Series Wrigley Field Detroit Tigers (4) 9–3 Chicago Cubs (1) World Series Sportsman's Park (III) St. Louis Cardinals (4) 4–3 Boston Red Sox (2) World Series Yankee Stadium (I) New York Yankees (2) 5–2 Brooklyn Dodgers (1) World Series Ebbets Field New York Yankees (3) 4–2 Brooklyn Dodgers (2) World Series Yankee Stadium (I) Brooklyn Dodgers (3) 2–0 New York Yankees (4) World Series Ebbets Field New York Yankees (5) 9–0 Brooklyn Dodgers (4) World Series Yankee Stadium (I) Milwaukee Braves (1) 5–0 New York Yankees (6) World Series Milwaukee County Stadium New York Yankees (7) 6–2 Milwaukee Braves (2) World Series Forbes Field Pittsburgh Pirates (3) 10–9 New York Yankees (8) World Series Candlestick Park New York Yankees (9) 1–0 San Francisco Giants (3) World Series Busch Stadium (I) St. Louis Cardinals (5) 7–5 New York Yankees (10) World Series Metropolitan Stadium Los Angeles Dodgers (5) 2–0 Minnesota Twins (3) World Series Fenway Park St. Louis Cardinals (6) 7–2 Boston Red Sox (3) World Series Busch Stadium (II) Detroit Tigers (5) 4–1 St. Louis Cardinals (7) World Series Memorial Stadium Pittsburgh Pirates (4) 2–1 Baltimore Orioles (1) World Series Riverfront Stadium Oakland Athletics (2) 3–2^ Cincinnati Reds (2) World Series Oakland–Alameda County Coliseum Oakland Athletics (3) 5–2 New York Mets (1) World Series Fenway Park Cincinnati Reds (3) 4–3 Boston Red Sox (4) World Series Memorial Stadium Pittsburgh Pirates (5) 4–1 Baltimore Orioles (2) World Series Busch Stadium (II) St. Louis Cardinals (8) 6–3 Milwaukee Brewers (1) ALCS Exhibition Stadium Kansas City Royals (1) 6–2 Toronto Blue Jays (1) World Series Royals Stadium Kansas City Royals (2) 11–0 St. Louis Cardinals (9) ALCS Fenway Park Boston Red Sox (5) 8–1 California Angels (1) World Series Shea Stadium New York Mets (2) 8–5 Boston Red Sox (6) NLCS Busch Stadium (II) St. Louis Cardinals (10) 6–0 San Francisco Giants (4) World Series Hubert H. Humphrey Metrodome Minnesota Twins (4) 4–2 St. Louis Cardinals (11) NLCS Dodger Stadium Los Angeles Dodgers (6) 6–0 New York Mets (3) NLCS Three Rivers Stadium Atlanta Braves (3) 4–0 Pittsburgh Pirates (6) World Series Hubert H. Humphrey Metrodome Minnesota Twins (5) 1–0 (10) Atlanta Braves (4) NLCS Atlanta–Fulton County Stadium Atlanta Braves (5) 3–2 Pittsburgh Pirates (7) NLCS Atlanta–Fulton County Stadium Atlanta Braves (6) 15–0 St. Louis Cardinals (12) World Series Pro Player Stadium Florida Marlins (1) 3–2 (11) Cleveland Indians (1) World Series Bank One Ballpark Arizona Diamondbacks (1) 3–2 New York Yankees (11) World Series Edison International Field Anaheim Angels (2) 4–1 San Francisco Giants (5) NLCS Wrigley Field Florida Marlins (2) 9–6 Chicago Cubs (2) ALCS Yankee Stadium (I) New York Yankees (12) 6–5 (11) Boston Red Sox (7) ALCS Yankee Stadium (I) Boston Red Sox (8)† 10–3 New York Yankees (13) NLCS Busch Stadium (II) St. Louis Cardinals (13) 5–2 Houston Astros (1) NLCS Shea Stadium St. Louis Cardinals (14) 3–1 New York Mets (4) ALCS Fenway Park Boston Red Sox (9) 11–2 Cleveland Indians (2) ALCS Tropicana Field Tampa Bay Rays (1) 3–1 Boston Red Sox (10) World Series Busch Stadium St. Louis Cardinals (15) 6–2 Texas Rangers (1) NLCS AT&T; Park San Francisco Giants (6) 9–0 St. Louis Cardinals (16) World Series Kaufmann Stadium San Francisco Giants (7) 3–2 Kansas City Royals (2) World Series Progressive Field Chicago Cubs (3) 8–7 (10) Cleveland Indians (3) ALCS Minute Maid Park Houston Astros (2) 4–0 New York Yankees (14) World Series Dodger Stadium Houston Astros (3) 5–1 Los Angeles Dodgers (7) NLCS Miller Park Los Angeles Dodgers (8) 5–1 Milwaukee Brewers (2) World Series Minute Maid Park Washington Nationals (1) 6–2 Houston Astros (4) ALCS Petco Park∞ Tampa Bay Rays (2) 4–2 Houston Astros (5)§ NLCS Globe Life Field∞ Los Angeles Dodgers (9) 4–3 Atlanta Braves (7) ==All-time standings== Team Games played Wins Losses Win–loss % St. Louis Cardinals 16 11 5 New York Yankees 14 6 8 Boston Red Sox 10 4 6 Brooklyn / Los Angeles Dodgers 9 5 4 Pittsburgh Pirates 7 5 2 Milwaukee / Atlanta Braves 7 4 3 New York / San Francisco Giants 7 2 5 Washington Senators / Minnesota Twins 5 3 2 Detroit Tigers 5 2 3 Houston Astros 5 2 3 New York Mets 4 1 3 Cincinnati Reds 3 2 1 Kansas City Royals 3 2 1 Chicago Cubs 3 1 2 Philadelphia / Oakland Athletics 3 1 2 Cleveland Indians / Guardians 3 0 3 Florida Marlins 2 2 0 Tampa Bay Rays 2 2 0 California / Anaheim Angels 2 1 1 Milwaukee Brewers 2 0 2 Arizona Diamondbacks 1 1 0 Washington Nationals 1 1 0 Texas Rangers 1 0 1 Toronto Blue Jays 1 0 1 Note: Five teams have never played a game seven: Philadelphia Phillies, Chicago White Sox, San Diego Padres, Seattle Mariners, Colorado Rockies. ESPN selected it as the ""Greatest of All Time"" in their ""World Series 100th Anniversary"" countdown, with five of its games being decided by a single run, four games decided in the final at-bat and three games going into extra innings. The remainder of the Postseason consists of the League Division Series, which is a best-of-5 series, and the League Championship Series, which is a best-of-7 series, followed by the World Series, a best-of-7 series to determine the Major League Baseball Champion. There are only two other occasions when a team has won at least three consecutive World Series: 1972 to 1974 by the Oakland Athletics, and 1998 to 2000 by the Yankees. ==== 1947–1964: New York City teams dominate World Series play ==== In an 18-year span from 1947 to 1964, except for 1948 and 1959, the World Series was played in New York City, featuring at least one of the three teams located in New York at the time. Source: MLB.com ;Notes American League (AL) teams have won 67 of the 118 World Series played (56.8%). Starting in 1976, the DH rule was used in the World Series held in even-numbered years. The World Series is the annual championship series of Major League Baseball (MLB) in the United States and Canada. Since then each league has conducted a League Championship Series (ALCS and NLCS) preceding the World Series to determine which teams will advance, while those series have been preceded in turn by Division Series (ALDS and NLDS) since 1995, and Wild Card games or series in each league since 2012. Historically and currently, professional baseball season revolves around a schedule of series, each typically lasting three or four games. The ""series"" schedule gives its name to the MLB championship series, the World Series. ",2,16,"""26.9""",1.6,4.8,A
 "Draw one card at random from a standard deck of cards. The sample space $S$ is the collection of the 52 cards. Assume that the probability set function assigns $1 / 52$ to each of the 52 outcomes. Let
 $$
 \begin{aligned}
@@ -870,166 +870,166 @@ C & =\{x: x \text { is a club }\}, \\
 D & =\{x: x \text { is a diamond, a heart, or a spade }\} .
 \end{aligned}
 $$
-Find $P(A)$","The queen of spades (Q) is one of 52 playing cards in a standard deck: the queen of the suit of spades (). 52 pickup or 52-card pickup is a humorous prank which consists only of picking up a scattered deck of playing cards. The player first turns the first pile up and looks for either an ace, a ten, a king, a queen, and a jack, cards which comprise a royal flush. Royal Flush is a solitaire card game which is played with a deck of 52 playing cards. First, the cards are dealt into nine columns in such a way that the first column contains nine cards, the second having eight cards, the third seven, and so on until the ninth column has a single card. King Albert is a patience or card solitaire using a deck of 52 playing cards of the open packer type. Royal Marriage is a patience or solitaire game using a deck of 52 playing cards. In Pinochle, the queen of spades and the jack of diamonds combine for a unique two-card meld known as a ""pinochle"". The remaining fifty cards are shuffled and placed on the top of the King to form the stock. Another version of the prank can be played where one player declares ""52-card pick up"" and is then granted power to throw each of the 52 cards individually at any of the opponents.. ==As a game== By introducing additional rules, the task of picking up the cards can be made into a solitaire game. In Old Maid and several games of the Hearts family, it serves as a single, powerful card in the deck. ==Roles by game== In the Hearts family of card games, the queen of spades is usually considered an unlucky card; it is the eponym of the Black Maria and Black Lady variants of Hearts. The game is won when the cards of the royal flush are the only ones left in the pile and are arranged in any order. They hold up the deck and take the cards one by one off the bottom as the other player(s) call out ""smoke"" ... ""smoke"" ... ""smoke"" ... and, with the first red card, ""fire!"" If there are any other cards sandwiched among the royal flush cards, the game is lost. ==Variation== To give the game some variation, Lee and Packard suggests the player to try other poker hands such as four-of- a-kinds, full houses, or straight flushes.Sloane Lee & Gabriel Packard, 100 Best Solitaire Games, The player can simply look for a specific hand or look for certain cards to include in their hand while playing the game. ==See also== * List of solitaire games * Glossary of solitaire terms ==References== ==External sources== Category:Single-deck patience card games Category:Closed non-builders Also, the suit of the first card found determines the suit of the entire royal flush. The game is won when the King and Queen are brought together -- that is, when only one or two cards remain in between them, which can then be discarded. ==Variations== Royal Marriage is possible to play in-hand, rather than on a surface such as a table. The other player must then pick them up.. ==Variations== Genuine card games sometimes end in 52 pickup. The dealer has a pack of cards, they then show the teams a card in the pack, e.g. two of spades. In the seven card stud poker variant known as ""The Bitch"", a face-up deal of the queen of spades results in the deal being abandoned, all cards being shuffled and a new deal starting with only those players who had not already folded when the queen of spades was dealt. Cards are played from the bottom of the deck onto the Queen, and fanned out to show all cards that could possibly affect play. It may be advantageous to retain as many cards of the Queen's suit as possible, as these may be easily eliminated by the King or Queen at any tie, but may be helpful in eliminating other cards. The discarded cards are set aside. ",0.72,29.36,2.57,0.4772,0.2307692308,E
-"An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?","Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. The probability that the second ball picked is red depends on whether the first ball was red or white. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. * Mixed replacement/non-replacement: the urn contains black and white balls. We want to calculate the probability that the red ball is not taken. The probability that the first ball picked is red is equal to the weight fraction of red balls: : p_1 = \frac{m_1 \omega_1}{m_1 \omega_1 + m_2 \omega_2}. This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. thumb|Two urns containing white and red balls. The probability that the red ball is not taken in the first draw is 1000/2000 = . While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. * Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour. What is the distribution of the number of black balls drawn after m draws? * multinomial distribution: there are balls of more than two colors. * Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector's problem and birthday problem. Various generalizations to this distribution exist for cases where the picking of colored balls is biased so that balls of one color are more likely to be picked than balls of another color. ",26.9,3920.70763168,0.2,35.2,2,C
-"Bowl $B_1$ contains two white chips, bowl $B_2$ contains two red chips, bowl $B_3$ contains two white and two red chips, and bowl $B_4$ contains three white chips and one red chip. The probabilities of selecting bowl $B_1, B_2, B_3$, or $B_4$ are $1 / 2,1 / 4,1 / 8$, and $1 / 8$, respectively. A bowl is selected using these probabilities and a chip is then drawn at random. Find $P(W)$, the probability of drawing a white chip.","thumb|Colorized photo of Chips. Eventually, Blumenthal developed the three-stage cooking process known as triple-cooked chips, which he identifies as ""the first recipe I could call my own"". The Sunday Times described triple-cooked chips as Blumenthal's most influential culinary innovation, which had given the chip ""a whole new lease of life"". ==History== Blumenthal said he was ""obsessed with the idea of the perfect chip"",Blumenthal, In Search of Perfection and described how, from 1992 onwards, he worked on a method for making ""chips with a glass-like crust and a soft, fluffy centre"". Triple-cooked chips are a type of chips developed by the English chef Heston Blumenthal. Bremner Wafers are made by the Bremner Biscuit Company. The Bowl of Baal is a 1975 science fiction novel by Robert Ames Bennet. Four color cards () is a game of the rummy family of card games, with a relatively long history in southern China. Chips served as a sentry dog for the Roosevelt-Churchill conference in 1943. The result is what Blumenthal calls ""chips with a glass-like crust and a soft, fluffy centre"". As of February 5, 2006, there are 8 variations of the original Bremner Wafer: * Original Wafers: suitable for fine wines as described above * Sesame Wafers: ""an excellent complement to cheese, pate, smoked fish or any spread"" * Cracked Wheat Wafers: for topping and spreads * Low Sodium Wafers * Caraway Wafers: designed for strong flavors such as Swiss Cheese * Crackers Made with Pure Sunflower Oil: for appetizers * Oyster Crackers Made with Pure Sunflower Oil: best suited for adding to soups. ==See also== * List of crackers ==References== ==External links== *Bremner Biscuit Company website Category:Brand name crackers Blumenthal describes moisture as the ""enemy"" of crisp chips. ;Meld: A group of one to four cards with specific matching conditions. Chips was a German Shepherd-Collie-Malamute mix owned by Edward J. Wren of Pleasantville, New York. In 2014, the London Fire Brigade attributed an increase in chip pan fires to the increased popularity of ""posh chips"", including triple-cooked chips. ==Preparation== ===Blumenthal's technique=== Previously, the traditional practice for cooking chips was a two-stage process, in which chipped potatoes were fried in oil first at a relatively low temperature to soften them and then at a higher temperature to crisp up the outside. Second, the cracks that develop in the chips provide places for oil to collect and harden during frying, making them crunchy.Blumenthal, Heston Blumenthal at Home Third, thoroughly drying out the chips drives off moisture that would otherwise keep the crust from becoming crisp. Bloomsbury. ==Further reading== * * ==External links== * Triple-Cooked Chips. The chips are first simmered, then cooled and drained using a sous-vide technique or by freezing; deep fried at and cooled again; and finally deep-fried again at . The dealer starts by taking 4 tiles initially from one of the walls, then the players proceed in anti-clockwise order by taking 4 tiles each from where the previous player left off; after each player has 20 tiles, the dealer takes one more tile as they must discard a tile to start play. Chips (1940–1946) was a trained sentry dog for United States Army, and reputedly the most decorated war dog from World War II. On July 10, 1943, Chips and his handler were pinned down on the beach by an Italian machine-gun team. The second of the three stages is frying the chips at for approximately 5 minutes, after which they are cooled once more in a freezer or sous-vide machine before the third and final stage: frying at for approximately 7 minutes until crunchy and golden. Blumenthal began work on the recipe in 1993, and eventually developed the three-stage cooking process. ",0.65625,2.74,0.5,35,5,A
-Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.,"To approximate the mean line segment length of a given shape, two points are randomly chosen in its interior and the distance is measured. In geometry, the mean line segment length is the average length of a line segment connecting two points chosen uniformly at random in a given shape. While the question may seem simple, it has a fairly complicated answer; the exact value for this is \frac{2 + \sqrt{2} + 5 \ln (1 + \sqrt{2})}{15}. == Formal definition == The mean line segment length for an n-dimensional shape S may formally be defined as the expected Euclidean distance ||⋅|| between two random points x and y, : \mathbb E[\|x-y\|]=\frac1{\lambda(S)^2}\int_S \int_S \|x-y\| \,d\lambda(x) \,d\lambda(y) where λ is the n-dimensional Lebesgue measure. The length of a line segment is given by the Euclidean distance between its endpoints. thumb|historical image – create a line segment (1699) In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. Length of line. For the two- dimensional case, this is defined using the distance formula for two points (x1, y1) and (x2, y2) : \frac1{\lambda(S)^2}\iint_S \iint_S \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \,dx_1 \,dy_1 \,dx_2 \,dy_2. == Approximation methods == Since computing the mean line segment length involves calculating multidimensional integrals, various methods for numerical integration can be used to approximate this value for any shape. Thus, the line segment can be expressed as a convex combination of the segment's two end points. A new approach to line and line segment clipping in homogeneous coordinates, The Visual Computer, ISSN 0178-2789, Vol. 21, No. 11, pp. 905–914, Springer Verlag, 2005. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. In other words, it is the expected Euclidean distance between two random points, where each point in the shape is equally likely to be chosen. However, in the random case, with high probability the longest edge has length approximately \sqrt{\frac{\log n}{\pi n}}, longer than the average by a non-constant factor. There are two common algorithms for line clipping: Cohen–Sutherland and Liang–Barsky. Line sampling is a method used in reliability engineering to compute small (i.e., rare event) failure probabilities encountered in engineering systems. At the same time, 100% of respondents selected either one of these quantities as being the least desirable. == Calculation methods == There are a few methods to calculate line length to fit the intended average count of characters that such lines should contain based on the factors listed above. The line or line segment p can be computed from points r1, r2 given in homogeneous coordinates directly using the cross product as :p = r1 × r2 = (x1, y1, w1) × (x2, y2, w2) or as :p = r1 × r2 = (x1, y1, 1) × (x2, y2, 1). ==See also== * Clipping (computer graphics) ==References== Category:Clipping (computer graphics) The global probability of failure is the mean of the probability of failure on the lines: : \tilde{p}_f = \frac{1}{N_L} \sum_{i=1}^{N_L} p_f^{(i)} where N_L is the total number of lines used in the analysis and the p_f^{(i)} are the partial probabilities of failure estimated along all the lines. 370x370px|right|Example of line clipping for a two-dimensional region In computer graphics, line clipping is the process of removing (clipping) lines or portions of lines outside an area of interest (a viewport or view volume). Even for simple shapes such as a square or a triangle, solving for the exact value of their mean line segment lengths can be difficult because their closed-form expressions can get quite complicated. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ). *More generally than above, the concept of a line segment can be defined in an ordered geometry. thumb|300px|right|Euclidean minimum spanning tree of 25 random points in the plane A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. ",26.9,0.66666666666,167.0,-191.2,2.3,B
-"In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select $6,7,8,9$.","A six-number lottery game is a form of lottery in which six numbers are drawn from a larger pool (for example, 6 out of 44). If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner—regardless of the order of the numbers. Pick 3 draws a 3 digit number and Pick 4 draws a 4 digit number. A lottery is a form of gambling which involves selling numbered tickets and giving prizes to the holders of numbers drawn at random. Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, e.g. (Rain:.70, No Rain:.30).Mas-Colell, Andreu, Michael Whinston and Jerry Green (1995). For example, in the 6 from 49 lottery, given 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in 56.66 × 10, or 566.6 (the probability would be divided by 10, to give an exact value of \frac{8815}{4994220}). The Missouri Lottery is the state-run lottery in Missouri. This can be written in a general form for all lotteries as: {K\choose B}{N-K\choose K-B}\over {N\choose K} where N is the number of balls in lottery, K is the number of balls in a single ticket, and B is the number of matching balls for a winning ticket. Winning the top prize, usually a progressive jackpot, requires a player to match all six regular numbers drawn; the order in which they are drawn is irrelevant. In the 5-from-90 lotto, the minimum number of tickets that can guarantee a ticket with at least 2 matches is 100. == Information theoretic results == As a discrete probability space, the probability of any particular lottery outcome is atomic, meaning it is greater than zero. If the player wagered an additional $1, they were eligible to win up to $25,000 in the Topper drawing, which was drawn by random number generator. ===Raffle=== The California Lottery offered two raffles; March 17, 2007 and one on January 1, 2008. Using X representing winning the 6-of-49 lottery, the Shannon entropy of 6-of-49 above is \begin{align} \Eta(X) &= -p\log(p) - q\log(q) = -\tfrac{1}{13,983,816}\log\\!{\tfrac{1}{13,983,816}} \- \tfrac{13,983,815}{13,983,816}\log\\!{\tfrac{13,983,815}{13,983,816}} \\\ & \approx 1.80065 \times 10^{-6} \text{ shannons.} \end{align} ==References== ==External links== * Euler's Analysis of the Genoese Lottery – Convergence (August 2010), Mathematical Association of America * Lottery Mathematics – INFAROM Publishing * 13,983,816 and the Lottery – YouTube video with James Clewett, Numberphile, March 2012 Mathematics Category:Combinatorics Category:Gambling mathematics A free ticket with 2 sets of numbers qualifying for the next Lotto draw is won by matching three numbers. Suppose the probabilities for lottery A are (Cured: .90, Uncured: .00, Dead: .10), and for lottery B are (Cured: .50, Uncured: .50, Dead: .00). That is to buy at least one lottery ticket for every possible number combination. There are two draws every day, televised at 1:29pm and 6:59pm.Televised Draw Results, California State Lottery ====Daily 4==== A ""pick 4"" type game premiered on May 19, 2008. This yields a final formula of :{n\choose k}={49\choose 6}={49\over 6} * {48\over 5} * {47\over 4} * {46\over 3} * {45\over 2} * {44\over 1} A 7th ball often is drawn as reserve ball, in the past only a second chance to get 5+1 numbers correct with 6 numbers played. ==Odds of getting other possibilities in choosing 6 from 49== One must divide the number of combinations producing the given result by the total number of possible combinations (for example, {49\choose 6} = 13,983,816 ). The prizes are smaller than other lottery games, but there are better odds (averaging 1:5). In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. ", 0.0024,-0.10,0.983,0.0245,313,A
-"Extend Example 1.4-6 to an $n$-sided die. That is, suppose that a fair $n$-sided die is rolled $n$ independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1,2, \ldots, n$. Find the limit of this probability as $n$ increases without bound.","More precisely, if E denotes the event in question, p its probability of occurrence, and Nn(E) the number of times E occurs in the first n trials, then with probability one,An Analytic Technique to Prove Borel's Strong Law of Large Numbers Wen, L. Am Math Month 1991 \frac{N_n(E)}{n}\to p\text{ as }n\to\infty. Therefore, while \lim_{n\to\infty} \sum_{i=1}^n \frac{X_i} n = \overline{X} other formulas that look similar are not verified, such as the raw deviation from ""theoretical results"": \sum_{i=1}^n X_i - n\times\overline{X} not only does it not converge toward zero as n increases, but it tends to increase in absolute value as n increases. ==Examples== For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability. What this means is that the probability that, as the number of trials n goes to infinity, the average of the observations converges to the expected value, is equal to one. Section XIII.7 that if this probability is written as p(n,k) then : \lim_{n\rightarrow \infty} p(n,k) \alpha_k^{n+1}=\beta_k where αk is the smallest positive real root of :x^{k+1}=2^{k+1}(x-1) and :\beta_k={2-\alpha_k \over k+1-k\alpha_k}. ==Values of the constants== k \alpha_k \beta_k 1 2 2 2 1.23606797... 1.44721359... 3 1.08737802... 1.23683983... 4 1.03758012... 1.13268577... According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. This result is useful to derive consistency of a large class of estimators (see Extremum estimator). ===Borel's law of large numbers=== Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. It does not converge in probability toward zero (or any other value) as n goes to infinity. The American Statistician, 56(3), pp.186-190 When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side: : \Pr \Bigl( \max_{1 \leq k \leq n} | S_k | \geq \alpha \Bigr) \leq \frac{27}{\alpha^2} \operatorname{var} (S_n). ==References== * (Theorem 22.5) * Category:Probabilistic inequalities Category:Statistical inequalities In particular, the proportion of heads after n flips will almost surely converge to as n approaches infinity. By Kolmogorov's zero–one law, for any fixed M, the probability that the event \limsup_n \frac{S_n}{\sqrt{n}} \geq M occurs is 0 or 1. Therefore, the expected value of the average of the rolls is: \frac{1+2+3+4+5+6}{6} = 3.5 According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) will approach 3.5, with the precision increasing as more dice are rolled. It says that: ::\Pr\left[|S_n-E_n| > t \right] \leq 2\exp\left[ - \frac{V_n}{C^2} h\left(\frac{C t}{V_n} \right)\right], where h(u) = (1+u)\log(1+u)-u 5\. It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. The law then states that this converges in probability to zero.) Thus, although the absolute value of the quantity S_n/\sqrt{2n\log\log n} is less than any predefined ε > 0 with probability approaching one, it will nevertheless almost surely be greater than ε infinitely often; in fact, the quantity will be visiting the neighborhoods of any point in the interval (-1,1) almost surely. thumb|Exhibition of Limit Theorems and their interrelationship ==Generalizations and variants== The law of the iterated logarithm (LIL) for a sum of independent and identically distributed (i.i.d.) random variables with zero mean and bounded increment dates back to Khinchin and Kolmogorov in the 1920s. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums Sn, scaled by n−1, converge to zero, respectively in probability and almost surely: : \frac{S_n}{n} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{n} \ \xrightarrow{a.s.} 0, \qquad \text{as}\ \ n\to\infty. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be ""balanced"" by the others (see the gambler's fallacy). In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. Let Sk denote the partial sum :S_k = X_1 + \cdots + X_k.\, Then :\Pr \Bigl( \max_{1 \leq k \leq n} | S_k | \geq 3 \alpha \Bigr) \leq 3 \max_{1 \leq k \leq n} \Pr \bigl( | S_k | \geq \alpha \bigr). ==Remark== Suppose that the random variables Xk have common expected value zero. In probability theory, Etemadi's inequality is a so-called ""maximal inequality"", an inequality that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. In probability theory, Cantelli's inequality (also called the Chebyshev- Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds.""Tail and Concentration Inequalities"" by Hung Q. Ngo""Concentration-of-measure inequalities"" by Gábor Lugosi The inequality states that, for \lambda > 0, : \Pr(X-\mathbb{E}[X]\ge\lambda) \le \frac{\sigma^2}{\sigma^2 + \lambda^2}, where :X is a real-valued random variable, :\Pr is the probability measure, :\mathbb{E}[X] is the expected value of X, :\sigma^2 is the variance of X. Applying the Cantelli inequality to -X gives a bound on the lower tail, : \Pr(X-\mathbb{E}[X]\le -\lambda) \le \frac{\sigma^2}{\sigma^2 + \lambda^2}. ",38,19.4,24.4,-1.32,0.6321205588,E
+Find $P(A)$","The queen of spades (Q) is one of 52 playing cards in a standard deck: the queen of the suit of spades (). 52 pickup or 52-card pickup is a humorous prank which consists only of picking up a scattered deck of playing cards. The player first turns the first pile up and looks for either an ace, a ten, a king, a queen, and a jack, cards which comprise a royal flush. Royal Flush is a solitaire card game which is played with a deck of 52 playing cards. First, the cards are dealt into nine columns in such a way that the first column contains nine cards, the second having eight cards, the third seven, and so on until the ninth column has a single card. King Albert is a patience or card solitaire using a deck of 52 playing cards of the open packer type. Royal Marriage is a patience or solitaire game using a deck of 52 playing cards. In Pinochle, the queen of spades and the jack of diamonds combine for a unique two-card meld known as a ""pinochle"". The remaining fifty cards are shuffled and placed on the top of the King to form the stock. Another version of the prank can be played where one player declares ""52-card pick up"" and is then granted power to throw each of the 52 cards individually at any of the opponents.. ==As a game== By introducing additional rules, the task of picking up the cards can be made into a solitaire game. In Old Maid and several games of the Hearts family, it serves as a single, powerful card in the deck. ==Roles by game== In the Hearts family of card games, the queen of spades is usually considered an unlucky card; it is the eponym of the Black Maria and Black Lady variants of Hearts. The game is won when the cards of the royal flush are the only ones left in the pile and are arranged in any order. They hold up the deck and take the cards one by one off the bottom as the other player(s) call out ""smoke"" ... ""smoke"" ... ""smoke"" ... and, with the first red card, ""fire!"" If there are any other cards sandwiched among the royal flush cards, the game is lost. ==Variation== To give the game some variation, Lee and Packard suggests the player to try other poker hands such as four-of- a-kinds, full houses, or straight flushes.Sloane Lee & Gabriel Packard, 100 Best Solitaire Games, The player can simply look for a specific hand or look for certain cards to include in their hand while playing the game. ==See also== * List of solitaire games * Glossary of solitaire terms ==References== ==External sources== Category:Single-deck patience card games Category:Closed non-builders Also, the suit of the first card found determines the suit of the entire royal flush. The game is won when the King and Queen are brought together -- that is, when only one or two cards remain in between them, which can then be discarded. ==Variations== Royal Marriage is possible to play in-hand, rather than on a surface such as a table. The other player must then pick them up.. ==Variations== Genuine card games sometimes end in 52 pickup. The dealer has a pack of cards, they then show the teams a card in the pack, e.g. two of spades. In the seven card stud poker variant known as ""The Bitch"", a face-up deal of the queen of spades results in the deal being abandoned, all cards being shuffled and a new deal starting with only those players who had not already folded when the queen of spades was dealt. Cards are played from the bottom of the deck onto the Queen, and fanned out to show all cards that could possibly affect play. It may be advantageous to retain as many cards of the Queen's suit as possible, as these may be easily eliminated by the King or Queen at any tie, but may be helpful in eliminating other cards. The discarded cards are set aside. ",0.72,29.36,"""2.57""",0.4772,0.2307692308,E
+"An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacement, and you are told that at least one of them is orange. What is the probability that the other ball is also orange?","Assume that an urn contains m_1 red balls and m_2 white balls, totalling N = m_1 + m_2 balls. n balls are drawn at random from the urn one by one without replacement. The probability that the second ball picked is red depends on whether the first ball was red or white. One pretends to remove one or more balls from the urn; the goal is to determine the probability of drawing one color or another, or some other properties. (A variation both on the first and the second question) ==Examples of urn problems== * beta- binomial distribution: as above, except that every time a ball is observed, an additional ball of the same color is added to the urn. When the mutator is drawn it is replaced along with an additional ball of an entirely new colour. * hypergeometric distribution: the balls are not returned to the urn once extracted. * Mixed replacement/non-replacement: the urn contains black and white balls. We want to calculate the probability that the red ball is not taken. The probability that the first ball picked is red is equal to the weight fraction of red balls: : p_1 = \frac{m_1 \omega_1}{m_1 \omega_1 + m_2 \omega_2}. This is referred to as ""drawing without replacement"", by opposition to ""drawing with replacement"". * multivariate hypergeometric distribution: the balls are not returned to the urn once extracted, but with balls of more than two colors. * geometric distribution: number of draws before the first successful (correctly colored) draw. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ . The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ . One ball is drawn randomly from the urn and its color observed; it is then placed back in the urn (or not), and the selection process is repeated.Urn Model: Simple Definition, Examples and Applications — The basic urn model Possible questions that can be answered in this model are: * Can I infer the proportion of white and black balls from n observations? In probability and statistics, an urn problem is an idealized mental exercise in which some objects of real interest (such as atoms, people, cars, etc.) are represented as colored balls in an urn or other container. thumb|Two urns containing white and red balls. The probability that the red ball is not taken in the first draw is 1000/2000 = . While black balls are set aside after a draw (non-replacement), white balls are returned to the urn after a draw (replacement). Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2−n as long as n is small compared to N. * Pólya urn: each time a ball of a particular colour is drawn, it is replaced along with an additional ball of the same colour. What is the distribution of the number of black balls drawn after m draws? * multinomial distribution: there are balls of more than two colors. * Occupancy problem: the distribution of the number of occupied urns after the random assignment of k balls into n urns, related to the coupon collector's problem and birthday problem. Various generalizations to this distribution exist for cases where the picking of colored balls is biased so that balls of one color are more likely to be picked than balls of another color. ",26.9,3920.70763168,"""0.2""",35.2,2,C
+"Bowl $B_1$ contains two white chips, bowl $B_2$ contains two red chips, bowl $B_3$ contains two white and two red chips, and bowl $B_4$ contains three white chips and one red chip. The probabilities of selecting bowl $B_1, B_2, B_3$, or $B_4$ are $1 / 2,1 / 4,1 / 8$, and $1 / 8$, respectively. A bowl is selected using these probabilities and a chip is then drawn at random. Find $P(W)$, the probability of drawing a white chip.","thumb|Colorized photo of Chips. Eventually, Blumenthal developed the three-stage cooking process known as triple-cooked chips, which he identifies as ""the first recipe I could call my own"". The Sunday Times described triple-cooked chips as Blumenthal's most influential culinary innovation, which had given the chip ""a whole new lease of life"". ==History== Blumenthal said he was ""obsessed with the idea of the perfect chip"",Blumenthal, In Search of Perfection and described how, from 1992 onwards, he worked on a method for making ""chips with a glass-like crust and a soft, fluffy centre"". Triple-cooked chips are a type of chips developed by the English chef Heston Blumenthal. Bremner Wafers are made by the Bremner Biscuit Company. The Bowl of Baal is a 1975 science fiction novel by Robert Ames Bennet. Four color cards () is a game of the rummy family of card games, with a relatively long history in southern China. Chips served as a sentry dog for the Roosevelt-Churchill conference in 1943. The result is what Blumenthal calls ""chips with a glass-like crust and a soft, fluffy centre"". As of February 5, 2006, there are 8 variations of the original Bremner Wafer: * Original Wafers: suitable for fine wines as described above * Sesame Wafers: ""an excellent complement to cheese, pate, smoked fish or any spread"" * Cracked Wheat Wafers: for topping and spreads * Low Sodium Wafers * Caraway Wafers: designed for strong flavors such as Swiss Cheese * Crackers Made with Pure Sunflower Oil: for appetizers * Oyster Crackers Made with Pure Sunflower Oil: best suited for adding to soups. ==See also== * List of crackers ==References== ==External links== *Bremner Biscuit Company website Category:Brand name crackers Blumenthal describes moisture as the ""enemy"" of crisp chips. ;Meld: A group of one to four cards with specific matching conditions. Chips was a German Shepherd-Collie-Malamute mix owned by Edward J. Wren of Pleasantville, New York. In 2014, the London Fire Brigade attributed an increase in chip pan fires to the increased popularity of ""posh chips"", including triple-cooked chips. ==Preparation== ===Blumenthal's technique=== Previously, the traditional practice for cooking chips was a two-stage process, in which chipped potatoes were fried in oil first at a relatively low temperature to soften them and then at a higher temperature to crisp up the outside. Second, the cracks that develop in the chips provide places for oil to collect and harden during frying, making them crunchy.Blumenthal, Heston Blumenthal at Home Third, thoroughly drying out the chips drives off moisture that would otherwise keep the crust from becoming crisp. Bloomsbury. ==Further reading== * * ==External links== * Triple-Cooked Chips. The chips are first simmered, then cooled and drained using a sous-vide technique or by freezing; deep fried at and cooled again; and finally deep-fried again at . The dealer starts by taking 4 tiles initially from one of the walls, then the players proceed in anti-clockwise order by taking 4 tiles each from where the previous player left off; after each player has 20 tiles, the dealer takes one more tile as they must discard a tile to start play. Chips (1940–1946) was a trained sentry dog for United States Army, and reputedly the most decorated war dog from World War II. On July 10, 1943, Chips and his handler were pinned down on the beach by an Italian machine-gun team. The second of the three stages is frying the chips at for approximately 5 minutes, after which they are cooled once more in a freezer or sous-vide machine before the third and final stage: frying at for approximately 7 minutes until crunchy and golden. Blumenthal began work on the recipe in 1993, and eventually developed the three-stage cooking process. ",0.65625,2.74,"""0.5""",35,5,A
+Divide a line segment into two parts by selecting a point at random. Use your intuition to assign a probability to the event that the longer segment is at least two times longer than the shorter segment.,"To approximate the mean line segment length of a given shape, two points are randomly chosen in its interior and the distance is measured. In geometry, the mean line segment length is the average length of a line segment connecting two points chosen uniformly at random in a given shape. While the question may seem simple, it has a fairly complicated answer; the exact value for this is \frac{2 + \sqrt{2} + 5 \ln (1 + \sqrt{2})}{15}. == Formal definition == The mean line segment length for an n-dimensional shape S may formally be defined as the expected Euclidean distance ||⋅|| between two random points x and y, : \mathbb E[\|x-y\|]=\frac1{\lambda(S)^2}\int_S \int_S \|x-y\| \,d\lambda(x) \,d\lambda(y) where λ is the n-dimensional Lebesgue measure. The length of a line segment is given by the Euclidean distance between its endpoints. thumb|historical image – create a line segment (1699) In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. Length of line. For the two- dimensional case, this is defined using the distance formula for two points (x1, y1) and (x2, y2) : \frac1{\lambda(S)^2}\iint_S \iint_S \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \,dx_1 \,dy_1 \,dx_2 \,dy_2. == Approximation methods == Since computing the mean line segment length involves calculating multidimensional integrals, various methods for numerical integration can be used to approximate this value for any shape. Thus, the line segment can be expressed as a convex combination of the segment's two end points. A new approach to line and line segment clipping in homogeneous coordinates, The Visual Computer, ISSN 0178-2789, Vol. 21, No. 11, pp. 905–914, Springer Verlag, 2005. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. In other words, it is the expected Euclidean distance between two random points, where each point in the shape is equally likely to be chosen. However, in the random case, with high probability the longest edge has length approximately \sqrt{\frac{\log n}{\pi n}}, longer than the average by a non-constant factor. There are two common algorithms for line clipping: Cohen–Sutherland and Liang–Barsky. Line sampling is a method used in reliability engineering to compute small (i.e., rare event) failure probabilities encountered in engineering systems. At the same time, 100% of respondents selected either one of these quantities as being the least desirable. == Calculation methods == There are a few methods to calculate line length to fit the intended average count of characters that such lines should contain based on the factors listed above. The line or line segment p can be computed from points r1, r2 given in homogeneous coordinates directly using the cross product as :p = r1 × r2 = (x1, y1, w1) × (x2, y2, w2) or as :p = r1 × r2 = (x1, y1, 1) × (x2, y2, 1). ==See also== * Clipping (computer graphics) ==References== Category:Clipping (computer graphics) The global probability of failure is the mean of the probability of failure on the lines: : \tilde{p}_f = \frac{1}{N_L} \sum_{i=1}^{N_L} p_f^{(i)} where N_L is the total number of lines used in the analysis and the p_f^{(i)} are the partial probabilities of failure estimated along all the lines. 370x370px|right|Example of line clipping for a two-dimensional region In computer graphics, line clipping is the process of removing (clipping) lines or portions of lines outside an area of interest (a viewport or view volume). Even for simple shapes such as a square or a triangle, solving for the exact value of their mean line segment lengths can be difficult because their closed-form expressions can get quite complicated. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ). *More generally than above, the concept of a line segment can be defined in an ordered geometry. thumb|300px|right|Euclidean minimum spanning tree of 25 random points in the plane A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. ",26.9,0.66666666666,"""167.0""",-191.2,2.3,B
+"In a state lottery, four digits are drawn at random one at a time with replacement from 0 to 9. Suppose that you win if any permutation of your selected integers is drawn. Give the probability of winning if you select $6,7,8,9$.","A six-number lottery game is a form of lottery in which six numbers are drawn from a larger pool (for example, 6 out of 44). If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner—regardless of the order of the numbers. Pick 3 draws a 3 digit number and Pick 4 draws a 4 digit number. A lottery is a form of gambling which involves selling numbered tickets and giving prizes to the holders of numbers drawn at random. Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, e.g. (Rain:.70, No Rain:.30).Mas-Colell, Andreu, Michael Whinston and Jerry Green (1995). For example, in the 6 from 49 lottery, given 10 powerball numbers, then the odds of getting a score of 3 and the powerball would be 1 in 56.66 × 10, or 566.6 (the probability would be divided by 10, to give an exact value of \frac{8815}{4994220}). The Missouri Lottery is the state-run lottery in Missouri. This can be written in a general form for all lotteries as: {K\choose B}{N-K\choose K-B}\over {N\choose K} where N is the number of balls in lottery, K is the number of balls in a single ticket, and B is the number of matching balls for a winning ticket. Winning the top prize, usually a progressive jackpot, requires a player to match all six regular numbers drawn; the order in which they are drawn is irrelevant. In the 5-from-90 lotto, the minimum number of tickets that can guarantee a ticket with at least 2 matches is 100. == Information theoretic results == As a discrete probability space, the probability of any particular lottery outcome is atomic, meaning it is greater than zero. If the player wagered an additional $1, they were eligible to win up to $25,000 in the Topper drawing, which was drawn by random number generator. ===Raffle=== The California Lottery offered two raffles; March 17, 2007 and one on January 1, 2008. Using X representing winning the 6-of-49 lottery, the Shannon entropy of 6-of-49 above is \begin{align} \Eta(X) &= -p\log(p) - q\log(q) = -\tfrac{1}{13,983,816}\log\\!{\tfrac{1}{13,983,816}} \- \tfrac{13,983,815}{13,983,816}\log\\!{\tfrac{13,983,815}{13,983,816}} \\\ & \approx 1.80065 \times 10^{-6} \text{ shannons.} \end{align} ==References== ==External links== * Euler's Analysis of the Genoese Lottery – Convergence (August 2010), Mathematical Association of America * Lottery Mathematics – INFAROM Publishing * 13,983,816 and the Lottery – YouTube video with James Clewett, Numberphile, March 2012 Mathematics Category:Combinatorics Category:Gambling mathematics A free ticket with 2 sets of numbers qualifying for the next Lotto draw is won by matching three numbers. Suppose the probabilities for lottery A are (Cured: .90, Uncured: .00, Dead: .10), and for lottery B are (Cured: .50, Uncured: .50, Dead: .00). That is to buy at least one lottery ticket for every possible number combination. There are two draws every day, televised at 1:29pm and 6:59pm.Televised Draw Results, California State Lottery ====Daily 4==== A ""pick 4"" type game premiered on May 19, 2008. This yields a final formula of :{n\choose k}={49\choose 6}={49\over 6} * {48\over 5} * {47\over 4} * {46\over 3} * {45\over 2} * {44\over 1} A 7th ball often is drawn as reserve ball, in the past only a second chance to get 5+1 numbers correct with 6 numbers played. ==Odds of getting other possibilities in choosing 6 from 49== One must divide the number of combinations producing the given result by the total number of possible combinations (for example, {49\choose 6} = 13,983,816 ). The prizes are smaller than other lottery games, but there are better odds (averaging 1:5). In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. ", 0.0024,-0.10,"""0.983""",0.0245,313,A
+"Extend Example 1.4-6 to an $n$-sided die. That is, suppose that a fair $n$-sided die is rolled $n$ independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1,2, \ldots, n$. Find the limit of this probability as $n$ increases without bound.","More precisely, if E denotes the event in question, p its probability of occurrence, and Nn(E) the number of times E occurs in the first n trials, then with probability one,An Analytic Technique to Prove Borel's Strong Law of Large Numbers Wen, L. Am Math Month 1991 \frac{N_n(E)}{n}\to p\text{ as }n\to\infty. Therefore, while \lim_{n\to\infty} \sum_{i=1}^n \frac{X_i} n = \overline{X} other formulas that look similar are not verified, such as the raw deviation from ""theoretical results"": \sum_{i=1}^n X_i - n\times\overline{X} not only does it not converge toward zero as n increases, but it tends to increase in absolute value as n increases. ==Examples== For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability. What this means is that the probability that, as the number of trials n goes to infinity, the average of the observations converges to the expected value, is equal to one. Section XIII.7 that if this probability is written as p(n,k) then : \lim_{n\rightarrow \infty} p(n,k) \alpha_k^{n+1}=\beta_k where αk is the smallest positive real root of :x^{k+1}=2^{k+1}(x-1) and :\beta_k={2-\alpha_k \over k+1-k\alpha_k}. ==Values of the constants== k \alpha_k \beta_k 1 2 2 2 1.23606797... 1.44721359... 3 1.08737802... 1.23683983... 4 1.03758012... 1.13268577... According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed. This result is useful to derive consistency of a large class of estimators (see Extremum estimator). ===Borel's law of large numbers=== Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. It does not converge in probability toward zero (or any other value) as n goes to infinity. The American Statistician, 56(3), pp.186-190 When bounding the event random variable deviates from its mean in only one direction (positive or negative), Cantelli's inequality gives an improvement over Chebyshev's inequality. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side: : \Pr \Bigl( \max_{1 \leq k \leq n} | S_k | \geq \alpha \Bigr) \leq \frac{27}{\alpha^2} \operatorname{var} (S_n). ==References== * (Theorem 22.5) * Category:Probabilistic inequalities Category:Statistical inequalities In particular, the proportion of heads after n flips will almost surely converge to as n approaches infinity. By Kolmogorov's zero–one law, for any fixed M, the probability that the event \limsup_n \frac{S_n}{\sqrt{n}} \geq M occurs is 0 or 1. Therefore, the expected value of the average of the rolls is: \frac{1+2+3+4+5+6}{6} = 3.5 According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean) will approach 3.5, with the precision increasing as more dice are rolled. It says that: ::\Pr\left[|S_n-E_n| > t \right] \leq 2\exp\left[ - \frac{V_n}{C^2} h\left(\frac{C t}{V_n} \right)\right], where h(u) = (1+u)\log(1+u)-u 5\. It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. The law then states that this converges in probability to zero.) Thus, although the absolute value of the quantity S_n/\sqrt{2n\log\log n} is less than any predefined ε > 0 with probability approaching one, it will nevertheless almost surely be greater than ε infinitely often; in fact, the quantity will be visiting the neighborhoods of any point in the interval (-1,1) almost surely. thumb|Exhibition of Limit Theorems and their interrelationship ==Generalizations and variants== The law of the iterated logarithm (LIL) for a sum of independent and identically distributed (i.i.d.) random variables with zero mean and bounded increment dates back to Khinchin and Kolmogorov in the 1920s. There are two versions of the law of large numbers — the weak and the strong — and they both state that the sums Sn, scaled by n−1, converge to zero, respectively in probability and almost surely: : \frac{S_n}{n} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{n} \ \xrightarrow{a.s.} 0, \qquad \text{as}\ \ n\to\infty. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be ""balanced"" by the others (see the gambler's fallacy). In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. Let Sk denote the partial sum :S_k = X_1 + \cdots + X_k.\, Then :\Pr \Bigl( \max_{1 \leq k \leq n} | S_k | \geq 3 \alpha \Bigr) \leq 3 \max_{1 \leq k \leq n} \Pr \bigl( | S_k | \geq \alpha \bigr). ==Remark== Suppose that the random variables Xk have common expected value zero. In probability theory, Etemadi's inequality is a so-called ""maximal inequality"", an inequality that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. In probability theory, Cantelli's inequality (also called the Chebyshev- Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds.""Tail and Concentration Inequalities"" by Hung Q. Ngo""Concentration-of-measure inequalities"" by Gábor Lugosi The inequality states that, for \lambda > 0, : \Pr(X-\mathbb{E}[X]\ge\lambda) \le \frac{\sigma^2}{\sigma^2 + \lambda^2}, where :X is a real-valued random variable, :\Pr is the probability measure, :\mathbb{E}[X] is the expected value of X, :\sigma^2 is the variance of X. Applying the Cantelli inequality to -X gives a bound on the lower tail, : \Pr(X-\mathbb{E}[X]\le -\lambda) \le \frac{\sigma^2}{\sigma^2 + \lambda^2}. ",38,19.4,"""24.4""",-1.32,0.6321205588,E
 "Calculating the maximum wavelength capable of photoejection
-A photon of radiation of wavelength $305 \mathrm{~nm}$ ejects an electron from a metal with a kinetic energy of $1.77 \mathrm{eV}$. Calculate the maximum wavelength of radiation capable of ejecting an electron from the metal.","However, because short-wavelength photons carry more energy per photon, the maximum amount of photosynthesis per incident unit of energy is at a longer wavelength, around 650 nm (deep red). A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For a black-body light source at 5800 K, such as the sun is approximately, a fraction 0.368 of its total emitted radiation is emitted as PAR. Using the expression above, the optimal efficiency or second law efficiency for the conversion of radiation to work in the PAR region (from \lambda_1 = 400 nm to \lambda_2 = 700 nm), for a blackbody at T = 5800 K and an organism at T_0 = 300 K is determined as: : \eta^{ex}_\text{PAR}(T) = \frac{\int_{\lambda_1}^{\lambda_2} Ex(\lambda,T)d\lambda}{\int_{0}^\infty L(\lambda, T)d\lambda} = 0.337563 about 8.3% lower than the value considered until now, as a direct consequence of the fact that the organisms which are using solar radiation are also emitting radiation as a consequence of their own temperature. Ultraviolet astronomy is the observation of electromagnetic radiation at ultraviolet wavelengths between approximately 10 and 320 nanometres; shorter wavelengths--higher energy photons--are studied by X-ray astronomy and gamma- ray astronomy. The red curve in the graph shows that photons around 610 nm (orange-red) have the highest amount of photosynthesis per photon. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. These wavelengths correspond to photon energies of down to . The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. Ultra-high-energy gamma rays are gamma rays with photon energies higher than 100 TeV (0.1 PeV). In a 18 May 2021 press release, China's Large High Altitude Air Shower Observatory (LHAASO) reported the detection of a dozen ultra-high- energy gamma rays with energies exceeding 1 peta-electron-volt (quadrillion electron-volts or PeV), including one at 1.4 PeV, the highest energy photon ever observed. thumb|upright=1.25|Photosynthetically active radiation (PAR) spans the visible light portion of the electromagnetic spectrum from 400 to 700 nanometers. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. Spectral irradiance of wavelengths in the solar spectrum. The following table shows the conversion factors from watts for black-body spectra that are truncated to the range 400-700 nm. The quantities in the table are calculated as :\eta_v(T) = \frac{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\, 683 \mathrm{~[lm/W]}\, y(\lambda)\,d\lambda}{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,d\lambda}, :\eta_{\mathrm{photon}}(T) = \frac{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,\frac{\lambda}{hcN_\text{A}} \,d\lambda}{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,d\lambda}, :\eta_{\mathrm{PAR}}(T) = \frac{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,d\lambda}{\int_0^{\infty} B(\lambda, T)\,d\lambda}, where B(\lambda,T) is the black-body spectrum according to Planck's law, y is the standard luminosity function, \lambda_1,\lambda_2 represent the wavelength range (400–700 nm) of PAR, and N_\text{A} is the Avogadro constant. == Second law PAR efficiency == Besides the amount of radiation reaching a plant in the PAR region of the spectrum, it is also important to consider the quality of such radiation. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. Class energy (TeV) energy (eV) energy (μJ) frequency (YHz) wavelength (am) comparison properties 10−12 1 1.602 × 10−13 2.418 × 10−12 1.2398 × 1012 near infrared photon (for comparison) 0.1 1 × 1011 0.01602 24.2 12 Z boson Very- high-energy gamma rays 1 1 × 1012 0.1602 242 1.2 flying mosquito produces Cherenkov light 10 1 × 1013 1.602 2.42 × 103 0.12 air shower reaches ground 100 1 × 1014 16.02 2.42 × 104 0.012 ping pong ball falling off a bat causes nitrogen to fluoresce Ultra-high-energy gamma rays 1000 1 × 1015 160.2 2.42 × 10 1.2 × 10−3 10 000 TeV 1 × 1016 1602 2.42 × 106 1.2 × 10−4 potential energy of golf ball on a tee 100 000 1 × 1017 1.602 × 104 2.42 × 107 1.2 × 10−5 1 000 000 1 × 1018 1.602 × 105 2.42 × 108 1.2 × 10−6 10 000 000 1 × 1019 1.602 × 106 2.42 × 109 1.2 × 10−7 air rifle shot 1.22091 × 1028 1.95611 × 109 1.855 × 1019 1.61623 × 10−17 explosion of a car tank full of gasoline Planck energy ==References== ==External links== *Search for Galactic PeV Gamma Rays with the IceCube Neutrino Observatory *Air shower detection of diffuse PeV gamma-rays Category:Gamma rays In the Earth's magnetic field, a 1021 eV photon is expected to interact about 5000 km above the earth's surface. Thus, in order for a rectifying antenna to be an efficient electromagnetic collector in the solar spectrum, it needs to be on the order of hundreds of nm in size. right|250px|thumb| Figure 3. Photons at shorter wavelengths tend to be so energetic that they can be damaging to cells and tissues, but are mostly filtered out by the ozone layer in the stratosphere. *Photo-dissociation fragments carry away some of the photon energy as kinetic energy, heating the gas. ",-3.8,418,0.375,2,540,E
-Estimate the molar volume of $\mathrm{CO}_2$ at $500 \mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.,"Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The molar van der Waals volume should not be confused with the molar volume of the substance. The density of solid helium at 1.1 K and 66 atm is , corresponding to a molar volume V = . However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",0.4772,2.6,3.2,0.14,0.366,E
-"The single electron in a certain excited state of a hydrogenic $\mathrm{He}^{+}$ion $(Z=2)$ is described by the wavefunction $R_{3,2}(r) \times$ $Y_{2,-1}(\theta, \phi)$. What is the energy of its electron?","An electron in the spherically symmetric Coulomb potential has potential energy: :U_\text{C} = -\dfrac{e^2}{4\pi\varepsilon_0r}. Additional terms in the potential energy expression for a Rydberg state, on top of the hydrogenic Coulomb potential energy require the introduction of a quantum defect, δl, into the expression for the binding energy: :E_\text{B} = -\frac{\rm Ry}{(n-\delta_l)^2}. === Electron wavefunctions === The long lifetimes of Rydberg states with high orbital angular momentum can be explained in terms of the overlapping of wavefunctions. This energy is assumed to equal the electron's rest energy, defined by special relativity (E = mc2). The three exceptions to the definition of a Rydberg atom as an atom with a hydrogenic potential, have an alternative, quantum mechanical description that can be characterized by the additional term(s) in the atomic Hamiltonian: *If a second electron is excited into a state ni, energetically close to the state of the outer electron no, then its wavefunction becomes almost as large as the first (a double Rydberg state). For an atom in a multiple Rydberg state, the additional term, Uee, includes a summation of each pair of highly excited electrons: :U_{ee} = \dfrac{e^2}{4\pi\varepsilon_0}\sum_{i < j}\dfrac{1}{|\mathbf{r}_i - \mathbf{r}_j|}. An atom in a Rydberg state has a valence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almost hydrogenic, Coulomb potential, UC from a compact ion core consisting of a nucleus with Z protons and the lower electron shells filled with Z-1 electrons. In hydrogen the binding energy is given by: : E_\text{B} = -\frac{\rm Ry}{n^2}, where Ry = 13.6 eV is the Rydberg constant. The electron (symbol e) is on the left. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. There are three notable exceptions that can be characterized by the additional term added to the potential energy: *An atom may have two (or more) electrons in highly excited states with comparable orbital radii. In atomic physics, a two-electron atom or helium-like ion is a quantum mechanical system consisting of one nucleus with a charge of Ze and just two electrons. The wavefunction is a function of the two electron's positions: \psi = \psi(\mathbf{r}_1,\mathbf{r}_2) There is no closed form solution for this equation. ==Spectrum== The optical spectrum of the two electron atom has two systems of lines. The helium hydride ion or hydridohelium(1+) ion or helonium is a cation (positively charged ion) with chemical formula HeH+. In this case, the electron-electron interaction gives rise to a significant deviation from the hydrogen potential. E\psi = -\hbar^2\left[\frac{1}{2\mu}\left( abla_1^2 + abla_2^2 \right) + \frac{1}{M} abla_1 \cdot abla_2\right] \psi + \frac{e^2}{4\pi\varepsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1} + \frac{1}{r_2} \right) \right] \psi where r1 is the position of one electron (r1 = |r1| is its magnitude), r2 is the position of the other electron (r2 = |r2| is the magnitude), r12 = |r12| is the magnitude of the separation between them given by |\mathbf{r}_{12}| = |\mathbf{r}_2 - \mathbf{r}_1 | μ is the two-body reduced mass of an electron with respect to the nucleus of mass M \mu = \frac{m_e M}{m_e+M} and Z is the atomic number for the element (not a quantum number). Stark - Coulomb potential for a Rydberg atom in a static electric field. The word electron is a combination of the words _electr_ ic and i _on_.""electron, n.2"". *If the valence electron has very low angular momentum (interpreted classically as an extremely eccentric elliptical orbit), then it may pass close enough to polarise the ion core, giving rise to a 1/r4 core polarization term in the potential. An electron-electron repulsion term must be included in the atomic Hamiltonian. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. The electron's mass is approximately 1/1836 that of the proton. It consists of a helium atom bonded to a hydrogen atom, with one electron removed. ",2.5151,1.60,6.9,30, -6.04697,E
-"Calculate the typical wavelength of neutrons after reaching thermal equilibrium with their surroundings at $373 \mathrm{~K}$. For simplicity, assume that the particles are travelling in one dimension.","This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Introduction of the Theory of Thermal Neutron Scattering. https://books.google.com/books?id=KUVD8KJt7_0C&dq;=thermal- neutron+reactor&pg;=PR9 thus scattering neutrons by nuclear forces, some nuclides are scattered large. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. The neutron fluence is defined as the neutron flux integrated over a certain time period, so its usual unit is cm−2 (neutrons per centimeter squared). The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. Earth atmospheric neutron flux, apparently from thunderstorms, can reach levels of 3·10−2 to 9·10+1 cm−2 s−1. A thermal-neutron reactor is a nuclear reactor that uses slow or thermal neutrons. The measured quantity is the difference in the number of gamma rays emitted within a solid angle between the two neutron spin states. The higher the neutron flux the greater the chance of a nuclear reaction occurring as there are more neutrons going through an area per unit time. === Reactor vessel wall neutron fluence === A reactor vessel of a typical nuclear power plant (PWR) endures in 40 years (32 full reactor years) of operation approximately 6.5×1019 cm−2 (E > 1 MeV) of neutron fluence.Nuclear Power Plant Borssele Reactor Pressure Vessel Safety Assessment, p. 29, 5.6 Neutron Fluence Calculation. A neutron may pass by a nucleus with a probability determined by the nuclear interaction distance, or be absorbed, or undergo scattering that may be either coherent or incoherent. Equivalently, it can be defined as the number of neutrons travelling through a small sphere of radius R in a time interval, divided by \pi R^2 (the cross section of the sphere) and by the time interval. Hence, \lambda_{\rm th} = \frac{h}{\sqrt{2\pi m k_{\mathrm B} T}} , where h is the Planck constant, is the mass of a gas particle, k_{\mathrm B} is the Boltzmann constant, and is the temperature of the gas. For example, when observing the long-wavelength spectrum of black body radiation, the classical Rayleigh–Jeans law can be applied, but when the observed wavelengths approach the thermal wavelength of the photons in the black body radiator, the quantum Planck's law must be used. ==General definition== A general definition of the thermal wavelength for an ideal gas of particles having an arbitrary power-law relationship between energy and momentum (dispersion relationship), in any number of dimensions, can be introduced. The polarization of the incoming neutron beam is alternated rapidly to study the spin correlation of the direction of the emitted gamma ray. TIme-dependent neutronics and temperatures (TINTE) is a two-group diffusion code for the study of nuclear and thermal behavior of high temperature reactors. (""Thermal"" does not mean hot in an absolute sense, but means in thermal equilibrium with the medium it is interacting with, the reactor's fuel, moderator and structure, which is much lower energy than the fast neutrons initially produced by fission.) This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. As with the thermal wavelength for massive particles, this is of the order of the average wavelength of the particles in the gas and defines a critical point at which quantum effects begin to dominate. The usual unit is cm−2s−1 (neutrons per centimeter squared per second). NPDGamma is an ongoing effort to measure the parity-violating asymmetry in polarized cold neutron capture on parahydrogen. :\vec n + p \to d + \gamma Polarized neutrons of energies 2 meV – 15 meV are incident on a liquid parahydrogen target. ",-45,7.27,17.0,2598960,226,E
+A photon of radiation of wavelength $305 \mathrm{~nm}$ ejects an electron from a metal with a kinetic energy of $1.77 \mathrm{eV}$. Calculate the maximum wavelength of radiation capable of ejecting an electron from the metal.","However, because short-wavelength photons carry more energy per photon, the maximum amount of photosynthesis per incident unit of energy is at a longer wavelength, around 650 nm (deep red). A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For a black-body light source at 5800 K, such as the sun is approximately, a fraction 0.368 of its total emitted radiation is emitted as PAR. Using the expression above, the optimal efficiency or second law efficiency for the conversion of radiation to work in the PAR region (from \lambda_1 = 400 nm to \lambda_2 = 700 nm), for a blackbody at T = 5800 K and an organism at T_0 = 300 K is determined as: : \eta^{ex}_\text{PAR}(T) = \frac{\int_{\lambda_1}^{\lambda_2} Ex(\lambda,T)d\lambda}{\int_{0}^\infty L(\lambda, T)d\lambda} = 0.337563 about 8.3% lower than the value considered until now, as a direct consequence of the fact that the organisms which are using solar radiation are also emitting radiation as a consequence of their own temperature. Ultraviolet astronomy is the observation of electromagnetic radiation at ultraviolet wavelengths between approximately 10 and 320 nanometres; shorter wavelengths--higher energy photons--are studied by X-ray astronomy and gamma- ray astronomy. The red curve in the graph shows that photons around 610 nm (orange-red) have the highest amount of photosynthesis per photon. This can be done by photoexcitation (PE), where the electron absorbs a photon and gains all its energy or by collisional excitation (CE), where the electron receives energy from a collision with another, energetic electron. These wavelengths correspond to photon energies of down to . The wavelengths in the solar spectrum range from approximately 0.3-2.0 μm. Ultra-high-energy gamma rays are gamma rays with photon energies higher than 100 TeV (0.1 PeV). In a 18 May 2021 press release, China's Large High Altitude Air Shower Observatory (LHAASO) reported the detection of a dozen ultra-high- energy gamma rays with energies exceeding 1 peta-electron-volt (quadrillion electron-volts or PeV), including one at 1.4 PeV, the highest energy photon ever observed. thumb|upright=1.25|Photosynthetically active radiation (PAR) spans the visible light portion of the electromagnetic spectrum from 400 to 700 nanometers. thumb|420x420px|A schematic of electron excitation, showing excitation by photon (left) and by particle collision (right) Electron excitation is the transfer of a bound electron to a more energetic, but still bound state. Spectral irradiance of wavelengths in the solar spectrum. The following table shows the conversion factors from watts for black-body spectra that are truncated to the range 400-700 nm. The quantities in the table are calculated as :\eta_v(T) = \frac{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\, 683 \mathrm{~[lm/W]}\, y(\lambda)\,d\lambda}{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,d\lambda}, :\eta_{\mathrm{photon}}(T) = \frac{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,\frac{\lambda}{hcN_\text{A}} \,d\lambda}{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,d\lambda}, :\eta_{\mathrm{PAR}}(T) = \frac{\int_{\lambda_1}^{\lambda_2} B(\lambda, T)\,d\lambda}{\int_0^{\infty} B(\lambda, T)\,d\lambda}, where B(\lambda,T) is the black-body spectrum according to Planck's law, y is the standard luminosity function, \lambda_1,\lambda_2 represent the wavelength range (400–700 nm) of PAR, and N_\text{A} is the Avogadro constant. == Second law PAR efficiency == Besides the amount of radiation reaching a plant in the PAR region of the spectrum, it is also important to consider the quality of such radiation. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. Class energy (TeV) energy (eV) energy (μJ) frequency (YHz) wavelength (am) comparison properties 10−12 1 1.602 × 10−13 2.418 × 10−12 1.2398 × 1012 near infrared photon (for comparison) 0.1 1 × 1011 0.01602 24.2 12 Z boson Very- high-energy gamma rays 1 1 × 1012 0.1602 242 1.2 flying mosquito produces Cherenkov light 10 1 × 1013 1.602 2.42 × 103 0.12 air shower reaches ground 100 1 × 1014 16.02 2.42 × 104 0.012 ping pong ball falling off a bat causes nitrogen to fluoresce Ultra-high-energy gamma rays 1000 1 × 1015 160.2 2.42 × 10 1.2 × 10−3 10 000 TeV 1 × 1016 1602 2.42 × 106 1.2 × 10−4 potential energy of golf ball on a tee 100 000 1 × 1017 1.602 × 104 2.42 × 107 1.2 × 10−5 1 000 000 1 × 1018 1.602 × 105 2.42 × 108 1.2 × 10−6 10 000 000 1 × 1019 1.602 × 106 2.42 × 109 1.2 × 10−7 air rifle shot 1.22091 × 1028 1.95611 × 109 1.855 × 1019 1.61623 × 10−17 explosion of a car tank full of gasoline Planck energy ==References== ==External links== *Search for Galactic PeV Gamma Rays with the IceCube Neutrino Observatory *Air shower detection of diffuse PeV gamma-rays Category:Gamma rays In the Earth's magnetic field, a 1021 eV photon is expected to interact about 5000 km above the earth's surface. Thus, in order for a rectifying antenna to be an efficient electromagnetic collector in the solar spectrum, it needs to be on the order of hundreds of nm in size. right|250px|thumb| Figure 3. Photons at shorter wavelengths tend to be so energetic that they can be damaging to cells and tissues, but are mostly filtered out by the ozone layer in the stratosphere. *Photo-dissociation fragments carry away some of the photon energy as kinetic energy, heating the gas. ",-3.8,418,"""0.375""",2,540,E
+Estimate the molar volume of $\mathrm{CO}_2$ at $500 \mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.,"Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The molar van der Waals volume should not be confused with the molar volume of the substance. The density of solid helium at 1.1 K and 66 atm is , corresponding to a molar volume V = . However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",0.4772,2.6,"""3.2""",0.14,0.366,E
+"The single electron in a certain excited state of a hydrogenic $\mathrm{He}^{+}$ion $(Z=2)$ is described by the wavefunction $R_{3,2}(r) \times$ $Y_{2,-1}(\theta, \phi)$. What is the energy of its electron?","An electron in the spherically symmetric Coulomb potential has potential energy: :U_\text{C} = -\dfrac{e^2}{4\pi\varepsilon_0r}. Additional terms in the potential energy expression for a Rydberg state, on top of the hydrogenic Coulomb potential energy require the introduction of a quantum defect, δl, into the expression for the binding energy: :E_\text{B} = -\frac{\rm Ry}{(n-\delta_l)^2}. === Electron wavefunctions === The long lifetimes of Rydberg states with high orbital angular momentum can be explained in terms of the overlapping of wavefunctions. This energy is assumed to equal the electron's rest energy, defined by special relativity (E = mc2). The three exceptions to the definition of a Rydberg atom as an atom with a hydrogenic potential, have an alternative, quantum mechanical description that can be characterized by the additional term(s) in the atomic Hamiltonian: *If a second electron is excited into a state ni, energetically close to the state of the outer electron no, then its wavefunction becomes almost as large as the first (a double Rydberg state). For an atom in a multiple Rydberg state, the additional term, Uee, includes a summation of each pair of highly excited electrons: :U_{ee} = \dfrac{e^2}{4\pi\varepsilon_0}\sum_{i < j}\dfrac{1}{|\mathbf{r}_i - \mathbf{r}_j|}. An atom in a Rydberg state has a valence electron in a large orbit far from the ion core; in such an orbit, the outermost electron feels an almost hydrogenic, Coulomb potential, UC from a compact ion core consisting of a nucleus with Z protons and the lower electron shells filled with Z-1 electrons. In hydrogen the binding energy is given by: : E_\text{B} = -\frac{\rm Ry}{n^2}, where Ry = 13.6 eV is the Rydberg constant. The electron (symbol e) is on the left. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. There are three notable exceptions that can be characterized by the additional term added to the potential energy: *An atom may have two (or more) electrons in highly excited states with comparable orbital radii. In atomic physics, a two-electron atom or helium-like ion is a quantum mechanical system consisting of one nucleus with a charge of Ze and just two electrons. The wavefunction is a function of the two electron's positions: \psi = \psi(\mathbf{r}_1,\mathbf{r}_2) There is no closed form solution for this equation. ==Spectrum== The optical spectrum of the two electron atom has two systems of lines. The helium hydride ion or hydridohelium(1+) ion or helonium is a cation (positively charged ion) with chemical formula HeH+. In this case, the electron-electron interaction gives rise to a significant deviation from the hydrogen potential. E\psi = -\hbar^2\left[\frac{1}{2\mu}\left( abla_1^2 + abla_2^2 \right) + \frac{1}{M} abla_1 \cdot abla_2\right] \psi + \frac{e^2}{4\pi\varepsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1} + \frac{1}{r_2} \right) \right] \psi where r1 is the position of one electron (r1 = |r1| is its magnitude), r2 is the position of the other electron (r2 = |r2| is the magnitude), r12 = |r12| is the magnitude of the separation between them given by |\mathbf{r}_{12}| = |\mathbf{r}_2 - \mathbf{r}_1 | μ is the two-body reduced mass of an electron with respect to the nucleus of mass M \mu = \frac{m_e M}{m_e+M} and Z is the atomic number for the element (not a quantum number). Stark - Coulomb potential for a Rydberg atom in a static electric field. The word electron is a combination of the words _electr_ ic and i _on_.""electron, n.2"". *If the valence electron has very low angular momentum (interpreted classically as an extremely eccentric elliptical orbit), then it may pass close enough to polarise the ion core, giving rise to a 1/r4 core polarization term in the potential. An electron-electron repulsion term must be included in the atomic Hamiltonian. A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. The electron's mass is approximately 1/1836 that of the proton. It consists of a helium atom bonded to a hydrogen atom, with one electron removed. ",2.5151,1.60,"""6.9""",30, -6.04697,E
+"Calculate the typical wavelength of neutrons after reaching thermal equilibrium with their surroundings at $373 \mathrm{~K}$. For simplicity, assume that the particles are travelling in one dimension.","This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. Introduction of the Theory of Thermal Neutron Scattering. https://books.google.com/books?id=KUVD8KJt7_0C&dq;=thermal- neutron+reactor&pg;=PR9 thus scattering neutrons by nuclear forces, some nuclides are scattered large. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. The neutron fluence is defined as the neutron flux integrated over a certain time period, so its usual unit is cm−2 (neutrons per centimeter squared). The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. Earth atmospheric neutron flux, apparently from thunderstorms, can reach levels of 3·10−2 to 9·10+1 cm−2 s−1. A thermal-neutron reactor is a nuclear reactor that uses slow or thermal neutrons. The measured quantity is the difference in the number of gamma rays emitted within a solid angle between the two neutron spin states. The higher the neutron flux the greater the chance of a nuclear reaction occurring as there are more neutrons going through an area per unit time. === Reactor vessel wall neutron fluence === A reactor vessel of a typical nuclear power plant (PWR) endures in 40 years (32 full reactor years) of operation approximately 6.5×1019 cm−2 (E > 1 MeV) of neutron fluence.Nuclear Power Plant Borssele Reactor Pressure Vessel Safety Assessment, p. 29, 5.6 Neutron Fluence Calculation. A neutron may pass by a nucleus with a probability determined by the nuclear interaction distance, or be absorbed, or undergo scattering that may be either coherent or incoherent. Equivalently, it can be defined as the number of neutrons travelling through a small sphere of radius R in a time interval, divided by \pi R^2 (the cross section of the sphere) and by the time interval. Hence, \lambda_{\rm th} = \frac{h}{\sqrt{2\pi m k_{\mathrm B} T}} , where h is the Planck constant, is the mass of a gas particle, k_{\mathrm B} is the Boltzmann constant, and is the temperature of the gas. For example, when observing the long-wavelength spectrum of black body radiation, the classical Rayleigh–Jeans law can be applied, but when the observed wavelengths approach the thermal wavelength of the photons in the black body radiator, the quantum Planck's law must be used. ==General definition== A general definition of the thermal wavelength for an ideal gas of particles having an arbitrary power-law relationship between energy and momentum (dispersion relationship), in any number of dimensions, can be introduced. The polarization of the incoming neutron beam is alternated rapidly to study the spin correlation of the direction of the emitted gamma ray. TIme-dependent neutronics and temperatures (TINTE) is a two-group diffusion code for the study of nuclear and thermal behavior of high temperature reactors. (""Thermal"" does not mean hot in an absolute sense, but means in thermal equilibrium with the medium it is interacting with, the reactor's fuel, moderator and structure, which is much lower energy than the fast neutrons initially produced by fission.) This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. As with the thermal wavelength for massive particles, this is of the order of the average wavelength of the particles in the gas and defines a critical point at which quantum effects begin to dominate. The usual unit is cm−2s−1 (neutrons per centimeter squared per second). NPDGamma is an ongoing effort to measure the parity-violating asymmetry in polarized cold neutron capture on parahydrogen. :\vec n + p \to d + \gamma Polarized neutrons of energies 2 meV – 15 meV are incident on a liquid parahydrogen target. ",-45,7.27,"""17.0""",2598960,226,E
 "Using the perfect gas equation
-Calculate the pressure in kilopascals exerted by $1.25 \mathrm{~g}$ of nitrogen gas in a flask of volume $250 \mathrm{~cm}^3$ at $20^{\circ} \mathrm{C}$.","The equation was developed by Martin Hans Christian Knudsen (1871–1949), a Danish physicist who taught and conducted research at the Technical University of Denmark. ==Cylindrical tube== For a cylindrical tube, the Knudsen equation is: :q = \frac16 \sqrt{2 \pi} \Delta P \frac{d^3}{ l \sqrt{\rho_1}}, where: Quantity Description q volume flow rate at unit pressure (volume×pressure/time) ΔP pressure drop from the beginning of the tube to the end d diameter of the tube l length of the tube ρ1 ratio of density and pressure For nitrogen (or air) at room temperature, the conductivity C (in liters per second) of a tube can be calculated from this equation: :\frac{C}{\mathrm{L}/\mathrm{s}} \approx 12 \, \frac{d^3/\mathrm{cm}^3}{{l/\mathrm{cm}}} == References == Category:Fluid dynamics In fluid dynamics, the Knudsen equation is used to describe how gas flows through a tube in free molecular flow. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. A newton is equal to 1 kg⋅m/s2, and a kilogram-force is 9.80665 N,The NIST Guide for the use of the International System of Units, National Institute of Standards and Technology, 18 Oct 2011 meaning that 1 kgf/cm2 equals 98.0665 kilopascals (kPa). For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. A Knudsen gas is a gas in a state of such low density that the average distance travelled by the gas molecules between collisions (mean free path) is greater than the diameter of the receptacle that contains it. If the diameter of the receptacle is less than 68nm, the Knudsen number would greater than 1, and this sample of air would be considered a Knudsen gas. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Note the ""square"" instead of 2. ( means ""oil"" in Swedish) A kilogram-force per centimetre square (kgf/cm2), often just kilogram per square centimetre (kg/cm2), or kilopond per centimetre square (kp/cm2) is a deprecated unit of pressure using metric units. It would not be a Knudsen gas if the diameter of the receptacle is greater than 68nm. ==References== == See also == * Free streaming * Kinetic theory Category:Gases Category:Phases of matter In all perfect gas models, intermolecular forces are neglected. In some older publications, kilogram-force per square centimetre is abbreviated ksc instead of kg/cm2. : 1 at = 98.0665 kPa 1 at ≈ standard atmospheres ==Ambiguity of at== The symbol ""at"" clashes with that of the katal (symbol: ""kat""), the SI unit of catalytic activity; a kilotechnical atmosphere would have the symbol ""kat"", indistinguishable from the symbol for the katal. When 10^{-1}<\rm{Kn}<10, the flow regime of the gas is transitional flow. It is not a part of the International System of Units (SI), the modern metric system. 1 kgf/cm2 equals 98.0665 kPa (kilopascals). There are more collisions between the gas molecules and the receptacle walls (shown in red) compared to collisions between gas molecules (shown in blue). == Knudsen number == For a Knudsen gas, the Knudsen number must be greater than 1. Pressure piling describes phenomena related to combustion of gases in a tube or long vessel. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. Dalton's law is not strictly followed by real gases, with the deviation increasing with pressure. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. ",15.425,25.6773,435.0,6.6,9.8,C
+Calculate the pressure in kilopascals exerted by $1.25 \mathrm{~g}$ of nitrogen gas in a flask of volume $250 \mathrm{~cm}^3$ at $20^{\circ} \mathrm{C}$.","The equation was developed by Martin Hans Christian Knudsen (1871–1949), a Danish physicist who taught and conducted research at the Technical University of Denmark. ==Cylindrical tube== For a cylindrical tube, the Knudsen equation is: :q = \frac16 \sqrt{2 \pi} \Delta P \frac{d^3}{ l \sqrt{\rho_1}}, where: Quantity Description q volume flow rate at unit pressure (volume×pressure/time) ΔP pressure drop from the beginning of the tube to the end d diameter of the tube l length of the tube ρ1 ratio of density and pressure For nitrogen (or air) at room temperature, the conductivity C (in liters per second) of a tube can be calculated from this equation: :\frac{C}{\mathrm{L}/\mathrm{s}} \approx 12 \, \frac{d^3/\mathrm{cm}^3}{{l/\mathrm{cm}}} == References == Category:Fluid dynamics In fluid dynamics, the Knudsen equation is used to describe how gas flows through a tube in free molecular flow. thumb|upright=1.75|An illustration of Dalton's law using the gases of air at sea level. A newton is equal to 1 kg⋅m/s2, and a kilogram-force is 9.80665 N,The NIST Guide for the use of the International System of Units, National Institute of Standards and Technology, 18 Oct 2011 meaning that 1 kgf/cm2 equals 98.0665 kilopascals (kPa). For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. Dalton's law (also called Dalton's law of partial pressures) states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. A Knudsen gas is a gas in a state of such low density that the average distance travelled by the gas molecules between collisions (mean free path) is greater than the diameter of the receptacle that contains it. If the diameter of the receptacle is less than 68nm, the Knudsen number would greater than 1, and this sample of air would be considered a Knudsen gas. In particular, the short average distances between molecules increases intermolecular forces between gas molecules enough to substantially change the pressure exerted by them, an effect not included in the ideal gas model. ==See also== * * * * * * * * * ==References== Category:Gas laws Category:Physical chemistry Category:Engineering thermodynamics de:Partialdruck#Dalton-Gesetz et:Daltoni seadus Dalton's law is related to the ideal gas laws. ==Formula== Mathematically, the pressure of a mixture of non-reactive gases can be defined as the summation: p_\text{total} = \sum_{i=1}^n p_i = p_1+p_2+p_3+\cdots+p_n where p1, p2, ..., pn represent the partial pressures of each component. p_{i} = p_\text{total} x_i where xi is the mole fraction of the ith component in the total mixture of n components . ==Volume-based concentration== The relationship below provides a way to determine the volume- based concentration of any individual gaseous component p_i = p_\text{total} c_i where ci is the concentration of component i. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Note the ""square"" instead of 2. ( means ""oil"" in Swedish) A kilogram-force per centimetre square (kgf/cm2), often just kilogram per square centimetre (kg/cm2), or kilopond per centimetre square (kp/cm2) is a deprecated unit of pressure using metric units. It would not be a Knudsen gas if the diameter of the receptacle is greater than 68nm. ==References== == See also == * Free streaming * Kinetic theory Category:Gases Category:Phases of matter In all perfect gas models, intermolecular forces are neglected. In some older publications, kilogram-force per square centimetre is abbreviated ksc instead of kg/cm2. : 1 at = 98.0665 kPa 1 at ≈ standard atmospheres ==Ambiguity of at== The symbol ""at"" clashes with that of the katal (symbol: ""kat""), the SI unit of catalytic activity; a kilotechnical atmosphere would have the symbol ""kat"", indistinguishable from the symbol for the katal. When 10^{-1}<\rm{Kn}<10, the flow regime of the gas is transitional flow. It is not a part of the International System of Units (SI), the modern metric system. 1 kgf/cm2 equals 98.0665 kPa (kilopascals). There are more collisions between the gas molecules and the receptacle walls (shown in red) compared to collisions between gas molecules (shown in blue). == Knudsen number == For a Knudsen gas, the Knudsen number must be greater than 1. Pressure piling describes phenomena related to combustion of gases in a tube or long vessel. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. Dalton's law is not strictly followed by real gases, with the deviation increasing with pressure. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. ",15.425,25.6773,"""435.0""",6.6,9.8,C
 "Determine the energies and degeneracies of the lowest four energy levels of an ${ }^1 \mathrm{H}^{35} \mathrm{Cl}$ molecule freely rotating in three dimensions. What is the frequency of the transition between the lowest two rotational levels? The moment of inertia of an ${ }^1 \mathrm{H}^{35} \mathrm{Cl}$ molecule is $2.6422 \times 10^{-47} \mathrm{~kg} \mathrm{~m}^2$.
-","For example, the rotational energy levels for linear molecules (in the rigid-rotor approximation) are :E_\text{rot} = hc BJ(J + 1). The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number, J, which defines the magnitude of the rotational angular momentum. For a linear molecule, analysis of the rotational spectrum provides values for the rotational constantThis article uses the molecular spectroscopist's convention of expressing the rotational constant B in cm��1. Fitting the spectra to the theoretical expressions gives numerical values of the angular moments of inertia from which very precise values of molecular bond lengths and angles can be derived in favorable cases. Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. Analytical expressions can be derived for the fourth category, asymmetric top, for rotational levels up to J=3, but higher energy levels need to be determined using numerical methods. Thus, by completing a Deslandres table it is easy to assign the correct vibrational quantum numbers v and v' for the transition, allowing important molecular properties to be calculated, such as the dissociation energy. == References == * Category:Spectroscopy The rotational energies are derived theoretically by considering the molecules to be rigid rotors and then applying extra terms to account for centrifugal distortion, fine structure, hyperfine structure and Coriolis coupling. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants. In the absence of an external electrical field, the rotational energy of a symmetric top is a function of only J and K and, in the rigid rotor approximation, the energy of each rotational state is given by : F\left( J,K \right) = B J \left( J+1 \right) + \left( A - B \right) K^2 \qquad J = 0, 1, 2, \ldots \quad \mbox{and}\quad K = +J, \ldots, 0, \ldots, -J where B = {h\over{8\pi^2cI_B}} and A = {h\over{8\pi^2cI_A}} for a prolate symmetric top molecule or A = {h\over{8\pi^2cI_C}} for an oblate molecule. When this is not possible, as with most asymmetric tops, all that can be done is to fit the spectra to three moments of inertia calculated from an assumed molecular structure. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, I_B = I_C, I_A=0 , so : B = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}} For a diatomic molecule : I=\frac{m_1m_2}{m_1 +m_2}d^2 where m1 and m2 are the masses of the atoms and d is the distance between them. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule. The third quantum number, K is associated with rotation about the principal rotation axis of the molecule. For rotational spectroscopy, molecules are classified according to symmetry into a spherical top, linear and symmetric top; analytical expressions can be derived for the rotational energy terms of these molecules. The second factor is the degeneracy of the rotational state, which is equal to . Under the rigid rotor model, the rotational energy levels, F(J), of the molecule can be expressed as, : F\left( J \right) = B J \left( J+1 \right) \qquad J = 0,1,2,... where B is the rotational constant of the molecule and is related to the moment of inertia of the molecule. For any molecule, there are three moments of inertia: I_A, I_B and I_C about three mutually orthogonal axes A, B, and C with the origin at the center of mass of the system. This spectrum is also interesting because it shows clear evidence of Coriolis coupling in the asymmetric structure of the band. ===Linear molecules=== right|thumb|300px|Energy levels and line positions calculated in the rigid rotor approximation The rigid rotor is a good starting point from which to construct a model of a rotating molecule. However, since only integer values of J are allowed, the maximum line intensity is observed for a neighboring integer J. :J = \sqrt{\frac{kT}{2hcB}} - \frac{1}{2} The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it. ====Centrifugal distortion==== When a molecule rotates, the centrifugal force pulls the atoms apart. In this approximation, the vibration-rotation wavenumbers of transitions are :\tilde u = \tilde u_\text{vib} + BJ(J + 1) - B'J'(J' + 1), where B and B' are rotational constants for the upper and lower vibrational state respectively, while J and J' are the rotational quantum numbers of the upper and lower levels. ",311875200,0.3359,12.0,635.7,0.9522,D
+","For example, the rotational energy levels for linear molecules (in the rigid-rotor approximation) are :E_\text{rot} = hc BJ(J + 1). The particular pattern of energy levels (and, hence, of transitions in the rotational spectrum) for a molecule is determined by its symmetry. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number, J, which defines the magnitude of the rotational angular momentum. For a linear molecule, analysis of the rotational spectrum provides values for the rotational constantThis article uses the molecular spectroscopist's convention of expressing the rotational constant B in cm−1. Fitting the spectra to the theoretical expressions gives numerical values of the angular moments of inertia from which very precise values of molecular bond lengths and angles can be derived in favorable cases. Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. Analytical expressions can be derived for the fourth category, asymmetric top, for rotational levels up to J=3, but higher energy levels need to be determined using numerical methods. Thus, by completing a Deslandres table it is easy to assign the correct vibrational quantum numbers v and v' for the transition, allowing important molecular properties to be calculated, such as the dissociation energy. == References == * Category:Spectroscopy The rotational energies are derived theoretically by considering the molecules to be rigid rotors and then applying extra terms to account for centrifugal distortion, fine structure, hyperfine structure and Coriolis coupling. Therefore, there should be three rotational diffusion constants - the eigenvalues of the rotational diffusion tensor - resulting in five rotational time constants. In the absence of an external electrical field, the rotational energy of a symmetric top is a function of only J and K and, in the rigid rotor approximation, the energy of each rotational state is given by : F\left( J,K \right) = B J \left( J+1 \right) + \left( A - B \right) K^2 \qquad J = 0, 1, 2, \ldots \quad \mbox{and}\quad K = +J, \ldots, 0, \ldots, -J where B = {h\over{8\pi^2cI_B}} and A = {h\over{8\pi^2cI_A}} for a prolate symmetric top molecule or A = {h\over{8\pi^2cI_C}} for an oblate molecule. When this is not possible, as with most asymmetric tops, all that can be done is to fit the spectra to three moments of inertia calculated from an assumed molecular structure. In a linear molecule the moment of inertia about an axis perpendicular to the molecular axis is unique, that is, I_B = I_C, I_A=0 , so : B = {h \over{8\pi^2cI_B}}= {h \over{8\pi^2cI_C}} For a diatomic molecule : I=\frac{m_1m_2}{m_1 +m_2}d^2 where m1 and m2 are the masses of the atoms and d is the distance between them. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the diatomic molecule. The third quantum number, K is associated with rotation about the principal rotation axis of the molecule. For rotational spectroscopy, molecules are classified according to symmetry into a spherical top, linear and symmetric top; analytical expressions can be derived for the rotational energy terms of these molecules. The second factor is the degeneracy of the rotational state, which is equal to . Under the rigid rotor model, the rotational energy levels, F(J), of the molecule can be expressed as, : F\left( J \right) = B J \left( J+1 \right) \qquad J = 0,1,2,... where B is the rotational constant of the molecule and is related to the moment of inertia of the molecule. For any molecule, there are three moments of inertia: I_A, I_B and I_C about three mutually orthogonal axes A, B, and C with the origin at the center of mass of the system. This spectrum is also interesting because it shows clear evidence of Coriolis coupling in the asymmetric structure of the band. ===Linear molecules=== right|thumb|300px|Energy levels and line positions calculated in the rigid rotor approximation The rigid rotor is a good starting point from which to construct a model of a rotating molecule. However, since only integer values of J are allowed, the maximum line intensity is observed for a neighboring integer J. :J = \sqrt{\frac{kT}{2hcB}} - \frac{1}{2} The diagram at the right shows an intensity pattern roughly corresponding to the spectrum above it. ====Centrifugal distortion==== When a molecule rotates, the centrifugal force pulls the atoms apart. In this approximation, the vibration-rotation wavenumbers of transitions are :\tilde u = \tilde u_\text{vib} + BJ(J + 1) - B'J'(J' + 1), where B and B' are rotational constants for the upper and lower vibrational state respectively, while J and J' are the rotational quantum numbers of the upper and lower levels. ",311875200,0.3359,"""12.0""",635.7,0.9522,D
 "Using the Planck distribution
 Compare the energy output of a black-body radiator (such as an incandescent lamp) at two different wavelengths by calculating the ratio of the energy output at $450 \mathrm{~nm}$ (blue light) to that at $700 \mathrm{~nm}$ (red light) at $298 \mathrm{~K}$.
-","Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law. thumb|upright=1.45|Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K. Notice that for a given temperature, different parameterizations imply different maximal wavelengths. The formula is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and is the Stefan–Boltzmann constant. ==Equations== ===Planck's law of black-body radiation=== Planck's law states that :B_ u(T) = \frac{2h u^3}{c^2}\frac{1}{e^{h u/kT} - 1}, where :B_{ u}(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency u radiation per unit frequency at thermal equilibrium at temperature T. Units: power / [area × solid angle × frequency]. :h is the Planck constant; :c is the speed of light in vacuum; :k is the Boltzmann constant; : u is the frequency of the electromagnetic radiation; :T is the absolute temperature of the body. Meanwhile, the average energy of a photon from a blackbody isE = \left[\frac{\pi^4}{30\ \zeta(3)}\right] k_\mathrm{B}T \approx 2.701\ k_\mathrm{B}T,where \zeta is the Riemann zeta function. ===Approximations=== In the limit of low frequencies (i.e. long wavelengths), Planck's law becomes the Rayleigh–Jeans law B_ u(T) \approx \frac{2 u^2 }{c^2} k_\mathrm{B} T or B_\lambda(T) \approx \frac{2c}{\lambda^4} k_\mathrm{B} T The radiance increases as the square of the frequency, illustrating the ultraviolet catastrophe. The relative spectral power distribution (SPD) of a Planckian radiator follows Planck's law, and depends on the second radiation constant, c_2=hc/k. Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation. The spectral radiance of Planckian radiation from a black body has the same value for every direction and angle of polarization, and so the black body is said to be a Lambertian radiator. ==Different forms== Planck's law can be encountered in several forms depending on the conventions and preferences of different scientific fields. The emitted energy flux density or irradiance B_ u(T,E), is related to the photon flux density b_ u(T,E) through :B_ u(T,E) = Eb_ u(T,E) ===Wien's displacement law=== Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. The Planckian locus is determined by substituting into the above equations the black body spectral radiant exitance, which is given by Planck's law: :M(\lambda,T) =\frac{c_{1}}{\lambda^5}\frac{1}{\exp\left(\frac{c_2}{{\lambda}T}\right)-1} where: :c1 = 2hc2 is the first radiation constant :c2 = hc/k is the second radiation constant and: :M is the black body spectral radiant exitance (power per unit area per unit wavelength: watt per square meter per meter (W/m3)) :T is the temperature of the black body :h is Planck's constant :c is the speed of light :k is Boltzmann's constant This will give the Planckian locus in CIE XYZ color space. Although the spectra of such lights are not accurately described by the black-body radiation curve, a color temperature (the correlated color temperature) is quoted for which black-body radiation would most closely match the subjective color of that source. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is: :u_{\lambda}(\lambda,T) = {2 h c^2\over \lambda^5}{1\over e^{h c/\lambda kT}-1}. They recommend that the Planck spectrum be plotted as a “spectral energy density per fractional bandwidth distribution,” using a logarithmic scale for the wavelength or frequency. ==See also== * Wien approximation * Emissivity * Sakuma–Hattori equation * Stefan–Boltzmann law * Thermometer * Ultraviolet catastrophe ==References== ==Further reading== * * ==External links== * Eric Weisstein's World of Physics Category:Statistical mechanics Category:Foundational quantum physics Category:Light Category:1893 in science Category:1893 in Germany Then for a perfectly black body, the wavelength- specific ratio of emissive power to absorption ratio is again just , with the dimensions of power. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the black-body curve. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also Wien's law). thumb|303px|As the temperature increases, the peak of the emitted black-body radiation curve moves to higher intensities and shorter wavelengths. From the Planck constant h and the Boltzmann constant k, Wien's constant b can be obtained. ==Peak differs according to parameterization== Constants for different parameterizations of Wien's law Parameterized by x_\mathrm{peak} b (μm⋅K) Wavelength, \lambda 2898 \log\lambda or \log u 3670 Frequency, u 5099 Other characterizations of spectrum Parameterized by x b (μm⋅K) Mean photon energy 5327 10% percentile 2195 25% percentile 2898 50% percentile 4107 70% percentile 5590 90% percentile 9376 The results in the tables above summarize results from other sections of this article. Additionally, for a given temperature the radiance consisting of all photons between two wavelengths must be the same regardless of which distribution you use. In the limit of high frequencies (i.e. small wavelengths) Planck's law tends to the Wien approximation: B_ u(T) \approx \frac{2 h u^3}{c^2} e^{-\frac{h u}{k_\mathrm{B}T}} or B_\lambda(T) \approx \frac{2 h c^2}{\lambda^5} e^{-\frac{hc}{\lambda k_\mathrm{B} T}}. ===Percentiles=== Percentile (μm·K) 0.01% 910 0.0632 0.1% 1110 0.0771 1% 1448 0.1006 10% 2195 0.1526 20% 2676 0.1860 25.0% 2898 0.2014 30% 3119 0.2168 40% 3582 0.2490 41.8% 3670 0.2551 50% 4107 0.2855 60% 4745 0.3298 64.6% 5099 0.3544 70% 5590 0.3885 80% 6864 0.4771 90% 9376 0.6517 99% 22884 1.5905 99.9% 51613 3.5873 99.99% 113374 7.8799 Wien's displacement law in its stronger form states that the shape of Planck's law is independent of temperature. According to Kirchhoff's law of thermal radiation, this entails that, for every frequency , at thermodynamic equilibrium at temperature , one has , so that the thermal radiation from a black body is always equal to the full amount specified by Planck's law. In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. thumb|right|upright=1.15|Planck's law accurately describes black-body radiation. UV-B lamps are lamps that emit a spectrum of ultraviolet light with wavelengths ranging from 290–320 nanometers. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity is again just , with the dimensions of power. ",358800, 7.42,0.14,2.10,0.11,D
-Lead has $T_{\mathrm{c}}=7.19 \mathrm{~K}$ and $\mathcal{H}_{\mathrm{c}}(0)=63.9 \mathrm{kA} \mathrm{m}^{-1}$. At what temperature does lead become superconducting in a magnetic field of $20 \mathrm{kA} \mathrm{m}^{-1}$ ?,"At that temperature even the weakest external magnetic field will destroy the superconducting state, so the strength of the critical field is zero. In 2007, the same group published results suggesting a superconducting transition temperature of 260 K. As of 2015, the highest critical temperature found for a conventional superconductor is 203 K for H2S, although high pressures of approximately 90 gigapascals were required. In 2020, a room-temperature superconductor (critical temperature 288 K) made from hydrogen, carbon and sulfur under pressures of around 270 gigapascals was described in a paper in Nature. It has been experimentally demonstrated that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material. Conventional superconductors usually have critical temperatures ranging from around 20 K to less than 1 K. Solid mercury, for example, has a critical temperature of 4.2 K. This material has critical temperature of 10 kelvins and can superconduct at up to about 15 teslas. Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external magnetic field is applied which is greater than the critical magnetic field. Cambridge University Press, Cambridge From about 1993, the highest-temperature superconductor known was a ceramic material consisting of mercury, barium, calcium, copper and oxygen (HgBa2Ca2Cu3O8+δ) with Tc = 133–138 K. Changes in either temperature or magnetic flux density can cause the phase transition between normal and superconducting states.High Temperature Superconductivity, Jeffrey W. Lynn Editor, Springer-Verlag (1990) The highest temperature under which the superconducting state is seen is known as the critical temperature. For a given temperature, the critical field refers to the maximum magnetic field strength below which a material remains superconducting. A room-temperature superconductor is a material that is capable of exhibiting superconductivity at operating temperatures above , that is, temperatures that can be reached and easily maintained in an everyday environment. , the material with the highest claimed superconducting temperature is an extremely pressurized carbonaceous sulfur hydride with a critical transition temperature of +15 °C at 267 GPa. One exception to this rule is the iron pnictide group of superconductors which display behaviour and properties typical of high-temperature superconductors, yet some of the group have critical temperatures below 30 K. === By material === thumb|""Top: Periodic table of superconducting elemental solids and their experimental critical temperature (T). Low temperature superconductors refer to materials with a critical temperature below 30 K, and are cooled mainly by liquid helium (Tc > 4.2 K). High-temperature superconductivity was discovered in the 1980s. In 2019, the material with the highest accepted superconducting temperature was highly pressurized lanthanum decahydride (), whose transition temperature is approximately . In 1913, lead was found to superconduct at 7 K, and in 1941 niobium nitride was found to superconduct at 16 K. Later, other substances with superconductivity at temperatures up to 30 K were found. For a type-I superconductor the discontinuity in heat capacity seen at the superconducting transition is generally related to the slope of the critical field (H_\text{c}) at the critical temperature (T_\text{c}):Superconductivity of Metals and Alloys, P. G. de Gennes, Addison-Wesley (1989) :C_\text{super} - C_\text{normal} = {T \over 4 \pi} \left(\frac{dH_\text{c}}{dT}\right)^2_{T=T_\text{c}} There is also a direct relation between the critical field and the critical current – the maximum electric current density that a given superconducting material can carry, before switching into the normal state. The upper critical field (at 0 K) can also be estimated from the coherence length () using the Ginzburg–Landau expression: .Introduction to Solid State Physics, Charles Kittel, John Wiley and Sons, Inc. ==Lower critical field== The lower critical field is the magnetic flux density at which the magnetic flux starts to penetrate a type-II superconductor. ==References== Category:Superconductivity Several hundred metals, compounds, alloys and ceramics possess the property of superconductivity at low temperatures. The results were strongly supported by Monte Carlo computer simulations. === Meissner effect === When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. ",41,6.0,91.7,2.6,61,B
-"When an electric discharge is passed through gaseous hydrogen, the $\mathrm{H}_2$ molecules are dissociated and energetically excited $\mathrm{H}$ atoms are produced. If the electron in an excited $\mathrm{H}$ atom makes a transition from $n=2$ to $n=1$, calculate the wavenumber of the corresponding line in the emission spectrum.","Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. It is emitted when the atomic electron transitions from an n = 2 orbital to the ground state (n = 1), where n is the principal quantum number. In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). These observed spectral lines are due to the electron making transitions between two energy levels in an atom. Here is an illustration of the first series of hydrogen emission lines: Historically, explaining the nature of the hydrogen spectrum was a considerable problem in physics. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. Also in . , vacuum (nm) 2 121.57 3 102.57 4 97.254 5 94.974 6 93.780 ∞ 91.175 Source: ===Balmer series ( = 2)=== 757px|thumb|center|The four visible hydrogen emission spectrum lines in the Balmer series. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For the Lyman series the naming convention is: *n = 2 to n = 1 is called Lyman- alpha, *n = 3 to n = 1 is called Lyman-beta, etc. H-alpha has a wavelength of 656.281 nm, is visible in the red part of the electromagnetic spectrum, and is the easiest way for astronomers to trace the ionized hydrogen content of gas clouds. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. Spectral emission occurs when an electron transitions, or jumps, from a higher energy state to a lower energy state. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. For example, the line is called ""Lyman-alpha"" (Ly-α), while the line is called ""Paschen-delta"" (Pa-δ). thumb|Energy level diagram of electrons in hydrogen atom There are emission lines from hydrogen that fall outside of these series, such as the 21 cm line. This two-step photodissociation process, known as the Solomon process, is one of the main mechanisms by which molecular hydrogen is destroyed in the interstellar medium. thumb|Electronic and vibrational levels of the hydrogen molecule In reference to the figure shown, Lyman-Werner photons are emitted as described below: *A hydrogen molecule can absorb a far- ultraviolet photon (11.2 eV < energy of the photon < 13.6 eV) and make a transition from the ground electronic state X to excited state B (Lyman) or C (Werner). In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. ",4.4,62.2,82258.0,0.05882352941,226,C
-Calculate the shielding constant for the proton in a free $\mathrm{H}$ atom.,"Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). The shielding constant for each group is formed as the sum of the following contributions: #An amount of 0.35 from each other electron within the same group except for the [1s] group, where the other electron contributes only 0.30. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. In January 2013, an updated value for the charge radius of a proton——was published. The constant is expressed for either hydrogen as R_\text{H}, or at the limit of infinite nuclear mass as R_\infty. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants according to his model of the atom. The radius of the proton is linked to the form factor and momentum-transfer cross section. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. The internationally accepted value of a proton's charge radius is . By measuring the energy required to excite hydrogen atoms from the 2S to the 2P state, the Rydberg constant could be calculated, and from this the proton radius inferred. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. Revised values of screening constants based on computations of atomic structure by the Hartree–Fock method were obtained by Enrico Clementi et al. in the 1960s. ==Rules== Firstly, the electrons are arranged into a sequence of groups in order of increasing principal quantum number n, and for equal n in order of increasing azimuthal quantum number l, except that s- and p- orbitals are kept together. :[1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s, 5p] [5d] etc. The result is again ~5% smaller than the previously-accepted proton radius. The nucleus of the most common isotope of the hydrogen atom (with the chemical symbol ""H"") is a lone proton. this opinion is not yet universally held. ==Problem== Prior to 2010, the proton charge radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering. ===Spectroscopy method=== The spectroscopy method uses the energy levels of electrons orbiting the nucleus. His personal assumption is that past measurements have misgauged the Rydberg constant and that the current official proton size is inaccurate. ===Quantum chromodynamic calculation=== In a paper by Belushkin et al. (2007), including different constraints and perturbative quantum chromodynamics, a smaller proton radius than the then-accepted 0.877 femtometres was predicted. ===Proton radius extrapolation=== Papers from 2016 suggested that the problem was with the extrapolations that had typically been used to extract the proton radius from the electron scattering data though these explanation would require that there was also a problem with the atomic Lamb shift measurements. ===Data analysis method=== In one of the attempts to resolve the puzzle without new physics, Alarcón et al. (2018) of Jefferson Lab have proposed that a different technique to fit the experimental scattering data, in a theoretically as well as analytically justified manner, produces a proton charge radius from the existing electron scattering data that is consistent with the muonic hydrogen measurement. The 2014 CODATA adjustment slightly reduced the recommended value for the proton radius (computed using electron measurements only) to , but this leaves the discrepancy at σ. The screening constant, and subsequently the shielded (or effective) nuclear charge for each electron is deduced as: : \begin{matrix} 4s &: s = 0.35 \times 1& \+ &0.85 \times 14 &+& 1.00 \times 10 &=& 22.25 &\Rightarrow& Z_{\mathrm{eff}}(4s) = 26.00 - 22.25 = 3.75\\\ 3d &: s = 0.35 \times 5& & &+& 1.00 \times 18 &=& 19.75 &\Rightarrow& Z_{\mathrm{eff}}(3d)= 26.00 - 19.75 =6.25\\\ 3s,3p &: s = 0.35 \times 7& \+ &0.85 \times 8 &+& 1.00 \times 2 &=& 11.25 &\Rightarrow& Z_{\mathrm{eff}}(3s,3p)= 26.00 - 11.25 =14.75\\\ 2s,2p &: s = 0.35 \times 7& \+ &0.85 \times 2 & & &=& 4.15 &\Rightarrow& Z_{\mathrm{eff}}(2s,2p)= 26.00 - 4.15 =21.85\\\ 1s &: s = 0.30 \times 1& & & & &=& 0.30 &\Rightarrow& Z_{\mathrm{eff}}(1s)= 26.00 - 0.30 =25.70 \end{matrix} Note that the effective nuclear charge is calculated by subtracting the screening constant from the atomic number, 26. ==Motivation== The rules were developed by John C. Slater in an attempt to construct simple analytic expressions for the atomic orbital of any electron in an atom. The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron: : R_\text{H} = R_\infty \frac{ m_\text{e} m_\text{p} }{ m_\text{e}+m_\text{p} } \approx 1.09678 \times 10^7 \text{ m}^{-1} , where * m_\text{e} is the mass of the electron, * m_\text{p} is the mass of the nucleus (a proton). === Rydberg unit of energy === The Rydberg unit of energy is equivalent to joules and electronvolts in the following manner: :1 \ \text{Ry} \equiv h c R_\infty = \frac{m_\text{e} e^4}{8 \varepsilon_{0}^{2} h^2} = \frac{e^2}{8 \pi \varepsilon_{0} a_0} = 2.179\;872\;361\;1035(42) \times 10^{-18}\ \text{J} \ = 13.605\;693\;122\;994(26)\ \text{eV}. === Rydberg frequency === :c R_\infty = 3.289\;841\;960\;2508(64) \times 10^{15}\ \text{Hz} . === Rydberg wavelength === :\frac 1 {R_\infty} = 9.112\;670\;505\;824(17) \times 10^{-8}\ \text{m}. The result is a protonated atom, which is a chemical compound of hydrogen. ",1.91,1.1,5275.0,1.775,1.7,D
-"An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let $A_1$ be those people with an auto policy only, $A_2$ those people with a homeowner policy only, and $A_3$ those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that $P\left(A_1\right)=0.3, P\left(A_2\right)=0.2$, and $P\left(A_3\right)=0.2$. Further, let $B$ be the event that the person will renew at least one of these policies. Say from past experience that we assign the conditional probabilities $P\left(B \mid A_1\right)=0.6, P\left(B \mid A_2\right)=0.7$, and $P\left(B \mid A_3\right)=0.8$. Given that the person selected at random has an auto or homeowner policy, what is the conditional probability that the person will renew at least one of those policies?","This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . * This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. In this event, the event B can be analyzed by a conditional probability with respect to A. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. The relationship between P(A|B) and P(B|A) is given by Bayes' theorem: :\begin{align} P(B\mid A) &= \frac{P(A\mid B) P(B)}{P(A)}\\\ \Leftrightarrow \frac{P(B\mid A)}{P(A\mid B)} &= \frac{P(B)}{P(A)} \end{align} That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B). === Assuming marginal and conditional probabilities are of similar size === In general, it cannot be assumed that P(A) ≈ P(A|B). Unconditionally (that is, without reference to C), A and B are independent of each other because \operatorname{P}(A)—the sum of the probabilities associated with a 1 in row A—is \tfrac{1}{2}, while \operatorname{P}(A\mid B) = \operatorname{P}(A \text{ and } B) / \operatorname{P}(B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} = \operatorname{P}(A). That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A after having accounted for evidence E or after having updated P(A). Thus, the conditional probability P(D1 = 2 | D1+D2 ≤ 5) = = 0.3: : Table 3 \+ + D2 D2 D2 D2 D2 D2 \+ + 1 2 3 4 5 6 D1 1 2 3 4 5 6 7 D1 2 3 4 5 6 7 8 D1 3 4 5 6 7 8 9 D1 4 5 6 7 8 9 10 D1 5 6 7 8 9 10 11 D1 6 7 8 9 10 11 12 Here, in the earlier notation for the definition of conditional probability, the conditioning event B is that D1 + D2 ≤ 5, and the event A is D1 = 2\. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. Since in the presence of C the probability of A is affected by the presence or absence of B, A and B are mutually dependent conditional on C. == See also == * * * == References == Category:Independence (probability theory) For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that = 5% and = 75 %. The existence of regular conditional probabilities: necessary and sufficient conditions. From the law of total probability, its expected value is equal to the unconditional probability of . === Partial conditional probability === The partial conditional probability P(A\mid B_1 \equiv b_1, \ldots, B_m \equiv b_m) is about the probability of event A given that each of the condition events B_i has occurred to a degree b_i (degree of belief, degree of experience) that might be different from 100%. The reverse, insufficient adjustment from the prior probability is conservatism. == Formal derivation == Formally, P(A | B) is defined as the probability of A according to a new probability function on the sample space, such that outcomes not in B have probability 0 and that it is consistent with all original probability measures.George Casella and Roger L. Berger (1990), Statistical Inference, Duxbury Press, (p. 18 et seq.)Grinstead and Snell's Introduction to Probability, p. 134 Let Ω be a discrete sample space with elementary events {ω}, and let P be the probability measure with respect to the σ-algebra of Ω. In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. Applying the law of total probability, we have: : \begin{align} P(A) & = P(A\mid B_X) \cdot P(B_X) + P(A\mid B_Y) \cdot P(B_Y) \\\\[4pt] & = {99 \over 100} \cdot {6 \over 10} + {95 \over 100} \cdot {4 \over 10} = {{594 + 380} \over 1000} = {974 \over 1000} \end{align} where * P(B_X)={6 \over 10} is the probability that the purchased bulb was manufactured by factory X; * P(B_Y)={4 \over 10} is the probability that the purchased bulb was manufactured by factory Y; * P(A\mid B_X)={99 \over 100} is the probability that a bulb manufactured by X will work for over 5000 hours; * P(A\mid B_Y)={95 \over 100} is the probability that a bulb manufactured by Y will work for over 5000 hours. In general, it cannot be assumed that P(A|B) ≈ P(B|A). ",0.686,7,27.0,2,-21.2,A
-What is the number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards?,"Each player is dealt thirteen cards from a standard 52-card deck. The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Note that all cards are dealt face up Fourteen Out (also known as Fourteen Off, Fourteen Puzzle, Take Fourteen, or just Fourteen) is a Patience card game played with a deck of 52 playing cards. There are 7,462 distinct poker hands. ===7-card poker hands=== In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall,https://www.casinodaniabeach.com/most-popular-types-of-poker/ a player uses the best five-card poker hand out of seven cards. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. thumb|left|180px| thumb|left|180px| In duplicate bridge, a board is an item of equipment that holds one deal, or one deck of 52 cards distributed in four hands of 13 cards each. Contract bridge, or simply bridge, is a trick-taking card game using a standard 52-card deck. *The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The diagram is typical of that used to illustrate a deal of 52 cards in four hands in the game of contract bridge.Bridge Writing Style Guide by Richard Pavlicek Each hand is designated by a point on the compass and so North–South are partners against East–West. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. # Player shuffle – before the start of play, each table receives a number of boards each containing 13 cards in each of its four pockets. The Total line also needs adjusting. ===7-card lowball poker hands=== In some variants of poker a player uses the best five-card low hand selected from seven cards. The frequencies are calculated in a manner similar to that shown for 5-card hands,https://www.pokerstrategy.com/strategy/various-poker/texas-holdem- probabilities/ except additional complications arise due to the extra two cards in the 7-card poker hand. The name refers to the goal of each turn to make pairs that add up to 14.""Take Fourteen"" (p.80) in The Little Book of Solitaire, Running Press, 2002. ==Rules== The cards are dealt face up into twelve columns, from left to right. The number of distinct poker hands is even smaller. The director is summoned if any player does not have exactly thirteen cards. The Total line also needs adjusting. ==See also== * Binomial coefficient * Combination * Combinatorial game theory * Effective hand strength algorithm * Event (probability theory) * Game complexity * Gaming mathematics * Odds * Permutation * Probability * Sample space * Set theory ==References== ==External links== * Brian Alspach's mathematics and poker page * MathWorld: Poker * Poker probabilities including conditional calculations * Numerous poker probability tables * 5, 6, and 7 card poker probabilities * Hold'em poker probabilities ",10.065778,635013559600,2.9,1.61,48,B
-"What is the number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons?","This is a list of fellows of the Royal Society elected in 1909.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1907.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1910.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1903.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1908.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1904.""Fellows of the Royal Society"", Royal Society. ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Edward Charles Cyril Baly (1871–1948) *Sir Thomas Barlow (1845–1945) *Ernest William Barnes (1874–1953) *Francis Arthur Bather (1863–1934) *Sir Robert Abbott Hadfield (1858–1940) *Sir Alfred Daniel Hall (1864–1942) *Sir Arthur Harden (1865–1940) *Alfred John Jukes-Browne (1851–1914) *Sir John Graham Kerr (1869–1957) *William James Lewis (1847–1926) *John Alexander McClelland (1870–1920) *William McFadden Orr (1866–1934) *Alfred Barton Rendle (1865–1938) *James Lorrain Smith (1862–1931) *James Thomas Wilson (1861–1945) ==Foreign members== *George Ellery Hale (1868–1938) *Hugo Kronecker (1839–1914) *Charles Emile Picard (1856–1941) *Santiago Ramon y Cajal (1852–1934) ==References== 1909 Category:1909 in the United Kingdom Category:1909 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Charles Jasper Joly (1864–1906) *Hugh Marshall (1868–1913) *Donald Alexander Smith Baron Strathcona and Mount Royal (1820–1914) *Thomas Gregor Brodie (1866–1916) *Alexander Muirhead (1848–1920) *Sir James Johnston Dobbie (1852–1924) *Sir Arthur Everett Shipley (1861–1927) *Harold William Taylor Wager (1862–1929) *Alfred Cardew Dixon (1865–1936) *George Henry Falkiner Nuttall (1862–1937) *Edward Meyrick (1854–1938) *Sir Sidney Gerald Burrard (1860–1943) *William Whitehead Watts (1860–1947) *Sir Thomas Henry Holland (1868–1947) *Sir Gilbert Thomas Walker (1868–1958) *Morris William Travers (1872–1961) ==References== 1904 Category:1904 in the United Kingdom Category:1904 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Frank Dawson Adams (1859–1942) *Sir Hugh Kerr Anderson (1865–1928) *Sir William Blaxland Benham (1860–1950) *Sir William Henry Bragg (1862–1942) *Archibald Campbell Campbell, 1st Baron Blythswood (1835–1908) *Frederick Daniel Chattaway (1860–1944) *Arthur William Crossley (1869–1927) *Arthur Robertson Cushny (1866–1926) *William Duddell (1872–1917) *Frederick William Gamble (1869–1926) *Sir Joseph Ernest Petavel (1873–1936) *Henry Cabourn Pocklington (1870–1952) *Henry Nicholas Ridley (1855–1956) *Sir Grafton Elliot Smith (1871–1937) *William Henry Young (1863–1942) ==Foreign members== *Ivan Petrovich Pavlov (1849–1936) *Edward Charles Pickering (1846–1919) *Magnus Gustaf Retzius (1842–1919) *Augusto Righi (1850–1920) ==References== 1907 Category:1907 in the United Kingdom Category:1907 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Thomas William Bridge (1848–1909) *John Edward Stead (1851–1923) *Johnson Symington (1851–1924) *Sir William Maddock Bayliss (1860–1924) *Sir Horace Darwin (1851–1928) *Sir Aubrey Strahan (1852–1928) *William Philip Hiern (1839–1929) *Henry Reginald Arnulph Mallock (1851–1933) *Sir David Orme Masson (1858–1937) *Arthur George Perkin (1861–1937) *Ernest Rutherford Baron Rutherford of Nelson (1871–1937) *Ralph Allen Sampson (1866–1939) *Alfred North Whitehead (1861–1947) *Sydney Arthur Monckton Copeman (1862–1947) *Sir John Sealy Edward Townsend (1868–1957) ==References== 1903 Category:1903 in the United Kingdom Category:1903 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== #Antoine Henri Becquerel (1852–1908) #David James Hamilton (1849–1909) #Silas Weir Mitchell (1829–1914) #Friedrich Robert Helmert (1843–1917) #William Gowland (1842–1922) #William Halse Rivers Rivers (1864–1922) #Charles Immanuel Forsyth Major (1843–1923) #Arthur Dendy (1865–1925) #H. H. Asquith (1852–1928) #Shibasaburo Kitasato (1852–1931) #Sir Dugald Clerk #Otto Stapf #William Barlow #Edmund Neville Nevill #Herbrand Russell, 11th Duke of Bedford #Sir Jocelyn Field Thorpe #Randal Thomas Mowbray Rawdon Berkeley #John Stanley Gardiner (1872–1946) #Henry Horatio Dixon #John Hilton Grace #Bertrand Russell ==References== 1908 Category:1908 in the United Kingdom Category:1908 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ",29.36,62.8318530718,1.6,5040,4.86,D
-"At a county fair carnival game there are 25 balloons on a board, of which 10 balloons 1.3-5 are yellow, 8 are red, and 7 are green. A player throws darts at the balloons to win a prize and randomly hits one of them. Given that the first balloon hit is yellow, what is the probability that the next balloon hit is also yellow?","The Topic International Darts League was a darts tournament held at the Triavium in Nijmegen, Netherlands. The festival began with approximately 15 balloons and to date has grown to about 30 balloons. The 2009 PartyPoker.com Grand Slam of Darts was the third staging of the darts tournament, the Grand Slam of Darts organised by the Professional Darts Corporation. thumb|various hot air balloons during the festival The Warren County Farmers' Fair Balloon Festival was started in 2001 and takes place during the week of the County Fair in Warren County, New Jersey. The 2003 Las Vegas Desert Classic was the second major Professional Darts Corporation Las Vegas Desert Classic darts tournament. The yellow-winged darter (Sympetrum flaveolum) is a dragonfly found in Europe and mid and northern China. The tournament was sponsored by PartyPoker.net, which has also sponsored other darts championships: the US Open, the Las Vegas Desert Classic and the German Darts Championship. ==References== ==External links== *Collated results of the 2008 European Championship Category:European Championship (darts) European Championship Darts Despite the presence of the PDC players in 2006 and 2007, the tournament was still a WDF/BDO ranking event, with all available points going only to the WDF/BDO players competing. ==International Darts League finals== Year Champion Each player's average score is based on the average for each 3-dart visit to the board (ie total points scored divided by darts thrown and multiplied by 3) Score Runner-up Prize money Prize money Prize money Sponsor Venue Year Champion Each player's average score is based on the average for each 3-dart visit to the board (ie total points scored divided by darts thrown and multiplied by 3) Score Runner-up Total Champion Runner-up Sponsor Venue 2003 Raymond van Barneveld (97.77) 8–5 Mervyn King (97.50) €134,000 €30,000 €15,000 Tempus Triavium, Nijmegen 2004 Raymond van Barneveld (101.64) 13–5 Tony David (95.04) €134,000 €30,000 €15,000 Tempus Triavium, Nijmegen 2005 Mervyn King (91.89) Tony O'Shea (91.74) €134,000 €30,000 €15,000 Tempus Triavium, Nijmegen 2006 Raymond van Barneveld (99.54) 13–5 Colin Lloyd (95.25) €134,000 €30,000 €15,000 Topic Triavium, Nijmegen 2007 Gary Anderson (95.85) 13–9 Mark Webster (94.54) €158,000 €30,000 €15,000 Topic Triavium, Nijmegen ==Sponsors== * 2003–2005 Tempus * 2006–2007 Topic ==References== ==External links== * International Darts League * IDL 2006 – A Review Category:2003 establishments in the Netherlands Category:2007 disestablishments in the Netherlands Category:Professional Darts Corporation tournaments Category:British Darts Organisation tournaments Category:Darts in the Netherlands Category:International sports competitions hosted by the Netherlands The 2008 PartyPoker.net European Championship was the inaugural edition of the Professional Darts Corporation tournament, which thereafter was promoted as the annual European Championship, matching top European players qualifying to play against the highest ranked players from the PDC Order of Merit. The event features some balloon races, including the typical hare and hound races, in addition to the Bicycle Balloon Race. The winner and the runner-up of the 2009 Championship League Darts would be invited, whilst it was announced that only the winner of the 2008 World Masters would be invited (though runner-up Scott Waites was invited anyway due to the withdrawal of Martin Adams). The case ended in failure on 21 February 2008, and the International Darts League was indefinitely postponed. An almost unmistakable darter, red-bodied in the male, with both sexes having large amounts of saffron-yellow colouration to the basal area of each wing, which is particularly noticeable on the hind-wings. The yellow-winged darter tends to make quite short flights when settled at a site, and frequently perches quite low down on vegetation. The future of the World Darts Trophy was also thrown into doubt as a result of the decision,IDL & WDT go to court Superstars of darts forum and both events were confirmed defunct by the failure of an appeal on April 29, 2008.IDL & WDT end Google translation from official web site ==Format== The format has changed slightly over the years – the 2006 competition had 8 round-robin groups of 4 players. Then the top 8 non-qualified players from the 2008 Players Championship Order of Merit after the October German Darts Trophy in Dinslaken, Germany joined them to make a field of 24. Played from 30 October–2 November 2008 at the Südbahnhof in Frankfurt, Germany, the inaugural tournament featured a field of 32 players and £200,000 in prize money, with a £50,000 winner's purse going to Phil Taylor.PDC website report - European Championship Details Confirmed from the Professional Darts Corporation obtained 12-08-2008 ==Format== First round — best of nine legs (by two legs) Second round — best of seventeen legs (ditto) Quarter-finals — best of seventeen legs (ditto) Semi-finals — best of twenty-one legs (ditto) Final — best of twenty-one legs (ditto) Each game had to be won by two clear legs, except that a game went to a sudden death leg if a further six legs did not separate the players; for example, a first round match played out to 7-7 is then decided with one sudden death leg. ==Prize money== A total of £200,000 was on offer to the players, divided based on the following performances: Position (no. of players) Position (no. of players) Prize money (Total: £200,000) Winner (1) £50,000 Runner-Up (1) £25,000 Semi-finalists (2) £12,500 Quarter-finalists (4) £8,500 Last 16 (second round) (8) £4,000 Last 32 (first round) (16) £2,000 Highest checkout (1) £2,000 ==Qualification== The top 16 players from the PDC Order of Merit after the 2008 Sky Poker World Grand Prix automatically qualified for the event. This was the second PDC darts tournament that ITV4 has broadcast, after the inaugural Grand Slam of Darts - after its rating success ITV chose to broadcast this event as well as the 2008 Grand Slam of Darts. It is the only major event that Phil Taylor has competed in at least once, but never won. ==End of event== Towards the end of 2007, the chairman of the PDC, Barry Hearn, announced that its players would not be competing in the 2008 International Darts League and World Darts Trophy events. The shootout occurred exactly one year to the day after a similar situation at the 2008 Grand Slam of Darts where Hamilton beat Alan Tabern. The yellow-winged darter has bred but is not established in the UK. Gary Anderson was the final champion, having claimed the title in 2007, when the tournament also became the first major event to witness two nine dart finishes. ",3,57.2,0.375,0.323,22,C
-What is the number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards?,"The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. *The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The deck is retrieved, and each player is dealt in turn from the deck the same number of cards they discarded so that each player again has five cards. Each player specifies how many of their cards they wish to replace and discards them. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. However, a rule used by many casinos is that a player is not allowed to draw five consecutive cards from the deck. The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. If the deck is depleted during the draw before all players have received their replacements, the last players can receive cards chosen randomly from among those discarded by previous players. This list arranges card games by the number of cards used. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Another common house rule is that the bottom card of the deck is never given as a replacement, to avoid the possibility of someone who might have seen it during the deal using that information. For example, if the last player to draw wants three replacements but there are only two cards remaining in the deck, the dealer gives the player the one top card he can give, then shuffles together the bottom card of the deck, the burn card, and the earlier players' discards (but not the player's own discards), and finally deals two more replacements to the last player. ==Sample deal== 200px|right The sample deal is being played by four players as shown to the right with Alice dealing. 52 pickup or 52-card pickup is a humorous prank which consists only of picking up a scattered deck of playing cards. In this case, if a player wishes to replace all five of their cards, that player is given four of them in turn, the other players are given their draws, and then the dealer returns to that player to give the fifth replacement card; if no other player draws it is necessary to deal a burn card first. Five-card draw (also known as Cantrell draw) is a poker variant that is considered the simplest variant of poker, and is the basis for video poker. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. With five players, the sixes are added to make a 36-card deck. Its ""Total"" represents the 95.4% of the time that a player can select a 5-card low hand without any pair. The other player must then pick them up.. ==Variations== Genuine card games sometimes end in 52 pickup. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. ",2.2,0.24995,4943.0,0.87,311875200,E
-A bowl contains seven blue chips and three red chips. Two chips are to be drawn successively at random and without replacement. We want to compute the probability that the first draw results in a red chip $(A)$ and the second draw results in a blue chip $(B)$. ,"The Sunday Times described triple-cooked chips as Blumenthal's most influential culinary innovation, which had given the chip ""a whole new lease of life"". ==History== Blumenthal said he was ""obsessed with the idea of the perfect chip"",Blumenthal, In Search of Perfection and described how, from 1992 onwards, he worked on a method for making ""chips with a glass-like crust and a soft, fluffy centre"". thumb|Colorized photo of Chips. The Bowl of Baal is a 1975 science fiction novel by Robert Ames Bennet. Eventually, Blumenthal developed the three-stage cooking process known as triple-cooked chips, which he identifies as ""the first recipe I could call my own"". Triple-cooked chips are a type of chips developed by the English chef Heston Blumenthal. 7 Colors (a.k.a. Filler) is a puzzle game, designed by Dmitry Pashkov. The result is what Blumenthal calls ""chips with a glass-like crust and a soft, fluffy centre"". The chips are first simmered, then cooled and drained using a sous-vide technique or by freezing; deep fried at and cooled again; and finally deep-fried again at . On July 10, 1943, Chips and his handler were pinned down on the beach by an Italian machine-gun team. In 2014, the London Fire Brigade attributed an increase in chip pan fires to the increased popularity of ""posh chips"", including triple-cooked chips. ==Preparation== ===Blumenthal's technique=== Previously, the traditional practice for cooking chips was a two-stage process, in which chipped potatoes were fried in oil first at a relatively low temperature to soften them and then at a higher temperature to crisp up the outside. thumb|A selection of Red Ribbon cakes on sale Red Ribbon Bakeshop, Inc. is a bakery chain based in the Philippines, which produces and distributes cakes and pastries. ==History== In 1979, Amalia Hizon Mercado, husband Renato Mercado, and their five children, Consuelo Tiutan, Teresita Moran, Renato Mercado, Ricky Mercado and Romy Mercado established Red Ribbon as a small cake shop along Timog Avenue in Quezon City. The second of the three stages is frying the chips at for approximately 5 minutes, after which they are cooled once more in a freezer or sous-vide machine before the third and final stage: frying at for approximately 7 minutes until crunchy and golden. Blumenthal describes moisture as the ""enemy"" of crisp chips. C.C. Moore eventually gifted Chips to the Wren family. Chips served as a sentry dog for the Roosevelt-Churchill conference in 1943. Bloomsbury. ==Further reading== * * ==External links== * Triple-Cooked Chips. Second, the cracks that develop in the chips provide places for oil to collect and harden during frying, making them crunchy.Blumenthal, Heston Blumenthal at Home Third, thoroughly drying out the chips drives off moisture that would otherwise keep the crust from becoming crisp. Blumenthal began work on the recipe in 1993, and eventually developed the three-stage cooking process. Chips (1940–1946) was a trained sentry dog for United States Army, and reputedly the most decorated war dog from World War II. Chips was a German Shepherd-Collie-Malamute mix owned by Edward J. Wren of Pleasantville, New York. Chips shipped out to the War Dog Training Center, Front Royal, Virginia, in 1942 for training as a sentry dog. ""A single frying at a high temperature leads to a thin crust that can easily be rendered soggy by whatever moisture remains in the chip’s interior."" ",-383,5,0.23333333333,0.66666666666,313,C
-"From an ordinary deck of playing cards, cards are to be drawn successively at random and without replacement. What is the probability that the third spade appears on the sixth draw?","*The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). Three Shuffles and a Draw is a solitaire game using one deck of playing cards. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand at least as good as three of a kind is about 2.87%. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. One would then expect to draw this hand about once in every 649,740 draws, or nearly 0.000154% of the time. The name ""Three Shuffles and a Draw"" comes from the fact that there are 3 shuffles (counting the original starting shuffle plus the 2 redeals, and then a draw, where you can free any one single buried card). Draw poker is any poker variant in which each player is dealt a complete hand before the first betting round, and then develops the hand for later rounds by replacing, or ""drawing"", cards. Then a third card is revealed, followed by a betting round, a fourth card, a betting round, and finally a showdown. As a bridge hand contains thirteen cards, only two hand patterns can be classified as three suiters: 4-4-4-1 and 5-4-4-0. right In the game of contract bridge a three suiter (or three-suited hand) denotes a hand containing at least four cards in three of the four suits. thumb|right|170px|Three of Cups from a deck of Italian cards Three of Cups is the third card on the suit of Cups. The object of the game is to move all of the cards to the Foundations. == Rules == Three Shuffles and a Draw has four foundations build up in suit from Ace to King, e.g. A♣, 2♣, 3♣, 4♣... The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. In the card game contract bridge, Gambling 3NT is a special of an opening of 3NT. Finally, each player draws as in normal draw poker, followed by a fourth betting round and showdown. The first betting round is then played, followed by a draw in which each player replaces cards from their hand with an equal number, so that each player still has only four cards in hand. Before the first betting round, each player examines their hand, removes exactly three cards from it, then places them on the table to their left. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. If any player opens, the game continues as traditional five-card draw poker. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. It is the , and this makes all 6-spot cards wild. ",313,0.064,122.0,19.4,0.123,B
-What is the probability of drawing three kings and two queens when drawing a five-card hand from a deck of 52 playing cards?,"*The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. The queen of spades (Q) is one of 52 playing cards in a standard deck: the queen of the suit of spades (). Probabilities are adjusted in the above table such that ""5-high"" is not listed, ""6-high"" has 781,824 distinct hands, and ""King-high"" has 21,457,920 distinct hands, respectively. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Royal Marriage is a patience or solitaire game using a deck of 52 playing cards. The remaining fifty cards are shuffled and placed on the top of the King to form the stock. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Royal Cotillion is a solitaire card game which uses two decks of 52 playing cards each. Royal Flush is a solitaire card game which is played with a deck of 52 playing cards. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. : Hand Distinct hands Frequency Probability Cumulative Odds against 5-high 1 1,024 0.0394% 0.0394% 2,537.05 : 1 6-high 5 5,120 0.197% 0.236% 506.61 : 1 7-high 15 15,360 0.591% 0.827% 168.20 : 1 8-high 35 35,840 1.38% 2.21% 71.52 : 1 9-high 70 71,680 2.76% 4.96% 35.26 : 1 10-high 126 129,024 4.96% 9.93% 19.14 : 1 Jack-high 210 215,040 8.27% 18.2% 11.09 : 1 Queen-high 330 337,920 13.0% 31.2% 6.69 : 1 King-high 495 506,880 19.5% 50.7% 4.13 : 1 Total 1,287 1,317,888 50.7% 50.7% 0.97 : 1 As can be seen from the table, just over half the time a player gets a hand that has no pairs, threes- or fours-of-a-kind. (50.7%) If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand. For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. Three Shuffles and a Draw is a solitaire game using one deck of playing cards. Probabilities are adjusted in the above table such that ""5-high"" is not listed"", ""6-high"" has one distinct hand, and ""King-high"" having 330 distinct hands, respectively. In this case, the deck is held face-down in one hand, with the King being uppermost face-down card and the Queen being held face-up above it. The game is won when the King and Queen are brought together -- that is, when only one or two cards remain in between them, which can then be discarded. ==Variations== Royal Marriage is possible to play in-hand, rather than on a surface such as a table. ",0.0000092,35,0.323,0.6321205588,14.5115,A
-"In an orchid show, seven orchids are to be placed along one side of the greenhouse. There are four lavender orchids and three white orchids. How many ways are there to lineup these orchids?","The Orchidoideae, or the orchidoid orchids, are a subfamily of the orchid family (Orchidaceae) that contains around 3630 species. Orchidales is an order of flowering plants. Genera Orchidacearum vol. 3: Orchidoideae part 2, Vanilloideae. Genera Orchidacearum 4. Genera Orchidacearum 5. Genera Orchidacearum 1. Genera Orchidacearum 3. This is a list of genera in the orchid family (Orchidaceae), originally according to The Families of Flowering Plants - L. Watson and M. J. Dallwitz. Genera Orchidacearum 2. This is a list of the orchids, sorted in alphabetical order, found in Metropolitan France. == A == * Anacamptis laxiflora * Anacamptis longicornu * Anacamptis morio * Anacamptis palustris == C == * Cephalanthera longifolia == D == * Dactylorhiza incarnata == E == * Epipactis phyllanthes == G == * Goodyera repens == O == * Ophrys aurelia * Ophrys catalaunica * Ophrys saratoi * Ophrys drumana * Orchis mascula == S == * Serapias lingua == References == France Phylogeny and Classification of the Orchid Family. She provided an English text, paintings, and drawings for the amateur reader, a mixture of impression and scientific illustration of the genera. ==Orchids of South Western Australia== Common name Genus No. species in southwest W.A. Remarks Babe-in-a-cradle Epiblema 1 Beard orchids Calochilus 6 Blue orchids Cyanicula 11 Bunny orchids Eriochilus 6 Donkey orchid Diuris ~36 Duck orchids Paracaleana 13 Elbow orchid Spiculaea 1 Enamel orchids Elythranthera 2 Fairy orchid Pheladenia 1 Fire orchids Pyrorchis 2 also Beak orchids Greenhoods Pterostylis ~90 Hammer orchids Drakaea 10 Hare orchid Leporella 1 Helmet orchids Corybas 4 Leafless orchid Praecoxanthus 1 Leek orchids Prasophyllum 25 Mignonette orchids Microtis 14 also Onion orchid Mosquito orchids Cyrtostylis 5 Potato orchids Gastrodia 1 also Bell orchid Pygmy orchid Corunastylis 1 Rabbit orchid Leptoceras 1 Rattle beaks Lyperanthus 1 Slipper orchids Cryptostylis 1 also Tongue orchid South African orchids Disa bracteata 1 introduced Spider orchids Caladenia 125 Sugar orchid Ericksonella 1 Sun orchids Thelymitra 37 Underground orchids Rhizanthella 1 * This table has its source as the Second Edition of Hoffman and Brown in 1992 ==References== thumb|Diuris plate III from West Australian Orchids, 1930 # ==Further reading== * * * * * * * == External links == * The Species Orchid Society of Western Australia (Inc) -- a gallery of orchids from Western Australia * Orchids from Western and South Australia * Terrestrial orchids of the south west western australia * Orchid Conservation Coalition List of orchids Western Australia Historically, the Orchidoideae have been partitioned into up to 6 tribes, including Orchideae, Diseae, Cranichideae, Chloraeeae, Diurideae, and Codonorchideae. Oxford Univ. Press == External links == *All recognized monocotiledons species (including Orchid family) - World Checklist of Selected Plant Families, Kew Botanic Garden - UK *Intergeneric orchid genus names (updated 11 Jan 2005) *List of orchid genera (updated 14 Jul 2004) *List of common names or *List of orchid hybrids - Royal Horticultural Society - UK *Orchid main page - eMonocot website Orchidaceae The first three orchids from Western Australia to be named were Caladenia menziesii (now Leptoceras menziesii), Caladenia flava, and Diuris longifolia. Dictionary of Orchid Names. This list is adapted regularly with the changes published in the Orchid Research Newsletter which is published twice a year by the Royal Botanic Gardens, Kew. This list is reflected on Wikispecies Orchidaceae and the new eMonocot website Orchidaceae Juss. Although mostly the order will consist of the orchids only (usually in one family only, but sometimes divided into more families, as in the Dahlgren system, see below), sometimes other families are added: ==Circumscription in the Takhtajan system== Takhtajan system: * order Orchidales *: family Orchidaceae ==Circumscription in the Cronquist system== Cronquist system (1981): * order Orchidales *: family Geosiridaceae *: family Burmanniaceae *: family Corsiaceae *: family Orchidaceae ==Circumscription in the Dahlgren system== Dahlgren system: * order Orchidales *: family Neuwiediaceae *: family Apostasiaceae *: family Cypripediaceae *: family Orchidaceae ==Circumscription in the Thorne system== Thorne system (1992): * order Orchidales *: family Orchidaceae ==APG system== The order is not recognized in the APG II system, which assigns the orchids to order Asparagales. ==See also== * Taxonomy of the orchid family Category:Monocots Category:Historically recognized angiosperm orders *Laeliopsis *Lanium *Lankesterella *Leaoa *Lecanorchis *Lemboglossum *Lemurella *Lemurorchis *Leochilus: smooth-lip orchid *Lepanthes: babyboot orchid *Lepanthopsis: tiny orchid *Lepidogyne *Leporella *Leptotes *Lesliea *Leucohyle *Ligeophila *Limodorum *Lindleyalis *Liparis: wide-lip orchid *Listrostachys *Lockhartia *Loefgrenianthus *Ludisia: jewel orchid *Lueddemannia *Luisia *Lycaste: bee orchid *Lycomormium *Lyperanthus *Lyroglossa ===M=== thumb|right|100px|Macodes lowii thumb|right|100px|Macodes petola thumb|right|100px|Maxillaria cucullata thumb|right|100px|Maxillaria picta thumb|right|100px|Mexicoa ghiesbrechtiana thumb|right|100px|Oncidium schroederianum *Macodes *Macradenia: long-gland orchid *Macroclinium *Macropodanthus *Malaxis: adder's mouth orchid *Malleola *Manniella *Margelliantha *Masdevallia *Mastigion *Maxillaria: tiger orchid, flame orchid *Mecopodum *Mediocalcar *Megalorchis *Megalotus *Megastylis *Meiracyllium *Meliorchis: extinct, 80-million-year-old orchid *Mendoncella *Mesadenella *Mesadenus: ladies'-tresses *Mesospinidium *Mexicoa *Microchilus *Microcoelia *Micropera *Microphytanthe *Microsaccus *Microtatorchis *Microterangis *Microthelys *Microtis *Miltonia Lindl.: pansy orchid *Miltoniopsis *Mischobulbum *Mixis *Mobilabium *Moerenhoutia *Monadenia *Monanthos *Monomeria *Monophyllorchis *Monosepalum *Mormodes *Mormolyca *Mycaranthes *Myoxanthus *Myrmechila D.L.Jones & M.A.Clem (2005) *Myrmechis *Myrmecophila *Myrosmodes *Mystacidium ===N=== *Nabaluia *Nageliella *Nematoceras *Neobathiea *Neobenthamia *Neobolusia *Neoclemensia *Neocogniauxia *Neodryas *Neoescobaria *Neofinetia *Neogardneria *Neogyna *Neomoorea *Neotinea *Neottia (including Listera) *Neowilliamsia *Nephelaphyllum *Nephrangis *Nervilia *Neuwiedia *Nidema: fairy orchid *Nigritella *Nitidobulbon *Nohawilliamsia *Nothodoritis *Nothostele *Notylia ===O=== thumb|right|100px|Oerstedella centropetalla thumb|right|100px|Ornithophora radicans *Oberonia *Oberonioides *Octarrhena *Octomeria *Odontochilus *Odontoglossum Kunth *Odontorrhynchus *Oeceoclades: monk orchid *Oeonia *Oeoniella *Oerstedella *Oestlundorchis *Olgasis *Oligochaetochilus *Oligophyton *Oliveriana *Omoea *Oncidium: dancing-lady orchid *Ophidion *Ophrys: ophrys *Orchipedum *Orchis: orchis *Oreorchis *Orestias *Orleanesia *Ornithidium *Ornithocephalus *Ornithochilus *Orthoceras *Osmoglossum *Ossiculum *Osyricera *Otochilus *Otoglossum *Otostylis *Oxystophyllum ===P=== thumb|right|100px|Phaius tankervilleae thumb|right|100px|Northern green orchid (Platanthera hyperborea) thumb|right|100px|Western prairie fringed orchid (Platanthera praeclara) thumb|right|100px|Polystachya pubescens thumb|right|100px|Prosthechea cochleata thumb|right|100px|Prosthechea garciana thumb|right|100px|Prosthechea radiata *Pabstia Garay *Pachites *Pachyphyllum *Pachyplectron *Pachystele *Pachystoma *Palmorchis *Panisea *Pantlingia *Paphinia *Paphiopedilum *Papilionanthe *Papillilabium *Paphiopedilum: Venus' slipper *Papperitzia *Papuaea *Paradisanthus *Paralophia P.J.Cribb & Hermans (2005) *Paraphalaenopsis *Parapteroceras *Pecteilis *Pedilochilus *Pedilonum *Pelatantheria *Pelexia: hachuela *Penkimia *Pennilabium *Peristeranthus *Peristeria *Peristylus *Pescatoria *Phaius: nun's-hood orchid *Phalaenopsis: moth orchid *Pheladenia *Pholidota *Phoringopsis *Phragmipedium *Phragmorchis *Phreatia *Phymatidium *Physoceras *Physogyne *Pilophyllum *Pinelia *Piperia: rein orchid *Pityphyllum *Platanthera: fringed orchid, bog orchid *Platantheroides *Platycoryne *Platyglottis *Platylepis *Platyrhiza *Platystele *Platythelys: jug orchid *Plectorrhiza *Plectrelminthus *Plectrophora *Pleione *Pleurothallis: bonnet orchid *Pleurothallopsis *Plexaure *Plocoglottis *Poaephyllum *Podangis *Podochilus *Pogonia: snake- mouth orchid *Pogoniopsis *Polycycnis *Polyotidium *Polyradicion: palmpolly *Polystachya *Pomatocalpa *Ponera *Ponerorchis *Ponthieva: shadow witch *Porpax *Porphyrodesme *Porphyroglottis *Porphyrostachys *Porroglossum *Porrorhachis *Potosia *Prasophyllum *Prescottia: Prescott orchid *Pristiglottis *Proctoria *Promenaea *Prosthechea *Pseudacoridium *Pseuderia *Pseudocentrum *Pseudocranichis *Pseudoeurystyles *Pseudogoodyera *Pseudolaelia *Pseudorchis *Pseudovanilla *Psilochilus: ragged-lip orchid *Psychilis: peacock orchid *Psychopsiella (sometimes included in Psychopsis) *Psychopsis: butterfly orchid *Psygmorchis *Pterichis *Pteroceras *Pteroglossa *Pteroglossaspis: giant orchid *Pterostemma *Pterostylis *Pterygodium *Pygmaeorchis *Pyrorchis ===Q=== *Quekettia *Quisqueya ===R=== thumb|right|100px|Rhyncholaelia glauca thumb|right|100px|Rhynchostele bictoniensis thumb|right|100px|Rhynchostele cordatum thumb|right|100px|Rossioglossum ampliatum *Rangaeris *Rauhiella *Raycadenco *Reichenbachanthus *Renanthera Lour. & Endl.: snail orchid *Comperia *Conchidium *Condylago Luer *Constantia *Corallorhiza (Haller) Chatelaine: coral root *Cordiglottis *Corunastylis *Coryanthes Hook.: bucket orchids *Corybas Salisb. *Grastidium *Greenwoodiella *Grobya *Grosourdya *Guarianthe Dressler & W.E.Higgins *Gunnarella *Gunnarorchis *Gymnadenia: fragrant orchid *Gymnadeniopsis *Gymnochilus *Gynoglottis ===H=== thumb|right|100px|Haraella retrocalla *Habenaria: bog orchid, false rein orchid *Hagsatera *Hammarbya *Hancockia *Hapalochilus *Hapalorchis *Haraella *Harrisella: airplant orchid *Hederorkis *Helcia *Helleriella: dotted orchid *Helonoma *Hemipilia *Herminium *Herpetophytum *Herpysma *Herschelianthe *Hetaeria *Heterotaxis *Heterozeuxine *Hexalectris: crested coralroot *Hexisea *Himantoglossum *Hintonella *Hippeophyllum *Hirtzia *Hispaniella *Hoehneella *Hoffmannseggella *Hofmeisterella *Holcoglossum *Holmesia *Holopogon *Holothrix *Homalopetalum *Horichia *Hormidium *Horvatia *Houlletia *Huntleya *Huttonaea *Hybochilus *Hydrorchis *Hygrochilus *Hylophila *Hymenorchis ===I=== *Imerinaea *Imerinorchis Szlach (2005) *Inobulbon *Ione *Ionopsis: violet orchid *Ipsea *Isabelia *Ischnocentrum *Ischnogyne *Isochilus: equal-lip orchid *Isotria: fiveleaf orchid *Ixyophora Dressler (2005) ===J=== *Jacquiniella: tufted orchid *Jejosephia *Jonesiopsis *Jostia *Jumellea ===K=== *Kalimpongia *Kaurorchis *Kefersteinia *Kegeliella *Kerigomnia *Kinetochilus *Kingidium *Kionophyton *Koellensteinia: grass-leaf orchid *Konantzia *Kraenzlinella *Kreodanthus *Kryptostoma *Kuhlhasseltia ===L=== thumb|right|100px|Leptotes bicolor thumb|right|100px|Ludisia discolor thumb|right|100px|Lycaste Cassiopeia (a cultivar) *Lacaena *Laelia Lindl. ",+4.1,35,-3.8,2.00,0.3085,B
-"If $P(A)=0.4, P(B)=0.5$, and $P(A \cap B)=0.3$, find $P(B \mid A)$.","* This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. In this event, the event B can be analyzed by a conditional probability with respect to A. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . If P(B) is not zero, then this is equivalent to the statement that :P(A\mid B) = P(A). For a value in and an event , the conditional probability is given by P(A \mid X=x) . More formally, P(A|B) is assumed to be approximately equal to P(B|A). ==Examples== ===Example 1=== Relative size Malignant Benign Total Test positive 0.8 (true positive) 9.9 (false positive) 10.7 Test negative 0.2 (false negative) 89.1 (true negative) 89.3 Total 1 99 100 In one study, physicians were asked to give the chances of malignancy with a 1% prior probability of occurring. In general, it cannot be assumed that P(A|B) ≈ P(B|A). For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that = 5% and = 75 %. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. Similarly, if P(A) is not zero, then :P(B\mid A) = P(B) is also equivalent. Similar reasoning can be used to show that P(Ā|B) = etc. The relationship between P(A|B) and P(B|A) is given by Bayes' theorem: :\begin{align} P(B\mid A) &= \frac{P(A\mid B) P(B)}{P(A)}\\\ \Leftrightarrow \frac{P(B\mid A)}{P(A\mid B)} &= \frac{P(B)}{P(A)} \end{align} That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B). === Assuming marginal and conditional probabilities are of similar size === In general, it cannot be assumed that P(A) ≈ P(A|B). It is tempting to define the undefined probability P(A \mid X=x) using this limit, but this cannot be done in a consistent manner. That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A after having accounted for evidence E or after having updated P(A). ", 0.01961, 7.0,311875200.0,0.02828,0.75,E
-What is the number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards?,"The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. *The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. There are 7,462 distinct poker hands. ===7-card poker hands=== In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall,https://www.casinodaniabeach.com/most-popular-types-of-poker/ a player uses the best five-card poker hand out of seven cards. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Hand The five cards (or less) dealt on the screen are known as a hand. ==See also== *Casino comps *Draw poker *Gambling *Gambling mathematics *Problem gambling *Video blackjack *Video Lottery Terminal ==References== ==External links== * Category:Arcade video games The Total line also needs adjusting. ===7-card lowball poker hands=== In some variants of poker a player uses the best five-card low hand selected from seven cards. Video poker is a casino game based on five-card draw poker. The Total line also needs adjusting. ==See also== * Binomial coefficient * Combination * Combinatorial game theory * Effective hand strength algorithm * Event (probability theory) * Game complexity * Gaming mathematics * Odds * Permutation * Probability * Sample space * Set theory ==References== ==External links== * Brian Alspach's mathematics and poker page * MathWorld: Poker * Poker probabilities including conditional calculations * Numerous poker probability tables * 5, 6, and 7 card poker probabilities * Hold'em poker probabilities The frequencies are calculated in a manner similar to that shown for 5-card hands,https://www.pokerstrategy.com/strategy/various-poker/texas-holdem- probabilities/ except additional complications arise due to the extra two cards in the 7-card poker hand. Note that all cards are dealt face up Fourteen Out (also known as Fourteen Off, Fourteen Puzzle, Take Fourteen, or just Fourteen) is a Patience card game played with a deck of 52 playing cards. This list of poker playing card nicknames has some nicknames for the playing cards in a 52-card deck, as used in poker. ==Poker hand nicknames== The following sets of playing cards can be referred to by the corresponding names in card games that include sets of three or more cards, particularly 3 and 5 card draw, Texas Hold 'em and Omaha Hold 'em. The number of distinct poker hands is even smaller. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands. Since poker is a game of incomplete information, the calculator is designed to evaluate the equity of ranges of hands that players can hold, instead of individual hands. The table does not extend to include five-card hands with at least one pair. (Wild cards substitute for any other card in the deck in order to make a better poker hand). ",14.80,655,0.375,2598960,8.44,D
-"A certain food service gives the following choices for dinner: $E_1$, soup or tomato 1.2-2 juice; $E_2$, steak or shrimp; $E_3$, French fried potatoes, mashed potatoes, or a baked potato; $E_4$, corn or peas; $E_5$, jello, tossed salad, cottage cheese, or coleslaw; $E_6$, cake, cookies, pudding, brownie, vanilla ice cream, chocolate ice cream, or orange sherbet; $E_7$, coffee, tea, milk, or punch. How many different dinner selections are possible if one of the listed choices is made for each of $E_1, E_2, \ldots$, and $E_7$ ?","The establishment of restaurants and restaurant menus allowed customers to choose from a list of unseen dishes, which were produced to order according to the customer's selection. A combination meal can also comprise a meal in which separate dishes are selected by consumers from an entire menu, and can include à la carte selections that are combined on a plate. It usually includes several dishes to pick in a fixed list: an entrée (introductory course), a main course (a choice between up to four dishes), a cheese, a dessert, bread, and sometimes beverage (wine) and coffee all for a set price fixed for the year between €15 and €55. In a restaurant, the menu is a list of food and beverages offered to customers and the prices. Combination meals may be priced lower compared to ordering items separately, but this is not always the case. A meat and three meal is one where the customer picks one meat and three side dishes as a fixed-price offering. A fast food combination meal can contain over . A combination meal is also a meal in which the consumer orders items à la carte to create their own meal combination. Other types of restaurants, such as fast-casual restaurants also offer combination meals. A 2010 study published in the Journal of Public Policy & Marketing found that some consumers may order a combination meal even if no price discount is applied compared to the price of ordering items separately. The study found that this behavior is based upon consumers perceiving an inherent value in combination meals, and also suggested that the ease and convenience of ordering, such as ordering a meal by number, plays a role compared to ordering items separately. Combination meals may be priced lower compared to ordering the items separately, and this lower pricing may serve to entice consumers that are budget-minded. This has a fixed menu and often comes with side dishes such as pickled vegetables and miso soup. * A wine list * A liquor and mixed drinks menu * A beer list * A dessert menu (which may also include a list of tea and coffee options) Some restaurants use only text in their menus. thumb|An example of foods served as a fast food combination meal thumb|A combination meal with chicken curry, rice and beef curry thumb|A Spanish combination meal, consisting of a hamburger, French fries and a beer A combination meal, often referred as a combo-meal, is a type of meal that typically includes food items and a beverage. The variation in Chinese cuisine from different regions led caterers to create a list or menu for their patrons. Fast food restaurants will often prepare variations on items already available, but to have them all on the menu would create clutter. Boston Market and Cracker Barrel chains of restaurants offer a similar style of food selection. == See also == * Garbage Plate * List of restaurant terminology == References == === Sources === * * * * * Category:Cuisine of the Southern United States Category:Restaurants by type Category:Restaurant terminology Category:Culture of Nashville, Tennessee Category:Food combinations This way, all of the patrons can see all of the choices, and the restaurant does not have to provide printed menus. Similar concepts include the Hawaiian plate lunch, which features a variety of entrée choices with fixed side items of white rice and macaroni salad, and the southern Louisiana plate lunch, which features menu options that change daily. Salad buffet, bread and butter and beverage are included, and sometimes also a simple starter, like a soup. Most commonly, there is a choice of two or three dishes: a meat/fish/poultry dish, a vegetarian alternative, and a pasta. ",1.51,49,4.5,2688,22,D
-"A rocket has a built-in redundant system. In this system, if component $K_1$ fails, it is bypassed and component $K_2$ is used. If component $K_2$ fails, it is bypassed and component $K_3$ is used. (An example of a system with these kinds of components is three computer systems.) Suppose that the probability of failure of any one component is 0.15 , and assume that the failures of these components are mutually independent events. Let $A_i$ denote the event that component $K_i$ fails for $i=1,2,3$. What is the probability that the system fails?","This can occur when a single part fails, increasing the probability that other portions of the system fail. Cascading failures may occur when one part of the system fails. :P_F = 1 - A_o \begin{cases} P_F = Probability \ of \ Mission \ Failure \\\ A_o = Operational \ Availability \end{cases} Apart from human error, mission failure results from the following causes. * Protection Strategies for Cascading Grid Failures — A Shortcut Approach * I. Dobson, B. A. Carreras, and D. E. Newman, preprint A loading-dependent model of probabilistic cascading failure, Probability in the Engineering and Informational Sciences, vol. 19, no. 1, January 2005, pp. 15–32. Those failures will occasionally combine in unforeseeable ways, and if they induce further failures in an operating environment of tightly interrelated processes, the failures will spin out of control, defeating all interventions."" Redundancy is a form of resilience that ensures system availability in the event of component failure. A system accident (or normal accident) is an ""unanticipated interaction of multiple failures"" in a complex system.Perrow (1999, p. 70). Physics of failure is a technique under the practice of reliability design that leverages the knowledge and understanding of the processes and mechanisms that induce failure to predict reliability and improve product performance. This is a concept which disagrees with that of system accident. == Scott Sagan == Scott Sagan has multiple publications discussing the reliability of complex systems, especially regarding nuclear weapons. If a system has no redundancy, then MTB is in return of failure rate, \lambda. : \lambda = \frac{1}{MTB} Systems with spare parts that are energized but that lack automatic fault bypass are to accept actually results because human action is required to restore operation after every failure. Software reliability is the probability of the software causing a system failure over some specified operating time. A system accident is one that requires many things to go wrong in a cascade. Another common technique is to calculate a safety margin for the system by computer simulation of possible failures, to establish safe operating levels below which none of the calculated scenarios is predicted to cause cascading failure, and to identify the parts of the network which are most likely to cause cascading failures. Such a failure may happen in many types of systems, including power transmission, computer networking, finance, transportation systems, organisms, the human body, and ecosystems. This failure process cascades through the elements of the system like a ripple on a pond and continues until substantially all of the elements in the system are compromised and/or the system becomes functionally disconnected from the source of its load. A cascading failure is a failure in a system of interconnected parts in which the failure of one or few parts leads to the failure of other parts, growing progressively as a result of positive feedback. They are often overworked or maintenance is deferred due to budget cuts, because managers know that they system will continue to operate without fixing the backup system.Perrow (1999). == General characterization == In 2012 Charles Perrow wrote, ""A normal accident [system accident] is where everyone tries very hard to play safe, but unexpected interaction of two or more failures (because of interactive complexity), causes a cascade of failures (because of tight coupling)."" Owing to this coupling, interdependent networks are extremely sensitive to random failures, and in particular to targeted attacks, such that a failure of a small fraction of nodes in one network can trigger an iterative cascade of failures in several interdependent networks. Crucitti, V. Latora and M. Marchiori, Model for cascading failures in complex networks, Physical Review E (Rapid Communications) 69, 045104 (2004). * Data centre power generators that activate when the normal power source is unavailable. === 1+1 redundancy === 1+1 redundancy typically offers the advantage of additional failover transparency in the event of component failure. Annual, vol., no., pp.285-289, 21-23 Jan 1992 using the algorithms for prognostic purposes,NASA.gov NASA Prognostic Center of Excellence and integrating physics of failure predictions into system-level reliability calculations.http://www.dfrsolutions.com/uploads/publications/2010_01_RAMS_Paper.pdf, McLeish, J.G.; ""Enhancing MIL-HDBK-217 reliability predictions with physics of failure methods,"" Reliability and Maintainability Symposium (RAMS), 2010 Proceedings - Annual, vol., no., pp.1-6, 25-28 Jan. 2010 ==Limitations== There are some limitations with the use of physics of failure in design assessments and reliability prediction. A cascade failure can affect large groups of people and systems. ",0.15,-1.49,0.9966,1,1.07,C
-"Suppose that $P(A)=0.7, P(B)=0.3$, and $P(A \cap B)=0.2$. These probabilities are 1.3-3 listed on the Venn diagram in Figure 1.3-1. Given that the outcome of the experiment belongs to $B$, what then is the probability of $A$ ? ","Each node on the diagram represents an event and is associated with the probability of that event. The region included in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by . == History == Venn diagrams were introduced in 1880 by John Venn in a paper entitled ""On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"" in the Philosophical Magazine and Journal of Science, about the different ways to represent propositions by diagrams. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). The probability associated with a node is the chance of that event occurring after the parent event occurs. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. A Venn diagram uses simple closed curves drawn on a plane to represent sets. Venn diagrams normally comprise overlapping circles. Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams. __NOTOC__ thumb|Tree diagram for events A and B. Venn viewed his diagrams as a pedagogical tool, analogous to verification of physical concepts through experiment. Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets. In probability theory, an outcome is a possible result of an experiment or trial. The book comes with a 3-page foldout of a seven-bit cylindrical Venn diagram.) * * * ==External links== * * Lewis Carroll's Logic Game – Venn vs. Euler at Cut-the-knot * Six sets Venn diagrams made from triangles * Interactive seven sets Venn diagram * VBVenn a free open source program for calculating and graphing quantitative two-circle Venn diagrams Category:Graphical concepts in set theory Category:Diagrams Category:Statistical charts and diagrams For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while the other circle may represent the set of all tables. Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. In 1866, Venn published The Logic of Chance, a groundbreaking book which espoused the frequency theory of probability, arguing that probability should be determined by how often something is forecast to occur as opposed to ""educated"" assumptions. In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets. thumb|The Venn Building, University of Hull thumb|alt=Plaque in the form of a Venn diagram with one set labelled 'Mathematician, Philosopher & Anglican priest', a second set labelled 'Really strong beard game' with the overlapping area labelled 'John Venn'|Alternative heritage plaque for John Venn in Hull John Venn, FRS, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. Venn did not use the term ""Venn diagram"" and referred to the concept as ""Eulerian Circles"". These diagrams were devised while designing a stained-glass window in memory of Venn. ===Other diagrams=== Edwards–Venn diagrams are topologically equivalent to diagrams devised by Branko Grünbaum, which were based around intersecting polygons with increasing numbers of sides. ",0.66666666666,200,8.0,17.4,-2,A
-A coin is flipped 10 times and the sequence of heads and tails is observed. What is the number of possible 10-tuplets that result in four heads and six tails?,"thumb|upright=1.35|Coin of Tennes. The Philippine ten-centavo coin (10¢) coin is a denomination of the Philippine peso. In the 1954/55 National Hunt season Four Ten won three of his first four races including two wide-margin victories at Warwick Racecourse in December and January. Four Ten won a total of nine races between his second Gold Cup attempt and his retirement. Commenting on the horse's tendency to jump to the left, Kirkpatrick explained that Four Ten would be better suited by a left-handed track and would probably contest both the Gold Cup and the Grand National. Four Ten (1946 - 1971) was a British Thoroughbred racehorse who won the 1954 Cheltenham Gold Cup. The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by . The horse ran six times in point-to-points winning four times consecutively before falling in a hunter chase. A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. In the early part of the following season, Four Ten further established himself as a high-class steeplechaser with an ""impressive"" win over three miles at Cheltenham in November, beating E.S.B. and Mariner's Log. The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice. A 6×4 or six-by-four is a vehicle with three axles, with a drivetrain delivering power to two wheel ends on two of them.International ProStar ES Class 8 truck: Axle configurations It is a form of four-wheel driveNACFE (North American Council for Freight Efficiency) Executive Report – 6x2 (Dead Axle) Tractors ""A typical three axle Class 8 tractor today is equipped with two rear drive axles (“live” tandem) and is commonly referred to as a 6 X 4 configuration meaning that it has four-wheel drive capability."" The name of the republic, the date and denomination are all on the obverse. ==== New Generation Currency Series ==== The BSP announced in 2017 that the ten-centavo coin would not be included in this series, and that it was dropping the coin from circulation. Mariner's Log took over the lead, but Four Ten, relishing the heavy ground, moved up to challenge at the last and drew away up the run-in to win by four lengths. In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: :P(n)=1-\sum_{x=0}^{n-1}\binom{6n}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{6n-x}\, . If r is the total number of dice selecting the 6 face, then P(r \ge k ; n, p) is the probability of having at least k correct selections when throwing exactly n dice. Strange originally used Four Ten as a hunter before training the horse himself for races on the amateur point-to-point circuit. These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). The BSP Series coin will still be used until that series is demonetized. ===Version history=== English Series (1958–1967) Pilipino Series (1969–1974) Ang Bagong Lipunan Series (1975–1983) Flora and Fauna Series (1983–1994) BSP Coin Series (1995–2017) Obverse centre|frameless|101x101px centre|frameless|101x101px centre|frameless|101x101px centre|frameless|102x102px centre|frameless|101x101px Reverse centre|frameless|100x100px centre|frameless|102x102px centre|frameless|102x102px centre|frameless|104x104px centre|frameless|101x101px ==References== Category:Currencies of the Philippines Category:Ten-cent coins The issues from 1979 to 1982 featured a mintmark underneath the 10 centavo. ==== Flora and Fauna Series ==== From 1983 to 1994, a new coin was issued with Baltazar again faced to the left in profile, and the denomination was moved to the reverse with the date on the front. Around it was the inscription 'Rey de Espana' (King of Spain) and the denomination as 10 Cs. de Po. (10 centimos of peso).http://worldcoingallery.com/countries/display.php?image=nmc2/142-148&desc;=Philippines km148 10 Centimos (1880-1885)&query;=Philippines === United States administration=== thumb|left|10 centavos issued 1907-1945 In 1903, the 10-centavo coin equivalent to was minted for the Philippines, weighing of 0.9 fine silver. ",210,0.020,7.0,72,0.405,A
-"Among nine orchids for a line of orchids along one wall, three are white, four lavender, and two yellow. How many color displays are there?","Bulbophyllum concolor is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia concolor Bulbophyllum bicolor is a species of orchid in the genus Bulbophyllum. Bulbophyllum tricolor is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia tricolor Bulbophyllum bicoloratum is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia *Photos of Bulbophyllum bicoloratum == External links == * * bicoloratum Category:Articles containing video clips Category:Plants described in 1924 Pabstiella tricolor is a species of orchid plant. == References == tricolor It is found only in Hong Kong and isolated parts of southeast China and northern Vietnam. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia bicolor Category:Plants described in 1830 7 Colors (a.k.a. Filler) is a puzzle game, designed by Dmitry Pashkov. The game was published by Infogrames for MS-DOS, Amiga, and NEC PC-9801. ==Reception== ==References== ==External links== * Category:1991 video games Category:Amiga games Category:DOS games Category:Hot B games Category:Infogrames games Category:Multiplayer and single-player video games Category:NEC PC-9801 games Category:Puzzle video games Category:Video games developed in Russia It was developed by the Russian company in 1991. Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor ",1.7,3.42,1260.0,0.84,0.333333333333333,C
-"A survey was taken of a group's viewing habits of sporting events on TV during I.I-5 the last year. Let $A=\{$ watched football $\}, B=\{$ watched basketball $\}, C=\{$ watched baseball $\}$. The results indicate that if a person is selected at random from the surveyed group, then $P(A)=0.43, P(B)=0.40, P(C)=0.32, P(A \cap B)=0.29$, $P(A \cap C)=0.22, P(B \cap C)=0.20$, and $P(A \cap B \cap C)=0.15$. Find $P(A \cup B \cup C)$.","The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. One may resolve this overlap by the principle of inclusion- exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: : Pr(at least one ""1"") = 1 − Pr(no ""1""s) := 1 − Pr([no ""1"" on 1st trial] and [no ""1"" on 2nd trial] and ... and [no ""1"" on 8th trial]) := 1 − Pr(no ""1"" on 1st trial) × Pr(no ""1"" on 2nd trial) × ... × Pr(no ""1"" on 8th trial) := 1 −(5/6) × (5/6) × ... × (5/6) := 1 − (5/6)8 := 0.7674... ==See also== *Logical complement *Exclusive disjunction *Binomial probability ==References== ==External links== *Complementary events - (free) page from probability book of McGraw-Hill Category:Experiment (probability theory) That is, for an event A, :P(A^c) = 1 - P(A). In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. Therefore, the probability of an event's complement must be unity minus the probability of the event. It may be tempting to say that : Pr([""1"" on 1st trial] or [""1"" on second trial] or ... or [""1"" on 8th trial]) := Pr(""1"" on 1st trial) + Pr(""1"" on second trial) + ... + P(""1"" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. Equivalently, the probabilities of an event and its complement must always total to 1. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. By contrast, in the example above the law of total probability applies, since the event X = 0.5 is included into a family of events X = x where x runs over (−1,1), and these events are a partition of the probability space. On the other hand, conditioning on an event B is well-defined, provided that \mathbb{P}(B) eq 0, irrespective of any partition that may contain B as one of several parts. ===Conditional distribution=== Given X = x, the conditional distribution of Y is : \mathbb{P} ( Y=y | X=x ) = \frac{ \binom 3 y \binom 7 {x-y} }{ \binom{10}x } = \frac{ \binom x y \binom{10-x}{3-y} }{ \binom{10}3 } for 0 ≤ y ≤ min ( 3, x ). The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). The 2010 Women's Junior World Handball Championship (17th tournament) took place in South Korea from July 17 to July 31. ==Preliminary round== ===Group A=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group B=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group C=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group D=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ==Main round== ===Group I=== \---- \---- \---- \---- \---- \---- \---- \---- ===Group II=== \---- \---- \---- \---- \---- \---- \---- \---- ==President's Cup== ===21st–24th=== \---- ====23rd/24th==== ====21st/22nd==== ===17th–20th=== \---- ====19th/20th==== ====17th/18th==== ===13th–16th=== \---- ====15th/16th==== ====13th/14th==== ==Placement matches== ===11th/12th=== ===9th/10th=== ===7th/8th=== ===5th/6th=== ==Final round== ===Semifinals=== \---- ===Bronze medal match=== ===Gold medal match=== ==Ranking and statistics== ===Final ranking=== Rank Team Image:Gold medal icon.svg Image:Silver medal icon.svg Image:Bronze medal icon.svg 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2010 Junior Women's World Champions 150px|border|Norway Norway First title ;Team roster Silje Solberg, Christine Homme, Veronica Kristiansen, Hanna Yttereng, Mai Marcussen, Stine Bredal Oftedal, Mari Molid, Maja Jakobsen, Sanna Solberg, Nora Mørk, Guro Rundbråten, Silje Katrine Svendsen, Ellen Marie Folkvord, Hilde Kamperud, Kristin Nørstebø, Susann Iren Hall. ===All Star Team=== *Goalkeeper: *Left wing: *Left back: *Pivot: *Centre back: *Right back: *Right wing: Chosen by team officials and IHF experts: IHF.info ===Other awards=== *Most Valuable Player: *Top Goalscorer: 75 goals ===Top goalkeepers=== Rank Name Team Saves Shots % 1 Jessica Oliveira 127 325 39.1% 2 Marta Žderić 122 314 38.9% 3 Nele Kurzke 93 285 32.6% 4 Marina Vukčević 85 220 38.6% 5 Guro Rundbråten 82 198 41.4% 6 Shuk Yee Wong 80 301 26.6% 7 Marija Colic 75 213 35.2% 8 Preeyanut Bureeruk 73 266 27.4% 9 Elena Fomina 71 194 36.6% 9 Sori Park 71 260 27.3% Source: ihf.info ===Top goalscorers=== Rank Name Team Goals Shots % 1 Nathalie Hagman 75 96 78.1% 2 Laura van den Heijden 72 109 66.1% 3 Ryu Eun-hee 63 109 57.8% 4 Milena Knežević 62 124 50.0% 5 Jelena Živković 59 103 57.3% 6 Lee Eun-bi 58 87 66.7% 7 Tatiana Khmyrova 55 80 68.8% 7 Luciana Mendoza 55 87 63.2% 9 Marta Lopez 52 91 57.1% 10 Ana Martinez 49 92 53.3% Source: ihf.info ==References== Tournament Summary ==External links== *XVII Women's Junior World Championship at IHF.info Category:International handball competitions hosted by South Korea Women's Junior World Handball Championship, 2010 2010 Category:Women's handball in South Korea Junior World Handball Championship Category:July 2010 sports events in South Korea Other events are proper subsets of the sample space that contain multiple elements. In the latter two examples the law of total probability is irrelevant, since only a single event (the condition) is given. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? The expectation of this random variable is equal to the (unconditional) probability, E ( P ( Y ≤ 1/3 | X ) ) = P ( Y ≤ 1/3 ), namely, : 1 \cdot \mathbb{P} (X<0.5) + 0 \cdot \mathbb{P} (X=0.5) + \frac13 \cdot \mathbb{P} (X>0.5) = 1 \cdot \frac16 + 0 \cdot \frac13 + \frac13 \cdot \left( \frac16 + \frac13 \right) = \frac13, which is an instance of the law of total probability E ( P ( A | X ) ) = P ( A ). American football win probability estimates often include whether a team is home or away, the down and distance, score difference, time remaining, and field position. Win probability added is the change in win probability, often how a play or team member affected the probable outcome of the game. ==Current research== Current research work involves measuring the accuracy of win probability estimates, as well as quantifying the uncertainty in individual estimates. ",0.65625,0.75,0.0,0.59,164,D
-"A grade school boy has five blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of his then drawing a blue marble from his right pocket?","thumb|right|Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12). thumb|Animation for the multiplication 2 × 3 = 6. thumb|right|4 × 5 = 20. At the end of each puzzle, the marbles that have been guided into their proper bins are returned to the player. Thus, likelihood of landing on a particular space in Blue Marble Game can't be exactly calculated like in Monopoly, as players can choose to go to different spaces on the board based on current game conditions. The aim is to ensure that each marble arrives in the bin of the same color as the marble. For example, four bags with three marbles each can be thought of as: :[4 bags] × [3 marbles per bag] = 12 marbles. Marble Drop is a puzzle video game published by Maxis on February 28, 1997. == Gameplay == Players are given an initial set of marbles that are divided evenly into six colors: red, orange, yellow, green, blue, and purple, with two more colors available to purchase: black and silver (steel). Steel (silver) balls are 20 percent of the price of colored marbles and can be used as test marbles or to help release a catch instead of using a valuable colored marble; additionally, there are steel-coloured exit bins in the final puzzle. Players must determine how the marble will travel through the puzzle, and how its journey will change the puzzle for the next marble. It can be played by 2 to 4 players.Blue Marble Game instruction booklet ==Gameplay== Players move around the board in order to buy property, build buildings on the properties, pay rent to other players, and earn a salary. Blue Marble Game (부루마불게임) is a Korean board game similar to Monopoly manufactured by Si-Yat-Sa. Black marbles are very expensive, but change to the correct color when they arrive in a bin. ==Reception== Marble Drop received lukewarm reception upon release. While Monopoly is traditionally played across locations in a single city, the Blue Marble Game features cities from across the world; its title is a reference to The Blue Marble photograph taken by the crew of Apollo 17, and its description of the Earth as seen from space. Lost marbles must be purchased when they are needed to complete a puzzle. There is no mortgage system in the Blue Marble Game. ===Statistics=== In Blue Marble Game, there is no space that sends the player to the deserted island other than the deserted island space itself. More marble has been extracted from the over 650 quarry sites near Carrara than from any other place. Marble died of cancer on September 11, 2015 at the age of 48.Roy Marble, Iowa's scoring leader, dies at 48 Marble's son, Devyn, followed in his father's footsteps to Iowa and the NBA. Devyn and his father were the first father-son duo in Big Ten history to each score 1,000 points.Iowa's Devyn Marble joins father Roy in scoring 1,000th point for Hawkeye Marble came into the news again in 2021 when his family expressed displeasure at the retirement of Luka Garza's jersey number (announced after the last game of the season on March 7), noting that they felt hurt and disrespected by the move upon the fact that Marble's number was not retired; Marble, alongside Murray Wier and Chuck Darling, are considered the best players to not have their jersey number retired by Iowa. The layers of marble are interbedded with schists and quartzites. These marbles are picked up and dropped by the players into funnels leading to a series of rails, switches, traps and other devices which grow more complex as the game progresses. Some of them, such as the Maffioli, who rented some quarries north of Carrara, in the Torano area, or, around 1490, Giovanni Pietro Buffa, who bought marble on credit from local quarrymen and then resold it on the Venetian market, were able to create a dense commercial network, exporting the marble even to distant locations. thumb|264x264px|The distinct green colour of the middle slab is a result of an abundance of serpentine minerals|alt=An image of some slabs of connemara marble against a wall. thumb|Sample sheets, 2016 Carrara marble, Luna marble to the Romans, is a type of white or blue-grey marble popular for use in sculpture and building decor. ",+10, 35.91,0.444444444444444,-1270,2.3,C
-"A faculty leader was meeting two students in Paris, one arriving by train from Amsterdam and the other arriving by train from Brussels at approximately the same time. Let $A$ and $B$ be the events that the respective trains are on time. Suppose we know from past experience that $P(A)=0.93, P(B)=0.89$, and $P(A \cap B)=0.87$. Find $P(A \cup B)$.","The probability of one or both events occurring is denoted P(A ∪ B) and in general, it equals P(A) + P(B) – P(A ∩ B). Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. When A and B are mutually exclusive, .Stats: Probability Rules. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.intmath.com; Mutually Exclusive Events. On 21 April 2012 at 18:30 local time (16:30 UTC), two trains were involved in a head-on collision at Westerpark, near Sloterdijk, in the west of Amsterdam, Netherlands. An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? Therefore, the probability of an event's complement must be unity minus the probability of the event. The probability that at least one of the events will occur is equal to one.Scott Bierman. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Formally said, the intersection of each two of them is empty (the null event): A ∩ B = ∅. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. It is estimated that at the moment of the collision the intercity was travelling at and the local train at about . The probabilities of the individual events (red, and club) are multiplied rather than added. The local train was travelling between Amsterdam and whilst the Intercity train was travelling between and . In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. That is, for an event A, :P(A^c) = 1 - P(A). Therefore, two mutually exclusive events cannot both occur. Given an event, the event and its complementary event define a Bernoulli trial: did the event occur or not? In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time). ",37.9,0.95,-3.141592,6.3,-167,B
-"What is the number of possible four-letter code words, selecting from the 26 letters in the alphabet?","The Code 39 specification defines 43 characters, consisting of uppercase letters (A through Z), numeric digits (0 through 9) and a number of special characters (-, ., $, /, +, %, and space). The alphabet for Modern English is a Latin-script alphabet consisting of 26 letters, each having an upper- and lower-case form. The English alphabet has 5 vowels, 19 consonants, and 2 letters (Y and W) that can function as consonants or vowels. 26 (twenty-six) is the natural number following 25 and preceding 27. == In mathematics == thumb|Poster designed to depict the speciality of the number 26 *26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). *26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections. A FourCC (""four-character code"") is a sequence of four bytes (typically ASCII) used to uniquely identify data formats. This code is traditionally mapped to the * character in barcode fonts and will often appear with the human-readable representation alongside the barcode. thumb|Code 39 Characters As a generality, the location of the two wide bars can be considered to encode a number between 1 and 10, and the location of the wide space (which has four possible positions) can be considered to classify the character into one of four groups (from left to right): Letters(+30) (U–Z), Digits(+0) (1–9,0), Letters(+10) (A–J), and Letters(+20) (K–T). Lower case letters, additional punctuation characters and control characters are represented by sequences of two characters of Code 39. Microsoft and Windows developers refer to their four-byte identifiers as FourCCs or Four-Character Codes. 262px|right|thumb|*WIKIPEDIA* encoded in Code 39 Code 39 (also known as Alpha39, Code 3 of 9, Code 3/9, Type 39, USS Code 39, or USD-3) is a variable length, discrete barcode symbology defined in ISO/IEC 16388:2007. Code Details Nr Character Encoding Nr Character Encoding Nr Character Encoding Nr Character Encoding 0 NUL %U 32 [space] [space] 64 @ %V 96 ` %W 1 SOH $A 33 ! * GOD=26=G7+O15+D4 in Simple6,74 English7,74 Gematria8,74 ('The Key': A=1, B2, C3, ..., Z26). Because there are only six letters in the Letters(+30) group (letters 30–35, or U–Z), the other four positions in this group (36–39) are used to represent three symbols (dash, period, space) as well as the start/stop character. However, Code 39 is still used by some postal services (although the Universal Postal Union recommends using Code 128 in all casesAs one example of an international standard, see ), and can be decoded with virtually any barcode reader. Depending on the way one counts, the West Frisian alphabet contains between 25 and 32 characters. ==Letters== West Frisian alphabet Upper case vowels and vowels with diacritics A Â E Ê É I/Y O Ô U Û Ú Lower case vowels and vowels with diacritics a â e ê é i/y o ô u û ú Vowel Pronunciation / Upper case letters B C D F G H J K L M N P Q R S T V W X Z Lower case letters b c d f g h j k l m n p q r s t v w x z Letter Pronunciation ==Alphabetical order== In alphabetical listings both I and Y are usually found between H and J. *The number of the last letter of the English alphabet, Z. In base ten, 26 is the smallest positive integer that is not a palindrome to have a square (262 = 676) that is a palindrome. == In science == *The atomic number of iron. This table outlines the Code 39 specification. Setting aside one of these characters as a start and stop pattern left 39 characters, which was the origin of the name Code 39. The two wide bars, out of five possible positions, encode a number between 1 and 10 using a two-out-of-five code with the following numeric equivalence: 1, 2, 4, 7, 0. RealMedia files also use four-character codes, however, the actual codes used differ from those found in AVI or QuickTime files. All these fields are four-character codes known as OSType. *In a normal deck of cards, there are 26 red cards and 26 black cards. ",358800,3.2,1068.0,0.0384,0.3333333,A
-"The change in molar internal energy when $\mathrm{CaCO}_3(\mathrm{~s})$ as calcite converts to another form, aragonite, is $+0.21 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the difference between the molar enthalpy and internal energy changes when the pressure is 1.0 bar given that the densities of the polymorphs are $2.71 \mathrm{~g} \mathrm{~cm}^{-3}$ and $2.93 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively.","However, the sea level, temperature, and calcium carbonate saturation state of the surrounding system also determine which polymorph of calcium carbonate (aragonite, low-magnesium calcite, high-magnesium calcite) will form. Aragonite will change to calcite over timescales of days or less at temperatures exceeding 300 °C, and vaterite is even less stable. ==Etymology== Calcite is derived from the German , a term from the 19th century that came from the Latin word for lime, (genitive ) with the suffix -ite used to name minerals. thumb|235px|Crystal structure of calcite Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate (CaCO3). Calcite, obtained from an 80 kg sample of Carrara marble, is used as the IAEA-603 isotopic standard in mass spectrometry for the calibration of δ18O and δ13C. Calcite defines hardness 3 on the Mohs scale of mineral hardness, based on scratch hardness comparison. The molecular formula SrCO3 (molar mass: 147.63 g/mol, exact mass: 147.8904 u) may refer to: * Strontianite * Strontium carbonate Other polymorphs of calcium carbonate are the minerals aragonite and vaterite. Twinning, cleavage and crystal forms are often given in morphological units. ==Properties == The diagnostic properties of calcite include a defining Mohs hardness of 3, a specific gravity of 2.71 and, in crystalline varieties, a vitreous luster. An aragonite sea contains aragonite and high-magnesium calcite as the primary inorganic calcium carbonate precipitates. The molecular formula C3H8O10P2 (molar mass: 266.035 g/mol) may refer to: * 1,3-Bisphosphoglyceric acid (1,3-BPG) * 2,3-Bisphosphoglyceric acid (2,3-BPG) Category:Molecular formulas The molecular formula C3H7O7P (molar mass: 186.06 g/mol, exact mass: 185.9929 u) may refer to: * 2-Phosphoglyceric acid, or 2-phosphoglycerate * 3-Phosphoglyceric acid The molecular formula C12H15N2O3PS (molar mass: 298.30 g/mol, exact mass: 298.0541 u) may refer to: * Phoxim * Quinalphos José María Patoni, San Juan del Río, Durango (Mexico) ==See also== *Carbonate rock *Ikaite, CaCO3·6H2O *List of minerals *Lysocline *Manganoan calcite, (Ca,Mn)CO3 *Monohydrocalcite, CaCO3·H2O *Nitratine *Ocean acidification *Ulexite ==References== ==Further reading== * Category:Calcium minerals Category:Carbonate minerals Category:Limestone Category:Optical materials Category:Transparent materials Category:Calcite group Category:Cave minerals Category:Trigonal minerals Category:Minerals in space group 167 Category:Evaporite Category:Luminescent minerals Category:Polymorphism (materials science) Category:Bastet Calcite is also more soluble at higher pressures. However, crystallization of calcite has been observed to be dependent on the starting pH and concentration of magnesium in solution. Calcite in limestone is divided into low-magnesium and high-magnesium calcite, with the dividing line placed at a composition of 4% magnesium. Due to its acidity, carbon dioxide has a slight solubilizing effect on calcite. These processes can be traced by the specific carbon isotope composition of the calcites, which are extremely depleted in the 13C isotope, by as much as −125 per mil PDB (δ13C). ==In Earth history== Calcite seas existed in Earth's history when the primary inorganic precipitate of calcium carbonate in marine waters was low-magnesium calcite (lmc), as opposed to the aragonite and high- magnesium calcite (hmc) precipitated today. The chemical conditions of the seawater must be notably high in magnesium content relative to calcium (high Mg/Ca ratio) for an aragonite sea to form. As ocean acidification causes pH to drop, carbonate ion concentrations will decline, potentially reducing natural calcite production. ==Gallery== File:Calcite-Mottramite-cktsu-45b.jpg|Calcite with mottramite File:Erbenochile eye.JPG|Trilobite eyes employed calcite File:CalciteEchinosphaerites.jpg|Calcite crystals inside a test of the cystoid Echinosphaerites aurantium (Middle Ordovician, northeastern Estonia) File:Calcite-Dolomite-Gypsum-159389.jpg|Rhombohedrons of calcite that appear almost as books of petals, piled up 3-dimensionally on the matrix File:Calcite-Hematite-Chalcopyrite-176263.jpg|Calcite crystal canted at an angle, with little balls of hematite and crystals of chalcopyrite both on its surface and included just inside the surface of the crystal File:GeopetalCarboniferousNV.jpg|Thin section of calcite crystals inside a recrystallized bivalve shell in a biopelsparite File:OoidSurface01.jpg|Grainstone with calcite ooids and sparry calcite cement; Carmel Formation, Middle Jurassic, of southern Utah, USA. File:Calcite-Aragonite-Sulphur-69380.jpg|Several well formed milky white casts, made up of many small sharp calcite crystals, from the sulfur mines at Agrigento, Sicily File:Calcite-tch21c.jpg|Reddish rhombohedral calcite crystals from China. Likewise, the occurrence of calcite seas is controlled by the same suite of factors controlling aragonite seas, with the most obvious being a low seawater Mg/Ca ratio (Mg/Ca < 2), which occurs during intervals of rapid seafloor spreading. ",-0.28,+2.35,1.16,83.81,28,A
-Estimate the molar volume of $\mathrm{CO}_2$ at $500 \mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.,"Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The molar van der Waals volume should not be confused with the molar volume of the substance. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The density of solid helium at 1.1 K and 66 atm is , corresponding to a molar volume V = . However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",432.07,7.82,0.66666666666,0.366,-0.347,D
-Suppose the concentration of a solute decays exponentially along the length of a container. Calculate the thermodynamic force on the solute at $25^{\circ} \mathrm{C}$ given that the concentration falls to half its value in $10 \mathrm{~cm}$.,"A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. In terms of separate decay constants, the total half-life T _{1/2} can be shown to be :T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2}. The exponential time-constant for the process is \tau = R \, C , so the half-life is R \, C \, \ln(2) . This plot shows decay for decay constant () of 25, 5, 1, 1/5, and 1/25 for from 0 to 5. The enthalpy change in this process, normalized by the mole number of solute, is evaluated as the molar integral heat of dissolution. The integral heat of dissolution is defined for a process of obtaining a certain amount of solution with a final concentration. For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. == Solution of the differential equation == The equation that describes exponential decay is :\frac{dN}{dt} = -\lambda N or, by rearranging (applying the technique called separation of variables), :\frac{dN}{N} = -\lambda dt. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where is the quantity at time , is the initial quantity, that is, the quantity at time . == Measuring rates of decay== === Mean lifetime === If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. The solution to this equation is given in the previous section, where the sum of \lambda _1 + \lambda _2\, is treated as a new total decay constant \lambda _c. :N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}. Concentration of X in solvent A/concentration of X in solvent B=Kď If C1 denotes the concentration of solute X in solvent A & C2 denotes the concentration of solute X in solvent B; Nernst's distribution law can be expressed as C1/C2 = Kd. The molar differential enthalpy change of dissolution is: :\Delta_\text{diss}^{d} H= \left(\frac{\partial \Delta_\text{diss} H}{\partial \Delta n_i}\right)_{T,p,n_B} where is the infinitesimal variation or differential of mole number of the solute during dissolution. For a non-ideal solution it is an excess molar quantity. ==Energetics== Dissolution by most gases is exothermic. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. In thermochemistry, the enthalpy of solution (heat of solution or enthalpy of solvation) is the enthalpy change associated with the dissolution of a substance in a solvent at constant pressure resulting in infinite dilution. The value of the enthalpy of solvation is the sum of these individual steps. The half-life can be written in terms of the decay constant, or the mean lifetime, as: :t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2). The temperature of the solution eventually decreases to match that of the surroundings. The dilution between two concentrations of the solute is associated to an intermediary heat of dilution by mole of solute. ==Dilution and Dissolution== The process of dissolution and the process of dilution are closely related to each other. In thermochemistry, the heat of dilution, or enthalpy of dilution, refers to the enthalpy change associated with the dilution process of a component in a solution at a constant pressure. The enthalpy change in this process, normalized by the mole number of solute, is evaluated as the molar integral heat of dilution. Thus, the amount of material left is 2−1 = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay. ",200,17,0.15,0.18162,0.14,B
+","Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law. thumb|upright=1.45|Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K. Notice that for a given temperature, different parameterizations imply different maximal wavelengths. The formula is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and is the Stefan–Boltzmann constant. ==Equations== ===Planck's law of black-body radiation=== Planck's law states that :B_ u(T) = \frac{2h u^3}{c^2}\frac{1}{e^{h u/kT} - 1}, where :B_{ u}(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency u radiation per unit frequency at thermal equilibrium at temperature T. Units: power / [area × solid angle × frequency]. :h is the Planck constant; :c is the speed of light in vacuum; :k is the Boltzmann constant; : u is the frequency of the electromagnetic radiation; :T is the absolute temperature of the body. Meanwhile, the average energy of a photon from a blackbody isE = \left[\frac{\pi^4}{30\ \zeta(3)}\right] k_\mathrm{B}T \approx 2.701\ k_\mathrm{B}T,where \zeta is the Riemann zeta function. ===Approximations=== In the limit of low frequencies (i.e. long wavelengths), Planck's law becomes the Rayleigh–Jeans law B_ u(T) \approx \frac{2 u^2 }{c^2} k_\mathrm{B} T or B_\lambda(T) \approx \frac{2c}{\lambda^4} k_\mathrm{B} T The radiance increases as the square of the frequency, illustrating the ultraviolet catastrophe. The relative spectral power distribution (SPD) of a Planckian radiator follows Planck's law, and depends on the second radiation constant, c_2=hc/k. Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (T) for the frequency in this equation. The spectral radiance of Planckian radiation from a black body has the same value for every direction and angle of polarization, and so the black body is said to be a Lambertian radiator. ==Different forms== Planck's law can be encountered in several forms depending on the conventions and preferences of different scientific fields. The emitted energy flux density or irradiance B_ u(T,E), is related to the photon flux density b_ u(T,E) through :B_ u(T,E) = Eb_ u(T,E) ===Wien's displacement law=== Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. The Planckian locus is determined by substituting into the above equations the black body spectral radiant exitance, which is given by Planck's law: :M(\lambda,T) =\frac{c_{1}}{\lambda^5}\frac{1}{\exp\left(\frac{c_2}{{\lambda}T}\right)-1} where: :c1 = 2hc2 is the first radiation constant :c2 = hc/k is the second radiation constant and: :M is the black body spectral radiant exitance (power per unit area per unit wavelength: watt per square meter per meter (W/m3)) :T is the temperature of the black body :h is Planck's constant :c is the speed of light :k is Boltzmann's constant This will give the Planckian locus in CIE XYZ color space. Although the spectra of such lights are not accurately described by the black-body radiation curve, a color temperature (the correlated color temperature) is quoted for which black-body radiation would most closely match the subjective color of that source. Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is: :u_{\lambda}(\lambda,T) = {2 h c^2\over \lambda^5}{1\over e^{h c/\lambda kT}-1}. They recommend that the Planck spectrum be plotted as a “spectral energy density per fractional bandwidth distribution,” using a logarithmic scale for the wavelength or frequency. ==See also== * Wien approximation * Emissivity * Sakuma–Hattori equation * Stefan–Boltzmann law * Thermometer * Ultraviolet catastrophe ==References== ==Further reading== * * ==External links== * Eric Weisstein's World of Physics Category:Statistical mechanics Category:Foundational quantum physics Category:Light Category:1893 in science Category:1893 in Germany Then for a perfectly black body, the wavelength- specific ratio of emissive power to absorption ratio is again just , with the dimensions of power. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength λ per unit time is strictly proportional to the black-body curve. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures (see also Wien's law). thumb|303px|As the temperature increases, the peak of the emitted black-body radiation curve moves to higher intensities and shorter wavelengths. From the Planck constant h and the Boltzmann constant k, Wien's constant b can be obtained. ==Peak differs according to parameterization== Constants for different parameterizations of Wien's law Parameterized by x_\mathrm{peak} b (μm⋅K) Wavelength, \lambda 2898 \log\lambda or \log u 3670 Frequency, u 5099 Other characterizations of spectrum Parameterized by x b (μm⋅K) Mean photon energy 5327 10% percentile 2195 25% percentile 2898 50% percentile 4107 70% percentile 5590 90% percentile 9376 The results in the tables above summarize results from other sections of this article. Additionally, for a given temperature the radiance consisting of all photons between two wavelengths must be the same regardless of which distribution you use. In the limit of high frequencies (i.e. small wavelengths) Planck's law tends to the Wien approximation: B_ u(T) \approx \frac{2 h u^3}{c^2} e^{-\frac{h u}{k_\mathrm{B}T}} or B_\lambda(T) \approx \frac{2 h c^2}{\lambda^5} e^{-\frac{hc}{\lambda k_\mathrm{B} T}}. ===Percentiles=== Percentile (μm·K) 0.01% 910 0.0632 0.1% 1110 0.0771 1% 1448 0.1006 10% 2195 0.1526 20% 2676 0.1860 25.0% 2898 0.2014 30% 3119 0.2168 40% 3582 0.2490 41.8% 3670 0.2551 50% 4107 0.2855 60% 4745 0.3298 64.6% 5099 0.3544 70% 5590 0.3885 80% 6864 0.4771 90% 9376 0.6517 99% 22884 1.5905 99.9% 51613 3.5873 99.99% 113374 7.8799 Wien's displacement law in its stronger form states that the shape of Planck's law is independent of temperature. According to Kirchhoff's law of thermal radiation, this entails that, for every frequency , at thermodynamic equilibrium at temperature , one has , so that the thermal radiation from a black body is always equal to the full amount specified by Planck's law. In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. thumb|right|upright=1.15|Planck's law accurately describes black-body radiation. UV-B lamps are lamps that emit a spectrum of ultraviolet light with wavelengths ranging from 290–320 nanometers. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorptivity is again just , with the dimensions of power. ",358800, 7.42,"""0.14""",2.10,0.11,D
+Lead has $T_{\mathrm{c}}=7.19 \mathrm{~K}$ and $\mathcal{H}_{\mathrm{c}}(0)=63.9 \mathrm{kA} \mathrm{m}^{-1}$. At what temperature does lead become superconducting in a magnetic field of $20 \mathrm{kA} \mathrm{m}^{-1}$ ?,"At that temperature even the weakest external magnetic field will destroy the superconducting state, so the strength of the critical field is zero. In 2007, the same group published results suggesting a superconducting transition temperature of 260 K. As of 2015, the highest critical temperature found for a conventional superconductor is 203 K for H2S, although high pressures of approximately 90 gigapascals were required. In 2020, a room-temperature superconductor (critical temperature 288 K) made from hydrogen, carbon and sulfur under pressures of around 270 gigapascals was described in a paper in Nature. It has been experimentally demonstrated that, as a consequence, when the magnetic field is increased beyond the critical field, the resulting phase transition leads to a decrease in the temperature of the superconducting material. Conventional superconductors usually have critical temperatures ranging from around 20 K to less than 1 K. Solid mercury, for example, has a critical temperature of 4.2 K. This material has critical temperature of 10 kelvins and can superconduct at up to about 15 teslas. Similarly, at a fixed temperature below the critical temperature, superconducting materials cease to superconduct when an external magnetic field is applied which is greater than the critical magnetic field. Cambridge University Press, Cambridge From about 1993, the highest-temperature superconductor known was a ceramic material consisting of mercury, barium, calcium, copper and oxygen (HgBa2Ca2Cu3O8+δ) with Tc = 133–138 K. Changes in either temperature or magnetic flux density can cause the phase transition between normal and superconducting states.High Temperature Superconductivity, Jeffrey W. Lynn Editor, Springer-Verlag (1990) The highest temperature under which the superconducting state is seen is known as the critical temperature. For a given temperature, the critical field refers to the maximum magnetic field strength below which a material remains superconducting. A room-temperature superconductor is a material that is capable of exhibiting superconductivity at operating temperatures above , that is, temperatures that can be reached and easily maintained in an everyday environment. , the material with the highest claimed superconducting temperature is an extremely pressurized carbonaceous sulfur hydride with a critical transition temperature of +15 °C at 267 GPa. One exception to this rule is the iron pnictide group of superconductors which display behaviour and properties typical of high-temperature superconductors, yet some of the group have critical temperatures below 30 K. === By material === thumb|""Top: Periodic table of superconducting elemental solids and their experimental critical temperature (T). Low temperature superconductors refer to materials with a critical temperature below 30 K, and are cooled mainly by liquid helium (Tc > 4.2 K). High-temperature superconductivity was discovered in the 1980s. In 2019, the material with the highest accepted superconducting temperature was highly pressurized lanthanum decahydride (), whose transition temperature is approximately . In 1913, lead was found to superconduct at 7 K, and in 1941 niobium nitride was found to superconduct at 16 K. Later, other substances with superconductivity at temperatures up to 30 K were found. For a type-I superconductor the discontinuity in heat capacity seen at the superconducting transition is generally related to the slope of the critical field (H_\text{c}) at the critical temperature (T_\text{c}):Superconductivity of Metals and Alloys, P. G. de Gennes, Addison-Wesley (1989) :C_\text{super} - C_\text{normal} = {T \over 4 \pi} \left(\frac{dH_\text{c}}{dT}\right)^2_{T=T_\text{c}} There is also a direct relation between the critical field and the critical current – the maximum electric current density that a given superconducting material can carry, before switching into the normal state. The upper critical field (at 0 K) can also be estimated from the coherence length () using the Ginzburg–Landau expression: .Introduction to Solid State Physics, Charles Kittel, John Wiley and Sons, Inc. ==Lower critical field== The lower critical field is the magnetic flux density at which the magnetic flux starts to penetrate a type-II superconductor. ==References== Category:Superconductivity Several hundred metals, compounds, alloys and ceramics possess the property of superconductivity at low temperatures. The results were strongly supported by Monte Carlo computer simulations. === Meissner effect === When a superconductor is placed in a weak external magnetic field H, and cooled below its transition temperature, the magnetic field is ejected. ",41,6.0,"""91.7""",2.6,61,B
+"When an electric discharge is passed through gaseous hydrogen, the $\mathrm{H}_2$ molecules are dissociated and energetically excited $\mathrm{H}$ atoms are produced. If the electron in an excited $\mathrm{H}$ atom makes a transition from $n=2$ to $n=1$, calculate the wavenumber of the corresponding line in the emission spectrum.","Therefore, each wavelength of the emission lines corresponds to an electron dropping from a certain energy level (greater than 1) to the first energy level. == See also == * Bohr model * H-alpha * Hydrogen spectral series * K-alpha * Lyman-alpha line * Lyman continuum photon * Moseley's law * Rydberg formula * Balmer series ==References== Category:Emission spectroscopy Category:Hydrogen physics The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. This equation is valid for all hydrogen-like species, i.e. atoms having only a single electron, and the particular case of hydrogen spectral lines is given by Z=1. ==Series== ===Lyman series ( = 1)=== In the Bohr model, the Lyman series includes the lines emitted by transitions of the electron from an outer orbit of quantum number n > 1 to the 1st orbit of quantum number n' = 1. It is emitted when the atomic electron transitions from an n = 2 orbital to the ground state (n = 1), where n is the principal quantum number. In physics and chemistry, the Lyman series is a hydrogen spectral series of transitions and resulting ultraviolet emission lines of the hydrogen atom as an electron goes from n ≥ 2 to n = 1 (where n is the principal quantum number), the lowest energy level of the electron. It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. Therefore the motion of the electron in the process of photon absorption or emission is always accompanied by motion of the nucleus, and, because the mass of the nucleus is always finite, the energy spectra of hydrogen-like atoms must depend on the nuclear mass. ==Rydberg formula== The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: : {1 \over \lambda} = Z^2 R_\infty \left( {1 \over {n'}^2} - {1 \over n^2} \right) where : is the atomic number, : (often written n_1) is the principal quantum number of the lower energy level, : (or n_2) is the principal quantum number of the upper energy level, and : R_\infty is the Rydberg constant. ( for hydrogen and for heavy metals). These observed spectral lines are due to the electron making transitions between two energy levels in an atom. Here is an illustration of the first series of hydrogen emission lines: Historically, explaining the nature of the hydrogen spectrum was a considerable problem in physics. thumb|A hydrogen atom with proton and electron spins aligned (top) undergoes a flip of the electron spin, resulting in emission of a photon with a 21 cm wavelength (bottom) The hydrogen line, 21 centimeter line, or H I line is a spectral line that is created by a change in the energy state of solitary, electrically neutral hydrogen atoms. Also in . , vacuum (nm) 2 121.57 3 102.57 4 97.254 5 94.974 6 93.780 ∞ 91.175 Source: ===Balmer series ( = 2)=== 757px|thumb|center|The four visible hydrogen emission spectrum lines in the Balmer series. A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. For the Lyman series the naming convention is: *n = 2 to n = 1 is called Lyman- alpha, *n = 3 to n = 1 is called Lyman-beta, etc. H-alpha has a wavelength of 656.281 nm, is visible in the red part of the electromagnetic spectrum, and is the easiest way for astronomers to trace the ionized hydrogen content of gas clouds. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. Replacing the energy in the above formula with the expression for the energy in the hydrogen atom where the initial energy corresponds to energy level n and the final energy corresponds to energy level m, : \frac{1}{\lambda} = \frac{E_\text{i} - E_\text{f}}{12398.4\,\text{eV Å}} = R_\text{H} \left(\frac{1}{m^2} - \frac{1}{n^2} \right) Where RH is the same Rydberg constant for hydrogen from Rydberg's long known formula. Spectral emission occurs when an electron transitions, or jumps, from a higher energy state to a lower energy state. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. For example, the line is called ""Lyman-alpha"" (Ly-α), while the line is called ""Paschen-delta"" (Pa-δ). thumb|Energy level diagram of electrons in hydrogen atom There are emission lines from hydrogen that fall outside of these series, such as the 21 cm line. This two-step photodissociation process, known as the Solomon process, is one of the main mechanisms by which molecular hydrogen is destroyed in the interstellar medium. thumb|Electronic and vibrational levels of the hydrogen molecule In reference to the figure shown, Lyman-Werner photons are emitted as described below: *A hydrogen molecule can absorb a far- ultraviolet photon (11.2 eV < energy of the photon < 13.6 eV) and make a transition from the ground electronic state X to excited state B (Lyman) or C (Werner). In a laboratory setting, the hydrogen line parameters have been more precisely measured as: : λ = 21.106114054160(30) cm : ν = 1420405751.768(2) Hz in a vacuum. ",4.4,62.2,"""82258.0""",0.05882352941,226,C
+Calculate the shielding constant for the proton in a free $\mathrm{H}$ atom.,"Any one of these constants can be written in terms of any of the others using the fine-structure constant \alpha : :r_{\mathrm{e}} = \alpha \frac{\lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0. ==Hydrogen atom and similar systems== The Bohr radius including the effect of reduced mass in the hydrogen atom is given by : \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 , where \mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p}) is the reduced mass of the electron–proton system (with m_\text{p} being the mass of proton). The shielding constant for each group is formed as the sum of the following contributions: #An amount of 0.35 from each other electron within the same group except for the [1s] group, where the other electron contributes only 0.30. This value is based on measurements involving a proton and an electron (namely, electron scattering measurements and complex calculation involving scattering cross section based on Rosenbluth equation for momentum-transfer cross section), and studies of the atomic energy levels of hydrogen and deuterium. In January 2013, an updated value for the charge radius of a proton——was published. The constant is expressed for either hydrogen as R_\text{H}, or at the limit of infinite nuclear mass as R_\infty. A resolution came in 2019, when two different studies, using different techniques involving the Lamb shift of the electron in hydrogen, and electron–proton scattering, found the radius of the proton to be 0.833 fm, with an uncertainty of ±0.010 fm, and 0.831 fm. The constant first arose as an empirical fitting parameter in the Rydberg formula for the hydrogen spectral series, but Niels Bohr later showed that its value could be calculated from more fundamental constants according to his model of the atom. The radius of the proton is linked to the form factor and momentum-transfer cross section. Consistent with the spectroscopy method, this produces a proton radius of about . ===2010 experiment=== In 2010, Pohl et al. published the results of an experiment relying on muonic hydrogen as opposed to normal hydrogen. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. The internationally accepted value of a proton's charge radius is . By measuring the energy required to excite hydrogen atoms from the 2S to the 2P state, the Rydberg constant could be calculated, and from this the proton radius inferred. For hydrogen, whose nucleus consists only of one proton, this indirectly measures the proton charge radius. Revised values of screening constants based on computations of atomic structure by the Hartree–Fock method were obtained by Enrico Clementi et al. in the 1960s. ==Rules== Firstly, the electrons are arranged into a sequence of groups in order of increasing principal quantum number n, and for equal n in order of increasing azimuthal quantum number l, except that s- and p- orbitals are kept together. :[1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s, 5p] [5d] etc. The result is again ~5% smaller than the previously-accepted proton radius. The nucleus of the most common isotope of the hydrogen atom (with the chemical symbol ""H"") is a lone proton. this opinion is not yet universally held. ==Problem== Prior to 2010, the proton charge radius was measured using one of two methods: one relying on spectroscopy, and one relying on nuclear scattering. ===Spectroscopy method=== The spectroscopy method uses the energy levels of electrons orbiting the nucleus. His personal assumption is that past measurements have misgauged the Rydberg constant and that the current official proton size is inaccurate. ===Quantum chromodynamic calculation=== In a paper by Belushkin et al. (2007), including different constraints and perturbative quantum chromodynamics, a smaller proton radius than the then-accepted 0.877 femtometres was predicted. ===Proton radius extrapolation=== Papers from 2016 suggested that the problem was with the extrapolations that had typically been used to extract the proton radius from the electron scattering data though these explanation would require that there was also a problem with the atomic Lamb shift measurements. ===Data analysis method=== In one of the attempts to resolve the puzzle without new physics, Alarcón et al. (2018) of Jefferson Lab have proposed that a different technique to fit the experimental scattering data, in a theoretically as well as analytically justified manner, produces a proton charge radius from the existing electron scattering data that is consistent with the muonic hydrogen measurement. The 2014 CODATA adjustment slightly reduced the recommended value for the proton radius (computed using electron measurements only) to , but this leaves the discrepancy at σ. The screening constant, and subsequently the shielded (or effective) nuclear charge for each electron is deduced as: : \begin{matrix} 4s &: s = 0.35 \times 1& \+ &0.85 \times 14 &+& 1.00 \times 10 &=& 22.25 &\Rightarrow& Z_{\mathrm{eff}}(4s) = 26.00 - 22.25 = 3.75\\\ 3d &: s = 0.35 \times 5& & &+& 1.00 \times 18 &=& 19.75 &\Rightarrow& Z_{\mathrm{eff}}(3d)= 26.00 - 19.75 =6.25\\\ 3s,3p &: s = 0.35 \times 7& \+ &0.85 \times 8 &+& 1.00 \times 2 &=& 11.25 &\Rightarrow& Z_{\mathrm{eff}}(3s,3p)= 26.00 - 11.25 =14.75\\\ 2s,2p &: s = 0.35 \times 7& \+ &0.85 \times 2 & & &=& 4.15 &\Rightarrow& Z_{\mathrm{eff}}(2s,2p)= 26.00 - 4.15 =21.85\\\ 1s &: s = 0.30 \times 1& & & & &=& 0.30 &\Rightarrow& Z_{\mathrm{eff}}(1s)= 26.00 - 0.30 =25.70 \end{matrix} Note that the effective nuclear charge is calculated by subtracting the screening constant from the atomic number, 26. ==Motivation== The rules were developed by John C. Slater in an attempt to construct simple analytic expressions for the atomic orbital of any electron in an atom. The Rydberg constant for hydrogen may be calculated from the reduced mass of the electron: : R_\text{H} = R_\infty \frac{ m_\text{e} m_\text{p} }{ m_\text{e}+m_\text{p} } \approx 1.09678 \times 10^7 \text{ m}^{-1} , where * m_\text{e} is the mass of the electron, * m_\text{p} is the mass of the nucleus (a proton). === Rydberg unit of energy === The Rydberg unit of energy is equivalent to joules and electronvolts in the following manner: :1 \ \text{Ry} \equiv h c R_\infty = \frac{m_\text{e} e^4}{8 \varepsilon_{0}^{2} h^2} = \frac{e^2}{8 \pi \varepsilon_{0} a_0} = 2.179\;872\;361\;1035(42) \times 10^{-18}\ \text{J} \ = 13.605\;693\;122\;994(26)\ \text{eV}. === Rydberg frequency === :c R_\infty = 3.289\;841\;960\;2508(64) \times 10^{15}\ \text{Hz} . === Rydberg wavelength === :\frac 1 {R_\infty} = 9.112\;670\;505\;824(17) \times 10^{-8}\ \text{m}. The result is a protonated atom, which is a chemical compound of hydrogen. ",1.91,1.1,"""5275.0""",1.775,1.7,D
+"An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let $A_1$ be those people with an auto policy only, $A_2$ those people with a homeowner policy only, and $A_3$ those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that $P\left(A_1\right)=0.3, P\left(A_2\right)=0.2$, and $P\left(A_3\right)=0.2$. Further, let $B$ be the event that the person will renew at least one of these policies. Say from past experience that we assign the conditional probabilities $P\left(B \mid A_1\right)=0.6, P\left(B \mid A_2\right)=0.7$, and $P\left(B \mid A_3\right)=0.8$. Given that the person selected at random has an auto or homeowner policy, what is the conditional probability that the person will renew at least one of those policies?","This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . * This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. In this event, the event B can be analyzed by a conditional probability with respect to A. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. The relationship between P(A|B) and P(B|A) is given by Bayes' theorem: :\begin{align} P(B\mid A) &= \frac{P(A\mid B) P(B)}{P(A)}\\\ \Leftrightarrow \frac{P(B\mid A)}{P(A\mid B)} &= \frac{P(B)}{P(A)} \end{align} That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B). === Assuming marginal and conditional probabilities are of similar size === In general, it cannot be assumed that P(A) ≈ P(A|B). Unconditionally (that is, without reference to C), A and B are independent of each other because \operatorname{P}(A)—the sum of the probabilities associated with a 1 in row A—is \tfrac{1}{2}, while \operatorname{P}(A\mid B) = \operatorname{P}(A \text{ and } B) / \operatorname{P}(B) = \tfrac{1/4}{1/2} = \tfrac{1}{2} = \operatorname{P}(A). That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A after having accounted for evidence E or after having updated P(A). Thus, the conditional probability P(D1 = 2 | D1+D2 ≤ 5) = = 0.3: : Table 3 \+ + D2 D2 D2 D2 D2 D2 \+ + 1 2 3 4 5 6 D1 1 2 3 4 5 6 7 D1 2 3 4 5 6 7 8 D1 3 4 5 6 7 8 9 D1 4 5 6 7 8 9 10 D1 5 6 7 8 9 10 11 D1 6 7 8 9 10 11 12 Here, in the earlier notation for the definition of conditional probability, the conditioning event B is that D1 + D2 ≤ 5, and the event A is D1 = 2\. It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. Since in the presence of C the probability of A is affected by the presence or absence of B, A and B are mutually dependent conditional on C. == See also == * * * == References == Category:Independence (probability theory) For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that = 5% and = 75 %. The existence of regular conditional probabilities: necessary and sufficient conditions. From the law of total probability, its expected value is equal to the unconditional probability of . === Partial conditional probability === The partial conditional probability P(A\mid B_1 \equiv b_1, \ldots, B_m \equiv b_m) is about the probability of event A given that each of the condition events B_i has occurred to a degree b_i (degree of belief, degree of experience) that might be different from 100%. The reverse, insufficient adjustment from the prior probability is conservatism. == Formal derivation == Formally, P(A | B) is defined as the probability of A according to a new probability function on the sample space, such that outcomes not in B have probability 0 and that it is consistent with all original probability measures.George Casella and Roger L. Berger (1990), Statistical Inference, Duxbury Press, (p. 18 et seq.)Grinstead and Snell's Introduction to Probability, p. 134 Let Ω be a discrete sample space with elementary events {ω}, and let P be the probability measure with respect to the σ-algebra of Ω. In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. Applying the law of total probability, we have: : \begin{align} P(A) & = P(A\mid B_X) \cdot P(B_X) + P(A\mid B_Y) \cdot P(B_Y) \\\\[4pt] & = {99 \over 100} \cdot {6 \over 10} + {95 \over 100} \cdot {4 \over 10} = {{594 + 380} \over 1000} = {974 \over 1000} \end{align} where * P(B_X)={6 \over 10} is the probability that the purchased bulb was manufactured by factory X; * P(B_Y)={4 \over 10} is the probability that the purchased bulb was manufactured by factory Y; * P(A\mid B_X)={99 \over 100} is the probability that a bulb manufactured by X will work for over 5000 hours; * P(A\mid B_Y)={95 \over 100} is the probability that a bulb manufactured by Y will work for over 5000 hours. In general, it cannot be assumed that P(A|B) ≈ P(B|A). ",0.686,7,"""27.0""",2,-21.2,A
+What is the number of possible 13-card hands (in bridge) that can be selected from a deck of 52 playing cards?,"Each player is dealt thirteen cards from a standard 52-card deck. The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Note that all cards are dealt face up Fourteen Out (also known as Fourteen Off, Fourteen Puzzle, Take Fourteen, or just Fourteen) is a Patience card game played with a deck of 52 playing cards. There are 7,462 distinct poker hands. ===7-card poker hands=== In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall,https://www.casinodaniabeach.com/most-popular-types-of-poker/ a player uses the best five-card poker hand out of seven cards. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. thumb|left|180px| thumb|left|180px| In duplicate bridge, a board is an item of equipment that holds one deal, or one deck of 52 cards distributed in four hands of 13 cards each. Contract bridge, or simply bridge, is a trick-taking card game using a standard 52-card deck. *The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The diagram is typical of that used to illustrate a deal of 52 cards in four hands in the game of contract bridge.Bridge Writing Style Guide by Richard Pavlicek Each hand is designated by a point on the compass and so North–South are partners against East–West. So eliminating identical hands that ignore relative suit values, there are only 134,459 distinct hands. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. # Player shuffle – before the start of play, each table receives a number of boards each containing 13 cards in each of its four pockets. The Total line also needs adjusting. ===7-card lowball poker hands=== In some variants of poker a player uses the best five-card low hand selected from seven cards. The frequencies are calculated in a manner similar to that shown for 5-card hands,https://www.pokerstrategy.com/strategy/various-poker/texas-holdem- probabilities/ except additional complications arise due to the extra two cards in the 7-card poker hand. The name refers to the goal of each turn to make pairs that add up to 14.""Take Fourteen"" (p.80) in The Little Book of Solitaire, Running Press, 2002. ==Rules== The cards are dealt face up into twelve columns, from left to right. The number of distinct poker hands is even smaller. The director is summoned if any player does not have exactly thirteen cards. The Total line also needs adjusting. ==See also== * Binomial coefficient * Combination * Combinatorial game theory * Effective hand strength algorithm * Event (probability theory) * Game complexity * Gaming mathematics * Odds * Permutation * Probability * Sample space * Set theory ==References== ==External links== * Brian Alspach's mathematics and poker page * MathWorld: Poker * Poker probabilities including conditional calculations * Numerous poker probability tables * 5, 6, and 7 card poker probabilities * Hold'em poker probabilities ",10.065778,635013559600,"""2.9""",1.61,48,B
+"What is the number of ways of selecting a president, a vice president, a secretary, and a treasurer in a club consisting of 10 persons?","This is a list of fellows of the Royal Society elected in 1909.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1907.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1910.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1903.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1908.""Fellows of the Royal Society"", Royal Society. This is a list of fellows of the Royal Society elected in 1904.""Fellows of the Royal Society"", Royal Society. ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Edward Charles Cyril Baly (1871–1948) *Sir Thomas Barlow (1845–1945) *Ernest William Barnes (1874–1953) *Francis Arthur Bather (1863–1934) *Sir Robert Abbott Hadfield (1858–1940) *Sir Alfred Daniel Hall (1864–1942) *Sir Arthur Harden (1865–1940) *Alfred John Jukes-Browne (1851–1914) *Sir John Graham Kerr (1869–1957) *William James Lewis (1847–1926) *John Alexander McClelland (1870–1920) *William McFadden Orr (1866–1934) *Alfred Barton Rendle (1865–1938) *James Lorrain Smith (1862–1931) *James Thomas Wilson (1861–1945) ==Foreign members== *George Ellery Hale (1868–1938) *Hugo Kronecker (1839–1914) *Charles Emile Picard (1856–1941) *Santiago Ramon y Cajal (1852–1934) ==References== 1909 Category:1909 in the United Kingdom Category:1909 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Charles Jasper Joly (1864–1906) *Hugh Marshall (1868–1913) *Donald Alexander Smith Baron Strathcona and Mount Royal (1820–1914) *Thomas Gregor Brodie (1866–1916) *Alexander Muirhead (1848–1920) *Sir James Johnston Dobbie (1852–1924) *Sir Arthur Everett Shipley (1861–1927) *Harold William Taylor Wager (1862–1929) *Alfred Cardew Dixon (1865–1936) *George Henry Falkiner Nuttall (1862–1937) *Edward Meyrick (1854–1938) *Sir Sidney Gerald Burrard (1860–1943) *William Whitehead Watts (1860–1947) *Sir Thomas Henry Holland (1868–1947) *Sir Gilbert Thomas Walker (1868–1958) *Morris William Travers (1872–1961) ==References== 1904 Category:1904 in the United Kingdom Category:1904 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Frank Dawson Adams (1859–1942) *Sir Hugh Kerr Anderson (1865–1928) *Sir William Blaxland Benham (1860–1950) *Sir William Henry Bragg (1862–1942) *Archibald Campbell Campbell, 1st Baron Blythswood (1835–1908) *Frederick Daniel Chattaway (1860–1944) *Arthur William Crossley (1869–1927) *Arthur Robertson Cushny (1866–1926) *William Duddell (1872–1917) *Frederick William Gamble (1869–1926) *Sir Joseph Ernest Petavel (1873–1936) *Henry Cabourn Pocklington (1870–1952) *Henry Nicholas Ridley (1855–1956) *Sir Grafton Elliot Smith (1871–1937) *William Henry Young (1863–1942) ==Foreign members== *Ivan Petrovich Pavlov (1849–1936) *Edward Charles Pickering (1846–1919) *Magnus Gustaf Retzius (1842–1919) *Augusto Righi (1850–1920) ==References== 1907 Category:1907 in the United Kingdom Category:1907 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *Thomas William Bridge (1848–1909) *John Edward Stead (1851–1923) *Johnson Symington (1851–1924) *Sir William Maddock Bayliss (1860–1924) *Sir Horace Darwin (1851–1928) *Sir Aubrey Strahan (1852–1928) *William Philip Hiern (1839–1929) *Henry Reginald Arnulph Mallock (1851–1933) *Sir David Orme Masson (1858–1937) *Arthur George Perkin (1861–1937) *Ernest Rutherford Baron Rutherford of Nelson (1871–1937) *Ralph Allen Sampson (1866–1939) *Alfred North Whitehead (1861–1947) *Sydney Arthur Monckton Copeman (1862–1947) *Sir John Sealy Edward Townsend (1868–1957) ==References== 1903 Category:1903 in the United Kingdom Category:1903 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== #Antoine Henri Becquerel (1852–1908) #David James Hamilton (1849–1909) #Silas Weir Mitchell (1829–1914) #Friedrich Robert Helmert (1843–1917) #William Gowland (1842–1922) #William Halse Rivers Rivers (1864–1922) #Charles Immanuel Forsyth Major (1843–1923) #Arthur Dendy (1865–1925) #H. H. Asquith (1852–1928) #Shibasaburo Kitasato (1852–1931) #Sir Dugald Clerk #Otto Stapf #William Barlow #Edmund Neville Nevill #Herbrand Russell, 11th Duke of Bedford #Sir Jocelyn Field Thorpe #Randal Thomas Mowbray Rawdon Berkeley #John Stanley Gardiner (1872–1946) #Henry Horatio Dixon #John Hilton Grace #Bertrand Russell ==References== 1908 Category:1908 in the United Kingdom Category:1908 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ""Fellowship from 1660 onwards"" (xlsx file on Google Docs via the Royal Society) ==Fellows== *August Friedrich Leopold Weismann (1834–1914) *Paul Ehrlich (1854–1915) *Henry George Plimmer (1856–1918) *Bertram Hopkinson (1874–1918) *John Allen Harker (1870–1923) *Sir William Boog Leishman (1865–1926) *Gilbert Charles Bourne (1861–1933) *Frederick Augustus Dixey (1855–1935) *Sir Archibald Edward Garrod (1857–1936) *Louis Napoleon George Filon (1875–1937) *Arthur Philemon Coleman (1852–1939) *Alfred Fowler (1868–1940) *Arthur Lapworth (1872–1941) *Sir Joseph Barcroft (187–1947) *Godfrey Harold Hardy (1877–1947) *John Theodore Hewitt (1868–;1954) *Frederick Soddy (1877–1956) ==Foreign members== # Svante August Arrhenius (1859-1927) ForMemRS # Jean-Baptiste Édouard Bornet (1828-1911) ForMemRS # Vito Volterra (1860-1940) ForMemRS # August Friedrich Leopold Weismann (1834-1914) ForMemRS ==References== 1909 Category:1910 in the United Kingdom Category:1910 in science ",29.36,62.8318530718,"""1.6""",5040,4.86,D
+"At a county fair carnival game there are 25 balloons on a board, of which 10 balloons 1.3-5 are yellow, 8 are red, and 7 are green. A player throws darts at the balloons to win a prize and randomly hits one of them. Given that the first balloon hit is yellow, what is the probability that the next balloon hit is also yellow?","The Topic International Darts League was a darts tournament held at the Triavium in Nijmegen, Netherlands. The festival began with approximately 15 balloons and to date has grown to about 30 balloons. The 2009 PartyPoker.com Grand Slam of Darts was the third staging of the darts tournament, the Grand Slam of Darts organised by the Professional Darts Corporation. thumb|various hot air balloons during the festival The Warren County Farmers' Fair Balloon Festival was started in 2001 and takes place during the week of the County Fair in Warren County, New Jersey. The 2003 Las Vegas Desert Classic was the second major Professional Darts Corporation Las Vegas Desert Classic darts tournament. The yellow-winged darter (Sympetrum flaveolum) is a dragonfly found in Europe and mid and northern China. The tournament was sponsored by PartyPoker.net, which has also sponsored other darts championships: the US Open, the Las Vegas Desert Classic and the German Darts Championship. ==References== ==External links== *Collated results of the 2008 European Championship Category:European Championship (darts) European Championship Darts Despite the presence of the PDC players in 2006 and 2007, the tournament was still a WDF/BDO ranking event, with all available points going only to the WDF/BDO players competing. ==International Darts League finals== Year Champion Each player's average score is based on the average for each 3-dart visit to the board (ie total points scored divided by darts thrown and multiplied by 3) Score Runner-up Prize money Prize money Prize money Sponsor Venue Year Champion Each player's average score is based on the average for each 3-dart visit to the board (ie total points scored divided by darts thrown and multiplied by 3) Score Runner-up Total Champion Runner-up Sponsor Venue 2003 Raymond van Barneveld (97.77) 8–5 Mervyn King (97.50) €134,000 €30,000 €15,000 Tempus Triavium, Nijmegen 2004 Raymond van Barneveld (101.64) 13–5 Tony David (95.04) €134,000 €30,000 €15,000 Tempus Triavium, Nijmegen 2005 Mervyn King (91.89) Tony O'Shea (91.74) €134,000 €30,000 €15,000 Tempus Triavium, Nijmegen 2006 Raymond van Barneveld (99.54) 13–5 Colin Lloyd (95.25) €134,000 €30,000 €15,000 Topic Triavium, Nijmegen 2007 Gary Anderson (95.85) 13–9 Mark Webster (94.54) €158,000 €30,000 €15,000 Topic Triavium, Nijmegen ==Sponsors== * 2003–2005 Tempus * 2006–2007 Topic ==References== ==External links== * International Darts League * IDL 2006 – A Review Category:2003 establishments in the Netherlands Category:2007 disestablishments in the Netherlands Category:Professional Darts Corporation tournaments Category:British Darts Organisation tournaments Category:Darts in the Netherlands Category:International sports competitions hosted by the Netherlands The 2008 PartyPoker.net European Championship was the inaugural edition of the Professional Darts Corporation tournament, which thereafter was promoted as the annual European Championship, matching top European players qualifying to play against the highest ranked players from the PDC Order of Merit. The event features some balloon races, including the typical hare and hound races, in addition to the Bicycle Balloon Race. The winner and the runner-up of the 2009 Championship League Darts would be invited, whilst it was announced that only the winner of the 2008 World Masters would be invited (though runner-up Scott Waites was invited anyway due to the withdrawal of Martin Adams). The case ended in failure on 21 February 2008, and the International Darts League was indefinitely postponed. An almost unmistakable darter, red-bodied in the male, with both sexes having large amounts of saffron-yellow colouration to the basal area of each wing, which is particularly noticeable on the hind-wings. The yellow-winged darter tends to make quite short flights when settled at a site, and frequently perches quite low down on vegetation. The future of the World Darts Trophy was also thrown into doubt as a result of the decision,IDL & WDT go to court Superstars of darts forum and both events were confirmed defunct by the failure of an appeal on April 29, 2008.IDL & WDT end Google translation from official web site ==Format== The format has changed slightly over the years – the 2006 competition had 8 round-robin groups of 4 players. Then the top 8 non-qualified players from the 2008 Players Championship Order of Merit after the October German Darts Trophy in Dinslaken, Germany joined them to make a field of 24. Played from 30 October–2 November 2008 at the Südbahnhof in Frankfurt, Germany, the inaugural tournament featured a field of 32 players and £200,000 in prize money, with a £50,000 winner's purse going to Phil Taylor.PDC website report - European Championship Details Confirmed from the Professional Darts Corporation obtained 12-08-2008 ==Format== First round — best of nine legs (by two legs) Second round — best of seventeen legs (ditto) Quarter-finals — best of seventeen legs (ditto) Semi-finals — best of twenty-one legs (ditto) Final — best of twenty-one legs (ditto) Each game had to be won by two clear legs, except that a game went to a sudden death leg if a further six legs did not separate the players; for example, a first round match played out to 7-7 is then decided with one sudden death leg. ==Prize money== A total of £200,000 was on offer to the players, divided based on the following performances: Position (no. of players) Position (no. of players) Prize money (Total: £200,000) Winner (1) £50,000 Runner-Up (1) £25,000 Semi-finalists (2) £12,500 Quarter-finalists (4) £8,500 Last 16 (second round) (8) £4,000 Last 32 (first round) (16) £2,000 Highest checkout (1) £2,000 ==Qualification== The top 16 players from the PDC Order of Merit after the 2008 Sky Poker World Grand Prix automatically qualified for the event. This was the second PDC darts tournament that ITV4 has broadcast, after the inaugural Grand Slam of Darts - after its rating success ITV chose to broadcast this event as well as the 2008 Grand Slam of Darts. It is the only major event that Phil Taylor has competed in at least once, but never won. ==End of event== Towards the end of 2007, the chairman of the PDC, Barry Hearn, announced that its players would not be competing in the 2008 International Darts League and World Darts Trophy events. The shootout occurred exactly one year to the day after a similar situation at the 2008 Grand Slam of Darts where Hamilton beat Alan Tabern. The yellow-winged darter has bred but is not established in the UK. Gary Anderson was the final champion, having claimed the title in 2007, when the tournament also became the first major event to witness two nine dart finishes. ",3,57.2,"""0.375""",0.323,22,C
+What is the number of ordered samples of 5 cards that can be drawn without replacement from a standard deck of 52 playing cards?,"The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. *The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The deck is retrieved, and each player is dealt in turn from the deck the same number of cards they discarded so that each player again has five cards. Each player specifies how many of their cards they wish to replace and discards them. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. However, a rule used by many casinos is that a player is not allowed to draw five consecutive cards from the deck. The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. If the deck is depleted during the draw before all players have received their replacements, the last players can receive cards chosen randomly from among those discarded by previous players. This list arranges card games by the number of cards used. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Another common house rule is that the bottom card of the deck is never given as a replacement, to avoid the possibility of someone who might have seen it during the deal using that information. For example, if the last player to draw wants three replacements but there are only two cards remaining in the deck, the dealer gives the player the one top card he can give, then shuffles together the bottom card of the deck, the burn card, and the earlier players' discards (but not the player's own discards), and finally deals two more replacements to the last player. ==Sample deal== 200px|right The sample deal is being played by four players as shown to the right with Alice dealing. 52 pickup or 52-card pickup is a humorous prank which consists only of picking up a scattered deck of playing cards. In this case, if a player wishes to replace all five of their cards, that player is given four of them in turn, the other players are given their draws, and then the dealer returns to that player to give the fifth replacement card; if no other player draws it is necessary to deal a burn card first. Five-card draw (also known as Cantrell draw) is a poker variant that is considered the simplest variant of poker, and is the basis for video poker. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. With five players, the sixes are added to make a 36-card deck. Its ""Total"" represents the 95.4% of the time that a player can select a 5-card low hand without any pair. The other player must then pick them up.. ==Variations== Genuine card games sometimes end in 52 pickup. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. ",2.2,0.24995,"""4943.0""",0.87,311875200,E
+A bowl contains seven blue chips and three red chips. Two chips are to be drawn successively at random and without replacement. We want to compute the probability that the first draw results in a red chip $(A)$ and the second draw results in a blue chip $(B)$. ,"The Sunday Times described triple-cooked chips as Blumenthal's most influential culinary innovation, which had given the chip ""a whole new lease of life"". ==History== Blumenthal said he was ""obsessed with the idea of the perfect chip"",Blumenthal, In Search of Perfection and described how, from 1992 onwards, he worked on a method for making ""chips with a glass-like crust and a soft, fluffy centre"". thumb|Colorized photo of Chips. The Bowl of Baal is a 1975 science fiction novel by Robert Ames Bennet. Eventually, Blumenthal developed the three-stage cooking process known as triple-cooked chips, which he identifies as ""the first recipe I could call my own"". Triple-cooked chips are a type of chips developed by the English chef Heston Blumenthal. 7 Colors (a.k.a. Filler) is a puzzle game, designed by Dmitry Pashkov. The result is what Blumenthal calls ""chips with a glass-like crust and a soft, fluffy centre"". The chips are first simmered, then cooled and drained using a sous-vide technique or by freezing; deep fried at and cooled again; and finally deep-fried again at . On July 10, 1943, Chips and his handler were pinned down on the beach by an Italian machine-gun team. In 2014, the London Fire Brigade attributed an increase in chip pan fires to the increased popularity of ""posh chips"", including triple-cooked chips. ==Preparation== ===Blumenthal's technique=== Previously, the traditional practice for cooking chips was a two-stage process, in which chipped potatoes were fried in oil first at a relatively low temperature to soften them and then at a higher temperature to crisp up the outside. thumb|A selection of Red Ribbon cakes on sale Red Ribbon Bakeshop, Inc. is a bakery chain based in the Philippines, which produces and distributes cakes and pastries. ==History== In 1979, Amalia Hizon Mercado, husband Renato Mercado, and their five children, Consuelo Tiutan, Teresita Moran, Renato Mercado, Ricky Mercado and Romy Mercado established Red Ribbon as a small cake shop along Timog Avenue in Quezon City. The second of the three stages is frying the chips at for approximately 5 minutes, after which they are cooled once more in a freezer or sous-vide machine before the third and final stage: frying at for approximately 7 minutes until crunchy and golden. Blumenthal describes moisture as the ""enemy"" of crisp chips. C.C. Moore eventually gifted Chips to the Wren family. Chips served as a sentry dog for the Roosevelt-Churchill conference in 1943. Bloomsbury. ==Further reading== * * ==External links== * Triple-Cooked Chips. Second, the cracks that develop in the chips provide places for oil to collect and harden during frying, making them crunchy.Blumenthal, Heston Blumenthal at Home Third, thoroughly drying out the chips drives off moisture that would otherwise keep the crust from becoming crisp. Blumenthal began work on the recipe in 1993, and eventually developed the three-stage cooking process. Chips (1940–1946) was a trained sentry dog for United States Army, and reputedly the most decorated war dog from World War II. Chips was a German Shepherd-Collie-Malamute mix owned by Edward J. Wren of Pleasantville, New York. Chips shipped out to the War Dog Training Center, Front Royal, Virginia, in 1942 for training as a sentry dog. ""A single frying at a high temperature leads to a thin crust that can easily be rendered soggy by whatever moisture remains in the chip’s interior."" ",-383,5,"""0.23333333333""",0.66666666666,313,C
+"From an ordinary deck of playing cards, cards are to be drawn successively at random and without replacement. What is the probability that the third spade appears on the sixth draw?","*The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). Three Shuffles and a Draw is a solitaire game using one deck of playing cards. For example, the probability of drawing three of a kind is approximately 2.11%, while the probability of drawing a hand at least as good as three of a kind is about 2.87%. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. One would then expect to draw this hand about once in every 649,740 draws, or nearly 0.000154% of the time. The name ""Three Shuffles and a Draw"" comes from the fact that there are 3 shuffles (counting the original starting shuffle plus the 2 redeals, and then a draw, where you can free any one single buried card). Draw poker is any poker variant in which each player is dealt a complete hand before the first betting round, and then develops the hand for later rounds by replacing, or ""drawing"", cards. Then a third card is revealed, followed by a betting round, a fourth card, a betting round, and finally a showdown. As a bridge hand contains thirteen cards, only two hand patterns can be classified as three suiters: 4-4-4-1 and 5-4-4-0. right In the game of contract bridge a three suiter (or three-suited hand) denotes a hand containing at least four cards in three of the four suits. thumb|right|170px|Three of Cups from a deck of Italian cards Three of Cups is the third card on the suit of Cups. The object of the game is to move all of the cards to the Foundations. == Rules == Three Shuffles and a Draw has four foundations build up in suit from Ace to King, e.g. A♣, 2♣, 3♣, 4♣... The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. In the card game contract bridge, Gambling 3NT is a special of an opening of 3NT. Finally, each player draws as in normal draw poker, followed by a fourth betting round and showdown. The first betting round is then played, followed by a draw in which each player replaces cards from their hand with an equal number, so that each player still has only four cards in hand. Before the first betting round, each player examines their hand, removes exactly three cards from it, then places them on the table to their left. For example, 3♣ 7♣ 8♣ Q♠ A♠ and 3♦ 7♣ 8♦ Q♥ A♥ are not identical hands when just ignoring suit assignments because one hand has three suits, while the other hand has only two—that difference could affect the relative value of each hand when there are more cards to come. If any player opens, the game continues as traditional five-card draw poker. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. It is the , and this makes all 6-spot cards wild. ",313,0.064,"""122.0""",19.4,0.123,B
+What is the probability of drawing three kings and two queens when drawing a five-card hand from a deck of 52 playing cards?,"*The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. The queen of spades (Q) is one of 52 playing cards in a standard deck: the queen of the suit of spades (). Probabilities are adjusted in the above table such that ""5-high"" is not listed, ""6-high"" has 781,824 distinct hands, and ""King-high"" has 21,457,920 distinct hands, respectively. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. Royal Marriage is a patience or solitaire game using a deck of 52 playing cards. The remaining fifty cards are shuffled and placed on the top of the King to form the stock. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Royal Cotillion is a solitaire card game which uses two decks of 52 playing cards each. Royal Flush is a solitaire card game which is played with a deck of 52 playing cards. For instance, with a royal flush, there are 4 ways to draw one, and 2,598,956 ways to draw something else, so the odds against drawing a royal flush are 2,598,956 : 4, or 649,739 : 1. : Hand Distinct hands Frequency Probability Cumulative Odds against 5-high 1 1,024 0.0394% 0.0394% 2,537.05 : 1 6-high 5 5,120 0.197% 0.236% 506.61 : 1 7-high 15 15,360 0.591% 0.827% 168.20 : 1 8-high 35 35,840 1.38% 2.21% 71.52 : 1 9-high 70 71,680 2.76% 4.96% 35.26 : 1 10-high 126 129,024 4.96% 9.93% 19.14 : 1 Jack-high 210 215,040 8.27% 18.2% 11.09 : 1 Queen-high 330 337,920 13.0% 31.2% 6.69 : 1 King-high 495 506,880 19.5% 50.7% 4.13 : 1 Total 1,287 1,317,888 50.7% 50.7% 0.97 : 1 As can be seen from the table, just over half the time a player gets a hand that has no pairs, threes- or fours-of-a-kind. (50.7%) If aces are not low, simply rotate the hand descriptions so that 6-high replaces 5-high for the best hand and ace-high replaces king-high as the worst hand. For example, there are 4 different ways to draw a royal flush (one for each suit), so the probability is , or one in 649,740. Three Shuffles and a Draw is a solitaire game using one deck of playing cards. Probabilities are adjusted in the above table such that ""5-high"" is not listed"", ""6-high"" has one distinct hand, and ""King-high"" having 330 distinct hands, respectively. In this case, the deck is held face-down in one hand, with the King being uppermost face-down card and the Queen being held face-up above it. The game is won when the King and Queen are brought together -- that is, when only one or two cards remain in between them, which can then be discarded. ==Variations== Royal Marriage is possible to play in-hand, rather than on a surface such as a table. ",0.0000092,35,"""0.323""",0.6321205588,14.5115,A
+"In an orchid show, seven orchids are to be placed along one side of the greenhouse. There are four lavender orchids and three white orchids. How many ways are there to lineup these orchids?","The Orchidoideae, or the orchidoid orchids, are a subfamily of the orchid family (Orchidaceae) that contains around 3630 species. Orchidales is an order of flowering plants. Genera Orchidacearum vol. 3: Orchidoideae part 2, Vanilloideae. Genera Orchidacearum 4. Genera Orchidacearum 5. Genera Orchidacearum 1. Genera Orchidacearum 3. This is a list of genera in the orchid family (Orchidaceae), originally according to The Families of Flowering Plants - L. Watson and M. J. Dallwitz. Genera Orchidacearum 2. This is a list of the orchids, sorted in alphabetical order, found in Metropolitan France. == A == * Anacamptis laxiflora * Anacamptis longicornu * Anacamptis morio * Anacamptis palustris == C == * Cephalanthera longifolia == D == * Dactylorhiza incarnata == E == * Epipactis phyllanthes == G == * Goodyera repens == O == * Ophrys aurelia * Ophrys catalaunica * Ophrys saratoi * Ophrys drumana * Orchis mascula == S == * Serapias lingua == References == France Phylogeny and Classification of the Orchid Family. She provided an English text, paintings, and drawings for the amateur reader, a mixture of impression and scientific illustration of the genera. ==Orchids of South Western Australia== Common name Genus No. species in southwest W.A. Remarks Babe-in-a-cradle Epiblema 1 Beard orchids Calochilus 6 Blue orchids Cyanicula 11 Bunny orchids Eriochilus 6 Donkey orchid Diuris ~36 Duck orchids Paracaleana 13 Elbow orchid Spiculaea 1 Enamel orchids Elythranthera 2 Fairy orchid Pheladenia 1 Fire orchids Pyrorchis 2 also Beak orchids Greenhoods Pterostylis ~90 Hammer orchids Drakaea 10 Hare orchid Leporella 1 Helmet orchids Corybas 4 Leafless orchid Praecoxanthus 1 Leek orchids Prasophyllum 25 Mignonette orchids Microtis 14 also Onion orchid Mosquito orchids Cyrtostylis 5 Potato orchids Gastrodia 1 also Bell orchid Pygmy orchid Corunastylis 1 Rabbit orchid Leptoceras 1 Rattle beaks Lyperanthus 1 Slipper orchids Cryptostylis 1 also Tongue orchid South African orchids Disa bracteata 1 introduced Spider orchids Caladenia 125 Sugar orchid Ericksonella 1 Sun orchids Thelymitra 37 Underground orchids Rhizanthella 1 * This table has its source as the Second Edition of Hoffman and Brown in 1992 ==References== thumb|Diuris plate III from West Australian Orchids, 1930 # ==Further reading== * * * * * * * == External links == * The Species Orchid Society of Western Australia (Inc) -- a gallery of orchids from Western Australia * Orchids from Western and South Australia * Terrestrial orchids of the south west western australia * Orchid Conservation Coalition List of orchids Western Australia Historically, the Orchidoideae have been partitioned into up to 6 tribes, including Orchideae, Diseae, Cranichideae, Chloraeeae, Diurideae, and Codonorchideae. Oxford Univ. Press == External links == *All recognized monocotiledons species (including Orchid family) - World Checklist of Selected Plant Families, Kew Botanic Garden - UK *Intergeneric orchid genus names (updated 11 Jan 2005) *List of orchid genera (updated 14 Jul 2004) *List of common names or *List of orchid hybrids - Royal Horticultural Society - UK *Orchid main page - eMonocot website Orchidaceae The first three orchids from Western Australia to be named were Caladenia menziesii (now Leptoceras menziesii), Caladenia flava, and Diuris longifolia. Dictionary of Orchid Names. This list is adapted regularly with the changes published in the Orchid Research Newsletter which is published twice a year by the Royal Botanic Gardens, Kew. This list is reflected on Wikispecies Orchidaceae and the new eMonocot website Orchidaceae Juss. Although mostly the order will consist of the orchids only (usually in one family only, but sometimes divided into more families, as in the Dahlgren system, see below), sometimes other families are added: ==Circumscription in the Takhtajan system== Takhtajan system: * order Orchidales *: family Orchidaceae ==Circumscription in the Cronquist system== Cronquist system (1981): * order Orchidales *: family Geosiridaceae *: family Burmanniaceae *: family Corsiaceae *: family Orchidaceae ==Circumscription in the Dahlgren system== Dahlgren system: * order Orchidales *: family Neuwiediaceae *: family Apostasiaceae *: family Cypripediaceae *: family Orchidaceae ==Circumscription in the Thorne system== Thorne system (1992): * order Orchidales *: family Orchidaceae ==APG system== The order is not recognized in the APG II system, which assigns the orchids to order Asparagales. ==See also== * Taxonomy of the orchid family Category:Monocots Category:Historically recognized angiosperm orders *Laeliopsis *Lanium *Lankesterella *Leaoa *Lecanorchis *Lemboglossum *Lemurella *Lemurorchis *Leochilus: smooth-lip orchid *Lepanthes: babyboot orchid *Lepanthopsis: tiny orchid *Lepidogyne *Leporella *Leptotes *Lesliea *Leucohyle *Ligeophila *Limodorum *Lindleyalis *Liparis: wide-lip orchid *Listrostachys *Lockhartia *Loefgrenianthus *Ludisia: jewel orchid *Lueddemannia *Luisia *Lycaste: bee orchid *Lycomormium *Lyperanthus *Lyroglossa ===M=== thumb|right|100px|Macodes lowii thumb|right|100px|Macodes petola thumb|right|100px|Maxillaria cucullata thumb|right|100px|Maxillaria picta thumb|right|100px|Mexicoa ghiesbrechtiana thumb|right|100px|Oncidium schroederianum *Macodes *Macradenia: long-gland orchid *Macroclinium *Macropodanthus *Malaxis: adder's mouth orchid *Malleola *Manniella *Margelliantha *Masdevallia *Mastigion *Maxillaria: tiger orchid, flame orchid *Mecopodum *Mediocalcar *Megalorchis *Megalotus *Megastylis *Meiracyllium *Meliorchis: extinct, 80-million-year-old orchid *Mendoncella *Mesadenella *Mesadenus: ladies'-tresses *Mesospinidium *Mexicoa *Microchilus *Microcoelia *Micropera *Microphytanthe *Microsaccus *Microtatorchis *Microterangis *Microthelys *Microtis *Miltonia Lindl.: pansy orchid *Miltoniopsis *Mischobulbum *Mixis *Mobilabium *Moerenhoutia *Monadenia *Monanthos *Monomeria *Monophyllorchis *Monosepalum *Mormodes *Mormolyca *Mycaranthes *Myoxanthus *Myrmechila D.L.Jones & M.A.Clem (2005) *Myrmechis *Myrmecophila *Myrosmodes *Mystacidium ===N=== *Nabaluia *Nageliella *Nematoceras *Neobathiea *Neobenthamia *Neobolusia *Neoclemensia *Neocogniauxia *Neodryas *Neoescobaria *Neofinetia *Neogardneria *Neogyna *Neomoorea *Neotinea *Neottia (including Listera) *Neowilliamsia *Nephelaphyllum *Nephrangis *Nervilia *Neuwiedia *Nidema: fairy orchid *Nigritella *Nitidobulbon *Nohawilliamsia *Nothodoritis *Nothostele *Notylia ===O=== thumb|right|100px|Oerstedella centropetalla thumb|right|100px|Ornithophora radicans *Oberonia *Oberonioides *Octarrhena *Octomeria *Odontochilus *Odontoglossum Kunth *Odontorrhynchus *Oeceoclades: monk orchid *Oeonia *Oeoniella *Oerstedella *Oestlundorchis *Olgasis *Oligochaetochilus *Oligophyton *Oliveriana *Omoea *Oncidium: dancing-lady orchid *Ophidion *Ophrys: ophrys *Orchipedum *Orchis: orchis *Oreorchis *Orestias *Orleanesia *Ornithidium *Ornithocephalus *Ornithochilus *Orthoceras *Osmoglossum *Ossiculum *Osyricera *Otochilus *Otoglossum *Otostylis *Oxystophyllum ===P=== thumb|right|100px|Phaius tankervilleae thumb|right|100px|Northern green orchid (Platanthera hyperborea) thumb|right|100px|Western prairie fringed orchid (Platanthera praeclara) thumb|right|100px|Polystachya pubescens thumb|right|100px|Prosthechea cochleata thumb|right|100px|Prosthechea garciana thumb|right|100px|Prosthechea radiata *Pabstia Garay *Pachites *Pachyphyllum *Pachyplectron *Pachystele *Pachystoma *Palmorchis *Panisea *Pantlingia *Paphinia *Paphiopedilum *Papilionanthe *Papillilabium *Paphiopedilum: Venus' slipper *Papperitzia *Papuaea *Paradisanthus *Paralophia P.J.Cribb & Hermans (2005) *Paraphalaenopsis *Parapteroceras *Pecteilis *Pedilochilus *Pedilonum *Pelatantheria *Pelexia: hachuela *Penkimia *Pennilabium *Peristeranthus *Peristeria *Peristylus *Pescatoria *Phaius: nun's-hood orchid *Phalaenopsis: moth orchid *Pheladenia *Pholidota *Phoringopsis *Phragmipedium *Phragmorchis *Phreatia *Phymatidium *Physoceras *Physogyne *Pilophyllum *Pinelia *Piperia: rein orchid *Pityphyllum *Platanthera: fringed orchid, bog orchid *Platantheroides *Platycoryne *Platyglottis *Platylepis *Platyrhiza *Platystele *Platythelys: jug orchid *Plectorrhiza *Plectrelminthus *Plectrophora *Pleione *Pleurothallis: bonnet orchid *Pleurothallopsis *Plexaure *Plocoglottis *Poaephyllum *Podangis *Podochilus *Pogonia: snake- mouth orchid *Pogoniopsis *Polycycnis *Polyotidium *Polyradicion: palmpolly *Polystachya *Pomatocalpa *Ponera *Ponerorchis *Ponthieva: shadow witch *Porpax *Porphyrodesme *Porphyroglottis *Porphyrostachys *Porroglossum *Porrorhachis *Potosia *Prasophyllum *Prescottia: Prescott orchid *Pristiglottis *Proctoria *Promenaea *Prosthechea *Pseudacoridium *Pseuderia *Pseudocentrum *Pseudocranichis *Pseudoeurystyles *Pseudogoodyera *Pseudolaelia *Pseudorchis *Pseudovanilla *Psilochilus: ragged-lip orchid *Psychilis: peacock orchid *Psychopsiella (sometimes included in Psychopsis) *Psychopsis: butterfly orchid *Psygmorchis *Pterichis *Pteroceras *Pteroglossa *Pteroglossaspis: giant orchid *Pterostemma *Pterostylis *Pterygodium *Pygmaeorchis *Pyrorchis ===Q=== *Quekettia *Quisqueya ===R=== thumb|right|100px|Rhyncholaelia glauca thumb|right|100px|Rhynchostele bictoniensis thumb|right|100px|Rhynchostele cordatum thumb|right|100px|Rossioglossum ampliatum *Rangaeris *Rauhiella *Raycadenco *Reichenbachanthus *Renanthera Lour. & Endl.: snail orchid *Comperia *Conchidium *Condylago Luer *Constantia *Corallorhiza (Haller) Chatelaine: coral root *Cordiglottis *Corunastylis *Coryanthes Hook.: bucket orchids *Corybas Salisb. *Grastidium *Greenwoodiella *Grobya *Grosourdya *Guarianthe Dressler & W.E.Higgins *Gunnarella *Gunnarorchis *Gymnadenia: fragrant orchid *Gymnadeniopsis *Gymnochilus *Gynoglottis ===H=== thumb|right|100px|Haraella retrocalla *Habenaria: bog orchid, false rein orchid *Hagsatera *Hammarbya *Hancockia *Hapalochilus *Hapalorchis *Haraella *Harrisella: airplant orchid *Hederorkis *Helcia *Helleriella: dotted orchid *Helonoma *Hemipilia *Herminium *Herpetophytum *Herpysma *Herschelianthe *Hetaeria *Heterotaxis *Heterozeuxine *Hexalectris: crested coralroot *Hexisea *Himantoglossum *Hintonella *Hippeophyllum *Hirtzia *Hispaniella *Hoehneella *Hoffmannseggella *Hofmeisterella *Holcoglossum *Holmesia *Holopogon *Holothrix *Homalopetalum *Horichia *Hormidium *Horvatia *Houlletia *Huntleya *Huttonaea *Hybochilus *Hydrorchis *Hygrochilus *Hylophila *Hymenorchis ===I=== *Imerinaea *Imerinorchis Szlach (2005) *Inobulbon *Ione *Ionopsis: violet orchid *Ipsea *Isabelia *Ischnocentrum *Ischnogyne *Isochilus: equal-lip orchid *Isotria: fiveleaf orchid *Ixyophora Dressler (2005) ===J=== *Jacquiniella: tufted orchid *Jejosephia *Jonesiopsis *Jostia *Jumellea ===K=== *Kalimpongia *Kaurorchis *Kefersteinia *Kegeliella *Kerigomnia *Kinetochilus *Kingidium *Kionophyton *Koellensteinia: grass-leaf orchid *Konantzia *Kraenzlinella *Kreodanthus *Kryptostoma *Kuhlhasseltia ===L=== thumb|right|100px|Leptotes bicolor thumb|right|100px|Ludisia discolor thumb|right|100px|Lycaste Cassiopeia (a cultivar) *Lacaena *Laelia Lindl. ",+4.1,35,"""-3.8""",2.00,0.3085,B
+"If $P(A)=0.4, P(B)=0.5$, and $P(A \cap B)=0.3$, find $P(B \mid A)$.","* This results in P(A \mid B) = P(A \cap B)/P(B) whenever P(B) > 0 and 0 otherwise. The conditional probability can be found by the quotient of the probability of the joint intersection of events and (P(A \cap B))—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of : :P(A \mid B) = \frac{P(A \cap B)}{P(B)}. This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the ""given"" one happening (how many times A occurs rather than not assuming B has occurred): P(A \mid B) = \frac{P(A \cap B)}{P(B)}. We denote the quantity \frac{P(A \cap B)}{P(B)} as P(A\mid B) and call it the ""conditional probability of given ."" It can be shown that :P(A_B)= \frac{P(A \cap B)}{P(B)} which meets the Kolmogorov definition of conditional probability. === Conditioning on an event of probability zero === If P(B)=0 , then according to the definition, P(A \mid B) is undefined. We have P(A\mid B)=\tfrac{P(A \cap B)}{P(B)} = \tfrac{3/36}{10/36}=\tfrac{3}{10}, as seen in the table. == Use in inference == In statistical inference, the conditional probability is an update of the probability of an event based on new information. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: P(A\mid B) = {{P(B\mid A) P(A)}\over{P(B)}}. * Without the knowledge of the occurrence of B, the information about the occurrence of A would simply be P(A) * The probability of A knowing that event B has or will have occurred, will be the probability of A \cap B relative to P(B), the probability that B has occurred. In this event, the event B can be analyzed by a conditional probability with respect to A. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = . If P(B) is not zero, then this is equivalent to the statement that :P(A\mid B) = P(A). For a value in and an event , the conditional probability is given by P(A \mid X=x) . More formally, P(A|B) is assumed to be approximately equal to P(B|A). ==Examples== ===Example 1=== Relative size Malignant Benign Total Test positive 0.8 (true positive) 9.9 (false positive) 10.7 Test negative 0.2 (false negative) 89.1 (true negative) 89.3 Total 1 99 100 In one study, physicians were asked to give the chances of malignancy with a 1% prior probability of occurring. In general, it cannot be assumed that P(A|B) ≈ P(B|A). For example, the conditional probability that someone unwell (sick) is coughing might be 75%, in which case we would have that = 5% and = 75 %. Substituting 1 and 2 into 3 to select α: :\begin{align} 1 &= \sum_{\omega \in \Omega} {P(\omega \mid B)} \\\ &= \sum_{\omega \in B} {P(\omega\mid B)} + \cancelto{0}{\sum_{\omega otin B} P(\omega\mid B)} \\\ &= \alpha \sum_{\omega \in B} {P(\omega)} \\\\[5pt] &= \alpha \cdot P(B) \\\\[5pt] \Rightarrow \alpha &= \frac{1}{P(B)} \end{align} So the new probability distribution is #\omega \in B: P(\omega\mid B) = \frac{P(\omega)}{P(B)} #\omega otin B: P(\omega\mid B) = 0 Now for a general event A, :\begin{align} P(A\mid B) &= \sum_{\omega \in A \cap B} {P(\omega \mid B)} + \cancelto{0}{\sum_{\omega \in A \cap B^c} P(\omega\mid B)} \\\ &= \sum_{\omega \in A \cap B} {\frac{P(\omega)}{P(B)}} \\\\[5pt] &= \frac{P(A \cap B)}{P(B)} \end{align} == See also == * Bayes' theorem * Bayesian epistemology * Borel–Kolmogorov paradox * Chain rule (probability) * Class membership probabilities * Conditional independence * Conditional probability distribution * Conditioning (probability) * Joint probability distribution * Monty Hall problem * Pairwise independent distribution * Posterior probability * Regular conditional probability == References == ==External links== * *Visual explanation of conditional probability Category:Mathematical fallacies Category:Statistical ratios For events in B, two conditions must be met: the probability of B is one and the relative magnitudes of the probabilities must be preserved. Similarly, if P(A) is not zero, then :P(B\mid A) = P(B) is also equivalent. Similar reasoning can be used to show that P(Ā|B) = etc. The relationship between P(A|B) and P(B|A) is given by Bayes' theorem: :\begin{align} P(B\mid A) &= \frac{P(A\mid B) P(B)}{P(A)}\\\ \Leftrightarrow \frac{P(B\mid A)}{P(A\mid B)} &= \frac{P(B)}{P(A)} \end{align} That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B). === Assuming marginal and conditional probabilities are of similar size === In general, it cannot be assumed that P(A) ≈ P(A|B). It is tempting to define the undefined probability P(A \mid X=x) using this limit, but this cannot be done in a consistent manner. That is, P(A) is the probability of A before accounting for evidence E, and P(A|E) is the probability of A after having accounted for evidence E or after having updated P(A). ", 0.01961, 7.0,"""311875200.0""",0.02828,0.75,E
+What is the number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards?,"The probability is calculated based on {52 \choose 5} = 2,598,960, the total number of 5-card combinations. The number of distinct 5-card poker hands that are possible from 7 cards is 4,824. *The Probability of drawing a given hand is calculated by dividing the number of ways of drawing the hand (Frequency) by the total number of 5-card hands (the sample space; {52 \choose 5} = 2,598,960). The total number of distinct 7-card hands is {52 \choose 7} = 133{,}784{,}560. In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands. == History == Probability and gambling have been ideas since long before the invention of poker. Perhaps surprisingly, this is fewer than the number of 5-card poker hands from 5 cards, as some 5-card hands are impossible with 7 cards (e.g. 7-high and 8-high). ===5-card lowball poker hands=== Some variants of poker, called lowball, use a low hand to determine the winning hand. The probability is calculated based on {52 \choose 7} = 133,784,560, the total number of 7-card combinations. There are 7,462 distinct poker hands. ===7-card poker hands=== In some popular variations of poker such as Texas hold 'em, the most widespread poker variant overall,https://www.casinodaniabeach.com/most-popular-types-of-poker/ a player uses the best five-card poker hand out of seven cards. The following chart enumerates the (absolute) frequency of each hand, given all combinations of five cards randomly drawn from a full deck of 52 without replacement. To this day, many gamblers still rely on the basic concepts of probability theory in order to make informed decisions while gambling. ==Frequencies== ===5-card poker hands=== In straight poker and five- card draw, where there are no hole cards, players are simply dealt five cards from a deck of 52. Hand The five cards (or less) dealt on the screen are known as a hand. ==See also== *Casino comps *Draw poker *Gambling *Gambling mathematics *Problem gambling *Video blackjack *Video Lottery Terminal ==References== ==External links== * Category:Arcade video games The Total line also needs adjusting. ===7-card lowball poker hands=== In some variants of poker a player uses the best five-card low hand selected from seven cards. Video poker is a casino game based on five-card draw poker. The Total line also needs adjusting. ==See also== * Binomial coefficient * Combination * Combinatorial game theory * Effective hand strength algorithm * Event (probability theory) * Game complexity * Gaming mathematics * Odds * Permutation * Probability * Sample space * Set theory ==References== ==External links== * Brian Alspach's mathematics and poker page * MathWorld: Poker * Poker probabilities including conditional calculations * Numerous poker probability tables * 5, 6, and 7 card poker probabilities * Hold'em poker probabilities The frequencies are calculated in a manner similar to that shown for 5-card hands,https://www.pokerstrategy.com/strategy/various-poker/texas-holdem- probabilities/ except additional complications arise due to the extra two cards in the 7-card poker hand. Note that all cards are dealt face up Fourteen Out (also known as Fourteen Off, Fourteen Puzzle, Take Fourteen, or just Fourteen) is a Patience card game played with a deck of 52 playing cards. This list of poker playing card nicknames has some nicknames for the playing cards in a 52-card deck, as used in poker. ==Poker hand nicknames== The following sets of playing cards can be referred to by the corresponding names in card games that include sets of three or more cards, particularly 3 and 5 card draw, Texas Hold 'em and Omaha Hold 'em. The number of distinct poker hands is even smaller. Eliminating identical hands that ignore relative suit values leaves 6,009,159 distinct 7-card hands. Since poker is a game of incomplete information, the calculator is designed to evaluate the equity of ranges of hands that players can hold, instead of individual hands. The table does not extend to include five-card hands with at least one pair. (Wild cards substitute for any other card in the deck in order to make a better poker hand). ",14.80,655,"""0.375""",2598960,8.44,D
+"A certain food service gives the following choices for dinner: $E_1$, soup or tomato 1.2-2 juice; $E_2$, steak or shrimp; $E_3$, French fried potatoes, mashed potatoes, or a baked potato; $E_4$, corn or peas; $E_5$, jello, tossed salad, cottage cheese, or coleslaw; $E_6$, cake, cookies, pudding, brownie, vanilla ice cream, chocolate ice cream, or orange sherbet; $E_7$, coffee, tea, milk, or punch. How many different dinner selections are possible if one of the listed choices is made for each of $E_1, E_2, \ldots$, and $E_7$ ?","The establishment of restaurants and restaurant menus allowed customers to choose from a list of unseen dishes, which were produced to order according to the customer's selection. A combination meal can also comprise a meal in which separate dishes are selected by consumers from an entire menu, and can include à la carte selections that are combined on a plate. It usually includes several dishes to pick in a fixed list: an entrée (introductory course), a main course (a choice between up to four dishes), a cheese, a dessert, bread, and sometimes beverage (wine) and coffee all for a set price fixed for the year between €15 and €55. In a restaurant, the menu is a list of food and beverages offered to customers and the prices. Combination meals may be priced lower compared to ordering items separately, but this is not always the case. A meat and three meal is one where the customer picks one meat and three side dishes as a fixed-price offering. A fast food combination meal can contain over . A combination meal is also a meal in which the consumer orders items à la carte to create their own meal combination. Other types of restaurants, such as fast-casual restaurants also offer combination meals. A 2010 study published in the Journal of Public Policy & Marketing found that some consumers may order a combination meal even if no price discount is applied compared to the price of ordering items separately. The study found that this behavior is based upon consumers perceiving an inherent value in combination meals, and also suggested that the ease and convenience of ordering, such as ordering a meal by number, plays a role compared to ordering items separately. Combination meals may be priced lower compared to ordering the items separately, and this lower pricing may serve to entice consumers that are budget-minded. This has a fixed menu and often comes with side dishes such as pickled vegetables and miso soup. * A wine list * A liquor and mixed drinks menu * A beer list * A dessert menu (which may also include a list of tea and coffee options) Some restaurants use only text in their menus. thumb|An example of foods served as a fast food combination meal thumb|A combination meal with chicken curry, rice and beef curry thumb|A Spanish combination meal, consisting of a hamburger, French fries and a beer A combination meal, often referred as a combo-meal, is a type of meal that typically includes food items and a beverage. The variation in Chinese cuisine from different regions led caterers to create a list or menu for their patrons. Fast food restaurants will often prepare variations on items already available, but to have them all on the menu would create clutter. Boston Market and Cracker Barrel chains of restaurants offer a similar style of food selection. == See also == * Garbage Plate * List of restaurant terminology == References == === Sources === * * * * * Category:Cuisine of the Southern United States Category:Restaurants by type Category:Restaurant terminology Category:Culture of Nashville, Tennessee Category:Food combinations This way, all of the patrons can see all of the choices, and the restaurant does not have to provide printed menus. Similar concepts include the Hawaiian plate lunch, which features a variety of entrée choices with fixed side items of white rice and macaroni salad, and the southern Louisiana plate lunch, which features menu options that change daily. Salad buffet, bread and butter and beverage are included, and sometimes also a simple starter, like a soup. Most commonly, there is a choice of two or three dishes: a meat/fish/poultry dish, a vegetarian alternative, and a pasta. ",1.51,49,"""4.5""",2688,22,D
+"A rocket has a built-in redundant system. In this system, if component $K_1$ fails, it is bypassed and component $K_2$ is used. If component $K_2$ fails, it is bypassed and component $K_3$ is used. (An example of a system with these kinds of components is three computer systems.) Suppose that the probability of failure of any one component is 0.15 , and assume that the failures of these components are mutually independent events. Let $A_i$ denote the event that component $K_i$ fails for $i=1,2,3$. What is the probability that the system fails?","This can occur when a single part fails, increasing the probability that other portions of the system fail. Cascading failures may occur when one part of the system fails. :P_F = 1 - A_o \begin{cases} P_F = Probability \ of \ Mission \ Failure \\\ A_o = Operational \ Availability \end{cases} Apart from human error, mission failure results from the following causes. * Protection Strategies for Cascading Grid Failures — A Shortcut Approach * I. Dobson, B. A. Carreras, and D. E. Newman, preprint A loading-dependent model of probabilistic cascading failure, Probability in the Engineering and Informational Sciences, vol. 19, no. 1, January 2005, pp. 15–32. Those failures will occasionally combine in unforeseeable ways, and if they induce further failures in an operating environment of tightly interrelated processes, the failures will spin out of control, defeating all interventions."" Redundancy is a form of resilience that ensures system availability in the event of component failure. A system accident (or normal accident) is an ""unanticipated interaction of multiple failures"" in a complex system.Perrow (1999, p. 70). Physics of failure is a technique under the practice of reliability design that leverages the knowledge and understanding of the processes and mechanisms that induce failure to predict reliability and improve product performance. This is a concept which disagrees with that of system accident. == Scott Sagan == Scott Sagan has multiple publications discussing the reliability of complex systems, especially regarding nuclear weapons. If a system has no redundancy, then MTB is in return of failure rate, \lambda. : \lambda = \frac{1}{MTB} Systems with spare parts that are energized but that lack automatic fault bypass are to accept actually results because human action is required to restore operation after every failure. Software reliability is the probability of the software causing a system failure over some specified operating time. A system accident is one that requires many things to go wrong in a cascade. Another common technique is to calculate a safety margin for the system by computer simulation of possible failures, to establish safe operating levels below which none of the calculated scenarios is predicted to cause cascading failure, and to identify the parts of the network which are most likely to cause cascading failures. Such a failure may happen in many types of systems, including power transmission, computer networking, finance, transportation systems, organisms, the human body, and ecosystems. This failure process cascades through the elements of the system like a ripple on a pond and continues until substantially all of the elements in the system are compromised and/or the system becomes functionally disconnected from the source of its load. A cascading failure is a failure in a system of interconnected parts in which the failure of one or few parts leads to the failure of other parts, growing progressively as a result of positive feedback. They are often overworked or maintenance is deferred due to budget cuts, because managers know that they system will continue to operate without fixing the backup system.Perrow (1999). == General characterization == In 2012 Charles Perrow wrote, ""A normal accident [system accident] is where everyone tries very hard to play safe, but unexpected interaction of two or more failures (because of interactive complexity), causes a cascade of failures (because of tight coupling)."" Owing to this coupling, interdependent networks are extremely sensitive to random failures, and in particular to targeted attacks, such that a failure of a small fraction of nodes in one network can trigger an iterative cascade of failures in several interdependent networks. Crucitti, V. Latora and M. Marchiori, Model for cascading failures in complex networks, Physical Review E (Rapid Communications) 69, 045104 (2004). * Data centre power generators that activate when the normal power source is unavailable. === 1+1 redundancy === 1+1 redundancy typically offers the advantage of additional failover transparency in the event of component failure. Annual, vol., no., pp.285-289, 21-23 Jan 1992 using the algorithms for prognostic purposes,NASA.gov NASA Prognostic Center of Excellence and integrating physics of failure predictions into system-level reliability calculations.http://www.dfrsolutions.com/uploads/publications/2010_01_RAMS_Paper.pdf, McLeish, J.G.; ""Enhancing MIL-HDBK-217 reliability predictions with physics of failure methods,"" Reliability and Maintainability Symposium (RAMS), 2010 Proceedings - Annual, vol., no., pp.1-6, 25-28 Jan. 2010 ==Limitations== There are some limitations with the use of physics of failure in design assessments and reliability prediction. A cascade failure can affect large groups of people and systems. ",0.15,-1.49,"""0.9966""",1,1.07,C
+"Suppose that $P(A)=0.7, P(B)=0.3$, and $P(A \cap B)=0.2$. These probabilities are 1.3-3 listed on the Venn diagram in Figure 1.3-1. Given that the outcome of the experiment belongs to $B$, what then is the probability of $A$ ? ","Each node on the diagram represents an event and is associated with the probability of that event. The region included in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by . == History == Venn diagrams were introduced in 1880 by John Venn in a paper entitled ""On the Diagrammatic and Mechanical Representation of Propositions and Reasonings"" in the Philosophical Magazine and Journal of Science, about the different ways to represent propositions by diagrams. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). The probability associated with a node is the chance of that event occurring after the parent event occurs. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. A Venn diagram uses simple closed curves drawn on a plane to represent sets. Venn diagrams normally comprise overlapping circles. Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams. __NOTOC__ thumb|Tree diagram for events A and B. Venn viewed his diagrams as a pedagogical tool, analogous to verification of physical concepts through experiment. Venn diagrams do not generally contain information on the relative or absolute sizes (cardinality) of sets. In probability theory, an outcome is a possible result of an experiment or trial. The book comes with a 3-page foldout of a seven-bit cylindrical Venn diagram.) * * * ==External links== * * Lewis Carroll's Logic Game – Venn vs. Euler at Cut-the-knot * Six sets Venn diagrams made from triangles * Interactive seven sets Venn diagram * VBVenn a free open source program for calculating and graphing quantitative two-circle Venn diagrams Category:Graphical concepts in set theory Category:Diagrams Category:Statistical charts and diagrams For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while the other circle may represent the set of all tables. Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. In 1866, Venn published The Logic of Chance, a groundbreaking book which espoused the frequency theory of probability, arguing that probability should be determined by how often something is forecast to occur as opposed to ""educated"" assumptions. In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets. thumb|The Venn Building, University of Hull thumb|alt=Plaque in the form of a Venn diagram with one set labelled 'Mathematician, Philosopher & Anglican priest', a second set labelled 'Really strong beard game' with the overlapping area labelled 'John Venn'|Alternative heritage plaque for John Venn in Hull John Venn, FRS, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. Venn did not use the term ""Venn diagram"" and referred to the concept as ""Eulerian Circles"". These diagrams were devised while designing a stained-glass window in memory of Venn. ===Other diagrams=== Edwards–Venn diagrams are topologically equivalent to diagrams devised by Branko Grünbaum, which were based around intersecting polygons with increasing numbers of sides. ",0.66666666666,200,"""8.0""",17.4,-2,A
+A coin is flipped 10 times and the sequence of heads and tails is observed. What is the number of possible 10-tuplets that result in four heads and six tails?,"thumb|upright=1.35|Coin of Tennes. The Philippine ten-centavo coin (10¢) coin is a denomination of the Philippine peso. In the 1954/55 National Hunt season Four Ten won three of his first four races including two wide-margin victories at Warwick Racecourse in December and January. Four Ten won a total of nine races between his second Gold Cup attempt and his retirement. Commenting on the horse's tendency to jump to the left, Kirkpatrick explained that Four Ten would be better suited by a left-handed track and would probably contest both the Gold Cup and the Grand National. Four Ten (1946 - 1971) was a British Thoroughbred racehorse who won the 1954 Cheltenham Gold Cup. The Hundred-dollar, Hundred-digit Challenge problems are 10 problems in numerical mathematics published in 2002 by . The horse ran six times in point-to-points winning four times consecutively before falling in a hunter chase. A $100 prize was offered to whoever produced the most accurate solutions, measured up to 10 significant digits. He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. In the early part of the following season, Four Ten further established himself as a high-class steeplechaser with an ""impressive"" win over three miles at Cheltenham in November, beating E.S.B. and Mariner's Log. The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice. A 6×4 or six-by-four is a vehicle with three axles, with a drivetrain delivering power to two wheel ends on two of them.International ProStar ES Class 8 truck: Axle configurations It is a form of four-wheel driveNACFE (North American Council for Freight Efficiency) Executive Report – 6x2 (Dead Axle) Tractors ""A typical three axle Class 8 tractor today is equipped with two rear drive axles (“live” tandem) and is commonly referred to as a 6 X 4 configuration meaning that it has four-wheel drive capability."" The name of the republic, the date and denomination are all on the obverse. ==== New Generation Currency Series ==== The BSP announced in 2017 that the ten-centavo coin would not be included in this series, and that it was dropping the coin from circulation. Mariner's Log took over the lead, but Four Ten, relishing the heavy ground, moved up to challenge at the last and drew away up the run-in to win by four lengths. In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: :P(n)=1-\sum_{x=0}^{n-1}\binom{6n}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{6n-x}\, . If r is the total number of dice selecting the 6 face, then P(r \ge k ; n, p) is the probability of having at least k correct selections when throwing exactly n dice. Strange originally used Four Ten as a hunter before training the horse himself for races on the amateur point-to-point circuit. These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). The BSP Series coin will still be used until that series is demonetized. ===Version history=== English Series (1958–1967) Pilipino Series (1969–1974) Ang Bagong Lipunan Series (1975–1983) Flora and Fauna Series (1983–1994) BSP Coin Series (1995–2017) Obverse centre|frameless|101x101px centre|frameless|101x101px centre|frameless|101x101px centre|frameless|102x102px centre|frameless|101x101px Reverse centre|frameless|100x100px centre|frameless|102x102px centre|frameless|102x102px centre|frameless|104x104px centre|frameless|101x101px ==References== Category:Currencies of the Philippines Category:Ten-cent coins The issues from 1979 to 1982 featured a mintmark underneath the 10 centavo. ==== Flora and Fauna Series ==== From 1983 to 1994, a new coin was issued with Baltazar again faced to the left in profile, and the denomination was moved to the reverse with the date on the front. Around it was the inscription 'Rey de Espana' (King of Spain) and the denomination as 10 Cs. de Po. (10 centimos of peso).http://worldcoingallery.com/countries/display.php?image=nmc2/142-148&desc;=Philippines km148 10 Centimos (1880-1885)&query;=Philippines === United States administration=== thumb|left|10 centavos issued 1907-1945 In 1903, the 10-centavo coin equivalent to was minted for the Philippines, weighing of 0.9 fine silver. ",210,0.020,"""7.0""",72,0.405,A
+"Among nine orchids for a line of orchids along one wall, three are white, four lavender, and two yellow. How many color displays are there?","Bulbophyllum concolor is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia concolor Bulbophyllum bicolor is a species of orchid in the genus Bulbophyllum. Bulbophyllum tricolor is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia tricolor Bulbophyllum bicoloratum is a species of orchid in the genus Bulbophyllum. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia *Photos of Bulbophyllum bicoloratum == External links == * * bicoloratum Category:Articles containing video clips Category:Plants described in 1924 Pabstiella tricolor is a species of orchid plant. == References == tricolor It is found only in Hong Kong and isolated parts of southeast China and northern Vietnam. ==References== *The Bulbophyllum-Checklist *The Internet Orchid Species Photo Encyclopedia bicolor Category:Plants described in 1830 7 Colors (a.k.a. Filler) is a puzzle game, designed by Dmitry Pashkov. The game was published by Infogrames for MS-DOS, Amiga, and NEC PC-9801. ==Reception== ==References== ==External links== * Category:1991 video games Category:Amiga games Category:DOS games Category:Hot B games Category:Infogrames games Category:Multiplayer and single-player video games Category:NEC PC-9801 games Category:Puzzle video games Category:Video games developed in Russia It was developed by the Russian company in 1991. Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor Pabstiella tricolor is a species of orchid plant. == References == tricolor ",1.7,3.42,"""1260.0""",0.84,0.333333333333333,C
+"A survey was taken of a group's viewing habits of sporting events on TV during I.I-5 the last year. Let $A=\{$ watched football $\}, B=\{$ watched basketball $\}, C=\{$ watched baseball $\}$. The results indicate that if a person is selected at random from the surveyed group, then $P(A)=0.43, P(B)=0.40, P(C)=0.32, P(A \cap B)=0.29$, $P(A \cap C)=0.22, P(B \cap C)=0.20$, and $P(A \cap B \cap C)=0.15$. Find $P(A \cup B \cup C)$.","The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. One may resolve this overlap by the principle of inclusion- exclusion, or, in this case, by simply finding the probability of the complementary event and subtracting it from 1, thus: : Pr(at least one ""1"") = 1 − Pr(no ""1""s) := 1 − Pr([no ""1"" on 1st trial] and [no ""1"" on 2nd trial] and ... and [no ""1"" on 8th trial]) := 1 − Pr(no ""1"" on 1st trial) × Pr(no ""1"" on 2nd trial) × ... × Pr(no ""1"" on 8th trial) := 1 −(5/6) × (5/6) × ... × (5/6) := 1 − (5/6)8 := 0.7674... ==See also== *Logical complement *Exclusive disjunction *Binomial probability ==References== ==External links== *Complementary events - (free) page from probability book of McGraw-Hill Category:Experiment (probability theory) That is, for an event A, :P(A^c) = 1 - P(A). In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. Therefore, the probability of an event's complement must be unity minus the probability of the event. It may be tempting to say that : Pr([""1"" on 1st trial] or [""1"" on second trial] or ... or [""1"" on 8th trial]) := Pr(""1"" on 1st trial) + Pr(""1"" on second trial) + ... + P(""1"" on 8th trial) := 1/6 + 1/6 + ... + 1/6 := 8/6 := 1.3333... For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. Equivalently, the probabilities of an event and its complement must always total to 1. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. By contrast, in the example above the law of total probability applies, since the event X = 0.5 is included into a family of events X = x where x runs over (−1,1), and these events are a partition of the probability space. On the other hand, conditioning on an event B is well-defined, provided that \mathbb{P}(B) eq 0, irrespective of any partition that may contain B as one of several parts. ===Conditional distribution=== Given X = x, the conditional distribution of Y is : \mathbb{P} ( Y=y | X=x ) = \frac{ \binom 3 y \binom 7 {x-y} }{ \binom{10}x } = \frac{ \binom x y \binom{10-x}{3-y} }{ \binom{10}3 } for 0 ≤ y ≤ min ( 3, x ). The probability (with respect to some probability measure) that an event S occurs is the probability that S contains the outcome x of an experiment (that is, it is the probability that x \in S). The 2010 Women's Junior World Handball Championship (17th tournament) took place in South Korea from July 17 to July 31. ==Preliminary round== ===Group A=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group B=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group C=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ===Group D=== \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- \---- ==Main round== ===Group I=== \---- \---- \---- \---- \---- \---- \---- \---- ===Group II=== \---- \---- \---- \---- \---- \---- \---- \---- ==President's Cup== ===21st–24th=== \---- ====23rd/24th==== ====21st/22nd==== ===17th–20th=== \---- ====19th/20th==== ====17th/18th==== ===13th–16th=== \---- ====15th/16th==== ====13th/14th==== ==Placement matches== ===11th/12th=== ===9th/10th=== ===7th/8th=== ===5th/6th=== ==Final round== ===Semifinals=== \---- ===Bronze medal match=== ===Gold medal match=== ==Ranking and statistics== ===Final ranking=== Rank Team Image:Gold medal icon.svg Image:Silver medal icon.svg Image:Bronze medal icon.svg 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2010 Junior Women's World Champions 150px|border|Norway Norway First title ;Team roster Silje Solberg, Christine Homme, Veronica Kristiansen, Hanna Yttereng, Mai Marcussen, Stine Bredal Oftedal, Mari Molid, Maja Jakobsen, Sanna Solberg, Nora Mørk, Guro Rundbråten, Silje Katrine Svendsen, Ellen Marie Folkvord, Hilde Kamperud, Kristin Nørstebø, Susann Iren Hall. ===All Star Team=== *Goalkeeper: *Left wing: *Left back: *Pivot: *Centre back: *Right back: *Right wing: Chosen by team officials and IHF experts: IHF.info ===Other awards=== *Most Valuable Player: *Top Goalscorer: 75 goals ===Top goalkeepers=== Rank Name Team Saves Shots % 1 Jessica Oliveira 127 325 39.1% 2 Marta Žderić 122 314 38.9% 3 Nele Kurzke 93 285 32.6% 4 Marina Vukčević 85 220 38.6% 5 Guro Rundbråten 82 198 41.4% 6 Shuk Yee Wong 80 301 26.6% 7 Marija Colic 75 213 35.2% 8 Preeyanut Bureeruk 73 266 27.4% 9 Elena Fomina 71 194 36.6% 9 Sori Park 71 260 27.3% Source: ihf.info ===Top goalscorers=== Rank Name Team Goals Shots % 1 Nathalie Hagman 75 96 78.1% 2 Laura van den Heijden 72 109 66.1% 3 Ryu Eun-hee 63 109 57.8% 4 Milena Knežević 62 124 50.0% 5 Jelena Živković 59 103 57.3% 6 Lee Eun-bi 58 87 66.7% 7 Tatiana Khmyrova 55 80 68.8% 7 Luciana Mendoza 55 87 63.2% 9 Marta Lopez 52 91 57.1% 10 Ana Martinez 49 92 53.3% Source: ihf.info ==References== Tournament Summary ==External links== *XVII Women's Junior World Championship at IHF.info Category:International handball competitions hosted by South Korea Women's Junior World Handball Championship, 2010 2010 Category:Women's handball in South Korea Junior World Handball Championship Category:July 2010 sports events in South Korea Other events are proper subsets of the sample space that contain multiple elements. In the latter two examples the law of total probability is irrelevant, since only a single event (the condition) is given. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? The expectation of this random variable is equal to the (unconditional) probability, E ( P ( Y ≤ 1/3 | X ) ) = P ( Y ≤ 1/3 ), namely, : 1 \cdot \mathbb{P} (X<0.5) + 0 \cdot \mathbb{P} (X=0.5) + \frac13 \cdot \mathbb{P} (X>0.5) = 1 \cdot \frac16 + 0 \cdot \frac13 + \frac13 \cdot \left( \frac16 + \frac13 \right) = \frac13, which is an instance of the law of total probability E ( P ( A | X ) ) = P ( A ). American football win probability estimates often include whether a team is home or away, the down and distance, score difference, time remaining, and field position. Win probability added is the change in win probability, often how a play or team member affected the probable outcome of the game. ==Current research== Current research work involves measuring the accuracy of win probability estimates, as well as quantifying the uncertainty in individual estimates. ",0.65625,0.75,"""0.0""",0.59,164,D
+"A grade school boy has five blue and four white marbles in his left pocket and four blue and five white marbles in his right pocket. If he transfers one marble at random from his left to his right pocket, what is the probability of his then drawing a blue marble from his right pocket?","thumb|right|Four bags with three marbles per bag gives twelve marbles (4 × 3 = 12). thumb|Animation for the multiplication 2 × 3 = 6. thumb|right|4 × 5 = 20. At the end of each puzzle, the marbles that have been guided into their proper bins are returned to the player. Thus, likelihood of landing on a particular space in Blue Marble Game can't be exactly calculated like in Monopoly, as players can choose to go to different spaces on the board based on current game conditions. The aim is to ensure that each marble arrives in the bin of the same color as the marble. For example, four bags with three marbles each can be thought of as: :[4 bags] × [3 marbles per bag] = 12 marbles. Marble Drop is a puzzle video game published by Maxis on February 28, 1997. == Gameplay == Players are given an initial set of marbles that are divided evenly into six colors: red, orange, yellow, green, blue, and purple, with two more colors available to purchase: black and silver (steel). Steel (silver) balls are 20 percent of the price of colored marbles and can be used as test marbles or to help release a catch instead of using a valuable colored marble; additionally, there are steel-coloured exit bins in the final puzzle. Players must determine how the marble will travel through the puzzle, and how its journey will change the puzzle for the next marble. It can be played by 2 to 4 players.Blue Marble Game instruction booklet ==Gameplay== Players move around the board in order to buy property, build buildings on the properties, pay rent to other players, and earn a salary. Blue Marble Game (부루마불게임) is a Korean board game similar to Monopoly manufactured by Si-Yat-Sa. Black marbles are very expensive, but change to the correct color when they arrive in a bin. ==Reception== Marble Drop received lukewarm reception upon release. While Monopoly is traditionally played across locations in a single city, the Blue Marble Game features cities from across the world; its title is a reference to The Blue Marble photograph taken by the crew of Apollo 17, and its description of the Earth as seen from space. Lost marbles must be purchased when they are needed to complete a puzzle. There is no mortgage system in the Blue Marble Game. ===Statistics=== In Blue Marble Game, there is no space that sends the player to the deserted island other than the deserted island space itself. More marble has been extracted from the over 650 quarry sites near Carrara than from any other place. Marble died of cancer on September 11, 2015 at the age of 48.Roy Marble, Iowa's scoring leader, dies at 48 Marble's son, Devyn, followed in his father's footsteps to Iowa and the NBA. Devyn and his father were the first father-son duo in Big Ten history to each score 1,000 points.Iowa's Devyn Marble joins father Roy in scoring 1,000th point for Hawkeye Marble came into the news again in 2021 when his family expressed displeasure at the retirement of Luka Garza's jersey number (announced after the last game of the season on March 7), noting that they felt hurt and disrespected by the move upon the fact that Marble's number was not retired; Marble, alongside Murray Wier and Chuck Darling, are considered the best players to not have their jersey number retired by Iowa. The layers of marble are interbedded with schists and quartzites. These marbles are picked up and dropped by the players into funnels leading to a series of rails, switches, traps and other devices which grow more complex as the game progresses. Some of them, such as the Maffioli, who rented some quarries north of Carrara, in the Torano area, or, around 1490, Giovanni Pietro Buffa, who bought marble on credit from local quarrymen and then resold it on the Venetian market, were able to create a dense commercial network, exporting the marble even to distant locations. thumb|264x264px|The distinct green colour of the middle slab is a result of an abundance of serpentine minerals|alt=An image of some slabs of connemara marble against a wall. thumb|Sample sheets, 2016 Carrara marble, Luna marble to the Romans, is a type of white or blue-grey marble popular for use in sculpture and building decor. ",+10, 35.91,"""0.444444444444444""",-1270,2.3,C
+"A faculty leader was meeting two students in Paris, one arriving by train from Amsterdam and the other arriving by train from Brussels at approximately the same time. Let $A$ and $B$ be the events that the respective trains are on time. Suppose we know from past experience that $P(A)=0.93, P(B)=0.89$, and $P(A \cap B)=0.87$. Find $P(A \cup B)$.","The probability of one or both events occurring is denoted P(A ∪ B) and in general, it equals P(A) + P(B) – P(A ∩ B). Generally, there is only one event B such that A and B are both mutually exclusive and exhaustive; that event is the complement of A. When A and B are mutually exclusive, .Stats: Probability Rules. Cengage Learning 2007, , p. 229 () The event A and its complement [not A] are mutually exclusive and exhaustive. For two events to be complements, they must be collectively exhaustive, together filling the entire sample space. In consequence, mutually exclusive events have the property: P(A ∩ B) = 0.intmath.com; Mutually Exclusive Events. On 21 April 2012 at 18:30 local time (16:30 UTC), two trains were involved in a head-on collision at Westerpark, near Sloterdijk, in the west of Amsterdam, Netherlands. An event defines a complementary event, namely the complementary set (the event occurring), and together these define a Bernoulli trial: did the event occur or not? Therefore, the probability of an event's complement must be unity minus the probability of the event. The probability that at least one of the events will occur is equal to one.Scott Bierman. In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur.Robert R. Johnson, Patricia J. Kuby: Elementary Statistics. Formally said, the intersection of each two of them is empty (the null event): A ∩ B = ∅. In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. It is estimated that at the moment of the collision the intercity was travelling at and the local train at about . The probabilities of the individual events (red, and club) are multiplied rather than added. The local train was travelling between Amsterdam and whilst the Intercity train was travelling between and . In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. The technique is wrong because the eight events whose probabilities got added are not mutually exclusive. That is, for an event A, :P(A^c) = 1 - P(A). Therefore, two mutually exclusive events cannot both occur. Given an event, the event and its complementary event define a Bernoulli trial: did the event occur or not? In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time). ",37.9,0.95,"""-3.141592""",6.3,-167,B
+"What is the number of possible four-letter code words, selecting from the 26 letters in the alphabet?","The Code 39 specification defines 43 characters, consisting of uppercase letters (A through Z), numeric digits (0 through 9) and a number of special characters (-, ., $, /, +, %, and space). The alphabet for Modern English is a Latin-script alphabet consisting of 26 letters, each having an upper- and lower-case form. The English alphabet has 5 vowels, 19 consonants, and 2 letters (Y and W) that can function as consonants or vowels. 26 (twenty-six) is the natural number following 25 and preceding 27. == In mathematics == thumb|Poster designed to depict the speciality of the number 26 *26 is the only integer that is one greater than a square (5 + 1) and one less than a cube (3 − 1). *26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise connections. A FourCC (""four-character code"") is a sequence of four bytes (typically ASCII) used to uniquely identify data formats. This code is traditionally mapped to the * character in barcode fonts and will often appear with the human-readable representation alongside the barcode. thumb|Code 39 Characters As a generality, the location of the two wide bars can be considered to encode a number between 1 and 10, and the location of the wide space (which has four possible positions) can be considered to classify the character into one of four groups (from left to right): Letters(+30) (U–Z), Digits(+0) (1–9,0), Letters(+10) (A–J), and Letters(+20) (K–T). Lower case letters, additional punctuation characters and control characters are represented by sequences of two characters of Code 39. Microsoft and Windows developers refer to their four-byte identifiers as FourCCs or Four-Character Codes. 262px|right|thumb|*WIKIPEDIA* encoded in Code 39 Code 39 (also known as Alpha39, Code 3 of 9, Code 3/9, Type 39, USS Code 39, or USD-3) is a variable length, discrete barcode symbology defined in ISO/IEC 16388:2007. Code Details Nr Character Encoding Nr Character Encoding Nr Character Encoding Nr Character Encoding 0 NUL %U 32 [space] [space] 64 @ %V 96 ` %W 1 SOH $A 33 ! * GOD=26=G7+O15+D4 in Simple6,74 English7,74 Gematria8,74 ('The Key': A=1, B2, C3, ..., Z26). Because there are only six letters in the Letters(+30) group (letters 30–35, or U–Z), the other four positions in this group (36–39) are used to represent three symbols (dash, period, space) as well as the start/stop character. However, Code 39 is still used by some postal services (although the Universal Postal Union recommends using Code 128 in all casesAs one example of an international standard, see ), and can be decoded with virtually any barcode reader. Depending on the way one counts, the West Frisian alphabet contains between 25 and 32 characters. ==Letters== West Frisian alphabet Upper case vowels and vowels with diacritics A Â E Ê É I/Y O Ô U Û Ú Lower case vowels and vowels with diacritics a â e ê é i/y o ô u û ú Vowel Pronunciation / Upper case letters B C D F G H J K L M N P Q R S T V W X Z Lower case letters b c d f g h j k l m n p q r s t v w x z Letter Pronunciation ==Alphabetical order== In alphabetical listings both I and Y are usually found between H and J. *The number of the last letter of the English alphabet, Z. In base ten, 26 is the smallest positive integer that is not a palindrome to have a square (262 = 676) that is a palindrome. == In science == *The atomic number of iron. This table outlines the Code 39 specification. Setting aside one of these characters as a start and stop pattern left 39 characters, which was the origin of the name Code 39. The two wide bars, out of five possible positions, encode a number between 1 and 10 using a two-out-of-five code with the following numeric equivalence: 1, 2, 4, 7, 0. RealMedia files also use four-character codes, however, the actual codes used differ from those found in AVI or QuickTime files. All these fields are four-character codes known as OSType. *In a normal deck of cards, there are 26 red cards and 26 black cards. ",358800,3.2,"""1068.0""",0.0384,0.3333333,A
+"The change in molar internal energy when $\mathrm{CaCO}_3(\mathrm{~s})$ as calcite converts to another form, aragonite, is $+0.21 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the difference between the molar enthalpy and internal energy changes when the pressure is 1.0 bar given that the densities of the polymorphs are $2.71 \mathrm{~g} \mathrm{~cm}^{-3}$ and $2.93 \mathrm{~g} \mathrm{~cm}^{-3}$, respectively.","However, the sea level, temperature, and calcium carbonate saturation state of the surrounding system also determine which polymorph of calcium carbonate (aragonite, low-magnesium calcite, high-magnesium calcite) will form. Aragonite will change to calcite over timescales of days or less at temperatures exceeding 300 °C, and vaterite is even less stable. ==Etymology== Calcite is derived from the German , a term from the 19th century that came from the Latin word for lime, (genitive ) with the suffix -ite used to name minerals. thumb|235px|Crystal structure of calcite Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate (CaCO3). Calcite, obtained from an 80 kg sample of Carrara marble, is used as the IAEA-603 isotopic standard in mass spectrometry for the calibration of δ18O and δ13C. Calcite defines hardness 3 on the Mohs scale of mineral hardness, based on scratch hardness comparison. The molecular formula SrCO3 (molar mass: 147.63 g/mol, exact mass: 147.8904 u) may refer to: * Strontianite * Strontium carbonate Other polymorphs of calcium carbonate are the minerals aragonite and vaterite. Twinning, cleavage and crystal forms are often given in morphological units. ==Properties == The diagnostic properties of calcite include a defining Mohs hardness of 3, a specific gravity of 2.71 and, in crystalline varieties, a vitreous luster. An aragonite sea contains aragonite and high-magnesium calcite as the primary inorganic calcium carbonate precipitates. The molecular formula C3H8O10P2 (molar mass: 266.035 g/mol) may refer to: * 1,3-Bisphosphoglyceric acid (1,3-BPG) * 2,3-Bisphosphoglyceric acid (2,3-BPG) Category:Molecular formulas The molecular formula C3H7O7P (molar mass: 186.06 g/mol, exact mass: 185.9929 u) may refer to: * 2-Phosphoglyceric acid, or 2-phosphoglycerate * 3-Phosphoglyceric acid The molecular formula C12H15N2O3PS (molar mass: 298.30 g/mol, exact mass: 298.0541 u) may refer to: * Phoxim * Quinalphos José María Patoni, San Juan del Río, Durango (Mexico) ==See also== *Carbonate rock *Ikaite, CaCO3·6H2O *List of minerals *Lysocline *Manganoan calcite, (Ca,Mn)CO3 *Monohydrocalcite, CaCO3·H2O *Nitratine *Ocean acidification *Ulexite ==References== ==Further reading== * Category:Calcium minerals Category:Carbonate minerals Category:Limestone Category:Optical materials Category:Transparent materials Category:Calcite group Category:Cave minerals Category:Trigonal minerals Category:Minerals in space group 167 Category:Evaporite Category:Luminescent minerals Category:Polymorphism (materials science) Category:Bastet Calcite is also more soluble at higher pressures. However, crystallization of calcite has been observed to be dependent on the starting pH and concentration of magnesium in solution. Calcite in limestone is divided into low-magnesium and high-magnesium calcite, with the dividing line placed at a composition of 4% magnesium. Due to its acidity, carbon dioxide has a slight solubilizing effect on calcite. These processes can be traced by the specific carbon isotope composition of the calcites, which are extremely depleted in the 13C isotope, by as much as −125 per mil PDB (δ13C). ==In Earth history== Calcite seas existed in Earth's history when the primary inorganic precipitate of calcium carbonate in marine waters was low-magnesium calcite (lmc), as opposed to the aragonite and high- magnesium calcite (hmc) precipitated today. The chemical conditions of the seawater must be notably high in magnesium content relative to calcium (high Mg/Ca ratio) for an aragonite sea to form. As ocean acidification causes pH to drop, carbonate ion concentrations will decline, potentially reducing natural calcite production. ==Gallery== File:Calcite-Mottramite-cktsu-45b.jpg|Calcite with mottramite File:Erbenochile eye.JPG|Trilobite eyes employed calcite File:CalciteEchinosphaerites.jpg|Calcite crystals inside a test of the cystoid Echinosphaerites aurantium (Middle Ordovician, northeastern Estonia) File:Calcite-Dolomite-Gypsum-159389.jpg|Rhombohedrons of calcite that appear almost as books of petals, piled up 3-dimensionally on the matrix File:Calcite-Hematite-Chalcopyrite-176263.jpg|Calcite crystal canted at an angle, with little balls of hematite and crystals of chalcopyrite both on its surface and included just inside the surface of the crystal File:GeopetalCarboniferousNV.jpg|Thin section of calcite crystals inside a recrystallized bivalve shell in a biopelsparite File:OoidSurface01.jpg|Grainstone with calcite ooids and sparry calcite cement; Carmel Formation, Middle Jurassic, of southern Utah, USA. File:Calcite-Aragonite-Sulphur-69380.jpg|Several well formed milky white casts, made up of many small sharp calcite crystals, from the sulfur mines at Agrigento, Sicily File:Calcite-tch21c.jpg|Reddish rhombohedral calcite crystals from China. Likewise, the occurrence of calcite seas is controlled by the same suite of factors controlling aragonite seas, with the most obvious being a low seawater Mg/Ca ratio (Mg/Ca < 2), which occurs during intervals of rapid seafloor spreading. ",-0.28,+2.35,"""1.16""",83.81,28,A
+Estimate the molar volume of $\mathrm{CO}_2$ at $500 \mathrm{~K}$ and 100 atm by treating it as a van der Waals gas.,"Hence the molar van der Waals volume, which only counts the volume occupied by the atoms or molecules, is usually about times smaller than the molar volume for a gas at standard temperature and pressure. ==Table of van der Waals radii== == Methods of determination == Van der Waals radii may be determined from the mechanical properties of gases (the original method), from the critical point, from measurements of atomic spacing between pairs of unbonded atoms in crystals or from measurements of electrical or optical properties (the polarizability and the molar refractivity). The complete Van der Waals equation is therefore: :\left(P+a\frac1{V_m^2}\right)(V_m-b)=R T For n moles of gas, it can also be written as: :\left(P+a \frac{n^2}{V^2}\right)(V-n b)=n R T When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the Van der Waals equation reduces to the ideal gas law, PVm=RT. The van der Waals volume is given by V_{\rm w} = \frac{\pi V_{\rm m}}{N_{\rm A}\sqrt{18}} where the factor of π/√18 arises from the packing of spheres: V = = 23.0 Å, corresponding to a van der Waals radius r = 1.76 Å. === Molar refractivity === The molar refractivity of a gas is related to its refractive index by the Lorentz–Lorenz equation: A = \frac{R T (n^2 - 1)}{3p} The refractive index of helium n = at 0 °C and 101.325 kPa,Kaye & Laby Tables, Refractive index of gases. which corresponds to a molar refractivity A = . The van der Waals constant b volume can be used to calculate the van der Waals volume of an atom or molecule with experimental data derived from measurements on gases. Gas d (Å) b (cmmol) V (Å) r (Å) Hydrogen 0.74611 26.61 44.19 2.02 Nitrogen 1.0975 39.13 64.98 2.25 Oxygen 1.208 31.83 52.86 2.06 Chlorine 1.988 56.22 93.36 2.39 van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. The van der Waals volume of an atom or molecule may also be determined by experimental measurements on gases, notably from the van der Waals constant b, the polarizability α, or the molar refractivity A. The van der Waals volume may be calculated if the van der Waals radii (and, for molecules, the inter-atomic distances, and angles) are known. D-166. b = 23.7 cm/mol. Helium is a monatomic gas, and each mole of helium contains atoms (the Avogadro constant, N): V_{\rm w} = {b\over{N_{\rm A}}} Therefore, the van der Waals volume of a single atom V = 39.36 Å, which corresponds to r = 2.11 Å (≈ 200 picometers). Indeed, there is no reason to assume that the van der Waals radius is a fixed property of the atom in all circumstances: rather, it tends to vary with the particular chemical environment of the atom in any given case. === Van der Waals equation of state === The van der Waals equation of state is the simplest and best-known modification of the ideal gas law to account for the behaviour of real gases: \left (p + a\left (\frac{n}{\tilde{V}}\right )^2\right ) (\tilde{V} - nb) = nRT, where is pressure, is the number of moles of the gas in question and and depend on the particular gas, \tilde{V} is the volume, is the specific gas constant on a unit mole basis and the absolute temperature; is a correction for intermolecular forces and corrects for finite atomic or molecular sizes; the value of equals the van der Waals volume per mole of the gas. The following table lists the Van der Waals constants (from the Van der Waals equation) for a number of common gases and volatile liquids. To find the van der Waals volume of a single atom or molecule, it is necessary to divide by the Avogadro constant N. These various methods give values for the van der Waals radius which are similar (1–2 Å, 100–200 pm) but not identical. The molar van der Waals volume should not be confused with the molar volume of the substance. An isotherm of the Van der Waals fluid taken at T r = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. == Other parameters, forms and applications == ===Other thermodynamic parameters=== We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nNA is the number of particles in the system. The Van der Waals equation includes intermolecular interaction by adding to the observed pressure P in the equation of state a term of the form a /V_m^2, where a is a constant whose value depends on the gas. The Van der Waals equation may be solved for VG and VL as functions of the temperature and the vapor pressure pV. The density of solid helium at 1.1 K and 66 atm is , corresponding to a molar volume V = . However, near the phase transitions between gas and liquid, in the range of p, V, and T where the liquid phase and the gas phase are in equilibrium, the Van der Waals equation fails to accurately model observed experimental behavior. The excluded volume b is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. Values of d and b from Weast (1981). van der Waals radii r in Å (or in 100 picometers) calculated from the van der Waals constants of some diatomic gases. ",432.07,7.82,"""0.66666666666""",0.366,-0.347,D
+Suppose the concentration of a solute decays exponentially along the length of a container. Calculate the thermodynamic force on the solute at $25^{\circ} \mathrm{C}$ given that the concentration falls to half its value in $10 \mathrm{~cm}$.,"A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. In terms of separate decay constants, the total half-life T _{1/2} can be shown to be :T_{1/2} = \frac{\ln 2}{\lambda _c} = \frac{\ln 2}{\lambda _1 + \lambda _2}. The exponential time-constant for the process is \tau = R \, C , so the half-life is R \, C \, \ln(2) . This plot shows decay for decay constant () of 25, 5, 1, 1/5, and 1/25 for from 0 to 5. The enthalpy change in this process, normalized by the mole number of solute, is evaluated as the molar integral heat of dissolution. The integral heat of dissolution is defined for a process of obtaining a certain amount of solution with a final concentration. For example, polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days. == Solution of the differential equation == The equation that describes exponential decay is :\frac{dN}{dt} = -\lambda N or, by rearranging (applying the technique called separation of variables), :\frac{dN}{N} = -\lambda dt. The solution to this equation (see derivation below) is: :N(t) = N_0 e^{-\lambda t}, where is the quantity at time , is the initial quantity, that is, the quantity at time . == Measuring rates of decay== === Mean lifetime === If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set. The solution to this equation is given in the previous section, where the sum of \lambda _1 + \lambda _2\, is treated as a new total decay constant \lambda _c. :N(t) = N_0 e^{-(\lambda _1 + \lambda _2) t} = N_0 e^{-(\lambda _c) t}. Concentration of X in solvent A/concentration of X in solvent B=Kď If C1 denotes the concentration of solute X in solvent A & C2 denotes the concentration of solute X in solvent B; Nernst's distribution law can be expressed as C1/C2 = Kd. The molar differential enthalpy change of dissolution is: :\Delta_\text{diss}^{d} H= \left(\frac{\partial \Delta_\text{diss} H}{\partial \Delta n_i}\right)_{T,p,n_B} where is the infinitesimal variation or differential of mole number of the solute during dissolution. For a non-ideal solution it is an excess molar quantity. ==Energetics== Dissolution by most gases is exothermic. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac{dN}{dt} = -\lambda N. In thermochemistry, the enthalpy of solution (heat of solution or enthalpy of solvation) is the enthalpy change associated with the dissolution of a substance in a solvent at constant pressure resulting in infinite dilution. The value of the enthalpy of solvation is the sum of these individual steps. The half-life can be written in terms of the decay constant, or the mean lifetime, as: :t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2). The temperature of the solution eventually decreases to match that of the surroundings. The dilution between two concentrations of the solute is associated to an intermediary heat of dilution by mole of solute. ==Dilution and Dissolution== The process of dissolution and the process of dilution are closely related to each other. In thermochemistry, the heat of dilution, or enthalpy of dilution, refers to the enthalpy change associated with the dilution process of a component in a solution at a constant pressure. The enthalpy change in this process, normalized by the mole number of solute, is evaluated as the molar integral heat of dilution. Thus, the amount of material left is 2−1 = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Reactions whose rate depends only on the concentration of one reactant (known as first-order reactions) consequently follow exponential decay. ",200,17,"""0.15""",0.18162,0.14,B
 "A container is divided into two equal compartments (Fig. 5.8). One contains $3.0 \mathrm{~mol} \mathrm{H}_2(\mathrm{~g})$ at $25^{\circ} \mathrm{C}$; the other contains $1.0 \mathrm{~mol} \mathrm{~N}_2(\mathrm{~g})$ at $25^{\circ} \mathrm{C}$. Calculate the Gibbs energy of mixing when the partition is removed. Assume perfect behaviour.
-","Upon removal of the dividing partition, they expand into a final common volume (the sum of the two initial volumes), and the entropy of mixing \Delta S_{mix} is given by :\Delta S_{mix} = -nR(x_1\ln x_1 + x_2\ln x_2)\,. where R is the gas constant, n the total number of moles and x_i the mole fraction of component i\,, which initially occupies volume V_i = x_iV\,. In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the thermodynamic operation of removal of impermeable partition(s) between them, followed by a time for establishment of a new thermodynamic state of internal equilibrium in the new unpartitioned closed system. Again, the same equations for the entropy of mixing apply, but only for homogeneous, uniform phases. ==Mixing under other constraints== ===Mixing with and without change of available volume=== In the established customary usage, expressed in the lead section of this article, the entropy of mixing comes from two mechanisms, the intermingling and possible interactions of the distinct molecular species, and the change in the volume available for each molecular species, or the change in concentration of each molecular species. This is not true for the corresponding Gibbs free energies however. == Ideal and regular mixtures == An ideal mixture is any in which the arithmetic mean (with respect to mole fraction) of the two pure substances is the same as that of the final mixture. We again obtain the entropy of mixing on multiplying by the Boltzmann constant k_\text{B}. This explains why enthalpy of mixing is typically experimentally determined. ==Relation to the Gibbs free energy of mixing== The excess Gibbs free energy of mixing can be related to the enthalpy of mixing by the ușe of the Gibbs-Helmholtz equation: :\left( \frac{\partial ( \Delta G^E/T ) } {\partial T} \right)_p = - \frac {\Delta H^E} {T^2} = - \frac {\Delta H_{mix}} {T^2} or equivalently :\left( \frac{\partial ( \Delta G^E/T ) } {\partial (1/T)} \right)_p = \Delta H^E = \Delta H_{mix} In these equations, the excess and total enthalpies of mixing are equal because the ideal enthalpy of mixing is zero. We can regard the mixing process as allowing the contents of the two originally separate contents to expand into the combined volume of the two conjoined containers. For an ideal gas mixture or an ideal solution, there is no enthalpy of mixing (\Delta H_\text{mix} \,), so that the Gibbs free energy of mixing is given by the entropy term only: :\Delta G_\text{mix} = - T\Delta S_\text{mix} For an ideal solution, the Gibbs free energy of mixing is always negative, meaning that mixing of ideal solutions is always spontaneous. The entropy of mixing is entirely accounted for by the diffusive expansion of each material into a final volume not initially accessible to it. Introduction to Modern Thermodynamics, Wiley, Chichester, , pages 197-199. ==See also== * CALPHAD * Enthalpy of mixing * Gibbs energy ==Notes== ==References== == External links == * Online lecture Category:Statistical mechanics Category:Thermodynamic entropy See also: Gibbs paradox, in which it would seem that ""mixing"" two samples of the same gas would produce entropy. ===Application to solutions=== If the solute is a crystalline solid, the argument is much the same. A version of Statistical Mechanics for Students of Physics and Chemistry, Cambridge University Press, Cambridge UK, pages 163-164 the conflating of the just-mentioned two mechanisms for the entropy of mixing is well established in customary terminology, but can be confusing unless it is borne in mind that the independent variables are the common initial and final temperature and total pressure; if the respective partial pressures or the total volume are chosen as independent variables instead of the total pressure, the description is different. ====Mixing with each gas kept at constant partial volume, with changing total volume==== In contrast to the established customary usage, ""mixing"" might be conducted reversibly at constant volume for each of two fixed masses of gases of equal volume, being mixed by gradually merging their initially separate volumes by use of two ideal semipermeable membranes, each permeable only to one of the respective gases, so that the respective volumes available to each gas remain constant during the merge. This quantity combines two physical effects—the enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing considered here. Therefore, \Delta S_\text{mix} = -\left(\frac{\partial \Delta G_\text{mix}}{\partial T}\right)_P is negative for mixing of these two equilibrium phases. In the general case of mixing non-ideal materials, however, the total final common volume may be different from the sum of the separate initial volumes, and there may occur transfer of work or heat, to or from the surroundings; also there may be a departure of the entropy of mixing from that of the corresponding ideal case. In ideal mixtures, the enthalpy of mixing is null. Enthalpy of mixing can often be ignored in calculations for mixtures where other heat terms exist, or in cases where the mixture is ideal. Among other important thermodynamic simplifications, this means that enthalpy of mixing is zero: \Delta H_{mix,ideal}=0. Principles and Applications.(MacMillan 1969) p.263 For binary mixtures the entropy of random mixing can be considered as a function of the mole fraction of one component. The lowest value is when the mole fraction is 0.5 for a mixture of two components, or 1/n for a mixture of n components. thumb|The entropy of mixing for an ideal solution of two species is maximized when the mole fraction of each species is 0.5. ==Solutions and temperature dependence of miscibility== ===Ideal and regular solutions=== The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or solid) solutions—those formed by completely random mixing so that the components move independently in the total volume. In non-ideal mixtures, the thermodynamic activity of each component is different from its concentration by multiplying with the activity coefficient. In this case, the increase in entropy is entirely due to the irreversible processes of expansion of the two gases, and involves no heat or work flow between the system and its surroundings. ===Gibbs free energy of mixing=== The Gibbs free energy change \Delta G_\text{mix} = \Delta H_\text{mix} - T\Delta S_\text{mix} determines whether mixing at constant (absolute) temperature T and pressure p is a spontaneous process. ",362880,-6.9,0.7854,-1368,-1.78,B
-"What is the mean speed, $\bar{c}$, of $\mathrm{N}_2$ molecules in air at $25^{\circ} \mathrm{C}$ ?","CO-0.40-0.22 is a high velocity compact gas cloud near the centre of the Milky Way. The gas is moving away from Earth at speeds ranging from 20 to 120 km/s. The name followed the precedent set by CO-0.02-0.02, which is another high velocity compact cloud in the central molecular zone. The molecular formula C20H24N2O (molar mass: 308.42 g/mol, exact mass: 308.1889 u) may refer to: * Affinisine * Indecainide Category:Molecular formulas The differences in the velocity, termed velocity dispersion, of the gas is unusually high at 100 km/s. Rotational correlation time (\tau_c) is the average time it takes for a molecule to rotate one radian. The Honda CM250 is a parallel twin cylinder air-cooled OHC four-stroke cruiser motorcycle produced by the Honda corporation from 1981–1983 with a top speed of 85 mph and delivering 70mpg.Secondhand Buying Guide: Honda CM125C/200T/250T (Motor Cycle News 9/01/91) The 234cc North American market variant was coded as the CM250C and was the precursor to the current Honda CMX250C, also known as the Honda Rebel 250. It is 200 light years away from the centre in the central molecular zone. The gas cloud includes carbon monoxide and hydrogen cyanide molecules. The cloud is 0.2° away from Sgr C to the galactic southeast. The spectral lines of carbon monoxide reveal that the gas is dense, and warm and fairly opaque. {m} | meanlimits = \bar c \pm 3\sqrt{\bar c} | meanstatistic = \bar c_i = \sum_{j=1}^n \mbox{no. of defects for } x_{ij} }} In statistical quality control, the c-chart is a type of control chart used to monitor ""count""-type data, typically total number of nonconformities per unit. Air-Speed, Inc. was a commuter airline in the United States from the 1970s based at Hanscom Field in Bedford, Massachusetts. Subsequent theoretical studies of the gas cloud and nearby IMBH candidates have re-opened the possibility, though no observational evidence for existence of an IMBH has been reported. The molecular cloud has a mass of 4,000 solar masses. For example, the \tau_c = 1.7 ps for water, and 100 ps for a pyrroline nitroxyl radical in a DMSO-water mixture. In solution, rotational correlation times are in the order of picoseconds. Another example of this naming convention is CO–0.30–0.07. ==References== Category:Molecular clouds Category:Sagittarius (constellation) Category:Intermediate-mass black holes The European market variant was identified as the CM250TB. ==Description== The CM250TB is based on the Honda Superdream CB250N engine but with a five-speed and not six-speed gearbox.Secondhand Buying Guide: Honda CM125C/200T/250T (Motor Cycle News 9/01/91) The model is instead characterised by its North American cruiser styling with stepped seat, high handlebars, 'megaphone' exhaust silencers, teardrop-shaped tank and many chromium-plated and polished alloy parts. Rotational correlation times of probe molecules in media have been measured by fluorescence lifetime or for radicals, from the linewidths of electron spin resonances. == References == Category:Molecular dynamics Category:Nuclear magnetic resonance Rotational correlation times are employed in the measurement of microviscosity (viscosity at the molecular level) and in protein characterization. Rotational correlation times may be measured by rotational (microwave), dielectric, and nuclear magnetic resonance (NMR) spectroscopy. ",164,475,0.9522,0.14,-11.2,B
-"Caesium (m.p. $29^{\circ} \mathrm{C}$, b.p. $686^{\circ} \mathrm{C}$ ) was introduced into a container and heated to $500^{\circ} \mathrm{C}$. When a hole of diameter $0.50 \mathrm{~mm}$ was opened in the container for $100 \mathrm{~s}$, a mass loss of $385 \mathrm{mg}$ was measured. Calculate the vapour pressure of liquid caesium at $500 \mathrm{~K}$.","The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. According to Monteith and Unsworth, ""Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C."" Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). On the computation of saturation vapour pressure. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Boca Raton, Florida, 2003; Section 4, Properties of the Elements and Inorganic Compounds; Vapor Pressure of the Metallic Elements :The equations reproduce the observed pressures to an accuracy of ±5% or better. Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. At boiling temperatures if Raoult's law applies, the total pressure becomes: :Ptot = x1 P o1T + x2 P o2T + ... etc. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. Recall from the first section that vapor pressures of liquids are very dependent on temperature. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. The concentration of a vapor in contact with its liquid, especially at equilibrium, is often expressed in terms of vapor pressure, which will be a partial pressure (a part of the total gas pressure) if any other gas(es) are present with the vapor. The chemical equation for this reaction is :CaCO3 \+ heat → CaO + CO2 This reaction can take place at anywhere above 840 °C (1544 °F), but is generally considered to occur at 900 °C(1655 °F) (at which temperature the partial pressure of CO2 is 1 atmosphere), but a temperature around 1000 °C (1832 °F) (at which temperature the partial pressure of CO2 is 3.8 atmospheresCRC Handbook of Chemistry and Physics, 54th Ed, p F-76) is usually used to make the reaction proceed quickly.Parkes, G.D. and Mellor, J.W. (1939). The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. An understanding of vapour pressure is also relevant in explaining high altitude breathing and cavitation. ==Approximation formulas == There are many published approximations for calculating saturated vapour pressure over water and over ice. AP, Amsterdam. http://store.elsevier.com/Principles-of-Environmental-Physics/John- Monteith/isbn-9780080924793/ ::P = 0.61078 \exp\left(\frac{17.27 T}{T + 237.3}\right), where temperature is in degrees Celsius (°C) and saturation vapor pressure is in kilopascals (kPa). With exhaust gas temperatures as low as 120 °C and lime temperature at kiln outlet in 80 °C range the heat loss of the regenerative kiln is minimal, fuel consumption is as low as 3.6 MJ/kg. ",8.7,650000,0.6957,4.5,1,A
-"To get an idea of the distance dependence of the tunnelling current in STM, suppose that the wavefunction of the electron in the gap between sample and needle is given by $\psi=B \mathrm{e}^{-\kappa x}$, where $\kappa=\left\{2 m_{\mathrm{e}}(V-E) / \hbar^2\right\}^{1 / 2}$; take $V-E=2.0 \mathrm{eV}$. By what factor would the current drop if the needle is moved from $L_1=0.50 \mathrm{~nm}$ to $L_2=0.60 \mathrm{~nm}$ from the surface?","Tunneling current is exponentially dependent on the separation of the sample and the tip, typically reducing by an order of magnitude when the separation is increased by 1 Å (0.1 nm). How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element : M_{\mu u} = \int_{z > z_0} \psi^S_\mu \,U_T\, {\psi^T_ u}^* \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. The current due to an applied voltage V (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level EF and EF − eV in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip. The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that leak through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material. Then the tunneling current is simply the convolution of the densities of states of the sample surface and the tip: : I_t \propto \int_0^{eV} \rho_S(E_F - eV + \varepsilon) \, \rho_T(E_F + \varepsilon) \, d \varepsilon. Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital. === Tunneling between two conductors === thumb|300x300px|Negative sample bias V raises its electronic levels by e⋅V. Using the modified Bardeen transfer Hamiltonian method, which treats tunneling as a perturbation, the tunneling current (I) is found to be where f\left(E\right) is the Fermi distribution function, \rho_s and \rho_T are the density of states (DOS) in the sample and tip, respectively, and M_{\mu u} is the tunneling matrix element between the modified wavefunctions of the tip and the sample surface. The tunneling electric current will be a small fraction of the impinging current. The main result was that the tunneling current is proportional to the local density of states of the sample at the Fermi level taken at the position of the center of curvature of a spherically symmetric tip (s-wave tip model). Using the WKB approximation, equations (5) and (7), we obtain:R. J. Hamers, “STM on Semiconductors,” from Scanning Tunneling Microscopy I, Springer Series in Surface Sciences 20, Ed. by H. -J. Güntherodt and R. Wiesendanger, Berlin: Springer-Verlag, 1992. The tunneling current can be related to the density of available or filled states in the sample. The tunneling current from a single level is therefore : j_t = \left[\frac{4k\kappa}{k^2 + \kappa^2}\right]^2 \, \frac{\hbar k}{m_e}\,e^{-2\kappa w}, where both wave vectors depend on the level's energy E, k = \tfrac{1}{\hbar} \sqrt{2m_eE}, and \kappa = \tfrac{1}{\hbar}\sqrt{2m_e(U - E)}. Since both the tunneling current, equation (5), and the conductance, equation (7), depend on the tip DOS and the tunneling transition probability, T, quantitative information about the sample DOS is very difficult to obtain. Since the tip-sample bias range in tunneling experiments is limited to \pm\phi/e, where \phi is the apparent barrier height, STM and STS only sample valence electron states. If at some point the tunneling current is below the set level, the tip is moved towards the sample, and conversely. For these ideal assumptions, the tunneling conductance is directly proportional to the sample DOS. With the height of the tip fixed, the electron tunneling current is then measured as a function of electron energy by varying the voltage between the tip and the sample (the tip to sample voltage sets the electron energy). The resulting tunneling current is a function of the tip position, applied voltage, and the local density of states (LDOS) of the sample. The current of tunneling electrons at each instance is therefore proportional to |c_ u(t + \mathrm{d}t)|^2 - |c_ u(t)|^2 divided by \mathrm{d}t, which is the time derivative of |c_ u(t)|^2, : \Gamma_{\mu \to u}\ \overset{\text{def}}{=}\ \frac{\mathrm{d}}{\mathrm{d}t} |c_ u(t)|^2 = \frac{2\pi}{\hbar} |M_{\mu u}|^2\frac{\sin\big[(E^S_\mu - E^T_ u) \tfrac{t}{\hbar}\big]}{\pi(E^S_\mu - E^T_ u)}. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: 2e \cdot \rho_S(E_F - eV + \varepsilon)\,\mathrm{d}\varepsilon representing available electrons, f(E_F - eV + \varepsilon) - f(E_F + \varepsilon) for those that are allowed to tunnel, and the probability factor \Gamma for those that will actually tunnel: : I_t = \frac{4 \pi e}{\hbar} \int_{-\infty}^{+\infty} [f(E_F - eV + \varepsilon) - f(E_F + \varepsilon)] \, \rho_S(E_F - eV + \varepsilon) \, \rho_T(E_F + \varepsilon) \, |M|^2 \, d \varepsilon. There is however also an important correction to the elastic component of the tunneling current at the onset. As the tip is moved across the surface in a discrete x–y matrix, the changes in surface height and population of the electronic states cause changes in the tunneling current. ",-30,0.14,0.59,0.23,0.333333,D
+","Upon removal of the dividing partition, they expand into a final common volume (the sum of the two initial volumes), and the entropy of mixing \Delta S_{mix} is given by :\Delta S_{mix} = -nR(x_1\ln x_1 + x_2\ln x_2)\,. where R is the gas constant, n the total number of moles and x_i the mole fraction of component i\,, which initially occupies volume V_i = x_iV\,. In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the thermodynamic operation of removal of impermeable partition(s) between them, followed by a time for establishment of a new thermodynamic state of internal equilibrium in the new unpartitioned closed system. Again, the same equations for the entropy of mixing apply, but only for homogeneous, uniform phases. ==Mixing under other constraints== ===Mixing with and without change of available volume=== In the established customary usage, expressed in the lead section of this article, the entropy of mixing comes from two mechanisms, the intermingling and possible interactions of the distinct molecular species, and the change in the volume available for each molecular species, or the change in concentration of each molecular species. This is not true for the corresponding Gibbs free energies however. == Ideal and regular mixtures == An ideal mixture is any in which the arithmetic mean (with respect to mole fraction) of the two pure substances is the same as that of the final mixture. We again obtain the entropy of mixing on multiplying by the Boltzmann constant k_\text{B}. This explains why enthalpy of mixing is typically experimentally determined. ==Relation to the Gibbs free energy of mixing== The excess Gibbs free energy of mixing can be related to the enthalpy of mixing by the ușe of the Gibbs-Helmholtz equation: :\left( \frac{\partial ( \Delta G^E/T ) } {\partial T} \right)_p = - \frac {\Delta H^E} {T^2} = - \frac {\Delta H_{mix}} {T^2} or equivalently :\left( \frac{\partial ( \Delta G^E/T ) } {\partial (1/T)} \right)_p = \Delta H^E = \Delta H_{mix} In these equations, the excess and total enthalpies of mixing are equal because the ideal enthalpy of mixing is zero. We can regard the mixing process as allowing the contents of the two originally separate contents to expand into the combined volume of the two conjoined containers. For an ideal gas mixture or an ideal solution, there is no enthalpy of mixing (\Delta H_\text{mix} \,), so that the Gibbs free energy of mixing is given by the entropy term only: :\Delta G_\text{mix} = - T\Delta S_\text{mix} For an ideal solution, the Gibbs free energy of mixing is always negative, meaning that mixing of ideal solutions is always spontaneous. The entropy of mixing is entirely accounted for by the diffusive expansion of each material into a final volume not initially accessible to it. Introduction to Modern Thermodynamics, Wiley, Chichester, , pages 197-199. ==See also== * CALPHAD * Enthalpy of mixing * Gibbs energy ==Notes== ==References== == External links == * Online lecture Category:Statistical mechanics Category:Thermodynamic entropy See also: Gibbs paradox, in which it would seem that ""mixing"" two samples of the same gas would produce entropy. ===Application to solutions=== If the solute is a crystalline solid, the argument is much the same. A version of Statistical Mechanics for Students of Physics and Chemistry, Cambridge University Press, Cambridge UK, pages 163-164 the conflating of the just-mentioned two mechanisms for the entropy of mixing is well established in customary terminology, but can be confusing unless it is borne in mind that the independent variables are the common initial and final temperature and total pressure; if the respective partial pressures or the total volume are chosen as independent variables instead of the total pressure, the description is different. ====Mixing with each gas kept at constant partial volume, with changing total volume==== In contrast to the established customary usage, ""mixing"" might be conducted reversibly at constant volume for each of two fixed masses of gases of equal volume, being mixed by gradually merging their initially separate volumes by use of two ideal semipermeable membranes, each permeable only to one of the respective gases, so that the respective volumes available to each gas remain constant during the merge. This quantity combines two physical effects—the enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing considered here. Therefore, \Delta S_\text{mix} = -\left(\frac{\partial \Delta G_\text{mix}}{\partial T}\right)_P is negative for mixing of these two equilibrium phases. In the general case of mixing non-ideal materials, however, the total final common volume may be different from the sum of the separate initial volumes, and there may occur transfer of work or heat, to or from the surroundings; also there may be a departure of the entropy of mixing from that of the corresponding ideal case. In ideal mixtures, the enthalpy of mixing is null. Enthalpy of mixing can often be ignored in calculations for mixtures where other heat terms exist, or in cases where the mixture is ideal. Among other important thermodynamic simplifications, this means that enthalpy of mixing is zero: \Delta H_{mix,ideal}=0. Principles and Applications.(MacMillan 1969) p.263 For binary mixtures the entropy of random mixing can be considered as a function of the mole fraction of one component. The lowest value is when the mole fraction is 0.5 for a mixture of two components, or 1/n for a mixture of n components. thumb|The entropy of mixing for an ideal solution of two species is maximized when the mole fraction of each species is 0.5. ==Solutions and temperature dependence of miscibility== ===Ideal and regular solutions=== The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or solid) solutions—those formed by completely random mixing so that the components move independently in the total volume. In non-ideal mixtures, the thermodynamic activity of each component is different from its concentration by multiplying with the activity coefficient. In this case, the increase in entropy is entirely due to the irreversible processes of expansion of the two gases, and involves no heat or work flow between the system and its surroundings. ===Gibbs free energy of mixing=== The Gibbs free energy change \Delta G_\text{mix} = \Delta H_\text{mix} - T\Delta S_\text{mix} determines whether mixing at constant (absolute) temperature T and pressure p is a spontaneous process. ",362880,-6.9,"""0.7854""",-1368,-1.78,B
+"What is the mean speed, $\bar{c}$, of $\mathrm{N}_2$ molecules in air at $25^{\circ} \mathrm{C}$ ?","CO-0.40-0.22 is a high velocity compact gas cloud near the centre of the Milky Way. The gas is moving away from Earth at speeds ranging from 20 to 120 km/s. The name followed the precedent set by CO-0.02-0.02, which is another high velocity compact cloud in the central molecular zone. The molecular formula C20H24N2O (molar mass: 308.42 g/mol, exact mass: 308.1889 u) may refer to: * Affinisine * Indecainide Category:Molecular formulas The differences in the velocity, termed velocity dispersion, of the gas is unusually high at 100 km/s. Rotational correlation time (\tau_c) is the average time it takes for a molecule to rotate one radian. The Honda CM250 is a parallel twin cylinder air-cooled OHC four-stroke cruiser motorcycle produced by the Honda corporation from 1981–1983 with a top speed of 85 mph and delivering 70mpg.Secondhand Buying Guide: Honda CM125C/200T/250T (Motor Cycle News 9/01/91) The 234cc North American market variant was coded as the CM250C and was the precursor to the current Honda CMX250C, also known as the Honda Rebel 250. It is 200 light years away from the centre in the central molecular zone. The gas cloud includes carbon monoxide and hydrogen cyanide molecules. The cloud is 0.2° away from Sgr C to the galactic southeast. The spectral lines of carbon monoxide reveal that the gas is dense, and warm and fairly opaque. {m} | meanlimits = \bar c \pm 3\sqrt{\bar c} | meanstatistic = \bar c_i = \sum_{j=1}^n \mbox{no. of defects for } x_{ij} }} In statistical quality control, the c-chart is a type of control chart used to monitor ""count""-type data, typically total number of nonconformities per unit. Air-Speed, Inc. was a commuter airline in the United States from the 1970s based at Hanscom Field in Bedford, Massachusetts. Subsequent theoretical studies of the gas cloud and nearby IMBH candidates have re-opened the possibility, though no observational evidence for existence of an IMBH has been reported. The molecular cloud has a mass of 4,000 solar masses. For example, the \tau_c = 1.7 ps for water, and 100 ps for a pyrroline nitroxyl radical in a DMSO-water mixture. In solution, rotational correlation times are in the order of picoseconds. Another example of this naming convention is CO–0.30–0.07. ==References== Category:Molecular clouds Category:Sagittarius (constellation) Category:Intermediate-mass black holes The European market variant was identified as the CM250TB. ==Description== The CM250TB is based on the Honda Superdream CB250N engine but with a five-speed and not six-speed gearbox.Secondhand Buying Guide: Honda CM125C/200T/250T (Motor Cycle News 9/01/91) The model is instead characterised by its North American cruiser styling with stepped seat, high handlebars, 'megaphone' exhaust silencers, teardrop-shaped tank and many chromium-plated and polished alloy parts. Rotational correlation times of probe molecules in media have been measured by fluorescence lifetime or for radicals, from the linewidths of electron spin resonances. == References == Category:Molecular dynamics Category:Nuclear magnetic resonance Rotational correlation times are employed in the measurement of microviscosity (viscosity at the molecular level) and in protein characterization. Rotational correlation times may be measured by rotational (microwave), dielectric, and nuclear magnetic resonance (NMR) spectroscopy. ",164,475,"""0.9522""",0.14,-11.2,B
+"Caesium (m.p. $29^{\circ} \mathrm{C}$, b.p. $686^{\circ} \mathrm{C}$ ) was introduced into a container and heated to $500^{\circ} \mathrm{C}$. When a hole of diameter $0.50 \mathrm{~mm}$ was opened in the container for $100 \mathrm{~s}$, a mass loss of $385 \mathrm{mg}$ was measured. Calculate the vapour pressure of liquid caesium at $500 \mathrm{~K}$.","The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Coefficients from this source: :* ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; Section 3.4.4, D. Ambrose, Vapour pressures from 0.2 to 101.325 kPa. According to Monteith and Unsworth, ""Values of saturation vapour pressure from Tetens' formula are within 1 Pa of exact values up to 35 °C."" Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). On the computation of saturation vapour pressure. The (unattributed) constants are given as {| class=""wikitable"" , °C , °C 8.07131 1730.63 233.426 1 99 8.14019 1810.94 244.485 100 374 ===Accuracy of different formulations=== Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapour pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005): : (°C) (Lide Table) (Eq 1) (Antoine) (Magnus) (Tetens) (Buck) (Goff-Gratch) 0 0.6113 0.6593 (+7.85%) 0.6056 (-0.93%) 0.6109 (-0.06%) 0.6108 (-0.09%) 0.6112 (-0.01%) 0.6089 (-0.40%) 20 2.3388 2.3755 (+1.57%) 2.3296 (-0.39%) 2.3334 (-0.23%) 2.3382 (+0.05%) 2.3383 (-0.02%) 2.3355 (-0.14%) 35 5.6267 5.5696 (-1.01%) 5.6090 (-0.31%) 5.6176 (-0.16%) 5.6225 (+0.04%) 5.6268 (+0.00%) 5.6221 (-0.08%) 50 12.344 12.065 (-2.26%) 12.306 (-0.31%) 12.361 (+0.13%) 12.336 (+0.08%) 12.349 (+0.04%) 12.338 (-0.05%) 75 38.563 37.738 (-2.14%) 38.463 (-0.26%) 39.000 (+1.13%) 38.646 (+0.40%) 38.595 (+0.08%) 38.555 (-0.02%) 100 101.32 101.31 (-0.01%) 101.34 (+0.02%) 104.077 (+2.72%) 102.21 (+1.10%) 101.31 (-0.01%) 101.32 (0.00%) A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). == Vapor pressure == P/(Pa) 1 10 100 200 1 k 2 k 5 k 10 k 20 k 50 k 100 k 101325 reference 1 H hydrogen use (T/K) 15 20 CRC.a (T/°C) -258.6 -252.8 KAL (T/K) 10 (s) 11.4 (s) 12.2 (s) 13.4 (s) 14.5 16.0 18.2 20.3 2 He helium use (T/K) 3 4 CRC.b (T/°C) -270.6 -268.9 KAL (T/K) 1.3 1.66 1.85 2.17 2.48 2.87 3.54 4.22 3 Li lithium use (T/K) 797 885 995 1144 1337 1610 CRC.f,k (T/°C) 524.3 612.3 722.1 871.2 1064.3 1337.1 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.673 - 8310 / (T/K) CR2 (T/K) liquid, m.p. to 1000 K: log (P/Pa) = 10.061 - 8023 / (T/K) 797.4 885.4 995.3 KAL (T/K) 1040 1145 1200 1275 1345 1415 1530 1630 SMI.a (T/K) liquid, 325..725 °C: log (P/Pa) = 9.625 - 7480 / (T/K) 777 867 981 4 Be beryllium use (T/K) 1462 1608 1791 2023 2327 2742 CRC.b (T/°C) 1189 (s) 1335 1518 1750 2054 2469 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.048 - 17020 / (T/K) - 0.4440 log (T/K) 1461.8 CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.792 - 15731 / (T/K) 1606.5 1789.2 SMI.c,g (T/K) solid, 942..1284 °C: log (P/Pa) = 12.115 - 18220 / (T/K) 1504 SMI.c,g (T/K) liquid, 1284..1582 °C: log (P/Pa) = 11.075 - 16590 / (T/K) 1647 1828 5 B boron use (T/K) 2348 2562 2822 3141 3545 4072 CRC.b (T/°C) 2075 2289 2549 2868 3272 3799 SMI.c (T/K) solid, 1052..1648 °C: log (P/Pa) = 13.255 - 21370 / (T/K) 1612 1744 1899 6 C carbon (graphite) use (T/K) 2839 3048 3289 3572 3908 CRC.h (T/°C) 2566 (s) 2775 (s) 3016 (s) 3299 (s) 3635 (s) 7 N nitrogen use (T/K) 37 41 46 53 62 77 CRC.a,d (T/°C) -236 (s) -232 (s) -226.8 (s) -220.2 (s) -211.1 (s) -195.9 KAL (T/K) 48.1 (s) 53.0 (s) 55.4 (s) 59.0 (s) 62.1 (s) 65.8 71.8 77.4 8 O oxygen use (T/K) 61 73 90 CRC.a,i (T/°C) -211.9 -200.5 -183.1 KAL (T/K) 55.4 61.3 64.3 68.8 72.7 77.1 83.9 90.2 9 F fluorine use (T/K) 38 44 50 58 69 85 CRC.a,d (T/°C) -235 (s) -229.5 (s) -222.9 (s) -214.8 -204.3 -188.3 KAL (T/K) 53 58 61 65.3 68.9 73.0 79.3 85.0 10 Ne neon use (T/K) 12 13 15 18 21 27 CRC.b (T/°C) -261 (s) -260 (s) -258 (s) -255 (s) -252 (s) -246.1 KAL (T/K) 16.3 (s) 18.1 (s) 19.0 (s) 20.4 (s) 21.6 (s) 22.9 (s) 24.9 27.1 11 Na sodium use (T/K) 554 617 697 802 946 1153 CRC.f,k (T/°C) 280.6 344.2 424.3 529 673 880.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.304 - 5603 / (T/K) CR2 (T/K) liquid, m.p. to 700 K: log (P/Pa) = 9.710 - 5377 / (T/K) 553.8 617.3 697.4 KAL (T/K) 729 807 846 904 954 1010 1095 1175 SMI.a (T/K) liquid, 158..437 °C: log (P/Pa) = 9.835 - 5480 / (T/K) 557 620 699 12 Mg magnesium use (T/K) 701 773 861 971 1132 1361 CRC.b (T/°C) 428 (s) 500 (s) 588 (s) 698 859 1088 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.495 - 7813 / (T/K) - 0.8253 log (T/K) 700.9 772.7 861.2 892.0 KAL (T/K) 891 979 1025 1085 1140 1205 1295 1375 SMI.a (T/K) solid, 287..605 °C: log (P/Pa) = 10.945 - 7741 / (T/K) 707 778 865 13 Al aluminium use (T/K) 1482 1632 1817 2054 2364 2790 CRC.b (T/°C) 1209 1359 1544 1781 2091 2517 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.465 - 17342 / (T/K) - 0.7927 log (T/K) CR2 (T/K) liquid, m.p. to 1800 K: log (P/Pa) = 10.917 - 16211 / (T/K) 1484.9 1634.7 1818.0 KAL (T/K) 1885 2060 2140 2260 2360 2430 2640 2790 SMI.c,l (T/K) liquid, 843..1465 °C: log (P/Pa) = 11.115 - 15630 / (T/K) 1406 1545 14 Si silicon use (T/K) 1908 2102 2339 2636 3021 3537 CRC.b (T/°C) 1635 1829 2066 2363 2748 3264 KAL (T/K) 2380 2590 2690 2840 2970 3100 3310 3490 SMI.c (T/K) solid, 1024..1410 °C: log (P/Pa) = 12.325 - 19720 / (T/K) 1600 SMI.c (T/K) liquid, 1410..1670 °C: log (P/Pa) = 11.675 - 18550 / (T/K) 1738 1917 15 P phosphorus (white) use (T/K) 279 307 342 388 453 549 CRC.c,e (T/°C) 6 (s) 34 (s) 69 115 180 276 KAL (T/K) 365 398 414 439 460 484 520.5 553.6 15 P phosphorus (red) use (T/K) 455 489 529 576 635 704 CRC.b,c (T/°C) 182 (s) 216 (s) 256 (s) 303 (s) 362 (s) 431 (s) 16 S sulfur use (T/K) 375 408 449 508 591 717 CRC.c (T/°C) 102 (s) 135 176 235 318 444 KAL (T/K) 462 505.7 528.0 561.3 590.1 622.5 672.4 717.8 17 Cl chlorine use (T/K) 128 139 153 170 197 239 CRC.a (T/°C) -145 (s) -133.7 (s) -120.2 (s) -103.6 (s) -76.1 -34.2 KAL (T/K) 158 (s) 170 (s) 176.3 187.4 197.0 207.8 224.3 239.2 18 Ar argon use (T/K) 47 53 61 71 87 CRC.a,d,l (T/°C) -226.4 (s) -220.3 (s) -212.4 (s) -201.7 (s) -186.0 KAL (T/K) 55.0 (s) 60.7 (s) 63.6 (s) 67.8 (s) 71.4 (s) 75.4 (s) 81.4 (s) 87.3 19 K potassium use (T/K) 473 530 601 697 832 1029 CRC.f,k (T/°C) 200.2 256.5 328 424 559 756.2 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.967 - 4646 / (T/K) CR2 (T/K) liquid, m.p. to 600 K: log (P/Pa) = 9.408 - 4453 / (T/K) 473.3 529.6 601.1 KAL (T/K) 633 704 740 794 841 894 975 1050 SMI.a (T/K) liquid, 91..338 °C: log (P/Pa) = 9.485 - 4503 / (T/K) 475 531 602 20 Ca calcium use (T/K) 864 956 1071 1227 1443 1755 CRC.b (T/°C) 591 (s) 683 (s) 798 (s) 954 1170 1482 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 15.133 - 9517 / (T/K) - 1.4030 log (T/K) 864.2 956.4 1071.5 1112.0 KAL (T/K) 1105 1220 1280 1365 1440 1525 1650 1765 SMI.a (T/K) 408..817 °C: log (P/Pa) = 10.425 - 9055 / (T/K) 869 961 1075 21 Sc scandium use (T/K) 1645 1804 (2006) (2266) (2613) (3101) CRC.c (T/°C) 1372 (s) 1531 (s) 1733 (i) 1993 (i) 2340 (i) 2828 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 11.656 - 19721 / (T/K) + 0.2885 log (T/K) - 0.3663 (T/K) 10−3 CR2 liquid, m.p. to 2000 K: log (P/Pa) = 10.801 - 17681 / (T/K) SMI.c (T/K) 1058..1804 °C: log (P/Pa) = 11.065 - 18570 / (T/K) 1678 1845 2049 22 Ti titanium use (T/K) 1982 2171 (2403) 2692 3064 3558 CRC.b (T/°C) 1709 1898 2130 (e) 2419 2791 3285 CR2 solid, 298 K to m.p.: log (P/Pa) = 16.931 - 24991 / (T/K) - 1.3376 log (T/K) CR2 (T/K) liquid, m.p. to 2400 K: log (P/Pa) = 11.364 - 22747 / (T/K) 2001.7 2194.8 SMI.c (T/K) solid, 1134..1727 °C: log (P/Pa) = 10.375 - 18640 / (T/K) 1797 1988 SMI.c (T/K) liquid, 1727..1965 °C: log (P/Pa) = 11.105 - 20110 / (T/K) 2209 23 V vanadium use (T/K) 2101 2289 2523 2814 3187 3679 CRC.b (T/°C) 1828 (s) 2016 2250 2541 2914 3406 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.750 - 27132 / (T/K) - 0.5501 log (T/K) 2099.6 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.935 - 25011 / (T/K) 2287.2 2517.5 SMI.c (T/K) 1465..2207 °C: log (P/Pa) = 12.445 - 26620 / (T/K) 2139 2326 24 Cr chromium use (T/K) 1656 1807 1991 2223 2530 2942 CRC.b (T/°C) 1383 (s) 1534 (s) 1718 (s) 1950 2257 2669 CR2 solid, 298 K to 2000 K: log (P/Pa) = 11.806 - 20733 / (T/K) + 0.4391 log (T/K) - 0.4094 (T/K)−3 KAL (T/K) 2020 2180 2260 2370 2470 2580 2730 2870 SMI.c,l (T/K) solid, 907..1504 °C: log (P/Pa) = 12.005 - 17560 / (T/K) 1463 1596 1755 25 Mn manganese use (T/K) 1228 1347 1493 1691 1955 2333 CRC.b (T/°C) 955 (s) 1074 (s) 1220 (s) 1418 1682 2060 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.811 - 15097 / (T/K) - 1.7896 log (T/K) 1229.2 1346.6 1490.0 KAL (T/K) 1560 1710 1780 1890 1980 2080 2240 2390 SMI.c (T/K) 717..1251 °C: log (P/Pa) = 11.375 - 14100 / (T/K) 1240 1359 1504 26 Fe iron use (T/K) 1728 1890 2091 2346 2679 3132 CRC.b (T/°C) 1455 (s) 1617 1818 2073 2406 2859 CR2 solid, 298 K to m.p.: log (P/Pa) = 12.106 - 21723 / (T/K) + 0.4536 log (T/K) - 0.5846 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 11.353 - 19574 / (T/K) 1890.7 2092.8 KAL (T/K) 2110 2290 2380 2500 2610 2720 2890 3030 SMI.c,f (T/K) solid, 1094..1535 °C: log (P/Pa) = 11.755 - 20000 / (T/K) 1701 SMI.c,f (T/K) liquid, 1535..1783 °C: log (P/Pa) = 12.535 - 21400 / (T/K) 1855 2031 27 Co cobalt use (T/K) 1790 1960 2165 2423 2755 3198 CRC.b (T/°C) 1517 1687 1892 2150 2482 2925 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.982 - 22576 / (T/K) - 1.0280 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.494 - 20578 / (T/K) 1790.3 1960.9 2167.5 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21960 / (T/K) 1900 2081 2298 28 Ni nickel use (T/K) 1783 1950 2154 2410 2741 3184 CRC.b (T/°C) 1510 1677 1881 2137 2468 2911 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.563 - 22606 / (T/K) - 0.8717 log (T/K) CR2 (T/K) liquid, m.p. to 2150 K: log (P/Pa) = 11.672 - 20765 / (T/K) 1779.0 1945.7 2146.9 KAL (T/K) 2230 2410 2500 2630 2740 2860 3030 3180 SMI.c,q solid, 1157..1455 °C: log (P/Pa) = 12.405 - 21840 / (T/K) SMI.c,q (T/K) liquid, 1455..1884 °C: log (P/Pa) = 11.675 - 20600 / (T/K) 1764 1930 2129 29 Cu copper use (T/K) 1509 1661 1850 2089 2404 2836 CRC.b (T/°C) 1236 1388 1577 1816 2131 2563 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.129 - 17748 / (T/K) - 0.7317 log (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.855 - 16415 / (T/K) 1512.2 1665.7 1853.8 KAL (T/K) 1940 2110 2200 2320 2420 2540 2710 2860 SMI.c,f solid, 946..1083 °C: log (P/Pa) = 11.935 - 18060 / (T/K) SMI.c,f (T/K) liquid, 1083..1628 °C: log (P/Pa) = 10.845 - 16580 / (T/K) 1529 1684 1875 30 Zn zinc use (T/K) 610 670 750 852 990 (1185) CRC.b (T/°C) 337 (s) 397 (s) 477 579 717 912 (e) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 11.108 - 6776 / (T/K) 610.0 670.4 CR2 (T/K) liquid, m.p. to 750 K: log (P/Pa) = 10.384 - 6286 / (T/K) 749.8 KAL (T/K) 780 854 891 945 991 1040 1120 1185 SMI.a (T/K) solid, 211..405 °C: log (P/Pa) = 11.065 - 6744 / (T/K) 609 670 31 Ga gallium use (T/K) 1310 1448 1620 1838 2125 2518 CRC.b (T/°C) 1037 1175 1347 1565 1852 2245 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.663 - 14208 / (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 11.760 - 13984 / (T/K) - 0.3413 log (T/K) 1307.4 1444.4 1613.8 KAL (T/K) 1665 1820 1900 2000 2100 2200 2350 2480 SMI.c (T/K) liquid, 771..1443 °C: log (P/Pa) = 9.915 - 13360 / (T/K) 1347 1499 1688 32 Ge germanium use (T/K) 1644 1814 2023 2287 2633 3104 CRC.b (T/°C) 1371 1541 1750 2014 2360 2831 SMI.c (T/K) 897..1635 °C: log (P/Pa) = 10.065 - 15150 / (T/K) 1505 1671 1878 33 As arsenic use (T/K) 553 596 646 706 781 874 CRC.c (T/°C) 280 (s) 323 (s) 373 (s) 433 (s) 508 (s) 601 (s) KAL (T/K) 656 (s) 701 (s) 723 (s) 754 (s) 780 (s) 808 (s) 849 (s) 883 (s) 34 Se selenium use (T/K) 500 552 617 704 813 958 CRC.c (T/°C) 227 279 344 431 540 685 KAL (T/K) 636 695 724 767 803 844 904 958 35 Br bromine use (T/K) 185 201 220 244 276 332 CRC.a (T/°C) -87.7 (s) -71.8 (s) -52.7 (s) -29.3 (s) 2.5 58.4 KAL (T/K) 227 (s) 244.1 (s) 252.1 (s) 263.6 (s) 275.7 290.0 312.0 332.0 36 Kr krypton use (T/K) 59 65 74 84 99 120 CRC.d (T/°C) -214.0 (s) -208.0 (s) -199.4 (s) -188.9 (s) -174.6 (s) -153.6 KAL (T/K) 77 (s) 84.3 (s) 88.1 (s) 93.8 (s) 98.6 (s) 103.9 (s) 112.0 (s) 119.7 37 Rb rubidium use (T/K) 434 486 552 641 769 958 CRC.f,k (T/°C) 160.4 212.5 278.9 368 496.1 685.3 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.863 - 4215 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.318 - 4040 / (T/K) 433.6 485.7 552.1 KAL (T/K) 583 649 684 735 779 829 907 978 SMI.d (T/K) liquid, 59.4..283 °C: log (P/Pa) = 9.545 - 4132 / (T/K) 433 484 548 38 Sr strontium use (T/K) 796 882 990 1139 1345 1646 CRC.b (T/°C) 523 (s) 609 (s) 717 (s) 866 1072 1373 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.232 - 8572 / (T/K) - 1.1926 log (T/K) 795.7 882.0 989.9 1028.0 KAL (T/K) 1040 (s) 1150 1205 1285 1355 1430 1550 1660 SMI.c (T/K) solid, 361..750 °C: log (P/Pa) = 10.255 - 8324 / (T/K) 812 899 1008 39 Y yttrium use (T/K) 1883 2075 (2320) (2627) (3036) (3607) CRC.c (T/°C) 1610.1 1802.3 2047 (i) 2354 (i) 2763 (i) 3334 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.741 - 22306 / (T/K) - 0.8705 log (T/K) CR2 (T/K) liquid, m.p. to 2300 K: log (P/Pa) = 10.801 - 20341 / (T/K) 1883.3 2075.4 SMI.c (T/K) 1249..2056 °C: log (P/Pa) = 11.555 - 21970 / (T/K) 1901 2081 2299 40 Zr zirconium use (T/K) 2639 2891 3197 3575 4053 4678 CRC.b (T/°C) 2366 2618 2924 3302 3780 4405 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.014 - 31512 / (T/K) - 0.7890 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.812 - 30295 / (T/K) SMI.c (T/K) solid, 1527..2127 °C: log (P/Pa) = 11.505 - 25870 / (T/K) 2249 SMI.c (T/K) liquid, 2127..2459 °C: log (P/Pa) = 12.165 - 27430 / (T/K) 2457 2698 41 Nb niobium use (T/K) 2942 3207 3524 3910 4393 5013 CRC.b (T/°C) 2669 2934 3251 3637 4120 4740 CR2 solid, 298 K to 2500 K: log (P/Pa) = 13.828 - 37818 / (T/K) - 0.2575 log (T/K) SMI.n 2194..2539 °C: log (P/Pa) = 13.495 - 40400 / (T/K) 42 Mo molybdenum use (T/K) 2742 2994 3312 3707 4212 4879 CRC.b (T/°C) 2469 (s) 2721 3039 3434 3939 4606 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.535 - 34626 / (T/K) - 1.1331 log (T/K) KAL (T/K) 3420 3720 3860 4060 4220 4410 4680 4900 SMI.r (T/K) solid, 1923..2533 °C: log (P/Pa) = 10.925 - 30310 / (T/K) 2774 43 Tc technetium use (T/K) (2727) (2998) (3324) (3726) (4234) (4894) CRC.b (T/°C) 2454 (e) 2725 (e) 3051 (e) 3453 (e) 3961 (e) 4621 (e) 44 Ru ruthenium use (T/K) 2588 2811 3087 3424 3845 4388 CRC.b (T/°C) 2315 (s) 2538 2814 3151 3572 4115 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.761 - 34154 / (T/K) - 0.4723 log (T/K) 2597.6 SMI.c (T/K) 1913..2946 °C: log (P/Pa) = 12.625 - 33800 / (T/K) 2677 2908 3181 45 Rh rhodium use (T/K) 2288 2496 2749 3063 3405 3997 CRC.b (T/°C) 2015 2223 2476 2790 3132 3724 CR2 solid, 298 K to m.p.: log (P/Pa) = 15.174 - 29010 / (T/K) - 0.7068 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.808 - 26792 / (T/K) 2269.0 2478.9 SMI.c (T/K) 1681..2607 °C: log (P/Pa) = 12.675 - 30400 / (T/K) 2398 2604 2848 46 Pd palladium use (T/K) 1721 1897 2117 2395 2753 3234 CRC.b (T/°C) 1448 (s) 1624 1844 2122 2480 2961 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.508 - 19813 / (T/K) - 0.9258 log (T/K) 1721.0 CR2 (T/K) liquid, m.p. to 2100 K: log (P/Pa) = 10.432 - 17899 / (T/K) 1897.7 SMI.c (T/K) 1156..2000 °C: log (P/Pa) = 10.585 - 19230 / (T/K) 1817 2006 2240 47 Ag silver use (T/K) 1283 1413 1575 1782 2055 2433 CRC.b (T/°C) 1010 1140 1302 1509 1782 2160 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.133 - 14999 / (T/K) - 0.7845 log (T/K) CR2 (T/K) liquid, m.p. to 1600 K: log (P/Pa) = 10.758 - 13827 / (T/K) 1285.3 1417.0 1578.8 KAL (T/K) 1640 1790 1865 1970 2060 2160 2310 2440 SMI.a solid, 767..961 °C: log (P/Pa) = 11.405 - 14850 / (T/K) SMI.a (T/K) liquid, 961..1353 °C: log (P/Pa) = 10.785 - 14090 / (T/K) 1306 1440 1604 48 Cd cadmium use (T/K) 530 583 654 745 867 1040 CRC.b (T/°C) 257 (s) 310 (s) 381 472 594 767 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 10.945 - 5799 / (T/K) 529.8 583.1 CR2 (T/K) liquid, m.p. to 650 K: log (P/Pa) = 10.248 - 5392 / (T/K) 653.7 KAL (T/K) 683 748 781 829 870 915 983 1045 SMI.a (T/K) solid, 148..321 °C: log (P/Pa) = 10.905 - 5798 / (T/K) 532 585 49 In indium use (T/K) 1196 1325 1485 1690 1962 2340 CRC.b (T/°C) 923 1052 1212 1417 1689 2067 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.997 - 12548 / (T/K) CR2 (T/K) liquid, m.p. to 1500 K: log (P/Pa) = 10.380 - 12276 / (T/K) 1182.7 1308.7 1464.9 SMI.c,m (T/K) liquid, 667..1260 °C: log (P/Pa) = 10.055 - 12150 / (T/K) 1208 1342 1508 50 Sn tin use (T/K) 1497 1657 1855 2107 2438 2893 CRC.b (T/°C) 1224 1384 1582 1834 2165 2620 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.042 - 15710 / (T/K) CR2 (T/K) liquid, m.p. to 1850 K: log (P/Pa) = 10.268 - 15332 / (T/K) 1493.2 1654.3 1854.4 KAL (T/K) 1930 2120 2210 2350 2470 2600 2800 2990 SMI.c (T/K) liquid, 823..1609 °C: log (P/Pa) = 9.095 - 13110 / (T/K) 1441 1620 1848 51 Sb antimony use (T/K) 807 876 1011 1219 1491 1858 CRC.b,c (T/°C) 534 (s) 603 (s) 738 946 1218 1585 KAL (T/K) 1065 1220 1295 1405 1495 1595 1740 1890 SMI.c (T/K) 466..904 °C: log (P/Pa) = 10.545 - 9913 / (T/K) 940 1039 1160 52 Te tellurium use (T/K) (775) (888) 1042 1266 CRC.d (T/°C) 502 (e) 615 (e) 768.8 992.4 KAL (T/K) 806 888 929 990 1040 1100 1190 1270 53 I iodine (rhombic) use (T/K) 260 282 309 342 381 457 CRC.a,b (T/°C) -12.8 (s) 9.3 (s) 35.9 (s) 68.7 (s) 108 (s) 184.0 KAL (T/K) 318 (s) 341.8 (s) 353.1 (s) 369.3 (s) 382.7 (s) 400.8 430.6 457.5 54 Xe xenon use (T/K) 83 92 103 117 137 165 CRC.d,m (T/°C) -190 (s) -181 (s) -170 (s) -155.8 (s) -136.6 (s) -108.4 KAL (T/K) 107 (s) 117.3 (s) 122.5 (s) 130.1 (s) 136.6 (s) 143.8 (s) 154.7 (s) 165.0 55 Cs caesium use (T/K) 418 469 534 623 750 940 CRC.f,k (T/°C) 144.5 195.6 260.9 350.0 477.1 667.0 CR2 solid, 298 K to m.p.: log (P/Pa) = 9.717 - 3999 / (T/K) CR2 (T/K) liquid, m.p. to 550 K: log (P/Pa) = 9.171 - 3830 / (T/K) 417.6 468.7 534.1 KAL (T/K) 559 624 657 708 752 802 883 959 SMI.e (T/K) liquid, 45..277 °C: log (P/Pa) = 8.985 - 3774 / (T/K) 420 473 540 56 Ba barium use (T/K) 911 1038 1185 1388 1686 2170 CRC.e (T/°C) 638 (s) 765 912 1115 1413 1897 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 17.411 - 9690 / (T/K) - 2.2890 log (T/K) 911.0 CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.013 - 8163 / (T/K) 1018.7 1164.0 KAL (T/K) 1165 1290 1360 1455 1540 1635 1780 1910 SMI.c (T/K) 418..858 °C: log (P/Pa) = 10.005 - 8908 / (T/K) 890 989 1113 57 La lanthanum use (T/K) (2005) (2208) (2458) (2772) (3178) (3726) CRC.c (T/°C) 1732 (i) 1935 (i) 2185 (i) 2499 (i) 2905 (i) 3453 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 12.469 - 22551 / (T/K) - 0.3142 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.917 - 21855 / (T/K) 2001.9 2203.8 2450.9 SMI.c (T/K) liquid, 1023..1754 °C: log (P/Pa) = 11.005 - 18000 / (T/K) 1636 1799 1999 58 Ce cerium use (T/K) 1992 2194 2442 2754 3159 3705 CRC.g (T/°C) 1719 1921 2169 2481 2886 3432 CR2 solid, 298 K to m.p.: log (P/Pa) = 11.145 - 21752 / (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 10.617 - 21200 / (T/K) 1996.8 2204.4 2460.3 SMI.c (T/K) liquid, 1004..1599 °C: log (P/Pa) = 12.865 - 20100 / (T/K) 1562 1694 1850 59 Pr praseodymium use (T/K) 1771 1973 (2227) (2571) (3054) (3779) CRC.c (T/°C) 1497.7 1699.4 1954 (i) 2298 (i) 2781 (i) 3506 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.865 - 18720 / (T/K) - 0.9512 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 9.778 - 17315 / (T/K) 1770.8 1972.5 60 Nd neodymium use (T/K) 1595 1774 1998 (2296) (2715) (3336) CRC.c (T/°C) 1322.3 1501.2 1725.3 2023 (i) 2442 (i) 3063 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.002 - 17264 / (T/K) - 0.9519 log (T/K) CR2 (T/K) liquid, m.p. to 2000 K: log (P/Pa) = 9.918 - 15824 / (T/K) 1595.5 1774.4 1998.5 62 Sm samarium use (T/K) 1001 1106 1240 (1421) (1675) (2061) CRC.c (T/°C) 728 (s) 833 (s) 967 (s) 1148 (i) 1402 (i) 1788 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.994 - 11034 / (T/K) - 1.3287 log (T/K) 1002.5 1109.2 1242.2 1289.0 63 Eu europium use (T/K) 863 957 1072 1234 1452 1796 CRC.g (T/°C) 590 (s) 684 (s) 799 (s) 961 1179 1523 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.246 - 9459 / (T/K) - 1.1661 log (T/K) 874.6 968.8 1086.5 64 Gd gadolinium use (T/K) (1836) (2028) (2267) (2573) (2976) (3535) CRC.c (T/°C) 1563 (i) 1755 (i) 1994 (i) 2300 (i) 2703 (i) 3262 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 13.350 - 20861 / (T/K) - 0.5775 log (T/K) CR2 (T/K) liquid, m.p. to 2250 K: log (P/Pa) = 10.563 - 19389 / (T/K) 1835.6 2027.5 2264.3 65 Tb terbium use (T/K) 1789 1979 (2201) (2505) (2913) (3491) CRC.c (T/°C) 1516.1 1706.1 1928 (i) 2232 (i) 2640 (i) 3218 (i) CR2 solid, 298 K to m.p.: log (P/Pa) = 14.516 - 20457 / (T/K) - 0.9247 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.417 - 18639 / (T/K) 1789.3 1979.3 2214.4 66 Dy dysprosium use (T/K) 1378 1523 (1704) (1954) (2304) (2831) CRC.c (T/°C) 1105 (s) 1250 (s) 1431 (i) 1681 (i) 2031 (i) 2558 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.585 - 15336 / (T/K) - 1.1114 log (T/K) 1382.3 1526.5 67 Ho holmium use (T/K) 1432 1584 (1775) (2040) (2410) (2964) CRC.c (T/°C) 1159 (s) 1311 (s) 1502 (i) 1767 (i) 2137 (i) 2691 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.791 - 15899 / (T/K) - 1.1753 log (T/K) 1434.8 1585.2 68 Er erbium use (T/K) 1504 1663 (1885) (2163) (2552) (3132) CRC.c (T/°C) 1231 (s) 1390 (s) 1612 (i) 1890 (i) 2279 (i) 2859 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 14.922 - 16642 / (T/K) - 1.2154 log (T/K) 1504.7 1663.0 CR2 (T/K) liquid, m.p. to 1900 K: log (P/Pa) = 9.674 - 14380 / (T/K) 1873.9 69 Tm thulium use (T/K) 1117 1235 1381 1570 (1821) (2217) CRC.c (T/°C) 844 (s) 962 (s) 1108 (s) 1297 (s) 1548 (i) 1944 (i) CR2 (T/K) solid, 298 K to 1400 K: log (P/Pa) = 13.888 - 12270 / (T/K) - 0.9564 log (T/K) 1118.3 1235.5 1381.0 70 Yb ytterbium use (T/K) 736 813 910 1047 (1266) (1465) CRC.c (T/°C) 463 (s) 540 (s) 637 (s) 774 (s) 993 (i) 1192 (i) CR2 (T/K) solid, 298 K to 900 K: log (P/Pa) = 14.117 - 8111 / (T/K) - 1.0849 log (T/K) 737.0 814.4 71 Lu lutetium use (T/K) 1906 2103 2346 (2653) (3072) (3663) CRC.c (T/°C) 1633 (s) 1829.8 2072.8 2380 (i) 2799 (i) 3390 (i) CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 13.799 - 22423 / (T/K) - 0.6200 log (T/K) 1905.9 CR2 (T/K) liquid, m.p. to 2350 K: log (P/Pa) = 10.654 - 20302 / (T/K) 2103.0 2346.0 72 Hf hafnium use (T/K) 2689 2954 3277 3679 4194 4876 CRC.e (T/°C) 2416 2681 3004 3406 3921 4603 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.451 - 32482 / (T/K) - 0.6735 log (T/K) 73 Ta tantalum use (T/K) 3297 3597 3957 4395 4939 5634 CRC.b (T/°C) 3024 3324 3684 4122 4666 5361 CR2 solid, 248 K to 2500 K: log (P/Pa) = 21.813 - 41346 / (T/K) - 3.2152 log (T/K) + 0.7437 (T/K) 10−3 SMI.o,p solid, 2407..2820 °C: log (P/Pa) = 12.125 - 40210 / (T/K) 74 W tungsten use (T/K) 3477 3773 4137 4579 5127 5823 CRC.b (T/°C) 3204 (s) 3500 3864 4306 4854 5550 CR2 solid, 298 K to 2350 K: log (P/Pa) = 7.951 - 44094 / (T/K) + 1.3677 log (T/K) CR2 solid, 2200 K to 2500 K: log (P/Pa) = -49.521 - 57687 / (T/K) - 12.2231 log (T/K) KAL (T/K) 4300 4630 4790 5020 5200 5400 5690 5940 SMI.s (T/K) solid, 2554..3309 °C: log (P/Pa) = 11.365 - 40260 / (T/K) 3542 75 Re rhenium use (T/K) 3303 3614 4009 4500 5127 5954 CRC.b (T/°C) 3030 (s) 3341 3736 4227 4854 5681 CR2 solid, 298 K to 2500 K: log (P/Pa) = 16.549 - 40726 / (T/K) - 1.1629 log (T/K) 76 Os osmium use (T/K) 3160 3423 3751 4148 4638 5256 CRC.b (T/°C) 2887 (s) 3150 3478 3875 4365 4983 CR2 solid, 298 K to 2500 K: log (P/Pa) = 14.425 - 41198 / (T/K) - 0.3896 log (T/K) SMI.c (T/K) 2101..3221 °C: log (P/Pa) = 12.715 - 37000 / (T/K) 2910 3158 3453 77 Ir iridium use (T/K) 2713 2957 3252 3614 4069 4659 CRC.b (T/°C) 2440 (s) 2684 2979 3341 3796 4386 CR2 solid, 298 K to 2500 K: log (P/Pa) = 15.512 - 35099 / (T/K) - 0.7500 log (T/K) SMI.c (T/K) 1993..3118 °C: log (P/Pa) = 12.185 - 34110 / (T/K) 2799 3050 3349 78 Pt platinum use (T/K) 2330 (2550) 2815 3143 3556 4094 CRC.b (T/°C) 2057 2277 (e) 2542 2870 3283 3821 CR2 solid, 298 K to m.p.: log (P/Pa) = 4.888 - 29387 / (T/K) + 1.1039 log (T/K) - 0.4527 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 11.392 - 26856 / (T/K) 2357.4 KAL (T/K) 2910 3150 3260 3420 3560 3700 3910 4090 SMI.c (T/K) 1606..2582 °C: log (P/Pa) = 11.758 - 27500 / (T/K) 2339 2556 2818 79 Au gold use (T/K) 1646 1814 2021 2281 2620 3078 CRC.b (T/°C) 1373 1541 1748 2008 2347 2805 CR2 solid, 298 K to m.p.: log (P/Pa) = 14.158 - 19343 / (T/K) - 0.7479 log (T/K) CR2 (T/K) liquid, m.p. to 2050 K: log (P/Pa) = 10.838 - 18024 / (T/K) 1663.0 1832.1 2039.4 KAL (T/K) 2100 2290 2390 2520 2640 2770 2960 3120 SMI.a (T/K) 1083..1867 °C: log (P/Pa) = 10.775 - 18520 / (T/K) 1719 1895 2111 80 Hg mercury use (T/K) 315 350 393 449 523 629 CRC.j,k (T/°C) 42.0 76.6 120.0 175.6 250.3 355.9 CR2 (T/K) liquid, 298 K to 400 K: log (P/Pa) = 10.122 - 3190 / (T/K) 315.2 349.7 392.8 KAL (T/K) 408 448.8 468.8 498.4 523.4 551.2 592.9 629.8 81 Tl thallium use (T/K) 882 977 1097 1252 1461 1758 CRC.b (T/°C) 609 704 824 979 1188 1485 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.977 - 9447 / (T/K) CR2 (T/K) liquid, m.p. to 1100 K: log (P/Pa) = 10.265 - 9037 / (T/K) 880.4 975.4 1093.4 KAL (T/K) 1125 1235 1290 1370 1440 1515 1630 1730 SMI.a (T/K) liquid, 405..821 °C: log (P/Pa) = 10.275 - 8920 / (T/K) 868 962 1078 82 Pb lead use (T/K) 978 1088 1229 1412 1660 2027 CRC.b (T/°C) 705 815 956 1139 1387 1754 CR2 solid, 298 K to m.p.: log (P/Pa) = 10.649 - 10143 / (T/K) CR2 (T/K) liquid, m.p. to 1200 K: log (P/Pa) = 9.917 - 9701 / (T/K) 978.2 1087.9 KAL (T/K) 1285 1420 1485 1585 1670 1760 1905 2030 SMI.a (T/K) liquid, 483..975 °C: log (P/Pa) = 9.815 - 9600 / (T/K) 978 1089 1228 83 Bi bismuth use (T/K) 941 1041 1165 1325 1538 1835 CRC.b (T/°C) 668 768 892 1052 1265 1562 KAL (T/K) 1225 1350 1410 1505 1580 1670 1800 1920 SMI.c (T/K) liquid, 474..934 °C: log (P/Pa) = 10.265 - 9824 / (T/K) 957 1060 1189 84 Po polonium use (T/K) (846) 1003 1236 CRC.d (T/°C) 573 (e) 730.2 963.3 85 At astatine use (T/K) 361 392 429 475 531 607 CRC.b (T/°C) 88 (s) 119 (s) 156 (s) 202 (s) 258 (s) 334 86 Rn radon use (T/K) 110 121 134 152 176 211 CRC.d (T/°C) -163 (s) -152 (s) -139 (s) -121.4 (s) -97.6 (s) -62.3 KAL (T/K) 139 (s) 152 (s) 158 (s) 168 (s) 176 (s) 184 (s) 198 (s) 211 87 Fr francium use (T/K) (404) (454) (519) (608) (738) (946) CRC.b (T/°C) 131 (e) 181 (e) 246 (e) 335 (e) 465 (e) 673 (e) 88 Ra radium use (T/K) 819 906 1037 1209 1446 1799 CRC.b (T/°C) 546 (s) 633 (s) 764 936 1173 1526 90 Th thorium use (T/K) 2633 2907 3248 3683 4259 5055 CRC.b (T/°C) 2360 2634 2975 3410 3986 4782 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.674 - 31483 / (T/K) - 0.5288 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = -13.447 - 24569 / (T/K) + 6.6473 log (T/K) SMI.c (T/K) 1686..2715 °C: log (P/Pa) = 11.645 - 28440 / (T/K) 2442 2672 2949 91 Pa protactinium CR2 solid, 298 K to m.p.: log (P/Pa) = 15.558 - 34869 / (T/K) - 1.0075 log (T/K) CR2 liquid, m.p. to 2500 K: log (P/Pa) = 11.183 - 32874 / (T/K) 92 U uranium use (T/K) 2325 2564 2859 3234 3727 4402 CRC.b (T/°C) 2052 2291 2586 2961 3454 4129 CR2 solid, 298 K to m.p.: log (P/Pa) = 5.776 - 27729 / (T/K) + 2.6982 log (T/K) - 1.5471 (T/K) 10−3 CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 25.741 - 28776 / (T/K) - 4.0962 log (T/K) 2422.6 SMI.c (T/K) liquid, 1461..2338 °C: log (P/Pa) = 12.005 - 25800 / (T/K) 2149 2344 2579 93 Np neptunium use (T/K) 2194 2437 CR2 solid, 298 K to m.p.: log (P/Pa) = 24.649 - 24886 / (T/K) - 3.9991 log (T/K) CR2 (T/K) liquid, m.p. to 2500 K: log (P/Pa) = 15.082 - 23378 / (T/K) - 1.3250 log (T/K) 2194.1 2436.6 94 Pu plutonium use (T/K) 1756 1953 2198 2511 2926 3499 CRC.b (T/°C) 1483 1680 1925 2238 2653 3226 CR2 solid, 500 K to m.p.: log (P/Pa) = 23.864 - 18460 / (T/K) - 4.4720 log (T/K) CR2 solid, 298 K to 600 K: log (P/Pa) = 31.166 - 19162 / (T/K) - 6.6675 log (T/K) CR2 (T/K) liquid, m.p. to 2450 K: log (P/Pa) = 8.672 - 16658 / (T/K) 1920.9 2171.3 95 Am americium use (T/K) 1239 1356 CR2 (T/K) solid, 298 K to m.p.: log (P/Pa) = 16.317 - 15059 / (T/K) - 1.3449 log (T/K) 1238.7 1356.1 96 Cm curium use (T/K) 1788 1982 CR2 solid, 298 K to m.p.: log (P/Pa) = 13.375 - 20364 / (T/K) - 0.5770 log (T/K) CR2 (T/K) liquid, m.p. to 2200 K: log (P/Pa) = 10.229 - 18292 / (T/K) 1788.2 1982.0 == Notes == *Values are given in terms of temperature necessary to reach the specified pressure. Boca Raton, Florida, 2003; Section 4, Properties of the Elements and Inorganic Compounds; Vapor Pressure of the Metallic Elements :The equations reproduce the observed pressures to an accuracy of ±5% or better. Applied Meteorology 6: 203-204. https://doi.org/10.1175/1520-0450(1967)006%3C0203:OTCOSV%3E2.0.CO;2 ::P = 0.61078 \exp\left(\frac{21.875 T}{T + 265.5}\right). ==See also== *Vapour pressure of water *Antoine equation *Arden Buck equation *Lee–Kesler method *Goff–Gratch equation ==References== Category:Meteorological concepts Category:Thermodynamic equations As quoted from these sources: :*a - Lide, D.R., and Kehiaian, H.V., CRC Handbook of Thermophysical and Thermochemical Data, CRC Press, Boca Raton, Florida, 1994. :*b - Stull, D., in American Institute of Physics Handbook, Third Edition, Gray, D.E., Ed., McGraw Hill, New York, 1972. :*c - Hultgren, R., Desai, P.D., Hawkins, D.T., Gleiser, M., Kelley, K.K., and Wagman, D.D., Selected Values of Thermodynamic Properties of the Elements, American Society for Metals, Metals Park, OH, 1973. :*d - TRCVP, Vapor Pressure Database, Version 2.2P, Thermodynamic Research Center, Texas A&M; University, College Station, TX. :*e - Barin, I., Thermochemical Data of Pure Substances, VCH Publishers, New York, 1993. :*f - Ohse, R.W. Handbook of Thermodynamic and Transport Properties of Alkali Metals, Blackwell Scientific Publications, Oxford, 1994. :*g - Gschneidner, K.A., in CRC Handbook of Chemistry and Physics, 77th Edition, p. 4-112, CRC Press, Boca Raton, Florida, 1996. :*h - . :*i - Wagner, W., and de Reuck, K.M., International Thermodynamic Tables of the Fluid State, No. 9. At boiling temperatures if Raoult's law applies, the total pressure becomes: :Ptot = x1 P o1T + x2 P o2T + ... etc. Table 363, Evaporation of Metals :The equations are described as reproducing the observed pressures to a satisfactory degree of approximation. Recall from the first section that vapor pressures of liquids are very dependent on temperature. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. *The temperature at standard pressure should be equal to the normal boiling point, but due to the considerable spread does not necessarily have to match values reported elsewhere. *log refers to log base 10 *(T/K) refers to temperature in Kelvin (K) *(P/Pa) refers to pressure in Pascal (Pa) == References == ===CRC.a-m=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 84th Edition. Calculations of the (saturation) vapour pressure of water are commonly used in meteorology. The concentration of a vapor in contact with its liquid, especially at equilibrium, is often expressed in terms of vapor pressure, which will be a partial pressure (a part of the total gas pressure) if any other gas(es) are present with the vapor. The chemical equation for this reaction is :CaCO3 \+ heat → CaO + CO2 This reaction can take place at anywhere above 840 °C (1544 °F), but is generally considered to occur at 900 °C(1655 °F) (at which temperature the partial pressure of CO2 is 1 atmosphere), but a temperature around 1000 °C (1832 °F) (at which temperature the partial pressure of CO2 is 3.8 atmospheresCRC Handbook of Chemistry and Physics, 54th Ed, p F-76) is usually used to make the reaction proceed quickly.Parkes, G.D. and Mellor, J.W. (1939). The saturation vapour pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation. An understanding of vapour pressure is also relevant in explaining high altitude breathing and cavitation. ==Approximation formulas == There are many published approximations for calculating saturated vapour pressure over water and over ice. AP, Amsterdam. http://store.elsevier.com/Principles-of-Environmental-Physics/John- Monteith/isbn-9780080924793/ ::P = 0.61078 \exp\left(\frac{17.27 T}{T + 237.3}\right), where temperature is in degrees Celsius (°C) and saturation vapor pressure is in kilopascals (kPa). With exhaust gas temperatures as low as 120 °C and lime temperature at kiln outlet in 80 °C range the heat loss of the regenerative kiln is minimal, fuel consumption is as low as 3.6 MJ/kg. ",8.7,650000,"""0.6957""",4.5,1,A
+"To get an idea of the distance dependence of the tunnelling current in STM, suppose that the wavefunction of the electron in the gap between sample and needle is given by $\psi=B \mathrm{e}^{-\kappa x}$, where $\kappa=\left\{2 m_{\mathrm{e}}(V-E) / \hbar^2\right\}^{1 / 2}$; take $V-E=2.0 \mathrm{eV}$. By what factor would the current drop if the needle is moved from $L_1=0.50 \mathrm{~nm}$ to $L_2=0.60 \mathrm{~nm}$ from the surface?","Tunneling current is exponentially dependent on the separation of the sample and the tip, typically reducing by an order of magnitude when the separation is increased by 1 Å (0.1 nm). How the tunneling current depends on distance between the two electrodes is contained in the tunneling matrix element : M_{\mu u} = \int_{z > z_0} \psi^S_\mu \,U_T\, {\psi^T_ u}^* \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. The current due to an applied voltage V (assume tunneling occurs from the sample to the tip) depends on two factors: 1) the number of electrons between the Fermi level EF and EF − eV in the sample, and 2) the number among them which have corresponding free states to tunnel into on the other side of the barrier at the tip. The exponential dependence of the tunneling current on the separation of the electrodes comes from the very wave functions that leak through the potential step at the surface and exhibit exponential decay into the classically forbidden region outside of the material. Then the tunneling current is simply the convolution of the densities of states of the sample surface and the tip: : I_t \propto \int_0^{eV} \rho_S(E_F - eV + \varepsilon) \, \rho_T(E_F + \varepsilon) \, d \varepsilon. Because of this, even when tunneling occurs from a non-ideally sharp tip, the dominant contribution to the current is from its most protruding atom or orbital. === Tunneling between two conductors === thumb|300x300px|Negative sample bias V raises its electronic levels by e⋅V. Using the modified Bardeen transfer Hamiltonian method, which treats tunneling as a perturbation, the tunneling current (I) is found to be where f\left(E\right) is the Fermi distribution function, \rho_s and \rho_T are the density of states (DOS) in the sample and tip, respectively, and M_{\mu u} is the tunneling matrix element between the modified wavefunctions of the tip and the sample surface. The tunneling electric current will be a small fraction of the impinging current. The main result was that the tunneling current is proportional to the local density of states of the sample at the Fermi level taken at the position of the center of curvature of a spherically symmetric tip (s-wave tip model). Using the WKB approximation, equations (5) and (7), we obtain:R. J. Hamers, “STM on Semiconductors,” from Scanning Tunneling Microscopy I, Springer Series in Surface Sciences 20, Ed. by H. -J. Güntherodt and R. Wiesendanger, Berlin: Springer-Verlag, 1992. The tunneling current can be related to the density of available or filled states in the sample. The tunneling current from a single level is therefore : j_t = \left[\frac{4k\kappa}{k^2 + \kappa^2}\right]^2 \, \frac{\hbar k}{m_e}\,e^{-2\kappa w}, where both wave vectors depend on the level's energy E, k = \tfrac{1}{\hbar} \sqrt{2m_eE}, and \kappa = \tfrac{1}{\hbar}\sqrt{2m_e(U - E)}. Since both the tunneling current, equation (5), and the conductance, equation (7), depend on the tip DOS and the tunneling transition probability, T, quantitative information about the sample DOS is very difficult to obtain. Since the tip-sample bias range in tunneling experiments is limited to \pm\phi/e, where \phi is the apparent barrier height, STM and STS only sample valence electron states. If at some point the tunneling current is below the set level, the tip is moved towards the sample, and conversely. For these ideal assumptions, the tunneling conductance is directly proportional to the sample DOS. With the height of the tip fixed, the electron tunneling current is then measured as a function of electron energy by varying the voltage between the tip and the sample (the tip to sample voltage sets the electron energy). The resulting tunneling current is a function of the tip position, applied voltage, and the local density of states (LDOS) of the sample. The current of tunneling electrons at each instance is therefore proportional to |c_ u(t + \mathrm{d}t)|^2 - |c_ u(t)|^2 divided by \mathrm{d}t, which is the time derivative of |c_ u(t)|^2, : \Gamma_{\mu \to u}\ \overset{\text{def}}{=}\ \frac{\mathrm{d}}{\mathrm{d}t} |c_ u(t)|^2 = \frac{2\pi}{\hbar} |M_{\mu u}|^2\frac{\sin\big[(E^S_\mu - E^T_ u) \tfrac{t}{\hbar}\big]}{\pi(E^S_\mu - E^T_ u)}. The tunneling current is therefore the sum of little contributions over all these energies of the product of three factors: 2e \cdot \rho_S(E_F - eV + \varepsilon)\,\mathrm{d}\varepsilon representing available electrons, f(E_F - eV + \varepsilon) - f(E_F + \varepsilon) for those that are allowed to tunnel, and the probability factor \Gamma for those that will actually tunnel: : I_t = \frac{4 \pi e}{\hbar} \int_{-\infty}^{+\infty} [f(E_F - eV + \varepsilon) - f(E_F + \varepsilon)] \, \rho_S(E_F - eV + \varepsilon) \, \rho_T(E_F + \varepsilon) \, |M|^2 \, d \varepsilon. There is however also an important correction to the elastic component of the tunneling current at the onset. As the tip is moved across the surface in a discrete x–y matrix, the changes in surface height and population of the electronic states cause changes in the tunneling current. ",-30,0.14,"""0.59""",0.23,0.333333,D
 "Calculate the typical wavelength of neutrons that have reached thermal equilibrium with their surroundings at $373 \mathrm{~K}$.
-","After a number of collisions with nuclei (scattering) in a medium (neutron moderator) at this temperature, those neutrons which are not absorbed reach about this energy level. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. The large wavelength of slow neutrons allows for the large cross section. == Neutron energy distribution ranges == Neutron energy range names Neutron energy Energy range 0.0 – 0.025 eV Cold (slow) neutrons 0.025 eV Thermal neutrons (at 20°C) 0.025–0.4 eV Epithermal neutrons 0.4–0.5 eV Cadmium neutrons 0.5–10 eV Epicadmium neutrons 10–300 eV Resonance neutrons 300 eV–1 MeV Intermediate neutrons 1–20 MeV Fast neutrons > 20 MeV Ultrafast neutrons But different ranges with different names are observed in other sources. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. The following is a detailed classification: === Thermal === A thermal neutron is a free neutron with a kinetic energy of about 0.025 eV (about 4.0×10−21 J or 2.4 MJ/kg, hence a speed of 2.19 km/s), which is the energy corresponding to the most probable speed at a temperature of 290 K (17 °C or 62 °F), the mode of the Maxwell–Boltzmann distribution for this temperature, Epeak = 1/2 k T. The term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. Introduction of the Theory of Thermal Neutron Scattering. https://books.google.com/books?id=KUVD8KJt7_0C&dq;=thermal- neutron+reactor&pg;=PR9 thus scattering neutrons by nuclear forces, some nuclides are scattered large. The neutron energy distribution is then adapted to the Maxwell distribution known for thermal motion. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. The neutron fluence is defined as the neutron flux integrated over a certain time period, so its usual unit is cm−2 (neutrons per centimeter squared). Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. Efficient neutron optical components are being developed and optimized to remedy this lack. ===Resonance=== :*Refers to neutrons which are strongly susceptible to non-fission capture by U-238. :*1 eV to 300 eV ===Intermediate=== :*Neutrons that are between slow and fast :*Few hundred eV to 0.5 MeV. ===Fast=== : A fast neutron is a free neutron with a kinetic energy level close to 1 MeV (100 TJ/kg), hence a speed of 14,000 km/s or higher. A thermal-neutron reactor is a nuclear reactor that uses slow or thermal neutrons. The measured quantity is the difference in the number of gamma rays emitted within a solid angle between the two neutron spin states. (""Thermal"" does not mean hot in an absolute sense, but means in thermal equilibrium with the medium it is interacting with, the reactor's fuel, moderator and structure, which is much lower energy than the fast neutrons initially produced by fission.) The momentum and wavelength of the neutron are related through the de Broglie relation. Thermal neutrons have a different and sometimes much larger effective neutron absorption cross-section for a given nuclide than fast neutrons, and can therefore often be absorbed more easily by an atomic nucleus, creating a heavier, often unstable isotope of the chemical element as a result. The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. Earth atmospheric neutron flux, apparently from thunderstorms, can reach levels of 3·10−2 to 9·10+1 cm−2 s−1. The usual unit is cm−2s−1 (neutrons per centimeter squared per second). ",226,48.6,16.0,27.211,6.283185307,A
-"Calculate the separation of the $\{123\}$ planes of an orthorhombic unit cell with $a=0.82 \mathrm{~nm}, b=0.94 \mathrm{~nm}$, and $c=0.75 \mathrm{~nm}$.","Bravais lattice Primitive orthorhombic Base-centered orthorhombic Body-centered orthorhombic Face-centered orthorhombic Pearson symbol oP oS oI oF Unit cell Orthohombic, simple Orthohombic, base-centered Orthohombic, body-centered Orthohombic, face- centered For the base-centered orthorhombic lattice, the primitive cell has the shape of a right rhombic prism;See , row oC, column Primitive, where the cell parameters are given as a1 = a2, α = β = 90° it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes. Assuming that the particle is spherical, a simple analysis to calculate critical separation particle sizes can be established. Carl Friedrich Gauss, in his treatise Allgemeine Theorie des Erdmagnetismus, presented a method, the Gauss separation algorithm, of partitioning the magnetic field vector, B(r, \theta, \phi), measured over the surface of a sphere into two components, internal and external, arising from electric currents (per the Biot–Savart law) flowing in the volumes interior and exterior to the spherical surface, respectively. thumb|Projections of K-cells onto the plane (from k=1 to 6). Only the edges of the higher-dimensional cells are shown. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. Note that the length a of the primitive cell below equals \frac{1}{2} \sqrt{a^2+b^2} of the conventional cell above. ==Crystal classes== The orthorhombic crystal system class names, examples, Schönflies notation, Hermann-Mauguin notation, point groups, International Tables for Crystallography space group number, orbifold notation, type, and space groups are listed in the table below. thumb|350px|right|The three possible plane-line relationships in three dimensions. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal. ==Bravais lattices== There are four orthorhombic Bravais lattices: primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic. In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Bravais lattice Rectangular Centered rectangular Pearson symbol op oc Unit cell 100px 100px ==See also== *Crystal structure *Crystal system *Overview of all space groups ==References== ==Further reading== * * Category:Crystal systems thumb|A partially demolished factory with dominating cyclonic separators Cyclonic separation is a method of removing particulates from an air, gas or liquid stream, without the use of filters, through vortex separation. The sides and edges of a k-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells. == Notes == == References == * * Category:Basic concepts in set theory Category:Compactness (mathematics) Intl Orb. Cox. Type Example Primitive Base- centered Face-centered Body-centered 16–24 Rhombic disphenoidal D2 (V) 222 222 [2,2]+ Enantiomorphic Epsomite P222, P2221, P21212, P212121 C2221, C222 F222 I222, I212121 25–46 Rhombic pyramidal C2v mm2 *22 [2] Polar Hemimorphite, bertrandite Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2 Cmm2, Cmc21, Ccc2 Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2 47–74 Rhombic dipyramidal D2h (Vh) mmm *222 [2,2] Centrosymmetric Olivine, aragonite, marcasite Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmca, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma == In two dimensions == In two dimensions there are two orthorhombic Bravais lattices: primitive rectangular and centered rectangular. One such method for separating cuticles from a rock matrix is acid maceration, which involves soaking the sample in agents such as dilute hydrogen peroxide or hydrochloric and hydrofluoric acid (known as the HCI/HF protocol) to break down the matrix. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid. (Shown in each case is only a portion of the plane, which extends infinitely far.) It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Substituting the equation for the line into the equation for the plane gives :((\mathbf{l_0} + \mathbf{l}\ d) - \mathbf{p_0})\cdot\mathbf{n} = 0. A k-cell is a higher-dimensional version of a rectangle or rectangular solid. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera. The method employs spherical harmonics. ",0.69,0.71,0.16,0.21,-1.78,D
-Calculate the moment of inertia of an $\mathrm{H}_2 \mathrm{O}$ molecule around the axis defined by the bisector of the $\mathrm{HOH}$ angle (3). The $\mathrm{HOH}$ bond angle is $104.5^{\circ}$ and the bond length is $95.7 \mathrm{pm}$.,"When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. Description Figure Moment(s) of inertia Point mass M at a distance r from the axis of rotation. The moment of inertia I_\mathbf{C} about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. Height (h) and breadth (b) are the linear measures, except for circles, which are effectively half-breadth derived, r === Sectional areas moment calculated thus === # Square: I_{xx}=I_{yy}=\frac{b^4}{12} # Rectangular: I_{xx}=\frac{bh^3}{12} and; I_{yy}=\frac{hb^3}{12} # Triangular: I_{xx}=\frac{bh^3}{36} # Circular: I_{xx}=I_{yy}=\frac{1}{4} {\pi} r^4=\frac{1}{64} {\pi} d^4 == Motion in a fixed plane == === Point mass === The moment of inertia about an axis of a body is calculated by summing mr^2 for every particle in the body, where r is the perpendicular distance to the specified axis. File:moment of inertia thick cylinder h.svg I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right) Classical Mechanics - Moment of inertia of a uniform hollow cylinder . A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. frameless|upright I = M r^2 Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles. frameless|upright I = \frac{ m_1 m_2 }{ m_1 \\! + \\! m_2 } x^2 = \mu x^2 Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center. The moment of inertia relative to the axis , which is at a distance from the center of mass along the x-axis, is :I = \int \left[(x - D)^2 + y^2\right] \, dm. Expanding the brackets yields :I = \int (x^2 + y^2) \, dm + D^2 \int dm - 2D\int x\, dm. The moment of inertia relative to the z-axis is then :I_\mathrm{cm} = \int (x^2 + y^2) \, dm. This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body. == Inertia tensor == For the same object, different axes of rotation will have different moments of inertia about those axes. The moment of inertia of a body with the shape of the cross- section is the second moment of this area about the z-axis perpendicular to the cross-section, weighted by its density. File:moment of inertia solid rectangular prism.png I_h = \frac{1}{12} m \left(w^2+d^2\right) I_w = \frac{1}{12} m \left(d^2+h^2\right) I_d = \frac{1}{12} m \left(w^2+h^2\right) Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. thumb|Lewis Structure of H2O indicating bond angle and bond length Water () is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. * The moment of inertia of a thin disc of constant thickness s, radius R, and density \rho about an axis through its center and perpendicular to its face (parallel to its axis of rotational symmetry) is determined by integration. Moment of inertia of potentially tilted cuboids. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. The parallel axis theorem, also known as Huygens-Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes. ==Mass moment of inertia== thumb|right|The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the center of mass. A product of inertia term such as I_{12} is obtained by the computation I_{12} = \mathbf{e}_1\cdot\mathbf{I}\cdot\mathbf{e}_2, and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis. This shows that the moment of inertia of the body is the sum of each of the mr^2 terms, that is I_P = \sum_{i=1}^N m_i r_i^2. In order to obtain the moment of inertia IS in terms of the moment of inertia IR, introduce the vector d from S to the center of mass R, : \begin{align} I_S & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}+\mathbf{d})\cdot (\mathbf{r}-\mathbf{R}+\mathbf{d}) \, dV \\\ & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R})\cdot (\mathbf{r}-\mathbf{R})dV + 2\mathbf{d}\cdot\left(\int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}) \, dV\right) + \left(\int_V \rho(\mathbf{r}) \, dV\right)\mathbf{d}\cdot\mathbf{d}. \end{align} The first term is the moment of inertia IR, the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector d. * The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. File:moment of inertia thin cylinder.png I \approx m r^2 \,\\! ",5.4,1.91,54.7,4,0.1800,B
+","After a number of collisions with nuclei (scattering) in a medium (neutron moderator) at this temperature, those neutrons which are not absorbed reach about this energy level. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. The large wavelength of slow neutrons allows for the large cross section. == Neutron energy distribution ranges == Neutron energy range names Neutron energy Energy range 0.0 – 0.025 eV Cold (slow) neutrons 0.025 eV Thermal neutrons (at 20°C) 0.025–0.4 eV Epithermal neutrons 0.4–0.5 eV Cadmium neutrons 0.5–10 eV Epicadmium neutrons 10–300 eV Resonance neutrons 300 eV–1 MeV Intermediate neutrons 1–20 MeV Fast neutrons > 20 MeV Ultrafast neutrons But different ranges with different names are observed in other sources. If is the number of dimensions, and the relationship between energy () and momentum () is given by E=ap^s (with and being constants), then the thermal wavelength is defined as \lambda_{\mathrm{th}}=\frac{h}{\sqrt{\pi}}\left(\frac{a}{k_{\mathrm B}T}\right)^{1/s} \left[\frac{\Gamma(n/2+1)}{\Gamma(n/s+1)}\right]^{1/n} , where is the Gamma function. The following is a detailed classification: === Thermal === A thermal neutron is a free neutron with a kinetic energy of about 0.025 eV (about 4.0×10−21 J or 2.4 MJ/kg, hence a speed of 2.19 km/s), which is the energy corresponding to the most probable speed at a temperature of 290 K (17 °C or 62 °F), the mode of the Maxwell–Boltzmann distribution for this temperature, Epeak = 1/2 k T. The term temperature is used, since hot, thermal and cold neutrons are moderated in a medium with a certain temperature. Such is the case for molecular or atomic gases at room temperature, and for thermal neutrons produced by a neutron source. ==Massive particles== For massive, non-interacting particles, the thermal de Broglie wavelength can be derived from the calculation of the partition function. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. Introduction of the Theory of Thermal Neutron Scattering. https://books.google.com/books?id=KUVD8KJt7_0C&dq;=thermal- neutron+reactor&pg;=PR9 thus scattering neutrons by nuclear forces, some nuclides are scattered large. The neutron energy distribution is then adapted to the Maxwell distribution known for thermal motion. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. The neutron fluence is defined as the neutron flux integrated over a certain time period, so its usual unit is cm−2 (neutrons per centimeter squared). Qualitatively, the higher the temperature, the higher the kinetic energy of the free neutrons. Efficient neutron optical components are being developed and optimized to remedy this lack. ===Resonance=== :*Refers to neutrons which are strongly susceptible to non-fission capture by U-238. :*1 eV to 300 eV ===Intermediate=== :*Neutrons that are between slow and fast :*Few hundred eV to 0.5 MeV. ===Fast=== : A fast neutron is a free neutron with a kinetic energy level close to 1 MeV (100 TJ/kg), hence a speed of 14,000 km/s or higher. A thermal-neutron reactor is a nuclear reactor that uses slow or thermal neutrons. The measured quantity is the difference in the number of gamma rays emitted within a solid angle between the two neutron spin states. (""Thermal"" does not mean hot in an absolute sense, but means in thermal equilibrium with the medium it is interacting with, the reactor's fuel, moderator and structure, which is much lower energy than the fast neutrons initially produced by fission.) The momentum and wavelength of the neutron are related through the de Broglie relation. Thermal neutrons have a different and sometimes much larger effective neutron absorption cross-section for a given nuclide than fast neutrons, and can therefore often be absorbed more easily by an atomic nucleus, creating a heavier, often unstable isotope of the chemical element as a result. The neutron detection temperature, also called the neutron energy, indicates a free neutron's kinetic energy, usually given in electron volts. Earth atmospheric neutron flux, apparently from thunderstorms, can reach levels of 3·10−2 to 9·10+1 cm−2 s−1. The usual unit is cm−2s−1 (neutrons per centimeter squared per second). ",226,48.6,"""16.0""",27.211,6.283185307,A
+"Calculate the separation of the $\{123\}$ planes of an orthorhombic unit cell with $a=0.82 \mathrm{~nm}, b=0.94 \mathrm{~nm}$, and $c=0.75 \mathrm{~nm}$.","Bravais lattice Primitive orthorhombic Base-centered orthorhombic Body-centered orthorhombic Face-centered orthorhombic Pearson symbol oP oS oI oF Unit cell Orthohombic, simple Orthohombic, base-centered Orthohombic, body-centered Orthohombic, face- centered For the base-centered orthorhombic lattice, the primitive cell has the shape of a right rhombic prism;See , row oC, column Primitive, where the cell parameters are given as a1 = a2, α = β = 90° it can be constructed because the two-dimensional centered rectangular base layer can also be described with primitive rhombic axes. Assuming that the particle is spherical, a simple analysis to calculate critical separation particle sizes can be established. Carl Friedrich Gauss, in his treatise Allgemeine Theorie des Erdmagnetismus, presented a method, the Gauss separation algorithm, of partitioning the magnetic field vector, B(r, \theta, \phi), measured over the surface of a sphere into two components, internal and external, arising from electric currents (per the Biot–Savart law) flowing in the volumes interior and exterior to the spherical surface, respectively. thumb|Projections of K-cells onto the plane (from k=1 to 6). Only the edges of the higher-dimensional cells are shown. Orthorhombic lattices result from stretching a cubic lattice along two of its orthogonal pairs by two different factors, resulting in a rectangular prism with a rectangular base (a by b) and height (c), such that a, b, and c are distinct. Note that the length a of the primitive cell below equals \frac{1}{2} \sqrt{a^2+b^2} of the conventional cell above. ==Crystal classes== The orthorhombic crystal system class names, examples, Schönflies notation, Hermann-Mauguin notation, point groups, International Tables for Crystallography space group number, orbifold notation, type, and space groups are listed in the table below. thumb|350px|right|The three possible plane-line relationships in three dimensions. All three bases intersect at 90° angles, so the three lattice vectors remain mutually orthogonal. ==Bravais lattices== There are four orthorhombic Bravais lattices: primitive orthorhombic, base-centered orthorhombic, body-centered orthorhombic, and face-centered orthorhombic. In crystallography, the orthorhombic crystal system is one of the 7 crystal systems. Bravais lattice Rectangular Centered rectangular Pearson symbol op oc Unit cell 100px 100px ==See also== *Crystal structure *Crystal system *Overview of all space groups ==References== ==Further reading== * * Category:Crystal systems thumb|A partially demolished factory with dominating cyclonic separators Cyclonic separation is a method of removing particulates from an air, gas or liquid stream, without the use of filters, through vortex separation. The sides and edges of a k-cell need not be equal in (Euclidean) length; although the unit cube (which has boundaries of equal Euclidean length) is a 3-cell, the set of all 3-cells with equal-length edges is a strict subset of the set of all 3-cells. == Notes == == References == * * Category:Basic concepts in set theory Category:Compactness (mathematics) Intl Orb. Cox. Type Example Primitive Base- centered Face-centered Body-centered 16–24 Rhombic disphenoidal D2 (V) 222 222 [2,2]+ Enantiomorphic Epsomite P222, P2221, P21212, P212121 C2221, C222 F222 I222, I212121 25–46 Rhombic pyramidal C2v mm2 *22 [2] Polar Hemimorphite, bertrandite Pmm2, Pmc21, Pcc2, Pma2, Pca21, Pnc2, Pmn21, Pba2, Pna21, Pnn2 Cmm2, Cmc21, Ccc2 Amm2, Aem2, Ama2, Aea2 Fmm2, Fdd2 Imm2, Iba2, Ima2 47–74 Rhombic dipyramidal D2h (Vh) mmm *222 [2,2] Centrosymmetric Olivine, aragonite, marcasite Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma Cmcm, Cmca, Cmmm, Cccm, Cmme, Ccce Fmmm, Fddd Immm, Ibam, Ibca, Imma == In two dimensions == In two dimensions there are two orthorhombic Bravais lattices: primitive rectangular and centered rectangular. One such method for separating cuticles from a rock matrix is acid maceration, which involves soaking the sample in agents such as dilute hydrogen peroxide or hydrochloric and hydrofluoric acid (known as the HCI/HF protocol) to break down the matrix. A 2-cell is the rectangle formed by the Cartesian product of two closed intervals, and a 3-cell is a rectangular solid. (Shown in each case is only a portion of the plane, which extends infinitely far.) It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Substituting the equation for the line into the equation for the plane gives :((\mathbf{l_0} + \mathbf{l}\ d) - \mathbf{p_0})\cdot\mathbf{n} = 0. A k-cell is a higher-dimensional version of a rectangle or rectangular solid. In vision-based 3D reconstruction, a subfield of computer vision, depth values are commonly measured by so-called triangulation method, which finds the intersection between light plane and ray reflected toward camera. The method employs spherical harmonics. ",0.69,0.71,"""0.16""",0.21,-1.78,D
+Calculate the moment of inertia of an $\mathrm{H}_2 \mathrm{O}$ molecule around the axis defined by the bisector of the $\mathrm{HOH}$ angle (3). The $\mathrm{HOH}$ bond angle is $104.5^{\circ}$ and the bond length is $95.7 \mathrm{pm}$.,"When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems. Description Figure Moment(s) of inertia Point mass M at a distance r from the axis of rotation. The moment of inertia I_\mathbf{C} about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. Height (h) and breadth (b) are the linear measures, except for circles, which are effectively half-breadth derived, r === Sectional areas moment calculated thus === # Square: I_{xx}=I_{yy}=\frac{b^4}{12} # Rectangular: I_{xx}=\frac{bh^3}{12} and; I_{yy}=\frac{hb^3}{12} # Triangular: I_{xx}=\frac{bh^3}{36} # Circular: I_{xx}=I_{yy}=\frac{1}{4} {\pi} r^4=\frac{1}{64} {\pi} d^4 == Motion in a fixed plane == === Point mass === The moment of inertia about an axis of a body is calculated by summing mr^2 for every particle in the body, where r is the perpendicular distance to the specified axis. File:moment of inertia thick cylinder h.svg I_z = \frac{1}{2} m \left(r_2^2 + r_1^2\right) = m r_2^2 \left(1-t+\frac{t^2}{2}\right) Classical Mechanics - Moment of inertia of a uniform hollow cylinder . A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved. frameless|upright I = M r^2 Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles. frameless|upright I = \frac{ m_1 m_2 }{ m_1 \\! + \\! m_2 } x^2 = \mu x^2 Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center. The moment of inertia relative to the axis , which is at a distance from the center of mass along the x-axis, is :I = \int \left[(x - D)^2 + y^2\right] \, dm. Expanding the brackets yields :I = \int (x^2 + y^2) \, dm + D^2 \int dm - 2D\int x\, dm. The moment of inertia relative to the z-axis is then :I_\mathrm{cm} = \int (x^2 + y^2) \, dm. This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body. == Inertia tensor == For the same object, different axes of rotation will have different moments of inertia about those axes. The moment of inertia of a body with the shape of the cross- section is the second moment of this area about the z-axis perpendicular to the cross-section, weighted by its density. File:moment of inertia solid rectangular prism.png I_h = \frac{1}{12} m \left(w^2+d^2\right) I_w = \frac{1}{12} m \left(d^2+h^2\right) I_d = \frac{1}{12} m \left(w^2+h^2\right) Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. thumb|Lewis Structure of H2O indicating bond angle and bond length Water () is a simple triatomic bent molecule with C2v molecular symmetry and bond angle of 104.5° between the central oxygen atom and the hydrogen atoms. * The moment of inertia of a thin disc of constant thickness s, radius R, and density \rho about an axis through its center and perpendicular to its face (parallel to its axis of rotational symmetry) is determined by integration. Moment of inertia of potentially tilted cuboids. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. The parallel axis theorem, also known as Huygens-Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes. ==Mass moment of inertia== thumb|right|The mass moment of inertia of a body around an axis can be determined from the mass moment of inertia around a parallel axis through the center of mass. A product of inertia term such as I_{12} is obtained by the computation I_{12} = \mathbf{e}_1\cdot\mathbf{I}\cdot\mathbf{e}_2, and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis. This shows that the moment of inertia of the body is the sum of each of the mr^2 terms, that is I_P = \sum_{i=1}^N m_i r_i^2. In order to obtain the moment of inertia IS in terms of the moment of inertia IR, introduce the vector d from S to the center of mass R, : \begin{align} I_S & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}+\mathbf{d})\cdot (\mathbf{r}-\mathbf{R}+\mathbf{d}) \, dV \\\ & = \int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R})\cdot (\mathbf{r}-\mathbf{R})dV + 2\mathbf{d}\cdot\left(\int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R}) \, dV\right) + \left(\int_V \rho(\mathbf{r}) \, dV\right)\mathbf{d}\cdot\mathbf{d}. \end{align} The first term is the moment of inertia IR, the second term is zero by definition of the center of mass, and the last term is the total mass of the body times the square magnitude of the vector d. * The moment of inertia of a thin rod with constant cross-section s and density \rho and with length \ell about a perpendicular axis through its center of mass is determined by integration. File:moment of inertia thin cylinder.png I \approx m r^2 \,\\! ",5.4,1.91,"""54.7""",4,0.1800,B
 "The data below show the temperature variation of the equilibrium constant of the reaction $\mathrm{Ag}_2 \mathrm{CO}_3(\mathrm{~s}) \rightleftharpoons \mathrm{Ag}_2 \mathrm{O}(\mathrm{s})+\mathrm{CO}_2(\mathrm{~g})$. Calculate the standard reaction enthalpy of the decomposition.
-$\begin{array}{lllll}T / \mathrm{K} & 350 & 400 & 450 & 500 \\ K & 3.98 \times 10^{-4} & 1.41 \times 10^{-2} & 1.86 \times 10^{-1} & 1.48\end{array}$","The standard enthalpy change ΔH⚬ is essentially the enthalpy change when the stoichiometric coefficients in the reaction are considered as the amounts of reactants and products (in mole); usually, the initial and final temperature is assumed to be 25 °C. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. The standard enthalpy of formation is then determined using Hess's law. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. The enthalpy of reaction can then be analyzed by applying Hess's Law, which states that the sum of the enthalpy changes for a number of individual reaction steps equals the enthalpy change of the overall reaction. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. For this reason it is important to note which standard concentration value is being used when consulting tables of enthalpies of formation. ==Introduction== Two initial thermodynamic systems, each isolated in their separate states of internal thermodynamic equilibrium, can, by a thermodynamic operation, be coalesced into a single new final isolated thermodynamic system. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions. ",+116.0,2600,-36.5,3.141592,+80,E
+$\begin{array}{lllll}T / \mathrm{K} & 350 & 400 & 450 & 500 \\ K & 3.98 \times 10^{-4} & 1.41 \times 10^{-2} & 1.86 \times 10^{-1} & 1.48\end{array}$","The standard enthalpy change ΔH⚬ is essentially the enthalpy change when the stoichiometric coefficients in the reaction are considered as the amounts of reactants and products (in mole); usually, the initial and final temperature is assumed to be 25 °C. Petrucci, Harwood and Herring, pages 241–243 Finally the reaction enthalpy may be estimated using bond energies for the bonds which are broken and formed in the reaction of interest. The standard enthalpy of formation is then determined using Hess's law. *The change in enthalpy for a reaction can be calculated from the enthalpies of formation of the reactants and the products *Elements in their standard states make no contribution to the enthalpy calculations for the reaction, since the enthalpy of an element in its standard state is zero. If the enthalpies for each step can be measured, then their sum gives the enthalpy of the overall single reaction. Solving for the standard of enthalpy of formation, :\Delta_\text{f} H^\ominus (\text{CH}_4) = [ \Delta_\text{f} H^\ominus (\text{CO}_2) + 2 \Delta_\text{f} H^\ominus (\text{H}_2 \text{O})] - \Delta_\text{comb} H^\ominus (\text{CH}_4). Accordingly, the calculation of standard enthalpy of reaction is the most established way of quantifying the conversion of chemical potential energy into thermal energy. ==Enthalpy of reaction for standard conditions defined and measured== The standard enthalpy of a reaction is defined so as to depend simply upon the standard conditions that are specified for it, not simply on the conditions under which the reactions actually occur. For tabulation purposes, standard formation enthalpies are all given at a single temperature: 298 K, represented by the symbol . == Hess's law == For many substances, the formation reaction may be considered as the sum of a number of simpler reactions, either real or fictitious. The standard enthalpy of reaction (denoted \Delta_{\text {rxn}} H^\ominus or \Delta H_{\text {reaction}}^\ominus) for a chemical reaction is the difference between total reactant and total product molar enthalpies, calculated for substances in their standard states. This calculation has a tacit assumption of ideal solution between reactants and products where the enthalpy of mixing is zero. For example, the standard enthalpy of combustion of ethane gas refers to the reaction C2H6 (g) + (7/2) O2 (g) → 2 CO2 (g) + 3 H2O (l). For a generic chemical reaction : u_{\text {A}} \text {A} + u_{\,\text {B}} \text {B} ~+ ~... \rightarrow u_{\,\text {X}} \text {X} + u_{\text {Y}} \text {Y} ~+ ~... the standard enthalpy of reaction \Delta_{\text {rxn}} H^\ominus is related to the standard enthalpy of formation \Delta_{\text {f}} H^\ominus values of the reactants and products by the following equation: : \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_p\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_r\Delta_{\text {f}} H_{r}^{\ominus} In this equation, u_p and u_r are the stoichiometric coefficients of each product p and reactant r. The enthalpy of reaction can then be analyzed by applying Hess's Law, which states that the sum of the enthalpy changes for a number of individual reaction steps equals the enthalpy change of the overall reaction. For example, the standard enthalpy of formation of carbon dioxide is the enthalpy of the following reaction under the above conditions: :C(s, graphite) + O2(g) -> CO2(g) All elements are written in their standard states, and one mole of product is formed. The enthalpy of reaction is then found from the van 't Hoff equation as \Delta_{\text {rxn}} H^\ominus = {RT^2}\frac{d}{dT} \ln K_\mathrm{eq}. However O2 is an element in its standard state, so that \Delta_{\text{f}} H^{\ominus }(\text{O}_2) = 0, and the heat of reaction is simplified to :\Delta_{\text{r}} H^{\ominus } = [\Delta_{\text{f}} H^{\ominus }(\text{CO}_2) + 2\Delta_{\text{f}} H^{\ominus } (\text{H}_2{}\text{O})] - \Delta_{\text{f}} H^{\ominus }(\text{CH}_4), which is the equation in the previous section for the enthalpy of combustion \Delta_{\text{comb}}H^{\ominus }. == Key concepts for enthalpy calculations == *When a reaction is reversed, the magnitude of ΔH stays the same, but the sign changes. Regardless, if each reactant and product can be prepared in its respective standard state, then the contribution of each species is equal to its molar enthalpy of formation multiplied by its stoichiometric coefficient in the reaction, and the enthalpy of reaction at constant (standard) pressure P^{\ominus} and constant temperature (usually 298 K) may be written as : Q_{P^{\ominus}} = \Delta_{\text {rxn}} H^\ominus = \sum_{products,~p} u_{p}\Delta_{\text {f}} H_{p}^{\ominus} - \sum_{reactants,~r} u_{r}\Delta_{\text {f}} H_{r}^{\ominus} As shown above, at constant pressure the heat of the reaction is exactly equal to the enthalpy change, \Delta_{\text {rxn}} H, of the reacting system. == Variation with temperature or pressure == The variation of the enthalpy of reaction with temperature is given by Kirchhoff's Law of Thermochemistry, which states that the temperature derivative of ΔH for a chemical reaction is given by the difference in heat capacity (at constant pressure) between products and reactants: :\left(\frac{\partial \Delta H}{\partial T}\right)_p = \Delta C_p. Since enthalpy is a state function, its value is the same for any path between given initial and final states, so that the measured ΔH is the same as if the temperature stayed constant during the combustion.Engel and Reid p.65 For reactions which are incomplete, the equilibrium constant can be determined as a function of temperature. * Standard enthalpy of combustion is the enthalpy change when one mole of an organic compound reacts with molecular oxygen (O2) to form carbon dioxide and liquid water. Since the pressure of the standard formation reaction is fixed at 1 bar, the standard formation enthalpy or reaction heat is a function of temperature. For this reason it is important to note which standard concentration value is being used when consulting tables of enthalpies of formation. ==Introduction== Two initial thermodynamic systems, each isolated in their separate states of internal thermodynamic equilibrium, can, by a thermodynamic operation, be coalesced into a single new final isolated thermodynamic system. It is also possible to evaluate the enthalpy of one reaction from the enthalpies of a number of other reactions whose sum is the reaction of interest, and these not need be formation reactions. ",+116.0,2600,"""-36.5""",3.141592,+80,E
 "The osmotic pressures of solutions of poly(vinyl chloride), PVC, in cyclohexanone at $298 \mathrm{~K}$ are given below. The pressures are expressed in terms of the heights of solution (of mass density $\rho=0.980 \mathrm{~g} \mathrm{~cm}^{-3}$ ) in balance with the osmotic pressure. Determine the molar mass of the polymer.
 $\begin{array}{llllll}c /\left(\mathrm{g} \mathrm{dm}^{-3}\right) & 1.00 & 2.00 & 4.00 & 7.00 & 9.00 \\ h / \mathrm{cm} & 0.28 & 0.71 & 2.01 & 5.10 & 8.00\end{array}$
-","The mass- average molecular mass, , is also related to the fractional monomer conversion, , in step-growth polymerization (for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per Carothers' equation: :\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}, where is the molecular mass of the repeating unit. ===Z-average molar mass=== The z-average molar mass is the third moment or third power average molar mass, which is calculated by \bar{M}_z=\frac{\sum M_i^3 N_i} {\sum M_i^2 N_i}\quad The z-average molar mass can be determined with ultracentrifugation. If the relationship between molar mass and the hydrodynamic volume changes (i.e., the polymer is not exactly the same shape as the standard) then the calibration for mass is in error. Combining these two equations gives an expression for the molar mass in terms of the vapour density for conditions of known pressure and temperature: :M = {{RT\rho}\over{p}} . === Freezing-point depression === The freezing point of a solution is lower than that of the pure solvent, and the freezing-point depression () is directly proportional to the amount concentration for dilute solutions. The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule: :\begin{array}{lll} M(\ce{H2}) &= 2\times 1.00797(7) \times M_\mathrm{u} &= 2.01588(14) \text{ g/mol} \\\ M(\ce{S8}) &= 8\times 32.065(5) \times M_\mathrm{u} &= 256.52(4) \text{ g/mol} \\\ M(\ce{Cl2}) &= 2\times 35.453(2) \times M_\mathrm{u} &= 70.906(4) \text{ g/mol} \end{array} == Molar masses of compounds == The molar mass of a compound is given by the sum of the relative atomic mass of the atoms which form the compound multiplied by the molar mass constant : :M = M_{\rm u} M_{\rm r} = M_{\rm u} \sum_i {A_{\rm r}}_i. It is determined by measuring the molecular mass of polymer molecules, summing the masses, and dividing by . \bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i} The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry, end-group determination or proton NMR.Polymer Molecular Weight Analysis by 1H NMR Spectroscopy Josephat U. Izunobi and Clement L. Higginbotham J. Chem. Educ., 2011, 88 (8), pp 1098–1104 High number-average molecular mass polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization, as per the Carothers' equation. ===Mass average molar mass=== The mass average molar mass (often loosely termed weight average molar mass) is another way of describing the molar mass of a polymer. The molar mass distribution of a polymer may be modified by polymer fractionation. == Definitions of molar mass average == Different average values can be defined, depending on the statistical method applied. The molar mass distribution of a polymer sample depends on factors such as chemical kinetics and work-up procedure. In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species () and the molar mass () of that species.I. Katime ""Química Física Macromolecular"". This is particularly important in polymer science, where different polymer molecules may contain different numbers of monomer units (non-uniform polymers). == Average molar mass of mixtures == The average molar mass of mixtures \overline{M} can be calculated from the mole fractions of the components and their molar masses : :\overline{M} = \sum_i x_i M_i. * Molar mass: chemistry second-level course. The molecular formula C3H7Cl (molar mass: 78.54 g/mol, exact mass: 78.0236 u) may refer to: * Isopropyl chloride * n-Propyl chloride, also known as 1-propyl chloride or 1-chloropropane Thus, for example, the average mass of a molecule of water is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol. The quantity a in the expression for the viscosity average molar mass varies from 0.5 to 0.8 and depends on the interaction between solvent and polymer in a dilute solution. If represents the mass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by :M = {{wK_\text{b}}\over{\Delta T}}.\ == See also == * Mole map (chemistry) == References == ==External links== * HTML5 Molar Mass Calculator web and mobile application. The molecular formula C2H3ClO (molar mass: 78.50 g/mol, exact mass: 77.9872 u) may refer to: * Acetyl chloride * Chloroacetaldehyde * Chloroethylene oxide In chemistry, the molar mass () of a chemical compound is defined as the ratio between the mass and the amount of substance (measured in moles) of any sample of said compound. About 57% of the mass of PVC is chlorine. Examples are: \begin{array}{ll} M(\ce{NaCl}) &= \bigl[22.98976928(2) + 35.453(2)\bigr] \times 1 \text{ g/mol} \\\ &= 58.443(2) \text{ g/mol} \\\\[4pt] M(\ce{C12H22O11}) &= \bigl[12 \times 12.0107(8) + 22 \times 1.00794(7) + 11 \times 15.9994(3)\bigr] \times 1 \text{ g/mol} \\\ &= 342.297(14) \text{ g/mol} \end{array} An average molar mass may be defined for mixtures of compounds. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in and a small reduction in , thus reducing dispersity. ===Number average molar mass=== The number average molar mass is a way of determining the molecular mass of a polymer. The molecular formula C3H4Cl2O2 (molar mass: 142.97 g/mol, exact mass: 141.9588 u) may refer to: * Chloroethyl chloroformate * Dalapon The mass average molar mass is calculated by \bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} where is the number of molecules of molecular mass . * Online Molar Mass Calculator with the uncertainty of M and all the calculations shown * Molar Mass Calculator Online Molar Mass and Elemental Composition Calculator * Stoichiometry Add-In for Microsoft Excel for calculation of molecular weights, reaction coefficients and stoichiometry. ",1.56,-0.0301,0.6321205588,48,1.2,E
-Estimate the wavelength of electrons that have been accelerated from rest through a potential difference of $40 \mathrm{kV}$.,"But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. The wavelength corresponding to the mean photon energy is given by :\lambda_{\langle E \rangle} = ( cm K)/T\;. ==Recommendation to de-emphasize== Marr and Wilkin (2012) contend that the widespread teaching of Wien's displacement law in introductory courses is undesirable, and it would be better replaced by alternate material. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. Solving for the wavelength \lambda in millimetres, and using kelvins for the temperature yields: : ===Parameterization by frequency=== Another common parameterization is by frequency. Using Wien's law, one finds a peak emission per nanometer (of wavelength) at a wavelength of about 500 nm, in the green portion of the spectrum near the peak sensitivity of the human eye.Walker, J. Fundamentals of Physics, 8th ed., John Wiley and Sons, 2008, p. 891. . thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). This is a consequence of the strong statement of Wien's law. ==Frequency-dependent formulation== For spectral flux considered per unit frequency d u (in hertz), Wien's displacement law describes a peak emission at the optical frequency u_\text{peak} given by: : u_\text{peak} = { x \over h} k\,T \approx (5.879 \times 10^{10} \ \mathrm{Hz/K}) \cdot T or equivalently :h u_\text{peak} = x\, k\, T \approx (2.431 \times 10^{-4} \ \mathrm{eV/K}) \cdot T where is a constant resulting from the maximization equation, is the Boltzmann constant, is the Planck constant, and is the absolute temperature. From the Planck constant h and the Boltzmann constant k, Wien's constant b can be obtained. ==Peak differs according to parameterization== Constants for different parameterizations of Wien's law Parameterized by x_\mathrm{peak} b (μm⋅K) Wavelength, \lambda 2898 \log\lambda or \log u 3670 Frequency, u 5099 Other characterizations of spectrum Parameterized by x b (μm⋅K) Mean photon energy 5327 10% percentile 2195 25% percentile 2898 50% percentile 4107 70% percentile 5590 90% percentile 9376 The results in the tables above summarize results from other sections of this article. For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides incorrect results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above. Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material. ==The interaction picture== Define the unperturbed Hamiltonian by H_0, the time dependent perturbing Hamiltonian by H_1 and total Hamiltonian by H. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. The relevant math is detailed in the next section. ==Derivation from Planck's law== ===Parameterization by wavelength=== Planck's law for the spectrum of black body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. In case of constant perturbation,c_{k'}^{(1)} is calculated by :c_{k'}^{(1)}=\frac{\lang\ k'|H_1|k\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\hbar}) :|c_{k'}(t)|^2= |\lang\ k'|H_1|k\rang |^2\frac {sin ^2(\frac {E_{k'}-E_k} {2 \hbar}t)} { ( \frac {E_{k'} -E_k} {2 \hbar} ) ^2 }\frac {1}{\hbar^2} Using the equation which is :\lim_{\alpha \rightarrow \infty} \frac{1}{\pi} \frac{sin^2(\alpha x)}{\alpha x^2}= \delta(x) The transition rate of an electron from the initial state k to final state k' is given by :P(k,k')=\frac {2 \pi} {\hbar} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) where E_k and E_{k'} are the energies of the initial and final states including the perturbation state and ensures the \delta-function indicate energy conservation. ==The scattering rate== The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by :w(k)=\sum_{k'}P(k,k')=\frac {2 \pi} {\hbar} \sum_{k'} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) The integral form is :w(k)=\frac {2 \pi} {\hbar} \frac {L^3} {(2 \pi)^3} \int d^3k' |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) ==References== * * Category:Semiconductor technology This is perhaps a more intuitive way of presenting ""wavelength of peak emission"". Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). This is an inverse relationship between wavelength and temperature. Contrary to the then known cathode rays which reached speeds only up to 0.3c, c being the speed of light, Becquerel rays reached velocities up to 0.9c. With the emission now considered per unit frequency, this peak now corresponds to a wavelength about 76% longer than the peak considered per unit wavelength. Formally, the wavelength version of Wien's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength \lambda_\text{peak} given by: :\lambda_\text{peak} = \frac{b}{T} where is the absolute temperature and is a constant of proportionality called Wien's displacement constant, equal to or . ",-1.00,35,-0.40864,6.1,0.139,D
-The standard enthalpy of formation of $\mathrm{H}_2 \mathrm{O}(\mathrm{g})$ at $298 \mathrm{~K}$ is $-241.82 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Estimate its value at $100^{\circ} \mathrm{C}$ given the following values of the molar heat capacities at constant pressure: $\mathrm{H}_2 \mathrm{O}(\mathrm{g}): 33.58 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \mathrm{H}_2(\mathrm{~g}): 28.82 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \mathrm{O}_2(\mathrm{~g})$ : $29.36 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Assume that the heat capacities are independent of temperature.,"In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . An ideal gas has the equation of state: P V = n R T\, where :P = pressure :V = volume :n = number of moles :R = universal gas constant(Gas constant) :T = temperature The ideal gas equation of state can be arranged to give: : V = n R T / P\, or \, n R = P V / T The following partial derivatives are obtained from the above equation of state: :\left(\frac{\partial V}{\partial T}\right)_{P}\ = \frac {n R}{P}\ = \left(\frac{V P}{T}\right)\left(\frac{1}{P}\right) = \frac{V}{T} :\left(\frac{\partial V}{\partial P}\right)_{T}\ = - \frac {n R T}{P^2}\ = - \frac {P V}{P^2}\ = - \frac{V}{P} The following simple expressions are obtained for thermal expansion coefficient \alpha : :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\ = \frac{1}{V}\left(\frac{V}{T}\right) :\alpha= 1 / T \, and for isothermal compressibility \beta_{T}: :\beta_{T}= - \frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\ = - \frac{1}{V}\left( - \frac{V}{P}\right) :\beta_{T}= 1 / P \, One can now calculate C_{P} - C_{V}\, for ideal gases from the previously-obtained general formula: :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\ = V T\frac{(1 / T)^2}{1 / P} = \frac{V P}{T} Substituting from the ideal gas equation gives finally: :C_{P} - C_{V} = n R\, where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? In an isenthalpic process, the enthalpy is constant. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size- dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? ",1.07,0.1792,2.567,-242.6,0.000226,D
+","The mass- average molecular mass, , is also related to the fractional monomer conversion, , in step-growth polymerization (for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per Carothers' equation: :\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}, where is the molecular mass of the repeating unit. ===Z-average molar mass=== The z-average molar mass is the third moment or third power average molar mass, which is calculated by \bar{M}_z=\frac{\sum M_i^3 N_i} {\sum M_i^2 N_i}\quad The z-average molar mass can be determined with ultracentrifugation. If the relationship between molar mass and the hydrodynamic volume changes (i.e., the polymer is not exactly the same shape as the standard) then the calibration for mass is in error. Combining these two equations gives an expression for the molar mass in terms of the vapour density for conditions of known pressure and temperature: :M = {{RT\rho}\over{p}} . === Freezing-point depression === The freezing point of a solution is lower than that of the pure solvent, and the freezing-point depression () is directly proportional to the amount concentration for dilute solutions. The molar mass of molecules of these elements is the molar mass of the atoms multiplied by the number of atoms in each molecule: :\begin{array}{lll} M(\ce{H2}) &= 2\times 1.00797(7) \times M_\mathrm{u} &= 2.01588(14) \text{ g/mol} \\\ M(\ce{S8}) &= 8\times 32.065(5) \times M_\mathrm{u} &= 256.52(4) \text{ g/mol} \\\ M(\ce{Cl2}) &= 2\times 35.453(2) \times M_\mathrm{u} &= 70.906(4) \text{ g/mol} \end{array} == Molar masses of compounds == The molar mass of a compound is given by the sum of the relative atomic mass of the atoms which form the compound multiplied by the molar mass constant : :M = M_{\rm u} M_{\rm r} = M_{\rm u} \sum_i {A_{\rm r}}_i. It is determined by measuring the molecular mass of polymer molecules, summing the masses, and dividing by . \bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i} The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry, end-group determination or proton NMR.Polymer Molecular Weight Analysis by 1H NMR Spectroscopy Josephat U. Izunobi and Clement L. Higginbotham J. Chem. Educ., 2011, 88 (8), pp 1098–1104 High number-average molecular mass polymers may be obtained only with a high fractional monomer conversion in the case of step-growth polymerization, as per the Carothers' equation. ===Mass average molar mass=== The mass average molar mass (often loosely termed weight average molar mass) is another way of describing the molar mass of a polymer. The molar mass distribution of a polymer may be modified by polymer fractionation. == Definitions of molar mass average == Different average values can be defined, depending on the statistical method applied. The molar mass distribution of a polymer sample depends on factors such as chemical kinetics and work-up procedure. In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species () and the molar mass () of that species.I. Katime ""Química Física Macromolecular"". This is particularly important in polymer science, where different polymer molecules may contain different numbers of monomer units (non-uniform polymers). == Average molar mass of mixtures == The average molar mass of mixtures \overline{M} can be calculated from the mole fractions of the components and their molar masses : :\overline{M} = \sum_i x_i M_i. * Molar mass: chemistry second-level course. The molecular formula C3H7Cl (molar mass: 78.54 g/mol, exact mass: 78.0236 u) may refer to: * Isopropyl chloride * n-Propyl chloride, also known as 1-propyl chloride or 1-chloropropane Thus, for example, the average mass of a molecule of water is about 18.0153 daltons, and the molar mass of water is about 18.0153 g/mol. The quantity a in the expression for the viscosity average molar mass varies from 0.5 to 0.8 and depends on the interaction between solvent and polymer in a dilute solution. If represents the mass fraction of the solute in solution, and assuming no dissociation of the solute, the molar mass is given by :M = {{wK_\text{b}}\over{\Delta T}}.\ == See also == * Mole map (chemistry) == References == ==External links== * HTML5 Molar Mass Calculator web and mobile application. The molecular formula C2H3ClO (molar mass: 78.50 g/mol, exact mass: 77.9872 u) may refer to: * Acetyl chloride * Chloroacetaldehyde * Chloroethylene oxide In chemistry, the molar mass () of a chemical compound is defined as the ratio between the mass and the amount of substance (measured in moles) of any sample of said compound. About 57% of the mass of PVC is chlorine. Examples are: \begin{array}{ll} M(\ce{NaCl}) &= \bigl[22.98976928(2) + 35.453(2)\bigr] \times 1 \text{ g/mol} \\\ &= 58.443(2) \text{ g/mol} \\\\[4pt] M(\ce{C12H22O11}) &= \bigl[12 \times 12.0107(8) + 22 \times 1.00794(7) + 11 \times 15.9994(3)\bigr] \times 1 \text{ g/mol} \\\ &= 342.297(14) \text{ g/mol} \end{array} An average molar mass may be defined for mixtures of compounds. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in and a small reduction in , thus reducing dispersity. ===Number average molar mass=== The number average molar mass is a way of determining the molecular mass of a polymer. The molecular formula C3H4Cl2O2 (molar mass: 142.97 g/mol, exact mass: 141.9588 u) may refer to: * Chloroethyl chloroformate * Dalapon The mass average molar mass is calculated by \bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i} where is the number of molecules of molecular mass . * Online Molar Mass Calculator with the uncertainty of M and all the calculations shown * Molar Mass Calculator Online Molar Mass and Elemental Composition Calculator * Stoichiometry Add-In for Microsoft Excel for calculation of molecular weights, reaction coefficients and stoichiometry. ",1.56,-0.0301,"""0.6321205588""",48,1.2,E
+Estimate the wavelength of electrons that have been accelerated from rest through a potential difference of $40 \mathrm{kV}$.,"But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. The wavelength corresponding to the mean photon energy is given by :\lambda_{\langle E \rangle} = ( cm K)/T\;. ==Recommendation to de-emphasize== Marr and Wilkin (2012) contend that the widespread teaching of Wien's displacement law in introductory courses is undesirable, and it would be better replaced by alternate material. Note that \scriptstyle \epsilon/m_{0}\sim1.95\times10^{7} emu/gm when the electron is at rest. For his final analysis, Bucherer recalculated the measured values of five runs with Lorentz's and Abraham's formulas respectively, in order to obtain the charge-to-mass ratio as if the electrons were at rest. Solving for the wavelength \lambda in millimetres, and using kelvins for the temperature yields: : ===Parameterization by frequency=== Another common parameterization is by frequency. Using Wien's law, one finds a peak emission per nanometer (of wavelength) at a wavelength of about 500 nm, in the green portion of the spectrum near the peak sensitivity of the human eye.Walker, J. Fundamentals of Physics, 8th ed., John Wiley and Sons, 2008, p. 891. . thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). This is a consequence of the strong statement of Wien's law. ==Frequency-dependent formulation== For spectral flux considered per unit frequency d u (in hertz), Wien's displacement law describes a peak emission at the optical frequency u_\text{peak} given by: : u_\text{peak} = { x \over h} k\,T \approx (5.879 \times 10^{10} \ \mathrm{Hz/K}) \cdot T or equivalently :h u_\text{peak} = x\, k\, T \approx (2.431 \times 10^{-4} \ \mathrm{eV/K}) \cdot T where is a constant resulting from the maximization equation, is the Boltzmann constant, is the Planck constant, and is the absolute temperature. From the Planck constant h and the Boltzmann constant k, Wien's constant b can be obtained. ==Peak differs according to parameterization== Constants for different parameterizations of Wien's law Parameterized by x_\mathrm{peak} b (μm⋅K) Wavelength, \lambda 2898 \log\lambda or \log u 3670 Frequency, u 5099 Other characterizations of spectrum Parameterized by x b (μm⋅K) Mean photon energy 5327 10% percentile 2195 25% percentile 2898 50% percentile 4107 70% percentile 5590 90% percentile 9376 The results in the tables above summarize results from other sections of this article. For other spectral transitions in multi-electron atoms, the Rydberg formula generally provides incorrect results, since the magnitude of the screening of inner electrons for outer-electron transitions is variable and not possible to compensate for in the simple manner above. Finally, with certain modifications (replacement of Z by Z − 1, and use of the integers 1 and 2 for the ns to give a numerical value of for the difference of their inverse squares), the Rydberg formula provides correct values in the special case of K-alpha lines, since the transition in question is the K-alpha transition of the electron from the 1s orbital to the 2p orbital. A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material. ==The interaction picture== Define the unperturbed Hamiltonian by H_0, the time dependent perturbing Hamiltonian by H_1 and total Hamiltonian by H. Walter Kaufmann's measurement of the electron charge-to- mass ratio for different velocities of the electron. The relevant math is detailed in the next section. ==Derivation from Planck's law== ===Parameterization by wavelength=== Planck's law for the spectrum of black body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization. In case of constant perturbation,c_{k'}^{(1)} is calculated by :c_{k'}^{(1)}=\frac{\lang\ k'|H_1|k\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\hbar}) :|c_{k'}(t)|^2= |\lang\ k'|H_1|k\rang |^2\frac {sin ^2(\frac {E_{k'}-E_k} {2 \hbar}t)} { ( \frac {E_{k'} -E_k} {2 \hbar} ) ^2 }\frac {1}{\hbar^2} Using the equation which is :\lim_{\alpha \rightarrow \infty} \frac{1}{\pi} \frac{sin^2(\alpha x)}{\alpha x^2}= \delta(x) The transition rate of an electron from the initial state k to final state k' is given by :P(k,k')=\frac {2 \pi} {\hbar} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) where E_k and E_{k'} are the energies of the initial and final states including the perturbation state and ensures the \delta-function indicate energy conservation. ==The scattering rate== The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by :w(k)=\sum_{k'}P(k,k')=\frac {2 \pi} {\hbar} \sum_{k'} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) The integral form is :w(k)=\frac {2 \pi} {\hbar} \frac {L^3} {(2 \pi)^3} \int d^3k' |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k) ==References== * * Category:Semiconductor technology This is perhaps a more intuitive way of presenting ""wavelength of peak emission"". Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). This is an inverse relationship between wavelength and temperature. Contrary to the then known cathode rays which reached speeds only up to 0.3c, c being the speed of light, Becquerel rays reached velocities up to 0.9c. With the emission now considered per unit frequency, this peak now corresponds to a wavelength about 76% longer than the peak considered per unit wavelength. Formally, the wavelength version of Wien's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength \lambda_\text{peak} given by: :\lambda_\text{peak} = \frac{b}{T} where is the absolute temperature and is a constant of proportionality called Wien's displacement constant, equal to or . ",-1.00,35,"""-0.40864""",6.1,0.139,D
+The standard enthalpy of formation of $\mathrm{H}_2 \mathrm{O}(\mathrm{g})$ at $298 \mathrm{~K}$ is $-241.82 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Estimate its value at $100^{\circ} \mathrm{C}$ given the following values of the molar heat capacities at constant pressure: $\mathrm{H}_2 \mathrm{O}(\mathrm{g}): 33.58 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \mathrm{H}_2(\mathrm{~g}): 28.82 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} ; \mathrm{O}_2(\mathrm{~g})$ : $29.36 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. Assume that the heat capacities are independent of temperature.,"In thermodynamics, the heat capacity at constant volume, C_{V}, and the heat capacity at constant pressure, C_{P}, are extensive properties that have the magnitude of energy divided by temperature. == Relations == The laws of thermodynamics imply the following relations between these two heat capacities (Gaskell 2003:23): :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\, :\frac{C_{P}}{C_{V}}=\frac{\beta_{T}}{\beta_{S}}\, Here \alpha is the thermal expansion coefficient: :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\, \beta_{T} is the isothermal compressibility (the inverse of the bulk modulus): :\beta_{T}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\, and \beta_{S} is the isentropic compressibility: :\beta_{S}=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{S}\, A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is: : c_p - c_v = \frac{T \alpha^2}{\rho \beta_T} where ρ is the density of the substance under the applicable conditions. On a per mole basis, the expression for difference in molar heat capacities becomes simply R for ideal gases as follows: : C_{P,m} - C_{V,m} = \frac{C_{P} - C_{V}}{n} = \frac{n R}{n} = R This result would be consistent if the specific difference were derived directly from the general expression for c_p - c_v\, . == See also == *Heat capacity ratio == References == * David R. Gaskell (2008), Introduction to the thermodynamics of materials, Fifth Edition, Taylor & Francis. . An ideal gas has the equation of state: P V = n R T\, where :P = pressure :V = volume :n = number of moles :R = universal gas constant(Gas constant) :T = temperature The ideal gas equation of state can be arranged to give: : V = n R T / P\, or \, n R = P V / T The following partial derivatives are obtained from the above equation of state: :\left(\frac{\partial V}{\partial T}\right)_{P}\ = \frac {n R}{P}\ = \left(\frac{V P}{T}\right)\left(\frac{1}{P}\right) = \frac{V}{T} :\left(\frac{\partial V}{\partial P}\right)_{T}\ = - \frac {n R T}{P^2}\ = - \frac {P V}{P^2}\ = - \frac{V}{P} The following simple expressions are obtained for thermal expansion coefficient \alpha : :\alpha=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}\ = \frac{1}{V}\left(\frac{V}{T}\right) :\alpha= 1 / T \, and for isothermal compressibility \beta_{T}: :\beta_{T}= - \frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T}\ = - \frac{1}{V}\left( - \frac{V}{P}\right) :\beta_{T}= 1 / P \, One can now calculate C_{P} - C_{V}\, for ideal gases from the previously-obtained general formula: :C_{P} - C_{V}= V T\frac{\alpha^{2}}{\beta_{T}}\ = V T\frac{(1 / T)^2}{1 / P} = \frac{V P}{T} Substituting from the ideal gas equation gives finally: :C_{P} - C_{V} = n R\, where n = number of moles of gas in the thermodynamic system under consideration and R = universal gas constant. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? ""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol·K) Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −118 kJ/mol Standard molar entropy S ~~o~~ solid ? In an isenthalpic process, the enthalpy is constant. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size- dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? ",1.07,0.1792,"""2.567""",-242.6,0.000226,D
 "The standard potential of the cell $\mathrm{Pt}(\mathrm{s})\left|\mathrm{H}_2(\mathrm{~g})\right| \mathrm{HBr}(\mathrm{aq})|\operatorname{AgBr}(\mathrm{s})| \mathrm{Ag}(\mathrm{s})$ was measured over a range of temperatures, and the data were found to fit the following polynomial:
 $$
 E_{\text {cell }}^{\bullet} / \mathrm{V}=0.07131-4.99 \times 10^{-4}(T / \mathrm{K}-298)-3.45 \times 10^{-6}(\mathrm{~T} / \mathrm{K}-298)^2
 $$
 The cell reaction is $\operatorname{AgBr}(\mathrm{s})+\frac{1}{2} \mathrm{H}_2(\mathrm{~g}) \rightarrow \mathrm{Ag}(\mathrm{s})+\mathrm{HBr}(\mathrm{aq})$. Evaluate the standard reaction Gibbs energy, enthalpy, and entropy at $298 \mathrm{~K}$.
-","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. -\beta\left[\log{\frac{x}{\sigma}}\right]^2} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right) | cdf =1-e^{-\alpha\log{\frac{x}{\sigma}}-\beta[\log{\frac{x}{\sigma}}]^2}| mean =\sigma+\tfrac{\sigma}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta}}\right) where H_n(x) is the ""probabilists' Hermite polynomials""| median =\sigma \left(e^{\frac{-\alpha+\sqrt{\alpha^2+\beta\log{16}}}{2\beta}}\right)| mode =| variance = \left(\sigma^2+\tfrac{2\sigma^2}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-2+\alpha}{\sqrt{2\beta}}\right)\right)-\mu^2 | skewness =| kurtosis =| entropy =| mgf =| char =| }} In probability, statistics, economics, and actuarial science, the Benini distribution is a continuous probability distribution that is a statistical size distribution often applied to model incomes, severity of claims or losses in actuarial applications, and other economic data.A. Sen and J. Silber (2001). {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup | cdf = F(x)=\frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{\theta}\right) | mean = k \theta | median = No simple closed form | mode = (k - 1)\theta \text{ for } k \geq 1, 0 \text{ for } k < 1 | variance = k \theta^2 | skewness = \frac{2}{\sqrt{k}} | kurtosis = \frac{6}{k} | entropy = \begin{align} k &\+ \ln\theta + \ln\Gamma(k)\\\ &\+ (1 - k)\psi(k) \end{align} | mgf = (1 - \theta t)^{-k} \text{ for } t < \frac{1}{\theta} | char = (1 - \theta it)^{-k} | parameters2 = | support2 = x \in (0, \infty) | pdf2 = f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x } | cdf2 = F(x)=\frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x) | mean2 = \frac{\alpha}{\beta} | median2 = No simple closed form | mode2 = \frac{\alpha - 1}{\beta} \text{ for } \alpha \geq 1\text{, }0 \text{ for } \alpha < 1 | variance2 = \frac{\alpha}{\beta^2} | skewness2 = \frac{2}{\sqrt{\alpha}} | kurtosis2 = \frac{6}{\alpha} | entropy2 = \begin{align} \alpha &\- \ln \beta + \ln\Gamma(\alpha)\\\ &\+ (1 - \alpha)\psi(\alpha) \end{align} | mgf2 = \left(1 - \frac{t}{\beta}\right)^{-\alpha} \text{ for } t < \beta | char2 = \left(1 - \frac{it}{\beta}\right)^{-\alpha} | moments = k = \frac{E[X]^2}{V[X]} \quad \quad \theta = \frac{V[X]}{E[X]} \quad \quad | moments2 = \alpha = \frac{E[X]^2}{V[X]} \beta = \frac{E[X]}{V[X]} | fisher = I(k, \theta) = \begin{pmatrix}\psi^{(1)}(k) & \theta^{-1} \\\ \theta^{-1} & k \theta^{-2}\end{pmatrix} | fisher2 = I(\alpha, \beta) = \begin{pmatrix}\psi^{(1)}(\alpha) & -\beta^{-1} \\\ -\beta^{-1} & \alpha \beta^{-2}\end{pmatrix} }} In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. This page provides supplementary chemical data on Ytterbium(III) chloride == Structure and properties data == Structure and properties Standard reduction potential, E° 1.620 eV Crystallographic constants a = 6.73 b = 11.65 c = 6.38 β = 110.4° Effective nuclear charge 3.290 Bond strength 1194±7 kJ/mol Bond length 2.434 (Yb-Cl) Bond angle 111.5° (Cl-Yb-Cl) Magnetic susceptibility 4.4 μB == Thermodynamic properties == Phase behavior Std enthalpy change of fusionΔfusH ~~o~~ 58.1±11.6 kJ/mol Std entropy change of fusionΔfusS ~~o~~ 50.3±10.1 J/(mol•K) Std enthalpy change of atomizationΔatH ~~o~~ 1166.5±4.3 kJ/mol Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −959.5±3.0 kJ/mol Standard molar entropy S ~~o~~ solid 163.5 J/(mol•K) Heat capacity cp 101.4 J/(mol•K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid −212.8 kJ/mol Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas −651.1±5.0 kJ/mol == Spectral data == UV-Vis λmax 268 nm Extinction coefficient 303 M−1cm−1 IR Major absorption bands ν1 = 368.0 cm−1 ν2 = 178.4 cm−1 ν3 = 330.7 cm−1 ν4 = 117.8 cm−1 MS Ionization potentials from electron impact YbCl3+ = 10.9±0.1 eV YbCl2+ = 11.6±0.1 eV YbCl+ = 14.3±0.1 eV ==References== Category:Chemical data pages Category:Chemical data pages cleanup :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Interpolated approximations and bounds are all of the form : u(k) \approx \tilde{g}(k) u_{L\infty}(k) + (1 - \tilde{g}(k)) u_U(k) where \tilde{g} is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator g(k): :g(k) = \frac{ u_U(k) - u(k)}{ u_U(k) - u_{L\infty}(k)} For the simplest interpolating function considered, a first-order rational function :\tilde{g}_1(k) = \frac{k}{b_0 + k} the tightest lower bound has :b_0 = \frac{\frac{8}{405} + e^{-\gamma} \log 2 - \frac{\log^2 2}{2}}{e^{-\gamma} - \log 2 + \frac{1}{3}} - \log 2 \approx 0.143472 and the tightest upper bound has :b_0 = \frac{e^{-\gamma} - \log 2 + \frac{1}{3}}{1 - \frac{e^{-\gamma} \pi^2}{12}} \approx 0.374654 The interpolated bounds are plotted (mostly inside the yellow region) in the log–log plot shown. Above 750 K Tc values may be in error by 10 K or more. Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. ==See also== * F-distribution * Folded-t and half-t distributions * Hotelling's T-squared distribution * Multivariate Student distribution * Standard normal table (Z-distribution table) * t-statistic * Tau distribution, for internally studentized residuals * Wilks' lambda distribution * Wishart distribution * Modified half-normal distribution with the pdf on (0, \infty) is given as f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\\\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function. ==Notes== ==References== * * * * ==External links== * *Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term ""Student's distribution"") * First Students on page 112. (DOE contract 95‑831). : p(k,\theta \mid p, q, r, s) = \frac{1}{Z} \frac{p^{k-1} e^{-\theta^{-1} q}}{\Gamma(k)^r \theta^{k s}}, where Z is the normalizing constant with no closed-form solution. Berg and Pedersen found more terms: : u(k) = k - \frac{1}{3} + \frac{8}{405 k} + \frac{184}{25515 k^2} + \frac{2248}{3444525 k^3} - \frac{19006408}{15345358875 k^4} - O\left(\frac{1}{k^5}\right) + \cdots 320px|thumb| Two gamma distribution median asymptotes which are conjectured to be bounds (upper solid red and lower dashed red), of the from u(k) \approx 2^{-1/k}(A + k), and an interpolation between them that makes an approximation (dotted red) that is exact at k = 1 and has maximum relative error of about 0.6%. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. Partial sums of these series are good approximations for high enough k; they are not plotted in the figure, which is focused on the low-k region that is less well approximated. If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion: :F(x;\alpha,\beta) = 1-\sum_{i=0}^{\alpha-1} \frac{(\beta x)^i}{i!} e^{-\beta x} = e^{-\beta x} \sum_{i=\alpha}^{\infty} \frac{(\beta x)^i}{i!}. === Characterization using shape k and scale θ === A random variable X that is gamma-distributed with shape k and scale θ is denoted by :X \sim \Gamma(k, \theta) \equiv \operatorname{Gamma}(k, \theta) [[Image:Gamma- PDF-3D.png|thumb|right|320px|Illustration of the gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5 and 6\. A bias-corrected variant of the estimator for the scale θ is : \tilde{\theta} = \frac{N}{N - 1} \hat{\theta} A bias correction for the shape parameter k is given as : \tilde{k} = \hat{k} - \frac{1}{N} \left(3 \hat{k} - \frac{2}{3} \left(\frac{\hat{k}}{1 + \hat{k}}\right) - \frac{4}{5} \frac{\hat{k}}{(1 + \hat{k})^2} \right) ====Bayesian minimum mean squared error==== With known k and unknown θ, the posterior density function for theta (using the standard scale-invariant prior for θ) is : P(\theta \mid k, x_1, \dots, x_N) \propto \frac 1 \theta \prod_{i=1}^N f(x_i; k, \theta) Denoting : y \equiv \sum_{i=1}^Nx_i , \qquad P(\theta \mid k, x_1, \dots, x_N) = C(x_i) \theta^{-N k-1} e^{-y/\theta} Integration with respect to θ can be carried out using a change of variables, revealing that 1/θ is gamma-distributed with parameters α = Nk, β = y. :\int_0^\infty \theta^{-Nk - 1 + m} e^{-y/\theta}\, d\theta = \int_0^\infty x^{Nk - 1 - m} e^{-xy} \, dx = y^{-(Nk - m)} \Gamma(Nk - m) \\! The Kullback–Leibler divergence (KL-divergence), of Gamma(αp, βp) (""true"" distribution) from Gamma(αq, βq) (""approximating"" distribution) is given byW.D. Penny, [www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities] : \begin{align} D_{\mathrm{KL}}(\alpha_p,\beta_p; \alpha_q, \beta_q) = {} & (\alpha_p-\alpha_q) \psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) \\\ & {} + \alpha_q(\log \beta_p - \log \beta_q) + \alpha_p\frac{\beta_q-\beta_p}{\beta_p}. \end{align} Written using the k, θ parameterization, the KL-divergence of Gamma(kp, θp) from Gamma(kq, θq) is given by : \begin{align} D_{\mathrm{KL}}(k_p,\theta_p; k_q, \theta_q) = {} & (k_p-k_q)\psi(k_p) - \log\Gamma(k_p) + \log\Gamma(k_q) \\\ & {} + k_q(\log \theta_q - \log \theta_p) + k_p \frac{\theta_p - \theta_q}{\theta_q}. \end{align} ===Laplace transform=== The Laplace transform of the gamma PDF is :F(s) = (1 + \theta s)^{-k} = \frac{\beta^\alpha}{(s + \beta)^\alpha} . ==Related distributions== ===General=== * Let X_1, X_2, \ldots, X_n be n independent and identically distributed random variables following an exponential distribution with rate parameter λ, then \sum_i X_i ~ Gamma(n, 1/λ) where n is the shape parameter and λ is the rate, and \bar{X} = \frac{1}{n} \sum_i X_i \sim \operatorname{Gamma}(n, n\lambda) where the rate changes nλ. For the two parameter model, the quantile function (inverse cdf) is : F^{-1}(u) = \sigma \exp \sqrt{-\frac{1}{\beta} \log(1-u)}, \quad 0 < u < 1\. ==Related distributions== * If X \sim \mathrm{Benini}(\alpha,0,\sigma)\,, then X has a Pareto distribution with x_\mathrm{m}=\sigma * If X \sim \mathrm{Benini}(0,\tfrac{1}{2\sigma^2},1), then X \sim e^U where U \sim \mathrm{Rayleigh}(\sigma) ==Software== The (two parameter) Benini distribution density, probability distribution, quantile function and random number generator is implemented in the VGAM package for R, which also provides maximum likelihood estimation of the shape parameter. In particular for integer valued degrees of freedom u we have: For u >1 even, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 5 \cdot 3} { 2 \sqrt{ u}( u -2)( u -4)\cdots 4 \cdot 2\,}\cdot For u >1 odd, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 4 \cdot 2} {\pi \sqrt{ u}( u -2)( u -4)\cdots 5 \cdot 3\,}\cdot\\! This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is ln(x). ===Information entropy=== The information entropy is : \begin{align} \operatorname{H}(X) & = \operatorname{E}[-\ln(p(X))] \\\\[4pt] & = \operatorname{E}[-\alpha \ln(\beta) + \ln(\Gamma(\alpha)) - (\alpha-1)\ln(X) + \beta X] \\\\[4pt] & = \alpha - \ln(\beta) + \ln(\Gamma(\alpha)) + (1-\alpha)\psi(\alpha). \end{align} In the k, θ parameterization, the information entropy is given by : \operatorname{H}(X) =k + \ln(\theta) + \ln(\Gamma(k)) + (1-k)\psi(k). ===Kullback–Leibler divergence=== thumb|right|320px|Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. ",1.3,-21.2,260.0,0.829,0.0547,B
-"In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ in a vessel of constant volume. If it enters the vessel at $100 \mathrm{~atm}$ and $300 \mathrm{~K}$, what pressure would it exert at the working temperature if it behaved as a perfect gas?","For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. thumb|Nitrogen is a liquid under -195.8 degrees Celsius (77K). thumb|A medium- sized dewar is being filled with liquid nitrogen by a larger cryogenic storage tank. thumb|right|A gas regulator attached to a nitrogen cylinder Industrial gases are the gaseous materials that are manufactured for use in industry. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. All of these substances are also provided as a gas (not a vapor) at the 200 bar pressure in a gas cylinder because that pressure is above their critical pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Springer Science+Business Media LLC (2007) This is a logical dividing line, since the normal boiling points of the so-called permanent gases (such as helium, hydrogen, neon, nitrogen, oxygen, and normal air) lie below 120K while the Freon refrigerants, hydrocarbons, and other common refrigerants have boiling points above 120K. The pressure melting point of ice is the temperature at which ice melts at a given pressure. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. A pressure reactor, sometimes referred to as a pressure tube, or a sealed tube, is a chemical reaction vessel which can conduct a reaction under pressure. Pressure vessels for gas storage may also be classified by volume. A pressure reactor is a special application of a pressure vessel. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. However, pressure can speed up the desired reaction and only impacts decomposition when it involves the release of a gas or a reaction with a gas in the vessel. With increasing pressure above 10 MPa, the pressure melting point decreases to a minimum of −21.9 °C at 209.9 MPa. The term “industrial gases” is sometimes narrowly defined as just the major gases sold, which are: nitrogen, oxygen, carbon dioxide, argon, hydrogen, acetylene and helium. This is because the internal energy of an ideal gas is at most a function of temperature, as shown by the thermodynamic equation \left({{\partial U} \over {\partial V}}\right)_T = T\left({{\partial S} \over {\partial V}}\right)_T - P = T\left({{\partial P} \over {\partial T}}\right)_V - P, which is exactly zero when P = nRT / V . A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. ",167,170,0.346,24,22,A
-"For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \%$, what will be the balance in the IRA at age 65 ?","For those 65 and over, 11.6% of retirement accounts have balances of at least $1 million, more than twice that of the $407,581 average (shown). For those 65 and over, 11.6% of retirement accounts have balances of at least $1 million, more than twice that of the $407,581 average (shown). The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual growth rate * Compound interest * Dollar cost averaging * Economic value added * Effective annual rate * Effective interest rate * Expected return * Holding period return * Internal rate of return * Modified Dietz method * Net present value * Rate of profit * Return of capital * Return on assets * Return on capital * Returns (economics) * Simple Dietz method * Time value of money * Time-weighted return * Value investing * Yield ==Notes== ==References== ==Further reading== * A. A. Groppelli and Ehsan Nikbakht. As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) − 1), assuming reinvestment at the end of the first year. thumb|Average balances of retirement accounts, for households having such accounts, exceed median net worth across all age groups. thumb|Average balances of retirement accounts, for households having such accounts, exceed median net worth across all age groups. Some highlights from the 2008 data follow: * The average and median IRA account balance was $54,863 and $15,756, respectively, while the average and median IRA individual balance (all accounts from the same person combined) was $69,498 and $20,046. The purpose of this report was to study individual retirement accounts (IRAs) and the account valuations. Also that there are an estimate of 791 taxpayers with IRA account balances between $10,000,000 and $25,000,000. Furthermore, the typical working household has virtually no retirement savings - the median retirement account balance is $2,500 for all working-age households and $14,500 for near-retirement households. * In 2014, the federal government will forgo an estimated $17.45 billion in tax revenue from IRAs, which Congress created to ensure equitable tax treatment for those not covered by employer-sponsored retirement plans. * 98.5% of taxpayers have IRA account balances at $1,000,000 or less. * 1.2% of tax payers have IRA account balances at $1,000,000 to $2,000,000. * 0.2% (or 83,529) taxpayers have IRA account balances of $2,000,000 to $3,000,000 * 0.1% (or 36,171) taxpayers have IRA account balances of $3,000,000 to $5,000,000 * <0.1% (or 7,952) taxpayers have IRA account balances of $5,000,000 to $10,000,000 * <0.1% (or 791) taxpayers have IRA account balances of $10,000,000 to $25,000,000 * <0.1% (or 314) taxpayers have IRA account balances of $25,000,000 or more ===Retirement savings=== While the average (mean) and median IRA individual balance in 2008 were approximately $70,000 and $20,000 respectively, higher balances are not rare. 6.3% of individuals had total balances of $250,000 or more (about 12.5 times the median), and in rare cases, individuals own IRAs with very substantial balances, in some cases $100 million or above (about 5,000 times the median individual balance). An individual retirement account is a type of individual retirement arrangementSee 26 C.F.R. sec. 1.408-4. as described in IRS Publication 590, Individual Retirement Arrangements (IRAs). Actuarial assumptions like 5% interest, 3% salary increases and the UP84 Life Table for mortality are used to calculate a level contribution rate that would create the needed lump sum at retirement age. For example, a person aged 45, who put $4,000 into a traditional IRA this year so far, can either put $2,000 more into this traditional IRA, or $2,000 in a Roth IRA, or some combination of those. The cash balance plan typically offers a lump sum at and often before normal retirement age. * Contributions are concentrated at the maximum amount – of those contributing to an IRA, approximately 40% contributed the maximum (whether contributing to traditional or Roth), and 46.7% contributed close to the maximum (in the $5,000–$6,000 range). * Excluding SEPs and SIMPLEs (i.e., concerning traditional, rollover, and Roth IRAs), 15.1% of individuals holding an IRA contributed to one. The Retirement Age as referred to in paragraph (2) is further increased by 1 (one) year for every subsequent 3 (three) years until it reaches the Retirement Age of 65 (sixty five) years. For example, if the logarithmic return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualized logarithmic rate of return is 0.14%/(1/250) = 0.14% x 250 = 35% ===Returns over multiple periods=== When the return is calculated over a series of sub- periods of time, the return in each sub-period is based on the investment value at the beginning of the sub-period. Cash flow example on $1,000 investment Year 1 Year 2 Year 3 Year 4 Dollar return $100 $55 $60 $50 ROI 10% 5.5% 6% 5% ==Uses== * Rates of return are useful for making investment decisions. For example, a return over one month of 1% converts to an annualized rate of return of 12.7% = ((1+0.01)12 − 1). * The continuously compounded rate of return in this example is: :\ln\left(\frac{103.02}{100}\right) = 2.98\%. ",588313,1.6,0.195,344,0.396,A
-"Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion.","Since the inertia of the beam can be found from its mass, the spring constant can be calculated. For waves on a string, or in a medium such as water, the amplitude is a displacement. Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). Thus, for g ≈ 9.8 ms−2, :L \approx \frac{894}{n^2} \; \text{m} | n 71 0.846 0.177 70 0.857 0.182 69 0.870 0.188 68 0.882 0.193 67 0.896 0.199 66 0.909 0.205 65 0.923 0.212 64 0.938 0.218 63 0.952 0.225 62 0.968 0.232 61 0.984 0.240 60 1.000 0.248 Parameters of the pendulum wave in the animation above ---|--- == See also == * Newton's cradle - a set of pendulums constrained to swing along the axis of the apparatus and collide with one another ==References== Category:Pendulums Category:Kinetic art RMS Amplitude . When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The amplitude of sound waves and audio signals (which relates to the volume) conventionally refers to the amplitude of the air pressure in the wave, but sometimes the amplitude of the displacement (movements of the air or the diaphragm of a speaker) is described. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by :\delta = \frac{v}{\omega}, :\delta = \frac{p}{\omega z(\mathbf{r},\, s)}. ==See also== *Sound *Sound particle *Particle velocity *Particle acceleration ==References and notes== Related Reading: * * * ==External links== *Acoustic Particle-Image Velocimetry. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . Hooke's law is the potential energy of the spring itself: :V_k=\frac{1}{2}kx^2 where k is the spring constant. Strictly speaking, this is no longer amplitude since there is the possibility that a constant (DC component) is included in the measurement. ===Peak-to-peak amplitude=== Peak-to-peak amplitude (abbreviated p–p) is the change between peak (highest amplitude value) and trough (lowest amplitude value, which can be negative). With appropriate circuitry, peak-to- peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. The square of the amplitude is proportional to the intensity of the wave. ",9.8,0.18162,5275.0,0.8561,0.444444444444444 ,B
-"At time $t=0$ a tank contains $Q_0 \mathrm{lb}$ of salt dissolved in 100 gal of water; see Figure 2.3.1. Assume that water containing $\frac{1}{4} \mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \mathrm{gal} / \mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \%$ of $Q_L$. ","Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). It may therefore be estimated by the steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. This case is referred to as diffusion control and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining. == See also == * Product-determining step * Rate-limiting step (biochemistry) ==References== * Category:Chemical kinetics ja:反応速度#律速段階 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. The observed rate law is :v = k \frac{\ce{[Cl2][H2C2O4]}}{[\ce{H+}]^2[\ce{Cl^-}]}, which implies an activated complex in which the reactants lose 2 + before the rate-determining step. This equation can be derived from the mass conservation equations of two-phase flow, under the assumptions listed below. ==Equation== In a quasi-1D domain, the Buckley–Leverett equation is given by: : \frac{\partial S_w}{\partial t} + \frac{\partial}{\partial x}\left( \frac{Q}{\phi A} f_w(S_w) \right) = 0, where S_w(x,t) is the wetting-phase (water) saturation, Q is the total flow rate, \phi is the rock porosity, A is the area of the cross-section in the sample volume, and f_w(S_w) is the fractional flow function of the wetting phase. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step. ===Pre- equilibrium: if the second step were rate-determining=== The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: r2 ≪ r−1. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram. Such a situation in which an intermediate (here ) forms an equilibrium with reactants prior to the rate-determining step is described as a pre-equilibriumPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W. H. Freeman 2006) p. 814–815. . In this example is formed in one step and reacts in two, so that : \frac{d\ce{[NO3]}}{dt} = r_1 - r_2 - r_{-1} \approx 0. A possible mechanism in two elementary steps that explains the rate equation is: # + → NO + (slow step, rate-determining) # + CO → + (fast step) In this mechanism the reactive intermediate species is formed in the first step with rate r1 and reacts with CO in the second step with rate r2. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. For a given reaction mechanism, the prediction of the corresponding rate equation (for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step. Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. Typically, f_w(S_w) is an 'S'-shaped, nonlinear function of the saturation S_w, which characterizes the relative mobilities of the two phases: : f_w(S_w) = \frac{\lambda_w}{\lambda_w + \lambda_n} = \frac{ \frac{k_{rw}}{\mu_w} }{ \frac{k_{rw}}{\mu_w} + \frac{k_{rn}}{\mu_n} }, where \lambda_w and \lambda_n denote the wetting and non-wetting phase mobilities. k_{rw}(S_w) and k_{rn}(S_w) denote the relative permeability functions of each phase and \mu_w and \mu_n represent the phase viscosities. ==Assumptions== The Buckley–Leverett equation is derived based on the following assumptions: * Flow is linear and horizontal * Both wetting and non-wetting phases are incompressible * Immiscible phases * Negligible capillary pressure effects (this implies that the pressures of the two phases are equal) * Negligible gravitational forces ==General solution== The characteristic velocity of the Buckley-Leverett equation is given by: :U(S_w) = \frac{Q}{\phi A} \frac{\mathrm{d} f_w}{\mathrm{d} S_w}. The Buckley–Leverett equation or the Buckley–Leverett displacement describes an immiscible displacement process, such as the displacement of oil by water, in a one-dimensional or quasi-one-dimensional reservoir. ",130.400766848,8.7,1.7,+80,0.4,A
-"Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion.","Since the inertia of the beam can be found from its mass, the spring constant can be calculated. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. thumb|Phase portrait of damped oscillator, with increasing damping strength. When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. In this case, the phase shift is simply the argument shift \tau, expressed as a fraction of the common period T (in terms of the modulo operation) of the two signals and then scaled to a full turn: :\varphi = 2\pi \left[\\!\\!\left[ \frac{\tau}{T} \right]\\!\\!\right]. The stiffness of the spring, its spring coefficient, \kappa\, in N·m/radian^2, along with the balance wheel's moment of inertia, I\, in kg·m2, determines the wheel's oscillation period T\,. The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. The term phase can refer to several different things: * It can refer to a specified reference, such as \textstyle \cos(2 \pi f t), in which case we would say the phase of \textstyle x(t) is \textstyle \varphi, and the phase of \textstyle y(t) is \textstyle \varphi - \frac{\pi}{2}. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . The formula above gives the phase as an angle in radians between 0 and 2\pi. A balance spring, or hairspring, is a spring attached to the balance wheel in mechanical timepieces. The term ""phase"" is also used when comparing a periodic function F with a shifted version G of it. If the shift in t is expressed as a fraction of the period, and then scaled to an angle \varphi spanning a whole turn, one gets the phase shift, phase offset, or phase difference of G relative to F. In physics and mathematics, the phase of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. Thus, for example, the sum of phase angles is 30° (, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (, plus one full turn). We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The phase \phi(t) is then the angle from the 12:00 position to the current position of the hand, at time t, measured clockwise. ",-0.40864,4.86,-6.8,+2.35, 9.73,A
-"The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \times 10^6 \mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later. ","Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is : \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right), and its solution is : N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}. ==References== * * * * * Category:Demography Category:Biostatistics Category:Fisheries science Category:Stochastic models As the population approaches its carrying capacity, the rate of growth decreases, and the population trend will become logistic. This S-shaped curve observed in logistic growth is a more accurate model than exponential growth for observing real- life population growth of organisms. ==See also== * Malthusian catastrophe * r/K selection theory ==References== ==Sources== John A. Miller and Stephen B. Harley zoology 4th edition ==External links== * Category:Biology Category:Biology articles needing attention Category:Population ecology Once the carrying capacity, or K, is incorporated to account for the finite resources that a population will be competing for within an environment, the aforementioned equation becomes the following: \frac{dN}{dt}=r_{max}\frac{dN}{dt}=r_{max}N\frac{K-N}{K} A graph of this equation creates an S-shaped curve, which demonstrates how initial population growth is exponential due to the abundance of resources and lack of competition. Here R0 is interpreted as the proliferation rate per generation and K = (R0 − 1) M is the carrying capacity of the environment. If, in a hypothetical population of size N, the birth rates (per capita) are represented as b and death rates (per capita) as d, then the increase or decrease in N during a time period t will be \frac{dN}{dt}=(b-d)N (b-d) is called the 'intrinsic rate of natural increase' and is a very important parameter chosen for assessing the impacts of any biotic or abiotic factor on population growth. The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number N t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,Ricker (1954) : N_{t+1} = N_t e^{r\left(1-\frac{N_t}{k}\right)}.\, Here r is interpreted as an intrinsic growth rate and k as the carrying capacity of the environment. The Monod equation is a mathematical model for the growth of microorganisms. In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation. The solution is : n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}. When the yield coefficient, being the ratio of mass of microorganisms to mass of substrate utilized, becomes very large this signifies that there is deficiency of substrate available for utilization. == Graphical determination of constants == As with the Michaelis–Menten equation graphical methods may be used to fit the coefficients of the Monod equation: * Eadie–Hofstee diagram * Hanes–Woolf plot * Lineweaver–Burk plot == See also == * Activated sludge model (uses the Monod equation to model bacterial growth and substrate utilization) * Bacterial growth * Hill equation (biochemistry) * Hill contribution to Langmuir equation * Langmuir adsorption model (equation with the same mathematical form) * Michaelis–Menten kinetics (equation with the same mathematical form) * Gompertz function * Victor Henry, who first wrote the general equation form in 1901 * Von Bertalanffy function == References == Category:Catalysis Category:Chemical kinetics Category:Environmental engineering Category:Enzyme kinetics Category:Ordinary differential equations Category:Sewerage They will differ between microorganism species and will also depend on the ambient environmental conditions, e.g., on the temperature, on the pH of the solution, and on the composition of the culture medium. == Application notes == The rate of substrate utilization is related to the specific growth rate as follows: : r_s = \mu X/Y where: * X is the total biomass (since the specific growth rate, μ is normalized to the total biomass) * Y is the yield coefficient rs is negative by convention. right|thumb|thumbtime=5|FVCOM simulation of hypersaline sea surface release and propagation under tidal conditions in the northern North Sea The Finite Volume Community Ocean Model (FVCOM; Formerly Finite Volume Coastal Ocean Model) is a prognostic, unstructured-grid, free-surface, 3-D primitive equation coastal ocean circulation model. The equation in discrete time is given by :1 = \sum_{a = 1}^\omega \lambda^{-a}\ell(a)b(a) where \lambda is the discrete growth rate, ℓ(a) is the fraction of individuals surviving to age a and b(a) is the number of offspring born to an individual of age a during the time step. First substitute the definition of the per-capita fertility and divide through by the left hand side: :1 = \frac{s_0b_1}{\lambda} + \frac{s_0s_1b_2}{\lambda^2} + \cdots + \frac{s_0\cdots s_{\omega - 1}b_{\omega}}{\lambda^{\omega}}. The model can be used to predict the number of fish that will be present in a fishery.de Vries et al.Marland Subsequent work has derived the model under other assumptions such as scramble competition,Brännström and Sumpter(2005) within-year resource limited competition or even as the outcome of source-sink Malthusian patches linked by density-dependent dispersal. When c = 1, the Hassell model is simply the Beverton–Holt model. ==See also== * Population dynamics of fisheries ==Notes== ==References== * Brännström A and Sumpter DJ (2005) ""The role of competition and clustering in population dynamics"" Proc Biol Sci., 272(1576): 2065-72\. Biological exponential growth is the unrestricted growth of a population of organisms, occurring when resources in its habitat are unlimited. The sum is taken over the entire life span of the organism. ==Derivations== ===Lotka's continuous model=== A.J. Lotka in 1911 developed a continuous model of population dynamics as follows. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an estimation of how a population is growing. Any species growing exponentially under unlimited resource conditions can reach enormous population densities in a short time. The empirical Monod equation is: : \mu = \mu_\max {[S] \over K_s + [S]} where: * μ is the growth rate of a considered microorganism * μmax is the maximum growth rate of this microorganism * [S] is the concentration of the limiting substrate S for growth * Ks is the ""half- velocity constant""—the value of [S] when μ/μmax = 0.5 μmax and Ks are empirical (experimental) coefficients to the Monod equation. ",46.7,38,0.9522,47,0,A
+","==Critical point== ref Tc(K) Tc(°C) Pc(MPa) Pc(other) Vc(cm3/mol) ρc(g/cm3) 1 H hydrogen use 32.97 −240.18 1.293 CRC.a 32.97 −240.18 1.293 65 KAL 33.2 1.297 65.0 SMI −239.9 13.2 kgf/cm² 0.0310 1 H hydrogen (equilibrium) LNG −240.17 1.294 12.77 atm 65.4 0.0308 1 H hydrogen (normal) LNG −239.91 1.297 12.8 atm 65.0 0.0310 1 D deuterium KAL 38.2 1.65 60 1 D deuterium (equilibrium) LNG −234.8 1.650 16.28 atm 60.4 0.0668 1 D deuterium (normal) LNG −234.7 1.665 16.43 atm 60.3 0.0669 2 He helium use 5.19 −267.96 0.227 CRC.a 5.19 −267.96 0.227 57 KAL 5.19 0.227 57.2 SMI −267.9 2.34 kgf/cm² 0.0693 2 He helium (equilibrium) LNG −267.96 0.2289 2.261 atm 0.06930 2 He helium-3 LNG −269.85 0.1182 1.13 atm 72.5 0.0414 2 He helium-4 LNG −267.96 0.227 2.24 atm 57.3 0.0698 3 Li lithium use (3223) (2950) (67) CRC.b (3223) 2950 (67) (66) 7 N nitrogen use 126.21 −146.94 3.39 CRC.a 126.21 −146.94 3.39 90 KAL 126.2 3.39 89.5 SMI −147.1 34.7 kgf/cm² 0.3110 7 N nitrogen-14 LNG −146.94 3.39 33.5 atm 89.5 0.313 7 N nitrogen-15 LNG −146.8 3.39 33.5 atm 90.4 0.332 8 O oxygen use 154.59 −118.56 5.043 CRC.a 154.59 −118.56 5.043 73 LNG −118.56 5.043 49.77 atm 73.4 0.436 KAL 154.6 5.04 73.4 SMI −118.8 51.4 kgf/cm² 0.430 9 F fluorine use 144.13 −129.02 5.172 CRC.a 144.13 −129.02 5.172 66 LNG −129.0 5.215 51.47 atm 66.2 0.574 KAL 144.3 5.22 66 10 Ne neon use 44.4 −228.7 2.76 CRC.a 44.4 −228.7 2.76 42 LNG −228.71 2.77 27.2 atm 41.7 0.4835 KAL 44.4 2.76 41.7 SMI −228.7 26.8 kgf/cm² 0.484 11 Na sodium use (2573) (2300) (35) CRC.b (2573) 2300 (35) (116) 15 P phosphorus use 994 721 CRC.a 994 721 LNG 721 16 S sulfur use 1314 1041 20.7 CRC.a 1314 1041 20.7 LNG 1041 11.7 116 atm KAL 1314 20.7 SMI 1040 17 Cl chlorine use 416.9 143.8 7.991 CRC.a 416.9 143.8 7.991 123 LNG 143.8 7.71 76.1 atm 124 0.573 KAL 416.9 7.98 124 SMI 144.0 78.7 kgf/cm² 0.573 18 Ar argon use 150.87 −122.28 4.898 CRC.a 150.87 −122.28 4.898 75 LNG −122.3 4.87 48.1 atm 74.6 0.536 KAL 150.9 4.90 74.6 SMI −122 49.7 kgf/cm² 0.531 19 K potassium use (2223) (1950) (16) CRC.b (2223) 1950 (16) (209) 33 As arsenic use 1673 1400 CRC.a 1673 1400 35 LNG 1400 34 Se selenium use 1766 1493 27.2 CRC.a 1766 1493 27.2 LNG 1493 35 Br bromine use 588 315 10.34 CRC.a 588 315 10.34 127 LNG 315 10.3 102 atm 135 1.184 KAL 588 10.3 127 SMI 302 1.18 36 Kr krypton use 209.41 −63.74 5.50 CRC.a 209.41 −63.74 5.50 91 LNG −63.75 5.50 54.3 atm 91.2 0.9085 KAL 209.4 5.50 91.2 37 Rb rubidium use (2093) (1820) (16) CRC.b (2093) 1820 (16) (247) LNG 1832 250 0.34 53 I iodine use 819 546 11.7 CRC.a 819 546 155 LNG 546 11.7 115 atm 155 0.164 KAL 819 155 SMI 553 54 Xe xenon use 289.77 16.62 5.841 CRC.c 289.77 16.62 5.841 118 LNG 16.583 5.84 57.64 atm 118 1.105 KAL 289.7 5.84 118 SMI 16.6 60.2 kgf/cm² 1.155 55 Cs caesium use 1938 1665 9.4 CRC.d 1938 1665 9.4 341 LNG 1806 300 0.44 80 Hg mercury use 1750 1477 172.00 CRC.a 1750 1477 172.00 43 LNG 1477 160.8 1587 atm KAL 1750 172 42.7 SMI 1460±20 1640±50 kgf/cm² 0.5 86 Rn radon use 377 104 6.28 CRC.a 377 104 6.28 LNG 104 6.28 62 atm 139 1.6 KAL 377 6.3 SMI 104 64.1 kgf/cm² ==References== ===CRC.a-d=== David R. Lide (ed), CRC Handbook of Chemistry and Physics, 85th Edition, online version. -\beta\left[\log{\frac{x}{\sigma}}\right]^2} \left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right) | cdf =1-e^{-\alpha\log{\frac{x}{\sigma}}-\beta[\log{\frac{x}{\sigma}}]^2}| mean =\sigma+\tfrac{\sigma}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta}}\right) where H_n(x) is the ""probabilists' Hermite polynomials""| median =\sigma \left(e^{\frac{-\alpha+\sqrt{\alpha^2+\beta\log{16}}}{2\beta}}\right)| mode =| variance = \left(\sigma^2+\tfrac{2\sigma^2}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-2+\alpha}{\sqrt{2\beta}}\right)\right)-\mu^2 | skewness =| kurtosis =| entropy =| mgf =| char =| }} In probability, statistics, economics, and actuarial science, the Benini distribution is a continuous probability distribution that is a statistical size distribution often applied to model incomes, severity of claims or losses in actuarial applications, and other economic data.A. Sen and J. Silber (2001). {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup | cdf = F(x)=\frac{1}{\Gamma(k)} \gamma\left(k, \frac{x}{\theta}\right) | mean = k \theta | median = No simple closed form | mode = (k - 1)\theta \text{ for } k \geq 1, 0 \text{ for } k < 1 | variance = k \theta^2 | skewness = \frac{2}{\sqrt{k}} | kurtosis = \frac{6}{k} | entropy = \begin{align} k &\+ \ln\theta + \ln\Gamma(k)\\\ &\+ (1 - k)\psi(k) \end{align} | mgf = (1 - \theta t)^{-k} \text{ for } t < \frac{1}{\theta} | char = (1 - \theta it)^{-k} | parameters2 = | support2 = x \in (0, \infty) | pdf2 = f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x } | cdf2 = F(x)=\frac{1}{\Gamma(\alpha)} \gamma(\alpha, \beta x) | mean2 = \frac{\alpha}{\beta} | median2 = No simple closed form | mode2 = \frac{\alpha - 1}{\beta} \text{ for } \alpha \geq 1\text{, }0 \text{ for } \alpha < 1 | variance2 = \frac{\alpha}{\beta^2} | skewness2 = \frac{2}{\sqrt{\alpha}} | kurtosis2 = \frac{6}{\alpha} | entropy2 = \begin{align} \alpha &\- \ln \beta + \ln\Gamma(\alpha)\\\ &\+ (1 - \alpha)\psi(\alpha) \end{align} | mgf2 = \left(1 - \frac{t}{\beta}\right)^{-\alpha} \text{ for } t < \beta | char2 = \left(1 - \frac{it}{\beta}\right)^{-\alpha} | moments = k = \frac{E[X]^2}{V[X]} \quad \quad \theta = \frac{V[X]}{E[X]} \quad \quad | moments2 = \alpha = \frac{E[X]^2}{V[X]} \beta = \frac{E[X]}{V[X]} | fisher = I(k, \theta) = \begin{pmatrix}\psi^{(1)}(k) & \theta^{-1} \\\ \theta^{-1} & k \theta^{-2}\end{pmatrix} | fisher2 = I(\alpha, \beta) = \begin{pmatrix}\psi^{(1)}(\alpha) & -\beta^{-1} \\\ -\beta^{-1} & \alpha \beta^{-2}\end{pmatrix} }} In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. :*(c) O. Sifner, J. Klomfar, J. Phys. Chem. Ref. Data 23, 63, (1994). \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. This page provides supplementary chemical data on Ytterbium(III) chloride == Structure and properties data == Structure and properties Standard reduction potential, E° 1.620 eV Crystallographic constants a = 6.73 b = 11.65 c = 6.38 β = 110.4° Effective nuclear charge 3.290 Bond strength 1194±7 kJ/mol Bond length 2.434 (Yb-Cl) Bond angle 111.5° (Cl-Yb-Cl) Magnetic susceptibility 4.4 μB == Thermodynamic properties == Phase behavior Std enthalpy change of fusionΔfusH ~~o~~ 58.1±11.6 kJ/mol Std entropy change of fusionΔfusS ~~o~~ 50.3±10.1 J/(mol•K) Std enthalpy change of atomizationΔatH ~~o~~ 1166.5±4.3 kJ/mol Solid properties Std enthalpy change of formation ΔfH ~~o~~ solid −959.5±3.0 kJ/mol Standard molar entropy S ~~o~~ solid 163.5 J/(mol•K) Heat capacity cp 101.4 J/(mol•K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid −212.8 kJ/mol Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas −651.1±5.0 kJ/mol == Spectral data == UV-Vis λmax 268 nm Extinction coefficient 303 M−1cm−1 IR Major absorption bands ν1 = 368.0 cm−1 ν2 = 178.4 cm−1 ν3 = 330.7 cm−1 ν4 = 117.8 cm−1 MS Ionization potentials from electron impact YbCl3+ = 10.9±0.1 eV YbCl2+ = 11.6±0.1 eV YbCl+ = 14.3±0.1 eV ==References== Category:Chemical data pages Category:Chemical data pages cleanup :*(d) N.B. Vargaftik, Int. J. Thermophys. 11, 467, (1990). ===LNG=== J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 6; Table 6.5 Critical Properties ===KAL=== National Physical Laboratory, Kaye and Laby Tables of Physical and Chemical Constants; D. Ambrose, M.B. Ewing, M.L. McGlashan, Critical constants and second virial coefficients of gases (retrieved Dec 2005) ===SMI=== W.E. Forsythe (ed.), Smithsonian Physical Tables 9th ed., online version (1954; Knovel 2003). :*(b) I.G. Dillon, P.A. Nelson, B.S. Swanson, J. Chem. Phys. 44, 4229, (1966). Interpolated approximations and bounds are all of the form : u(k) \approx \tilde{g}(k) u_{L\infty}(k) + (1 - \tilde{g}(k)) u_U(k) where \tilde{g} is an interpolating function running monotonically from 0 at low k to 1 at high k, approximating an ideal, or exact, interpolator g(k): :g(k) = \frac{ u_U(k) - u(k)}{ u_U(k) - u_{L\infty}(k)} For the simplest interpolating function considered, a first-order rational function :\tilde{g}_1(k) = \frac{k}{b_0 + k} the tightest lower bound has :b_0 = \frac{\frac{8}{405} + e^{-\gamma} \log 2 - \frac{\log^2 2}{2}}{e^{-\gamma} - \log 2 + \frac{1}{3}} - \log 2 \approx 0.143472 and the tightest upper bound has :b_0 = \frac{e^{-\gamma} - \log 2 + \frac{1}{3}}{1 - \frac{e^{-\gamma} \pi^2}{12}} \approx 0.374654 The interpolated bounds are plotted (mostly inside the yellow region) in the log–log plot shown. Above 750 K Tc values may be in error by 10 K or more. Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. ==See also== * F-distribution * Folded-t and half-t distributions * Hotelling's T-squared distribution * Multivariate Student distribution * Standard normal table (Z-distribution table) * t-statistic * Tau distribution, for internally studentized residuals * Wilks' lambda distribution * Wishart distribution * Modified half-normal distribution with the pdf on (0, \infty) is given as f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\\\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function. ==Notes== ==References== * * * * ==External links== * *Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term ""Student's distribution"") * First Students on page 112. (DOE contract 95‑831). : p(k,\theta \mid p, q, r, s) = \frac{1}{Z} \frac{p^{k-1} e^{-\theta^{-1} q}}{\Gamma(k)^r \theta^{k s}}, where Z is the normalizing constant with no closed-form solution. Berg and Pedersen found more terms: : u(k) = k - \frac{1}{3} + \frac{8}{405 k} + \frac{184}{25515 k^2} + \frac{2248}{3444525 k^3} - \frac{19006408}{15345358875 k^4} - O\left(\frac{1}{k^5}\right) + \cdots 320px|thumb| Two gamma distribution median asymptotes which are conjectured to be bounds (upper solid red and lower dashed red), of the from u(k) \approx 2^{-1/k}(A + k), and an interpolation between them that makes an approximation (dotted red) that is exact at k = 1 and has maximum relative error of about 0.6%. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. Partial sums of these series are good approximations for high enough k; they are not plotted in the figure, which is focused on the low-k region that is less well approximated. If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion: :F(x;\alpha,\beta) = 1-\sum_{i=0}^{\alpha-1} \frac{(\beta x)^i}{i!} e^{-\beta x} = e^{-\beta x} \sum_{i=\alpha}^{\infty} \frac{(\beta x)^i}{i!}. === Characterization using shape k and scale θ === A random variable X that is gamma-distributed with shape k and scale θ is denoted by :X \sim \Gamma(k, \theta) \equiv \operatorname{Gamma}(k, \theta) [[Image:Gamma- PDF-3D.png|thumb|right|320px|Illustration of the gamma PDF for parameter values over k and x with θ set to 1, 2, 3, 4, 5 and 6\. A bias-corrected variant of the estimator for the scale θ is : \tilde{\theta} = \frac{N}{N - 1} \hat{\theta} A bias correction for the shape parameter k is given as : \tilde{k} = \hat{k} - \frac{1}{N} \left(3 \hat{k} - \frac{2}{3} \left(\frac{\hat{k}}{1 + \hat{k}}\right) - \frac{4}{5} \frac{\hat{k}}{(1 + \hat{k})^2} \right) ====Bayesian minimum mean squared error==== With known k and unknown θ, the posterior density function for theta (using the standard scale-invariant prior for θ) is : P(\theta \mid k, x_1, \dots, x_N) \propto \frac 1 \theta \prod_{i=1}^N f(x_i; k, \theta) Denoting : y \equiv \sum_{i=1}^Nx_i , \qquad P(\theta \mid k, x_1, \dots, x_N) = C(x_i) \theta^{-N k-1} e^{-y/\theta} Integration with respect to θ can be carried out using a change of variables, revealing that 1/θ is gamma-distributed with parameters α = Nk, β = y. :\int_0^\infty \theta^{-Nk - 1 + m} e^{-y/\theta}\, d\theta = \int_0^\infty x^{Nk - 1 - m} e^{-xy} \, dx = y^{-(Nk - m)} \Gamma(Nk - m) \\! The Kullback–Leibler divergence (KL-divergence), of Gamma(αp, βp) (""true"" distribution) from Gamma(αq, βq) (""approximating"" distribution) is given byW.D. Penny, [www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities] : \begin{align} D_{\mathrm{KL}}(\alpha_p,\beta_p; \alpha_q, \beta_q) = {} & (\alpha_p-\alpha_q) \psi(\alpha_p) - \log\Gamma(\alpha_p) + \log\Gamma(\alpha_q) \\\ & {} + \alpha_q(\log \beta_p - \log \beta_q) + \alpha_p\frac{\beta_q-\beta_p}{\beta_p}. \end{align} Written using the k, θ parameterization, the KL-divergence of Gamma(kp, θp) from Gamma(kq, θq) is given by : \begin{align} D_{\mathrm{KL}}(k_p,\theta_p; k_q, \theta_q) = {} & (k_p-k_q)\psi(k_p) - \log\Gamma(k_p) + \log\Gamma(k_q) \\\ & {} + k_q(\log \theta_q - \log \theta_p) + k_p \frac{\theta_p - \theta_q}{\theta_q}. \end{align} ===Laplace transform=== The Laplace transform of the gamma PDF is :F(s) = (1 + \theta s)^{-k} = \frac{\beta^\alpha}{(s + \beta)^\alpha} . ==Related distributions== ===General=== * Let X_1, X_2, \ldots, X_n be n independent and identically distributed random variables following an exponential distribution with rate parameter λ, then \sum_i X_i ~ Gamma(n, 1/λ) where n is the shape parameter and λ is the rate, and \bar{X} = \frac{1}{n} \sum_i X_i \sim \operatorname{Gamma}(n, n\lambda) where the rate changes nλ. For the two parameter model, the quantile function (inverse cdf) is : F^{-1}(u) = \sigma \exp \sqrt{-\frac{1}{\beta} \log(1-u)}, \quad 0 < u < 1\. ==Related distributions== * If X \sim \mathrm{Benini}(\alpha,0,\sigma)\,, then X has a Pareto distribution with x_\mathrm{m}=\sigma * If X \sim \mathrm{Benini}(0,\tfrac{1}{2\sigma^2},1), then X \sim e^U where U \sim \mathrm{Rayleigh}(\sigma) ==Software== The (two parameter) Benini distribution density, probability distribution, quantile function and random number generator is implemented in the VGAM package for R, which also provides maximum likelihood estimation of the shape parameter. In particular for integer valued degrees of freedom u we have: For u >1 even, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 5 \cdot 3} { 2 \sqrt{ u}( u -2)( u -4)\cdots 4 \cdot 2\,}\cdot For u >1 odd, : \frac{\Gamma(\frac{ u+1}{2})} {\sqrt{ u\pi}\,\Gamma(\frac{ u}{2})} = \frac{( u -1)( u -3)\cdots 4 \cdot 2} {\pi \sqrt{ u}( u -2)( u -4)\cdots 5 \cdot 3\,}\cdot\\! This can be derived using the exponential family formula for the moment generating function of the sufficient statistic, because one of the sufficient statistics of the gamma distribution is ln(x). ===Information entropy=== The information entropy is : \begin{align} \operatorname{H}(X) & = \operatorname{E}[-\ln(p(X))] \\\\[4pt] & = \operatorname{E}[-\alpha \ln(\beta) + \ln(\Gamma(\alpha)) - (\alpha-1)\ln(X) + \beta X] \\\\[4pt] & = \alpha - \ln(\beta) + \ln(\Gamma(\alpha)) + (1-\alpha)\psi(\alpha). \end{align} In the k, θ parameterization, the information entropy is given by : \operatorname{H}(X) =k + \ln(\theta) + \ln(\Gamma(k)) + (1-k)\psi(k). ===Kullback–Leibler divergence=== thumb|right|320px|Illustration of the Kullback–Leibler (KL) divergence for two gamma PDFs. ",1.3,-21.2,"""260.0""",0.829,0.0547,B
+"In an industrial process, nitrogen is heated to $500 \mathrm{~K}$ in a vessel of constant volume. If it enters the vessel at $100 \mathrm{~atm}$ and $300 \mathrm{~K}$, what pressure would it exert at the working temperature if it behaved as a perfect gas?","For a fixed number of moles of gas n, a thermally perfect gas * is in thermodynamic equilibrium * is not chemically reacting * has internal energy U, enthalpy H, and constant volume / constant pressure heat capacities C_V, C_P that are solely functions of temperature and not of pressure P or volume V, i.e., U = U(T), H = H(T), dU = C_V (T) dT, dH = C_P (T) dT. thumb|Nitrogen is a liquid under -195.8 degrees Celsius (77K). thumb|A medium- sized dewar is being filled with liquid nitrogen by a larger cryogenic storage tank. thumb|right|A gas regulator attached to a nitrogen cylinder Industrial gases are the gaseous materials that are manufactured for use in industry. If the internal pressure exceeds the mechanical limitations of the cylinder and there are no means to safely vent the pressurized gas to the atmosphere, the vessel will fail mechanically. The pressure melting point is nearly a constant 0 °C at pressures above the triple point at 611.7 Pa, where water can exist in only the solid or liquid phases, through atmospheric pressure (100 kPa) until about 10 MPa. All of these substances are also provided as a gas (not a vapor) at the 200 bar pressure in a gas cylinder because that pressure is above their critical pressure. In a fire, the pressure in a gas cylinder rises in direct proportion to its temperature. Springer Science+Business Media LLC (2007) This is a logical dividing line, since the normal boiling points of the so-called permanent gases (such as helium, hydrogen, neon, nitrogen, oxygen, and normal air) lie below 120K while the Freon refrigerants, hydrocarbons, and other common refrigerants have boiling points above 120K. The pressure melting point of ice is the temperature at which ice melts at a given pressure. In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. Nomenclature 1 Nomenclature 2 Heat capacity at constant V, C_V, or constant P, C_P Ideal-gas law PV = nRT and C_P - C_V = nR Calorically perfect Perfect Thermally perfect Semi-perfect Ideal Imperfect Imperfect, or non-ideal === Thermally and calorically perfect gas === Along with the definition of a perfect gas, there are also two more simplifications that can be made although various textbooks either omit or combine the following simplifications into a general ""perfect gas"" definition. For gases that are liquid under storage, e.g., propane, the outlet pressure is dependent on the vapor pressure of the gas, and does not fall until the cylinder is nearly exhausted, although it will vary according to the temperature of the cylinder contents. A pressure reactor, sometimes referred to as a pressure tube, or a sealed tube, is a chemical reaction vessel which can conduct a reaction under pressure. Pressure vessels for gas storage may also be classified by volume. A pressure reactor is a special application of a pressure vessel. However, the idea of a perfect gas model is often invoked as a combination of the ideal gas equation of state with specific additional assumptions regarding the variation (or nonvariation) of the heat capacity with temperature. == Perfect gas nomenclature == The terms perfect gas and ideal gas are sometimes used interchangeably, depending on the particular field of physics and engineering. However, pressure can speed up the desired reaction and only impacts decomposition when it involves the release of a gas or a reaction with a gas in the vessel. With increasing pressure above 10 MPa, the pressure melting point decreases to a minimum of −21.9 °C at 209.9 MPa. The term “industrial gases” is sometimes narrowly defined as just the major gases sold, which are: nitrogen, oxygen, carbon dioxide, argon, hydrogen, acetylene and helium. This is because the internal energy of an ideal gas is at most a function of temperature, as shown by the thermodynamic equation \left({{\partial U} \over {\partial V}}\right)_T = T\left({{\partial S} \over {\partial V}}\right)_T - P = T\left({{\partial P} \over {\partial T}}\right)_V - P, which is exactly zero when P = nRT / V . A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. It can be proved that an ideal gas (i.e. satisfying the ideal gas equation of state, PV = nRT ) is either calorically perfect or thermally perfect. ",167,170,"""0.346""",24,22,A
+"For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \%$, what will be the balance in the IRA at age 65 ?","For those 65 and over, 11.6% of retirement accounts have balances of at least $1 million, more than twice that of the $407,581 average (shown). For those 65 and over, 11.6% of retirement accounts have balances of at least $1 million, more than twice that of the $407,581 average (shown). The return over the five-year period for such an investor would be ($19.90 + $5.78) / $14.21 − 1 = 80.72%, and the arithmetic average rate of return would be 80.72%/5 = 16.14% per year. ==See also== * Annual percentage yield * Average for a discussion of annualization of returns * Capital budgeting * Compound annual growth rate * Compound interest * Dollar cost averaging * Economic value added * Effective annual rate * Effective interest rate * Expected return * Holding period return * Internal rate of return * Modified Dietz method * Net present value * Rate of profit * Return of capital * Return on assets * Return on capital * Returns (economics) * Simple Dietz method * Time value of money * Time-weighted return * Value investing * Yield ==Notes== ==References== ==Further reading== * A. A. Groppelli and Ehsan Nikbakht. As another example, a two-year return of 10% converts to an annualized rate of return of 4.88% = ((1+0.1)(12/24) − 1), assuming reinvestment at the end of the first year. thumb|Average balances of retirement accounts, for households having such accounts, exceed median net worth across all age groups. thumb|Average balances of retirement accounts, for households having such accounts, exceed median net worth across all age groups. Some highlights from the 2008 data follow: * The average and median IRA account balance was $54,863 and $15,756, respectively, while the average and median IRA individual balance (all accounts from the same person combined) was $69,498 and $20,046. The purpose of this report was to study individual retirement accounts (IRAs) and the account valuations. Also that there are an estimate of 791 taxpayers with IRA account balances between $10,000,000 and $25,000,000. Furthermore, the typical working household has virtually no retirement savings - the median retirement account balance is $2,500 for all working-age households and $14,500 for near-retirement households. * In 2014, the federal government will forgo an estimated $17.45 billion in tax revenue from IRAs, which Congress created to ensure equitable tax treatment for those not covered by employer-sponsored retirement plans. * 98.5% of taxpayers have IRA account balances at $1,000,000 or less. * 1.2% of tax payers have IRA account balances at $1,000,000 to $2,000,000. * 0.2% (or 83,529) taxpayers have IRA account balances of $2,000,000 to $3,000,000 * 0.1% (or 36,171) taxpayers have IRA account balances of $3,000,000 to $5,000,000 * <0.1% (or 7,952) taxpayers have IRA account balances of $5,000,000 to $10,000,000 * <0.1% (or 791) taxpayers have IRA account balances of $10,000,000 to $25,000,000 * <0.1% (or 314) taxpayers have IRA account balances of $25,000,000 or more ===Retirement savings=== While the average (mean) and median IRA individual balance in 2008 were approximately $70,000 and $20,000 respectively, higher balances are not rare. 6.3% of individuals had total balances of $250,000 or more (about 12.5 times the median), and in rare cases, individuals own IRAs with very substantial balances, in some cases $100 million or above (about 5,000 times the median individual balance). An individual retirement account is a type of individual retirement arrangementSee 26 C.F.R. sec. 1.408-4. as described in IRS Publication 590, Individual Retirement Arrangements (IRAs). Actuarial assumptions like 5% interest, 3% salary increases and the UP84 Life Table for mortality are used to calculate a level contribution rate that would create the needed lump sum at retirement age. For example, a person aged 45, who put $4,000 into a traditional IRA this year so far, can either put $2,000 more into this traditional IRA, or $2,000 in a Roth IRA, or some combination of those. The cash balance plan typically offers a lump sum at and often before normal retirement age. * Contributions are concentrated at the maximum amount – of those contributing to an IRA, approximately 40% contributed the maximum (whether contributing to traditional or Roth), and 46.7% contributed close to the maximum (in the $5,000–$6,000 range). * Excluding SEPs and SIMPLEs (i.e., concerning traditional, rollover, and Roth IRAs), 15.1% of individuals holding an IRA contributed to one. The Retirement Age as referred to in paragraph (2) is further increased by 1 (one) year for every subsequent 3 (three) years until it reaches the Retirement Age of 65 (sixty five) years. For example, if the logarithmic return of a security per trading day is 0.14%, assuming 250 trading days in a year, then the annualized logarithmic rate of return is 0.14%/(1/250) = 0.14% x 250 = 35% ===Returns over multiple periods=== When the return is calculated over a series of sub- periods of time, the return in each sub-period is based on the investment value at the beginning of the sub-period. Cash flow example on $1,000 investment Year 1 Year 2 Year 3 Year 4 Dollar return $100 $55 $60 $50 ROI 10% 5.5% 6% 5% ==Uses== * Rates of return are useful for making investment decisions. For example, a return over one month of 1% converts to an annualized rate of return of 12.7% = ((1+0.01)12 − 1). * The continuously compounded rate of return in this example is: :\ln\left(\frac{103.02}{100}\right) = 2.98\%. ",588313,1.6,"""0.195""",344,0.396,A
+"Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion.","Since the inertia of the beam can be found from its mass, the spring constant can be calculated. For waves on a string, or in a medium such as water, the amplitude is a displacement. Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). Thus, for g ≈ 9.8 ms−2, :L \approx \frac{894}{n^2} \; \text{m} | n 71 0.846 0.177 70 0.857 0.182 69 0.870 0.188 68 0.882 0.193 67 0.896 0.199 66 0.909 0.205 65 0.923 0.212 64 0.938 0.218 63 0.952 0.225 62 0.968 0.232 61 0.984 0.240 60 1.000 0.248 Parameters of the pendulum wave in the animation above ---|--- == See also == * Newton's cradle - a set of pendulums constrained to swing along the axis of the apparatus and collide with one another ==References== Category:Pendulums Category:Kinetic art RMS Amplitude . When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The amplitude of sound waves and audio signals (which relates to the volume) conventionally refers to the amplitude of the air pressure in the wave, but sometimes the amplitude of the displacement (movements of the air or the diaphragm of a speaker) is described. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations. ==Analysis and interpretation== thumb|300px|2 DOF elastic pendulum with polar coordinate plots. Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by :\delta = \frac{v}{\omega}, :\delta = \frac{p}{\omega z(\mathbf{r},\, s)}. ==See also== *Sound *Sound particle *Particle velocity *Particle acceleration ==References and notes== Related Reading: * * * ==External links== *Acoustic Particle-Image Velocimetry. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . Hooke's law is the potential energy of the spring itself: :V_k=\frac{1}{2}kx^2 where k is the spring constant. Strictly speaking, this is no longer amplitude since there is the possibility that a constant (DC component) is included in the measurement. ===Peak-to-peak amplitude=== Peak-to-peak amplitude (abbreviated p–p) is the change between peak (highest amplitude value) and trough (lowest amplitude value, which can be negative). With appropriate circuitry, peak-to- peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an oscilloscope. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. The square of the amplitude is proportional to the intensity of the wave. ",9.8,0.18162,"""5275.0""",0.8561,0.444444444444444 ,B
+"At time $t=0$ a tank contains $Q_0 \mathrm{lb}$ of salt dissolved in 100 gal of water; see Figure 2.3.1. Assume that water containing $\frac{1}{4} \mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \mathrm{gal} / \mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \%$ of $Q_L$. ","Renewal time simply becomes a question how quickly could the inflows of the lake fill the entire volume of the basin (this does still assume the outflows are unchanged). It may therefore be estimated by the steady-state approximation, which specifies that the rate at which it is formed equals the (total) rate at which it is consumed. This case is referred to as diffusion control and, in general, occurs when the formation of product from the activated complex is very rapid and thus the provision of the supply of reactants is rate-determining. == See also == * Product-determining step * Rate-limiting step (biochemistry) ==References== * Category:Chemical kinetics ja:反応速度#律速段階 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. Instead, residence time models developed for gas and fluid dynamics, chemical engineering, and bio-hydrodynamics can be adapted to generate residence times for sub-volumes of lakes. == Renewal time == One useful mathematical model the measurement of how quickly inflows are able to refill a lake. In principle, the time evolution of the reactant and product concentrations can be determined from the set of simultaneous rate equations for the individual steps of the mechanism, one for each step. The observed rate law is :v = k \frac{\ce{[Cl2][H2C2O4]}}{[\ce{H+}]^2[\ce{Cl^-}]}, which implies an activated complex in which the reactants lose 2 + before the rate-determining step. This equation can be derived from the mass conservation equations of two-phase flow, under the assumptions listed below. ==Equation== In a quasi-1D domain, the Buckley–Leverett equation is given by: : \frac{\partial S_w}{\partial t} + \frac{\partial}{\partial x}\left( \frac{Q}{\phi A} f_w(S_w) \right) = 0, where S_w(x,t) is the wetting-phase (water) saturation, Q is the total flow rate, \phi is the rock porosity, A is the area of the cross-section in the sample volume, and f_w(S_w) is the fractional flow function of the wetting phase. That is, the overall rate is determined by the rate of the first step, and (almost) all molecules that react at the first step continue to the fast second step. ===Pre- equilibrium: if the second step were rate-determining=== The other possible case would be that the second step is slow and rate-determining, meaning that it is slower than the first step in the reverse direction: r2 ≪ r−1. In the simplest case the initial step is the slowest, and the overall rate is just the rate of the first step. The rate-determining step is then the step with the largest Gibbs energy difference relative either to the starting material or to any previous intermediate on the diagram. Such a situation in which an intermediate (here ) forms an equilibrium with reactants prior to the rate-determining step is described as a pre-equilibriumPeter Atkins and Julio de Paula, Physical Chemistry (8th ed., W. H. Freeman 2006) p. 814–815. . In this example is formed in one step and reacts in two, so that : \frac{d\ce{[NO3]}}{dt} = r_1 - r_2 - r_{-1} \approx 0. A possible mechanism in two elementary steps that explains the rate equation is: # + → NO + (slow step, rate-determining) # + CO → + (fast step) In this mechanism the reactive intermediate species is formed in the first step with rate r1 and reacts with CO in the second step with rate r2. It roughly expresses the amount of time taken for a substance introduced into a lake to flow out of it again. For a given reaction mechanism, the prediction of the corresponding rate equation (for comparison with the experimental rate law) is often simplified by using this approximation of the rate-determining step. Time of concentration is a concept used in hydrology to measure the response of a watershed to a rain event. The retention time is particularly important where downstream flooding or pollutants are concerned. ==Global retention time== The global retention time for a lake (the overall mean time that water spends in the lake) is calculated by dividing the lake volume by either the mean rate of inflow of all tributaries, or by the mean rate of outflow (ideally including evaporation and seepage). A number of methods can be used to calculate time of concentration, including the Kirpich (1940) and NRCS (1997) methods. This can be important for infrastructure development (design of bridges, culverts, etc.) and management, as well as to assess flood risk such as the ARkStorm-scenario. ==Example== thumb|left|400px|NOAA diagram illustrating the concept underlying time of concentration This image shows the basic principle which leads to determination of the time of concentration. Typically, f_w(S_w) is an 'S'-shaped, nonlinear function of the saturation S_w, which characterizes the relative mobilities of the two phases: : f_w(S_w) = \frac{\lambda_w}{\lambda_w + \lambda_n} = \frac{ \frac{k_{rw}}{\mu_w} }{ \frac{k_{rw}}{\mu_w} + \frac{k_{rn}}{\mu_n} }, where \lambda_w and \lambda_n denote the wetting and non-wetting phase mobilities. k_{rw}(S_w) and k_{rn}(S_w) denote the relative permeability functions of each phase and \mu_w and \mu_n represent the phase viscosities. ==Assumptions== The Buckley–Leverett equation is derived based on the following assumptions: * Flow is linear and horizontal * Both wetting and non-wetting phases are incompressible * Immiscible phases * Negligible capillary pressure effects (this implies that the pressures of the two phases are equal) * Negligible gravitational forces ==General solution== The characteristic velocity of the Buckley-Leverett equation is given by: :U(S_w) = \frac{Q}{\phi A} \frac{\mathrm{d} f_w}{\mathrm{d} S_w}. The Buckley–Leverett equation or the Buckley–Leverett displacement describes an immiscible displacement process, such as the displacement of oil by water, in a one-dimensional or quasi-one-dimensional reservoir. ",130.400766848,8.7,"""1.7""",+80,0.4,A
+"Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion.","Since the inertia of the beam can be found from its mass, the spring constant can be calculated. thumb In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. When this property is exactly satisfied, the balance spring is said to be isochronous, and the period of oscillation is independent of the amplitude of oscillation. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. thumb|Phase portrait of damped oscillator, with increasing damping strength. When the oscillatory motion of the balance dies out, the deflection will be proportional to the force: :\theta = FL/\kappa\, To determine F\, it is necessary to find the torsion spring constant \kappa\,. In this case, the phase shift is simply the argument shift \tau, expressed as a fraction of the common period T (in terms of the modulo operation) of the two signals and then scaled to a full turn: :\varphi = 2\pi \left[\\!\\!\left[ \frac{\tau}{T} \right]\\!\\!\right]. The stiffness of the spring, its spring coefficient, \kappa\, in N·m/radian^2, along with the balance wheel's moment of inertia, I\, in kg·m2, determines the wheel's oscillation period T\,. The general differential equation of motion is: :I\frac{d^2\theta}{dt^2} + C\frac{d\theta}{dt} + \kappa\theta = \tau(t) If the damping is small, C \ll \sqrt{\kappa I}\,, as is the case with torsion pendulums and balance wheels, the frequency of vibration is very near the natural resonant frequency of the system: :f_n = \frac{\omega_n}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\, Therefore, the period is represented by: :T_n = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{I}{\kappa}}\, The general solution in the case of no drive force (\tau = 0\,), called the transient solution, is: :\theta = Ae^{-\alpha t} \cos{(\omega t + \phi)}\, where: ::\alpha = C/2I\, ::\omega = \sqrt{\omega_n^2 - \alpha^2} = \sqrt{\kappa/I - (C/2I)^2}\, ===Applications=== thumb|Animation of a torsion spring oscillating The balance wheel of a mechanical watch is a harmonic oscillator whose resonant frequency f_n\, sets the rate of the watch. The term phase can refer to several different things: * It can refer to a specified reference, such as \textstyle \cos(2 \pi f t), in which case we would say the phase of \textstyle x(t) is \textstyle \varphi, and the phase of \textstyle y(t) is \textstyle \varphi - \frac{\pi}{2}. The weight rotates about the axis of the spring, twisting it, instead of swinging like an ordinary pendulum. The value of damping that causes the oscillatory motion to settle quickest is called the critical dampingC_c\,: :C_c = 2 \sqrt{\kappa I}\, ==See also== * Beam (structure) * Slinky, helical toy spring ==References== ==Bibliography== * . The formula above gives the phase as an angle in radians between 0 and 2\pi. A balance spring, or hairspring, is a spring attached to the balance wheel in mechanical timepieces. The term ""phase"" is also used when comparing a periodic function F with a shifted version G of it. If the shift in t is expressed as a fraction of the period, and then scaled to an angle \varphi spanning a whole turn, one gets the phase shift, phase offset, or phase difference of G relative to F. In physics and mathematics, the phase of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum. ===Lagrangian=== The spring has the rest length l_0 and can be stretched by a length x. Thus, for example, the sum of phase angles is 30° (, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (, plus one full turn). We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. thumb|Video of a model torsion pendulum oscillating A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. The phase \phi(t) is then the angle from the 12:00 position to the current position of the hand, at time t, measured clockwise. ",-0.40864,4.86,"""-6.8""",+2.35, 9.73,A
+"The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \times 10^6 \mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later. ","Because of this structure, the model can be considered as the discrete-time analogue of the continuous-time logistic equation for population growth introduced by Verhulst; for comparison, the logistic equation is : \frac{dN}{dt} = rN \left( 1 - \frac{N}{K} \right), and its solution is : N(t) = \frac{K N(0)}{N(0) + (K - N(0)) e^{-rt}}. ==References== * * * * * Category:Demography Category:Biostatistics Category:Fisheries science Category:Stochastic models As the population approaches its carrying capacity, the rate of growth decreases, and the population trend will become logistic. This S-shaped curve observed in logistic growth is a more accurate model than exponential growth for observing real- life population growth of organisms. ==See also== * Malthusian catastrophe * r/K selection theory ==References== ==Sources== John A. Miller and Stephen B. Harley zoology 4th edition ==External links== * Category:Biology Category:Biology articles needing attention Category:Population ecology Once the carrying capacity, or K, is incorporated to account for the finite resources that a population will be competing for within an environment, the aforementioned equation becomes the following: \frac{dN}{dt}=r_{max}\frac{dN}{dt}=r_{max}N\frac{K-N}{K} A graph of this equation creates an S-shaped curve, which demonstrates how initial population growth is exponential due to the abundance of resources and lack of competition. Here R0 is interpreted as the proliferation rate per generation and K = (R0 − 1) M is the carrying capacity of the environment. If, in a hypothetical population of size N, the birth rates (per capita) are represented as b and death rates (per capita) as d, then the increase or decrease in N during a time period t will be \frac{dN}{dt}=(b-d)N (b-d) is called the 'intrinsic rate of natural increase' and is a very important parameter chosen for assessing the impacts of any biotic or abiotic factor on population growth. The Ricker model, named after Bill Ricker, is a classic discrete population model which gives the expected number N t+1 (or density) of individuals in generation t + 1 as a function of the number of individuals in the previous generation,Ricker (1954) : N_{t+1} = N_t e^{r\left(1-\frac{N_t}{k}\right)}.\, Here r is interpreted as an intrinsic growth rate and k as the carrying capacity of the environment. The Monod equation is a mathematical model for the growth of microorganisms. In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation. The solution is : n_t = \frac{K n_0}{n_0 + (K - n_0) R_0^{-t}}. When the yield coefficient, being the ratio of mass of microorganisms to mass of substrate utilized, becomes very large this signifies that there is deficiency of substrate available for utilization. == Graphical determination of constants == As with the Michaelis–Menten equation graphical methods may be used to fit the coefficients of the Monod equation: * Eadie–Hofstee diagram * Hanes–Woolf plot * Lineweaver–Burk plot == See also == * Activated sludge model (uses the Monod equation to model bacterial growth and substrate utilization) * Bacterial growth * Hill equation (biochemistry) * Hill contribution to Langmuir equation * Langmuir adsorption model (equation with the same mathematical form) * Michaelis–Menten kinetics (equation with the same mathematical form) * Gompertz function * Victor Henry, who first wrote the general equation form in 1901 * Von Bertalanffy function == References == Category:Catalysis Category:Chemical kinetics Category:Environmental engineering Category:Enzyme kinetics Category:Ordinary differential equations Category:Sewerage They will differ between microorganism species and will also depend on the ambient environmental conditions, e.g., on the temperature, on the pH of the solution, and on the composition of the culture medium. == Application notes == The rate of substrate utilization is related to the specific growth rate as follows: : r_s = \mu X/Y where: * X is the total biomass (since the specific growth rate, μ is normalized to the total biomass) * Y is the yield coefficient rs is negative by convention. right|thumb|thumbtime=5|FVCOM simulation of hypersaline sea surface release and propagation under tidal conditions in the northern North Sea The Finite Volume Community Ocean Model (FVCOM; Formerly Finite Volume Coastal Ocean Model) is a prognostic, unstructured-grid, free-surface, 3-D primitive equation coastal ocean circulation model. The equation in discrete time is given by :1 = \sum_{a = 1}^\omega \lambda^{-a}\ell(a)b(a) where \lambda is the discrete growth rate, ℓ(a) is the fraction of individuals surviving to age a and b(a) is the number of offspring born to an individual of age a during the time step. First substitute the definition of the per-capita fertility and divide through by the left hand side: :1 = \frac{s_0b_1}{\lambda} + \frac{s_0s_1b_2}{\lambda^2} + \cdots + \frac{s_0\cdots s_{\omega - 1}b_{\omega}}{\lambda^{\omega}}. The model can be used to predict the number of fish that will be present in a fishery.de Vries et al.Marland Subsequent work has derived the model under other assumptions such as scramble competition,Brännström and Sumpter(2005) within-year resource limited competition or even as the outcome of source-sink Malthusian patches linked by density-dependent dispersal. When c = 1, the Hassell model is simply the Beverton–Holt model. ==See also== * Population dynamics of fisheries ==Notes== ==References== * Brännström A and Sumpter DJ (2005) ""The role of competition and clustering in population dynamics"" Proc Biol Sci., 272(1576): 2065-72\. Biological exponential growth is the unrestricted growth of a population of organisms, occurring when resources in its habitat are unlimited. The sum is taken over the entire life span of the organism. ==Derivations== ===Lotka's continuous model=== A.J. Lotka in 1911 developed a continuous model of population dynamics as follows. Based on the age demographic of females in the population and female births (since in many cases it is the females that are more limited in the ability to reproduce), this equation allows for an estimation of how a population is growing. Any species growing exponentially under unlimited resource conditions can reach enormous population densities in a short time. The empirical Monod equation is: : \mu = \mu_\max {[S] \over K_s + [S]} where: * μ is the growth rate of a considered microorganism * μmax is the maximum growth rate of this microorganism * [S] is the concentration of the limiting substrate S for growth * Ks is the ""half- velocity constant""—the value of [S] when μ/μmax = 0.5 μmax and Ks are empirical (experimental) coefficients to the Monod equation. ",46.7,38,"""0.9522""",47,0,A
 "A fluid has density $870 \mathrm{~kg} / \mathrm{m}^3$ and flows with velocity $\mathbf{v}=z \mathbf{i}+y^2 \mathbf{j}+x^2 \mathbf{k}$, where $x, y$, and $z$ are measured in meters and the components of $\mathbf{v}$ in meters per second. Find the rate of flow outward through the cylinder $x^2+y^2=4$, $0 \leqslant z \leqslant 1$.
-","Thus we find the maximum speed in the flow, , in the low pressure on the sides of the cylinder. The goal is to find the steady velocity vector and pressure in a plane, subject to the condition that far from the cylinder the velocity vector (relative to unit vectors and ) is: :\mathbf{V}=U\mathbf{i}+0\mathbf{j} \,, where is a constant, and at the boundary of the cylinder :\mathbf{V}\cdot\mathbf{\hat n}=0 \,, where is the vector normal to the cylinder surface. Far from the cylinder, the flow is unidirectional and uniform. Then the solution to first-order approximation in terms of the velocity potential is :\phi(r,\theta) = Ur\left(1+ \frac{a^2}{r^2}\right)\cos\theta - \mathrm{M}^2 \frac{Ur}{12} \left[\left( \frac{13 a^2}{r^2} - \frac{6 a^4}{r^4} + \frac{a^6}{r^6}\right) \cos\theta + \left(\frac{a^4}{r^4} - \frac{3a^2}{r^2} \right) \cos 3\theta\right]+ \mathrm{O}\left(\mathrm{M}^4\right) \, where a is the radius of the cylinder. ==Potential flow over a circular cylinder with slight variations== Regular perturbation analysis for a flow around a cylinder with slight perturbation in the configurations can be found in Milton Van Dyke (1975). The low pressure on sides on the cylinder is needed to provide the centripetal acceleration of the flow: :\frac{\partial p}{\partial r}=\frac{\rho V^2}{L} \,, where is the radius of curvature of the flow. The pressure at each point on the wake side of the cylinder will be lower than on the upstream side, resulting in a drag force in the downstream direction. ==Janzen–Rayleigh expansion== The problem of potential compressible flow over circular cylinder was first studied by O. Janzen in 1913O. Now let be the internal pressure coefficient inside the cylinder, then a slight normal velocity due to the slight porousness is given by :\frac{1}{r}\frac{\partial \psi}{\partial \theta} = \varepsilon U \left(C_\mathrm{pi} - C_\mathrm{ps}\right) = \varepsilon U \left(C_\mathrm{pi} +1 - 2\cos 2\theta\right) \quad \text{at } r=a \,, but the zero net flux condition :\int_0^{2\pi} \frac{1}{r}\frac{\partial \psi}{\partial \theta} \,\mathrm{d}\theta = 0 requires that . Velocity vectors. thumb|350px|Close-up view of one quadrant of the flow. Then the solution to first-order approximation is :\psi(r,\theta) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta + \varepsilon \frac{Ur}{2} \left( \frac{3a^2}{r^2}\sin \theta - \frac{a^4}{r^4} \sin 3 \theta \right) + \mathrm{O}\left(\varepsilon^2\right) ===Slightly pulsating circle=== Here the radius of the cylinder varies with time slightly so . Then the solution to the first-order approximation is :\psi(r,\theta) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta - \varepsilon U \frac{a^3}{r^2} \sin 2\theta+ \mathrm{O}\left(\varepsilon^2\right) \,. ===Corrugated quasi-cylinder=== If the cylinder has variable radius in the axial direction, the -axis, , then the solution to the first-order approximation in terms of the three-dimensional velocity potential is :\phi(r,\theta,z) = Ur\left(1+ \frac{a^2}{r^2}\right)\cos\theta - 2\varepsilon U b \frac{K_1\left(\frac{r}{b}\right)}{K_1'\left(\frac{r}{b}\right)} \cos\theta \sin \frac{z}{b} + \mathrm{O}\left(\varepsilon^2\right) \,, where is the modified Bessel function of the first kind of order one. ==See also== *Joukowsky transform *Kutta condition *Magnus effect ==References== Category:Fluid dynamics A cylinder (or disk) of radius is placed in a two-dimensional, incompressible, inviscid flow. The velocity components (v_r,v_\theta,v_z) of the Rankine vortex, expressed in terms of the cylindrical-coordinate system (r,\theta,z) are given by :v_r=0,\quad v_\theta(r) = \frac{\Gamma}{2\pi}\begin{cases} r/a^2 & r \le a, \\\ 1/ r & r > a \end{cases}, \quad v_z = 0 where \Gamma is the circulation strength of the Rankine vortex. The fluid velocity in the pores \mathbf{v}_a (or short but inaccurately called pore velocity) is related to Darcy velocity by the relation :\mathbf{v}_a = \phi^{-1} \mathbf{q}_a = \phi^{-1} \mathbf{u}_a where a = w, o, g The volumetric flux is an intensive quantity, so it is not good at describing how much fluid is coming per time. The flow is inviscid, incompressible and has constant mass density . In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. This equation is also called the Churchill–Bernstein correlation. ==Heat transfer definition== :\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 where: * \overline{\mathrm{Nu}}_D is the surface averaged Nusselt number with characteristic length of diameter; * \mathrm{Re}_D\,\\! is the Reynolds number with the cylinder diameter as its characteristic length; * \Pr is the Prandtl number. Nomenclature Symbol Description SI Units subscript: phase a, vector component \sigma A{_{\sigma}} vector component \sigma of directional contact surface between two grid cells m2 \mathbf{A} directional contact surface between two (usually neighboring) grid cells m2 {\mathbf{e}_z}{^{3}} unit vector along 3rd axis (z is a reminder here: 3 is z-direction) 1 g acceleration of gravity m/s2 \mathbf{g} acceleration of gravity with direction m/s2 \mathbf{K} absolute permeability as a 3x3 tensor m2 K_{ra} relative permeability of phase a= w, o, g fraction \mathbf{K}_{ra} directional relative permeability (i.e. 3x3 tensor) fraction P_a pressure Pa \mathbf{q}_a volumetric flux (Darcy velocity) through grid cell contact surface m/s {Q_a} volumetric flow rate through grid cell contact surface m3/s \mathbf{v}_a pore (fluid) flow velocity m/s {u_a}^\sigma Darcy (fluid) velocity along axis \sigma m/s \mathbf{u}_a Darcy (fluid) velocity m/s abla gradient operator m−1 \mu_a dynamic viscosity Pa \cdot s \rho_a mass density kg/m3 :\mathbf{u}_a = -\mu_a^{-1} K_{ra} \mathbf{K} \cdot \left( abla P - \rho_a \mathbf{g} \right) where a = w, o, g The present fluid phases are water, oil and gas, and they are represented by the subscript a = w,o,g respectively. With the cylinder blocking some of the flow, must be greater than somewhere in the plane through the center of the cylinder and transverse to the flow. == Comparison with flow of a real fluid past a cylinder == The symmetry of this ideal solution has a stagnation point on the rear side of the cylinder, as well as on the front side. The flow equation in component form (using summation convention) is :u{_a}^\sigma = -\mu_a^{-1} K_{ra}{^\sigma}_{\beta} K{^\beta}_{\gamma} \left( abla^{\gamma} P_a - \rho_a g {e_z}{^{\gamma}} \right) where a = w, o, g where \sigma = 1,2,3 The Darcy velocity \mathbf{u}_a is not the velocity of a fluid particle, but the volumetric flux (frequently represented by the symbol \mathbf{q}_a ) of the fluid stream. In the following, will represent a small positive parameter and is the radius of the cylinder. Then the solution to first-order approximation is :\psi(r,\theta,t) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta + \varepsilon Ur\left( \frac{a^2}{Ur} \theta f'(t) - \frac{2 a^2}{r^2} f(t) \sin \theta\right) + \mathrm{O}\left(\varepsilon^2\right) ===Flow with slight vorticity=== In general, the free-stream velocity is uniform, in other words , but here a small vorticity is imposed in the outer flow. ====Linear shear==== Here a linear shear in the velocity is introduced. :\begin{align} \psi &= U \left(y + \frac{1}{2} \varepsilon \frac{y^2}{a}\right)\,, \\\\[3pt] \omega &= - abla^2 \psi = - \varepsilon \frac{U}{a} \quad \text{as } x\rightarrow -\infty\,, \end{align} where is the small parameter. One should not expect much more than 20% accuracy from the above equation due to the wide range of flow conditions that the equation encompasses. ",355.1,0.4772,0.0,-1.0,122,C
-Suppose that $2 \mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \mathrm{~cm}$ to a length of $42 \mathrm{~cm}$. How far beyond its natural length will a force of $30 \mathrm{~N}$ keep the spring stretched?,"Let be the amount by which the free end of the spring was displaced from its ""relaxed"" position (when it is not being stretched). The force an ideal spring would exert is exactly proportional to its extension or compression. An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. Force of fully compressed spring : F_{max} = \frac{E d^4 (L-n d)}{16 (1+ u) (D-d)^3 n} \ where : E – Young's modulus : d – spring wire diameter : L – free length of spring : n – number of active windings : u – Poisson ratio : D – spring outer diameter ==Zero-length springs== thumb|left|120px|Simplified LaCoste suspension using a zero-length spring thumb|upright|Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant ""Zero-length spring"" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. thumb|Hooke's law: the force is proportional to the extension In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Obviously a coil spring cannot contract to zero length, because at some point the coils touch each other and the spring can't shorten any more. Zero length springs are made by manufacturing a coil spring with built-in tension (A twist is introduced into the wire as it is coiled during manufacture; this works because a coiled spring ""unwinds"" as it stretches), so if it could contract further, the equilibrium point of the spring, the point at which its restoring force is zero, occurs at a length of zero. Explaining the Power of Springing Bodies, London, 1678. as: (""as the extension, so the force"" or ""the extension is proportional to the force""). The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. That is, in a line graph of the spring's force versus its length, the line passes through the origin. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. In practice, zero length springs are made by combining a ""negative length"" spring, made with even more tension so its equilibrium point would be at a ""negative"" length, with a piece of inelastic material of the proper length so the zero force point would occur at zero length. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. : F is the resulting force vector – the magnitude and direction of the restoring force the spring exerts : k is the rate, spring constant or force constant of the spring, a constant that depends on the spring's material and construction. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. Big Spring has been explored to a length of 270 m and a depth of 25 m, and Little Spring to a length of 305 m and depth of 45 m. ==References== ==External links== *Retovje Springs on Geopedia * Category:Municipality of Vrhnika Category:Springs of Slovenia Category:Karst springs SRetovje Since the inertia of the beam can be found from its mass, the spring constant can be calculated. ",1410,-273,2.81,6.283185307,10.8,E
-"Find the work done by a force $\mathbf{F}=8 \mathbf{i}-6 \mathbf{j}+9 \mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons.","In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The work done is given by the dot product of the two vectors. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . In physics, work is the energy transferred to or from an object via the application of force along a displacement. The work of the net force is calculated as the product of its magnitude and the particle displacement. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Notice that the work done by gravity depends only on the vertical movement of the object. The dot product of two perpendicular vectors is always zero, so the work , and the magnetic force does not do work. In this case the dot product , where is the angle between the force vector and the direction of movement, that is W = \int_C \mathbf{F} \cdot d\mathbf{s} = Fs\cos\theta. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism Therefore, work need only be computed for the gravitational forces acting on the bodies. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. If the force is always directed along this line, and the magnitude of the force is , then this integral simplifies to W = \int_C F\,ds where is displacement along the line. The derivation of the work–energy principle begins with Newton’s second law of motion and the resultant force on a particle. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . ",2.3,292,144.0,+93.4,310,C
+","Thus we find the maximum speed in the flow, , in the low pressure on the sides of the cylinder. The goal is to find the steady velocity vector and pressure in a plane, subject to the condition that far from the cylinder the velocity vector (relative to unit vectors and ) is: :\mathbf{V}=U\mathbf{i}+0\mathbf{j} \,, where is a constant, and at the boundary of the cylinder :\mathbf{V}\cdot\mathbf{\hat n}=0 \,, where is the vector normal to the cylinder surface. Far from the cylinder, the flow is unidirectional and uniform. Then the solution to first-order approximation in terms of the velocity potential is :\phi(r,\theta) = Ur\left(1+ \frac{a^2}{r^2}\right)\cos\theta - \mathrm{M}^2 \frac{Ur}{12} \left[\left( \frac{13 a^2}{r^2} - \frac{6 a^4}{r^4} + \frac{a^6}{r^6}\right) \cos\theta + \left(\frac{a^4}{r^4} - \frac{3a^2}{r^2} \right) \cos 3\theta\right]+ \mathrm{O}\left(\mathrm{M}^4\right) \, where a is the radius of the cylinder. ==Potential flow over a circular cylinder with slight variations== Regular perturbation analysis for a flow around a cylinder with slight perturbation in the configurations can be found in Milton Van Dyke (1975). The low pressure on sides on the cylinder is needed to provide the centripetal acceleration of the flow: :\frac{\partial p}{\partial r}=\frac{\rho V^2}{L} \,, where is the radius of curvature of the flow. The pressure at each point on the wake side of the cylinder will be lower than on the upstream side, resulting in a drag force in the downstream direction. ==Janzen–Rayleigh expansion== The problem of potential compressible flow over circular cylinder was first studied by O. Janzen in 1913O. Now let be the internal pressure coefficient inside the cylinder, then a slight normal velocity due to the slight porousness is given by :\frac{1}{r}\frac{\partial \psi}{\partial \theta} = \varepsilon U \left(C_\mathrm{pi} - C_\mathrm{ps}\right) = \varepsilon U \left(C_\mathrm{pi} +1 - 2\cos 2\theta\right) \quad \text{at } r=a \,, but the zero net flux condition :\int_0^{2\pi} \frac{1}{r}\frac{\partial \psi}{\partial \theta} \,\mathrm{d}\theta = 0 requires that . Velocity vectors. thumb|350px|Close-up view of one quadrant of the flow. Then the solution to first-order approximation is :\psi(r,\theta) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta + \varepsilon \frac{Ur}{2} \left( \frac{3a^2}{r^2}\sin \theta - \frac{a^4}{r^4} \sin 3 \theta \right) + \mathrm{O}\left(\varepsilon^2\right) ===Slightly pulsating circle=== Here the radius of the cylinder varies with time slightly so . Then the solution to the first-order approximation is :\psi(r,\theta) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta - \varepsilon U \frac{a^3}{r^2} \sin 2\theta+ \mathrm{O}\left(\varepsilon^2\right) \,. ===Corrugated quasi-cylinder=== If the cylinder has variable radius in the axial direction, the -axis, , then the solution to the first-order approximation in terms of the three-dimensional velocity potential is :\phi(r,\theta,z) = Ur\left(1+ \frac{a^2}{r^2}\right)\cos\theta - 2\varepsilon U b \frac{K_1\left(\frac{r}{b}\right)}{K_1'\left(\frac{r}{b}\right)} \cos\theta \sin \frac{z}{b} + \mathrm{O}\left(\varepsilon^2\right) \,, where is the modified Bessel function of the first kind of order one. ==See also== *Joukowsky transform *Kutta condition *Magnus effect ==References== Category:Fluid dynamics A cylinder (or disk) of radius is placed in a two-dimensional, incompressible, inviscid flow. The velocity components (v_r,v_\theta,v_z) of the Rankine vortex, expressed in terms of the cylindrical-coordinate system (r,\theta,z) are given by :v_r=0,\quad v_\theta(r) = \frac{\Gamma}{2\pi}\begin{cases} r/a^2 & r \le a, \\\ 1/ r & r > a \end{cases}, \quad v_z = 0 where \Gamma is the circulation strength of the Rankine vortex. The fluid velocity in the pores \mathbf{v}_a (or short but inaccurately called pore velocity) is related to Darcy velocity by the relation :\mathbf{v}_a = \phi^{-1} \mathbf{q}_a = \phi^{-1} \mathbf{u}_a where a = w, o, g The volumetric flux is an intensive quantity, so it is not good at describing how much fluid is coming per time. The flow is inviscid, incompressible and has constant mass density . In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. This equation is also called the Churchill–Bernstein correlation. ==Heat transfer definition== :\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 where: * \overline{\mathrm{Nu}}_D is the surface averaged Nusselt number with characteristic length of diameter; * \mathrm{Re}_D\,\\! is the Reynolds number with the cylinder diameter as its characteristic length; * \Pr is the Prandtl number. Nomenclature Symbol Description SI Units subscript: phase a, vector component \sigma A{_{\sigma}} vector component \sigma of directional contact surface between two grid cells m2 \mathbf{A} directional contact surface between two (usually neighboring) grid cells m2 {\mathbf{e}_z}{^{3}} unit vector along 3rd axis (z is a reminder here: 3 is z-direction) 1 g acceleration of gravity m/s2 \mathbf{g} acceleration of gravity with direction m/s2 \mathbf{K} absolute permeability as a 3x3 tensor m2 K_{ra} relative permeability of phase a= w, o, g fraction \mathbf{K}_{ra} directional relative permeability (i.e. 3x3 tensor) fraction P_a pressure Pa \mathbf{q}_a volumetric flux (Darcy velocity) through grid cell contact surface m/s {Q_a} volumetric flow rate through grid cell contact surface m3/s \mathbf{v}_a pore (fluid) flow velocity m/s {u_a}^\sigma Darcy (fluid) velocity along axis \sigma m/s \mathbf{u}_a Darcy (fluid) velocity m/s abla gradient operator m−1 \mu_a dynamic viscosity Pa \cdot s \rho_a mass density kg/m3 :\mathbf{u}_a = -\mu_a^{-1} K_{ra} \mathbf{K} \cdot \left( abla P - \rho_a \mathbf{g} \right) where a = w, o, g The present fluid phases are water, oil and gas, and they are represented by the subscript a = w,o,g respectively. With the cylinder blocking some of the flow, must be greater than somewhere in the plane through the center of the cylinder and transverse to the flow. == Comparison with flow of a real fluid past a cylinder == The symmetry of this ideal solution has a stagnation point on the rear side of the cylinder, as well as on the front side. The flow equation in component form (using summation convention) is :u{_a}^\sigma = -\mu_a^{-1} K_{ra}{^\sigma}_{\beta} K{^\beta}_{\gamma} \left( abla^{\gamma} P_a - \rho_a g {e_z}{^{\gamma}} \right) where a = w, o, g where \sigma = 1,2,3 The Darcy velocity \mathbf{u}_a is not the velocity of a fluid particle, but the volumetric flux (frequently represented by the symbol \mathbf{q}_a ) of the fluid stream. In the following, will represent a small positive parameter and is the radius of the cylinder. Then the solution to first-order approximation is :\psi(r,\theta,t) = Ur\left(1- \frac{a^2}{r^2}\right)\sin\theta + \varepsilon Ur\left( \frac{a^2}{Ur} \theta f'(t) - \frac{2 a^2}{r^2} f(t) \sin \theta\right) + \mathrm{O}\left(\varepsilon^2\right) ===Flow with slight vorticity=== In general, the free-stream velocity is uniform, in other words , but here a small vorticity is imposed in the outer flow. ====Linear shear==== Here a linear shear in the velocity is introduced. :\begin{align} \psi &= U \left(y + \frac{1}{2} \varepsilon \frac{y^2}{a}\right)\,, \\\\[3pt] \omega &= - abla^2 \psi = - \varepsilon \frac{U}{a} \quad \text{as } x\rightarrow -\infty\,, \end{align} where is the small parameter. One should not expect much more than 20% accuracy from the above equation due to the wide range of flow conditions that the equation encompasses. ",355.1,0.4772,"""0.0""",-1.0,122,C
+Suppose that $2 \mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \mathrm{~cm}$ to a length of $42 \mathrm{~cm}$. How far beyond its natural length will a force of $30 \mathrm{~N}$ keep the spring stretched?,"Let be the amount by which the free end of the spring was displaced from its ""relaxed"" position (when it is not being stretched). The force an ideal spring would exert is exactly proportional to its extension or compression. An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. Force of fully compressed spring : F_{max} = \frac{E d^4 (L-n d)}{16 (1+ u) (D-d)^3 n} \ where : E – Young's modulus : d – spring wire diameter : L – free length of spring : n – number of active windings : u – Poisson ratio : D – spring outer diameter ==Zero-length springs== thumb|left|120px|Simplified LaCoste suspension using a zero-length spring thumb|upright|Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant ""Zero-length spring"" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. thumb|Hooke's law: the force is proportional to the extension In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Obviously a coil spring cannot contract to zero length, because at some point the coils touch each other and the spring can't shorten any more. Zero length springs are made by manufacturing a coil spring with built-in tension (A twist is introduced into the wire as it is coiled during manufacture; this works because a coiled spring ""unwinds"" as it stretches), so if it could contract further, the equilibrium point of the spring, the point at which its restoring force is zero, occurs at a length of zero. Explaining the Power of Springing Bodies, London, 1678. as: (""as the extension, so the force"" or ""the extension is proportional to the force""). The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. That is, in a line graph of the spring's force versus its length, the line passes through the origin. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. We will use the word ""torsion"" in the following for a torsion spring according to the definition given above, whether the material it is made of actually works by torsion or by bending. ==Torsion coefficient== As long as they are not twisted beyond their elastic limit, torsion springs obey an angular form of Hooke's law: : \tau = -\kappa\theta\, where \tau\, is the torque exerted by the spring in newton- meters, and \theta\, is the angle of twist from its equilibrium position in radians. \kappa\, is a constant with units of newton-meters / radian, variously called the spring's torsion coefficient, torsion elastic modulus, rate, or just spring constant, equal to the change in torque required to twist the spring through an angle of 1 radian. In practice, zero length springs are made by combining a ""negative length"" spring, made with even more tension so its equilibrium point would be at a ""negative"" length, with a piece of inelastic material of the proper length so the zero force point would occur at zero length. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. : F is the resulting force vector – the magnitude and direction of the restoring force the spring exerts : k is the rate, spring constant or force constant of the spring, a constant that depends on the spring's material and construction. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. Big Spring has been explored to a length of 270 m and a depth of 25 m, and Little Spring to a length of 305 m and depth of 45 m. ==References== ==External links== *Retovje Springs on Geopedia * Category:Municipality of Vrhnika Category:Springs of Slovenia Category:Karst springs SRetovje Since the inertia of the beam can be found from its mass, the spring constant can be calculated. ",1410,-273,"""2.81""",6.283185307,10.8,E
+"Find the work done by a force $\mathbf{F}=8 \mathbf{i}-6 \mathbf{j}+9 \mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons.","In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. The work done is given by the dot product of the two vectors. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . In physics, work is the energy transferred to or from an object via the application of force along a displacement. The work of the net force is calculated as the product of its magnitude and the particle displacement. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Notice that the work done by gravity depends only on the vertical movement of the object. The dot product of two perpendicular vectors is always zero, so the work , and the magnetic force does not do work. In this case the dot product , where is the angle between the force vector and the direction of movement, that is W = \int_C \mathbf{F} \cdot d\mathbf{s} = Fs\cos\theta. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism Therefore, work need only be computed for the gravitational forces acting on the bodies. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. If the force is always directed along this line, and the magnitude of the force is , then this integral simplifies to W = \int_C F\,ds where is displacement along the line. The derivation of the work–energy principle begins with Newton’s second law of motion and the resultant force on a particle. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . ",2.3,292,"""144.0""",+93.4,310,C
 "A ball is thrown at an angle of $45^{\circ}$ to the ground. If the ball lands $90 \mathrm{~m}$ away, what was the initial speed of the ball?
-","thumb|upright=1.5|Spherical pendulum: angles and velocities. 50 meter running target or 50 meter running boar is an ISSF shooting event, shot with a .22-calibre rifle at a target depicting a boar moving sideways across a 10-meter wide opening. If thrown correctly, the changeup will confuse the batter because the human eye cannot discern that the ball is coming significantly slower until it is around 30 feet from the plate. thumb|The target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm, height 0.75 m above the floor 50 meter rifle prone (formerly known as one of four free rifle disciplines) is an International Shooting Sport Federation event consisting of 60 shots from the prone position with a .22 Long Rifle (5.6 mm) caliber rifle. A ball moves due to the changes in the pressure of the air surrounding the ball as a result of the kind of pitch thrown. Trajectory of the mass on the sphere can be obtained from the expression for the total energy :E=\underbrace{\left[\frac{1}{2}ml^2\dot\theta^2 + \frac{1}{2}ml^2\sin^2\theta \dot \phi^2\right]}_{T}+\underbrace{ \bigg[-mgl\cos\theta\bigg]}_{V} by noting that the horizontal component of angular momentum L_z = ml^2\sin^2\\!\theta \,\dot\phi is a constant of motion, independent of time. thumb|420px|The typical motion of a pitcher. thumb|Demonstration of pitching techniques In baseball, the pitch is the act of throwing the baseball toward home plate to start a play. Therefore, the ball keeps moving in the path of least resistance, which constantly changes. A changeup is generally thrown 8–15 miles per hour slower than a fastball. That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. It is thrown the same as a fastball, but simply farther back in the hand, which makes it release from the hand slower but still retaining the look of a fastball. While throwing the fastball it is very important to have proper mechanics, because this increases the chance of getting the ball to its highest velocity, making it difficult for the opposing player to hit the pitch. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. ==Lagrangian mechanics== Routinely, in order to write down the kinetic T=\tfrac{1}{2}mv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. thumb|Standard 50m target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm International Rifle events that occur in three positions are conducted with an equal number of shots fired from the Kneeling, Prone and Standing positions, although the order has changed over the years. It consists of a mass moving without friction on the surface of a sphere. thumb|The target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm, height 0.75 m above the floor 50 meter rifle three positions (formerly known as one of four free rifle disciplines) is an International Shooting Sport Federation event, a miniature version of 300 meter rifle three positions. The two Olympic events are shot with a rimfire rifle at 50m. Typically, pitchers from the set use a high leg kick, but may instead release the ball more quickly by using the slide step. == See also == * First-pitch strike * Bowling – pitching a cricket ball ** Throwing (cricket), a type of bowling more similar to baseball pitching * Pitch (softball) ==References== == External links == * Category:Baseball terminology Category:Articles containing video clips The most common fastball pitches are: * Cutter * Four-seam fastball * Sinker * Split-finger fastball * Two-seam fastball ==Breaking balls== thumb|130px|A common grip of a slider Well-thrown breaking balls have movement, usually sideways or downward. For example, a batter swings at the ball as if it was a 90 mph fastball but it is coming at 75 mph which means he is swinging too early to hit the ball well, making the changeup very effective. In most cases junior shooting is done at either 10m or 50 ft. distances. Most breaking balls are considered off-speed pitches. ",0.333333333333333,2,2.84367,30,588313,D
+","thumb|upright=1.5|Spherical pendulum: angles and velocities. 50 meter running target or 50 meter running boar is an ISSF shooting event, shot with a .22-calibre rifle at a target depicting a boar moving sideways across a 10-meter wide opening. If thrown correctly, the changeup will confuse the batter because the human eye cannot discern that the ball is coming significantly slower until it is around 30 feet from the plate. thumb|The target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm, height 0.75 m above the floor 50 meter rifle prone (formerly known as one of four free rifle disciplines) is an International Shooting Sport Federation event consisting of 60 shots from the prone position with a .22 Long Rifle (5.6 mm) caliber rifle. A ball moves due to the changes in the pressure of the air surrounding the ball as a result of the kind of pitch thrown. Trajectory of the mass on the sphere can be obtained from the expression for the total energy :E=\underbrace{\left[\frac{1}{2}ml^2\dot\theta^2 + \frac{1}{2}ml^2\sin^2\theta \dot \phi^2\right]}_{T}+\underbrace{ \bigg[-mgl\cos\theta\bigg]}_{V} by noting that the horizontal component of angular momentum L_z = ml^2\sin^2\\!\theta \,\dot\phi is a constant of motion, independent of time. thumb|420px|The typical motion of a pitcher. thumb|Demonstration of pitching techniques In baseball, the pitch is the act of throwing the baseball toward home plate to start a play. Therefore, the ball keeps moving in the path of least resistance, which constantly changes. A changeup is generally thrown 8–15 miles per hour slower than a fastball. That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. It is thrown the same as a fastball, but simply farther back in the hand, which makes it release from the hand slower but still retaining the look of a fastball. While throwing the fastball it is very important to have proper mechanics, because this increases the chance of getting the ball to its highest velocity, making it difficult for the opposing player to hit the pitch. Owing to the spherical geometry of the problem, spherical coordinates are used to describe the position of the mass in terms of , where is fixed such that r = l. ==Lagrangian mechanics== Routinely, in order to write down the kinetic T=\tfrac{1}{2}mv^2 and potential V parts of the Lagrangian L=T-V in arbitrary generalized coordinates the position of the mass is expressed along Cartesian axes. thumb|Standard 50m target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm International Rifle events that occur in three positions are conducted with an equal number of shots fired from the Kneeling, Prone and Standing positions, although the order has changed over the years. It consists of a mass moving without friction on the surface of a sphere. thumb|The target: total Ø = 154.4 mm. 4 ring Ø = 106.4 mm. 9 ring Ø = 26.4 mm. 10 ring Ø = 10.4 mm, height 0.75 m above the floor 50 meter rifle three positions (formerly known as one of four free rifle disciplines) is an International Shooting Sport Federation event, a miniature version of 300 meter rifle three positions. The two Olympic events are shot with a rimfire rifle at 50m. Typically, pitchers from the set use a high leg kick, but may instead release the ball more quickly by using the slide step. == See also == * First-pitch strike * Bowling – pitching a cricket ball ** Throwing (cricket), a type of bowling more similar to baseball pitching * Pitch (softball) ==References== == External links == * Category:Baseball terminology Category:Articles containing video clips The most common fastball pitches are: * Cutter * Four-seam fastball * Sinker * Split-finger fastball * Two-seam fastball ==Breaking balls== thumb|130px|A common grip of a slider Well-thrown breaking balls have movement, usually sideways or downward. For example, a batter swings at the ball as if it was a 90 mph fastball but it is coming at 75 mph which means he is swinging too early to hit the ball well, making the changeup very effective. In most cases junior shooting is done at either 10m or 50 ft. distances. Most breaking balls are considered off-speed pitches. ",0.333333333333333,2,"""2.84367""",30,588313,D
 "Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of $12 \mathrm{~m}$ at a constant speed with a rope that weighs $0.8 \mathrm{~kg} / \mathrm{m}$. Initially the bucket contains $36 \mathrm{~kg}$ of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?
-","If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work of the net force is calculated as the product of its magnitude and the particle displacement. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. Therefore, work need only be computed for the gravitational forces acting on the bodies. The small amount of work that occurs over an instant of time is calculated as \delta W = \mathbf{F} \cdot d\mathbf{s} = \mathbf{F} \cdot \mathbf{v}dt where the is the power over the instant . The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. Integral energy is the amount of energy required to remove water from soil with an initial water content \theta_i to water content of \theta_f (where \theta_i > \theta_f). Use this to simplify the formula for work of gravity to, W = -\int^{t_2}_{t_1}\frac{GmM}{r^3}(r\mathbf{e}_r) \cdot \left(\dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_t\right) dt = -\int^{t_2}_{t_1}\frac{GmM}{r^3}r\dot{r}dt = \frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . In addition to completing the dam, work needed was the construction of shipping locks and discharge sluices at the ends of the dam. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. The work done is given by the dot product of the two vectors. In physics, work is the energy transferred to or from an object via the application of force along a displacement. After the war, work was started on draining the Flevolands, a massive project totalling almost 1000 km2. ", 0.01961,4.68,3857.0,460.5,2,C
-Find the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base.,"In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. After integrating these two functions with the disk method we would subtract them to yield the desired volume. Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. The horizontal plane shows the four quadrants between x- and y-axis. Recycling the subducted slab presents volcanism by flux melting from the mantle wedge. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). Wallis's Conical Edge with right|thumb|600px| Figure 2. thumb|400px|Three axial planes (x=0, y=0, z=0) divide space into eight octants. thumb|right|300px|A volume is approximated by a collection of hollow cylinders. The slab affects the convection and evolution of the Earth's mantle due to the insertion of the hydrous oceanic lithosphere. Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. right|280px|Femisphere The femisphere is a solid that has one single surface, two edges, and four vertices. == Description == The form of the femisphere is reminiscent of that of a sphericon but without straight lines. An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. thumb|upright=1.75|The figure is a schematic diagram depicting a subduction zone. Dense oceanic lithosphere retreats into the Earth's mantle, while lightweight continental lithospheric material produces active continental margins and volcanic arcs, generating volcanism. For this reason, when rolled over a sphere, it contacts the whole surface area of it in a single revolution.Sphericon Homepage: Femisphere The area of a femisphere of unit radius is S = 4 \pi. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution. ==Definition== The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the -plane around the -axis. Temperature gradients of subducted slabs depend on the oceanic plate's time and thermal structures. In geology, the slab is a significant constituent of subduction zones . Schaum's Outlines: Calculus. The polar circles of the triangles of a complete quadrilateral form a coaxal system. ",0.7071067812,3.0,0.1792,6.6,24,E
-A swimming pool is circular with a $40-\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \mathrm{ft}$ at the south end to $7 \mathrm{ft}$ at the north end. Find the volume of water in the pool.,"thumb|491px|right|Output from a shallow-water equation model of water in a bathtub. A stream pool, in hydrology, is a stretch of a river or stream in which the water depth is above average and the water velocity is below average.Matthew Chasse, Riffle characteristics in stream investigations == Formation == right|thumb|250px|Stream pool formation. The instantaneous water depth is , with zb(x) the bed level (i.e. elevation of the lowest point in the bed above datum, see the cross-section figure). The shallow- water equations are thus derived. The Swimming Pool (, translit. Therefore, the diver floats at the water's surface. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. The trough structure is 7 ft in height, with a width of 7.5 ft. Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. If the diver rises, by even the most minuscule amount, the pressure on the bubble will decrease, it will expand, it will displace more water, and the diver will become more positively buoyant, rising still more quickly. The trapped air in the straw makes the diver slightly buoyant, and it thus floats. 2\. A stream pool may be bedded with sediment or armoured with gravel, and in some cases the pool formations may have been formed as basins in exposed bedrock formations. This water in turn exerts additional pressure on the air bubble inside the diver; because the air inside the diver is compressible but the water is an incompressible fluid, the air's volume is decreased but the water's volume does not expand, such that the pressure external to the diver a) forces the water already in the diver further inward and b) drives water from outside the diver into the diver. The x-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the x-direction – can be written as: \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}+ w \frac{\partial u}{\partial z}= -\frac{\partial p}{\partial x} \frac{1}{\rho} + u \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)+ f_x, where u is the velocity in the x-direction, v is the velocity in the y-direction, w is the velocity in the z-direction, t is time, p is the pressure, ρ is the density of water, ν is the kinematic viscosity, and fx is the body force in the x-direction. For non-moving channel walls the cross-sectional area A in equation () can be written as: A(x,t) = \int_0^{h(x,t)} b(x,h')\, dh', with b(x,h) the effective width of the channel cross section at location x when the fluid depth is h – so for rectangular channels. Pools are often formed on the outside of a bend in a meandering river.http://mostreamteam.org/assets/habitat.pdf == Dynamics == The depth and lack of water velocity often leads to stratification in stream pools, especially in warmer regions. Assuming such a state were to exist at some point, any departure of the diver from its current depth, however small, will alter the pressure exerted on the bubble in the diver due to the change in the weight of the water above it in the vessel. It might be thought that if the weight of displaced water exactly matched the weight of the diver, it would neither rise nor sink, but float in the middle of the container; however, this does not occur in practice. Conversely, should the diver drop by the smallest amount, the pressure will increase, the bubble contract, additional water enter, the diver will become less buoyant, and the rate of the drop will accelerate as the pressure from the water rises still further. In order for shallow-water equations to be valid, the wavelength of the phenomenon they are supposed to model has to be much larger than the depth of the basin where the phenomenon takes place. When the pressure on the container is released, the air expands again, increasing the weight of water displaced and the diver again becomes positively buoyant and floats. ",5654.86677646,0.318,30.0,140,6,A
-"The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?","Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. The comet came to perihelion on 18 September 2012, and reached about apparent magnitude 17. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 160P on Seiichi Yoshida's comet list * Elements and Ephemeris for 160P/LINEAR – Minor Planet Center Category:Periodic comets 0160 # Category:Astronomical objects discovered in 2004 170P/Christensen is a periodic comet in the Solar System. For 0 this is an ellipse with = a \cdot \sqrt{1-e^2}|}} For e = 1 this is a parabola with focal length \tfrac{p}{2} For e > 1 this is a hyperbola with = a \cdot \sqrt{e^2-1}|}} The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue) thumb|300px|A diagram of the various forms of the Kepler Orbit and their eccentricities. Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). 1308 Halleria, provisional designation , is a carbonaceous Charis asteroid from the outer regions of the asteroid belt, approximately 43 kilometers in diameter. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value \tfrac{p}{1-e}. 160P/LINEAR is a periodic comet in the Solar System. It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 Its orbit has an eccentricity of 0.01 and an inclination of 6° with respect to the ecliptic. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 Mathematically, the distance between a central body and an orbiting body can be expressed as: r(\theta) = \frac{a(1-e^2)}{1+e\cos(\theta)} where: *r is the distance *a is the semi- major axis, which defines the size of the orbit *e is the eccentricity, which defines the shape of the orbit *\theta is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis). The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation () In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. Red is an elliptical orbit (0 < e < 1). thumb|An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. Alternately, the equation can be expressed as: r(\theta) = \frac{p}{1+e\cos(\theta)} Where p is called the semi-latus rectum of the curve. Lightcurve analysis gave a consolidated rotation period of 6.028 hours with a brightness amplitude between 0.14 and 0.17 magnitude. === Diameter and albedo === According to the surveys carried out by the Infrared Astronomical Satellite IRAS, the Japanese Akari satellite and the NEOWISE mission of NASA's Wide-field Infrared Survey Explorer, Halleria measures between 39.33 and 50.046 kilometers in diameter and its surface has an albedo between 0.0338 and 0.05. Making the substitutions p=\tfrac{|\mathbf{H}|^2}{\alpha} and e=\tfrac{c}{\alpha}, we again arrive at the equation This is the equation in polar coordinates for a conic section with origin in a focal point. We can then define the eccentricity vector associated with the orbit as: \mathbf{e} \triangleq \frac{\mathbf{c}}{\alpha} = \frac{\dot{\mathbf{r}}\times\mathbf{H}}{\alpha} - \mathbf{u} = \frac{\mathbf{v}\times\mathbf{H}}{\alpha} - \frac{\mathbf{r}}{r} = \frac{\mathbf{v}\times(\mathbf{r} \times \mathbf{v})}{\alpha} - \frac{\mathbf{r}}{r} where \mathbf{H} = \mathbf{r} \times \dot{\mathbf{r}} = \mathbf{r} \times \mathbf{v} is the constant angular momentum vector of the orbit, and \mathbf{v} is the velocity vector associated with the position vector \mathbf{r}. For the hyperbola the range for \theta is -\cos^{-1}\left(-\frac{1}{e}\right) < \theta < \cos^{-1}\left(-\frac{1}{e}\right) and for a parabola the range is -\pi < \theta < \pi Using the chain rule for differentiation (), the equation () and the definition of p as \frac {H^2}{\alpha} one gets that the radial velocity component is e \sin \theta|}} and that the tangential component (velocity component perpendicular to V_r) is \cdot (1 + e \cdot \cos \theta)|}} The connection between the polar argument \theta and time t is slightly different for elliptic and hyperbolic orbits. ",2.0,-273,35.64,"89,034.79", 10.7598,C
-" If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.","(If the time variable is continuous, the average value during the time period is the integral over the period divided by the length of the duration of the period.) ==See also== Moving average ==References== Category:Means 20 Hrs. 40 Min.: The temporal mean is the arithmetic mean of a series of values over a time period. A simple moving average can be considered to be a sequence of temporal means over periods of equal duration. Timeslip is a horizontally scrolling shooter written by Jon Williams for the Commodore 16 / Commodore Plus/4 computers and published by English Software in 1985. 40+ is a 2019 Maldivian horror comedy film written and directed by Yoosuf Shafeeu and the sequel to his previous comedy film Naughty 40 (2017). 40 Minutes was a BBC TV documentary strand broadcast on BBC Two between 1981 and 1994.BFI | Film & TV Database | 40 MINUTES Some documentaries in the original series were revisited and updated in a 2006 version, Forty Minutes On.BBC Four - Forty Minutes On ==See also== * Sixty Minutes (British TV programme) ==References== Category:BBC television documentaries Category:1981 British television series debuts Category:1994 British television series endings Category:1980s British documentary television series Category:1990s British documentary television series Category:English-language television shows 40 under 40 or Forty under 40 etc. may refer to: * 40 Under 40, annual list published in Fortune magazine * Business Journals Forty Under 40, annual list published by American City Business Journals If a player is hit, they receive a 30 minute penalty. Assuming equidistant measuring or sampling times, it can be computed as the sum of the values over a period divided by the number of values. In addition, if a player is hit five times, a ""timeslip"" occurs, which is a desynchronisation of all clocks. Your Commodore reviewer summed up Timeslip as the best game he had seen on the C16, and he recommended it without hesitation. The object of the game is to destroy 36 orbs placed within the three sections and synchronize the clocks in all three zones to 00.00 hours. The film ends with everyone laughing. == Cast == * Yoosuf Shafeeu as Ashvani * Ali Seezan as Zahidh * Mohamed Manik as Ajwad * Ahmed Saeed as Ahsan * Sheela Najeeb as Zarifa * Fathimath Azifa as Thaniya * Mohamed Faisal as Akram * Ali Azim as Nadheem * Ahmed Easa as Saiman * Mariyam Shakeela as Gumeyra * Ali Shahid as Zubeiru * Mariyam Shifa as Laila * Irufana Ibrahim as Shamra * Hunaisha Adam Naseer as Mishka * Aminath Ziyadha as Fazna * Aishath Sam'aa as Shaira * Ahmed Bassam as Fairooz * Mariyam Azza in the item number ""Lailaa"" (Special appearance) ==Development== A sequel to Yoosuf Shafeeu's commercially successful comedy film Naughty 40 (2017) was announced on 31 October 2017. Sections are played one at a time and the player can switch zones at will, leaving the other two frozen in time. ==Reception== Timeslip received mostly positive reviews. I think aviation has a chance to increase intimacy, > understanding, and far-flung friendships thus. 20 Hrs. 40 Min. was the first of two books Earhart would write in her lifetime; the other being 1932's The Fun of It. Filming was commenced on 23 December 2017 in Th. Burunee scheduled to be completed within twenty five days. The game was described by reviewers as ""three versions of Scramble rolled into one"". ==Gameplay== thumb|left|Atari 8-bit screenshot In Timeslip the player is presented with the screen divided into three sections or time zones. The review in Computer and Video Games magazine was equally positive: ""Timeslip's designer and programmer, Jon Williams, has come up with a nifty and exciting little game. C16 owners should raise three cheers for him "" ==References== ==External links== *Timeslip at Atari Mania * Category:1985 video games Category:Atari 8-bit family games Category:Commodore 16 and Plus/4 games Category:Horizontally scrolling shooters Category:English Software games Atari 8-bit version followed a year later. Shooting of the film was completed on 21 January 2018. ==Soundtrack== ==Release== The film was initially planned to release on 1 August 2018 though they pushed the release date to the following year citing the political instability in the country in relation to 2018 Maldivian presidential election. ",0.118,0.0526315789,9.14,7.58,-32,E
+","If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work of the net force is calculated as the product of its magnitude and the particle displacement. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. Therefore, work need only be computed for the gravitational forces acting on the bodies. The small amount of work that occurs over an instant of time is calculated as \delta W = \mathbf{F} \cdot d\mathbf{s} = \mathbf{F} \cdot \mathbf{v}dt where the is the power over the instant . The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. Integral energy is the amount of energy required to remove water from soil with an initial water content \theta_i to water content of \theta_f (where \theta_i > \theta_f). Use this to simplify the formula for work of gravity to, W = -\int^{t_2}_{t_1}\frac{GmM}{r^3}(r\mathbf{e}_r) \cdot \left(\dot{r}\mathbf{e}_r + r\dot{\theta}\mathbf{e}_t\right) dt = -\int^{t_2}_{t_1}\frac{GmM}{r^3}r\dot{r}dt = \frac{GMm}{r(t_2)}-\frac{GMm}{r(t_1)}. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . In addition to completing the dam, work needed was the construction of shipping locks and discharge sluices at the ends of the dam. This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . 500px|right|thumb|The lake retention time for a body of water with the volume and the exit flow of is 20 hours. Substituting the above equations, one obtains: W = Fs = mas = ma\frac{v_2^2-v_1^2}{2a} = \frac{mv_2^2}{2}- \frac{mv_1^2}{2} = \Delta E_\text{k} Other derivation: W = Fs = mas = m\frac{v_2^2 - v_1^2}{2s}s = \frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2 = \Delta E_\text{k} In the general case of rectilinear motion, when the net force is not constant in magnitude, but is constant in direction, and parallel to the velocity of the particle, the work must be integrated along the path of the particle: W = \int_{t_1}^{t_2} \mathbf{F}\cdot \mathbf{v}dt = \int_{t_1}^{t_2} F \,v \, dt = \int_{t_1}^{t_2} ma \,v \, dt = m \int_{t_1}^{t_2} v \,\frac{dv}{dt}\,dt = m \int_{v_1}^{v_2} v\,dv = \tfrac12 m \left(v_2^2 - v_1^2\right) . ===General derivation of the work–energy principle for a particle=== For any net force acting on a particle moving along any curvilinear path, it can be demonstrated that its work equals the change in the kinetic energy of the particle by a simple derivation analogous to the equation above. The work done is given by the dot product of the two vectors. In physics, work is the energy transferred to or from an object via the application of force along a displacement. After the war, work was started on draining the Flevolands, a massive project totalling almost 1000 km2. ", 0.01961,4.68,"""3857.0""",460.5,2,C
+Find the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base.,"In the end we find the volume is cubic units. ==See also== *Solid of revolution *Disc integration ==References== * *Frank Ayres, Elliott Mendelson. Then the formula for the volume will be: :2 \pi \int_a^b x f(x)\, dx If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes: :2 \pi \int_a^b y f(y)\, dy If the function is rotating around the line then the formula becomes: :\begin{cases} \displaystyle 2 \pi \int_a^b (x-h) f(x)\,dx, & \text{if}\ h \le a < b\\\ \displaystyle 2 \pi \int_a^b (h-x) f(x)\,dx, & \text{if}\ a < b \le h, \end{cases} and for rotations around it becomes :\begin{cases} \displaystyle 2 \pi \int_a^b (y-k) f(y)\,dy, & \text{if}\ k \le a < b\\\ \displaystyle 2 \pi \int_a^b (k-y) f(y)\,dy, & \text{if}\ a < b \le k. \end{cases} The formula is derived by computing the double integral in polar coordinates. ==Example== Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by: :y = (x-1)^2(x-2)^2 In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. After integrating these two functions with the disk method we would subtract them to yield the desired volume. Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. The horizontal plane shows the four quadrants between x- and y-axis. Recycling the subducted slab presents volcanism by flux melting from the mantle wedge. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). Wallis's Conical Edge with right|thumb|600px| Figure 2. thumb|400px|Three axial planes (x=0, y=0, z=0) divide space into eight octants. thumb|right|300px|A volume is approximated by a collection of hollow cylinders. The slab affects the convection and evolution of the Earth's mantle due to the insertion of the hydrous oceanic lithosphere. Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. right|280px|Femisphere The femisphere is a solid that has one single surface, two edges, and four vertices. == Description == The form of the femisphere is reminiscent of that of a sphericon but without straight lines. An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. thumb|upright=1.75|The figure is a schematic diagram depicting a subduction zone. Dense oceanic lithosphere retreats into the Earth's mantle, while lightweight continental lithospheric material produces active continental margins and volcanic arcs, generating volcanism. For this reason, when rolled over a sphere, it contacts the whole surface area of it in a single revolution.Sphericon Homepage: Femisphere The area of a femisphere of unit radius is S = 4 \pi. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution. ==Definition== The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the -plane around the -axis. Temperature gradients of subducted slabs depend on the oceanic plate's time and thermal structures. In geology, the slab is a significant constituent of subduction zones . Schaum's Outlines: Calculus. The polar circles of the triangles of a complete quadrilateral form a coaxal system. ",0.7071067812,3.0,"""0.1792""",6.6,24,E
+A swimming pool is circular with a $40-\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \mathrm{ft}$ at the south end to $7 \mathrm{ft}$ at the north end. Find the volume of water in the pool.,"thumb|491px|right|Output from a shallow-water equation model of water in a bathtub. A stream pool, in hydrology, is a stretch of a river or stream in which the water depth is above average and the water velocity is below average.Matthew Chasse, Riffle characteristics in stream investigations == Formation == right|thumb|250px|Stream pool formation. The instantaneous water depth is , with zb(x) the bed level (i.e. elevation of the lowest point in the bed above datum, see the cross-section figure). The shallow- water equations are thus derived. The Swimming Pool (, translit. Therefore, the diver floats at the water's surface. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: \begin{align} \frac{\partial (\rho \eta) }{\partial t} &\+ \frac{\partial (\rho \eta u)}{\partial x} + \frac{\partial (\rho \eta v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta u)}{\partial t} &\+ \frac{\partial}{\partial x}\left( \rho \eta u^2 + \frac{1}{2}\rho g \eta^2 \right) + \frac{\partial (\rho \eta u v)}{\partial y} = 0,\\\\[3pt] \frac{\partial (\rho \eta v)}{\partial t} &\+ \frac{\partial}{\partial y}\left(\rho \eta v^2 + \frac{1}{2}\rho g \eta ^2\right) + \frac{\partial (\rho \eta uv)}{\partial x} = 0. \end{align} Here η is the total fluid column height (instantaneous fluid depth as a function of x, y and t), and the 2D vector (u,v) is the fluid's horizontal flow velocity, averaged across the vertical column. While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. The trough structure is 7 ft in height, with a width of 7.5 ft. Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. If the diver rises, by even the most minuscule amount, the pressure on the bubble will decrease, it will expand, it will displace more water, and the diver will become more positively buoyant, rising still more quickly. The trapped air in the straw makes the diver slightly buoyant, and it thus floats. 2\. A stream pool may be bedded with sediment or armoured with gravel, and in some cases the pool formations may have been formed as basins in exposed bedrock formations. This water in turn exerts additional pressure on the air bubble inside the diver; because the air inside the diver is compressible but the water is an incompressible fluid, the air's volume is decreased but the water's volume does not expand, such that the pressure external to the diver a) forces the water already in the diver further inward and b) drives water from outside the diver into the diver. The x-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the x-direction – can be written as: \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}+ w \frac{\partial u}{\partial z}= -\frac{\partial p}{\partial x} \frac{1}{\rho} + u \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right)+ f_x, where u is the velocity in the x-direction, v is the velocity in the y-direction, w is the velocity in the z-direction, t is time, p is the pressure, ρ is the density of water, ν is the kinematic viscosity, and fx is the body force in the x-direction. For non-moving channel walls the cross-sectional area A in equation () can be written as: A(x,t) = \int_0^{h(x,t)} b(x,h')\, dh', with b(x,h) the effective width of the channel cross section at location x when the fluid depth is h – so for rectangular channels. Pools are often formed on the outside of a bend in a meandering river.http://mostreamteam.org/assets/habitat.pdf == Dynamics == The depth and lack of water velocity often leads to stratification in stream pools, especially in warmer regions. Assuming such a state were to exist at some point, any departure of the diver from its current depth, however small, will alter the pressure exerted on the bubble in the diver due to the change in the weight of the water above it in the vessel. It might be thought that if the weight of displaced water exactly matched the weight of the diver, it would neither rise nor sink, but float in the middle of the container; however, this does not occur in practice. Conversely, should the diver drop by the smallest amount, the pressure will increase, the bubble contract, additional water enter, the diver will become less buoyant, and the rate of the drop will accelerate as the pressure from the water rises still further. In order for shallow-water equations to be valid, the wavelength of the phenomenon they are supposed to model has to be much larger than the depth of the basin where the phenomenon takes place. When the pressure on the container is released, the air expands again, increasing the weight of water displaced and the diver again becomes positively buoyant and floats. ",5654.86677646,0.318,"""30.0""",140,6,A
+"The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] Find a polar equation for the orbit of Halley's comet. What is the maximum distance from the comet to the sun?","Perihelion distance at recent epochs Epoch Perihelion (AU) 2028 1.310 2022 1.306 2015 1.349 2008 1.355 Comet Borrelly or Borrelly's Comet (official designation: 19P/Borrelly) is a periodic comet, which was visited by the spacecraft Deep Space 1 in 2001. The comet came to perihelion on 18 September 2012, and reached about apparent magnitude 17. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 160P on Seiichi Yoshida's comet list * Elements and Ephemeris for 160P/LINEAR – Minor Planet Center Category:Periodic comets 0160 # Category:Astronomical objects discovered in 2004 170P/Christensen is a periodic comet in the Solar System. For 0 this is an ellipse with = a \cdot \sqrt{1-e^2}|}} For e = 1 this is a parabola with focal length \tfrac{p}{2} For e > 1 this is a hyperbola with = a \cdot \sqrt{e^2-1}|}} The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue) thumb|300px|A diagram of the various forms of the Kepler Orbit and their eccentricities. Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). 1308 Halleria, provisional designation , is a carbonaceous Charis asteroid from the outer regions of the asteroid belt, approximately 43 kilometers in diameter. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value \tfrac{p}{1-e}. 160P/LINEAR is a periodic comet in the Solar System. It came to perihelion in September 2014 at about apparent magnitude 18. == References == == External links == * Orbital simulation from JPL (Java) / Horizons Ephemeris * 170P on Seiichi Yoshida's comet list * Elements and Ephemeris for 170P/Christensen – Minor Planet Center Category:Periodic comets 0170 170P 20050617 Its orbit has an eccentricity of 0.01 and an inclination of 6° with respect to the ecliptic. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 Mathematically, the distance between a central body and an orbiting body can be expressed as: r(\theta) = \frac{a(1-e^2)}{1+e\cos(\theta)} where: *r is the distance *a is the semi- major axis, which defines the size of the orbit *e is the eccentricity, which defines the shape of the orbit *\theta is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis). The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation () In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. The comet last came to perihelion (closest approach to the Sun) on February 1, 2022 and will next come to perihelion on December 11, 2028. Deep Space 1 returned images of the comet's nucleus from 3400 kilometers away. Red is an elliptical orbit (0 < e < 1). thumb|An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. Alternately, the equation can be expressed as: r(\theta) = \frac{p}{1+e\cos(\theta)} Where p is called the semi-latus rectum of the curve. Lightcurve analysis gave a consolidated rotation period of 6.028 hours with a brightness amplitude between 0.14 and 0.17 magnitude. === Diameter and albedo === According to the surveys carried out by the Infrared Astronomical Satellite IRAS, the Japanese Akari satellite and the NEOWISE mission of NASA's Wide-field Infrared Survey Explorer, Halleria measures between 39.33 and 50.046 kilometers in diameter and its surface has an albedo between 0.0338 and 0.05. Making the substitutions p=\tfrac{|\mathbf{H}|^2}{\alpha} and e=\tfrac{c}{\alpha}, we again arrive at the equation This is the equation in polar coordinates for a conic section with origin in a focal point. We can then define the eccentricity vector associated with the orbit as: \mathbf{e} \triangleq \frac{\mathbf{c}}{\alpha} = \frac{\dot{\mathbf{r}}\times\mathbf{H}}{\alpha} - \mathbf{u} = \frac{\mathbf{v}\times\mathbf{H}}{\alpha} - \frac{\mathbf{r}}{r} = \frac{\mathbf{v}\times(\mathbf{r} \times \mathbf{v})}{\alpha} - \frac{\mathbf{r}}{r} where \mathbf{H} = \mathbf{r} \times \dot{\mathbf{r}} = \mathbf{r} \times \mathbf{v} is the constant angular momentum vector of the orbit, and \mathbf{v} is the velocity vector associated with the position vector \mathbf{r}. For the hyperbola the range for \theta is -\cos^{-1}\left(-\frac{1}{e}\right) < \theta < \cos^{-1}\left(-\frac{1}{e}\right) and for a parabola the range is -\pi < \theta < \pi Using the chain rule for differentiation (), the equation () and the definition of p as \frac {H^2}{\alpha} one gets that the radial velocity component is e \sin \theta|}} and that the tangential component (velocity component perpendicular to V_r) is \cdot (1 + e \cdot \cos \theta)|}} The connection between the polar argument \theta and time t is slightly different for elliptic and hyperbolic orbits. ",2.0,-273,"""35.64""","89,034.79", 10.7598,C
+" If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.","(If the time variable is continuous, the average value during the time period is the integral over the period divided by the length of the duration of the period.) ==See also== Moving average ==References== Category:Means 20 Hrs. 40 Min.: The temporal mean is the arithmetic mean of a series of values over a time period. A simple moving average can be considered to be a sequence of temporal means over periods of equal duration. Timeslip is a horizontally scrolling shooter written by Jon Williams for the Commodore 16 / Commodore Plus/4 computers and published by English Software in 1985. 40+ is a 2019 Maldivian horror comedy film written and directed by Yoosuf Shafeeu and the sequel to his previous comedy film Naughty 40 (2017). 40 Minutes was a BBC TV documentary strand broadcast on BBC Two between 1981 and 1994.BFI | Film & TV Database | 40 MINUTES Some documentaries in the original series were revisited and updated in a 2006 version, Forty Minutes On.BBC Four - Forty Minutes On ==See also== * Sixty Minutes (British TV programme) ==References== Category:BBC television documentaries Category:1981 British television series debuts Category:1994 British television series endings Category:1980s British documentary television series Category:1990s British documentary television series Category:English-language television shows 40 under 40 or Forty under 40 etc. may refer to: * 40 Under 40, annual list published in Fortune magazine * Business Journals Forty Under 40, annual list published by American City Business Journals If a player is hit, they receive a 30 minute penalty. Assuming equidistant measuring or sampling times, it can be computed as the sum of the values over a period divided by the number of values. In addition, if a player is hit five times, a ""timeslip"" occurs, which is a desynchronisation of all clocks. Your Commodore reviewer summed up Timeslip as the best game he had seen on the C16, and he recommended it without hesitation. The object of the game is to destroy 36 orbs placed within the three sections and synchronize the clocks in all three zones to 00.00 hours. The film ends with everyone laughing. == Cast == * Yoosuf Shafeeu as Ashvani * Ali Seezan as Zahidh * Mohamed Manik as Ajwad * Ahmed Saeed as Ahsan * Sheela Najeeb as Zarifa * Fathimath Azifa as Thaniya * Mohamed Faisal as Akram * Ali Azim as Nadheem * Ahmed Easa as Saiman * Mariyam Shakeela as Gumeyra * Ali Shahid as Zubeiru * Mariyam Shifa as Laila * Irufana Ibrahim as Shamra * Hunaisha Adam Naseer as Mishka * Aminath Ziyadha as Fazna * Aishath Sam'aa as Shaira * Ahmed Bassam as Fairooz * Mariyam Azza in the item number ""Lailaa"" (Special appearance) ==Development== A sequel to Yoosuf Shafeeu's commercially successful comedy film Naughty 40 (2017) was announced on 31 October 2017. Sections are played one at a time and the player can switch zones at will, leaving the other two frozen in time. ==Reception== Timeslip received mostly positive reviews. I think aviation has a chance to increase intimacy, > understanding, and far-flung friendships thus. 20 Hrs. 40 Min. was the first of two books Earhart would write in her lifetime; the other being 1932's The Fun of It. Filming was commenced on 23 December 2017 in Th. Burunee scheduled to be completed within twenty five days. The game was described by reviewers as ""three versions of Scramble rolled into one"". ==Gameplay== thumb|left|Atari 8-bit screenshot In Timeslip the player is presented with the screen divided into three sections or time zones. The review in Computer and Video Games magazine was equally positive: ""Timeslip's designer and programmer, Jon Williams, has come up with a nifty and exciting little game. C16 owners should raise three cheers for him "" ==References== ==External links== *Timeslip at Atari Mania * Category:1985 video games Category:Atari 8-bit family games Category:Commodore 16 and Plus/4 games Category:Horizontally scrolling shooters Category:English Software games Atari 8-bit version followed a year later. Shooting of the film was completed on 21 January 2018. ==Soundtrack== ==Release== The film was initially planned to release on 1 August 2018 though they pushed the release date to the following year citing the political instability in the country in relation to 2018 Maldivian presidential election. ",0.118,0.0526315789,"""9.14""",7.58,-32,E
 "A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \mathrm{~cm}$ apart. The liver is $15 \mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63$, 39 , and 0 . Use the Midpoint Rule to estimate the volume of the liver.
-","thumb|Maximum intensity projection of a PET/CT with choline: Note the physiologic accumulation in the liver, pancreas, kidney, bladder, spleen, bone marrow and salivary glands. The human abdomen is divided into quadrants and regions by anatomists and physicians for the purposes of study, diagnosis, and treatment. The nine regions offer more detailed anatomy and are delineated by two vertical and two horizontal lines. == Quadrants == thumb|right|upright=0.8|Quadrants of the abdomen thumb|right|upright=0.8|Diagram showing which organs (or parts of organs) are in each quadrant of the abdomen The left lower quadrant (LLQ) of the human abdomen is the area left of the midline and below the umbilicus. thumb|Relationship between number of feet, octave and size of an open flue pipe (1′ = 1 foot = about 32 cm) Scaling is the ratio of an organ pipe's diameter to its length. Töpfer reasoned that the cross-sectional area of the pipe was the critical factor, and he chose to vary this by the geometric mean of the ratios 1:2 and 1:4 per octave. The median aperture (also known as the medial aperture, and foramen of Magendie) is an opening of the fourth ventricle at the caudal portion of the roof of the fourth ventricle. Nine regions of the abdomen can be marked using two horizontal and two vertical dividing lines. One of the first authors to publish data on the scaling of organ pipes was Dom Bédos de Celles. The left upper quadrant extends from the umbilical plane to the left ribcage. thumb|upright=1.5|Conical scanning concept. The right upper quadrant extends from umbilical plane to the right ribcage. He established this as a standard scale, or in German, Normalmensur, with the additional stipulation that the internal diameter be at 8′ C (the lowest note of the modern organ compass) and the mouth width one-quarter of the circumference of such a pipe. Nonetheless, the median aperture accounts for most of the outflow of CSF out of the fourt ventricle. This meant that the cross-sectional area varied as 1 : \sqrt{8}. Important organs here are: *Cecum *Appendix *Ascending colon *Right ovary and Fallopian tube *Right ureter ==Regions== thumb|left|upright=0.8|Regions of abdomen thumb|upright=1.8|Regions shown on left in side-by-side comparison with quadrants. The division into four quadrants allows the localisation of pain and tenderness, scars, lumps, and other items of interest, narrowing in on which organs and tissues may be involved. This ratio does not concern the muzzle or face, and thus is distinct from the craniofacial ratio, which compares the size of the cranium to the length of the muzzle. Important organs here are: *the descending colon and sigmoid colon *the left ovary and fallopian tube *the left ureter The left upper quadrant (LUQ) extends from the median plane to the left of the patient, and from the umbilical plane to the left ribcage. The cephalic index of a vertebrate is the ratio between the width (side to side) and length (front to back) of its cranium (skull). Important organs here are: *Stomach *Spleen *Left lobe of liver *Body of pancreas *Left kidney and adrenal gland *Splenic flexure of colon *Parts of transverse and descending colon The right upper quadrant (RUQ) extends from the median plane to the right of the patient, and from the umbilical plane to the right ribcage. The median aperture varies in size. == Anatomy == === Relations === The median foramen on axial images is posterior to the pons and anterior to the caudal cerebellum. The equivalent in other animals is right anterior quadrant. ",22,0,1110.0,+116.0,210,C
+","thumb|Maximum intensity projection of a PET/CT with choline: Note the physiologic accumulation in the liver, pancreas, kidney, bladder, spleen, bone marrow and salivary glands. The human abdomen is divided into quadrants and regions by anatomists and physicians for the purposes of study, diagnosis, and treatment. The nine regions offer more detailed anatomy and are delineated by two vertical and two horizontal lines. == Quadrants == thumb|right|upright=0.8|Quadrants of the abdomen thumb|right|upright=0.8|Diagram showing which organs (or parts of organs) are in each quadrant of the abdomen The left lower quadrant (LLQ) of the human abdomen is the area left of the midline and below the umbilicus. thumb|Relationship between number of feet, octave and size of an open flue pipe (1′ = 1 foot = about 32 cm) Scaling is the ratio of an organ pipe's diameter to its length. Töpfer reasoned that the cross-sectional area of the pipe was the critical factor, and he chose to vary this by the geometric mean of the ratios 1:2 and 1:4 per octave. The median aperture (also known as the medial aperture, and foramen of Magendie) is an opening of the fourth ventricle at the caudal portion of the roof of the fourth ventricle. Nine regions of the abdomen can be marked using two horizontal and two vertical dividing lines. One of the first authors to publish data on the scaling of organ pipes was Dom Bédos de Celles. The left upper quadrant extends from the umbilical plane to the left ribcage. thumb|upright=1.5|Conical scanning concept. The right upper quadrant extends from umbilical plane to the right ribcage. He established this as a standard scale, or in German, Normalmensur, with the additional stipulation that the internal diameter be at 8′ C (the lowest note of the modern organ compass) and the mouth width one-quarter of the circumference of such a pipe. Nonetheless, the median aperture accounts for most of the outflow of CSF out of the fourt ventricle. This meant that the cross-sectional area varied as 1 : \sqrt{8}. Important organs here are: *Cecum *Appendix *Ascending colon *Right ovary and Fallopian tube *Right ureter ==Regions== thumb|left|upright=0.8|Regions of abdomen thumb|upright=1.8|Regions shown on left in side-by-side comparison with quadrants. The division into four quadrants allows the localisation of pain and tenderness, scars, lumps, and other items of interest, narrowing in on which organs and tissues may be involved. This ratio does not concern the muzzle or face, and thus is distinct from the craniofacial ratio, which compares the size of the cranium to the length of the muzzle. Important organs here are: *the descending colon and sigmoid colon *the left ovary and fallopian tube *the left ureter The left upper quadrant (LUQ) extends from the median plane to the left of the patient, and from the umbilical plane to the left ribcage. The cephalic index of a vertebrate is the ratio between the width (side to side) and length (front to back) of its cranium (skull). Important organs here are: *Stomach *Spleen *Left lobe of liver *Body of pancreas *Left kidney and adrenal gland *Splenic flexure of colon *Parts of transverse and descending colon The right upper quadrant (RUQ) extends from the median plane to the right of the patient, and from the umbilical plane to the right ribcage. The median aperture varies in size. == Anatomy == === Relations === The median foramen on axial images is posterior to the pons and anterior to the caudal cerebellum. The equivalent in other animals is right anterior quadrant. ",22,0,"""1110.0""",+116.0,210,C
 "A manufacturer of corrugated metal roofing wants to produce panels that are $28 \mathrm{in}$. wide and $2 \mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\sin (\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)
-","A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is independent of the circle radius; * a sine function (over a whole number n of half-periods), which can be calculated by computing the sine curve's arclength on those periods, is S = \textstyle \tfrac{1}{n\pi} \int_{0}^{n\pi} \sqrt{1 + (\cos x)^2} dx \approx 1.216... thumb|right|Example with 270° angle With similar opposite arcs joints in the same plane, continuously differentiable: Central angle Sinuosity Degrees Radians Exact Decimal 30° \frac{\pi}{6} \frac{\pi}{3(\sqrt{6}-\sqrt{2})} 1.0115 60° \frac{\pi}{3} \frac{\pi}{3} 1.0472 90° \frac{\pi}{2} \frac{\pi}{2\sqrt{2}} 1.1107 120° \frac{2\cdot\pi}{3} \frac{2\cdot\pi}{3\sqrt{3}} 1.2092 150° \frac{5\cdot\pi}{6} \frac{5\cdot\pi}{3(\sqrt{6}+\sqrt{2})} 1.3552 180° \pi \frac{\pi}{2} 1.5708 210° \frac{7\cdot\pi}{6} \frac{7\cdot\pi}{3(\sqrt{6}+\sqrt{2})} 1.8972 240° \frac{4\cdot\pi}{3} \frac{4\cdot\pi}{3\sqrt{3}} 2.4184 270° \frac{3\cdot\pi}{2} \frac{3\cdot\pi}{2\sqrt{2}} 3.3322 300° \frac{5\cdot\pi}{3} \frac{5\cdot\pi}{3} 5.2360 330° \frac{11\cdot\pi}{6} \frac{11\cdot\pi}{3(\sqrt{6}-\sqrt{2})} 11.1267 ==Rivers== In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by: : \text{SI} = \frac{{\text{channel length}}}{{\text{downvalley length}}} The difference from the general form happens because the downvalley path is not perfectly straight. At this position, the top surface of the sine bar is inclined the same amount as the wedge. Some engineering and metalworking reference books contain tables showing the dimension required to obtain an angle from 0-90 degrees, incremented by 1 minute intervals. \sin \left(angle \right) = {perpendicular \over hypotenuse} Angles may be measured or set with this tool. ==Principle== thumb|10-inch and 100-millimetre sine bars. When a sine bar is placed on a level surface the top edge will be parallel to that surface. This is why the frequency of the sine wave increases as one moves to the left in the graph. The sine of the angle of inclination of the wedge is the ratio of the height of the gauge blocks used and the distance between the centers of the cylinders. ==Types== The simplest type consists of a lapped steel bar, at each end of which is attached an accurate cylinder, the axes of the cylinders being mutually parallel and parallel to the upper surface of the bar. These alternative functions are usually known as normalized Fresnel integrals. == Euler spiral == 250px|thumb| Euler spiral . The calculation of the sinuosity is valid in a 3-dimensional space (e.g. for the central axis of the small intestine), although it is often performed in a plane (with then a possible orthogonal projection of the curve in the selected plan; ""classic"" sinuosity on the horizontal plane, longitudinal profile sinuosity on the vertical plane). thumb|250px|Calculation of sinuosity for an oscillating curve. thumb|Two ski tracks with different degrees of sinuosity on the same slope Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. 300px 300px The Keulegan–Carpenter number is important for the computation of the wave forces on offshore platforms. From the definitions of Fresnel integrals, the infinitesimals and are thus: \begin{align} dx &= C'(t)\,dt = \cos\left(t^2\right)\,dt, \\\ dy &= S'(t)\,dt = \sin\left(t^2\right)\,dt. \end{align} Thus the length of the spiral measured from the origin can be expressed as L = \int_0^{t_0} \sqrt {dx^2 + dy^2} = \int_0^{t_0} dt = t_0. The sine bar is placed over the inclined surface of the wedge. This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. The resonant frequencies of a string are proportional to: the length between the fixed ends; the tension of the string; and inversely proportional to the mass per unit length of the string. == See also == * Crest (physics) * Damped sine wave * Fourier transform * Harmonic analysis * Harmonic series (mathematics) * Harmonic series (music) * Helmholtz equation * Instantaneous phase * In-phase and quadrature components * Least-squares spectral analysis * Oscilloscope * Phasor * Pure tone * Simple harmonic motion * Sinusoidal model * Wave (physics) * Wave equation * ∿ the sine wave symbol (U+223F) == References == ==Further reading== * Category:Trigonometry Category:Wave mechanics Category:Waves Category:Waveforms Category:Sound Category:Acoustics * The angle is calculated by using the sine rule (a trigonometric function from mathematics). It cannot measure the angle more than 60 degrees. ===Sine table=== A sine table (or sine plate) is a large and wide sine bar, typically equipped with a mechanism for locking it in place after positioning, which is used to hold workpieces during operations. ===Compound sine table=== It is used to measure compound angles of large workpiece. A sine bar consists of a hardened, precision ground body with two precision ground cylinders fixed at the ends. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. ==Sinusoidal plane wave== == Cosine == The term sinusoid describes any wave with characteristics of a sine wave. The Fresnel integrals and are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (). The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). == Definition == 250px|thumb| Fresnel integrals with arguments instead of converge to instead of . ",2.72,1.5,130.400766848,12,29.36,E
-"The dye dilution method is used to measure cardiac output with $6 \mathrm{mg}$ of dye. The dye concentrations, in $\mathrm{mg} / \mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \leqslant t \leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.","In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). The concentration of this admixture should be small and the gradient of this concentration should be also small. Dye transfer is a continuous-tone color photographic printing process. Fluorescent dye Color mass (g/mol) Absorb (nm) Emit (nm) ε (M−1cm−1) FluoProbes 390 violet 343 390 479 24 000 FluoProbes 488 green 804 493 519 85 000 FluoProbes 532 yellow 765 532 553 117 000 FluoProbes547H orange 736 557 574 150 000 FluoProbes 594 red 1137 601 627 120 000 FluoProbes647H far-red 761 653 674 250 000 FluoProbes 682 far-red 853 690 709 140 000 FluoProbes 752 near-IR 879 748 772 270 000 FluoProbes 782 near-IR 976 783 800 170 000 Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient The FluoProbes series of fluorescent dyes were developed by Interchim to improve performances of standard fluorophores. Dilution is reduction of concentration, e.g. by adding solvent to a solution. The dye transfer process possesses a larger color gamut and tonal scale than any other process, including inkjet. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. The dye is absorbed by the paper for one minute after which the matrix is picked up by the farthermost corners and peeled off the paper. Colorimetric analysis is a method of determining the concentration of a chemical element or chemical compound in a solution with the aid of a color reagent. In the test tube with dark blue liquid (in front), the blue dye is dissolved in a high concentration. Another important characteristic of dye transfer is that it allows the practitioner the highest degree of photographic control compared to any other photochemical color print process. * FluoProbes dyes that have comparable excitation and emission spectra to standard fluorophores such as fluoresceins, rhodamines, cyanines Cy2/3/5/5.5/7, are claimed to solve limiting issues observed in some applications such as too high background, insufficient polarity, photobleaching, insufficient brightness, or pH- sensitivity. Recently, colorimetric analyses developed for colorimeters have been adapted for use with plate readers to speed up analysis and reduce the waste stream.Greenan, N. S., R.L. Mulvaney, and G.K. Sims. 1995. The dyes used in the process are very spectrally pure compared to normal coupler-induced photographic dyes, with the exception of the Kodak cyan. The Dyecrete Process is a method of adding dye to permanently color concrete. ==References== *Cited in the Academic Press Dictionary of Science and Technology. Concentrations are often called levels, reflecting the mental schema of levels on the vertical axis of a graph, which can be high or low (for example, ""high serum levels of bilirubin"" are concentrations of bilirubin in the blood serum that are greater than normal). ==Quantitative notation== There are four quantities that describe concentration: ===Mass concentration=== The mass concentration \rho_i is defined as the mass of a constituent m_i divided by the volume of the mixture V: :\rho_i = \frac {m_i}{V}. This is the diffusion coefficient. thumb|Test tubes with liquid in which a blue dye is dissolved in different concentrations. Note that these should not be called concentrations. ===Normality=== Normality is defined as the molar concentration c_i divided by an equivalence factor f_\mathrm{eq}. Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript ""eq"", q is the charge and φ is the electric potential. The method is widely used in medical laboratories and for industrial purposes, e.g. the analysis of water samples in connection with industrial water treatment. ==Equipment== The equipment required is a colorimeter, some cuvettes and a suitable color reagent. By contrast, to dilute a solution, one must add more solvent, or reduce the amount of solute. ",650000,35.64,0.000216,0.166666666,6.6,E
-"A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \mathrm{ft}$ high with a radius of $200 \mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?","For bauxites having more than 10% silica, the Bayer process becomes uneconomic because of the formation of insoluble sodium aluminium silicate, which reduces yield, so another process must be chosen. 1.9-3.6 tons of bauxite is required to produce 1 ton of aluminium oxide. The metal contents in ore shoots are distributed in areas that vary in deposit sizes. == Sizes and structure == The circumference of deposit sizes can range from a few meters, to many kilometres. The extraction process (digestion) converts the aluminium oxide in the ore to soluble sodium aluminate, NaAlO2, according to the chemical equation: :Al2O3 \+ 2 NaOH → 2 NaAlO2 \+ H2O This treatment also dissolves silica, forming sodium silicate : :2 NaOH + SiO2 → Na2SiO3 \+ H2O The other components of Bauxite, however, do not dissolve. thumb|Pouring smelter slag onto the dump, El Teniente thumb|upright|""The ore is mined by a highly developed caving system and carried down to the main transportation level through an elaborate system of ore passes."" In the Bayer process, bauxite ore is heated in a pressure vessel along with a sodium hydroxide solution (caustic soda) at a temperature of 150 to 200 °C. A structure may consist of multiple ore shoots with some veins or lodes being as thick as , and extending to thousands of feet horizontally and vertically. == Locations == There are complex stratigraphic historical parameters required in understanding how ore shoots are formed. An ore shoot is a mass of ore deposited in a vein. This is due to a majority of the aluminium in the ore being dissolved in the process. The undissolved waste after the aluminium compounds are extracted, bauxite tailings, contains iron oxides, silica, calcia, titania and some unreacted alumina. The aluminium oxide must be further purified before it can be refined into aluminium metal. == Process == Bauxite ore is a mixture of hydrated aluminium oxides and compounds of other elements such as iron. Over 90% (95-96%) of the aluminium oxide produced is used in the Hall–Héroult process to produce aluminium. == Waste == Red mud is the waste product that is produced in the digestion of bauxite with sodium hydroxide. Year Tonnes Aluminium Price Net Profit Employees 1979 153,537 1575 -1,172,000 1,252 1980 154,740 1770 17,470,000 1,258 1981 153,979 1302 2,941,000 1,359 1982 163,419 1026 -20,698,000 1,452 1983 218,609 1478 -9,665,000 1,651 1984 242,850 1281 1,766,000 1,631 1985 240,835 1072 -24,772,000 1,529 1986 236,332 1160 -18,188,000 1,506 1987 248,365 1496 92,570,000 1,429 1988 257,006 2367 173,040,000 1,770 1989 258,359 1915 118,500,000 1,820 1990 259,408 1635 42,051,000 1,720 1991 258,790 1333 -34,122,000 1,465 1992 241,775 1279 -18,649,000 1,415 1993 267,200 1161 -18,016,000 1,465 The smelter production is in tonnes of saleable metal, the aluminium price is the average London Metal Exchange 3 month in US$/tonne, the Nett Profit/Loss is after tax and NZ$. In 2011 the smelter produced 354,030 saleable tonnes of aluminium, which was its highest ever output at the time. Estimates of the waste stockpiled at the site range up to a quarter of million tonnes. The ore body surrounds the Braden Pipe in a continuous ring with a width of 2000 feet. thumb|Tiwai Point Aluminium Smelter as seen from the top of Bluff Hill The Tiwai Point Aluminium Smelter is an aluminium smelter owned by Rio Tinto Group (79.36%) and the Sumitomo Group (20.64%), via a joint venture called New Zealand Aluminium Smelters (NZAS) Limited. The ore shoot consists of the most valuable part of the ore deposit. Bauxite, the most important ore of aluminium, contains only 30–60% aluminium oxide (Al2O3), the rest being a mixture of silica, various iron oxides, and titanium dioxide. The company investigated sources of large quantities of cheap electricity needed to reduce the alumina recovered from the bauxite into aluminium. The Bayer process is the principal industrial means of refining bauxite to produce alumina (aluminium oxide) and was developed by Carl Josef Bayer. In 2016, an analyst at First New Zealand Capital (FCNZ) utilities said that the smelter was thought to be breaking even, helped by favourable currency rates and low alumina prices. ===Price negotiations, 2019 to 2021=== In October 2019, Rio Tinto announced a strategic review of the Tiwai Point Aluminium Smelter, including a wide range of issues associated with closure. The Deal–Grove model mathematically describes the growth of an oxide layer on the surface of a material. ",11,0.396,9.8,0.95,5.0,C
+","A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is independent of the circle radius; * a sine function (over a whole number n of half-periods), which can be calculated by computing the sine curve's arclength on those periods, is S = \textstyle \tfrac{1}{n\pi} \int_{0}^{n\pi} \sqrt{1 + (\cos x)^2} dx \approx 1.216... thumb|right|Example with 270° angle With similar opposite arcs joints in the same plane, continuously differentiable: Central angle Sinuosity Degrees Radians Exact Decimal 30° \frac{\pi}{6} \frac{\pi}{3(\sqrt{6}-\sqrt{2})} 1.0115 60° \frac{\pi}{3} \frac{\pi}{3} 1.0472 90° \frac{\pi}{2} \frac{\pi}{2\sqrt{2}} 1.1107 120° \frac{2\cdot\pi}{3} \frac{2\cdot\pi}{3\sqrt{3}} 1.2092 150° \frac{5\cdot\pi}{6} \frac{5\cdot\pi}{3(\sqrt{6}+\sqrt{2})} 1.3552 180° \pi \frac{\pi}{2} 1.5708 210° \frac{7\cdot\pi}{6} \frac{7\cdot\pi}{3(\sqrt{6}+\sqrt{2})} 1.8972 240° \frac{4\cdot\pi}{3} \frac{4\cdot\pi}{3\sqrt{3}} 2.4184 270° \frac{3\cdot\pi}{2} \frac{3\cdot\pi}{2\sqrt{2}} 3.3322 300° \frac{5\cdot\pi}{3} \frac{5\cdot\pi}{3} 5.2360 330° \frac{11\cdot\pi}{6} \frac{11\cdot\pi}{3(\sqrt{6}-\sqrt{2})} 11.1267 ==Rivers== In studies of rivers, the sinuosity index is similar but not identical to the general form given above, being given by: : \text{SI} = \frac{{\text{channel length}}}{{\text{downvalley length}}} The difference from the general form happens because the downvalley path is not perfectly straight. At this position, the top surface of the sine bar is inclined the same amount as the wedge. Some engineering and metalworking reference books contain tables showing the dimension required to obtain an angle from 0-90 degrees, incremented by 1 minute intervals. \sin \left(angle \right) = {perpendicular \over hypotenuse} Angles may be measured or set with this tool. ==Principle== thumb|10-inch and 100-millimetre sine bars. When a sine bar is placed on a level surface the top edge will be parallel to that surface. This is why the frequency of the sine wave increases as one moves to the left in the graph. The sine of the angle of inclination of the wedge is the ratio of the height of the gauge blocks used and the distance between the centers of the cylinders. ==Types== The simplest type consists of a lapped steel bar, at each end of which is attached an accurate cylinder, the axes of the cylinders being mutually parallel and parallel to the upper surface of the bar. These alternative functions are usually known as normalized Fresnel integrals. == Euler spiral == 250px|thumb| Euler spiral . The calculation of the sinuosity is valid in a 3-dimensional space (e.g. for the central axis of the small intestine), although it is often performed in a plane (with then a possible orthogonal projection of the curve in the selected plan; ""classic"" sinuosity on the horizontal plane, longitudinal profile sinuosity on the vertical plane). thumb|250px|Calculation of sinuosity for an oscillating curve. thumb|Two ski tracks with different degrees of sinuosity on the same slope Sinuosity, sinuosity index, or sinuosity coefficient of a continuously differentiable curve having at least one inflection point is the ratio of the curvilinear length (along the curve) and the Euclidean distance (straight line) between the end points of the curve. 300px 300px The Keulegan–Carpenter number is important for the computation of the wave forces on offshore platforms. From the definitions of Fresnel integrals, the infinitesimals and are thus: \begin{align} dx &= C'(t)\,dt = \cos\left(t^2\right)\,dt, \\\ dy &= S'(t)\,dt = \sin\left(t^2\right)\,dt. \end{align} Thus the length of the spiral measured from the origin can be expressed as L = \int_0^{t_0} \sqrt {dx^2 + dy^2} = \int_0^{t_0} dt = t_0. The sine bar is placed over the inclined surface of the wedge. This equation gives a sine wave for a single dimension; thus the generalized equation given above gives the displacement of the wave at a position x at time t along a single line. The resonant frequencies of a string are proportional to: the length between the fixed ends; the tension of the string; and inversely proportional to the mass per unit length of the string. == See also == * Crest (physics) * Damped sine wave * Fourier transform * Harmonic analysis * Harmonic series (mathematics) * Harmonic series (music) * Helmholtz equation * Instantaneous phase * In-phase and quadrature components * Least-squares spectral analysis * Oscilloscope * Phasor * Pure tone * Simple harmonic motion * Sinusoidal model * Wave (physics) * Wave equation * ∿ the sine wave symbol (U+223F) == References == ==Further reading== * Category:Trigonometry Category:Wave mechanics Category:Waves Category:Waveforms Category:Sound Category:Acoustics * The angle is calculated by using the sine rule (a trigonometric function from mathematics). It cannot measure the angle more than 60 degrees. ===Sine table=== A sine table (or sine plate) is a large and wide sine bar, typically equipped with a mechanism for locking it in place after positioning, which is used to hold workpieces during operations. ===Compound sine table=== It is used to measure compound angles of large workpiece. A sine bar consists of a hardened, precision ground body with two precision ground cylinders fixed at the ends. For more complex waves such as the height of a water wave in a pond after a stone has been dropped in, more complex equations are needed. ==Sinusoidal plane wave== == Cosine == The term sinusoid describes any wave with characteristics of a sine wave. The Fresnel integrals and are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (). The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). == Definition == 250px|thumb| Fresnel integrals with arguments instead of converge to instead of . ",2.72,1.5,"""130.400766848""",12,29.36,E
+"The dye dilution method is used to measure cardiac output with $6 \mathrm{mg}$ of dye. The dye concentrations, in $\mathrm{mg} / \mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \leqslant t \leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.","In the following test tubes, the blue dye is dissolved in a lower concentration (and at the same time in a smaller amount, since the volume is approximately the same). The concentration of this admixture should be small and the gradient of this concentration should be also small. Dye transfer is a continuous-tone color photographic printing process. Fluorescent dye Color mass (g/mol) Absorb (nm) Emit (nm) ε (M−1cm−1) FluoProbes 390 violet 343 390 479 24 000 FluoProbes 488 green 804 493 519 85 000 FluoProbes 532 yellow 765 532 553 117 000 FluoProbes547H orange 736 557 574 150 000 FluoProbes 594 red 1137 601 627 120 000 FluoProbes647H far-red 761 653 674 250 000 FluoProbes 682 far-red 853 690 709 140 000 FluoProbes 752 near-IR 879 748 772 270 000 FluoProbes 782 near-IR 976 783 800 170 000 Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient Abs = absorption maximum#, Em = emission maximum# .................................. ε = molar extinction coefficient The FluoProbes series of fluorescent dyes were developed by Interchim to improve performances of standard fluorophores. Dilution is reduction of concentration, e.g. by adding solvent to a solution. The dye transfer process possesses a larger color gamut and tonal scale than any other process, including inkjet. thumb|Time lapse video of diffusion of a dye dissolved in water into a gel. thumb|Three-dimensional rendering of diffusion of purple dye in water. The dye is absorbed by the paper for one minute after which the matrix is picked up by the farthermost corners and peeled off the paper. Colorimetric analysis is a method of determining the concentration of a chemical element or chemical compound in a solution with the aid of a color reagent. In the test tube with dark blue liquid (in front), the blue dye is dissolved in a high concentration. Another important characteristic of dye transfer is that it allows the practitioner the highest degree of photographic control compared to any other photochemical color print process. * FluoProbes dyes that have comparable excitation and emission spectra to standard fluorophores such as fluoresceins, rhodamines, cyanines Cy2/3/5/5.5/7, are claimed to solve limiting issues observed in some applications such as too high background, insufficient polarity, photobleaching, insufficient brightness, or pH- sensitivity. Recently, colorimetric analyses developed for colorimeters have been adapted for use with plate readers to speed up analysis and reduce the waste stream.Greenan, N. S., R.L. Mulvaney, and G.K. Sims. 1995. The dyes used in the process are very spectrally pure compared to normal coupler-induced photographic dyes, with the exception of the Kodak cyan. The Dyecrete Process is a method of adding dye to permanently color concrete. ==References== *Cited in the Academic Press Dictionary of Science and Technology. Concentrations are often called levels, reflecting the mental schema of levels on the vertical axis of a graph, which can be high or low (for example, ""high serum levels of bilirubin"" are concentrations of bilirubin in the blood serum that are greater than normal). ==Quantitative notation== There are four quantities that describe concentration: ===Mass concentration=== The mass concentration \rho_i is defined as the mass of a constituent m_i divided by the volume of the mixture V: :\rho_i = \frac {m_i}{V}. This is the diffusion coefficient. thumb|Test tubes with liquid in which a blue dye is dissolved in different concentrations. Note that these should not be called concentrations. ===Normality=== Normality is defined as the molar concentration c_i divided by an equivalence factor f_\mathrm{eq}. Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript ""eq"", q is the charge and φ is the electric potential. The method is widely used in medical laboratories and for industrial purposes, e.g. the analysis of water samples in connection with industrial water treatment. ==Equipment== The equipment required is a colorimeter, some cuvettes and a suitable color reagent. By contrast, to dilute a solution, one must add more solvent, or reduce the amount of solute. ",650000,35.64,"""0.000216""",0.166666666,6.6,E
+"A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \mathrm{ft}$ high with a radius of $200 \mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?","For bauxites having more than 10% silica, the Bayer process becomes uneconomic because of the formation of insoluble sodium aluminium silicate, which reduces yield, so another process must be chosen. 1.9-3.6 tons of bauxite is required to produce 1 ton of aluminium oxide. The metal contents in ore shoots are distributed in areas that vary in deposit sizes. == Sizes and structure == The circumference of deposit sizes can range from a few meters, to many kilometres. The extraction process (digestion) converts the aluminium oxide in the ore to soluble sodium aluminate, NaAlO2, according to the chemical equation: :Al2O3 \+ 2 NaOH → 2 NaAlO2 \+ H2O This treatment also dissolves silica, forming sodium silicate : :2 NaOH + SiO2 → Na2SiO3 \+ H2O The other components of Bauxite, however, do not dissolve. thumb|Pouring smelter slag onto the dump, El Teniente thumb|upright|""The ore is mined by a highly developed caving system and carried down to the main transportation level through an elaborate system of ore passes."" In the Bayer process, bauxite ore is heated in a pressure vessel along with a sodium hydroxide solution (caustic soda) at a temperature of 150 to 200 °C. A structure may consist of multiple ore shoots with some veins or lodes being as thick as , and extending to thousands of feet horizontally and vertically. == Locations == There are complex stratigraphic historical parameters required in understanding how ore shoots are formed. An ore shoot is a mass of ore deposited in a vein. This is due to a majority of the aluminium in the ore being dissolved in the process. The undissolved waste after the aluminium compounds are extracted, bauxite tailings, contains iron oxides, silica, calcia, titania and some unreacted alumina. The aluminium oxide must be further purified before it can be refined into aluminium metal. == Process == Bauxite ore is a mixture of hydrated aluminium oxides and compounds of other elements such as iron. Over 90% (95-96%) of the aluminium oxide produced is used in the Hall–Héroult process to produce aluminium. == Waste == Red mud is the waste product that is produced in the digestion of bauxite with sodium hydroxide. Year Tonnes Aluminium Price Net Profit Employees 1979 153,537 1575 -1,172,000 1,252 1980 154,740 1770 17,470,000 1,258 1981 153,979 1302 2,941,000 1,359 1982 163,419 1026 -20,698,000 1,452 1983 218,609 1478 -9,665,000 1,651 1984 242,850 1281 1,766,000 1,631 1985 240,835 1072 -24,772,000 1,529 1986 236,332 1160 -18,188,000 1,506 1987 248,365 1496 92,570,000 1,429 1988 257,006 2367 173,040,000 1,770 1989 258,359 1915 118,500,000 1,820 1990 259,408 1635 42,051,000 1,720 1991 258,790 1333 -34,122,000 1,465 1992 241,775 1279 -18,649,000 1,415 1993 267,200 1161 -18,016,000 1,465 The smelter production is in tonnes of saleable metal, the aluminium price is the average London Metal Exchange 3 month in US$/tonne, the Nett Profit/Loss is after tax and NZ$. In 2011 the smelter produced 354,030 saleable tonnes of aluminium, which was its highest ever output at the time. Estimates of the waste stockpiled at the site range up to a quarter of million tonnes. The ore body surrounds the Braden Pipe in a continuous ring with a width of 2000 feet. thumb|Tiwai Point Aluminium Smelter as seen from the top of Bluff Hill The Tiwai Point Aluminium Smelter is an aluminium smelter owned by Rio Tinto Group (79.36%) and the Sumitomo Group (20.64%), via a joint venture called New Zealand Aluminium Smelters (NZAS) Limited. The ore shoot consists of the most valuable part of the ore deposit. Bauxite, the most important ore of aluminium, contains only 30–60% aluminium oxide (Al2O3), the rest being a mixture of silica, various iron oxides, and titanium dioxide. The company investigated sources of large quantities of cheap electricity needed to reduce the alumina recovered from the bauxite into aluminium. The Bayer process is the principal industrial means of refining bauxite to produce alumina (aluminium oxide) and was developed by Carl Josef Bayer. In 2016, an analyst at First New Zealand Capital (FCNZ) utilities said that the smelter was thought to be breaking even, helped by favourable currency rates and low alumina prices. ===Price negotiations, 2019 to 2021=== In October 2019, Rio Tinto announced a strategic review of the Tiwai Point Aluminium Smelter, including a wide range of issues associated with closure. The Deal–Grove model mathematically describes the growth of an oxide layer on the surface of a material. ",11,0.396,"""9.8""",0.95,5.0,C
 "A boatman wants to cross a canal that is $3 \mathrm{~km}$ wide and wants to land at a point $2 \mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \mathrm{~km} / \mathrm{h}$ and the speed of his boat is $13 \mathrm{~km} / \mathrm{h}$. How long will the trip take?
-","This is often called the hull speed and is a function of the length of the ship V=k\sqrt{L} where constant (k) should be taken as: 2.43 for velocity (V) in kn and length (L) in metres (m) or, 1.34 for velocity (V) in kn and length (L) in feet (ft). This procedure meant a delay of at least 20 minutes for the boat travelling upstream, while the ship heading downstream suffered a delay of about 45 minutes as a result of the manoeuvre. The smaller boat, which is travelling downstream, is moving very fast, driven by the large water sails on either side and is thereby hauling the larger boat upstream against the current. Practical experience showed that free-running, paddle tugboats capable of could achieve a speed of about against a river current of . On the Neckar river with seven chain boats that meant six passing manoeuvres costing at least five hours for those travelling downstream. The boat is pulling itself upstream on a cable laid along the river. So variations in river depth complicated the handling of the vessel considerably.Carl Victor Suppán: Wasserstrassen und Binnenschiffahrt. However, fast currents when rivers were in spate could also be problematic for chain boats. Chain-boat navigationDocument, Volume 1, Issues 1-9, The Commission, US National Waterways Commission, 1909, p. right|thumb|300px|The canal's route is close to the dashed line of the railway across the neck of the peninsula. thumb|293x293px|a ship crossing the Karakum Canal. By the time the attached barges entered the faster-flowing area, the steamer had already sailed past it and was able to generate its full traction again. When travelling downstream the barges were usually just allowed to drift with the current in order to save money, In strong currents, operating a long string of barges was quite dangerous. The electric powered bridge is lifted several times a day to let boats pass through it. == Activities == thumb|left|Sunset on the Shark River Inlet looking west from the drawbridge. The number of trips a boat could make increased, for example, on the Elbe almost three times.Erich Pleißner: ""Konzentration der Güterschiffahrt auf der Elbe"". These sections of river could be negotiated by anchoring a rope ahead of the boat and then using the crew to haul it upstream.Sigbert Zesewitz, Helmut Düntzsch, Theodor Grötschel: Kettenschiffahrt. On the remaining section with its strong currents the chain boats made a profit in many years of about 30%. The canal tolls reflected these improvements, but if a boat missed the tide it would have to wait in the canal basin for longer than the journey round Hoo would have taken. ==Higham and Strood tunnel== The Higham and Strood tunnel is long, and was the second longest canal tunnel built in the UK (the longest is the Standedge Canal Tunnel). When travelling downstream, boats were either simply propelled along by the current or sails would employ wind power. When heading downstream, however, they were faster and could also haul barges with them. At this point, by the Domfelsen, the river flowed particularly fast. Above 0.3‰ the chain boat has the advantage. ",-111.92, 6.07,20.2,38,205,C
-Find the area bounded by the curves $y=\cos x$ and $y=\cos ^2 x$ between $x=0$ and $x=\pi$.,"thumb|Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1 In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. This function has no global maximum or minimum. x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. cos(x) Infinitely many global maxima at 0, ±2, ±4, ..., and infinitely many global minima at ±, ±3, ±5, .... 2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum. cos(3x)/x with Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. thumb|right|A graph of the function f(x) = e^{-x^2} and the area between it and the x-axis, (i.e. the entire real line) which is equal to \sqrt{\pi}. In mathematics, the definite integral :\int_a^b f(x)\, dx is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total. The limits of these functions as goes to infinity are known: \int_0^\infty \cos \left(t^2\right)\,dt = \int_0^\infty \sin \left(t^2\right) \, dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}} \approx 0.6267. right|250px|thumb|The sector contour used to calculate the limits of the Fresnel integrals This can be derived with any one of several methods. thumb|300px|Graph y=ƒ(x) with the x-axis as the horizontal axis and the y-axis as the vertical axis. It is easy to see that the volume of the region under f(x,y) and above z=0 , which is 1 , can be obtained by integrating the area, which is -2\pi\log(z/c^2) , of the circle with radius of value x>0 such that f(x,0)=z between z=0 and z=c^2 . In mathematics, tables of trigonometric functions are useful in a number of areas. As such, these points satisfy x = 0. ==Using equations== If the curve in question is given as y= f(x), the y-coordinate of the y-intercept is found by calculating f(0). 250px|thumb| Plots of and . The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1). The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). == Definition == 250px|thumb| Fresnel integrals with arguments instead of converge to instead of . Let \begin{align} y & = xs \\\ dy & = x\,ds. \end{align} Since the limits on as depend on the sign of , it simplifies the calculation to use the fact that is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity. Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Since width is positive, then x>0, and since that implies that Plug in critical point as well as endpoints 0 and into and the results are 2500, 0, and 0 respectively. They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations: S(x) = \int_0^x \sin\left(t^2\right)\,dt, \quad C(x) = \int_0^x \cos\left(t^2\right)\,dt. Category:Trigonometry Category:Numerical analysis Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is ==Functions of more than one variable== thumb|right|The global maximum is the point at the top thumb|right|Counterexample: The red dot shows a local minimum that is not a global minimum For functions of more than one variable, similar conditions apply. The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals. Combining these yields \left ( \int_{-\infty}^\infty e^{-x^2}\,dx \right )^2=\pi, so \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}. ====Complete proof==== To justify the improper double integrals and equating the two expressions, we begin with an approximating function: I(a) = \int_{-a}^a e^{-x^2}dx. As such, these points satisfy y=0. Now retrieve the endpoints by determining the interval to which x is restricted. ",3.29527,1,355.1,2,-0.75,D
-A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\circ}$ above the horizontal moves the sled $80 \mathrm{ft}$. Find the work done by the force.,"This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. In physics, work is the energy transferred to or from an object via the application of force along a displacement. The work of the net force is calculated as the product of its magnitude and the particle displacement. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Notice that the work done by gravity depends only on the vertical movement of the object. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. The work done is given by the dot product of the two vectors. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. Therefore, work need only be computed for the gravitational forces acting on the bodies. Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. The work of the torque is calculated as \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt, where the is the power over the instant . Work transfers energy from one place to another or one form to another. ===Derivation for a particle moving along a straight line=== In the case the resultant force is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line. In this case the dot product , where is the angle between the force vector and the direction of movement, that is W = \int_C \mathbf{F} \cdot d\mathbf{s} = Fs\cos\theta. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. ",6.9,1838.50666349,4152.0,0.6247,12,B
+","This is often called the hull speed and is a function of the length of the ship V=k\sqrt{L} where constant (k) should be taken as: 2.43 for velocity (V) in kn and length (L) in metres (m) or, 1.34 for velocity (V) in kn and length (L) in feet (ft). This procedure meant a delay of at least 20 minutes for the boat travelling upstream, while the ship heading downstream suffered a delay of about 45 minutes as a result of the manoeuvre. The smaller boat, which is travelling downstream, is moving very fast, driven by the large water sails on either side and is thereby hauling the larger boat upstream against the current. Practical experience showed that free-running, paddle tugboats capable of could achieve a speed of about against a river current of . On the Neckar river with seven chain boats that meant six passing manoeuvres costing at least five hours for those travelling downstream. The boat is pulling itself upstream on a cable laid along the river. So variations in river depth complicated the handling of the vessel considerably.Carl Victor Suppán: Wasserstrassen und Binnenschiffahrt. However, fast currents when rivers were in spate could also be problematic for chain boats. Chain-boat navigationDocument, Volume 1, Issues 1-9, The Commission, US National Waterways Commission, 1909, p. right|thumb|300px|The canal's route is close to the dashed line of the railway across the neck of the peninsula. thumb|293x293px|a ship crossing the Karakum Canal. By the time the attached barges entered the faster-flowing area, the steamer had already sailed past it and was able to generate its full traction again. When travelling downstream the barges were usually just allowed to drift with the current in order to save money, In strong currents, operating a long string of barges was quite dangerous. The electric powered bridge is lifted several times a day to let boats pass through it. == Activities == thumb|left|Sunset on the Shark River Inlet looking west from the drawbridge. The number of trips a boat could make increased, for example, on the Elbe almost three times.Erich Pleißner: ""Konzentration der Güterschiffahrt auf der Elbe"". These sections of river could be negotiated by anchoring a rope ahead of the boat and then using the crew to haul it upstream.Sigbert Zesewitz, Helmut Düntzsch, Theodor Grötschel: Kettenschiffahrt. On the remaining section with its strong currents the chain boats made a profit in many years of about 30%. The canal tolls reflected these improvements, but if a boat missed the tide it would have to wait in the canal basin for longer than the journey round Hoo would have taken. ==Higham and Strood tunnel== The Higham and Strood tunnel is long, and was the second longest canal tunnel built in the UK (the longest is the Standedge Canal Tunnel). When travelling downstream, boats were either simply propelled along by the current or sails would employ wind power. When heading downstream, however, they were faster and could also haul barges with them. At this point, by the Domfelsen, the river flowed particularly fast. Above 0.3‰ the chain boat has the advantage. ",-111.92, 6.07,"""20.2""",38,205,C
+Find the area bounded by the curves $y=\cos x$ and $y=\cos ^2 x$ between $x=0$ and $x=\pi$.,"thumb|Local and global maxima and minima for cos(3πx)/x, 0.1≤ x ≤1.1 In mathematical analysis, the maximum and minimum of a function are, respectively, the largest and smallest value taken by the function. This function has no global maximum or minimum. x| Global minimum at x = 0 that cannot be found by taking derivatives, because the derivative does not exist at x = 0. cos(x) Infinitely many global maxima at 0, ±2, ±4, ..., and infinitely many global minima at ±, ±3, ±5, .... 2 cos(x) − x Infinitely many local maxima and minima, but no global maximum or minimum. cos(3x)/x with Global maximum at x = 0.1 (a boundary), a global minimum near x = 0.3, a local maximum near x = 0.6, and a local minimum near x = 1.0. thumb|right|A graph of the function f(x) = e^{-x^2} and the area between it and the x-axis, (i.e. the entire real line) which is equal to \sqrt{\pi}. In mathematics, the definite integral :\int_a^b f(x)\, dx is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total. The limits of these functions as goes to infinity are known: \int_0^\infty \cos \left(t^2\right)\,dt = \int_0^\infty \sin \left(t^2\right) \, dt = \frac{\sqrt{2\pi}}{4} = \sqrt{\frac{\pi}{8}} \approx 0.6267. right|250px|thumb|The sector contour used to calculate the limits of the Fresnel integrals This can be derived with any one of several methods. thumb|300px|Graph y=ƒ(x) with the x-axis as the horizontal axis and the y-axis as the vertical axis. It is easy to see that the volume of the region under f(x,y) and above z=0 , which is 1 , can be obtained by integrating the area, which is -2\pi\log(z/c^2) , of the circle with radius of value x>0 such that f(x,0)=z between z=0 and z=c^2 . In mathematics, tables of trigonometric functions are useful in a number of areas. As such, these points satisfy x = 0. ==Using equations== If the curve in question is given as y= f(x), the y-coordinate of the y-intercept is found by calculating f(0). 250px|thumb| Plots of and . The y-intercept of ƒ(x) is indicated by the red dot at (x=0, y=1). The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). == Definition == 250px|thumb| Fresnel integrals with arguments instead of converge to instead of . Let \begin{align} y & = xs \\\ dy & = x\,ds. \end{align} Since the limits on as depend on the sign of , it simplifies the calculation to use the fact that is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity. Modern computers and pocket calculators now generate trigonometric function values on demand, using special libraries of mathematical code. Since width is positive, then x>0, and since that implies that Plug in critical point as well as endpoints 0 and into and the results are 2500, 0, and 0 respectively. They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations: S(x) = \int_0^x \sin\left(t^2\right)\,dt, \quad C(x) = \int_0^x \cos\left(t^2\right)\,dt. Category:Trigonometry Category:Numerical analysis Therefore, the greatest area attainable with a rectangle of 200 feet of fencing is ==Functions of more than one variable== thumb|right|The global maximum is the point at the top thumb|right|Counterexample: The red dot shows a local minimum that is not a global minimum For functions of more than one variable, similar conditions apply. The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals. Combining these yields \left ( \int_{-\infty}^\infty e^{-x^2}\,dx \right )^2=\pi, so \int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt{\pi}. ====Complete proof==== To justify the improper double integrals and equating the two expressions, we begin with an approximating function: I(a) = \int_{-a}^a e^{-x^2}dx. As such, these points satisfy y=0. Now retrieve the endpoints by determining the interval to which x is restricted. ",3.29527,1,"""355.1""",2,-0.75,D
+A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\circ}$ above the horizontal moves the sled $80 \mathrm{ft}$. Find the work done by the force.,"This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. In physics, work is the energy transferred to or from an object via the application of force along a displacement. The work of the net force is calculated as the product of its magnitude and the particle displacement. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Notice that the work done by gravity depends only on the vertical movement of the object. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. The work done is given by the dot product of the two vectors. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. Therefore, work need only be computed for the gravitational forces acting on the bodies. Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. The work of the torque is calculated as \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt, where the is the power over the instant . Work transfers energy from one place to another or one form to another. ===Derivation for a particle moving along a straight line=== In the case the resultant force is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line. In this case the dot product , where is the angle between the force vector and the direction of movement, that is W = \int_C \mathbf{F} \cdot d\mathbf{s} = Fs\cos\theta. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. ",6.9,1838.50666349,"""4152.0""",0.6247,12,B
 " If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then
 $$
 \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}
 $$
-If the resistances are measured in ohms as $R_1=25 \Omega$, $R_2=40 \Omega$, and $R_3=50 \Omega$, with a possible error of $0.5 \%$ in each case, estimate the maximum error in the calculated value of $R$.","The unit was based upon the ohm equal to 109 units of resistance of the C.G.S. system of electromagnetic units. The ohm (symbol: Ω) is the unit of electrical resistance in the International System of Units (SI). Residual-resistivity ratio (also known as Residual-resistance ratio or just RRR) is usually defined as the ratio of the resistivity of a material at room temperature and at 0 K. The formula is a combination of Ohm's law and Joule's law: P=V I =\frac{V^2}{R} = I^2 R, where is the power, is the resistance, is the voltage across the resistor, and is the current through the resistor. The B.A. ohm was intended to be 109 CGS units but owing to an error in calculations the definition was 1.3% too small. A legal ohm, a reproducible standard, was defined by the international conference of electricians at Paris in 1884 as the resistance of a mercury column of specified weight and 106 cm long; this was a compromise value between the B. A. unit (equivalent to 104.7 cm), the Siemens unit (100 cm by definition), and the CGS unit. Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. The 3ω-method (3 omega method) or 3ω-technique, is a measurement method for determining the thermal conductivities of bulk material (i.e. solid or liquid) and thin layers. The international ohm is represented by the resistance offered to an unvarying electric current in a mercury column of constant cross-sectional area 106.3 cm long of mass 14.4521 grams and 0 °C. On 21 September 1881 the Congrès internationale des électriciens (international conference of electricians) defined a practical unit of ohm for the resistance, based on CGS units, using a mercury column 1 mm2 in cross-section, approximately 104.9 cm in length at 0 °C, similar to the apparatus suggested by Siemens. In the electronics industry it is common to use the character R instead of the Ω symbol, thus, a 10 Ω resistor may be represented as 10R. Multiple drug resistance (MDR), multidrug resistance or multiresistance is antimicrobial resistance shown by a species of microorganism to at least one antimicrobial drug in three or more antimicrobial categories. Multiple resistance may refer to: * Multiple drug resistance ** including Antimicrobial resistance * The measured voltage will contain both the fundamental and third harmonic components (ω and 3ω respectively), because the Joule heating of the metal structure induces oscillations in its resistance with frequency 2ω due to the temperature coefficient of resistance (TCR) of the metal heater/sensor as stated in the following equation: :V=IR=I_0e^{i\omega t}\left (R_0+\frac{\partial R}{\partial T}\Delta T \right )=I_0e^{i\omega t}\left (R_0+C_0e^{i2\omega t} \right )=I_0R_0e^{i\omega t} + I_0C_0e^{i3\omega t}, where C0 is constant. In Mac OS, does the same. == See also == * Electronic color code * History of measurement * International Committee for Weights and Measures * Orders of magnitude (resistance) * Resistivity == Notes and references == == External links == * Scanned books of Georg Simon Ohm at the library of the University of Applied Sciences Nuernberg * Official SI brochure * NIST Special Publication 811 * History of the ohm at sizes.com * History of the electrical units. :\Omega = \dfrac{\text{V}}{\text{A}} = \dfrac{1}{\text{S}} = \dfrac{\text{W}}{\text{A}^2} = \dfrac{\text{V}^2}{\text{W}} = \dfrac{\text{s}}{\text{F}} = \dfrac{\text{H}}{\text{s}} = \dfrac{\text{J} {\cdot} \text{s}}{\text{C}^2} = \dfrac{\text{kg} {\cdot} \text{m}^2}{\text{s} {\cdot} \text{C}^2} = \dfrac{\text{J}}{\text{s} {\cdot} \text{A}^2}=\dfrac{\text{kg}{\cdot}\text{m}^2}{\text{s}^3 {\cdot} \text{A}^2} in which the following units appear: volt (V), ampere (A), siemens (S), watt (W), second (s), farad (F), henry (H), joule (J), coulomb (C), kilogram (kg), and meter (m). Since resistivity usually increases as defect prevalence increases, a large RRR is associated with a pure sample. Since the ohm belongs to a coherent system of units, when each of these quantities has its corresponding SI unit (watt for , ohm for , volt for and ampere for , which are related as in ) this formula remains valid numerically when these units are used (and thought of as being cancelled or omitted). == History == The rapid rise of electrotechnology in the last half of the 19th century created a demand for a rational, coherent, consistent, and international system of units for electrical quantities. Various artifact standards were proposed as the definition of the unit of resistance. The quantum Hall experiments are used to check the stability of working standards that have convenient values for comparison.R. Dzuiba and others, Stability of Double-Walled Maganin Resistors in NIST Special Publication Proceedings of SPIE, The Institute, 1988 pp. 63–64 Following the 2019 redefinition of the SI base units, in which the ampere and the kilogram were redefined in terms of fundamental constants, the ohm is now also defined in terms of these constants. == Symbol == The symbol Ω was suggested, because of the similar sound of ohm and omega, by William Henry Preece in 1867. Note that since it is a unitless ratio there is no difference between a residual resistivity and residual-resistance ratio. ==Background== Usually at ""warm"" temperatures the resistivity of a metal varies linearly with temperature. Advances in metrology allowed definitions to be formulated with a high degree of precision and repeatability. === Historical units of resistance === UnitGordon Wigan (trans. and ed.), Electrician's Pocket Book, Cassel and Company, London, 1884 Definition Value in B.A. ohms Remarks Absolute foot/second × 107 using imperial units 0.3048 considered obsolete even in 1884 Thomson's unit using imperial units 0.3202 , considered obsolete even in 1884 Jacobi copper unit A specified copper wire long weighing 0.6367 Used in 1850s Weber's absolute unit × 107 Based on the meter and the second 0.9191 Siemens mercury unit 1860. ",313,0.05882352941,82258.0,49,-167,B
+If the resistances are measured in ohms as $R_1=25 \Omega$, $R_2=40 \Omega$, and $R_3=50 \Omega$, with a possible error of $0.5 \%$ in each case, estimate the maximum error in the calculated value of $R$.","The unit was based upon the ohm equal to 109 units of resistance of the C.G.S. system of electromagnetic units. The ohm (symbol: Ω) is the unit of electrical resistance in the International System of Units (SI). Residual-resistivity ratio (also known as Residual-resistance ratio or just RRR) is usually defined as the ratio of the resistivity of a material at room temperature and at 0 K. The formula is a combination of Ohm's law and Joule's law: P=V I =\frac{V^2}{R} = I^2 R, where is the power, is the resistance, is the voltage across the resistor, and is the current through the resistor. The B.A. ohm was intended to be 109 CGS units but owing to an error in calculations the definition was 1.3% too small. A legal ohm, a reproducible standard, was defined by the international conference of electricians at Paris in 1884 as the resistance of a mercury column of specified weight and 106 cm long; this was a compromise value between the B. A. unit (equivalent to 104.7 cm), the Siemens unit (100 cm by definition), and the CGS unit. Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. The 3ω-method (3 omega method) or 3ω-technique, is a measurement method for determining the thermal conductivities of bulk material (i.e. solid or liquid) and thin layers. The international ohm is represented by the resistance offered to an unvarying electric current in a mercury column of constant cross-sectional area 106.3 cm long of mass 14.4521 grams and 0 °C. On 21 September 1881 the Congrès internationale des électriciens (international conference of electricians) defined a practical unit of ohm for the resistance, based on CGS units, using a mercury column 1 mm2 in cross-section, approximately 104.9 cm in length at 0 °C, similar to the apparatus suggested by Siemens. In the electronics industry it is common to use the character R instead of the Ω symbol, thus, a 10 Ω resistor may be represented as 10R. Multiple drug resistance (MDR), multidrug resistance or multiresistance is antimicrobial resistance shown by a species of microorganism to at least one antimicrobial drug in three or more antimicrobial categories. Multiple resistance may refer to: * Multiple drug resistance ** including Antimicrobial resistance * The measured voltage will contain both the fundamental and third harmonic components (ω and 3ω respectively), because the Joule heating of the metal structure induces oscillations in its resistance with frequency 2ω due to the temperature coefficient of resistance (TCR) of the metal heater/sensor as stated in the following equation: :V=IR=I_0e^{i\omega t}\left (R_0+\frac{\partial R}{\partial T}\Delta T \right )=I_0e^{i\omega t}\left (R_0+C_0e^{i2\omega t} \right )=I_0R_0e^{i\omega t} + I_0C_0e^{i3\omega t}, where C0 is constant. In Mac OS, does the same. == See also == * Electronic color code * History of measurement * International Committee for Weights and Measures * Orders of magnitude (resistance) * Resistivity == Notes and references == == External links == * Scanned books of Georg Simon Ohm at the library of the University of Applied Sciences Nuernberg * Official SI brochure * NIST Special Publication 811 * History of the ohm at sizes.com * History of the electrical units. :\Omega = \dfrac{\text{V}}{\text{A}} = \dfrac{1}{\text{S}} = \dfrac{\text{W}}{\text{A}^2} = \dfrac{\text{V}^2}{\text{W}} = \dfrac{\text{s}}{\text{F}} = \dfrac{\text{H}}{\text{s}} = \dfrac{\text{J} {\cdot} \text{s}}{\text{C}^2} = \dfrac{\text{kg} {\cdot} \text{m}^2}{\text{s} {\cdot} \text{C}^2} = \dfrac{\text{J}}{\text{s} {\cdot} \text{A}^2}=\dfrac{\text{kg}{\cdot}\text{m}^2}{\text{s}^3 {\cdot} \text{A}^2} in which the following units appear: volt (V), ampere (A), siemens (S), watt (W), second (s), farad (F), henry (H), joule (J), coulomb (C), kilogram (kg), and meter (m). Since resistivity usually increases as defect prevalence increases, a large RRR is associated with a pure sample. Since the ohm belongs to a coherent system of units, when each of these quantities has its corresponding SI unit (watt for , ohm for , volt for and ampere for , which are related as in ) this formula remains valid numerically when these units are used (and thought of as being cancelled or omitted). == History == The rapid rise of electrotechnology in the last half of the 19th century created a demand for a rational, coherent, consistent, and international system of units for electrical quantities. Various artifact standards were proposed as the definition of the unit of resistance. The quantum Hall experiments are used to check the stability of working standards that have convenient values for comparison.R. Dzuiba and others, Stability of Double-Walled Maganin Resistors in NIST Special Publication Proceedings of SPIE, The Institute, 1988 pp. 63–64 Following the 2019 redefinition of the SI base units, in which the ampere and the kilogram were redefined in terms of fundamental constants, the ohm is now also defined in terms of these constants. == Symbol == The symbol Ω was suggested, because of the similar sound of ohm and omega, by William Henry Preece in 1867. Note that since it is a unitless ratio there is no difference between a residual resistivity and residual-resistance ratio. ==Background== Usually at ""warm"" temperatures the resistivity of a metal varies linearly with temperature. Advances in metrology allowed definitions to be formulated with a high degree of precision and repeatability. === Historical units of resistance === UnitGordon Wigan (trans. and ed.), Electrician's Pocket Book, Cassel and Company, London, 1884 Definition Value in B.A. ohms Remarks Absolute foot/second × 107 using imperial units 0.3048 considered obsolete even in 1884 Thomson's unit using imperial units 0.3202 , considered obsolete even in 1884 Jacobi copper unit A specified copper wire long weighing 0.6367 Used in 1850s Weber's absolute unit × 107 Based on the meter and the second 0.9191 Siemens mercury unit 1860. ",313,0.05882352941,"""82258.0""",49,-167,B
 "The length and width of a rectangle are measured as $30 \mathrm{~cm}$ and $24 \mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.
-","The big rectangle has width m and length T + 3m. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. There are a number of variants of the problem, depending on the particularities of this generic formulation, in particular, depending on the measure of the ""size"", domain (type of obstacles), and the orientation of the rectangle. In particular, for the case of points within rectangle an optimal algorithm of time complexity \Theta(n \log n) is known. ===Domain: rectangle containing points=== A problem first discussed by Naamad, Lee and Hsu in 1983 is stated as follows: given a rectangle A containing n points, find a largest-area rectangle with sides parallel to those of A which lies within A and does not contain any of the given points. Each side of a maximal empty rectangle abuts an obstacle (otherwise the side may be shifted outwards, increasing the empty rectangle). E shows a maximal empty rectangle with arbitrary orientation In computational geometry, the largest empty rectangle problem, maximal empty rectangle problem or maximum empty rectangle problem, is the problem of finding a rectangle of maximal size to be placed among obstacles in the plane. In contrast, in rectangle packing (as in real-life packing problems) the sizes of the rectangles are given, but their locations are flexible. In this more general case, it is not clear if the problem is in NP, since it is much harder to verify a solution. == Packing different rectangles in a minimum-area rectangle == In this variant, the small rectangles can have varying lengths and widths, and their orientation is fixed (they cannot be rotated). This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. In the contexts of many algorithms for largest empty rectangles, ""maximal empty rectangles"" are candidate solutions to be considered by the algorithm, since it is easily proven that, e.g., a maximum-area empty rectangle is a maximal empty rectangle. ==Classification== In terms of size measure, the two most common cases are the largest-area empty rectangle and largest-perimeter empty rectangle. Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon, such that no two small rectangles overlap. * Maximum disjoint set (or Maximum independent set) is a problem in which both the sizes and the locations of the input rectangles are fixed, and the goal is to select a largest sum of non-overlapping rectangles. Another major classification is whether the rectangle is sought among axis-oriented or arbitrarily oriented rectangles. ==Special cases == ===Maximum-area square=== The case when the sought rectangle is an axis-oriented square may be treated using Voronoi diagrams in L_1metrics for the corresponding obstacle set, similarly to the largest empty circle problem. Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles, find the area of their union. Finding a largest square-packing is NP-hard; one may prove this by reducing from 3SAT. == Packing different rectangles in a given rectangle == In this variant, the small rectangles can have varying lengths and widths, and they should be packed in a given large rectangle. The light green rectangle would be suboptimal (non- maximal) solution. Probabilistic bounding analysis in the quantification of margins and uncertainties. The goal is to pack them in an enclosing rectangle of minimum area, with no boundaries on the enclosing rectangle's width or height. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle. Therefore, the packing must involve exactly m rows where each row contains rectangles with a total length of exactly T. A maximal empty rectangle is a rectangle which is not contained in another empty rectangle. ",29.36,5.4,2.0,27,-87.8,B
-The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \times 10^7 \mathrm{~km}$. Find its maximum distance from the sun.,"Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). thumb|An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. The following chart of the perihelion and aphelion of the planets, dwarf planets and Halley's Comet demonstrates the variation of the eccentricity of their elliptical orbits. 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halley's Comet and Eris. ==Radial elliptic trajectory== A radial trajectory can be a double line segment, which is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1. Mathematically, the distance between a central body and an orbiting body can be expressed as: r(\theta) = \frac{a(1-e^2)}{1+e\cos(\theta)} where: *r is the distance *a is the semi- major axis, which defines the size of the orbit *e is the eccentricity, which defines the shape of the orbit *\theta is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis). This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac{Mm}{r_1 + r_2} Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result. ==Flight path angle== The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Kepler-1704b goes on a highly eccentric 2.7 year-long (988.88 days) orbit around its star as well as transiting. The extreme eccentricity yields a temperature difference of up to 700 K. == Star == The star, Kepler-1704, is a G2, 5745-kelvin star from Earth and the sun. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. thumb|right|Kepler's equation solutions for five different eccentricities between 0 and 1 In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. Alternatively, Kepler's equation can be solved numerically. This includes the radial elliptic orbit, with eccentricity equal to 1. Kepler-1704b is a super-Jupiter on a highly eccentric orbit around the star Kepler-1704. The planet's distance from its star varies from 0.16 to 3.9 AU. For 0 this is an ellipse with = a \cdot \sqrt{1-e^2}|}} For e = 1 this is a parabola with focal length \tfrac{p}{2} For e > 1 this is a hyperbola with = a \cdot \sqrt{e^2-1}|}} The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue) thumb|300px|A diagram of the various forms of the Kepler Orbit and their eccentricities. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. The standard Kepler equation is used for elliptic orbits (0 \le e < 1). ",0.086,7,30.0,54.7,322,B
-Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \mathrm{~cm}$ and height $12 \mathrm{~cm}$ if the tin is $0.04 \mathrm{~cm}$ thick.,"A standard size tin can holds roughly 400 g; though the weight can vary between 385 g and 425 g depending on the density of the contents. Estimates of tin production have historically varied with the market and mining technology. Most of the world's tin is produced from placer deposits, which can contain as little as 0.015% tin. The springs on the tinning machine can be set to different forces to give different thicknesses of tin. thumb|upright|An empty tin can A steel can, tin can, tin (especially in British English, Australian English, Canadian English and South African English), steel packaging, or can is a container for the distribution or storage of goods, made of thin metal. The US Occupational Safety and Health Administration (OSHA) set the permissible exposure limit for tin exposure in the workplace as 2 mg/m3 over an 8-hour workday. Tin melts at about , the lowest in group 14\. Recovery of tin through recycling is increasing rapidly. I Tall 3 × 4 16.70 0.813 No. 303 3 × 4 16.88 0.821 Fruits, Vegetables, Soups No. 303 Cylinder 3 × 5 21.86 1.060 No. 2 Vacuum 3 × 3 14.71 0.716 No. 2 3 × 4 20.55 1.000 Juices, Soups, Vegetables Jumbo 3 × 5 25.80 1.2537 No. 2 Cylinder 3 × 5 26.40 1.284 No. 1.25 4 × 2 13.81 0.672 No. 2.5 4 × 4 29.79 1.450 Fruits, Vegetables No. 3 Vacuum 4 × 3 23.90 1.162 No. 3 Cylinder 4 × 7 51.70 2.515 No. 5 5 × 5 59.10 2.8744 Fruit Juice, Soups No. 10 6 × 7 109.43 5.325 Fruits, Vegetables In parts of the world using the metric system, tins are made in 250, 500, 750 ml (millilitre) and 1 L (litre) sizes (250 ml is approximately 1 cup or 8 ounces). For instance, if a plate is desired the tin bar is cut to a length and width that is divisible by 14 and 20. Because of the higher specific gravity of tin dioxide, about 80% of mined tin is from secondary deposits found downstream from the primary lodes. The flat surfaces of rimmed cans are recessed from the edge of any rim (toward the middle of the can) by about the width of the rim; the inside diameter of a rim, adjacent to this recessed surface, is slightly smaller than the inside diameter of the rest of the can. (See BPA controversy#Chemical manufacturers reactions to bans.) ==See also== * Albion metal *Can collecting * Drink can * Oil can * Tin box * Tin can wall ==References== ===General references, further reading=== * Nicolas Appert * Guide to Tinplate * History of the Tin Can on About.com * Yam, K. L., Encyclopedia of Packaging Technology, John Wiley & Sons, 2009, * Soroka, W, Fundamentals of Packaging Technology, Institute of Packaging Professionals (IoPP), 2002, ==External links== *Steeluniversity Packaging Module *Steel industry fact sheet on food cans *Standard U.S. can sizes at GourmetSleuth Category:Containers Category:Packaging Category:Food storage containers Category:British inventions Category:1810 introductions Category:Food packaging Category:Steel Category:19th-century inventions Category:Metallic objects The Can Manufacturers Institute defines these sizes, expressing them in three-digit numbers, as measured in whole and sixteenths of an inch for the container's nominal outside dimensions: a 307 × 512 would thus measure 3 and 7/16"" in diameter by 5 and 3/4"" (12/16"") in height. Depending on contents and available coatings, some canneries still use tin-free steel. The bar is then rolled and doubled over, with the number of times being doubled over dependent on how large the tin bar is and what the final thickness is. If the starting tin bar is then it must be at least finished on the fours, or doubled over twice, and if a thin gauge is required then it may be finished on the eights, or doubled over three times. The tin layer is usually applied by electroplating. ===Two-piece steel can design=== Most steel beverage cans are two-piece designs, made from 1) a disc re-formed into a cylinder with an integral end, double-seamed after filling and 2) a loose end to close it. The International Tin Association estimated that global refined tin consumption will grow 7.2 percent in 2021, after losing 1.6 percent in 2020 as the COVID-19 pandemic disrupted global manufacturing industries. ==Applications== thumb|right|World consumption of refined tin by end-use, 2006 In 2018, just under half of all tin produced was used in solder. A 2002 study showed that 99.5% of 1200 tested cans contained below the UK regulatory limit of 200 mg/kg of tin, an improvement over most previous studies largely attributed to the increased use of fully lacquered cans for acidic foods, and concluded that the results do not raise any long term food safety concerns for consumers. Because of the low toxicity of inorganic tin, tin-plated steel is widely used for food packaging as tin cans. File:Inside of a tin platted can.jpg|Inside of a tin can. ==Design and manufacture== ===Steel for can making=== The majority of steel used in packaging is tinplate, which is steel that has been coated with a thin layer of tin, whose functionality is required for the production process. ",0.69,1.8763,16.0,24,0.000226,C
+","The big rectangle has width m and length T + 3m. For a circle with slightly smaller radius, the area is nearly the same, but the circle contains only 69 points, producing a larger error of approximately 9.54. thumb|A circle of radius 5 centered at the origin has area 25, approximately 78.54, but it contains 81 integer points, so the error in estimating its area by counting grid points is approximately 2.46. There are a number of variants of the problem, depending on the particularities of this generic formulation, in particular, depending on the measure of the ""size"", domain (type of obstacles), and the orientation of the rectangle. In particular, for the case of points within rectangle an optimal algorithm of time complexity \Theta(n \log n) is known. ===Domain: rectangle containing points=== A problem first discussed by Naamad, Lee and Hsu in 1983 is stated as follows: given a rectangle A containing n points, find a largest-area rectangle with sides parallel to those of A which lies within A and does not contain any of the given points. Each side of a maximal empty rectangle abuts an obstacle (otherwise the side may be shifted outwards, increasing the empty rectangle). E shows a maximal empty rectangle with arbitrary orientation In computational geometry, the largest empty rectangle problem, maximal empty rectangle problem or maximum empty rectangle problem, is the problem of finding a rectangle of maximal size to be placed among obstacles in the plane. In contrast, in rectangle packing (as in real-life packing problems) the sizes of the rectangles are given, but their locations are flexible. In this more general case, it is not clear if the problem is in NP, since it is much harder to verify a solution. == Packing different rectangles in a minimum-area rectangle == In this variant, the small rectangles can have varying lengths and widths, and their orientation is fixed (they cannot be rotated). This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. In the contexts of many algorithms for largest empty rectangles, ""maximal empty rectangles"" are candidate solutions to be considered by the algorithm, since it is easily proven that, e.g., a maximum-area empty rectangle is a maximal empty rectangle. ==Classification== In terms of size measure, the two most common cases are the largest-area empty rectangle and largest-perimeter empty rectangle. Rectangle packing is a packing problem where the objective is to determine whether a given set of small rectangles can be placed inside a given large polygon, such that no two small rectangles overlap. * Maximum disjoint set (or Maximum independent set) is a problem in which both the sizes and the locations of the input rectangles are fixed, and the goal is to select a largest sum of non-overlapping rectangles. Another major classification is whether the rectangle is sought among axis-oriented or arbitrarily oriented rectangles. ==Special cases == ===Maximum-area square=== The case when the sought rectangle is an axis-oriented square may be treated using Voronoi diagrams in L_1metrics for the corresponding obstacle set, similarly to the largest empty circle problem. Later in 1977, Jon Bentley considered a 2-dimensional analogue of this problem: given a collection of n rectangles, find the area of their union. Finding a largest square-packing is NP-hard; one may prove this by reducing from 3SAT. == Packing different rectangles in a given rectangle == In this variant, the small rectangles can have varying lengths and widths, and they should be packed in a given large rectangle. The light green rectangle would be suboptimal (non- maximal) solution. Probabilistic bounding analysis in the quantification of margins and uncertainties. The goal is to pack them in an enclosing rectangle of minimum area, with no boundaries on the enclosing rectangle's width or height. The Gauss circle problem concerns bounding this error more generally, as a function of the radius of the circle. Therefore, the packing must involve exactly m rows where each row contains rectangles with a total length of exactly T. A maximal empty rectangle is a rectangle which is not contained in another empty rectangle. ",29.36,5.4,"""2.0""",27,-87.8,B
+The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \times 10^7 \mathrm{~km}$. Find its maximum distance from the sun.,"Its extreme orbital eccentricity brings it to within 0.071 AU of the Sun (23% of Mercury's perihelion) and takes it as far as 3.562 AU from the Sun (well beyond the orbit of Mars). thumb|An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. Conclusions: *The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (a\,\\!), *For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law). ==Energy== Under standard assumptions, the specific orbital energy (\epsilon) of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form: :{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0 where: *v\, is the orbital speed of the orbiting body, *r\, is the distance of the orbiting body from the central body, *a\, is the length of the semi-major axis, *\mu\, is the standard gravitational parameter. The following chart of the perihelion and aphelion of the planets, dwarf planets and Halley's Comet demonstrates the variation of the eccentricity of their elliptical orbits. 250px|thumb|Animation of Orbit by eccentricity In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. If the eccentricity is less than 1 then the equation of motion describes an elliptical orbit. Note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halley's Comet and Eris. ==Radial elliptic trajectory== A radial trajectory can be a double line segment, which is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1. Mathematically, the distance between a central body and an orbiting body can be expressed as: r(\theta) = \frac{a(1-e^2)}{1+e\cos(\theta)} where: *r is the distance *a is the semi- major axis, which defines the size of the orbit *e is the eccentricity, which defines the shape of the orbit *\theta is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis). This is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: :E = - G \frac{Mm}{r_1 + r_2} Since r_1 = a + a \epsilon and r_2 = a - a \epsilon, where epsilon is the eccentricity of the orbit, we finally have the stated result. ==Flight path angle== The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Kepler-1704b goes on a highly eccentric 2.7 year-long (988.88 days) orbit around its star as well as transiting. The extreme eccentricity yields a temperature difference of up to 700 K. == Star == The star, Kepler-1704, is a G2, 5745-kelvin star from Earth and the sun. Due to its very small perihelion and comparably large aphelion, achieves the fastest speed of any known asteroid bound to the Solar System with a velocity of 157 km/s (565,000 km/h; 351,000 mi/h) at perihelionAs calculated with the vis-viva-equation : v^2 = GM \left({ 2 \over r} - {1 \over a}\right) where: * v is the relative speed of the two bodies * r is the distance between the two bodies * a is the semi-major axis (a > 0 for ellipses, a = ∞ or 1/a = 0 for parabolas, and a < 0 for hyperbolas) * G is the gravitational constant * M is the mass of the central body (there are comets, however, which obtain much higher speeds). == See also== * List of Mercury- crossing minor planets * List of Venus-crossing minor planets * Apollo asteroids * List of Mars-crossing minor planets == References == == External links == * * * # Category:Minor planet object articles (unnumbered) Category:Mercury-crossing asteroids Category:Venus-crossing asteroids 20050430 The total energy of the orbit is given by :E = - G \frac{M m}{2a}, where a is the semi major axis. ==== Derivation ==== Since gravity is a central force, the angular momentum is constant: :\dot{\mathbf{L}} = \mathbf{r} \times \mathbf{F} = \mathbf{r} \times F(r)\mathbf{\hat{r}} = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: :L = r p = r m v. thumb|right|Kepler's equation solutions for five different eccentricities between 0 and 1 In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. Alternatively, Kepler's equation can be solved numerically. This includes the radial elliptic orbit, with eccentricity equal to 1. Kepler-1704b is a super-Jupiter on a highly eccentric orbit around the star Kepler-1704. The planet's distance from its star varies from 0.16 to 3.9 AU. For 0 this is an ellipse with = a \cdot \sqrt{1-e^2}|}} For e = 1 this is a parabola with focal length \tfrac{p}{2} For e > 1 this is a hyperbola with = a \cdot \sqrt{e^2-1}|}} The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue) thumb|300px|A diagram of the various forms of the Kepler Orbit and their eccentricities. Conclusions: *For a given semi-major axis the specific orbital energy is independent of the eccentricity. The total energy of the orbit is given by :E = \frac{1}{2}m v^2 - G \frac{Mm}{r}. The standard Kepler equation is used for elliptic orbits (0 \le e < 1). ",0.086,7,"""30.0""",54.7,322,B
+Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \mathrm{~cm}$ and height $12 \mathrm{~cm}$ if the tin is $0.04 \mathrm{~cm}$ thick.,"A standard size tin can holds roughly 400 g; though the weight can vary between 385 g and 425 g depending on the density of the contents. Estimates of tin production have historically varied with the market and mining technology. Most of the world's tin is produced from placer deposits, which can contain as little as 0.015% tin. The springs on the tinning machine can be set to different forces to give different thicknesses of tin. thumb|upright|An empty tin can A steel can, tin can, tin (especially in British English, Australian English, Canadian English and South African English), steel packaging, or can is a container for the distribution or storage of goods, made of thin metal. The US Occupational Safety and Health Administration (OSHA) set the permissible exposure limit for tin exposure in the workplace as 2 mg/m3 over an 8-hour workday. Tin melts at about , the lowest in group 14\. Recovery of tin through recycling is increasing rapidly. I Tall 3 × 4 16.70 0.813 No. 303 3 × 4 16.88 0.821 Fruits, Vegetables, Soups No. 303 Cylinder 3 × 5 21.86 1.060 No. 2 Vacuum 3 × 3 14.71 0.716 No. 2 3 × 4 20.55 1.000 Juices, Soups, Vegetables Jumbo 3 × 5 25.80 1.2537 No. 2 Cylinder 3 × 5 26.40 1.284 No. 1.25 4 × 2 13.81 0.672 No. 2.5 4 × 4 29.79 1.450 Fruits, Vegetables No. 3 Vacuum 4 × 3 23.90 1.162 No. 3 Cylinder 4 × 7 51.70 2.515 No. 5 5 × 5 59.10 2.8744 Fruit Juice, Soups No. 10 6 × 7 109.43 5.325 Fruits, Vegetables In parts of the world using the metric system, tins are made in 250, 500, 750 ml (millilitre) and 1 L (litre) sizes (250 ml is approximately 1 cup or 8 ounces). For instance, if a plate is desired the tin bar is cut to a length and width that is divisible by 14 and 20. Because of the higher specific gravity of tin dioxide, about 80% of mined tin is from secondary deposits found downstream from the primary lodes. The flat surfaces of rimmed cans are recessed from the edge of any rim (toward the middle of the can) by about the width of the rim; the inside diameter of a rim, adjacent to this recessed surface, is slightly smaller than the inside diameter of the rest of the can. (See BPA controversy#Chemical manufacturers reactions to bans.) ==See also== * Albion metal *Can collecting * Drink can * Oil can * Tin box * Tin can wall ==References== ===General references, further reading=== * Nicolas Appert * Guide to Tinplate * History of the Tin Can on About.com * Yam, K. L., Encyclopedia of Packaging Technology, John Wiley & Sons, 2009, * Soroka, W, Fundamentals of Packaging Technology, Institute of Packaging Professionals (IoPP), 2002, ==External links== *Steeluniversity Packaging Module *Steel industry fact sheet on food cans *Standard U.S. can sizes at GourmetSleuth Category:Containers Category:Packaging Category:Food storage containers Category:British inventions Category:1810 introductions Category:Food packaging Category:Steel Category:19th-century inventions Category:Metallic objects The Can Manufacturers Institute defines these sizes, expressing them in three-digit numbers, as measured in whole and sixteenths of an inch for the container's nominal outside dimensions: a 307 × 512 would thus measure 3 and 7/16"" in diameter by 5 and 3/4"" (12/16"") in height. Depending on contents and available coatings, some canneries still use tin-free steel. The bar is then rolled and doubled over, with the number of times being doubled over dependent on how large the tin bar is and what the final thickness is. If the starting tin bar is then it must be at least finished on the fours, or doubled over twice, and if a thin gauge is required then it may be finished on the eights, or doubled over three times. The tin layer is usually applied by electroplating. ===Two-piece steel can design=== Most steel beverage cans are two-piece designs, made from 1) a disc re-formed into a cylinder with an integral end, double-seamed after filling and 2) a loose end to close it. The International Tin Association estimated that global refined tin consumption will grow 7.2 percent in 2021, after losing 1.6 percent in 2020 as the COVID-19 pandemic disrupted global manufacturing industries. ==Applications== thumb|right|World consumption of refined tin by end-use, 2006 In 2018, just under half of all tin produced was used in solder. A 2002 study showed that 99.5% of 1200 tested cans contained below the UK regulatory limit of 200 mg/kg of tin, an improvement over most previous studies largely attributed to the increased use of fully lacquered cans for acidic foods, and concluded that the results do not raise any long term food safety concerns for consumers. Because of the low toxicity of inorganic tin, tin-plated steel is widely used for food packaging as tin cans. File:Inside of a tin platted can.jpg|Inside of a tin can. ==Design and manufacture== ===Steel for can making=== The majority of steel used in packaging is tinplate, which is steel that has been coated with a thin layer of tin, whose functionality is required for the production process. ",0.69,1.8763,"""16.0""",24,0.000226,C
 "Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it.  A cable that weighs $2 \mathrm{~lb} / \mathrm{ft}$ is used to lift $800 \mathrm{~lb}$ of coal up a mine shaft $500 \mathrm{~ft}$ deep. Find the work done.
-","If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . The work of the net force is calculated as the product of its magnitude and the particle displacement. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work done is given by the dot product of the two vectors. To show that the external work done to move a point charge q+ from infinity to a distance r is: :W_{ext} = \frac{q_1q_2}{4\pi\varepsilon_0}\frac{1}{r} This could have been obtained equally by using the definition of W and integrating F with respect to r, which will prove the above relationship. This is easy to see mathematically, as reversing the boundaries of integration reverses the sign. === Uniform electric field === Where the electric field is constant (i.e. not a function of displacement, r), the work equation simplifies to: :W = Q (\mathbf{E} \cdot \, \mathbf{r})=\mathbf{F_E} \cdot \, \mathbf{r} or 'force times distance' (times the cosine of the angle between them). ==Electric power== The electric power is the rate of energy transferred in an electric circuit. Therefore, work need only be computed for the gravitational forces acting on the bodies. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Notice that the work done by gravity depends only on the vertical movement of the object. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Rope access or industrial climbing or commercial climbing, is a form of work positioning, initially developed from techniques used in climbing and caving, which applies practical ropework to allow workers to access difficult-to-reach locations without the use of scaffolding, cradles or an aerial work platform. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The work of the torque is calculated as \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt, where the is the power over the instant . If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. ",524,0.33333333,650000.0,-1,57.2,C
-" A patient takes $150 \mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? ","Such drugs need only a low maintenance dose in order to keep the amount of the drug in the body at the appropriate level, but this also means that, without an initial higher dose, it would take a long time for the amount of the drug in the body to reach that level. == Calculating the maintenance dose == The required maintenance dose may be calculated as: :\mbox{MD} = \frac{C_p CL}{F } Where: : MD is the maintenance dose rate [mg/h] Cp = desired peak concentration of drug [mg/L] CL = clearance of drug in body [L/h] F = bioavailability For an intravenously administered drug, the bioavailability F will equal 1, since the drug is directly introduced to the bloodstream. If the pill removed is a half pill, then it is simply consumed and nothing is returned to the jar. ==Mathematical derivation== The problem becomes very easy to solve once a binary variable Xk defined as Xk = 1, if the kth half pill remains inside the jar after all the whole pills are removed. One half pill is consumed and the other one is returned to the jar. 150 Milligrams (, lit. The sales of generic drugs dominate in the prescription category at 64.5%. Continuing the maintenance dose for about 4 to 5 half-lives (t½) of the drug will approximate the steady state level. The prescription drugs sales historically took the biggest share of the market, capturing 61% of the market in 2016. The pill jar puzzle is a probability puzzle, which asks the expected value of the number of half-pills remaining when the last whole pill is popped from a jar initially containing whole pills and the way to proceed is by removing a pill from the bottle at random. In pharmacokinetics, a maintenance dose is the maintenance rate [mg/h] of drug administration equal to the rate of elimination at steady state. But for companies that approved between eight and 13 drugs over 10 years, the cost per drug went as high as $5.5 billion. Percentage solution may refer to: * Mass fraction (or ""% w/w"" or ""wt.%""), for percent mass * Volume fraction (or ""% v/v"" or ""vol.%""), volume concentration, for percent volume * ""Mass/volume percentage"" (or ""% m/v"") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). The pharmaceutical industry in Russia had a turnover of $16.5 billion in 2016, which was equal to 1.3 % of GDP and 19.9% of health spending. Thus, profits made from one drug need to cover the costs of previous ""failed drugs"". === Relationship === Overall, research and development expenses relating to a pharmaceutical drug amount to the billions. Therefore, an appropriate figure like $60 billion would be approximate sales figure that a pharmaceutical company like AstraZeneca would aim to generate to cover these costs and make a profit at the same time. Out of all pharmaceutical sales they constitute only 39.4%. If the patient requires an oral dose, bioavailability will be less than 1 (depending upon absorption, first pass metabolism etc.), requiring a larger loading dose. == See also == * Therapeutic index ==References== Category:Pharmacokinetics Xk = 1 if out of the n − k + 1 pills (n − k whole pills + kth half pill), the one half pill is removed at the very end. If the pill removed is a whole pill, it is broken into two half pills. This is not to be confused with dose regimen, which is a type of drug therapy in which the dose [mg] of a drug is given at a regular dosing interval on a repetitive basis. In an analysis of the drug development costs for 98 companies over a decade, the average cost per drug developed and approved by a single- drug company was $350 million. This is important in setting projected profit goals for a particular drug and thus, is one of the most necessary steps pharmaceutical companies take in pricing a particular drug. ==Research on costs== Tufts Center for the Study of Drug Development has published numerous studies estimating the cost of developing new pharmaceutical drugs. A 2022 study invalidated the common argument for high medication costs that research and development investments are reflected in and necessitate the treatment costs, finding no correlation for investments in drugs (for cases where transparency was sufficient) and their costs. == References == == Further reading == * * * * * Category:Drug pricing Category:Drug discovery ",8.8,35,157.875,7.58,1.06,C
-A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.,"If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Therefore, work need only be computed for the gravitational forces acting on the bodies. Therefore, the distance in feet down a 6% grade to reach the velocity is at least s = \frac{\Delta z}{0.06}= 8.3\frac{V^2}{g},\quad\text{or}\quad s=8.3\frac{88^2}{32.2}\approx 2000\mathrm{ft}. The work of the net force is calculated as the product of its magnitude and the particle displacement. Step 3) Obtain net work (a single number) by numerical integration of muscle power data. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In physics, work is the energy transferred to or from an object via the application of force along a displacement. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as ""force times straight path segment"" would only apply in the most simple of circumstances, as noted above. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. ",30,1.51,-0.1,1.88,7200,E
+","If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . The work of the net force is calculated as the product of its magnitude and the particle displacement. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Work is defined by: : \delta W = \mathbf{F}\cdot\mathbf{v}\delta t, Therefore :\frac{\partial W}{\partial t}=\mathbf{F_E} \cdot \,\mathbf{v} ==References== Category:Electromagnetism This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work done is given by the dot product of the two vectors. To show that the external work done to move a point charge q+ from infinity to a distance r is: :W_{ext} = \frac{q_1q_2}{4\pi\varepsilon_0}\frac{1}{r} This could have been obtained equally by using the definition of W and integrating F with respect to r, which will prove the above relationship. This is easy to see mathematically, as reversing the boundaries of integration reverses the sign. === Uniform electric field === Where the electric field is constant (i.e. not a function of displacement, r), the work equation simplifies to: :W = Q (\mathbf{E} \cdot \, \mathbf{r})=\mathbf{F_E} \cdot \, \mathbf{r} or 'force times distance' (times the cosine of the angle between them). ==Electric power== The electric power is the rate of energy transferred in an electric circuit. Therefore, work need only be computed for the gravitational forces acting on the bodies. In physics, work is the energy transferred to or from an object via the application of force along a displacement. Notice that the work done by gravity depends only on the vertical movement of the object. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Rope access or industrial climbing or commercial climbing, is a form of work positioning, initially developed from techniques used in climbing and caving, which applies practical ropework to allow workers to access difficult-to-reach locations without the use of scaffolding, cradles or an aerial work platform. The work is the product of the distance times the spring force, which is also dependent on distance; hence the result. ===Work by a gas=== The work W done by a body of gas on its surroundings is: W = \int_a^b P \, dV where is pressure, is volume, and and are initial and final volumes. ==Work–energy principle== The principle of work and kinetic energy (also known as the work–energy principle) states that the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle. The work of the torque is calculated as \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt, where the is the power over the instant . If the torque \tau is aligned with the angular velocity vector so that, \mathbf{T} = \tau \mathbf{S}, and both the torque and angular velocity are constant, then the work takes the form, W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1). 250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm This result can be understood more simply by considering the torque as arising from a force of constant magnitude , being applied perpendicularly to a lever arm at a distance r, as shown in the figure. In general this integral requires that the path along which the velocity is defined, so the evaluation of work is said to be path dependent. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. ",524,0.33333333,"""650000.0""",-1,57.2,C
+" A patient takes $150 \mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? ","Such drugs need only a low maintenance dose in order to keep the amount of the drug in the body at the appropriate level, but this also means that, without an initial higher dose, it would take a long time for the amount of the drug in the body to reach that level. == Calculating the maintenance dose == The required maintenance dose may be calculated as: :\mbox{MD} = \frac{C_p CL}{F } Where: : MD is the maintenance dose rate [mg/h] Cp = desired peak concentration of drug [mg/L] CL = clearance of drug in body [L/h] F = bioavailability For an intravenously administered drug, the bioavailability F will equal 1, since the drug is directly introduced to the bloodstream. If the pill removed is a half pill, then it is simply consumed and nothing is returned to the jar. ==Mathematical derivation== The problem becomes very easy to solve once a binary variable Xk defined as Xk = 1, if the kth half pill remains inside the jar after all the whole pills are removed. One half pill is consumed and the other one is returned to the jar. 150 Milligrams (, lit. The sales of generic drugs dominate in the prescription category at 64.5%. Continuing the maintenance dose for about 4 to 5 half-lives (t½) of the drug will approximate the steady state level. The prescription drugs sales historically took the biggest share of the market, capturing 61% of the market in 2016. The pill jar puzzle is a probability puzzle, which asks the expected value of the number of half-pills remaining when the last whole pill is popped from a jar initially containing whole pills and the way to proceed is by removing a pill from the bottle at random. In pharmacokinetics, a maintenance dose is the maintenance rate [mg/h] of drug administration equal to the rate of elimination at steady state. But for companies that approved between eight and 13 drugs over 10 years, the cost per drug went as high as $5.5 billion. Percentage solution may refer to: * Mass fraction (or ""% w/w"" or ""wt.%""), for percent mass * Volume fraction (or ""% v/v"" or ""vol.%""), volume concentration, for percent volume * ""Mass/volume percentage"" (or ""% m/v"") in biology, for mass per unit volume; incorrectly used to denote mass concentration (chemistry). The pharmaceutical industry in Russia had a turnover of $16.5 billion in 2016, which was equal to 1.3 % of GDP and 19.9% of health spending. Thus, profits made from one drug need to cover the costs of previous ""failed drugs"". === Relationship === Overall, research and development expenses relating to a pharmaceutical drug amount to the billions. Therefore, an appropriate figure like $60 billion would be approximate sales figure that a pharmaceutical company like AstraZeneca would aim to generate to cover these costs and make a profit at the same time. Out of all pharmaceutical sales they constitute only 39.4%. If the patient requires an oral dose, bioavailability will be less than 1 (depending upon absorption, first pass metabolism etc.), requiring a larger loading dose. == See also == * Therapeutic index ==References== Category:Pharmacokinetics Xk = 1 if out of the n − k + 1 pills (n − k whole pills + kth half pill), the one half pill is removed at the very end. If the pill removed is a whole pill, it is broken into two half pills. This is not to be confused with dose regimen, which is a type of drug therapy in which the dose [mg] of a drug is given at a regular dosing interval on a repetitive basis. In an analysis of the drug development costs for 98 companies over a decade, the average cost per drug developed and approved by a single- drug company was $350 million. This is important in setting projected profit goals for a particular drug and thus, is one of the most necessary steps pharmaceutical companies take in pricing a particular drug. ==Research on costs== Tufts Center for the Study of Drug Development has published numerous studies estimating the cost of developing new pharmaceutical drugs. A 2022 study invalidated the common argument for high medication costs that research and development investments are reflected in and necessitate the treatment costs, finding no correlation for investments in drugs (for cases where transparency was sufficient) and their costs. == References == == Further reading == * * * * * Category:Drug pricing Category:Drug discovery ",8.8,35,"""157.875""",7.58,1.06,C
+A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.,"If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Therefore, work need only be computed for the gravitational forces acting on the bodies. Therefore, the distance in feet down a 6% grade to reach the velocity is at least s = \frac{\Delta z}{0.06}= 8.3\frac{V^2}{g},\quad\text{or}\quad s=8.3\frac{88^2}{32.2}\approx 2000\mathrm{ft}. The work of the net force is calculated as the product of its magnitude and the particle displacement. Step 3) Obtain net work (a single number) by numerical integration of muscle power data. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In physics, work is the energy transferred to or from an object via the application of force along a displacement. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as ""force times straight path segment"" would only apply in the most simple of circumstances, as noted above. The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. ",30,1.51,"""-0.1""",1.88,7200,E
 "Find the area of triangle $A B C$, correct to five decimal places, if
 $$
 |A B|=10 \mathrm{~cm} \quad|B C|=3 \mathrm{~cm} \quad \angle A B C=107^{\circ}
-$$","Substituting this in the formula T=\tfrac12 bh derived above, the area of the triangle can be expressed as: :T = \tfrac12 ab\sin \gamma = \tfrac12 bc\sin \alpha = \tfrac12 ca\sin \beta (where α is the interior angle at A, β is the interior angle at B, \gamma is the interior angle at C and c is the line AB). In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. thumb|The area of the pink triangle is one-seventh of the area of the large triangle ABC. The reader is advised that several of the formulas in this source are not correct. gave a collection of over a hundred distinct area formulas for the triangle. thumb|Triangle with the area 6, a congruent number. thumb|upright=1.5|\begin{align}&\text{all inner angles} < 120^\circ : \\\ &\text{grey area} = 3 \Delta \leq \Delta_a+\Delta_b+\Delta_c \end{align} thumb|upright=1.5|\begin{align}&\text{one inner angle} \geq 120^\circ : \\\ &\text{grey area} = 3 \Delta \leq \Delta_c < \Delta_a+\Delta_b+\Delta_c \end{align} In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt{3}\, \Delta. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). Then a, b and c are the legs and hypotenuse of a right triangle with area n. In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where :p connects A to a point on BC that is one-third the distance from B to C, :q connects B to a point on CA that is one-third the distance from C to A, :r connects C to a point on AB that is one-third the distance from A to B. :T = D^{2} \sqrt{S(S-\sin \alpha)(S-\sin \beta)(S-\sin \gamma)} where D is the diameter of the circumcircle: D=\tfrac{a}{\sin \alpha} = \tfrac{b}{\sin \beta} = \tfrac{c}{\sin \gamma}. ==Using vectors== The area of triangle ABC is half of the area of a parallelogram: :\tfrac12|\mathbf{AB}\times\mathbf{AC}|. The values (1201, 140, 1151, 1249) give (a, b, c) = (7/10, 120/7, 1201/70). 300px|thumb|A graphic derivation of the formula T=\frac{h}{2}b that avoids the usual procedure of doubling the area of the triangle and then halving it. The area can also be expressed asPathan, Alex, and Tony Collyer, ""Area properties of triangles revisited,"" Mathematical Gazette 89, November 2005, 495–497. By Heron's formula: :T = \sqrt{s(s-a)(s-b)(s-c)} where s= \tfrac12(a+b+c) is the semiperimeter, or half of the triangle's perimeter. This can now can be shown by replicating area of the triangle three times within the equilateral triangles. The area of a triangle then falls out as the case of a polygon with three sides. Three other area bisectors are parallel to the triangle's sides. The area of triangle ABC can also be expressed in terms of dot products as follows: :\tfrac12 \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\tfrac12 \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2}.\, In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as: :\tfrac12|x_1 y_2 - x_2 y_1|. ==Using coordinates== If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by and , then the area can be computed as times the absolute value of the determinant :T = \tfrac12\left|\det\begin{pmatrix}x_B & x_C \\\ y_B & y_C \end{pmatrix}\right| = \tfrac12 |x_B y_C - x_C y_B|. There can be one, two, or three of these for any given triangle. ==See also== *Area of a circle *Congruence of triangles ==References== Category:Area Category:Triangles The height of a triangle can be found through the application of trigonometry. The area of parallelogram ABDC is then :|\mathbf{AB}\times\mathbf{AC}|, which is the magnitude of the cross product of vectors AB and AC. However if one angle is greater or equal to 120^\circ it is possible to replicate the whole triangle three times within the largest equilateral triangle, so the sum of areas of all equilateral triangles stays greater than the threefold area of the triangle anyhow. == Further proofs == The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. ",-87.8,14.34457,1.6,-24,0.2115,B
+$$","Substituting this in the formula T=\tfrac12 bh derived above, the area of the triangle can be expressed as: :T = \tfrac12 ab\sin \gamma = \tfrac12 bc\sin \alpha = \tfrac12 ca\sin \beta (where α is the interior angle at A, β is the interior angle at B, \gamma is the interior angle at C and c is the line AB). In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. thumb|The area of the pink triangle is one-seventh of the area of the large triangle ABC. The reader is advised that several of the formulas in this source are not correct. gave a collection of over a hundred distinct area formulas for the triangle. thumb|Triangle with the area 6, a congruent number. thumb|upright=1.5|\begin{align}&\text{all inner angles} < 120^\circ : \\\ &\text{grey area} = 3 \Delta \leq \Delta_a+\Delta_b+\Delta_c \end{align} thumb|upright=1.5|\begin{align}&\text{one inner angle} \geq 120^\circ : \\\ &\text{grey area} = 3 \Delta \leq \Delta_c < \Delta_a+\Delta_b+\Delta_c \end{align} In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt{3}\, \Delta. thumb|upright=1.25|Polar circle (red) of a triangle ABC thumb|upright=1.25|polar circle (d), nine point circle (t), circumcircle (e), circumcircle of the tangential triangle (s) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is : \begin{align} r^2 & = HA\times HD=HB\times HE=HC\times HF \\\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac{1}{2}(a^2+b^2+c^2), \end{align} where A, B, C denote both the triangle's vertices and the angle measures at those vertices, H is the orthocenter (the intersection of the triangle's altitudes), D, E, F are the feet of the altitudes from vertices A, B, C respectively, R is the triangle's circumradius (the radius of its circumscribed circle), and a, b, c are the lengths of the triangle's sides opposite vertices A, B, C respectively.Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). Then a, b and c are the legs and hypotenuse of a right triangle with area n. In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where :p connects A to a point on BC that is one-third the distance from B to C, :q connects B to a point on CA that is one-third the distance from C to A, :r connects C to a point on AB that is one-third the distance from A to B. :T = D^{2} \sqrt{S(S-\sin \alpha)(S-\sin \beta)(S-\sin \gamma)} where D is the diameter of the circumcircle: D=\tfrac{a}{\sin \alpha} = \tfrac{b}{\sin \beta} = \tfrac{c}{\sin \gamma}. ==Using vectors== The area of triangle ABC is half of the area of a parallelogram: :\tfrac12|\mathbf{AB}\times\mathbf{AC}|. The values (1201, 140, 1151, 1249) give (a, b, c) = (7/10, 120/7, 1201/70). 300px|thumb|A graphic derivation of the formula T=\frac{h}{2}b that avoids the usual procedure of doubling the area of the triangle and then halving it. The area can also be expressed asPathan, Alex, and Tony Collyer, ""Area properties of triangles revisited,"" Mathematical Gazette 89, November 2005, 495–497. By Heron's formula: :T = \sqrt{s(s-a)(s-b)(s-c)} where s= \tfrac12(a+b+c) is the semiperimeter, or half of the triangle's perimeter. This can now can be shown by replicating area of the triangle three times within the equilateral triangles. The area of a triangle then falls out as the case of a polygon with three sides. Three other area bisectors are parallel to the triangle's sides. The area of triangle ABC can also be expressed in terms of dot products as follows: :\tfrac12 \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\tfrac12 \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2}.\, In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as: :\tfrac12|x_1 y_2 - x_2 y_1|. ==Using coordinates== If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by and , then the area can be computed as times the absolute value of the determinant :T = \tfrac12\left|\det\begin{pmatrix}x_B & x_C \\\ y_B & y_C \end{pmatrix}\right| = \tfrac12 |x_B y_C - x_C y_B|. There can be one, two, or three of these for any given triangle. ==See also== *Area of a circle *Congruence of triangles ==References== Category:Area Category:Triangles The height of a triangle can be found through the application of trigonometry. The area of parallelogram ABDC is then :|\mathbf{AB}\times\mathbf{AC}|, which is the magnitude of the cross product of vectors AB and AC. However if one angle is greater or equal to 120^\circ it is possible to replicate the whole triangle three times within the largest equilateral triangle, so the sum of areas of all equilateral triangles stays greater than the threefold area of the triangle anyhow. == Further proofs == The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. ",-87.8,14.34457,"""1.6""",-24,0.2115,B
 "Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.
-","The Kelvin–Stokes theorem: \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S abla \times \mathbf{F} \cdot \mathbf{\hat n} \, dS. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface \Sigma in Euclidean three-space to the line integral of the vector field over its boundary. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. More explicitly, the equality says that \begin{align} &\iint_\Sigma \left(\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \right)\,\mathrm{d}y\, \mathrm{d}z +\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\, \mathrm{d}z\, \mathrm{d}x +\left (\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\, \mathrm{d}x\, \mathrm{d}y\right) \\\ & = \oint_{\partial\Sigma} \Bigl(F_x\, \mathrm{d}x+F_y\, \mathrm{d}y+F_z\, \mathrm{d}z\Bigr). \end{align} The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. Start with the left side of Green's theorem: \oint_C (L\, dx + M\, dy) = \oint_C (L, M, 0) \cdot (dx, dy, dz) = \oint_C \mathbf{F} \cdot d\mathbf{r}. Stokes' theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:""Bi-Bun-Seki-Bun-Gaku"" Sho-Ka-Bou(jp) 1983/12 (Written in Japanese)Atsuo Fujimoto;""Vector-Kai-Seki Gendai su-gaku rekucha zu. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. This classical case relates the surface integral of the curl of a vector field \textbf{F} over a surface (that is, the flux of \text{curl}\,\textbf{F}) in Euclidean three- space to the line integral of the vector field over the surface boundary. == Introduction == The second fundamental theorem of calculus states that the integral of a function f over the interval [a,b] can be calculated by finding an antiderivative F of f: \int_a^b f(x)\,dx = F(b) - F(a)\,. The expression inside the integral becomes abla \times \mathbf{F} \cdot \mathbf{\hat n} = \left[ \left(\frac{\partial 0}{\partial y} - \frac{\partial M}{\partial z}\right) \mathbf{i} + \left(\frac{\partial L}{\partial z} - \frac{\partial 0}{\partial x}\right) \mathbf{j} + \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) \mathbf{k} \right] \cdot \mathbf{k} = \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right). Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: \oint_C L \,dx + M \,dy = \oint_{\partial D} \\! \omega = \int_D d\omega = \int_D \frac{\partial L}{\partial y} \,dy \wedge \,dx + \frac{\partial M}{\partial x} \,dx \wedge \,dy = \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \,dx \,dy. ==Relationship to the divergence theorem== Considering only two- dimensional vector fields, Green's theorem is equivalent to the two- dimensional version of the divergence theorem: :\iint_D\left( abla\cdot\mathbf{F}\right)dA=\oint_C \mathbf{F} \cdot \mathbf{\hat n} \, ds, where abla\cdot\mathbf{F} is the divergence on the two-dimensional vector field \mathbf{F}, and \mathbf{\hat n} is the outward- pointing unit normal vector on the boundary. Then \begin{align} 0 &= \int_\Omega \vec{ abla} \cdot \vec{c} f - \int_{\partial \Omega} \vec{n} \cdot \vec{c} f & \text{by the divergence theorem} \\\ &= \int_\Omega \vec{c} \cdot \vec{ abla} f - \int_{\partial \Omega} \vec{c} \cdot \vec{n} f \\\ &= \vec{c} \cdot \int_\Omega \vec{ abla} f - \vec{c} \cdot \int_{\partial \Omega} \vec{n} f \\\ &= \vec{c} \cdot \left( \int_\Omega \vec{ abla} f - \int_{\partial \Omega} \vec{n} f \right) \end{align} Since this holds for any \vec{c} (in particular, for every basis vector), the result follows. ==See also== *Chandrasekhar–Wentzel lemma ==Footnotes== ==References== ==Further reading== * * * * * * * * * * * ==External links== * * * Proof of the Divergence Theorem and Stokes' Theorem * Calculus 3 – Stokes Theorem from lamar.edu – an expository explanation Category:Differential topology Category:Differential forms Category:Duality theories Category:Integration on manifolds Category:Theorems in calculus Category:Theorems in differential geometry Stokes' theorem says that the integral of a differential form \omega over the boundary \partial\Omega of some orientable manifold \Omega is equal to the integral of its exterior derivative d\omega over the whole of \Omega, i.e., \int_{\partial \Omega} \omega = \int_\Omega d\omega\,. Stokes is a census-designated place in Pitt County, North Carolina, United States. Stokes' theorem is a special case of the generalized Stokes theorem. For Faraday's law, Stokes's theorem is applied to the electric field, \mathbf{E}: \oint_{\partial\Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{ abla}\times \mathbf{E} \cdot \mathrm{d} \mathbf{S} . Thus, by generalized Stokes's theorem, \oint_{\partial\Sigma}{\mathbf{F}\cdot\,\mathrm{d}\mathbf{\gamma}} =\oint_{\partial\Sigma}{\omega_{\mathbf{F}}} =\int_{\Sigma}{\mathrm{d}\omega_{\mathbf{F}}} =\int_{\Sigma}{\star\omega_{ abla\times\mathbf{F}}} =\iint_{\Sigma}{ abla\times\mathbf{F}\cdot\,\mathrm{d}\mathbf{\Sigma}} ==Applications== ===Irrotational fields=== In this section, we will discuss the irrotational field (lamellar vector field) based on Stokes's theorem. Stokes' theorem is a vast generalization of this theorem in the following sense. The circulation of a vector field around a closed curve is the line integral: \Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. ",0.05882352941,41,7.82,1.70,-0.5,E
+","The Kelvin–Stokes theorem: \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S abla \times \mathbf{F} \cdot \mathbf{\hat n} \, dS. thumb|right|An illustration of Stokes' theorem, with surface , its boundary and the normal vector . The classical Stokes' theorem relates the surface integral of the curl of a vector field over a surface \Sigma in Euclidean three-space to the line integral of the vector field over its boundary. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. More explicitly, the equality says that \begin{align} &\iint_\Sigma \left(\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \right)\,\mathrm{d}y\, \mathrm{d}z +\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)\, \mathrm{d}z\, \mathrm{d}x +\left (\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)\, \mathrm{d}x\, \mathrm{d}y\right) \\\ & = \oint_{\partial\Sigma} \Bigl(F_x\, \mathrm{d}x+F_y\, \mathrm{d}y+F_z\, \mathrm{d}z\Bigr). \end{align} The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. Start with the left side of Green's theorem: \oint_C (L\, dx + M\, dy) = \oint_C (L, M, 0) \cdot (dx, dy, dz) = \oint_C \mathbf{F} \cdot d\mathbf{r}. Stokes' theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:""Bi-Bun-Seki-Bun-Gaku"" Sho-Ka-Bou(jp) 1983/12 (Written in Japanese)Atsuo Fujimoto;""Vector-Kai-Seki Gendai su-gaku rekucha zu. By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter, \Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S abla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S} Here, the closed integration path is the boundary or perimeter of an open surface , whose infinitesimal element normal is oriented according to the right-hand rule. This classical case relates the surface integral of the curl of a vector field \textbf{F} over a surface (that is, the flux of \text{curl}\,\textbf{F}) in Euclidean three- space to the line integral of the vector field over the surface boundary. == Introduction == The second fundamental theorem of calculus states that the integral of a function f over the interval [a,b] can be calculated by finding an antiderivative F of f: \int_a^b f(x)\,dx = F(b) - F(a)\,. The expression inside the integral becomes abla \times \mathbf{F} \cdot \mathbf{\hat n} = \left[ \left(\frac{\partial 0}{\partial y} - \frac{\partial M}{\partial z}\right) \mathbf{i} + \left(\frac{\partial L}{\partial z} - \frac{\partial 0}{\partial x}\right) \mathbf{j} + \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right) \mathbf{k} \right] \cdot \mathbf{k} = \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y}\right). Green's theorem is also a straightforward result of the general Stokes' theorem using differential forms and exterior derivatives: \oint_C L \,dx + M \,dy = \oint_{\partial D} \\! \omega = \int_D d\omega = \int_D \frac{\partial L}{\partial y} \,dy \wedge \,dx + \frac{\partial M}{\partial x} \,dx \wedge \,dy = \iint_D \left(\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) \,dx \,dy. ==Relationship to the divergence theorem== Considering only two- dimensional vector fields, Green's theorem is equivalent to the two- dimensional version of the divergence theorem: :\iint_D\left( abla\cdot\mathbf{F}\right)dA=\oint_C \mathbf{F} \cdot \mathbf{\hat n} \, ds, where abla\cdot\mathbf{F} is the divergence on the two-dimensional vector field \mathbf{F}, and \mathbf{\hat n} is the outward- pointing unit normal vector on the boundary. Then \begin{align} 0 &= \int_\Omega \vec{ abla} \cdot \vec{c} f - \int_{\partial \Omega} \vec{n} \cdot \vec{c} f & \text{by the divergence theorem} \\\ &= \int_\Omega \vec{c} \cdot \vec{ abla} f - \int_{\partial \Omega} \vec{c} \cdot \vec{n} f \\\ &= \vec{c} \cdot \int_\Omega \vec{ abla} f - \vec{c} \cdot \int_{\partial \Omega} \vec{n} f \\\ &= \vec{c} \cdot \left( \int_\Omega \vec{ abla} f - \int_{\partial \Omega} \vec{n} f \right) \end{align} Since this holds for any \vec{c} (in particular, for every basis vector), the result follows. ==See also== *Chandrasekhar–Wentzel lemma ==Footnotes== ==References== ==Further reading== * * * * * * * * * * * ==External links== * * * Proof of the Divergence Theorem and Stokes' Theorem * Calculus 3 – Stokes Theorem from lamar.edu – an expository explanation Category:Differential topology Category:Differential forms Category:Duality theories Category:Integration on manifolds Category:Theorems in calculus Category:Theorems in differential geometry Stokes' theorem says that the integral of a differential form \omega over the boundary \partial\Omega of some orientable manifold \Omega is equal to the integral of its exterior derivative d\omega over the whole of \Omega, i.e., \int_{\partial \Omega} \omega = \int_\Omega d\omega\,. Stokes is a census-designated place in Pitt County, North Carolina, United States. Stokes' theorem is a special case of the generalized Stokes theorem. For Faraday's law, Stokes's theorem is applied to the electric field, \mathbf{E}: \oint_{\partial\Sigma} \mathbf{E} \cdot \mathrm{d}\boldsymbol{l}= \iint_\Sigma \mathbf{ abla}\times \mathbf{E} \cdot \mathrm{d} \mathbf{S} . Thus, by generalized Stokes's theorem, \oint_{\partial\Sigma}{\mathbf{F}\cdot\,\mathrm{d}\mathbf{\gamma}} =\oint_{\partial\Sigma}{\omega_{\mathbf{F}}} =\int_{\Sigma}{\mathrm{d}\omega_{\mathbf{F}}} =\int_{\Sigma}{\star\omega_{ abla\times\mathbf{F}}} =\iint_{\Sigma}{ abla\times\mathbf{F}\cdot\,\mathrm{d}\mathbf{\Sigma}} ==Applications== ===Irrotational fields=== In this section, we will discuss the irrotational field (lamellar vector field) based on Stokes's theorem. Stokes' theorem is a vast generalization of this theorem in the following sense. The circulation of a vector field around a closed curve is the line integral: \Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}. With the above notation, if \mathbf{F} is any smooth vector field on \R^3, thenRobert Scheichl, lecture notes for University of Bath mathematics course \oint_{\partial\Sigma} \mathbf{F}\, \cdot\, \mathrm{d}{\mathbf{\Gamma}} = \iint_{\Sigma} abla\times\mathbf{F}\, \cdot\, \mathrm{d}\mathbf{\Sigma}. ",0.05882352941,41,"""7.82""",1.70,-0.5,E
 "A hawk flying at $15 \mathrm{~m} / \mathrm{s}$ at an altitude of $180 \mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation
 $$
 y=180-\frac{x^2}{45}
 $$
-until it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.","The short-tailed hawk hunts from a soaring flight, often at the borders between wooded and open areas. While following this path, the aircraft and its payload are in free fall at certain points of its flight path. The clutch of one or two eggs is incubated for around 37 days, beginning after the first egg is laid. ===Food and feeding=== The roadside hawk's diet consists mainly of insects, squamates, and small mammals, such as young common marmosets and similar small monkeys which are hunted quite often. The roadside hawk is the smallest hawk in the widespread genus Buteo; although Ridgway's hawk and the white-rumped hawk are scarcely larger. The short-tailed hawk (Buteo brachyurus) is an American bird of prey in the family Accipitridae, which also includes the eagles and Old World vultures. When approached on the nest, the adults will get airborne and observe the intruder from above, unlike related hawks, which usually wait much longer to flush and then launch a determined attack. ==Footnotes== ==References== * eNature (2007): White-tailed Hawk. In one case, 95% of a single hawk's prey selection was found to consist of red-winged blackbirds. The semiplumbeous hawk (Leucopternis semiplumbeus) is a species of bird of prey in the family Accipitridae. The white-tailed hawk (Geranoaetus albicaudatus) is a large bird of prey species found in tropical and subtropical environments of the Americas. ==Description== The white-tailed hawk is a large, stocky hawk. It has been reported up to 1600 m in altitude in one instance. == Population and research == Semiplumbeous hawks is currently categorized as ""Least Concern"" in terms of global threat level and is listed under CITES II, but was previously classified as ""Near Threatened."" Semiplumbeous Hawk Leucopternis semiplumbeus at Birdlife.org. In flight, the relatively long tail and disproportionately short wings of the roadside hawk are distinctive. The diet of the white-tailed hawk varies with its environment. It frequently soars, but does not hover. ==Distribution and habitat== The roadside hawk is common throughout its range: from Mexico through Central America to most of South America east of the Andes Cordillera. Like many Accipitridae, white- tailed hawks do not like to abandon a nest site, and nests built up over the years can thus reach sizes of up to three feet (1 m) across. ""The dynamics of parabolic flight: flight characteristics and passenger percepts"". griseocauda eating speckled racer, Belize The roadside hawk (Rupornis magnirostris) is a relatively small bird of prey found in the Americas. PDF fulltext * Hawk Conservancy Trust (HCT) (2008): White-tailed Hawk - Buteo albicaudatus. Breeding pairs of white- tailed hawks build nests out of freshly broken twigs, often of thorny plants, 5–15 ft (1.5–5 m) or more above the ground on top of a tree or yucca, preferably one growing in an elevated location giving good visibility from the nest. There are isolated records of short-tails preying on sharp-shinned hawks (Accipiter striatus) and American kestrels (Falco sparverius). Its natural habitat is subtropical or tropical moist lowland forests. ==Morphology== The semiplumbeous hawk is a small bird, averaging about in lengthHenderson, Carrol L. ""Birds of Costa Rica."" Space Hawk is a multidirectional shooter released by Mattel for its Intellivision console in 1982. ",209.1,0.6957,0.44,35,+0.60,A
-"The intensity of light with wavelength $\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\theta$ is given by $I(\theta)=N^2 \sin ^2 k / k^2$, where $k=(\pi N d \sin \theta) / \lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\lambda=632.8 \times 10^{-9} \mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ emerging from the grating.","Defining the relative intensity I_\text{rel}as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation: I_\text{rel}(w) = J_0^2\left(\frac{w R \pi}{g \lambda}\right) + J_1^2\left(\frac{w R \pi}{g \lambda}\right) where J_0and J_1are the Bessel functions of the first kind. It is worth to notice that the parameter N depends on the excitation wavelength. == Impact of ambient pressure == The atmospheric pressure strongly influences the LIWE intensity. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. The dependence of intensity on power is described by the formula: , where N is the number of near infrared photons absorbed for LIWE generation. So, the diffraction formula becomes U(P) = -\frac{i}{2\lambda} \frac{ae^{ikr_0}}{r_0} \int_{S} \frac{e^{iks}}{s} (1 + \cos\chi) dS, where the integral is done over the part of the wavefront at r0 which is the closest to the aperture in the diagram. If, however, we assume that the light from the source at each point in the aperture has a well-defined direction, which is the case if the distance between the source and the aperture is significantly greater than the wavelength, then we can write U_0(r) \approx a(r) e^{ikr}, where a(r) is the magnitude of the disturbance at the point r in the aperture. In order to derive the intensity behind the circular obstacle using this integral one assumes that the experimental parameters fulfill the requirements of the near-field diffraction regime (the size of the circular obstacle is large compared to the wavelength and small compared to the distances g = P0C and b = CP1). thumb|Photo of the Arago spot in a shadow of a 5.8 mm circular obstacle thumb|700px|Numerical simulation of the intensity of monochromatic light of wavelength λ = 0.5 µm behind a circular obstacle of radius . thumb|Formation of the Arago spot (select ""WebM source"" for good quality) thumb|Arago spot forming in the shadow In optics, the Arago spot, Poisson spot, or Fresnel spot""Although this phenomenon is often called Poisson's spot, Poisson probably was not happy to have seen it because it supported the wave model of light. Kirchhoff's diffraction formula (also called Fresnel–Kirchhoff diffraction formula) approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The lateral intensity distribution on the screen has in fact the shape of a squared zeroth Bessel function of the first kind when close to the optical axis and using a plane wave source (point source at infinity): U(P_1, r) \propto J_0^2 \left(\frac{\pi r d}{\lambda b}\right) where * r is the distance of the point P1 on the screen from the optical axis * d is the diameter of circular object * λ is the wavelength * b is the distance between circular object and screen. This is not the case, and this is _one of the approximations_ used in deriving the Kirchhoff's diffraction formula.J.Z. Buchwald & C.-P. Yeang, ""Kirchhoff's theory for optical diffraction, its predecessor and subsequent development: the resilience of an inconsistent theory"" , Archive for History of Exact Sciences, vol.70, no.5 (Sep.2016), pp.463–511; .J. Saatsi & P. Vickers, ""Miraculous success? The following images show the radial intensity distribution of the simulated Arago spot images above: 200px 200px 200px The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above. ===Finite source size and spatial coherence=== The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. The source intensity, which is the square of the field amplitude, is I_0 = \left|\frac{1}{g} A e^{\mathbf{i} k g}\right|^2 and the intensity at the screen I = \left| U(P_1) \right|^2. Generally when light of a certain wavelength falls on a subwavelength aperture, it is diffracted isotropically in all directions evenly, with minimal far-field transmission. Each image is 16 mm wide. ==Experimental aspects== ===Intensity and size=== For an ideal point source, the intensity of the Arago spot equals that of the undisturbed wave front. The on-axis intensity as a function of the distance b is hence given by: I = \frac{b^2}{b^2 + a^2} I_0. The white light emission intensity is exponentially dependent on excitation power density and pressure surrounding the samples. Laser-induced white emission (LIWE) is a broadband light in the visible spectral range. Going to polar coordinates then yields the integral for a circular object of radius a (see for example Born and Wolf): U(P_1) = - \frac{\mathbf{i}}{\lambda} \frac{A e^{\mathbf{i} k (g + b)}}{g b} 2\pi \int_a^\infty e^{\mathbf{i} k \frac{1}{2} \left(\frac{1}{g} + \frac{1}{b}\right) r^2} r \, dr. 300px|thumb|right|The on- axis intensity at the center of the shadow of a small circular obstacle converges to the unobstructed intensity. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. The adjacent Fresnel zone is approximately given by: \Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g + b}} - r. However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity. ",59.4,1.19,3.51,0.2307692308,24,A
-"A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \%$, use differentials to estimate the maximum percentage error in the calculated surface area.","It is [4W (kg) + 7]/[90 + W (kg)].Costeff H, ""A simple empirical formula for calculating approximate surface area in children.,"" The Growth of the Surface Area of the Human Body. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. In physiology and medicine, the body surface area (BSA) is the measured or calculated surface area of a human body. The surface area of a solid object is a measure of the total area that the surface of the object occupies. Thus, the surface area falls off steeply with increasing volume. == See also == * Perimeter length * Projected area * BET theory, technique for the measurement of the specific surface area of materials * Spherical area * Surface integral == References == * ==External links== *Surface Area Video at Thinkwell Category:Area The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. For different applications a minimal or maximal surface area may be desired. == In biology == The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion. The Sail Area-Displacement ratio (SA/D) is a calculation used to express how much sail a boat carries relative to its weight. :\mathit{SA/D} = \frac{\mathit{Sail Area}(\text{ft}^2)} {[\mathit{Displacement}(\text{lb})/64]^{\frac{2}{3}}} = \frac{\mathit{Sail Area}(\text{m}^2)} {\mathit{Displacement}(\text{m}^3)^{\frac{2}{3}}} In the first equation, the denominator in pounds is divided by 64 to convert it to cubic feet (because 1 cubic foot of salt water weights 64 pounds). The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. Surface area is important in chemical kinetics. Increased surface area can also lead to biological problems. thumb|300px|Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. The resulting surface area to volume ratio is therefore . thumb|upright=1.3|Measurement of volume by displacement, (a) before and (b) after an object has been submerged. An increased surface area to volume ratio also means increased exposure to the environment. Thus the area of SD is obtained by integrating the length of the normal vector \vec{r}_u\times\vec{r}_v to the surface over the appropriate region D in the parametric uv plane. Let the radius be r and the height be h (which is 2r for the sphere). \begin{array}{rlll} \text{Sphere surface area} & = 4 \pi r^2 & & = (2 \pi r^2) \times 2 \\\ \text{Cylinder surface area} & = 2 \pi r (h + r) & = 2 \pi r (2r + r) & = (2 \pi r^2) \times 3 \end{array} The discovery of this ratio is credited to Archimedes. == In chemistry == thumb|Surface area of particles of different sizes. ",-501,9.30,0.0384,22.2036033112,2.3,E
-"The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \leqslant x \leqslant 4$","Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. Thus the rate of heat flow into V is also given by the surface integral q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS where n(x) is the outward pointing normal vector at x. Rate of heat flow = - (heat transfer coefficient) * (area of the body) * (variation of the temperature) / (length of the material) The formula for the rate of heat flow is: :\frac{Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x} where * Q is the net heat (energy) transfer, * \Delta t is the time taken, * \Delta T is the difference in temperature between the cold and hot sides, * \Delta x is the thickness of the material conducting heat (distance between hot and cold sides), * k is the thermal conductivity, and * A is the surface area of the surface emitting heat. Consider a fluid of uniform temperature T_o and velocity u_o impinging onto a stationary plate uniformly heated to a temperature T_s. This equation is also called the Churchill–Bernstein correlation. ==Heat transfer definition== :\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 where: * \overline{\mathrm{Nu}}_D is the surface averaged Nusselt number with characteristic length of diameter; * \mathrm{Re}_D\,\\! is the Reynolds number with the cylinder diameter as its characteristic length; * \Pr is the Prandtl number. This form is more general and particularly useful to recognize which property (e.g. cp or \rho) influences which term. :\rho c_p \frac{\partial T}{\partial t} - abla \cdot \left( k abla T \right) = \dot q_V where \dot q_V is the volumetric heat source. ===Three-dimensional problem=== In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is : \frac{\partial u}{\partial t} = \alpha abla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) where: * u = u(x, y, z, t) is temperature as a function of space and time; * \tfrac{\partial u}{\partial t} is the rate of change of temperature at a point over time; * u_{xx} , u_{yy} , and u_{zz} are the second spatial derivatives (thermal conductions) of temperature in the x , y , and z directions, respectively; * \alpha \equiv \tfrac{k}{c_p\rho} is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k , the specific heat capacity c_p , and the mass density \rho . One then says that is a solution of the heat equation if :\frac{\partial u}{\partial t} = \alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) in which is a positive coefficient called the thermal diffusivity of the medium. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The equation yields the surface averaged Nusselt number, which is used to determine the average convective heat transfer coefficient. Steady-state condition: :\frac{\partial u}{\partial t} = 0 The steady-state heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poisson's equation: :-k abla^2 u = q where u is the temperature, k is the thermal conductivity and q is the rate of heat generation per unit volume. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The general solution of the heat equation on Rn is then obtained by a convolution, so that to solve the initial value problem with u(x, 0) = g(x), one has :u(\mathbf{x},t) = \int_{\R^n}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y})d\mathbf{y}. That is, :\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x} From the above equations it follows that :\frac{\partial u}{\partial t} \;=\; - \frac{1}{c \rho} \frac{\partial q}{\partial x} \;=\; - \frac{1}{c \rho} \frac{\partial}{\partial x} \left(-k \,\frac{\partial u}{\partial x} \right) \;=\; \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2} which is the heat equation in one dimension, with diffusivity coefficient :\alpha = \frac{k}{c\rho} This quantity is called the thermal diffusivity of the medium. ====Accounting for radiative loss==== An additional term may be introduced into the equation to account for radiative loss of heat. In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, \partial Q/\partial t, is proportional to the rate of change of its temperature, \partial u/\partial t. One further variation is that some of these solve the inhomogeneous equation :u_{t}=ku_{xx}+f. where f is some given function of x and t. ==== Homogeneous heat equation ==== ;Initial value problem on (−∞,∞) :\begin{cases} u_{t}=ku_{xx} & (x, t) \in \R \times (0, \infty) \\\ u(x,0)=g(x) & \text{Initial condition} \end{cases} :u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_{-\infty}^{\infty} \exp\left(-\frac{(x-y)^2}{4kt}\right)g(y)\,dy right|thumb|upright=2|Fundamental solution of the one-dimensional heat equation. * The time rate of heat flow into a region V is given by a time- dependent quantity qt(V). If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. The rate of heat flow is the amount of heat that is transferred per unit of time in some material, usually measured in watt (joules per second). * By the divergence theorem, the previous surface integral for heat flow into V can be transformed into the volume integral \begin{align} q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\\ &= \int_{\partial V} \mathbf{A}(x) \cdot abla u (x) \cdot \mathbf{n}(x) \, dS \\\ &= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx \end{align} * The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ \partial_t u(x,t) = \kappa(x) Q(x,t) Putting these equations together gives the general equation of heat flow: : \partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr) Remarks. By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it: :\mathbf{q} = - k \, abla u where k is the thermal conductivity of the material, u=u(\mathbf{x},t) is the temperature, and \mathbf{q} = \mathbf{q}(\mathbf{x},t) is a vector field that represents the magnitude and direction of the heat flow at the point \mathbf{x} of space and time t. The temperature at the solid wall is T_s and gradually changes to T_o as one moves toward the free stream of the fluid. ",3920.70763168,14.34457,22.0,0.2553,1.92,A
-"If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.","Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s). By combining this equation with the suvat equation , it is possible to relate the displacement and the average velocity by \boldsymbol{x} = \frac{(\boldsymbol{u} + \boldsymbol{v})}{2} t = \boldsymbol{\bar{v}}t. Velocity is the speed and the direction of motion of an object. From there, we can obtain an expression for velocity as the area under an acceleration vs. time graph. Average velocity can be calculated as: \boldsymbol{\bar{v}} = \frac{\Delta\boldsymbol{x}}{\Delta t} . Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. thumb|Forty Foot changing rooms and clubhouse kitchen, 2008 thumb|Sunrise at the Forty Foot, 2018 The Forty Foot () is a promontory on the southern tip of Dublin Bay at Sandycove, County Dublin, Ireland, from which people have been swimming in the Irish Sea all year round for some 250 years.as of 2008 * * ==Name== The name ""Forty Foot"" is somewhat obscure. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows: v^{2} = \boldsymbol{v}\cdot\boldsymbol{v} = (\boldsymbol{u}+\boldsymbol{a}t) \cdot (\boldsymbol{u}+\boldsymbol{a}t) = u^{2} + 2t(\boldsymbol{a}\cdot\boldsymbol{u})+a^{2}t^{2} (2\boldsymbol{a})\cdot\boldsymbol{x} = (2\boldsymbol{a})\cdot(\boldsymbol{u}t + \tfrac{1}{2} \boldsymbol{a} t^2) = 2t (\boldsymbol{a} \cdot \boldsymbol{u}) + a^2 t^2 = v^{2} - u^{2} \therefore v^2 = u^2 + 2(\boldsymbol{a}\cdot\boldsymbol{x}) where etc. In calculus terms, the integral of the velocity function is the displacement function . In other words, only relative velocity can be calculated. ===Quantities that are dependent on velocity=== The kinetic energy of a moving object is dependent on its velocity and is given by the equation E_{\text{k}} = \tfrac{1}{2} m v^2 ignoring special relativity, where Ek is the kinetic energy and m is the mass. 20 Hrs. 40 Min.: The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes ""escape velocity"" somewhat of a misnomer, as the more correct term would be ""escape speed"": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path. ==Relative velocity== Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. In terms of a displacement-time ( vs. ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point, and the average velocity as the slope of the secant line between two points with coordinates equal to the boundaries of the time period for the average velocity. Hence, the car is considered to be undergoing an acceleration. ==Difference between speed and velocity== thumb|300px|Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. The average velocity is always less than or equal to the average speed of an object. Rocket Racing Composite Corp. Acquires Velocity Aircraft, Parabolic Arc, 2008-04-14, accessed 2010-12-05. Velocity Aircraft Receives Purchase Order For 20 Velocity XL-5's, Space Fellowship, 2008-10-08, accessed 2010-12-11. As above, this is done using the concept of the integral: \boldsymbol{v} = \int \boldsymbol{a} \ dt . ====Constant acceleration==== In the special case of constant acceleration, velocity can be studied using the suvat equations. Escape velocity is the minimum speed a ballistic object needs to escape from a massive body such as Earth. Velocity is a physical vector quantity; both magnitude and direction are needed to define it. ",-24,0.63,0.0,144, 672.4,A
-A woman walks due west on the deck of a ship at $3 \mathrm{mi} / \mathrm{h}$. The ship is moving north at a speed of $22 \mathrm{mi} / \mathrm{h}$. Find the speed of the woman relative to the surface of the water.,"This is often called the hull speed and is a function of the length of the ship V=k\sqrt{L} where constant (k) should be taken as: 2.43 for velocity (V) in kn and length (L) in metres (m) or, 1.34 for velocity (V) in kn and length (L) in feet (ft). right|thumb|upright=1.3|Load lines, by showing how low a ship is sitting in the water, make it possible to determine displacement. A ship must be designed to move efficiently through the water with a minimum of external force. thumb|A close up quartering view of Grey Lady at dock MV Grey Lady is a high speed catamaran ferry operated by Hy-Line Cruises that travels on a route between Hyannis and Nantucket. Denise ""Dee"" Caffari MBE (born 23 January 1973) is a British sailor, and in 2006 became the first woman to sail single-handedly and non-stop around the world ""the wrong way""; westward against the prevailing winds and currents. The relationship between the velocity of ships and that of the transverse waves can be found by equating the wave celerity and the ship's velocity. ==Propulsion== (Main article: Marine propulsion) Ships can be propelled by numerous sources of power: human, animal, or wind power (sails, kites, rotors and turbines), water currents, chemical or atomic fuels and stored electricity, pressure, heat or solar power supplying engines and motors. ==Gallery== Image:Archimedes principle.svg|A floating ship's displacement Fp and buoyancy Fa must be equal. Seawater (1,025 kg/m3) is more dense than fresh water (1,000 kg/m3);Turpin and McEwen, 1980. so a ship will ride higher in salt water than in fresh. The ship's hydrostatic tables show the corresponding volume displaced.George, 2005. p. 465. For a displacement vessel, that is the usual type of ship, three main types of resistance are considered: that due to wave-making, that due to the pressure of the moving water on the form, often not calculated or measured separately, and that due to friction of moving water on the wetted surface of the hull. As the term indicates, it is measured indirectly, using Archimedes' principle, by first calculating the volume of water displaced by the ship, then converting that value into weight. Ship's boats have always provided transport between the shore and other ships. The displacement or displacement tonnage of a ship is its weight. Grunnslep fra 1932.jpg|Principle of sea surveying with two boats, Norwegian Sea Survey, 1932. A ship's boat is a utility boat carried by a larger vessel. These waves were first studied by William Thomson, 1st Baron Kelvin, who found that regardless of the speed of the ship, they were always contained within the 39° wedge shape (19.5° on each side) following the ship. Froude had observed that when a ship or model was at its so-called Hull speed the wave pattern of the transverse waves (the waves along the hull) have a wavelength equal to the length of the waterline. The total (upward) force due to this buoyancy is equal to the (downward) weight of the displaced water. thumb|A ship for updating nautical charts. thumb|300px|HMS Thetis aground. The divergent waves do not cause much resistance against the ship's forward motion. The boundary layer undergoes shear at different rates extending from the hull surface until it reaches the field flow of the water. ==Wave-making resistance== (Main article: Wave-making resistance) A ship moving over the surface of undisturbed water sets up waves emanating mainly from the bow and stern of the ship. ",0.18,22.2036033112,7.0, 10.7598,92,B
-A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. ,"If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Therefore, the distance in feet down a 6% grade to reach the velocity is at least s = \frac{\Delta z}{0.06}= 8.3\frac{V^2}{g},\quad\text{or}\quad s=8.3\frac{88^2}{32.2}\approx 2000\mathrm{ft}. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Step 3) Obtain net work (a single number) by numerical integration of muscle power data. Therefore, work need only be computed for the gravitational forces acting on the bodies. The work of the net force is calculated as the product of its magnitude and the particle displacement. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). In physics, work is the energy transferred to or from an object via the application of force along a displacement. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as ""force times straight path segment"" would only apply in the most simple of circumstances, as noted above. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. Notice that this formula uses the fact that the mass of the vehicle is . thumb|Lotus type 119B gravity racer at Lotus 60th celebration thumb|Gravity racing championship in Campos Novos, Santa Catarina, Brazil, 8 September 2010 ===Coasting down an inclined surface (gravity racing)=== Consider the case of a vehicle that starts at rest and coasts down an inclined surface (such as mountain road), the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity , of say 60 mph (88 fps). ",0,7200,0.95,71,0,B
+until it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.","The short-tailed hawk hunts from a soaring flight, often at the borders between wooded and open areas. While following this path, the aircraft and its payload are in free fall at certain points of its flight path. The clutch of one or two eggs is incubated for around 37 days, beginning after the first egg is laid. ===Food and feeding=== The roadside hawk's diet consists mainly of insects, squamates, and small mammals, such as young common marmosets and similar small monkeys which are hunted quite often. The roadside hawk is the smallest hawk in the widespread genus Buteo; although Ridgway's hawk and the white-rumped hawk are scarcely larger. The short-tailed hawk (Buteo brachyurus) is an American bird of prey in the family Accipitridae, which also includes the eagles and Old World vultures. When approached on the nest, the adults will get airborne and observe the intruder from above, unlike related hawks, which usually wait much longer to flush and then launch a determined attack. ==Footnotes== ==References== * eNature (2007): White-tailed Hawk. In one case, 95% of a single hawk's prey selection was found to consist of red-winged blackbirds. The semiplumbeous hawk (Leucopternis semiplumbeus) is a species of bird of prey in the family Accipitridae. The white-tailed hawk (Geranoaetus albicaudatus) is a large bird of prey species found in tropical and subtropical environments of the Americas. ==Description== The white-tailed hawk is a large, stocky hawk. It has been reported up to 1600 m in altitude in one instance. == Population and research == Semiplumbeous hawks is currently categorized as ""Least Concern"" in terms of global threat level and is listed under CITES II, but was previously classified as ""Near Threatened."" Semiplumbeous Hawk Leucopternis semiplumbeus at Birdlife.org. In flight, the relatively long tail and disproportionately short wings of the roadside hawk are distinctive. The diet of the white-tailed hawk varies with its environment. It frequently soars, but does not hover. ==Distribution and habitat== The roadside hawk is common throughout its range: from Mexico through Central America to most of South America east of the Andes Cordillera. Like many Accipitridae, white- tailed hawks do not like to abandon a nest site, and nests built up over the years can thus reach sizes of up to three feet (1 m) across. ""The dynamics of parabolic flight: flight characteristics and passenger percepts"". griseocauda eating speckled racer, Belize The roadside hawk (Rupornis magnirostris) is a relatively small bird of prey found in the Americas. PDF fulltext * Hawk Conservancy Trust (HCT) (2008): White-tailed Hawk - Buteo albicaudatus. Breeding pairs of white- tailed hawks build nests out of freshly broken twigs, often of thorny plants, 5–15 ft (1.5–5 m) or more above the ground on top of a tree or yucca, preferably one growing in an elevated location giving good visibility from the nest. There are isolated records of short-tails preying on sharp-shinned hawks (Accipiter striatus) and American kestrels (Falco sparverius). Its natural habitat is subtropical or tropical moist lowland forests. ==Morphology== The semiplumbeous hawk is a small bird, averaging about in lengthHenderson, Carrol L. ""Birds of Costa Rica."" Space Hawk is a multidirectional shooter released by Mattel for its Intellivision console in 1982. ",209.1,0.6957,"""0.44""",35,+0.60,A
+"The intensity of light with wavelength $\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\theta$ is given by $I(\theta)=N^2 \sin ^2 k / k^2$, where $k=(\pi N d \sin \theta) / \lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\lambda=632.8 \times 10^{-9} \mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ emerging from the grating.","Defining the relative intensity I_\text{rel}as the intensity divided by the intensity of the undisturbed wavefront, the relative intensity for an extended circular source of diameter w can be expressed exactly using the following equation: I_\text{rel}(w) = J_0^2\left(\frac{w R \pi}{g \lambda}\right) + J_1^2\left(\frac{w R \pi}{g \lambda}\right) where J_0and J_1are the Bessel functions of the first kind. It is worth to notice that the parameter N depends on the excitation wavelength. == Impact of ambient pressure == The atmospheric pressure strongly influences the LIWE intensity. Only the width of the Arago spot intensity peak depends on the distances between source, circular object and screen, as well as the source's wavelength and the diameter of the circular object. The dependence of intensity on power is described by the formula: , where N is the number of near infrared photons absorbed for LIWE generation. So, the diffraction formula becomes U(P) = -\frac{i}{2\lambda} \frac{ae^{ikr_0}}{r_0} \int_{S} \frac{e^{iks}}{s} (1 + \cos\chi) dS, where the integral is done over the part of the wavefront at r0 which is the closest to the aperture in the diagram. If, however, we assume that the light from the source at each point in the aperture has a well-defined direction, which is the case if the distance between the source and the aperture is significantly greater than the wavelength, then we can write U_0(r) \approx a(r) e^{ikr}, where a(r) is the magnitude of the disturbance at the point r in the aperture. In order to derive the intensity behind the circular obstacle using this integral one assumes that the experimental parameters fulfill the requirements of the near-field diffraction regime (the size of the circular obstacle is large compared to the wavelength and small compared to the distances g = P0C and b = CP1). thumb|Photo of the Arago spot in a shadow of a 5.8 mm circular obstacle thumb|700px|Numerical simulation of the intensity of monochromatic light of wavelength λ = 0.5 µm behind a circular obstacle of radius . thumb|Formation of the Arago spot (select ""WebM source"" for good quality) thumb|Arago spot forming in the shadow In optics, the Arago spot, Poisson spot, or Fresnel spot""Although this phenomenon is often called Poisson's spot, Poisson probably was not happy to have seen it because it supported the wave model of light. Kirchhoff's diffraction formula (also called Fresnel–Kirchhoff diffraction formula) approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The lateral intensity distribution on the screen has in fact the shape of a squared zeroth Bessel function of the first kind when close to the optical axis and using a plane wave source (point source at infinity): U(P_1, r) \propto J_0^2 \left(\frac{\pi r d}{\lambda b}\right) where * r is the distance of the point P1 on the screen from the optical axis * d is the diameter of circular object * λ is the wavelength * b is the distance between circular object and screen. This is not the case, and this is _one of the approximations_ used in deriving the Kirchhoff's diffraction formula.J.Z. Buchwald & C.-P. Yeang, ""Kirchhoff's theory for optical diffraction, its predecessor and subsequent development: the resilience of an inconsistent theory"" , Archive for History of Exact Sciences, vol.70, no.5 (Sep.2016), pp.463–511; .J. Saatsi & P. Vickers, ""Miraculous success? The following images show the radial intensity distribution of the simulated Arago spot images above: 200px 200px 200px The red lines in these three graphs correspond to the simulated images above, and the green lines were computed by applying the corresponding parameters to the squared Bessel function given above. ===Finite source size and spatial coherence=== The main reason why the Arago spot is hard to observe in circular shadows from conventional light sources is that such light sources are bad approximations of point sources. The source intensity, which is the square of the field amplitude, is I_0 = \left|\frac{1}{g} A e^{\mathbf{i} k g}\right|^2 and the intensity at the screen I = \left| U(P_1) \right|^2. Generally when light of a certain wavelength falls on a subwavelength aperture, it is diffracted isotropically in all directions evenly, with minimal far-field transmission. Each image is 16 mm wide. ==Experimental aspects== ===Intensity and size=== For an ideal point source, the intensity of the Arago spot equals that of the undisturbed wave front. The on-axis intensity as a function of the distance b is hence given by: I = \frac{b^2}{b^2 + a^2} I_0. The white light emission intensity is exponentially dependent on excitation power density and pressure surrounding the samples. Laser-induced white emission (LIWE) is a broadband light in the visible spectral range. Going to polar coordinates then yields the integral for a circular object of radius a (see for example Born and Wolf): U(P_1) = - \frac{\mathbf{i}}{\lambda} \frac{A e^{\mathbf{i} k (g + b)}}{g b} 2\pi \int_a^\infty e^{\mathbf{i} k \frac{1}{2} \left(\frac{1}{g} + \frac{1}{b}\right) r^2} r \, dr. 300px|thumb|right|The on- axis intensity at the center of the shadow of a small circular obstacle converges to the unobstructed intensity. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. The adjacent Fresnel zone is approximately given by: \Delta r \approx \sqrt{r^2 + \lambda \frac{g b}{g + b}} - r. However, if random edge corrugation have amplitude comparable to or greater than the width of that adjacent Fresnel zone, the contributions from radial segments are no longer in phase and cancel each other reducing the Arago spot intensity. ",59.4,1.19,"""3.51""",0.2307692308,24,A
+"A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \%$, use differentials to estimate the maximum percentage error in the calculated surface area.","It is [4W (kg) + 7]/[90 + W (kg)].Costeff H, ""A simple empirical formula for calculating approximate surface area in children.,"" The Growth of the Surface Area of the Human Body. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. In physiology and medicine, the body surface area (BSA) is the measured or calculated surface area of a human body. The surface area of a solid object is a measure of the total area that the surface of the object occupies. Thus, the surface area falls off steeply with increasing volume. == See also == * Perimeter length * Projected area * BET theory, technique for the measurement of the specific surface area of materials * Spherical area * Surface integral == References == * ==External links== *Surface Area Video at Thinkwell Category:Area The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. One of the subtleties of surface area, as compared to arc length of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. For different applications a minimal or maximal surface area may be desired. == In biology == The surface area of an organism is important in several considerations, such as regulation of body temperature and digestion. The Sail Area-Displacement ratio (SA/D) is a calculation used to express how much sail a boat carries relative to its weight. :\mathit{SA/D} = \frac{\mathit{Sail Area}(\text{ft}^2)} {[\mathit{Displacement}(\text{lb})/64]^{\frac{2}{3}}} = \frac{\mathit{Sail Area}(\text{m}^2)} {\mathit{Displacement}(\text{m}^3)^{\frac{2}{3}}} In the first equation, the denominator in pounds is divided by 64 to convert it to cubic feet (because 1 cubic foot of salt water weights 64 pounds). The below given formulas can be used to show that the surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3, as follows. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. Surface area is important in chemical kinetics. Increased surface area can also lead to biological problems. thumb|300px|Graphs of surface area, A against volume, V of the Platonic solids and a sphere, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. The resulting surface area to volume ratio is therefore . thumb|upright=1.3|Measurement of volume by displacement, (a) before and (b) after an object has been submerged. An increased surface area to volume ratio also means increased exposure to the environment. Thus the area of SD is obtained by integrating the length of the normal vector \vec{r}_u\times\vec{r}_v to the surface over the appropriate region D in the parametric uv plane. Let the radius be r and the height be h (which is 2r for the sphere). \begin{array}{rlll} \text{Sphere surface area} & = 4 \pi r^2 & & = (2 \pi r^2) \times 2 \\\ \text{Cylinder surface area} & = 2 \pi r (h + r) & = 2 \pi r (2r + r) & = (2 \pi r^2) \times 3 \end{array} The discovery of this ratio is credited to Archimedes. == In chemistry == thumb|Surface area of particles of different sizes. ",-501,9.30,"""0.0384""",22.2036033112,2.3,E
+"The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \leqslant x \leqslant 4$","Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. In convective heat transfer, the Churchill–Bernstein equation is used to estimate the surface averaged Nusselt number for a cylinder in cross flow at various velocities. Thus the rate of heat flow into V is also given by the surface integral q_t(V)= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS where n(x) is the outward pointing normal vector at x. Rate of heat flow = - (heat transfer coefficient) * (area of the body) * (variation of the temperature) / (length of the material) The formula for the rate of heat flow is: :\frac{Q}{\Delta t} = -kA \frac{\Delta T}{\Delta x} where * Q is the net heat (energy) transfer, * \Delta t is the time taken, * \Delta T is the difference in temperature between the cold and hot sides, * \Delta x is the thickness of the material conducting heat (distance between hot and cold sides), * k is the thermal conductivity, and * A is the surface area of the surface emitting heat. Consider a fluid of uniform temperature T_o and velocity u_o impinging onto a stationary plate uniformly heated to a temperature T_s. This equation is also called the Churchill–Bernstein correlation. ==Heat transfer definition== :\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad \Pr\mathrm{Re}_D \ge 0.2 where: * \overline{\mathrm{Nu}}_D is the surface averaged Nusselt number with characteristic length of diameter; * \mathrm{Re}_D\,\\! is the Reynolds number with the cylinder diameter as its characteristic length; * \Pr is the Prandtl number. This form is more general and particularly useful to recognize which property (e.g. cp or \rho) influences which term. :\rho c_p \frac{\partial T}{\partial t} - abla \cdot \left( k abla T \right) = \dot q_V where \dot q_V is the volumetric heat source. ===Three-dimensional problem=== In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is : \frac{\partial u}{\partial t} = \alpha abla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) where: * u = u(x, y, z, t) is temperature as a function of space and time; * \tfrac{\partial u}{\partial t} is the rate of change of temperature at a point over time; * u_{xx} , u_{yy} , and u_{zz} are the second spatial derivatives (thermal conductions) of temperature in the x , y , and z directions, respectively; * \alpha \equiv \tfrac{k}{c_p\rho} is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k , the specific heat capacity c_p , and the mass density \rho . One then says that is a solution of the heat equation if :\frac{\partial u}{\partial t} = \alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right) in which is a positive coefficient called the thermal diffusivity of the medium. The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The equation yields the surface averaged Nusselt number, which is used to determine the average convective heat transfer coefficient. Steady-state condition: :\frac{\partial u}{\partial t} = 0 The steady-state heat equation for a volume that contains a heat source (the inhomogeneous case), is the Poisson's equation: :-k abla^2 u = q where u is the temperature, k is the thermal conductivity and q is the rate of heat generation per unit volume. By the combination of these observations, the heat equation says the rate \dot u at which the material at a point will heat up (or cool down) is proportional to how much hotter (or cooler) the surrounding material is. The general solution of the heat equation on Rn is then obtained by a convolution, so that to solve the initial value problem with u(x, 0) = g(x), one has :u(\mathbf{x},t) = \int_{\R^n}\Phi(\mathbf{x}-\mathbf{y},t)g(\mathbf{y})d\mathbf{y}. That is, :\frac{\partial Q}{\partial t} = - \frac{\partial q}{\partial x} From the above equations it follows that :\frac{\partial u}{\partial t} \;=\; - \frac{1}{c \rho} \frac{\partial q}{\partial x} \;=\; - \frac{1}{c \rho} \frac{\partial}{\partial x} \left(-k \,\frac{\partial u}{\partial x} \right) \;=\; \frac{k}{c \rho} \frac{\partial^2 u}{\partial x^2} which is the heat equation in one dimension, with diffusivity coefficient :\alpha = \frac{k}{c\rho} This quantity is called the thermal diffusivity of the medium. ====Accounting for radiative loss==== An additional term may be introduced into the equation to account for radiative loss of heat. In the absence of heat energy generation, from external or internal sources, the rate of change in internal heat energy per unit volume in the material, \partial Q/\partial t, is proportional to the rate of change of its temperature, \partial u/\partial t. One further variation is that some of these solve the inhomogeneous equation :u_{t}=ku_{xx}+f. where f is some given function of x and t. ==== Homogeneous heat equation ==== ;Initial value problem on (−∞,∞) :\begin{cases} u_{t}=ku_{xx} & (x, t) \in \R \times (0, \infty) \\\ u(x,0)=g(x) & \text{Initial condition} \end{cases} :u(x,t) = \frac{1}{\sqrt{4\pi kt}} \int_{-\infty}^{\infty} \exp\left(-\frac{(x-y)^2}{4kt}\right)g(y)\,dy right|thumb|upright=2|Fundamental solution of the one-dimensional heat equation. * The time rate of heat flow into a region V is given by a time- dependent quantity qt(V). If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. The rate of heat flow is the amount of heat that is transferred per unit of time in some material, usually measured in watt (joules per second). * By the divergence theorem, the previous surface integral for heat flow into V can be transformed into the volume integral \begin{align} q_t(V) &= - \int_{\partial V} \mathbf{H}(x) \cdot \mathbf{n}(x) \, dS \\\ &= \int_{\partial V} \mathbf{A}(x) \cdot abla u (x) \cdot \mathbf{n}(x) \, dS \\\ &= \int_V \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t) \bigr)\,dx \end{align} * The time rate of temperature change at x is proportional to the heat flowing into an infinitesimal volume element, where the constant of proportionality is dependent on a constant κ \partial_t u(x,t) = \kappa(x) Q(x,t) Putting these equations together gives the general equation of heat flow: : \partial_t u(x,t) = \kappa(x) \sum_{i, j} \partial_{x_i} \bigl( a_{i j}(x) \partial_{x_j} u (x,t)\bigr) Remarks. By Fourier's law for an isotropic medium, the rate of flow of heat energy per unit area through a surface is proportional to the negative temperature gradient across it: :\mathbf{q} = - k \, abla u where k is the thermal conductivity of the material, u=u(\mathbf{x},t) is the temperature, and \mathbf{q} = \mathbf{q}(\mathbf{x},t) is a vector field that represents the magnitude and direction of the heat flow at the point \mathbf{x} of space and time t. The temperature at the solid wall is T_s and gradually changes to T_o as one moves toward the free stream of the fluid. ",3920.70763168,14.34457,"""22.0""",0.2553,1.92,A
+"If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.","Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s). By combining this equation with the suvat equation , it is possible to relate the displacement and the average velocity by \boldsymbol{x} = \frac{(\boldsymbol{u} + \boldsymbol{v})}{2} t = \boldsymbol{\bar{v}}t. Velocity is the speed and the direction of motion of an object. From there, we can obtain an expression for velocity as the area under an acceleration vs. time graph. Average velocity can be calculated as: \boldsymbol{\bar{v}} = \frac{\Delta\boldsymbol{x}}{\Delta t} . Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. thumb|Forty Foot changing rooms and clubhouse kitchen, 2008 thumb|Sunrise at the Forty Foot, 2018 The Forty Foot () is a promontory on the southern tip of Dublin Bay at Sandycove, County Dublin, Ireland, from which people have been swimming in the Irish Sea all year round for some 250 years.as of 2008 * * ==Name== The name ""Forty Foot"" is somewhat obscure. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows: v^{2} = \boldsymbol{v}\cdot\boldsymbol{v} = (\boldsymbol{u}+\boldsymbol{a}t) \cdot (\boldsymbol{u}+\boldsymbol{a}t) = u^{2} + 2t(\boldsymbol{a}\cdot\boldsymbol{u})+a^{2}t^{2} (2\boldsymbol{a})\cdot\boldsymbol{x} = (2\boldsymbol{a})\cdot(\boldsymbol{u}t + \tfrac{1}{2} \boldsymbol{a} t^2) = 2t (\boldsymbol{a} \cdot \boldsymbol{u}) + a^2 t^2 = v^{2} - u^{2} \therefore v^2 = u^2 + 2(\boldsymbol{a}\cdot\boldsymbol{x}) where etc. In calculus terms, the integral of the velocity function is the displacement function . In other words, only relative velocity can be calculated. ===Quantities that are dependent on velocity=== The kinetic energy of a moving object is dependent on its velocity and is given by the equation E_{\text{k}} = \tfrac{1}{2} m v^2 ignoring special relativity, where Ek is the kinetic energy and m is the mass. 20 Hrs. 40 Min.: The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes ""escape velocity"" somewhat of a misnomer, as the more correct term would be ""escape speed"": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path. ==Relative velocity== Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. In terms of a displacement-time ( vs. ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point, and the average velocity as the slope of the secant line between two points with coordinates equal to the boundaries of the time period for the average velocity. Hence, the car is considered to be undergoing an acceleration. ==Difference between speed and velocity== thumb|300px|Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. The average velocity is always less than or equal to the average speed of an object. Rocket Racing Composite Corp. Acquires Velocity Aircraft, Parabolic Arc, 2008-04-14, accessed 2010-12-05. Velocity Aircraft Receives Purchase Order For 20 Velocity XL-5's, Space Fellowship, 2008-10-08, accessed 2010-12-11. As above, this is done using the concept of the integral: \boldsymbol{v} = \int \boldsymbol{a} \ dt . ====Constant acceleration==== In the special case of constant acceleration, velocity can be studied using the suvat equations. Escape velocity is the minimum speed a ballistic object needs to escape from a massive body such as Earth. Velocity is a physical vector quantity; both magnitude and direction are needed to define it. ",-24,0.63,"""0.0""",144, 672.4,A
+A woman walks due west on the deck of a ship at $3 \mathrm{mi} / \mathrm{h}$. The ship is moving north at a speed of $22 \mathrm{mi} / \mathrm{h}$. Find the speed of the woman relative to the surface of the water.,"This is often called the hull speed and is a function of the length of the ship V=k\sqrt{L} where constant (k) should be taken as: 2.43 for velocity (V) in kn and length (L) in metres (m) or, 1.34 for velocity (V) in kn and length (L) in feet (ft). right|thumb|upright=1.3|Load lines, by showing how low a ship is sitting in the water, make it possible to determine displacement. A ship must be designed to move efficiently through the water with a minimum of external force. thumb|A close up quartering view of Grey Lady at dock MV Grey Lady is a high speed catamaran ferry operated by Hy-Line Cruises that travels on a route between Hyannis and Nantucket. Denise ""Dee"" Caffari MBE (born 23 January 1973) is a British sailor, and in 2006 became the first woman to sail single-handedly and non-stop around the world ""the wrong way""; westward against the prevailing winds and currents. The relationship between the velocity of ships and that of the transverse waves can be found by equating the wave celerity and the ship's velocity. ==Propulsion== (Main article: Marine propulsion) Ships can be propelled by numerous sources of power: human, animal, or wind power (sails, kites, rotors and turbines), water currents, chemical or atomic fuels and stored electricity, pressure, heat or solar power supplying engines and motors. ==Gallery== Image:Archimedes principle.svg|A floating ship's displacement Fp and buoyancy Fa must be equal. Seawater (1,025 kg/m3) is more dense than fresh water (1,000 kg/m3);Turpin and McEwen, 1980. so a ship will ride higher in salt water than in fresh. The ship's hydrostatic tables show the corresponding volume displaced.George, 2005. p. 465. For a displacement vessel, that is the usual type of ship, three main types of resistance are considered: that due to wave-making, that due to the pressure of the moving water on the form, often not calculated or measured separately, and that due to friction of moving water on the wetted surface of the hull. As the term indicates, it is measured indirectly, using Archimedes' principle, by first calculating the volume of water displaced by the ship, then converting that value into weight. Ship's boats have always provided transport between the shore and other ships. The displacement or displacement tonnage of a ship is its weight. Grunnslep fra 1932.jpg|Principle of sea surveying with two boats, Norwegian Sea Survey, 1932. A ship's boat is a utility boat carried by a larger vessel. These waves were first studied by William Thomson, 1st Baron Kelvin, who found that regardless of the speed of the ship, they were always contained within the 39° wedge shape (19.5° on each side) following the ship. Froude had observed that when a ship or model was at its so-called Hull speed the wave pattern of the transverse waves (the waves along the hull) have a wavelength equal to the length of the waterline. The total (upward) force due to this buoyancy is equal to the (downward) weight of the displaced water. thumb|A ship for updating nautical charts. thumb|300px|HMS Thetis aground. The divergent waves do not cause much resistance against the ship's forward motion. The boundary layer undergoes shear at different rates extending from the hull surface until it reaches the field flow of the water. ==Wave-making resistance== (Main article: Wave-making resistance) A ship moving over the surface of undisturbed water sets up waves emanating mainly from the bow and stern of the ship. ",0.18,22.2036033112,"""7.0""", 10.7598,92,B
+A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. ,"If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity. Therefore, the distance in feet down a 6% grade to reach the velocity is at least s = \frac{\Delta z}{0.06}= 8.3\frac{V^2}{g},\quad\text{or}\quad s=8.3\frac{88^2}{32.2}\approx 2000\mathrm{ft}. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. The work is doubled either by lifting twice the weight the same distance or by lifting the same weight twice the distance. Step 3) Obtain net work (a single number) by numerical integration of muscle power data. Therefore, work need only be computed for the gravitational forces acting on the bodies. The work of the net force is calculated as the product of its magnitude and the particle displacement. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. That is, the work W done by the resultant force on a particle equals the change in the particle's kinetic energy E_\text{k}, W = \Delta E_\text{k} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 where v_1 and v_2 are the speeds of the particle before and after the work is done, and is its mass. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). In physics, work is the energy transferred to or from an object via the application of force along a displacement. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. The sum of these small amounts of work over the trajectory of the rigid body yields the work, W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as ""force times straight path segment"" would only apply in the most simple of circumstances, as noted above. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.Resnick, Robert, Halliday, David (1966), Physics, Section 1–3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Work is the result of a force on a point that follows a curve , with a velocity , at each instant. Notice that this formula uses the fact that the mass of the vehicle is . thumb|Lotus type 119B gravity racer at Lotus 60th celebration thumb|Gravity racing championship in Campos Novos, Santa Catarina, Brazil, 8 September 2010 ===Coasting down an inclined surface (gravity racing)=== Consider the case of a vehicle that starts at rest and coasts down an inclined surface (such as mountain road), the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity , of say 60 mph (88 fps). ",0,7200,"""0.95""",71,0,B
 " A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \mathrm{i}+80 \mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \mathrm{ft} / \mathrm{s}^2$, so the acceleration vector is $\mathbf{a}=-4 \mathbf{j}-32 \mathbf{k}$. What speed does the ball land?
-","The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. Ground speed is the horizontal speed of an aircraft relative to the Earth’s surface. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. Velocity is the speed and the direction of motion of an object. thumb|150px|A thrust ball bearing A thrust bearing is a particular type of rotary bearing. That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. The material acceleration at P is: \mathbf a_P = \frac{d \mathbf v_P}{dt} = \mathbf a_C + \boldsymbol \alpha \times (\mathbf r_P - \mathbf r_C) + \boldsymbol \omega \times (\mathbf v_P - \mathbf v_C) where \boldsymbol \alpha is the angular acceleration vector. thumb|upright=1.5|Spherical pendulum: angles and velocities. Hence, the car is considered to be undergoing an acceleration. ==Difference between speed and velocity== thumb|300px|Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. In other words, only relative velocity can be calculated. ===Quantities that are dependent on velocity=== The kinetic energy of a moving object is dependent on its velocity and is given by the equation E_{\text{k}} = \tfrac{1}{2} m v^2 ignoring special relativity, where Ek is the kinetic energy and m is the mass. If an object A is moving with velocity vector v and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors: \boldsymbol{v}_{A\text{ relative to }B} = \boldsymbol{v} - \boldsymbol{w} Similarly, the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is: \boldsymbol{v}_{B\text{ relative to }A} = \boldsymbol{w} - \boldsymbol{v} Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest. ===Scalar velocities=== In the one-dimensional case,Basic principle the velocities are scalars and the equation is either: v_\text{rel} = v - (-w), if the two objects are moving in opposite directions, or: v_\text{rel} = v -(+w), if the two objects are moving in the same direction. ==Polar coordinates== In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving. The transverse velocity is the component of velocity along a circle centered at the origin. \boldsymbol{v}=\boldsymbol{v}_T+\boldsymbol{v}_R where *\boldsymbol{v}_T is the transverse velocity *\boldsymbol{v}_R is the radial velocity. This means that the outer race groove exerts less force inward against the ball as the bearing spins. His was the first modern ball-bearing design, with the ball running along a groove in the axle assembly. The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement. v_R = \frac{\boldsymbol{v} \cdot \boldsymbol{r}}{\left|\boldsymbol{r}\right|} where \boldsymbol{r} is displacement. This makes ""escape velocity"" somewhat of a misnomer, as the more correct term would be ""escape speed"": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path. ==Relative velocity== Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. The linear velocity vector \mathbf v_P at P is expressed in terms of the velocity vector \mathbf v_C at C as: \mathbf v_P = \mathbf v_C + \boldsymbol \omega \times (\mathbf r_P - \mathbf r_C) where \boldsymbol \omega is the angular velocity vector. The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. In high speed applications, such as turbines, jet engines, and dentistry equipment, the centrifugal forces generated by the balls changes the contact angle at the inner and outer race. ",0.33333333,+93.4,96.4365076099,+17.7,130.41,C
+","The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. Ground speed is the horizontal speed of an aircraft relative to the Earth’s surface. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. Velocity is the speed and the direction of motion of an object. thumb|150px|A thrust ball bearing A thrust bearing is a particular type of rotary bearing. That is a consequence of the rotational symmetry of the system around the vertical axis. == Trajectory == thumb|250x250px|Trajectory of a spherical pendulum. The material acceleration at P is: \mathbf a_P = \frac{d \mathbf v_P}{dt} = \mathbf a_C + \boldsymbol \alpha \times (\mathbf r_P - \mathbf r_C) + \boldsymbol \omega \times (\mathbf v_P - \mathbf v_C) where \boldsymbol \alpha is the angular acceleration vector. thumb|upright=1.5|Spherical pendulum: angles and velocities. Hence, the car is considered to be undergoing an acceleration. ==Difference between speed and velocity== thumb|300px|Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a. In other words, only relative velocity can be calculated. ===Quantities that are dependent on velocity=== The kinetic energy of a moving object is dependent on its velocity and is given by the equation E_{\text{k}} = \tfrac{1}{2} m v^2 ignoring special relativity, where Ek is the kinetic energy and m is the mass. If an object A is moving with velocity vector v and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors: \boldsymbol{v}_{A\text{ relative to }B} = \boldsymbol{v} - \boldsymbol{w} Similarly, the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is: \boldsymbol{v}_{B\text{ relative to }A} = \boldsymbol{w} - \boldsymbol{v} Usually, the inertial frame chosen is that in which the latter of the two mentioned objects is in rest. ===Scalar velocities=== In the one-dimensional case,Basic principle the velocities are scalars and the equation is either: v_\text{rel} = v - (-w), if the two objects are moving in opposite directions, or: v_\text{rel} = v -(+w), if the two objects are moving in the same direction. ==Polar coordinates== In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). Velocity is a physical vector quantity; both magnitude and direction are needed to define it. The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. Speed, the scalar magnitude of a velocity vector, denotes only how fast an object is moving. The transverse velocity is the component of velocity along a circle centered at the origin. \boldsymbol{v}=\boldsymbol{v}_T+\boldsymbol{v}_R where *\boldsymbol{v}_T is the transverse velocity *\boldsymbol{v}_R is the radial velocity. This means that the outer race groove exerts less force inward against the ball as the bearing spins. His was the first modern ball-bearing design, with the ball running along a groove in the axle assembly. The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement. v_R = \frac{\boldsymbol{v} \cdot \boldsymbol{r}}{\left|\boldsymbol{r}\right|} where \boldsymbol{r} is displacement. This makes ""escape velocity"" somewhat of a misnomer, as the more correct term would be ""escape speed"": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of the base body as long as it does not intersect with something in its path. ==Relative velocity== Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. The linear velocity vector \mathbf v_P at P is expressed in terms of the velocity vector \mathbf v_C at C as: \mathbf v_P = \mathbf v_C + \boldsymbol \omega \times (\mathbf r_P - \mathbf r_C) where \boldsymbol \omega is the angular velocity vector. The purpose of a ball bearing is to reduce rotational friction and support radial and axial loads. In high speed applications, such as turbines, jet engines, and dentistry equipment, the centrifugal forces generated by the balls changes the contact angle at the inner and outer race. ",0.33333333,+93.4,"""96.4365076099""",+17.7,130.41,C
 "The demand function for a commodity is given by
 $$
 p=2000-0.1 x-0.01 x^2
 $$
-Find the consumer surplus when the sales level is 100 .","McGraw-Hill 2005 To compute the inverse demand function, simply solve for P from the demand function. Specifying values for the non-price determinants, Prg = 4.00 and Y = 50, results in demand equation Q = 325 - P - 30(4) +1.4(50) or Q = 275 - P. A simple example of a demand equation is Qd = 325 - P - 30Prg \+ 1.4Y. To compute the inverse demand equation, simply solve for P from the demand equation.The form of the inverse linear demand equation is P = a/b - 1/bQ. Pe = 80 is the equilibrium price at which quantity demanded is equal to the quantity supplied. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². For example, if the demand equation is Q = 240 - 2P then the inverse demand equation would be P = 120 - .5Q, the right side of which is the inverse demand function.Samuelson, W & Marks, S. Managerial Economics 4th ed. p. 37. So 20 is the profit- maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P. ==See also== *Supply and demand *Demand *Law of demand *Profit (economics) ==References== Category:Mathematical finance Category:Demand For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would be P = 120 - .5Q.Samuelson & Marks, Managerial Economics 4th ed. (Wiley 2003) Note that although price is the dependent variable in the inverse demand function, it is still the case that the equation represents how the price determines the quantity demanded, not the reverse. ==Relation to marginal revenue== There is a close relationship between any inverse demand function for a linear demand equation and the marginal revenue function. *The x intercept of the marginal revenue function is one- half the x intercept of the inverse demand function. The value P in the inverse demand function is the highest price that could be charged and still generate the quantity demanded Q.Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London This is useful because economists typically place price (P) on the vertical axis and quantity (Q) on the horizontal axis in supply-and-demand diagrams, so it is the inverse demand function that depicts the graphed demand curve in the way the reader expects to see. The supply curve, shown in orange, intersects with the demand curve at price (Pe) = 80 and quantity (Qe)= 120. If good i is a Giffen good whose price increases while other goods' prices are held fixed (so that p_j'-p_j=0 \; \forall j eq i), the law of demand is clearly violated, as we have both p_i'-p_i>0 (as price increased) and q_i'-q_i>0 (as we consider a Giffen good), so that (p'-p)(x'-x)=(p_i'-p_i)(x_i'-x_i)>0. == Demand versus quantity demanded == It is very important to apprehend the difference between demand and quantity demanded as they are used to mean different things in the economic jargon. If income were to increase to 55, the new demand equation would be Q = 282 - P. Graphically, this change in a non-price determinant of demand would be reflected in an outward shift of the demand function caused by a change in the x-intercept. ==Demand curve== In economics the demand curve is the graphical representation of the relationship between the price and the quantity that consumers are willing to purchase. The mathematical relationship between the price of the substitute and the demand for the good in question is positive. The above equation, when plotted with quantity demanded (Q_x) on the x-axis and price (P_x) on the y-axis, gives the demand curve, which is also known as the demand schedule. The inverse demand equation, or price equation, treats price as a function f of quantity demanded: P = f(Q). The law of demand states that \frac{\partial f}{\partial P_x} < 0. The number of consumers in a market: The market demand for a good is obtained by adding individual demands of the present, as well as prospective consumers of a good at various possible prices. Demand vacuum in economics and marketing is the effect created by consumer demand on the supply chain. The formula to solve for the coefficient of price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in Price. ",1.3,0.66666666666,5040.0,-4.37 ,7166.67,E
-"The linear density in a rod $8 \mathrm{~m}$ long is $12 / \sqrt{x+1} \mathrm{~kg} / \mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.","upright=1.4|thumb|The deformation of a thin straight rod into a closed loop. Knowing the volume of the unit cell of a crystalline material and its formula weight (in daltons), the density can be calculated. In the United States until 1 January 2023, the rod was often defined as 16.5 US survey feet, or approximately 5.029 210 058 m. ==History== In England, the perch was officially discouraged in favour of the rod as early as the 15th century;Encyclopædia Britannica, English measure however, local customs maintained its use. The rod is useful as a unit of length because integer multiples of it can form one acre of square measure (area). * Determination of Density of Solid, instructions for performing classroom experiment. Standing at arm's length from the tree, estimate its average diameter by taking a note on the rod's markings. A cruising rod is a simple device used to quickly estimate the number of pieces of lumber yielded by a given piece of timber. In the US, the rod, along with the chain, furlong, and statute mile (as well as the survey inch and survey foot) were based on the pre-1959 values for United States customary units of linear measurement until 1 January 2023. Counting rods () are small bars, typically 3–14 cm (1"" to 6"") long, that were used by mathematicians for calculation in ancient East Asia. Mathematically, density is defined as mass divided by volume: \rho = \frac{m}{V} where ρ is the density, m is the mass, and V is the volume. In that case the density around any given location is determined by calculating the density of a small volume around that location. The rod as a survey measure was standardized by Edmund Gunter in England in 1607 as a quarter of a chain (of ), or long. ===In ancient cultures=== The perch as a lineal measure in Rome (also decempeda) was 10 Roman feet (2.96 metres), and in France varied from 10 feet (perche romanie) to 22 feet (perche d'arpent—apparently of ""the range of an arrow""—about 220 feet). Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. Rod was also sometimes used as a unit of area to refer to a rood. * Gas density calculator Calculate density of a gas for as a function of temperature and pressure. See below for a list of some of the most common units of density. ===Homogeneous materials=== The density at all points of a homogeneous object equals its total mass divided by its total volume. The Railway Operating Division (ROD) ROD 2-8-0 is a type of 2-8-0 steam locomotive which was the standard heavy freight locomotive operated in Europe by the ROD during the First World War. ==ROD need for a standard locomotive== During the First World War the Railway Operating Division of the Royal Engineers requisitioned about 600 locomotives of various types from thirteen United Kingdom railway companies; the first arrived in France in late 1916. Rods can also be found on the older legal descriptions of tracts of land in the United States, following the ""metes and bounds"" method of land survey; as shown in this actual legal description of rural real estate: ==Area and volume== The terms pole, perch, rod and rood have been used as units of area, and perch is also used as a unit of volume. Bars of metal one rod long were used as standards of length when surveying land. The rod, perch, or pole (sometimes also lug) is a surveyor's tool and unit of length of various historical definitions. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more specifically called specific weight. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. ",6,90,7.0,0.082,0.000226,A
-A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \mathrm{~ft}$ to $x=10 \mathrm{~ft}$.,"If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. In physics, work is the energy transferred to or from an object via the application of force along a displacement. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work of the net force is calculated as the product of its magnitude and the particle displacement. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. Therefore, work need only be computed for the gravitational forces acting on the bodies. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as ""force times straight path segment"" would only apply in the most simple of circumstances, as noted above. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. Work transfers energy from one place to another or one form to another. ===Derivation for a particle moving along a straight line=== In the case the resultant force is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line. The work done is given by the dot product of the two vectors. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. ",-0.55,3.07,4.5,17,0.42,C
-"One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \%$ of the population to become infected?","Now, the epidemic model is : \frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x), where V_i(x)= [V^-_i(x)-V^+_i(x)] In the above equations, F_i(x) represents the rate of appearance of new infections in compartment i . An epidemic (from Greek ἐπί epi ""upon or above"" and δῆμος demos ""people"") is the rapid spread of disease to a large number of hosts in a given population within a short period of time. * Epidemic – when this disease is found to infect a significantly larger number of people at the same time than is common at that time, and among that population, and may spread through one or several communities. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model. ==Sub-exponential growth== A common explanation for the growth of epidemics holds that 1 person infects 2, those 2 infect 4 and so on and so on with the number of infected doubling every generation. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow sub-exponentially and there will be an epidemic, any less and the disease will die out). Classic epidemic models of disease transmission are described in Compartmental models in epidemiology. Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. There is another variation, both as regards the number of people affected and the number who die in successive epidemics: the severity of successive epidemics rises and falls over periods of five or ten years. ==Types== ===Common source outbreak=== In a common source outbreak epidemic, the affected individuals had an exposure to a common agent. This quantity determines whether the infection will increase sub- exponentially, die out, or remain constant: if R0 > 1, then each person on average infects more than one other person so the disease will spread; if R0 < 1, then each person infects fewer than one person on average so the disease will die out; and if R0 = 1, then each person will infect on average exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease. ==Endemic steady state== An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. Thus this model of an epidemic leads to a curve that grows exponentially until it crashes to zero as all the population have been infected. i.e. no herd immunity and no peak and gradual decline as seen in reality. == Reproduction number == The basic reproduction number (denoted by R0) is a measure of how transferable a disease is. For example, in meningococcal infections, an attack rate in excess of 15 cases per 100,000 people for two consecutive weeks is considered an epidemic. Research topics include: * antigenic shift * epidemiological networks * evolution and spread of resistance * immuno- epidemiology * intra-host dynamics * Pandemic * pathogen population genetics * persistence of pathogens within hosts * phylodynamics * role and identification of infection reservoirs * role of host genetic factors * spatial epidemiology * statistical and mathematical tools and innovations * Strain (biology) structure and interactions * transmission, spread and control of infection * virulence ==Mathematics of mass vaccination== If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population and its transmission dies out. The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. In epidemiology, particularly in the discussion of infectious disease dynamics (mathematical modeling of disease spread), the infectious period is the time interval during which a host (individual or patient) is infectious, i.e. capable of directly or indirectly transmitting pathogenic infectious agents or pathogens to another susceptible host. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. Epidemics of infectious disease are generally caused by several factors including a change in the ecology of the host population (e.g., increased stress or increase in the density of a vector species), a genetic change in the pathogen reservoir or the introduction of an emerging pathogen to a host population (by movement of pathogen or host). It is the average number of people that a single infectious person will infect over the course of their infection. Statistical agent-level disease dissemination in small or large populations can be determined by stochastic methods. ===Deterministic=== When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. thumb|upright=1.5|Example of an epidemic showing the number of new infections over time. A classic compartmental model in epidemiology is the SIR model, which may be used as a simple model for modelling epidemics. The Centers for Disease Control and Prevention defines epidemic broadly: ""the occurrence of more cases of disease, injury, or other health condition than expected in a given area or among a specific group of persons during a particular period. ",2.3613, 1.16,71.0,15,2.8108,D
-"Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $5 \mathrm{~cm}$","The volume of this tetrahedron is one-third the volume of the cube. Let be the volume of the tetrahedron; then :V=\frac{\sqrt{4 a^2 b^2 c^2-a^2 X^2-b^2 Y^2-c^2 Z^2+X Y Z}}{12} where :\begin{align}X&=b^2+c^2-x^2, \\\ Y&=a^2+c^2-y^2, \\\ Z&=a^2+b^2-z^2. \end{align} The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles. Let V be the volume of the tetrahedron. Since the four subtetrahedra fill the volume, we have V = \frac13A_1r+\frac13A_2r+\frac13A_3r+\frac13A_4r. ===Circumradius=== Denote the circumradius of a tetrahedron as R. Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to of the volume of any parallelepiped that shares three converging edges with it. thumb|3D model of regular tetrahedron. Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant: :288 \cdot V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix} where the subscripts represent the vertices and d is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. ===Coordinates for a regular tetrahedron=== The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges: :\left(\pm 1, 0, -\frac{1}{\sqrt{2}}\right) \quad \mbox{and} \quad \left(0, \pm 1, \frac{1}{\sqrt{2}}\right) Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face parallel to the xy plane, the vertices are: v_1 = \left(\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right) v_2 = \left(-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right) v_3 = \left(-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right) v_4 = (0,0,1) with the edge length of \sqrt{\frac{8}{3}}. Then another volume formula is given by :V = \frac {d |(\mathbf{a} \times \mathbf{(b-c)})| } {6}. ===Properties analogous to those of a triangle=== The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. :V = \frac {abc} {6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}} ==== Heron-type formula for the volume of a tetrahedron ==== right|thumb|240px|Six edge-lengths of Tetrahedron If , , , , , are lengths of edges of the tetrahedron (first three form a triangle; with opposite , opposite , opposite ), then : V = \frac{\sqrt {\,( - p + q + r + s)\,(p - q + r + s)\,(p + q - r + s)\,(p + q + r - s)}}{192\,u\,v\,w} where : \begin{align} p & = \sqrt {xYZ}, & q & = \sqrt {yZX}, & r & = \sqrt {zXY}, & s & = \sqrt {xyz}, \end{align} : \begin{align} X & = (w - U + v)\,(U + v + w), & x & = (U - v + w)\,(v - w + U), \\\ Y & = (u - V + w)\,(V + w + u), & y & = (V - w + u)\,(w - u + V), \\\ Z & = (v - W + u)\,(W + u + v), & z & = (W - u + v)\,(u - v + W). \end{align} ====Volume divider==== Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron.Bottema, O. The three faces interior to the tetrahedron are: a right triangle with edges 1, \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{2}}, a right triangle with edges \sqrt{\tfrac{1}{3}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}}, and a right triangle with edges \sqrt{\tfrac{4}{3}}, \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{6}}. ===Space-filling tetrahedra=== A space-filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space. Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron edge arc arc dihedral dihedral 𝒍 2 109°28′16″ \pi - 2\text{𝜿} 70°31′44″ \pi - 2\text{𝟁} 𝟀 \sqrt{\tfrac{4}{3}} \approx 1.155 70°31′44″ 2\text{𝜿} 60° \tfrac{\pi}{3} 𝝓 1 54°44′8″ \tfrac{\pi}{2} - \text{𝜿} 60° \tfrac{\pi}{3} 𝟁 \sqrt{\tfrac{1}{3}} \approx 0.577 54°44′8″ \tfrac{\pi}{2} - \text{𝜿} 60° \tfrac{\pi}{3} _0R/l \sqrt{\tfrac{3}{2}} \approx 1.225 _1R/l \sqrt{\tfrac{1}{2}} \approx 0.707 _2R/l \sqrt{\tfrac{1}{6}} \approx 0.408 \text{𝜿} 35°15′52″ \tfrac{\text{arc sec }3}{2} If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths \sqrt{\tfrac{4}{3}}, 1, \sqrt{\tfrac{1}{3}} (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}} (edges that are the characteristic radii of the regular tetrahedron). The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: : \Delta_i^2 = \Delta_j^2 + \Delta_k^2 + \Delta_l^2 - 2(\Delta_j\Delta_k\cos\theta_{il} + \Delta_j\Delta_l \cos\theta_{ik} + \Delta_k\Delta_l \cos\theta_{ij}) === Interior point === Let P be any interior point of a tetrahedron of volume V for which the vertices are A, B, C, and D, and for which the areas of the opposite faces are Fa, Fb, Fc, and Fd. The exterior face is a 60-90-30 triangle which is one-sixth of a tetrahedron face. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube. (As a side-note: these two kinds of tetrahedron have the same volume.) This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. The only two isometries are 1 and the rotation (12)(34), giving the group C2 isomorphic to the cyclic group, Z2. ==General properties== ===Volume=== The volume of a tetrahedron is given by the pyramid volume formula: :V = \frac13 A_0\,h \, where A0 is the area of the base and h is the height from the base to the apex. :Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1) :Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1) thumb|right|300px|Regular tetrahedron ABCD and its circumscribed sphere ===Angles and distances=== For a regular tetrahedron of edge length a: Face area A_0=\frac{\sqrt{3}}{4}a^2\, Surface areaCoxeter, Harold Scott MacDonald; Regular Polytopes, Methuen and Co., 1948, Table I(i) A=4\,A_0={\sqrt{3}}a^2\, Height of pyramidKöller, Jürgen, ""Tetrahedron"", Mathematische Basteleien, 2001 h=\frac{\sqrt{6}}{3}a=\sqrt{\frac23}\,a\, Centroid to vertex distance \frac34\,h = \frac{\sqrt{6}}{4}\,a = \sqrt{\frac{3}{8}}\,a\, Edge to opposite edge distance l=\frac{1}{\sqrt{2}}\,a\, Volume V=\frac13 A_0h =\frac{\sqrt{2}}{12}a^3=\frac{a^3}{6\sqrt{2}}\, Face-vertex-edge angle \arccos\left(\frac{1}{\sqrt{3}}\right) = \arctan\left(\sqrt{2}\right)\, (approx. 54.7356°) Face-edge-face angle, i.e., ""dihedral angle"" \arccos\left(\frac13\right) = \arctan\left(2\sqrt{2}\right)\, (approx. 70.5288°) Vertex-Center-Vertex angle, the angle between lines from the tetrahedron center to any two vertices. thumb|3D model of a truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron. ",0.011,9,10.0,3.29527,1.4,C
-The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.,"The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry It is well known that the relations of the volumes of a right circular cone, one half of a sphere and a right circular cylinder with same radii and heights are 1 : 2 : 3. The intersection of three cylinders with perpendicularly intersecting axes generates a surface of a solid with vertices where 3 edges meet and vertices where 4 edges meet. thumb|right|170px Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis ""parallel"" to the axis of revolution. The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The volume of the 8 pyramids is: \textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 , and then we can calculate that the bicylinder volume is \textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3 == Tricylinder == 450px|thumb|Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. The key for the determination of volume and surface area is the observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). One simply must solve each equation for before one inserts them into the integration formula. ==See also== *Solid of revolution *Shell integration ==References== * * *Frank Ayres, Elliott Mendelson. Hence the area of this development is thumb|cloister vault :B = \int_{0}^{\pi r} r\sin\left(\frac{\xi}{r}\right) \mathrm{d}\xi = 2r^2 and the total surface area is: :A=8\cdot B=16r^2. === Alternate proof of the volume formula === Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). The concept of included angle is discussed at: * Congruence of triangles * Solution of triangles The Euler line of an isosceles triangle is perpendicular to the triangle's base. The altitudes of a triangle are perpendicular to their respective bases. thumb|Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. File:Sphere volume derivation using bicylinder.jpg|Zu Chongzhi's method (similar to Cavalieri's principle) for calculating a sphere's volume includes calculating the volume of a bicylinder. For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. For example, the next figure shows the rotation along the -axis of the red ""leaf"" enclosed between the square- root and quadratic curves: thumb|Rotation about x-axis The volume of this solid is: :\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,. If is the value of a horizontal axis, then the volume equals :\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. thumb|right|150px|Animated depiction of a bicylinder == Bicylinder == 300px|thumb|The generation of a bicylinder 180px|thumb|Calculating the volume of a bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac{16}{3} r^3 and the ;surface area :A=16 r^2. Wallis's Conical Edge with right|thumb|600px| Figure 2. File:Bicylinder and cube sections related by pyramids.png|Relationship of the area of a bicylinder section with a cube section The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. ",140,157.875,-0.16,36,-1,D
-A projectile is fired with an initial speed of $200 \mathrm{~m} / \mathrm{s}$ and angle of elevation $60^{\circ}$. Find the speed at impact.,"Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. Composites Science and Technology 64:35-54. . is as follows: V_b=\frac{\pi\,\Gamma\,\sqrt{\rho_t\,\sigma_e}\,D^2\,T}{4\,m} \left [1+\sqrt{1+\frac{8\,m}{\pi\,\Gamma^2\,\rho_t\,D^2\,T}}\, \right ] where *V_b\, is the ballistic limit *\Gamma\, is a projectile constant determined experimentally *\rho_t\, is the density of the laminate *\sigma_e\, is the static linear elastic compression limit *D\, is the diameter of the projectile *T\, is the thickness of the laminate *m\, is the mass of the projectile Additionally, the ballistic limit for small-caliber into homogeneous armor by TM5-855-1 is: V_1= 19.72 \left [ \frac{7800 d^3 \left [ \left ( \frac{e_h}{d} \right) \sec \theta \right ]^{1.6}}{W_T} \right ]^{0.5} where *V_1 is the ballistic limit velocity in fps *d is the caliber of the projectile, in inches *e_h is the thickness of the homogeneous armor (valid from BHN 360 - 440) in inches *\theta is the angle of obliquity *W_T is the weight of the projectile, in lbs == References == Category:Ballistics thumb|The aftermath of a hypervelocity impact, with a projectile the same size as the one that impacted for scale Hypervelocity is very high velocity, approximately over 3,000 meters per second (6,700 mph, 11,000 km/h, 10,000 ft/s, or Mach 8.8). The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. right|288px|thumb|Impact parameter and scattering angle In physics, the impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). After several steps of algebraic manipulation : t = \frac {v \sin \theta} {g} \pm \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} The square root must be a positive number, and since the velocity and the sine of the launch angle can also be assumed to be positive, the solution with the greater time will occur when the positive of the plus or minus sign is used. Ballistic impact is a high velocity impact by a small mass object, analogous to runway debris or small arms fire. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. Here, the object that the projectile is approaching is a hard sphere with radius R. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Mathematical equations of motion are used to analyze projectile trajectory. The ballistic limit or limit velocity is the velocity required for a particular projectile to reliably (at least 50% of the time) penetrate a particular piece of material. (And see Trajectory of a projectile.) (And see Trajectory of a projectile.) The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance. The impact parameter is related to the scattering angle byLandau L. D. and Lifshitz E. M. (1976) Mechanics, 3rd. ed., Pergamon Press. (hardcover) and (softcover). : \theta = \pi - 2b\int_{r_\text{min}}^\infty \frac{dr}{r^2\sqrt{1 - (b/r)^2 - 2U/(mv_\infty^2)}}, where is the velocity of the projectile when it is far from the center, and is its closest distance from the center. ==Scattering from a hard sphere== The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. When b > R , the projectile misses the hard sphere. ",7,2.50,200.0,4943,1.775,C
+Find the consumer surplus when the sales level is 100 .","McGraw-Hill 2005 To compute the inverse demand function, simply solve for P from the demand function. Specifying values for the non-price determinants, Prg = 4.00 and Y = 50, results in demand equation Q = 325 - P - 30(4) +1.4(50) or Q = 275 - P. A simple example of a demand equation is Qd = 325 - P - 30Prg \+ 1.4Y. To compute the inverse demand equation, simply solve for P from the demand equation.The form of the inverse linear demand equation is P = a/b - 1/bQ. Pe = 80 is the equilibrium price at which quantity demanded is equal to the quantity supplied. Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². Multiply the inverse demand function by Q to derive the total revenue function: TR = (120 - .5Q) × Q = 120Q - 0.5Q². For example, if the demand equation is Q = 240 - 2P then the inverse demand equation would be P = 120 - .5Q, the right side of which is the inverse demand function.Samuelson, W & Marks, S. Managerial Economics 4th ed. p. 37. So 20 is the profit- maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P. ==See also== *Supply and demand *Demand *Law of demand *Profit (economics) ==References== Category:Mathematical finance Category:Demand For example, if the demand function has the form Q = 240 - 2P then the inverse demand function would be P = 120 - .5Q.Samuelson & Marks, Managerial Economics 4th ed. (Wiley 2003) Note that although price is the dependent variable in the inverse demand function, it is still the case that the equation represents how the price determines the quantity demanded, not the reverse. ==Relation to marginal revenue== There is a close relationship between any inverse demand function for a linear demand equation and the marginal revenue function. *The x intercept of the marginal revenue function is one- half the x intercept of the inverse demand function. The value P in the inverse demand function is the highest price that could be charged and still generate the quantity demanded Q.Varian, H.R (2006) Intermediate Microeconomics, Seventh Edition, W.W Norton & Company: London This is useful because economists typically place price (P) on the vertical axis and quantity (Q) on the horizontal axis in supply-and-demand diagrams, so it is the inverse demand function that depicts the graphed demand curve in the way the reader expects to see. The supply curve, shown in orange, intersects with the demand curve at price (Pe) = 80 and quantity (Qe)= 120. If good i is a Giffen good whose price increases while other goods' prices are held fixed (so that p_j'-p_j=0 \; \forall j eq i), the law of demand is clearly violated, as we have both p_i'-p_i>0 (as price increased) and q_i'-q_i>0 (as we consider a Giffen good), so that (p'-p)(x'-x)=(p_i'-p_i)(x_i'-x_i)>0. == Demand versus quantity demanded == It is very important to apprehend the difference between demand and quantity demanded as they are used to mean different things in the economic jargon. If income were to increase to 55, the new demand equation would be Q = 282 - P. Graphically, this change in a non-price determinant of demand would be reflected in an outward shift of the demand function caused by a change in the x-intercept. ==Demand curve== In economics the demand curve is the graphical representation of the relationship between the price and the quantity that consumers are willing to purchase. The mathematical relationship between the price of the substitute and the demand for the good in question is positive. The above equation, when plotted with quantity demanded (Q_x) on the x-axis and price (P_x) on the y-axis, gives the demand curve, which is also known as the demand schedule. The inverse demand equation, or price equation, treats price as a function f of quantity demanded: P = f(Q). The law of demand states that \frac{\partial f}{\partial P_x} < 0. The number of consumers in a market: The market demand for a good is obtained by adding individual demands of the present, as well as prospective consumers of a good at various possible prices. Demand vacuum in economics and marketing is the effect created by consumer demand on the supply chain. The formula to solve for the coefficient of price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in Price. ",1.3,0.66666666666,"""5040.0""",-4.37 ,7166.67,E
+"The linear density in a rod $8 \mathrm{~m}$ long is $12 / \sqrt{x+1} \mathrm{~kg} / \mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.","upright=1.4|thumb|The deformation of a thin straight rod into a closed loop. Knowing the volume of the unit cell of a crystalline material and its formula weight (in daltons), the density can be calculated. In the United States until 1 January 2023, the rod was often defined as 16.5 US survey feet, or approximately 5.029 210 058 m. ==History== In England, the perch was officially discouraged in favour of the rod as early as the 15th century;Encyclopædia Britannica, English measure however, local customs maintained its use. The rod is useful as a unit of length because integer multiples of it can form one acre of square measure (area). * Determination of Density of Solid, instructions for performing classroom experiment. Standing at arm's length from the tree, estimate its average diameter by taking a note on the rod's markings. A cruising rod is a simple device used to quickly estimate the number of pieces of lumber yielded by a given piece of timber. In the US, the rod, along with the chain, furlong, and statute mile (as well as the survey inch and survey foot) were based on the pre-1959 values for United States customary units of linear measurement until 1 January 2023. Counting rods () are small bars, typically 3–14 cm (1"" to 6"") long, that were used by mathematicians for calculation in ancient East Asia. Mathematically, density is defined as mass divided by volume: \rho = \frac{m}{V} where ρ is the density, m is the mass, and V is the volume. In that case the density around any given location is determined by calculating the density of a small volume around that location. The rod as a survey measure was standardized by Edmund Gunter in England in 1607 as a quarter of a chain (of ), or long. ===In ancient cultures=== The perch as a lineal measure in Rome (also decempeda) was 10 Roman feet (2.96 metres), and in France varied from 10 feet (perche romanie) to 22 feet (perche d'arpent—apparently of ""the range of an arrow""—about 220 feet). Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. Rod was also sometimes used as a unit of area to refer to a rood. * Gas density calculator Calculate density of a gas for as a function of temperature and pressure. See below for a list of some of the most common units of density. ===Homogeneous materials=== The density at all points of a homogeneous object equals its total mass divided by its total volume. The Railway Operating Division (ROD) ROD 2-8-0 is a type of 2-8-0 steam locomotive which was the standard heavy freight locomotive operated in Europe by the ROD during the First World War. ==ROD need for a standard locomotive== During the First World War the Railway Operating Division of the Royal Engineers requisitioned about 600 locomotives of various types from thirteen United Kingdom railway companies; the first arrived in France in late 1916. Rods can also be found on the older legal descriptions of tracts of land in the United States, following the ""metes and bounds"" method of land survey; as shown in this actual legal description of rural real estate: ==Area and volume== The terms pole, perch, rod and rood have been used as units of area, and perch is also used as a unit of volume. Bars of metal one rod long were used as standards of length when surveying land. The rod, perch, or pole (sometimes also lug) is a surveyor's tool and unit of length of various historical definitions. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate – this quantity is more specifically called specific weight. The length of the rod remains almost unchanged during the deformation, which indicates that the strain is small. ",6,90,"""7.0""",0.082,0.000226,A
+A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \mathrm{~ft}$ to $x=10 \mathrm{~ft}$.,"If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is: W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y where Fg is weight (pounds in imperial units, and newtons in SI units), and Δy is the change in height y. Notice that the work done by gravity depends only on the vertical movement of the object. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. In physics, work is the energy transferred to or from an object via the application of force along a displacement. This force will act through the distance along the circular arc l=s=r\phi, so the work done is W = F s = F r \phi . Therefore, work on an object that is merely displaced in a conservative force field, without change in velocity or rotation, is equal to minus the change of potential energy of the object, W = -\Delta E_\text{p}. The presence of friction does not affect the work done on the object by its weight. ===Work by gravity in space=== The force of gravity exerted by a mass on another mass is given by \mathbf{F} = -\frac{GMm}{r^2} \hat\mathbf{r} = -\frac{GMm}{r^3}\mathbf{r}, where is the position vector from to and is the unit vector in the direction of . The sum of these small amounts of work over the trajectory of the point yields the work, W = \int_{t_1}^{t_2}\mathbf{F} \cdot \mathbf{v} \, dt = \int_{t_1}^{t_2}\mathbf{F} \cdot \tfrac{d\mathbf{s}}{dt} \, dt =\int_C \mathbf{F} \cdot d\mathbf{s}, where C is the trajectory from x(t1) to x(t2). The work of the net force is calculated as the product of its magnitude and the particle displacement. When a force component is perpendicular to the displacement of the object (such as when a body moves in a circular path under a central force), no work is done, since the cosine of 90° is zero. When the force is constant and the angle between the force and the displacement is also constant, then the work done is given by: W = F s \cos{\theta} Work is a scalar quantity, so it has only magnitude and no direction. Therefore, work need only be computed for the gravitational forces acting on the bodies. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force strength and the distance traveled. For example, when a ball is held above the ground and then dropped, the work done by the gravitational force on the ball as it falls is positive, and is equal to the weight of the ball (a force) multiplied by the distance to the ground (a displacement). The work of this spring on a body moving along the space with the curve , is calculated using its velocity, , to obtain W=\int_0^t\mathbf{F}\cdot\mathbf{v}dt =-\int_0^tkx v_x dt = -\frac{1}{2}kx^2. If force is changing, or if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). Also, no work is done on a body moving circularly at a constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge. === Work done by a variable force === Calculating the work as ""force times straight path segment"" would only apply in the most simple of circumstances, as noted above. Let the mass move at the velocity ; then the work of gravity on this mass as it moves from position to is given by W = -\int^{\mathbf{r}(t_2)}_{\mathbf{r}(t_1)} \frac{GMm}{r^3} \mathbf{r} \cdot d\mathbf{r} = -\int^{t_2}_{t_1} \frac{GMm}{r^3}\mathbf{r} \cdot \mathbf{v} \, dt. Net work is typically calculated either from instantaneous power (muscle force x muscle velocity) or from the area enclosed by the work loop on a force vs. length plot. Work transfers energy from one place to another or one form to another. ===Derivation for a particle moving along a straight line=== In the case the resultant force is constant in both magnitude and direction, and parallel to the velocity of the particle, the particle is moving with constant acceleration a along a straight line. The work done is given by the dot product of the two vectors. If the ball is thrown upwards, the work done by the gravitational force is negative, and is equal to the weight multiplied by the displacement in the upwards direction. ",-0.55,3.07,"""4.5""",17,0.42,C
+"One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \%$ of the population to become infected?","Now, the epidemic model is : \frac{\mathrm{d} x_i}{\mathrm{d}t}= F_i (x)-V_i(x), where V_i(x)= [V^-_i(x)-V^+_i(x)] In the above equations, F_i(x) represents the rate of appearance of new infections in compartment i . An epidemic (from Greek ἐπί epi ""upon or above"" and δῆμος demos ""people"") is the rapid spread of disease to a large number of hosts in a given population within a short period of time. * Epidemic – when this disease is found to infect a significantly larger number of people at the same time than is common at that time, and among that population, and may spread through one or several communities. In other words, the changes in population of a compartment can be calculated using only the history that was used to develop the model. ==Sub-exponential growth== A common explanation for the growth of epidemics holds that 1 person infects 2, those 2 infect 4 and so on and so on with the number of infected doubling every generation. This means that, on average, each infected person is infecting exactly one other person (any more and the number of people infected will grow sub-exponentially and there will be an epidemic, any less and the disease will die out). Classic epidemic models of disease transmission are described in Compartmental models in epidemiology. Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. There is another variation, both as regards the number of people affected and the number who die in successive epidemics: the severity of successive epidemics rises and falls over periods of five or ten years. ==Types== ===Common source outbreak=== In a common source outbreak epidemic, the affected individuals had an exposure to a common agent. This quantity determines whether the infection will increase sub- exponentially, die out, or remain constant: if R0 > 1, then each person on average infects more than one other person so the disease will spread; if R0 < 1, then each person infects fewer than one person on average so the disease will die out; and if R0 = 1, then each person will infect on average exactly one other person, so the disease will become endemic: it will move throughout the population but not increase or decrease. ==Endemic steady state== An infectious disease is said to be endemic when it can be sustained in a population without the need for external inputs. In epidemiology, the next-generation matrix is used to derive the basic reproduction number, for a compartmental model of the spread of infectious diseases. Thus this model of an epidemic leads to a curve that grows exponentially until it crashes to zero as all the population have been infected. i.e. no herd immunity and no peak and gradual decline as seen in reality. == Reproduction number == The basic reproduction number (denoted by R0) is a measure of how transferable a disease is. For example, in meningococcal infections, an attack rate in excess of 15 cases per 100,000 people for two consecutive weeks is considered an epidemic. Research topics include: * antigenic shift * epidemiological networks * evolution and spread of resistance * immuno- epidemiology * intra-host dynamics * Pandemic * pathogen population genetics * persistence of pathogens within hosts * phylodynamics * role and identification of infection reservoirs * role of host genetic factors * spatial epidemiology * statistical and mathematical tools and innovations * Strain (biology) structure and interactions * transmission, spread and control of infection * virulence ==Mathematics of mass vaccination== If the proportion of the population that is immune exceeds the herd immunity level for the disease, then the disease can no longer persist in the population and its transmission dies out. The earliest account of mathematical modelling of spread of disease was carried out in 1760 by Daniel Bernoulli. In epidemiology, particularly in the discussion of infectious disease dynamics (mathematical modeling of disease spread), the infectious period is the time interval during which a host (individual or patient) is infectious, i.e. capable of directly or indirectly transmitting pathogenic infectious agents or pathogens to another susceptible host. While building such models, it must be assumed that the population size in a compartment is differentiable with respect to time and that the epidemic process is deterministic. Epidemics of infectious disease are generally caused by several factors including a change in the ecology of the host population (e.g., increased stress or increase in the density of a vector species), a genetic change in the pathogen reservoir or the introduction of an emerging pathogen to a host population (by movement of pathogen or host). It is the average number of people that a single infectious person will infect over the course of their infection. Statistical agent-level disease dissemination in small or large populations can be determined by stochastic methods. ===Deterministic=== When dealing with large populations, as in the case of tuberculosis, deterministic or compartmental mathematical models are often used. thumb|upright=1.5|Example of an epidemic showing the number of new infections over time. A classic compartmental model in epidemiology is the SIR model, which may be used as a simple model for modelling epidemics. The Centers for Disease Control and Prevention defines epidemic broadly: ""the occurrence of more cases of disease, injury, or other health condition than expected in a given area or among a specific group of persons during a particular period. ",2.3613, 1.16,"""71.0""",15,2.8108,D
+"Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $5 \mathrm{~cm}$","The volume of this tetrahedron is one-third the volume of the cube. Let be the volume of the tetrahedron; then :V=\frac{\sqrt{4 a^2 b^2 c^2-a^2 X^2-b^2 Y^2-c^2 Z^2+X Y Z}}{12} where :\begin{align}X&=b^2+c^2-x^2, \\\ Y&=a^2+c^2-y^2, \\\ Z&=a^2+b^2-z^2. \end{align} The above formula uses six lengths of edges, and the following formula uses three lengths of edges and three angles. Let V be the volume of the tetrahedron. Since the four subtetrahedra fill the volume, we have V = \frac13A_1r+\frac13A_2r+\frac13A_3r+\frac13A_4r. ===Circumradius=== Denote the circumradius of a tetrahedron as R. Comparing this formula with that used to compute the volume of a parallelepiped, we conclude that the volume of a tetrahedron is equal to of the volume of any parallelepiped that shares three converging edges with it. thumb|3D model of regular tetrahedron. Given the distances between the vertices of a tetrahedron the volume can be computed using the Cayley–Menger determinant: :288 \cdot V^2 = \begin{vmatrix} 0 & 1 & 1 & 1 & 1 \\\ 1 & 0 & d_{12}^2 & d_{13}^2 & d_{14}^2 \\\ 1 & d_{12}^2 & 0 & d_{23}^2 & d_{24}^2 \\\ 1 & d_{13}^2 & d_{23}^2 & 0 & d_{34}^2 \\\ 1 & d_{14}^2 & d_{24}^2 & d_{34}^2 & 0 \end{vmatrix} where the subscripts represent the vertices and d is the pairwise distance between them – i.e., the length of the edge connecting the two vertices. The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. ===Coordinates for a regular tetrahedron=== The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges: :\left(\pm 1, 0, -\frac{1}{\sqrt{2}}\right) \quad \mbox{and} \quad \left(0, \pm 1, \frac{1}{\sqrt{2}}\right) Expressed symmetrically as 4 points on the unit sphere, centroid at the origin, with lower face parallel to the xy plane, the vertices are: v_1 = \left(\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right) v_2 = \left(-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right) v_3 = \left(-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right) v_4 = (0,0,1) with the edge length of \sqrt{\frac{8}{3}}. Then another volume formula is given by :V = \frac {d |(\mathbf{a} \times \mathbf{(b-c)})| } {6}. ===Properties analogous to those of a triangle=== The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. :V = \frac {abc} {6} \sqrt{1 + 2\cos{\alpha}\cos{\beta}\cos{\gamma}-\cos^2{\alpha}-\cos^2{\beta}-\cos^2{\gamma}} ==== Heron-type formula for the volume of a tetrahedron ==== right|thumb|240px|Six edge-lengths of Tetrahedron If , , , , , are lengths of edges of the tetrahedron (first three form a triangle; with opposite , opposite , opposite ), then : V = \frac{\sqrt {\,( - p + q + r + s)\,(p - q + r + s)\,(p + q - r + s)\,(p + q + r - s)}}{192\,u\,v\,w} where : \begin{align} p & = \sqrt {xYZ}, & q & = \sqrt {yZX}, & r & = \sqrt {zXY}, & s & = \sqrt {xyz}, \end{align} : \begin{align} X & = (w - U + v)\,(U + v + w), & x & = (U - v + w)\,(v - w + U), \\\ Y & = (u - V + w)\,(V + w + u), & y & = (V - w + u)\,(w - u + V), \\\ Z & = (v - W + u)\,(W + u + v), & z & = (W - u + v)\,(u - v + W). \end{align} ====Volume divider==== Any plane containing a bimedian (connector of opposite edges' midpoints) of a tetrahedron bisects the volume of the tetrahedron.Bottema, O. The three faces interior to the tetrahedron are: a right triangle with edges 1, \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{2}}, a right triangle with edges \sqrt{\tfrac{1}{3}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}}, and a right triangle with edges \sqrt{\tfrac{4}{3}}, \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{6}}. ===Space-filling tetrahedra=== A space-filling tetrahedron packs with directly congruent or enantiomorphous (mirror image) copies of itself to tile space. Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron Characteristics of the regular tetrahedron edge arc arc dihedral dihedral 𝒍 2 109°28′16″ \pi - 2\text{𝜿} 70°31′44″ \pi - 2\text{𝟁} 𝟀 \sqrt{\tfrac{4}{3}} \approx 1.155 70°31′44″ 2\text{𝜿} 60° \tfrac{\pi}{3} 𝝓 1 54°44′8″ \tfrac{\pi}{2} - \text{𝜿} 60° \tfrac{\pi}{3} 𝟁 \sqrt{\tfrac{1}{3}} \approx 0.577 54°44′8″ \tfrac{\pi}{2} - \text{𝜿} 60° \tfrac{\pi}{3} _0R/l \sqrt{\tfrac{3}{2}} \approx 1.225 _1R/l \sqrt{\tfrac{1}{2}} \approx 0.707 _2R/l \sqrt{\tfrac{1}{6}} \approx 0.408 \text{𝜿} 35°15′52″ \tfrac{\text{arc sec }3}{2} If the regular tetrahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths \sqrt{\tfrac{4}{3}}, 1, \sqrt{\tfrac{1}{3}} (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus \sqrt{\tfrac{3}{2}}, \sqrt{\tfrac{1}{2}}, \sqrt{\tfrac{1}{6}} (edges that are the characteristic radii of the regular tetrahedron). The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles about a vertex, is given by the following relation: : \Delta_i^2 = \Delta_j^2 + \Delta_k^2 + \Delta_l^2 - 2(\Delta_j\Delta_k\cos\theta_{il} + \Delta_j\Delta_l \cos\theta_{ik} + \Delta_k\Delta_l \cos\theta_{ij}) === Interior point === Let P be any interior point of a tetrahedron of volume V for which the vertices are A, B, C, and D, and for which the areas of the opposite faces are Fa, Fb, Fc, and Fd. The exterior face is a 60-90-30 triangle which is one-sixth of a tetrahedron face. These two tetrahedra's vertices combined are the vertices of a cube, demonstrating that the regular tetrahedron is the 3-demicube. (As a side-note: these two kinds of tetrahedron have the same volume.) This formula is obtained from dividing the tetrahedron into four tetrahedra whose points are the three points of one of the original faces and the incenter. The only two isometries are 1 and the rotation (12)(34), giving the group C2 isomorphic to the cyclic group, Z2. ==General properties== ===Volume=== The volume of a tetrahedron is given by the pyramid volume formula: :V = \frac13 A_0\,h \, where A0 is the area of the base and h is the height from the base to the apex. :Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1) :Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1) thumb|right|300px|Regular tetrahedron ABCD and its circumscribed sphere ===Angles and distances=== For a regular tetrahedron of edge length a: Face area A_0=\frac{\sqrt{3}}{4}a^2\, Surface areaCoxeter, Harold Scott MacDonald; Regular Polytopes, Methuen and Co., 1948, Table I(i) A=4\,A_0={\sqrt{3}}a^2\, Height of pyramidKöller, Jürgen, ""Tetrahedron"", Mathematische Basteleien, 2001 h=\frac{\sqrt{6}}{3}a=\sqrt{\frac23}\,a\, Centroid to vertex distance \frac34\,h = \frac{\sqrt{6}}{4}\,a = \sqrt{\frac{3}{8}}\,a\, Edge to opposite edge distance l=\frac{1}{\sqrt{2}}\,a\, Volume V=\frac13 A_0h =\frac{\sqrt{2}}{12}a^3=\frac{a^3}{6\sqrt{2}}\, Face-vertex-edge angle \arccos\left(\frac{1}{\sqrt{3}}\right) = \arctan\left(\sqrt{2}\right)\, (approx. 54.7356°) Face-edge-face angle, i.e., ""dihedral angle"" \arccos\left(\frac13\right) = \arctan\left(2\sqrt{2}\right)\, (approx. 70.5288°) Vertex-Center-Vertex angle, the angle between lines from the tetrahedron center to any two vertices. thumb|3D model of a truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. If all three pairs of opposite edges of a tetrahedron are perpendicular, then it is called an orthocentric tetrahedron. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron. ",0.011,9,"""10.0""",3.29527,1.4,C
+The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.,"The volume of a tricylinder is :V=8(2 - \sqrt{2}) r^3 and the surface area is :A=24(2 - \sqrt{2}) r^2. == More cylinders == With four cylinders, with axes connecting the vertices of a tetrahedron to the corresponding points on the other side of the solid, the volume is :V_4=12 \left( 2\sqrt{2} - \sqrt{6} \right) r^3 \, With six cylinders, with axes parallel to the diagonals of the faces of a cube, the volume is: :V_6=\frac{16}{3} \left( 3 + 2\sqrt{3} - 4\sqrt{2} \right) r^3 \, == See also == * Ungula == References == == External links == *A 3D model of Steinmetz solid in Google 3D Warehouse Category:Euclidean solid geometry It is well known that the relations of the volumes of a right circular cone, one half of a sphere and a right circular cylinder with same radii and heights are 1 : 2 : 3. The intersection of three cylinders with perpendicularly intersecting axes generates a surface of a solid with vertices where 3 edges meet and vertices where 4 edges meet. thumb|right|170px Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis ""parallel"" to the axis of revolution. The volume and the surface area of the curved triangles can be determined by similar considerations as it is done for the bicylinder above. right|thumb|236px|The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The volume of the 8 pyramids is: \textstyle 8 \times \frac{1}{3} r^2 \times r = \frac{8}{3} r^3 , and then we can calculate that the bicylinder volume is \textstyle (2 r)^3 - \frac{8}{3} r^3 = \frac{16}{3} r^3 == Tricylinder == 450px|thumb|Generating the surface of a tricylinder: At first two cylinders (red, blue) are cut. The key for the determination of volume and surface area is the observation that the tricylinder can be resampled by the cube with the vertices where 3 edges meet (s. diagram) and 6 curved pyramids (the triangles are parts of cylinder surfaces). One simply must solve each equation for before one inserts them into the integration formula. ==See also== *Solid of revolution *Shell integration ==References== * * *Frank Ayres, Elliott Mendelson. Hence the area of this development is thumb|cloister vault :B = \int_{0}^{\pi r} r\sin\left(\frac{\xi}{r}\right) \mathrm{d}\xi = 2r^2 and the total surface area is: :A=8\cdot B=16r^2. === Alternate proof of the volume formula === Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). The concept of included angle is discussed at: * Congruence of triangles * Solution of triangles The Euler line of an isosceles triangle is perpendicular to the triangle's base. The altitudes of a triangle are perpendicular to their respective bases. thumb|Steinmetz solid (intersection of two cylinders) In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. File:Sphere volume derivation using bicylinder.jpg|Zu Chongzhi's method (similar to Cavalieri's principle) for calculating a sphere's volume includes calculating the volume of a bicylinder. For one half of a bicylinder a similar statement is true: * The relations of the volumes of the inscribed square pyramid (a=2r, h=r, V=\frac{4}{3}r^3), the half bicylinder (V=\frac{8}{3} r^3) and the surrounding squared cuboid ( a= 2r, h=r, V=4r^3) are 1 : 2 : 3. ==== Using Multivariable Calculus ==== Consider the equations of the cylinders: x^2+z^2=r^2 x^2+y^2=r^2 The volume will be given by: V = \iiint_V \mathrm{d}z\mathrm{d}y\mathrm{d}x With the limits of integration: -\sqrt{r^2-x^2} \leqslant z \leqslant \sqrt{r^2-x^2} -\sqrt{r^2-x^2} \leqslant y \leqslant \sqrt{r^2-x^2} -r \leqslant x \leqslant r Substituting, we have: V = \int_{-r}^{r}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}}\int_{-\sqrt{r^2-x^2}}^{\sqrt{r^2-x^2}} \mathrm{d}z\mathrm{d}y\mathrm{d}x = 8r^3-\frac{8r^3}{3} = \frac{16r^3}{3} === Proof of the area formula === The surface area consists of two red and two blue cylindrical biangles. For example, the next figure shows the rotation along the -axis of the red ""leaf"" enclosed between the square- root and quadratic curves: thumb|Rotation about x-axis The volume of this solid is: :\pi\int_0^1\left(\left(\sqrt{x}\right)^2 - \left(x^2\right)^2 \right)\,dx\,. If is the value of a horizontal axis, then the volume equals :\pi\int_a^b\left(\left(h-R_\mathrm{O}(x)\right)^2 - \left(h-R_\mathrm{I}(x)\right)^2\right)\,dx\,. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. thumb|right|150px|Animated depiction of a bicylinder == Bicylinder == 300px|thumb|The generation of a bicylinder 180px|thumb|Calculating the volume of a bicylinder A bicylinder generated by two cylinders with radius r has the ;volume :V=\frac{16}{3} r^3 and the ;surface area :A=16 r^2. Wallis's Conical Edge with right|thumb|600px| Figure 2. File:Bicylinder and cube sections related by pyramids.png|Relationship of the area of a bicylinder section with a cube section The volume of the cube (red) minus the volume of the eight pyramids (blue) is the volume of the bicylinder (white). Wallis's Conical Edge with , In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt{a^2-b^2\cos^2u} where , and are constants. ",140,157.875,"""-0.16""",36,-1,D
+A projectile is fired with an initial speed of $200 \mathrm{~m} / \mathrm{s}$ and angle of elevation $60^{\circ}$. Find the speed at impact.,"Our equations of motion are now : x(t) = v t \cos \theta and : y(t) = y_0 + v t \sin \theta - \frac{1}{2} g t^2 Once again we solve for (t) in the case where the (y) position of the projectile is at zero (since this is how we defined our starting height to begin with) : 0 = y_0 + v t \sin \theta - \frac{1} {2} g t^2 Again by applying the quadratic formula we find two solutions for the time. Note that the source's y-y0 is replaced with the article's y0 : d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right) where * d is the total horizontal distance travelled by the projectile. * v is the velocity at which the projectile is launched * g is the gravitational acceleration--usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface * θ is the angle at which the projectile is launched * y0 is the initial height of the projectile If y0 is taken to be zero, meaning that the object is being launched on flat ground, the range of the projectile will simplify to: : d = \frac{v^2}{g} \sin 2\theta == Ideal projectile motion == Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. Let tg be any time when the height of the projectile is equal to its initial value. : 0 = v t \sin \theta - \frac{1} {2} g t^2 By factoring: : t = 0 or : t = \frac{2 v \sin \theta} {g} but t = T = time of flight : T = \frac{2 v \sin \theta} {g} The first solution corresponds to when the projectile is first launched. Composites Science and Technology 64:35-54. . is as follows: V_b=\frac{\pi\,\Gamma\,\sqrt{\rho_t\,\sigma_e}\,D^2\,T}{4\,m} \left [1+\sqrt{1+\frac{8\,m}{\pi\,\Gamma^2\,\rho_t\,D^2\,T}}\, \right ] where *V_b\, is the ballistic limit *\Gamma\, is a projectile constant determined experimentally *\rho_t\, is the density of the laminate *\sigma_e\, is the static linear elastic compression limit *D\, is the diameter of the projectile *T\, is the thickness of the laminate *m\, is the mass of the projectile Additionally, the ballistic limit for small-caliber into homogeneous armor by TM5-855-1 is: V_1= 19.72 \left [ \frac{7800 d^3 \left [ \left ( \frac{e_h}{d} \right) \sec \theta \right ]^{1.6}}{W_T} \right ]^{0.5} where *V_1 is the ballistic limit velocity in fps *d is the caliber of the projectile, in inches *e_h is the thickness of the homogeneous armor (valid from BHN 360 - 440) in inches *\theta is the angle of obliquity *W_T is the weight of the projectile, in lbs == References == Category:Ballistics thumb|The aftermath of a hypervelocity impact, with a projectile the same size as the one that impacted for scale Hypervelocity is very high velocity, approximately over 3,000 meters per second (6,700 mph, 11,000 km/h, 10,000 ft/s, or Mach 8.8). The horizontal position of the projectile is : x(t) = v t \cos \theta In the vertical direction : y(t) = v t \sin \theta - \frac{1} {2} g t^2 We are interested in the time when the projectile returns to the same height it originated. right|288px|thumb|Impact parameter and scattering angle In physics, the impact parameter is defined as the perpendicular distance between the path of a projectile and the center of a potential field created by an object that the projectile is approaching (see diagram). After several steps of algebraic manipulation : t = \frac {v \sin \theta} {g} \pm \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} The square root must be a positive number, and since the velocity and the sine of the launch angle can also be assumed to be positive, the solution with the greater time will occur when the positive of the plus or minus sign is used. Ballistic impact is a high velocity impact by a small mass object, analogous to runway debris or small arms fire. The maximum horizontal distance traveled by the projectile, neglecting air resistance, can be calculated as follows: Extract of page 132. Thus, the solution is : t = \frac {v \sin \theta} {g} + \frac {\sqrt{v^2 \sin^2 \theta + 2 g y_0}} {g} Solving for the range once again : d = \frac {v \cos \theta} {g} \left ( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2 g y_0} \right) To maximize the range at any height : \theta = \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} Checking the limit as y_0 approaches 0 : \lim_{y_0 \to 0} \arccos \sqrt{ \frac {2 g y_0 + v^2} {2 g y_0 + 2v^2}} = \frac {\pi} {4} ====Angle of impact==== The angle ψ at which the projectile lands is given by: : \tan \psi = \frac {-v_y(t_d)} {v_x(t_d)} = \frac {\sqrt { v^2 \sin^2 \theta + 2 g y_0 }} { v \cos \theta} For maximum range, this results in the following equation: : \tan^2 \psi = \frac { 2 g y_0 + v^2 } { v^2 } = C+1 Rewriting the original solution for θ, we get: : \tan^2 \theta = \frac { 1 - \cos^2 \theta } { \cos^2 \theta } = \frac { v^2 } { 2 g y_0 + v^2 } = \frac { 1 } { C + 1 } Multiplying with the equation for (tan ψ)^2 gives: : \tan^2 \psi \, \tan^2 \theta = \frac { 2 g y_0 + v^2 } { v^2 } \frac { v^2 } { 2 g y_0 + v^2 } = 1 Because of the trigonometric identity : \tan (\theta + \psi) = \frac { \tan \theta + \tan \psi } { 1 - \tan \theta \tan \psi } , this means that θ + ψ must be 90 degrees. == Actual projectile motion == In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered. === Projectile characteristics === Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. Here, the object that the projectile is approaching is a hard sphere with radius R. right|thumb|250 px|The path of this projectile launched from a height y0 has a range d. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance. === Derivations === A launch angle of 45 degrees displaces the projectile the farthest horizontally. Mathematical equations of motion are used to analyze projectile trajectory. The ballistic limit or limit velocity is the velocity required for a particular projectile to reliably (at least 50% of the time) penetrate a particular piece of material. (And see Trajectory of a projectile.) (And see Trajectory of a projectile.) The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity. Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially ranged weapon munitions such as bullets, unguided bombs, rockets or the like; the science or art of designing and accelerating projectiles so as to achieve a desired performance. The impact parameter is related to the scattering angle byLandau L. D. and Lifshitz E. M. (1976) Mechanics, 3rd. ed., Pergamon Press. (hardcover) and (softcover). : \theta = \pi - 2b\int_{r_\text{min}}^\infty \frac{dr}{r^2\sqrt{1 - (b/r)^2 - 2U/(mv_\infty^2)}}, where is the velocity of the projectile when it is far from the center, and is its closest distance from the center. ==Scattering from a hard sphere== The simplest example illustrating the use of the impact parameter is in the case of scattering from a sphere. When b > R , the projectile misses the hard sphere. ",7,2.50,"""200.0""",4943,1.775,C
 "A force of $30 \mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $12 \mathrm{~cm}$ to $20 \mathrm{~cm}$ ?
-","Let be the amount by which the free end of the spring was displaced from its ""relaxed"" position (when it is not being stretched). An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. The force an ideal spring would exert is exactly proportional to its extension or compression. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. The force is applied through the ends of the spring. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. The manufacture normally specifies the spring rate. Explaining the Power of Springing Bodies, London, 1678. as: (""as the extension, so the force"" or ""the extension is proportional to the force""). The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. : F is the resulting force vector – the magnitude and direction of the restoring force the spring exerts : k is the rate, spring constant or force constant of the spring, a constant that depends on the spring's material and construction. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. thumb|Hooke's law: the force is proportional to the extension In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. Force of fully compressed spring : F_{max} = \frac{E d^4 (L-n d)}{16 (1+ u) (D-d)^3 n} \ where : E – Young's modulus : d – spring wire diameter : L – free length of spring : n – number of active windings : u – Poisson ratio : D – spring outer diameter ==Zero-length springs== thumb|left|120px|Simplified LaCoste suspension using a zero-length spring thumb|upright|Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant ""Zero-length spring"" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. The work can be split into two terms \delta W = \delta W_\mathrm{s} + \delta W_\mathrm{b} where is the work done by surface forces while is the work done by body forces. Coil springs can be made from various materials, including steel, brass, and bronze. == Spring rate == Spring rate is the measurement of how much a coil spring can hold until it compresses . To meet the demands of today's consumers, spring manufacturers must be able to produce springs in a wide range of sizes and shapes. A torsion spring's rate is in units of torque divided by angle, such as N·m/rad or ft·lbf/degree. A spring (made by winding a wire around a cylinder) is of two types: * Tension or extension springs are designed to become longer under load. ",418,7.00,3.51,30,3.2,E
-"Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\eta=0.027, R=0.008 \mathrm{~cm}$, $I=2 \mathrm{~cm}$, and $P=4000$ dynes $/ \mathrm{cm}^2$.","As , Poiseuille solution is recovered. ==Poiseuille flow in an annular section== thumb|Poiseuille flow in annular section If is the inner cylinder radii and is the outer cylinder radii, with constant applied pressure gradient between the two ends , the velocity distribution and the volume flux through the annular pipe are : \begin{align} u(r) &= \frac{G}{4\mu}\left(R_1^2-r^2\right) + \frac{G}{4\mu}\left(R_2^2-R_1^2\right) \frac{\ln r/R_1}{\ln R_2/R_1},\\\\[6pt] Q &= \frac{G \pi}{8\mu}\left[R_2^4-R_1^4- \frac{\left(R_2^2-R_1^2\right)^2}{\ln R_2/R_1}\right] . \end{align} When , , the original problem is recovered. ==Poiseuille flow in a pipe with an oscillating pressure gradient== Flow through pipes with an oscillating pressure gradient finds applications in blood flow through large arteries. Both Ohm's law and Poiseuille's law illustrate transport phenomena. == Medical applications – intravenous access and fluid delivery == The Hagen–Poiseuille equation is useful in determining the vascular resistance and hence flow rate of intravenous (IV) fluids that may be achieved using various sizes of peripheral and central cannulas. Hence the volumetric flow rate at the pipe outlet is given by : Q_2 =\frac{\pi R^4}{16 \mu L} \left( \frac{ p_1^2-p_2^2}{p_2}\right) = \frac{\pi R^4 \left( p_1-p_2\right)}{8 \mu L} \frac{\left( p_1+p_2\right)}{2 p_2}. However, the viscosity of blood will cause additional pressure drop along the direction of flow, which is proportional to length traveled (as per Poiseuille's law). The volumetric flow rate is usually expressed at the outlet pressure. It is also useful to understand that viscous fluids will flow slower (e.g. in blood transfusion). ==See also== * Couette flow * Darcy's law * Pulse * Wave * Hydraulic circuit ==Cited references== ==References== *. *. *. == External links == *Poiseuille's law for power-law non-Newtonian fluid *Poiseuille's law in a slightly tapered tube *Hagen–Poiseuille equation calculator Category:Equations of fluid dynamics Category:Mathematics in medicine Mass flow rate can be calculated by multiplying the volume flow rate by the mass density of the fluid, ρ. Finally, put this expression in the form of a differential equation, dropping the term quadratic in . : \frac{1}{\mu} \frac{\Delta p}{\Delta x} = \frac{\mathrm{d}^2 v}{\mathrm{d}r^2} + \frac{1}{r} \frac{\mathrm{d}v}{\mathrm{d}r} The above equation is the same as the one obtained from the Navier–Stokes equations and the derivation from here on follows as before. ===Startup of Poiseuille flow in a pipe=== When a constant pressure gradient is applied between two ends of a long pipe, the flow will not immediately obtain Poiseuille profile, rather it develops through time and reaches the Poiseuille profile at steady state. The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen–Poiseuille law to be valid. Over a short section of the pipe, the gas flowing through the pipe can be assumed to be incompressible so that Poiseuille law can be used locally, :-\frac{\mathrm{d}p}{\mathrm{d}x} = \frac{8\mu Q}{\pi R^4} = \frac{8\mu Q_2p_2}{\pi p R^4} \quad \Rightarrow \quad -p\frac{\mathrm{d}p}{\mathrm{d}x} = \frac{8\mu Q_2p_2}{\pi R^4}. An artery (plural arteries) ()ἀρτηρία, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus is a blood vessel in humans and most animals that takes blood away from the heart to one or more parts of the body (tissues, lungs, brain etc.). The Navier–Stokes equations reduce to :\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = - \frac{G}{\mu} with no-slip condition on both walls :u(0)=0, \quad u(h)=0 Therefore, the velocity distribution and the volume flow rate per unit length are :u(y) = \frac{G}{2\mu} y(h-y), \quad Q = \frac{Gh^3}{12\mu}. ==Poiseuille flow through some non-circular cross-sections== Joseph Boussinesq derived the velocity profile and volume flow rate in 1868 for rectangular channel and tubes of equilateral triangular cross-section and for elliptical cross-section. Poiseuille's law was later in 1891 extended to turbulent flow by L. R. Wilberforce, based on Hagenbach's work. == Derivation == The Hagen–Poiseuille equation can be derived from the Navier–Stokes equations. In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. The volume flow rate is calculated by multiplying the flow velocity of the mass elements, v, by the cross-sectional vector area, A. Mass flow rate can also be calculated by: :\dot m = \rho \cdot \dot V = \rho \cdot \mathbf{v} \cdot \mathbf{A} = \mathbf{j}_{\rm m} \cdot \mathbf{A} where: *\dot V or Q = Volume flow rate, *ρ = mass density of the fluid, *v = Flow velocity of the mass elements, *A = cross-sectional vector area/surface, * jm = mass flux. Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. The reason why Poiseuille's law leads to a different formula for the resistance is the difference between the fluid flow and the electric current. Mass flow rate can be used to calculate the energy flow rate of a fluid: :\dot{E}=\dot{m}e where: * e = unit mass energy of a system Energy flow rate has SI units of kilojoule per second or kilowatt. == See also == * Continuity equation * Fluid dynamics * Mass flow controller * Mass flow meter * Mass flux * Orifice plate * Standard cubic centimetres per minute * Thermal mass flow meter * Volumetric flow rate ==References== Category:Fluid dynamics Category:Temporal rates Category:Mass If we introduce a new dependent variable as :U = u +\frac{G}{4\mu}\left(y^2+z^2\right), then it is easy to see that the problem reduces to that integrating a Laplace equation :\frac{\partial^2 U}{\partial y^2}+\frac{\partial^2 U}{\partial z^2}=0 satisfying the condition :U = \frac{G}{4\mu}\left(y^2+z^2\right) on the wall. == Poiseuille's equation for an ideal isothermal gas== For a compressible fluid in a tube the volumetric flow rate and the axial velocity are not constant along the tube; but the mass flow rate is constant along the tube length. Since the net force acting on the fluid is equal to , where , i.e. , then from Poiseuille's law, it follows that :\Delta F = \frac{8 \mu LQ}{r^2}. Sometimes these equations are used to define the mass flow rate. If the pipe is too short, the Hagen–Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli's principle, under less restrictive conditions, by :\begin{align} \Delta p = \frac{1}{2} \rho \overline{v}_\text{max}^2 &= \frac{1}{2} \rho \left(\frac{Q_\text{max}}{\pi R^2}\right)^2 \\\\[6pt] \rightarrow \, \, \, Q_\max{} &= \pi R^2 \sqrt\frac{2 \Delta p}{\rho}, \end{align} because it is impossible to have negative (absolute) pressure (not to be confused with gauge pressure) in an incompressible flow. ==Relation to the Darcy–Weisbach equation== Normally, Hagen–Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic. ",1.19,169,-1.46,0.0526315789, 135.36,A
+","Let be the amount by which the free end of the spring was displaced from its ""relaxed"" position (when it is not being stretched). An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. The force an ideal spring would exert is exactly proportional to its extension or compression. The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. If a spring has a rate of 100 then the spring would compress 1 inch (2.54 cm) with of load. == Types== Types of coil spring are: * Tension/extension coil springs, designed to resist stretching. The force is applied through the ends of the spring. When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The change in length may be expressed as \Delta L = \varepsilon L = \frac{F L}{A E}\,. === Spring energy === The potential energy stored in a spring is given by U_\mathrm{el}(x) = \tfrac 1 2 kx^2 which comes from adding up the energy it takes to incrementally compress the spring. The manufacture normally specifies the spring rate. Explaining the Power of Springing Bodies, London, 1678. as: (""as the extension, so the force"" or ""the extension is proportional to the force""). The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. : F is the resulting force vector – the magnitude and direction of the restoring force the spring exerts : k is the rate, spring constant or force constant of the spring, a constant that depends on the spring's material and construction. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) system of linear equations or equivalently as an energy minimization problem. == Known spring lengths == If the nominal lengths, L, of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. ===Simple harmonic motion=== Since force is equal to mass, m, times acceleration, a, the force equation for a spring obeying Hooke's law looks like: :F = m a \quad \Rightarrow \quad -k x = m a. thumb|Hooke's law: the force is proportional to the extension In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of the spring (i.e., its stiffness), and is small compared to the total possible deformation of the spring. Force of fully compressed spring : F_{max} = \frac{E d^4 (L-n d)}{16 (1+ u) (D-d)^3 n} \ where : E – Young's modulus : d – spring wire diameter : L – free length of spring : n – number of active windings : u – Poisson ratio : D – spring outer diameter ==Zero-length springs== thumb|left|120px|Simplified LaCoste suspension using a zero-length spring thumb|upright|Spring length L vs force F graph of ordinary (+), zero-length (0) and negative-length (−) springs with the same minimum length L0 and spring constant ""Zero-length spring"" is a term for a specially designed coil spring that would exert zero force if it had zero length; if there were no constraint due to the finite wire diameter of such a helical spring, it would have zero length in the unstretched condition. \, The mass of the spring is small in comparison to the mass of the attached mass and is ignored. The work can be split into two terms \delta W = \delta W_\mathrm{s} + \delta W_\mathrm{b} where is the work done by surface forces while is the work done by body forces. Coil springs can be made from various materials, including steel, brass, and bronze. == Spring rate == Spring rate is the measurement of how much a coil spring can hold until it compresses . To meet the demands of today's consumers, spring manufacturers must be able to produce springs in a wide range of sizes and shapes. A torsion spring's rate is in units of torque divided by angle, such as N·m/rad or ft·lbf/degree. A spring (made by winding a wire around a cylinder) is of two types: * Tension or extension springs are designed to become longer under load. ",418,7.00,"""3.51""",30,3.2,E
+"Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\eta=0.027, R=0.008 \mathrm{~cm}$, $I=2 \mathrm{~cm}$, and $P=4000$ dynes $/ \mathrm{cm}^2$.","As , Poiseuille solution is recovered. ==Poiseuille flow in an annular section== thumb|Poiseuille flow in annular section If is the inner cylinder radii and is the outer cylinder radii, with constant applied pressure gradient between the two ends , the velocity distribution and the volume flux through the annular pipe are : \begin{align} u(r) &= \frac{G}{4\mu}\left(R_1^2-r^2\right) + \frac{G}{4\mu}\left(R_2^2-R_1^2\right) \frac{\ln r/R_1}{\ln R_2/R_1},\\\\[6pt] Q &= \frac{G \pi}{8\mu}\left[R_2^4-R_1^4- \frac{\left(R_2^2-R_1^2\right)^2}{\ln R_2/R_1}\right] . \end{align} When , , the original problem is recovered. ==Poiseuille flow in a pipe with an oscillating pressure gradient== Flow through pipes with an oscillating pressure gradient finds applications in blood flow through large arteries. Both Ohm's law and Poiseuille's law illustrate transport phenomena. == Medical applications – intravenous access and fluid delivery == The Hagen–Poiseuille equation is useful in determining the vascular resistance and hence flow rate of intravenous (IV) fluids that may be achieved using various sizes of peripheral and central cannulas. Hence the volumetric flow rate at the pipe outlet is given by : Q_2 =\frac{\pi R^4}{16 \mu L} \left( \frac{ p_1^2-p_2^2}{p_2}\right) = \frac{\pi R^4 \left( p_1-p_2\right)}{8 \mu L} \frac{\left( p_1+p_2\right)}{2 p_2}. However, the viscosity of blood will cause additional pressure drop along the direction of flow, which is proportional to length traveled (as per Poiseuille's law). The volumetric flow rate is usually expressed at the outlet pressure. It is also useful to understand that viscous fluids will flow slower (e.g. in blood transfusion). ==See also== * Couette flow * Darcy's law * Pulse * Wave * Hydraulic circuit ==Cited references== ==References== *. *. *. == External links == *Poiseuille's law for power-law non-Newtonian fluid *Poiseuille's law in a slightly tapered tube *Hagen–Poiseuille equation calculator Category:Equations of fluid dynamics Category:Mathematics in medicine Mass flow rate can be calculated by multiplying the volume flow rate by the mass density of the fluid, ρ. Finally, put this expression in the form of a differential equation, dropping the term quadratic in . : \frac{1}{\mu} \frac{\Delta p}{\Delta x} = \frac{\mathrm{d}^2 v}{\mathrm{d}r^2} + \frac{1}{r} \frac{\mathrm{d}v}{\mathrm{d}r} The above equation is the same as the one obtained from the Navier–Stokes equations and the derivation from here on follows as before. ===Startup of Poiseuille flow in a pipe=== When a constant pressure gradient is applied between two ends of a long pipe, the flow will not immediately obtain Poiseuille profile, rather it develops through time and reaches the Poiseuille profile at steady state. The ratio of length to radius of a pipe should be greater than one forty-eighth of the Reynolds number for the Hagen–Poiseuille law to be valid. Over a short section of the pipe, the gas flowing through the pipe can be assumed to be incompressible so that Poiseuille law can be used locally, :-\frac{\mathrm{d}p}{\mathrm{d}x} = \frac{8\mu Q}{\pi R^4} = \frac{8\mu Q_2p_2}{\pi p R^4} \quad \Rightarrow \quad -p\frac{\mathrm{d}p}{\mathrm{d}x} = \frac{8\mu Q_2p_2}{\pi R^4}. An artery (plural arteries) ()ἀρτηρία, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus is a blood vessel in humans and most animals that takes blood away from the heart to one or more parts of the body (tissues, lungs, brain etc.). The Navier–Stokes equations reduce to :\frac{\mathrm{d}^2 u}{\mathrm{d}y^2} = - \frac{G}{\mu} with no-slip condition on both walls :u(0)=0, \quad u(h)=0 Therefore, the velocity distribution and the volume flow rate per unit length are :u(y) = \frac{G}{2\mu} y(h-y), \quad Q = \frac{Gh^3}{12\mu}. ==Poiseuille flow through some non-circular cross-sections== Joseph Boussinesq derived the velocity profile and volume flow rate in 1868 for rectangular channel and tubes of equilateral triangular cross-section and for elliptical cross-section. Poiseuille's law was later in 1891 extended to turbulent flow by L. R. Wilberforce, based on Hagenbach's work. == Derivation == The Hagen–Poiseuille equation can be derived from the Navier–Stokes equations. In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. The volume flow rate is calculated by multiplying the flow velocity of the mass elements, v, by the cross-sectional vector area, A. Mass flow rate can also be calculated by: :\dot m = \rho \cdot \dot V = \rho \cdot \mathbf{v} \cdot \mathbf{A} = \mathbf{j}_{\rm m} \cdot \mathbf{A} where: *\dot V or Q = Volume flow rate, *ρ = mass density of the fluid, *v = Flow velocity of the mass elements, *A = cross-sectional vector area/surface, * jm = mass flux. Considering flow through porous media, a special quantity, superficial mass flow rate, can be introduced. The reason why Poiseuille's law leads to a different formula for the resistance is the difference between the fluid flow and the electric current. Mass flow rate can be used to calculate the energy flow rate of a fluid: :\dot{E}=\dot{m}e where: * e = unit mass energy of a system Energy flow rate has SI units of kilojoule per second or kilowatt. == See also == * Continuity equation * Fluid dynamics * Mass flow controller * Mass flow meter * Mass flux * Orifice plate * Standard cubic centimetres per minute * Thermal mass flow meter * Volumetric flow rate ==References== Category:Fluid dynamics Category:Temporal rates Category:Mass If we introduce a new dependent variable as :U = u +\frac{G}{4\mu}\left(y^2+z^2\right), then it is easy to see that the problem reduces to that integrating a Laplace equation :\frac{\partial^2 U}{\partial y^2}+\frac{\partial^2 U}{\partial z^2}=0 satisfying the condition :U = \frac{G}{4\mu}\left(y^2+z^2\right) on the wall. == Poiseuille's equation for an ideal isothermal gas== For a compressible fluid in a tube the volumetric flow rate and the axial velocity are not constant along the tube; but the mass flow rate is constant along the tube length. Since the net force acting on the fluid is equal to , where , i.e. , then from Poiseuille's law, it follows that :\Delta F = \frac{8 \mu LQ}{r^2}. Sometimes these equations are used to define the mass flow rate. If the pipe is too short, the Hagen–Poiseuille equation may result in unphysically high flow rates; the flow is bounded by Bernoulli's principle, under less restrictive conditions, by :\begin{align} \Delta p = \frac{1}{2} \rho \overline{v}_\text{max}^2 &= \frac{1}{2} \rho \left(\frac{Q_\text{max}}{\pi R^2}\right)^2 \\\\[6pt] \rightarrow \, \, \, Q_\max{} &= \pi R^2 \sqrt\frac{2 \Delta p}{\rho}, \end{align} because it is impossible to have negative (absolute) pressure (not to be confused with gauge pressure) in an incompressible flow. ==Relation to the Darcy–Weisbach equation== Normally, Hagen–Poiseuille flow implies not just the relation for the pressure drop, above, but also the full solution for the laminar flow profile, which is parabolic. ",1.19,169,"""-1.46""",0.0526315789, 135.36,A
 "In this problem, 3.00 mol of liquid mercury is transformed from an initial state characterized by $T_i=300 . \mathrm{K}$ and $P_i=1.00$ bar to a final state characterized by $T_f=600 . \mathrm{K}$ and $P_f=3.00$ bar.
-a. Calculate $\Delta S$ for this process; $\beta=1.81 \times 10^{-4} \mathrm{~K}^{-1}, \rho=13.54 \mathrm{~g} \mathrm{~cm}^{-3}$, and $C_{P, m}$ for $\mathrm{Hg}(l)=27.98 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$.","System A12 A21 Acetone(1)-Chloroform(2) -0.8404 -0.5610 Acetone(1)-Methanol(2) 0.6184 0.5788 Acetone(1)-Water(2) 2.0400 1.5461 Carbon tetrachloride(1)-Benzene (2) 0.0948 0.0922 Chloroform(1)-Methanol(2) 0.8320 1.7365 Ethanol(1)-Benzene(2) 1.8362 1.4717 Ethanol(1)-Water(2) 1.6022 0.7947 ==See also== * Van Laar equation ==Literature== ==External links== *Ternary systems Margules Category:Physical chemistry Category:Thermodynamic models When A>2 the system starts to demix in two liquids at 50/50 composition; i.e. plait point is at 50 mol%. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. Since: :A = \ln \gamma_1^\infty = \ln \gamma_2^\infty : \gamma_1^\infty = \gamma_2^\infty > \exp(2) \approx 7.38 For asymmetric binary systems, A12≠A21, the liquid-liquid separation always occurs for : :A_{21} + A_{12} > 4 Or equivalently: :\gamma_1^\infty \gamma_2^\infty > \exp(4) \approx 54.6 The plait point is not located at 50 mol%. ΔP (Delta P) is a mathematical term symbolizing a change (Δ) in pressure (P). ==Uses== *Young–Laplace equation ===Darcy–Weisbach equation=== Given that the head loss hf expresses the pressure loss Δp as the height of a column of fluid, :\Delta p = \rho \cdot g \cdot h_f where ρ is the density of the fluid. This yields, when applied only to the first term and using the Gibbs–Duhem equation,:Phase Equilibria in Chemical Engineering, Stanley M. Walas, (1985) p180 Butterworth Publ. : \left\\{\begin{matrix} \ln\ \gamma_1=[A_{12}+2(A_{21}-A_{12})x_1]x^2_2 \\\ \ln\ \gamma_2=[A_{21}+2(A_{12}-A_{21})x_2]x^2_1 \end{matrix}\right. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup In chemical engineering the Margules Gibbs free energy model for liquid mixtures is better known as the Margules activity or activity coefficient model. thumb|The Mollier enthalpy–entropy diagram for water and steam. In an isenthalpic process, the enthalpy is constant. Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Calculating compliance on minute volume (VE: ΔV is always defined by tidal volume (VT), but ΔP is different for the measurement of dynamic vs. static compliance. ====Dynamic compliance (Cdyn)==== :C_{dyn} = \frac{V_T}\mathrm{PIP-PEEP} where PIP = peak inspiratory pressure (the maximum pressure during inspiration), and PEEP = positive end expiratory pressure. So the expansion process in a turbine can be easily calculated using the h–s chart when the process is considered to be ideal (which is the case normally when calculating enthalpies, entropies, etc. The enthalpy coordinate is skewed and the constant enthalpy lines are parallel and evenly spaced. ==See also== *Thermodynamic diagrams *Contour line *Phase diagram == References == Category:Thermodynamics Category:Entropy de:Wasserdampf#h-s- Diagramm The leading term X_1X_2 assures that the excess Gibbs energy becomes zero at x1=0 and x1=1. === Activity coefficient === The activity coefficient of component i is found by differentiation of the excess Gibbs energy towards xi. Although the model is old it has the characteristic feature to describe extrema in the activity coefficient, which modern models like NRTL and Wilson cannot. ==Equations== === Excess Gibbs free energy === Margules expressed the intensive excess Gibbs free energy of a binary liquid mixture as a power series of the mole fractions xi: : \frac{G^{ex}}{RT}=X_1 X_2 (A_{21} X_1 +A_{12} X_2) + X_1^2 X_2^2 (B_{21}X_1+ B_{12} X_2) + ... + X_1^m X_2^m (M_{21}X_1+ M_{12} X_2) In here the A, B are constants, which are derived from regressing experimental phase equilibria data. When A_{12}=A_{21}=A, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model: : \left\\{\begin{matrix} \ln\ \gamma_1=Ax^2_2 \\\ \ln\ \gamma_2=Ax^2_1 \end{matrix}\right. The work done in a process on vapor cycles is represented by length of , so it can be measured directly, whereas in a T–s diagram it has to be computed using thermodynamic relationship between thermodynamic properties. The Margules activity model is a simple thermodynamic model for the excess Gibbs free energy of a liquid mixture introduced in 1895 by Max Margules.https://archive.org/details/sitzungsbericht10wiengoog After Lewis had introduced the concept of the activity coefficient, the model could be used to derive an expression for the activity coefficients \gamma_i of a compound i in a liquid, a measure for the deviation from ideal solubility, also known as Raoult's law. The Mollier diagram coordinates are enthalpy h and humidity ratio x. The parameters for a description at 20 °C are A12=0.6298 and A21=1.9522. ",58.2, -1,3.54,358800,6.9,A
-"For an ensemble consisting of 1.00 moles of particles having two energy levels separated by $h v=1.00 \times 10^{-20} \mathrm{~J}$, at what temperature will the internal energy of this system equal $1.00 \mathrm{~kJ}$ ?","The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T : U = C_V n T, where C_V is the isochoric (at constant volume) molar heat capacity of the gas. The internal energy relative to the mass with unit J/kg is the specific internal energy. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U(n,T). The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree: : U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots), where \alpha is a factor describing the growth of the system. The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence: : U = \sum_{i=1}^N p_i \,E_i. At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). The differential internal energy may be written as :\mathrm{d} U = \frac{\partial U}{\partial S} \mathrm{d} S + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - P \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i, which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure P to be the negative of the similar derivative with respect to volume V, : T = \frac{\partial U}{\partial S}, : P = -\frac{\partial U}{\partial V}, and where the coefficients \mu_{i} are the chemical potentials for the components of type i in the system. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy. MJ/kg may refer to: * megajoules per kilogram * Specific kinetic energy * Heat of fusion * Heat of combustion In statistical mechanics, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from translations, rotations, and vibrations, and of the potential energies associated with microscopic forces, including chemical bonds. The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains. kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties S, V, n (entropy, volume, mass). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be sensible. Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. The expression relating changes in internal energy to changes in temperature and volume is : \mathrm{d}U =C_{V} \, \mathrm{d}T +\left[T\left(\frac{\partial P}{\partial T}\right)_{V} - P\right] \mathrm{d}V. For a closed system, with transfers only as heat and work, the change in the internal energy is : \mathrm{d} U = \delta Q - \delta W, expressing the first law of thermodynamics. When a closed system receives energy as heat, this energy increases the internal energy. ",8.8,8,7.0,3.8,449,E
-"The sedimentation coefficient of lysozyme $\left(\mathrm{M}=14,100 \mathrm{~g} \mathrm{~mol}^{-1}\right)$ in water at $20^{\circ} \mathrm{C}$ is $1.91 \times 10^{-13} \mathrm{~s}$ and the specific volume is $0.703 \mathrm{~cm}^3 \mathrm{~g}^{-1}$. The density of water at this temperature is $0.998 \mathrm{~g} \mathrm{~cm}^{-3}$ and $\eta=1.002 \mathrm{cP}$. Assuming lysozyme is spherical, what is the radius of this protein?","The commonly used molar ratio of lysozyme and mPEG-aldehyde is 1:6 or 1:6.67. The hydrodynamic radius of a macromolecule or colloid particle is R_{\rm hyd}. The protein and mPEG-aldehyde are dissolved using a sodium phosphate buffer with sodium cyanoborohydride, which acts as a reducing agent and conditions the aldehyde group of mPEG-aldehyde to have a strong affinity towards the lysine residue on the N-terminal of lysozyme. The CD spectra range from 189 - 260 nm with a pitch of 0.1 nm showed no significant change in the secondary structure of the intact and PEGylated lysozyme. === Enzymatic activity assay === ==== Glycol chitosan ==== Enzymatic activity of intact and PEGylated lysozyme can be evaluated using glycol chitosan by reacting 1 mL of 0.05% (w/v) glycol chitosan in 100 mM of pH 5.5 acetate buffer and 100 μL of the intact or PEGylated protein at 40 °C for 30 min and subsequently adding 2 mL of 0.5 M sodium carbonate with 1 μg of potassium ferricyanide. There is a negative correlation between molecular weight and the retention time of the PEGylated protein in the chromatogram; larger protein, or more PEGylated protein elutes first, and smaller protein, or intact protein the latest. == Characterization == === Identification === The most common analyses for identifying intact and PEGylated lysozyme can be achieved via size-exclusion chromatography (high-performance liquid chromatography or HPLC), SDS-PAGE and Matrix-assisted laser desorption/ionization (MALDI). === Conformation === The secondary structure of intact and PEGylated lysozyme can be characterized by circular dichroism (CD) spectroscopy. Previous works on lysozyme PEGylation showed various chromatographic schemes in order to purify PEGylated lysozyme, which included ion exchange chromatography, hydrophobic interaction chromatography, and size-exclusion chromatography (fast protein liquid chromatography), and proved its stable conformation via circular dichroism and improved thermal stability by enzymatic activity assays, SDS-PAGE, and size- exclusion chromatography (high-performance liquid chromatography). == Methodology == === PEGylation === The chemical modification of lysozyme by PEGylation involves the addition of methoxy-PEG-aldehyde (mPEG-aldehyde) with varying molecular sizes, ranging from 2 kDa to 40 kDa, to the protein. Due to the high pI of lysozyme (pI = 10.7), cation exchange chromatography is used. As the enzymatic activity to hydrolyze β-1,4- N-acetylglucosamine linkage was retained after PEGylation, there was no decay in the enzymatic activity by increasing the degree of PEGylation. ==== Micrococcus lysodeikticus ==== By the measurement of decrease in turbidity of M. lysodeikticus by incubating it with lysozyme, enzymatic activity can be evaluated. 7.5 μL of 0.1 - 1 mg/mL proteins is added to 200 μL of M. lysodeikticus at its optical density (OD) of 1.7 AU, and the mixture is measured at 450 nm periodically for reaction rate calculation. Inside the Vainshtein radius; see also :r_V = l_\text{P}\left( \frac{m_\text{P}^3M}{m^4_G} \right)^\frac{1}{5} :with Planck length l_\text{P} and Planck mass m_\text{P} the gravitational field around a body of mass M is the same in a theory where the graviton mass m_G is zero and where it's very small because the helicity 0 degree of freedom becomes effective on distance scales r \gg r_V. ==See also== * == References == Category:Quantum gravity In its full form, the Flory–Rehner equation is written as: : -\left[ \ln{\left(1 - u_2\right)}+ u_2+ \chi_1 u_2^2 \right] = \frac{V_1}{\bar{ u}M_c} \left(1-\frac{2M_c}{M}\right) \left( u_2^\frac{1}{3}-\frac{ u_2}{2}\right) where, \bar{ u} is the specific volume of the polymer, M is the primary molecular mass, and M_c is the average molecular mass between crosslinks or the network parameter. == Flory–Rehner theory == The Flory–Rehner theory gives the change of free energy upon swelling of the polymer gel similar to the Flory–Huggins solution theory: :\Delta F = \Delta F_\mathrm{mix} + \Delta F_\mathrm{elastic}. thumb|right| T4 lysozyme ribbon schematic (from PDB 1LZM) Brian W. Matthews is a biochemist and biophysicist educated at the University of Adelaide, contributor to x-ray crystallographic methodology at the University of Cambridge, and since 1970 at the University of Oregon as Professor of Physics and HHMI investigator in the Institute of Molecular Biology. Lysozyme has six lysine residues which are accessible for PEGylation reactions. Note that in biophysics, hydrodynamic radius refers to the Stokes radius, or commonly to the apparent Stokes radius obtained from size exclusion chromatography. Lysozyme PEGylation is the covalent attachment of Polyethylene glycol (PEG) to Lysozyme, which is one of the most widely investigated PEGylated proteins. The theoretical hydrodynamic radius R_{\rm hyd} arises in the study of the dynamic properties of polymers moving in a solvent. In the continuum limit, where the mean free path of the particle is negligible compared to a characteristic length scale of the particle, the hydrodynamic radius is defined as the radius that gives the same magnitude of the frictional force, \boldsymbol{F}_d as that of a sphere with that radius, i.e. :\boldsymbol{F}_d = 6\pi\mu R_{hyd}\boldsymbol{v} where \mu is the viscosity of the surrounding fluid, and \boldsymbol{v} is the velocity of the particle. He created hundreds of mutants of T4 lysozyme (making it the commonest structure in the PDB), determined their structure by x-ray crystallography and measured their melting temperatures. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The theoretical hydrodynamic radius R_{\rm hyd} was originally an estimate by John Gamble Kirkwood of the Stokes radius of a polymer, and some sources still use hydrodynamic radius as a synonym for the Stokes radius. Thus, the PEGylation of lysozyme, or lysozyme PEGylation, can be a good model system for the PEGylation of other proteins with enzymatic activities by showing the enhancement of its physical and thermal stability while retaining its activity. In polymer science Flory–Rehner equation is an equation that describes the mixing of polymer and liquid molecules as predicted by the equilibrium swelling theory of Flory and Rehner. The Flory–Rehner equation is written as: : -\left[ \ln{\left(1 - u_2\right)}+ u_2+ \chi_1 u_2^2 \right] = V_1 n \left( u_2^\frac{1}{3}-\frac{ u_2}{2}\right) where, u_2 is the volume fraction of polymer in the swollen mass, V_1 the molar volume of the solvent, n is the number of network chain segments bounded on both ends by crosslinks, and \chi_1 is the Flory solvent-polymer interaction term. ",41.40,0.332,2.534324263,14,1.94,E
-Determine the standard molar entropy of $\mathrm{Ne}$ and $\mathrm{Kr}$ under standard thermodynamic conditions.,"The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. Factor (J K) Value Item 10 9.5699 J K Entropy equivalent of one bit of information, equal to k times ln(2) page 72 (page 5 of pdf) 10 1.381 Boltzmann Constant 10 5.74 J K Standard entropy of 1 mole of graphite 10 ≈ 10 J K Entropy of the Sun (given as ≈ 10 erg K in Bekenstein (1973)) 10 1.5 × 10 J K Entropy of a black hole of one solar mass (given as ≈ 10 erg K in Bekenstein (1973)) 10 4.3 × 10 J K One estimate of the theoretical maximum entropy of the universeCalculated: 3.1e104 * k = 3.1e104 * 1.381e-23 J/K = 4.3e81 J/K ==See also== *Orders of magnitude (data), relates to information entropy *Order of magnitude (terminology) ==References== Entropy Category:Entropy The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system by k. Approximate values of kT at 298 K Units kT = J kT = pN⋅nm kT = cal kT = meV kT=-174 dBm/Hz kT/hc ≈ cm−1 kT/e = 25.7 mV RT = kT ⋅ NA = kJ⋅mol−1 RT = 0.593 kcal⋅mol−1 h/kT = 0.16 ps kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. The following list shows different orders of magnitude of entropy. This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. Negative values for indicate that entropy decreases on forming the transition state, which often indicates an associative mechanism in which two reaction partners form a single activated complex.James H. Espenson Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002), p.156-160 == Derivation == It is possible to obtain entropy of activation using Eyring equation. Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. Chemical equations make use of the standard molar entropy of reactants and products to find the standard entropy of reaction: :{\Delta S^\circ}_{rxn} = S^\circ_{products} - S^\circ_{reactants} The standard entropy of reaction helps determine whether the reaction will take place spontaneously. To calculate the conformational entropy, the possible conformations of the molecule may first be discretized into a finite number of states, usually characterized by unique combinations of certain structural parameters, each of which has been assigned an energy. It differs from kT only by a factor of the Avogadro constant, NA. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The quantity \frac{dQ_{k}}{T} represents the ratio of a very small exchange of heat energy to the temperature . In chemical thermodynamics, conformational entropy is the entropy associated with the number of conformations of a molecule. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). The factor is needed because of the pressure dependence of the reaction rate. (bar·L)/(mol·K).Laidler, K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p.381-2 The value of provides clues about the molecularity of the rate determining step in a reaction, i.e. the number of molecules that enter this step.Laidler and Meiser p.365 Positive values suggest that entropy increases upon achieving the transition state, which often indicates a dissociative mechanism in which the activated complex is loosely bound and about to dissociate. ",0.0000092,52,475.0,0.2307692308,164,E
-"Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \mathrm{C}$ decay events per minute. How old is the wood?","The 774–775 carbon-14 spike is an observed increase of around 1.2% in the concentration of the radioactive carbon-14 isotope in tree rings dated to 774 or 775 CE, which is about 20 times higher than the normal year-to-year variation of radiocarbon in the atmosphere. The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon. Carbon-14 decays into nitrogen-14 () through beta decay. A calculation or (more accurately) a direct comparison of carbon-14 levels in a sample, with tree ring or cave-deposit carbon-14 levels of a known age, then gives the wood or animal sample age-since-formation. Carbon-14, C-14, or radiocarbon, is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. The presence of carbon-14 in the isotopic signature of a sample of carbonaceous material possibly indicates its contamination by biogenic sources or the decay of radioactive material in surrounding geologic strata. The following inventory of carbon-14 has been given: * Global inventory: ~8500 PBq (about 50 t) ** Atmosphere: 140 PBq (840 kg) ** Terrestrial materials: the balance * From nuclear testing (until 1990): 220 PBq (1.3 t) ===In fossil fuels=== Many man-made chemicals are derived from fossil fuels (such as petroleum or coal) in which is greatly depleted because the age of fossils far exceeds the half-life of . Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. Contamination with old carbon, with no remaining , causes an error in the other direction, which does not depend on age—a sample that has been contaminated with 1% old carbon will appear to be about 80 years older than it really is, regardless of the date of the sample.Aitken (1990), pp. 85–86. As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment, it has a / ratio lower than that of the biosphere. ==Dating considerations== The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. If a sample that is in fact 17,000 years old is contaminated so that 1% of the sample is actually modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old the same amount of contamination would cause an error of 4,000 years. Libby estimated that the radioactivity of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram of pure carbon, and this is still used as the activity of the modern radiocarbon standard. If a sample that is 17,000 years old is contaminated so that 1% of the sample is modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old, the same amount of contamination would cause an error of 4,000 years. Cosmogenic nuclides are also used as proxy data to characterize cosmic particle and solar activity of the distant past. ==Origin== ===Natural production in the atmosphere=== right|thumb| 1: Formation of carbon-14 2: Decay of carbon-14 3: The ""equal"" equation is for living organisms, and the unequal one is for dead organisms, in which the C-14 then decays (See 2). This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes.Bowman, Radiocarbon Dating, pp. 38-39.Taylor, Radiocarbon Dating, p. 124. This resemblance is used in chemical and biological research, in a technique called carbon labeling: carbon-14 atoms can be used to replace nonradioactive carbon, in order to trace chemical and biochemical reactions involving carbon atoms from any given organic compound. ==Radioactive decay and detection== Carbon-14 goes through radioactive beta decay: : → + + + 156.5 keV By emitting an electron and an electron antineutrino, one of the neutrons in the carbon-14 atom decays to a proton and the carbon-14 (half-life of 5,730 ± 40 years) decays into the stable (non- radioactive) isotope nitrogen-14. Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. Calculating radiocarbon ages also requires the value of the half-life for . Carbon-12 and carbon-13 are both stable, while carbon-14 is unstable and has a half-life of years. ",4.86,432,0.15,1.33,0.0384,A
-"Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \mathrm{C}$ decay events per minute. How old is the wood?","The 774–775 carbon-14 spike is an observed increase of around 1.2% in the concentration of the radioactive carbon-14 isotope in tree rings dated to 774 or 775 CE, which is about 20 times higher than the normal year-to-year variation of radiocarbon in the atmosphere. The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon. Carbon-14 decays into nitrogen-14 () through beta decay. A calculation or (more accurately) a direct comparison of carbon-14 levels in a sample, with tree ring or cave-deposit carbon-14 levels of a known age, then gives the wood or animal sample age-since-formation. Carbon-14, C-14, or radiocarbon, is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. The presence of carbon-14 in the isotopic signature of a sample of carbonaceous material possibly indicates its contamination by biogenic sources or the decay of radioactive material in surrounding geologic strata. The following inventory of carbon-14 has been given: * Global inventory: ~8500 PBq (about 50 t) ** Atmosphere: 140 PBq (840 kg) ** Terrestrial materials: the balance * From nuclear testing (until 1990): 220 PBq (1.3 t) ===In fossil fuels=== Many man-made chemicals are derived from fossil fuels (such as petroleum or coal) in which is greatly depleted because the age of fossils far exceeds the half-life of . As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment, it has a / ratio lower than that of the biosphere. ==Dating considerations== The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. Cosmogenic nuclides are also used as proxy data to characterize cosmic particle and solar activity of the distant past. ==Origin== ===Natural production in the atmosphere=== right|thumb| 1: Formation of carbon-14 2: Decay of carbon-14 3: The ""equal"" equation is for living organisms, and the unequal one is for dead organisms, in which the C-14 then decays (See 2). Libby estimated that the radioactivity of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram of pure carbon, and this is still used as the activity of the modern radiocarbon standard. Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. If a sample that is 17,000 years old is contaminated so that 1% of the sample is modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old, the same amount of contamination would cause an error of 4,000 years. This resemblance is used in chemical and biological research, in a technique called carbon labeling: carbon-14 atoms can be used to replace nonradioactive carbon, in order to trace chemical and biochemical reactions involving carbon atoms from any given organic compound. ==Radioactive decay and detection== Carbon-14 goes through radioactive beta decay: : → + + + 156.5 keV By emitting an electron and an electron antineutrino, one of the neutrons in the carbon-14 atom decays to a proton and the carbon-14 (half-life of 5,730 ± 40 years) decays into the stable (non- radioactive) isotope nitrogen-14. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes.Bowman, Radiocarbon Dating, pp. 38-39.Taylor, Radiocarbon Dating, p. 124. Carbon-12 and carbon-13 are both stable, while carbon-14 is unstable and has a half-life of years. The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. Calculating radiocarbon ages also requires the value of the half-life for . The old wood effect or old wood problem is a pitfall encountered in the archaeological technique of radiocarbon dating. The globally averaged production of carbon-14 for this event is . ==Hypotheses== Several possible causes of the event have been considered. ",4.86, 135.36,3.8,1.5377,3920.70763168,A
-Determine the diffusion coefficient for Ar at $298 \mathrm{~K}$ and a pressure of $1.00 \mathrm{~atm}$.,"thumb|Thermal diffusion coefficients vs. temperature, for air at normal pressure The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Rearranging yields: \ln k = \frac{-E_{\rm a}}{R}\left(\frac{1}{T}\right) + \ln A. ""On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)"". Alternatively, the equation may be expressed as k = Ae^\frac{-E_{\rm a}}{k_{\rm B}T}, where * is the activation energy for the reaction (in the same units as kBT), * is the Boltzmann constant. Suction pressure is also called Diffusion Pressure Deficit. The Boltzmann–Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. DPD decreases with dilution of the solution. == History == The term diffusion pressure deficit (DPD) was coined by B.S. Meyer in 1938. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. The difference between diffusion pressure of pure solvent and solution is called diffusion pressure deficit (DPD). The diffusion equation is a parabolic partial differential equation. McGraw-Hill == External links == * Diffusion Calculator for Impurities & Dopants in Silicon * A tutorial on the theory behind and solution of the Diffusion Equation. The Diffusion Handbook: Applied Solutions for Engineers. This form is significantly easier to solve numerically, and one only needs to perform a back-substitution of t or x into the definition of ξ to find the value of the other variable. === The parabolic law === Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. The Mathematics of Diffusion. If some solute is dissolved in solvent, its diffusion pressure decreases. The suction pressure, along with the suction temperature the wet bulb temperature of the discharge air are used to determine the correct refrigerant charge in a system. == Further reading == # The measurement of Diffusion Pressure Deficit in plants by the method of Vapour Equilibrium (By R. O. SLATYER, 1958) == References == Category:Diffusion This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law. == Matano’s method == Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys. ",+2.35,0.011,1.1,35.2,0.132,C
-Gas cylinders of $\mathrm{CO}_2$ are sold in terms of weight of $\mathrm{CO}_2$. A cylinder contains $50 \mathrm{lb}$ (22.7 $\mathrm{kg}$ ) of $\mathrm{CO}_2$. How long can this cylinder be used in an experiment that requires flowing $\mathrm{CO}_2$ at $293 \mathrm{~K}(\eta=146 \mu \mathrm{P})$ through a 1.00-m-long tube (diameter $\left.=0.75 \mathrm{~mm}\right)$ with an input pressure of $1.05 \mathrm{~atm}$ and output pressure of $1.00 \mathrm{~atm}$ ? The flow is measured at the tube output.,"Fill the gas collecting tube with one of those fluids of given mass density and measure the overall mass, do the same with the second one giving the two mass values m_{fullgas}, m_{fullliquid}. Measure the overall mass m_{full} to calculate the mass of the fluid inside the tube m=m_{full}-m_{evactube} yielding the desired mass density \rho=\frac{m}{V}. ==Molar Mass from the Mass Density of a Gas== If the gas is a pure gaseous chemical substance (and not a mixture), with the mass density \rho=\frac{m}{V}, then using the ideal gas law permits to calculate the molar mass M of the gaseous chemical substance: :M = \frac{ m \cdot R \cdot T }{ p \cdot V}\, Where R represents the universal gas constant, T the absolute temperature at which the measurements took place. == References == ==External links== * Source of the notion ""gas collecting tube"", among others Category:Measuring instruments Category:Laboratory equipment Category:Laboratory glassware The gas collecting tube is weighed for a third and last time containing the liquid yielding the value m_{containing liquid}. The mass and volume of a displaced amount of gas are determined: At atmospheric pressure p, the gas collecting tube is filled with the gas to be investigated and the overall mass m_{full} is measured. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. ""Pressure Vessel Handbook, 14th Edition."" For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: Thickness has to be less than 0.356 times inner radius :\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE} Cylindrical shells: Thickness has to be less than 0.5 times inner radius :\sigma_\theta = \frac{p(r + 0.6t)}{tE} :\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE} where E is the joint efficiency, and all others variables as stated above. Consequently, for those two fluids, the definition of mass density can be rewritten: :\rho_{gas}=\frac{m_{fullgas}-m_{evactube}}{V}\, :\rho_{liquid}=\frac{m_{fullliquid}-m_{evactube}}{V} These two equations with two unknowns m_{evactube} and V can be solved by using elementary algebra: :V=\frac{m_{fullliquid}-m_{fullgas}}{\rho_{liquid}-\rho_{gas}}\, :m_{evactube}=\frac{\rho_{gas}\cdot m_{fullliquid}-\rho_{liquid} \cdot m_{fullgas}}{\rho_{liquid}-\rho_{gas}} (The relative error of the result significantly depends on the relative proportions of the given mass densities and the measured masses.) Each cylinder shall be painted externally in the colours corresponding to its gaseous contents. == Common sizes == The below are example cylinder sizes and do not constitute an industry standard. size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) Internal volume, 70 °F (21 °C), 1atm Internal volume, 70 °F (21 °C), 1atm U.S. DOT specs size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) (liters) (cu.ft) U.S. DOT specs 2HP 43.3 1.53 3AA3500 K 110 49.9 1.76 3AA2400 A 96 43.8 1.55 3AA2015 B 37.9 17.2 0.61 3AA2015 C 15.2 6.88 0.24 3AA2015 D 4.9 2.24 0.08 3AA2015 AL 64.8 29.5 1.04 3AL2015 BL 34.6 15.7 0.55 3AL2216 CL 13 5.9 0.21 3AL2216 XL 238 108 3.83 4BA240 SSB 41.6 18.9 0.67 3A1800 10S 8.3 3.8 0.13 3A1800 LB 1 0.44 0.016 3E1800 XF 60.9 2.15 8AL XG 278 126.3 4.46 4AA480 XM 120 54.3 1.92 3A480 XP 124 55.7 1.98 4BA300 QT (includes 4.5 inches for valve) (includes 1.5 lb for valve) 2.0 0.900 0.0318 4B-240ET LP5 47.7 21.68 0.76 4BW240 Medical E (excludes valve and cap) (excludes valve and cap) 9.9 4.5 0.16 3AA2015 (US DOT specs define material, making, and maximum pressure in psi. (The difference of masses of the nearly evacuated tube and the liquid-containing tube gives the mass (m_{liquid}=m_{containing liquid}-m_{sucked}) of the sucked-in liquid, that took the place of the extracted amount of gas.) Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. For a cylinder with hemispherical ends, :M = 2 \pi R^2 (R + W) P {\rho \over \sigma}, where *R is the Radius (m) *W is the middle cylinder width only, and the overall width is W + 2R (m) ====Cylindrical vessel with semi-elliptical ends==== In a vessel with an aspect ratio of middle cylinder width to radius of 2:1, :M = 6 \pi R^3 P {\rho \over \sigma}. ====Gas storage==== In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. * ISO 11439: Compressed natural gas (CNG) cylinders. For gases that remain gaseous under ambient storage conditions, the upstream pressure gauge can be used to estimate how much gas is left in the cylinder according to pressure. thumb|A 20 lb () steel propane cylinder. upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. Because (for a given pressure) the thickness of the walls scales with the radius of the tank, the mass of a tank (which scales as the length times radius times thickness of the wall for a cylindrical tank) scales with the volume of the gas held (which scales as length times radius squared). Further the volume of the gas is (4πr3)/3. The vapor pressure in the cylinder is a function of temperature. Now fill the gas collecting tube with the fluid to be investigated. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. ",3,0.166666666,4.738,4.49,1260,D
-The vibrational frequency of $I_2$ is $208 \mathrm{~cm}^{-1}$. What is the probability of $I_2$ populating the $n=2$ vibrational level if the molecular temperature is $298 \mathrm{~K}$ ?,"A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is Planck's constant. The probability of resonance absorption is called the resonance factor \psi, and the sum of the two factors is p + \psi = 1. The excess energy of the excited vibrational mode is transferred to the kinetic modes in the same molecule or to the surrounding molecules. The vibration frequencies, νi, are obtained from the eigenvalues, λi, of the matrix product GF. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics For a diatomic molecule A−B, the vibrational frequency in s−1 is given by u = \frac{1}{2 \pi} \sqrt{k / \mu} , where k is the force constant in dyne/cm or erg/cm2 and μ is the reduced mass given by \frac{1}{\mu} = \frac{1}{m_A}+\frac{1}{m_B}. The vibrational temperature is used commonly when finding the vibrational partition function. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. Vibrational energy relaxation, or vibrational population relaxation, is a process in which the population distribution of molecules in quantum states of high energy level caused by an external perturbation returns to the Maxwell–Boltzmann distribution. Through this process, the initially excited vibrational mode moves to a vibrational state of a lower energy. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency u (in the harmonic oscillator approximation). Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1 and wavelengths of approximately 30 to 3 µm. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state. Intramolecular vibrational energy redistribution (IVR) is a process in which energy is redistributed between different quantum states of a vibrationally excited molecule, which is required by successful theories explaining unimolecular reaction rates such as RRKM theory. Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. In addition, by the electronic transition, the molecule often moves to the vibrationally excited state of the electronic excited state. ",0,0.082,-2.5,0.086,0.66666666666,D
-"The value of $\Delta G_f^{\circ}$ for $\mathrm{Fe}(g)$ is $370.7 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298.15 \mathrm{~K}$, and $\Delta H_f^{\circ}$ for $\mathrm{Fe}(g)$ is $416.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at the same temperature. Assuming that $\Delta H_f^{\circ}$ is constant in the interval $250-400 \mathrm{~K}$, calculate $\Delta G_f^{\circ}$ for $\mathrm{Fe}(g)$ at 400. K.","The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of . The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The same effect can be achieved with a low temperature and a long holding time, or with a higher temperature and a short holding time. ==Formula== In the Hollomon–Jaffe parameter, this exchangeability of time and temperature can be described by the following formula: :H_p = \frac {(273.15 + T)}{1000} \cdot (C + \log(t)) This formula is not consistent concerning the units; the parameters must be entered in a certain manner. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. T is in degrees Celsius. The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well. The ""diffusivity constant"" is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. Then, according to the chain rule, one has Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of and solutions of the heat equation with . The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The vibrational temperature is used commonly when finding the vibrational partition function. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. The Hollomon–Jaffe parameter (HP), also generally known as the Larson–Miller parameter, describes the effect of a heat treatment at a temperature for a certain time. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. thumb|250px|The plot of the specific heat capacity versus temperature. The temperature gradient is a dimensional quantity expressed in units of degrees (on a particular temperature scale) per unit length. This form is more general and particularly useful to recognize which property (e.g. cp or \rho) influences which term. :\rho c_p \frac{\partial T}{\partial t} - abla \cdot \left( k abla T \right) = \dot q_V where \dot q_V is the volumetric heat source. ===Three-dimensional problem=== In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is : \frac{\partial u}{\partial t} = \alpha abla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) where: * u = u(x, y, z, t) is temperature as a function of space and time; * \tfrac{\partial u}{\partial t} is the rate of change of temperature at a point over time; * u_{xx} , u_{yy} , and u_{zz} are the second spatial derivatives (thermal conductions) of temperature in the x , y , and z directions, respectively; * \alpha \equiv \tfrac{k}{c_p\rho} is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k , the specific heat capacity c_p , and the mass density \rho . Holloman and Jaffe determined the value of C experimentally by plotting hardness versus tempering time for a series of tempering temperatures of interest and interpolating the data to obtain the time necessary to yield a number of different hardness values. These authors derived an expression for the temperature at the center of a sphere :\frac{T_C - T_S}{T_0 - T_S} =2 \sum_{n = 1}^{\infty} (-1)^{n+1} \exp\left({-\frac{n^2 \pi^2 \alpha t}{L^2}}\right) where is the initial temperature of the sphere and the temperature at the surface of the sphere, of radius . A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. ",+10,1.61,2598960.0,92,355.1,E
-"The reactant 1,3-cyclohexadiene can be photochemically converted to cis-hexatriene. In an experiment, $2.5 \mathrm{mmol}$ of cyclohexadiene are converted to cis-hexatriene when irradiated with 100. W of 280. nm light for $27.0 \mathrm{~s}$. All of the light is absorbed by the sample. What is the overall quantum yield for this photochemical process?","A quantum yield of 1.0 (100%) describes a process where each photon absorbed results in a photon emitted. Since not all photons are absorbed productively, the typical quantum yield will be less than 1. The quantum yield of ANS is ~0.002 in aqueous buffer, but near 0.4 when bound to serum albumin. === Photochemical reactions === The quantum yield of a photochemical reaction describes the number of molecules undergoing a photochemical event per absorbed photon: \Phi=\frac{\rm \\#\ molecules\ undergoing\ reaction\ of\ interest}{\rm \\#\ photons\ absorbed\ by\ photoreactive\ substance} In a chemical photodegradation process, when a molecule dissociates after absorbing a light quantum, the quantum yield is the number of destroyed molecules divided by the number of photons absorbed by the system. In optical spectroscopy, the quantum yield is the probability that a given quantum state is formed from the system initially prepared in some other quantum state. Such effects can be studied with wavelength-tunable lasers and the resulting quantum yield data can help predict conversion and selectivity of photochemical reactions. For example, a singlet to triplet transition quantum yield is the fraction of molecules that, after being photoexcited into a singlet state, cross over to the triplet state. === Photosynthesis === Quantum yield is used in modeling photosynthesis: \Phi = \frac {\rm \mu mol\ CO_2 \ fixed} {\rm \mu mol\ photons \ absorbed} ==See also== *Quantum dot *Quantum efficiency == References == Category:Radiation Category:Spectroscopy Category:Photochemistry Photoexcitation is the production of an excited state of a quantum system by photon absorption. The quantum yield is then calculated by: \Phi = \Phi_\mathrm{R} \times \frac{\mathit{Int}}{\mathit{Int}_\mathrm{R}} \times \frac{1-10^{-A_\mathrm{R}}}{1-10^{-A}} \times \frac{{n}^2}{{n_\mathrm{R}}^2} where * is the quantum yield, * is the area under the emission peak (on a wavelength scale), * is absorbance (also called ""optical density"") at the excitation wavelength, * is the refractive index of the solvent. Quantum photoelectrochemistry in particular provides fundamental insight into basic light-harvesting and photoinduced electro-optical processes in several emerging solar energy conversion technologies for generation of both electricity (photovoltaics) and solar fuels.Ponseca Jr., Carlito S.; Chábera, Pavel; Uhlig, Jens; Persson, Petter; Sundström, Villy (August 2017). The absorption of photons with energy equal to or greater than the band gap of the semiconductor initiates photocatalytic reactions. Quantum photoelectrochemistry is the investigation of the quantum mechanical nature of photoelectrochemistry, the subfield of study within physical chemistry concerned with the interaction of light with electrochemical systems, typically through the application of quantum chemical calculations.Quantum Photoelectrochemistry - Theoretical Studies of Organic Adsorbates on Metal Oxide Surfaces, Petter Persson, Acta Univ. Upsaliensis., Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 544, 53 pp. Uppsala. . On the atomic and molecular scale photoexcitation is the photoelectrochemical process of electron excitation by photon absorption, when the energy of the photon is too low to cause photoionization. Fluorescence quantum yields are measured by comparison to a standard of known quantum yield. One example is the reaction of hydrogen with chlorine, in which as many as 106 molecules of hydrogen chloride can be formed per quantum of blue light absorbed.Laidler K.J., Chemical Kinetics (3rd ed., Harper & Row 1987) p.289 Quantum yields of photochemical reactions can be highly dependent on the structure, proximity and concentration of the reactive chromophores, the type of solvent environment as well as the wavelength of the incident light. Quantum photoelectrochemistry provides an expansion of quantum electrochemistry to processes involving also the interaction with light (photons). Key aspects of quantum photoelectrochemistry are calculations of optical excitations, photoinduced electron and energy transfer processes, excited state evolution, as well as interfacial charge separation and charge transport in nanoscale energy conversion systems.Multiscale Modelling of Interfacial Electron Transfer, Petter Persson, Chapter 3 in: Solar Energy Conversion – Dynamics of Electron and Excitation Transfer P. Piotrowiak (Ed.), RSC Energy and Environment Series (2013) thumbnail|Quantum photoelectrochemistry calculation of photoinduced interfacial electron transfer in a dye-sensitized solar cell. In particle physics, the quantum yield (denoted ) of a radiation-induced process is the number of times a specific event occurs per photon absorbed by the system. The term quantum efficiency (QE) may apply to incident photon to converted electron (IPCE) ratio of a photosensitive device, or it may refer to the TMR effect of a Magnetic Tunnel Junction. Thus, the fluorescence quantum yield is affected if the rate of any non-radiative pathway changes. Quantum yield is defined by the fraction of excited state fluorophores that decay through fluorescence: \Phi_f = \frac{k_f}{k_f + \sum k_\mathrm{nr}} where * is the fluorescence quantum yield, * is the rate constant for radiative relaxation (fluorescence), * is the rate constant for all non-radiative relaxation processes. This synthesis demonstrates that the thermal Diels–Alder reaction favors the undesired regioisomer, but the photoredox-catalyzed reaction gives the desired regioisomer in improved yield. 400px|frameless|center|Key photoredox cycloaddition in total synthesis of Heitziamide A === Photoredox organocatalysis === Organocatalysis is a subfield of catalysis that explores the potential of organic small molecules as catalysts, particularly for the enantioselective creation of chiral molecules. The photoexcitation causes the electrons in atoms to go to an excited state. ",1.4,0.5,243.0,0.396,0.33333333,D
+a. Calculate $\Delta S$ for this process; $\beta=1.81 \times 10^{-4} \mathrm{~K}^{-1}, \rho=13.54 \mathrm{~g} \mathrm{~cm}^{-3}$, and $C_{P, m}$ for $\mathrm{Hg}(l)=27.98 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$.","System A12 A21 Acetone(1)-Chloroform(2) -0.8404 -0.5610 Acetone(1)-Methanol(2) 0.6184 0.5788 Acetone(1)-Water(2) 2.0400 1.5461 Carbon tetrachloride(1)-Benzene (2) 0.0948 0.0922 Chloroform(1)-Methanol(2) 0.8320 1.7365 Ethanol(1)-Benzene(2) 1.8362 1.4717 Ethanol(1)-Water(2) 1.6022 0.7947 ==See also== * Van Laar equation ==Literature== ==External links== *Ternary systems Margules Category:Physical chemistry Category:Thermodynamic models When A>2 the system starts to demix in two liquids at 50/50 composition; i.e. plait point is at 50 mol%. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. Since: :A = \ln \gamma_1^\infty = \ln \gamma_2^\infty : \gamma_1^\infty = \gamma_2^\infty > \exp(2) \approx 7.38 For asymmetric binary systems, A12≠A21, the liquid-liquid separation always occurs for : :A_{21} + A_{12} > 4 Or equivalently: :\gamma_1^\infty \gamma_2^\infty > \exp(4) \approx 54.6 The plait point is not located at 50 mol%. ΔP (Delta P) is a mathematical term symbolizing a change (Δ) in pressure (P). ==Uses== *Young–Laplace equation ===Darcy–Weisbach equation=== Given that the head loss hf expresses the pressure loss Δp as the height of a column of fluid, :\Delta p = \rho \cdot g \cdot h_f where ρ is the density of the fluid. This yields, when applied only to the first term and using the Gibbs–Duhem equation,:Phase Equilibria in Chemical Engineering, Stanley M. Walas, (1985) p180 Butterworth Publ. : \left\\{\begin{matrix} \ln\ \gamma_1=[A_{12}+2(A_{21}-A_{12})x_1]x^2_2 \\\ \ln\ \gamma_2=[A_{21}+2(A_{12}-A_{21})x_2]x^2_1 \end{matrix}\right. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup In chemical engineering the Margules Gibbs free energy model for liquid mixtures is better known as the Margules activity or activity coefficient model. thumb|The Mollier enthalpy–entropy diagram for water and steam. In an isenthalpic process, the enthalpy is constant. Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Calculating compliance on minute volume (VE: ΔV is always defined by tidal volume (VT), but ΔP is different for the measurement of dynamic vs. static compliance. ====Dynamic compliance (Cdyn)==== :C_{dyn} = \frac{V_T}\mathrm{PIP-PEEP} where PIP = peak inspiratory pressure (the maximum pressure during inspiration), and PEEP = positive end expiratory pressure. So the expansion process in a turbine can be easily calculated using the h–s chart when the process is considered to be ideal (which is the case normally when calculating enthalpies, entropies, etc. The enthalpy coordinate is skewed and the constant enthalpy lines are parallel and evenly spaced. ==See also== *Thermodynamic diagrams *Contour line *Phase diagram == References == Category:Thermodynamics Category:Entropy de:Wasserdampf#h-s- Diagramm The leading term X_1X_2 assures that the excess Gibbs energy becomes zero at x1=0 and x1=1. === Activity coefficient === The activity coefficient of component i is found by differentiation of the excess Gibbs energy towards xi. Although the model is old it has the characteristic feature to describe extrema in the activity coefficient, which modern models like NRTL and Wilson cannot. ==Equations== === Excess Gibbs free energy === Margules expressed the intensive excess Gibbs free energy of a binary liquid mixture as a power series of the mole fractions xi: : \frac{G^{ex}}{RT}=X_1 X_2 (A_{21} X_1 +A_{12} X_2) + X_1^2 X_2^2 (B_{21}X_1+ B_{12} X_2) + ... + X_1^m X_2^m (M_{21}X_1+ M_{12} X_2) In here the A, B are constants, which are derived from regressing experimental phase equilibria data. When A_{12}=A_{21}=A, which implies molecules of same molecular size but different polarity, the equations reduce to the one-parameter Margules activity model: : \left\\{\begin{matrix} \ln\ \gamma_1=Ax^2_2 \\\ \ln\ \gamma_2=Ax^2_1 \end{matrix}\right. The work done in a process on vapor cycles is represented by length of , so it can be measured directly, whereas in a T–s diagram it has to be computed using thermodynamic relationship between thermodynamic properties. The Margules activity model is a simple thermodynamic model for the excess Gibbs free energy of a liquid mixture introduced in 1895 by Max Margules.https://archive.org/details/sitzungsbericht10wiengoog After Lewis had introduced the concept of the activity coefficient, the model could be used to derive an expression for the activity coefficients \gamma_i of a compound i in a liquid, a measure for the deviation from ideal solubility, also known as Raoult's law. The Mollier diagram coordinates are enthalpy h and humidity ratio x. The parameters for a description at 20 °C are A12=0.6298 and A21=1.9522. ",58.2, -1,"""3.54""",358800,6.9,A
+"For an ensemble consisting of 1.00 moles of particles having two energy levels separated by $h v=1.00 \times 10^{-20} \mathrm{~J}$, at what temperature will the internal energy of this system equal $1.00 \mathrm{~kJ}$ ?","The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. The internal energy of an ideal gas is proportional to its mass (number of moles) n and to its temperature T : U = C_V n T, where C_V is the isochoric (at constant volume) molar heat capacity of the gas. The internal energy relative to the mass with unit J/kg is the specific internal energy. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): U = U(n,T). The internal energy is an extensive function of the extensive variables S, V, and the amounts N_j, the internal energy may be written as a linearly homogeneous function of first degree: : U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots ) = \alpha U(S,V,N_{1},N_{2},\ldots), where \alpha is a factor describing the growth of the system. The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence: : U = \sum_{i=1}^N p_i \,E_i. At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). The differential internal energy may be written as :\mathrm{d} U = \frac{\partial U}{\partial S} \mathrm{d} S + \frac{\partial U}{\partial V} \mathrm{d} V + \sum_i\ \frac{\partial U}{\partial N_i} \mathrm{d} N_i\ = T \,\mathrm{d} S - P \,\mathrm{d} V + \sum_i\mu_i \mathrm{d} N_i, which shows (or defines) temperature T to be the partial derivative of U with respect to entropy S and pressure P to be the negative of the similar derivative with respect to volume V, : T = \frac{\partial U}{\partial S}, : P = -\frac{\partial U}{\partial V}, and where the coefficients \mu_{i} are the chemical potentials for the components of type i in the system. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the internal energy. MJ/kg may refer to: * megajoules per kilogram * Specific kinetic energy * Heat of fusion * Heat of combustion In statistical mechanics, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from translations, rotations, and vibrations, and of the potential energies associated with microscopic forces, including chemical bonds. The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains. kJ/kg may refer to: * kilojoules per kilogram * The SI derived units of specific energy * Specific Internal energy * Specific kinetic energy * Heat of fusion * Heat of combustion Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties S, V, n (entropy, volume, mass). In statistical mechanics, the excitation temperature () is defined for a population of particles via the Boltzmann factor. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be sensible. Thus the excitation temperature is the temperature at which we would expect to find a system with this ratio of level populations. The expression relating changes in internal energy to changes in temperature and volume is : \mathrm{d}U =C_{V} \, \mathrm{d}T +\left[T\left(\frac{\partial P}{\partial T}\right)_{V} - P\right] \mathrm{d}V. For a closed system, with transfers only as heat and work, the change in the internal energy is : \mathrm{d} U = \delta Q - \delta W, expressing the first law of thermodynamics. When a closed system receives energy as heat, this energy increases the internal energy. ",8.8,8,"""7.0""",3.8,449,E
+"The sedimentation coefficient of lysozyme $\left(\mathrm{M}=14,100 \mathrm{~g} \mathrm{~mol}^{-1}\right)$ in water at $20^{\circ} \mathrm{C}$ is $1.91 \times 10^{-13} \mathrm{~s}$ and the specific volume is $0.703 \mathrm{~cm}^3 \mathrm{~g}^{-1}$. The density of water at this temperature is $0.998 \mathrm{~g} \mathrm{~cm}^{-3}$ and $\eta=1.002 \mathrm{cP}$. Assuming lysozyme is spherical, what is the radius of this protein?","The commonly used molar ratio of lysozyme and mPEG-aldehyde is 1:6 or 1:6.67. The hydrodynamic radius of a macromolecule or colloid particle is R_{\rm hyd}. The protein and mPEG-aldehyde are dissolved using a sodium phosphate buffer with sodium cyanoborohydride, which acts as a reducing agent and conditions the aldehyde group of mPEG-aldehyde to have a strong affinity towards the lysine residue on the N-terminal of lysozyme. The CD spectra range from 189 - 260 nm with a pitch of 0.1 nm showed no significant change in the secondary structure of the intact and PEGylated lysozyme. === Enzymatic activity assay === ==== Glycol chitosan ==== Enzymatic activity of intact and PEGylated lysozyme can be evaluated using glycol chitosan by reacting 1 mL of 0.05% (w/v) glycol chitosan in 100 mM of pH 5.5 acetate buffer and 100 μL of the intact or PEGylated protein at 40 °C for 30 min and subsequently adding 2 mL of 0.5 M sodium carbonate with 1 μg of potassium ferricyanide. There is a negative correlation between molecular weight and the retention time of the PEGylated protein in the chromatogram; larger protein, or more PEGylated protein elutes first, and smaller protein, or intact protein the latest. == Characterization == === Identification === The most common analyses for identifying intact and PEGylated lysozyme can be achieved via size-exclusion chromatography (high-performance liquid chromatography or HPLC), SDS-PAGE and Matrix-assisted laser desorption/ionization (MALDI). === Conformation === The secondary structure of intact and PEGylated lysozyme can be characterized by circular dichroism (CD) spectroscopy. Previous works on lysozyme PEGylation showed various chromatographic schemes in order to purify PEGylated lysozyme, which included ion exchange chromatography, hydrophobic interaction chromatography, and size-exclusion chromatography (fast protein liquid chromatography), and proved its stable conformation via circular dichroism and improved thermal stability by enzymatic activity assays, SDS-PAGE, and size- exclusion chromatography (high-performance liquid chromatography). == Methodology == === PEGylation === The chemical modification of lysozyme by PEGylation involves the addition of methoxy-PEG-aldehyde (mPEG-aldehyde) with varying molecular sizes, ranging from 2 kDa to 40 kDa, to the protein. Due to the high pI of lysozyme (pI = 10.7), cation exchange chromatography is used. As the enzymatic activity to hydrolyze β-1,4- N-acetylglucosamine linkage was retained after PEGylation, there was no decay in the enzymatic activity by increasing the degree of PEGylation. ==== Micrococcus lysodeikticus ==== By the measurement of decrease in turbidity of M. lysodeikticus by incubating it with lysozyme, enzymatic activity can be evaluated. 7.5 μL of 0.1 - 1 mg/mL proteins is added to 200 μL of M. lysodeikticus at its optical density (OD) of 1.7 AU, and the mixture is measured at 450 nm periodically for reaction rate calculation. Inside the Vainshtein radius; see also :r_V = l_\text{P}\left( \frac{m_\text{P}^3M}{m^4_G} \right)^\frac{1}{5} :with Planck length l_\text{P} and Planck mass m_\text{P} the gravitational field around a body of mass M is the same in a theory where the graviton mass m_G is zero and where it's very small because the helicity 0 degree of freedom becomes effective on distance scales r \gg r_V. ==See also== * == References == Category:Quantum gravity In its full form, the Flory–Rehner equation is written as: : -\left[ \ln{\left(1 - u_2\right)}+ u_2+ \chi_1 u_2^2 \right] = \frac{V_1}{\bar{ u}M_c} \left(1-\frac{2M_c}{M}\right) \left( u_2^\frac{1}{3}-\frac{ u_2}{2}\right) where, \bar{ u} is the specific volume of the polymer, M is the primary molecular mass, and M_c is the average molecular mass between crosslinks or the network parameter. == Flory–Rehner theory == The Flory–Rehner theory gives the change of free energy upon swelling of the polymer gel similar to the Flory–Huggins solution theory: :\Delta F = \Delta F_\mathrm{mix} + \Delta F_\mathrm{elastic}. thumb|right| T4 lysozyme ribbon schematic (from PDB 1LZM) Brian W. Matthews is a biochemist and biophysicist educated at the University of Adelaide, contributor to x-ray crystallographic methodology at the University of Cambridge, and since 1970 at the University of Oregon as Professor of Physics and HHMI investigator in the Institute of Molecular Biology. Lysozyme has six lysine residues which are accessible for PEGylation reactions. Note that in biophysics, hydrodynamic radius refers to the Stokes radius, or commonly to the apparent Stokes radius obtained from size exclusion chromatography. Lysozyme PEGylation is the covalent attachment of Polyethylene glycol (PEG) to Lysozyme, which is one of the most widely investigated PEGylated proteins. The theoretical hydrodynamic radius R_{\rm hyd} arises in the study of the dynamic properties of polymers moving in a solvent. In the continuum limit, where the mean free path of the particle is negligible compared to a characteristic length scale of the particle, the hydrodynamic radius is defined as the radius that gives the same magnitude of the frictional force, \boldsymbol{F}_d as that of a sphere with that radius, i.e. :\boldsymbol{F}_d = 6\pi\mu R_{hyd}\boldsymbol{v} where \mu is the viscosity of the surrounding fluid, and \boldsymbol{v} is the velocity of the particle. He created hundreds of mutants of T4 lysozyme (making it the commonest structure in the PDB), determined their structure by x-ray crystallography and measured their melting temperatures. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup The theoretical hydrodynamic radius R_{\rm hyd} was originally an estimate by John Gamble Kirkwood of the Stokes radius of a polymer, and some sources still use hydrodynamic radius as a synonym for the Stokes radius. Thus, the PEGylation of lysozyme, or lysozyme PEGylation, can be a good model system for the PEGylation of other proteins with enzymatic activities by showing the enhancement of its physical and thermal stability while retaining its activity. In polymer science Flory–Rehner equation is an equation that describes the mixing of polymer and liquid molecules as predicted by the equilibrium swelling theory of Flory and Rehner. The Flory–Rehner equation is written as: : -\left[ \ln{\left(1 - u_2\right)}+ u_2+ \chi_1 u_2^2 \right] = V_1 n \left( u_2^\frac{1}{3}-\frac{ u_2}{2}\right) where, u_2 is the volume fraction of polymer in the swollen mass, V_1 the molar volume of the solvent, n is the number of network chain segments bounded on both ends by crosslinks, and \chi_1 is the Flory solvent-polymer interaction term. ",41.40,0.332,"""2.534324263""",14,1.94,E
+Determine the standard molar entropy of $\mathrm{Ne}$ and $\mathrm{Kr}$ under standard thermodynamic conditions.,"The standard molar entropy at pressure = P^0 is usually given the symbol , and has units of joules per mole per kelvin (J⋅mol−1⋅K−1). In chemistry, the standard molar entropy is the entropy content of one mole of pure substance at a standard state of pressure and any temperature of interest. Therefore, a table of values for \frac{C_{p_k}}{T} is required to find the total molar entropy. The entropy of a pure crystalline structure can be 0J⋅mol−1⋅K−1 only at 0K, according to the third law of thermodynamics. Factor (J K) Value Item 10 9.5699 J K Entropy equivalent of one bit of information, equal to k times ln(2) page 72 (page 5 of pdf) 10 1.381 Boltzmann Constant 10 5.74 J K Standard entropy of 1 mole of graphite 10 ≈ 10 J K Entropy of the Sun (given as ≈ 10 erg K in Bekenstein (1973)) 10 1.5 × 10 J K Entropy of a black hole of one solar mass (given as ≈ 10 erg K in Bekenstein (1973)) 10 4.3 × 10 J K One estimate of the theoretical maximum entropy of the universeCalculated: 3.1e104 * k = 3.1e104 * 1.381e-23 J/K = 4.3e81 J/K ==See also== *Orders of magnitude (data), relates to information entropy *Order of magnitude (terminology) ==References== Entropy Category:Entropy The total molar entropy is the sum of many small changes in molar entropy, where each small change can be considered a reversible process. ==Chemistry== The standard molar entropy of a gas at STP includes contributions from: * The heat capacity of one mole of the solid from 0K to the melting point (including heat absorbed in any changes between different crystal structures). More fundamentally, kT is the amount of heat required to increase the thermodynamic entropy of a system by k. Approximate values of kT at 298 K Units kT = J kT = pN⋅nm kT = cal kT = meV kT=-174 dBm/Hz kT/hc ≈ cm−1 kT/e = 25.7 mV RT = kT ⋅ NA = kJ⋅mol−1 RT = 0.593 kcal⋅mol−1 h/kT = 0.16 ps kT (also written as kBT) is the product of the Boltzmann constant, k (or kB), and the temperature, T. The following list shows different orders of magnitude of entropy. This equation is of the form k = \frac{\kappa k_\mathrm{B}T}{h} e^{\frac{\Delta S^\ddagger }{R}} e^{-\frac{\Delta H^\ddagger}{RT}}where: * k = reaction rate constant * T = absolute temperature * \Delta H^\ddagger = enthalpy of activation * R = gas constant * \kappa = transmission coefficient * k_\mathrm{B} = Boltzmann constant = R/NA, NA = Avogadro constant * h = Planck's constant * \Delta S^\ddagger = entropy of activation This equation can be turned into the form \ln \frac{k}{T} = \frac{-\Delta H^\ddagger}{R} \cdot \frac{1}{T} + \ln \frac{\kappa k_\mathrm{B}}{h} + \frac{\Delta S^\ddagger}{R} The plot of \ln(k/T) versus 1/T gives a straight line with slope -\Delta H^\ddagger/ R from which the enthalpy of activation can be derived and with intercept \ln(\kappa k_\mathrm{B} / h) + \Delta S^\ddagger/ R from which the entropy of activation is derived. ==References== Category:Chemical kinetics However, the residual entropy is often quite negligible and can be accounted for when it occurs using statistical mechanics. ==Thermodynamics== If a mole of a solid substance is a perfectly ordered solid at 0K, then if the solid is warmed by its surroundings to 298.15K without melting, its absolute molar entropy would be the sum of a series of stepwise and reversible entropy changes. In chemical kinetics, the entropy of activation of a reaction is one of the two parameters (along with the enthalpy of activation) which are typically obtained from the temperature dependence of a reaction rate constant, when these data are analyzed using the Eyring equation of the transition state theory. Negative values for indicate that entropy decreases on forming the transition state, which often indicates an associative mechanism in which two reaction partners form a single activated complex.James H. Espenson Chemical Kinetics and Reaction Mechanisms (2nd ed., McGraw-Hill 2002), p.156-160 == Derivation == It is possible to obtain entropy of activation using Eyring equation. Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. Chemical equations make use of the standard molar entropy of reactants and products to find the standard entropy of reaction: :{\Delta S^\circ}_{rxn} = S^\circ_{products} - S^\circ_{reactants} The standard entropy of reaction helps determine whether the reaction will take place spontaneously. To calculate the conformational entropy, the possible conformations of the molecule may first be discretized into a finite number of states, usually characterized by unique combinations of certain structural parameters, each of which has been assigned an energy. It differs from kT only by a factor of the Avogadro constant, NA. According to the second law of thermodynamics, a spontaneous reaction always results in an increase in total entropy of the system and its surroundings: :(\Delta S_{total} = \Delta S_{system} + \Delta S_{surroundings})>0 Molar entropy is not the same for all gases. The quantity \frac{dQ_{k}}{T} represents the ratio of a very small exchange of heat energy to the temperature . In chemical thermodynamics, conformational entropy is the entropy associated with the number of conformations of a molecule. In these equations is the base of natural logarithms, is the Planck constant, is the Boltzmann constant and the absolute temperature. ' is the ideal gas constant in units of (bar·L)/(mol·K). The factor is needed because of the pressure dependence of the reaction rate. (bar·L)/(mol·K).Laidler, K.J. and Meiser J.H. Physical Chemistry (Benjamin/Cummings 1982) p.381-2 The value of provides clues about the molecularity of the rate determining step in a reaction, i.e. the number of molecules that enter this step.Laidler and Meiser p.365 Positive values suggest that entropy increases upon achieving the transition state, which often indicates a dissociative mechanism in which the activated complex is loosely bound and about to dissociate. ",0.0000092,52,"""475.0""",0.2307692308,164,E
+"Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \mathrm{C}$ decay events per minute. How old is the wood?","The 774–775 carbon-14 spike is an observed increase of around 1.2% in the concentration of the radioactive carbon-14 isotope in tree rings dated to 774 or 775 CE, which is about 20 times higher than the normal year-to-year variation of radiocarbon in the atmosphere. The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon. Carbon-14 decays into nitrogen-14 () through beta decay. A calculation or (more accurately) a direct comparison of carbon-14 levels in a sample, with tree ring or cave-deposit carbon-14 levels of a known age, then gives the wood or animal sample age-since-formation. Carbon-14, C-14, or radiocarbon, is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. The presence of carbon-14 in the isotopic signature of a sample of carbonaceous material possibly indicates its contamination by biogenic sources or the decay of radioactive material in surrounding geologic strata. The following inventory of carbon-14 has been given: * Global inventory: ~8500 PBq (about 50 t) ** Atmosphere: 140 PBq (840 kg) ** Terrestrial materials: the balance * From nuclear testing (until 1990): 220 PBq (1.3 t) ===In fossil fuels=== Many man-made chemicals are derived from fossil fuels (such as petroleum or coal) in which is greatly depleted because the age of fossils far exceeds the half-life of . Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. Contamination with old carbon, with no remaining , causes an error in the other direction, which does not depend on age—a sample that has been contaminated with 1% old carbon will appear to be about 80 years older than it really is, regardless of the date of the sample.Aitken (1990), pp. 85–86. As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment, it has a / ratio lower than that of the biosphere. ==Dating considerations== The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. If a sample that is in fact 17,000 years old is contaminated so that 1% of the sample is actually modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old the same amount of contamination would cause an error of 4,000 years. Libby estimated that the radioactivity of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram of pure carbon, and this is still used as the activity of the modern radiocarbon standard. If a sample that is 17,000 years old is contaminated so that 1% of the sample is modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old, the same amount of contamination would cause an error of 4,000 years. Cosmogenic nuclides are also used as proxy data to characterize cosmic particle and solar activity of the distant past. ==Origin== ===Natural production in the atmosphere=== right|thumb| 1: Formation of carbon-14 2: Decay of carbon-14 3: The ""equal"" equation is for living organisms, and the unequal one is for dead organisms, in which the C-14 then decays (See 2). This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes.Bowman, Radiocarbon Dating, pp. 38-39.Taylor, Radiocarbon Dating, p. 124. This resemblance is used in chemical and biological research, in a technique called carbon labeling: carbon-14 atoms can be used to replace nonradioactive carbon, in order to trace chemical and biochemical reactions involving carbon atoms from any given organic compound. ==Radioactive decay and detection== Carbon-14 goes through radioactive beta decay: : → + + + 156.5 keV By emitting an electron and an electron antineutrino, one of the neutrons in the carbon-14 atom decays to a proton and the carbon-14 (half-life of 5,730 ± 40 years) decays into the stable (non- radioactive) isotope nitrogen-14. Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. Calculating radiocarbon ages also requires the value of the half-life for . Carbon-12 and carbon-13 are both stable, while carbon-14 is unstable and has a half-life of years. ",4.86,432,"""0.15""",1.33,0.0384,A
+"Carbon-14 is a radioactive nucleus with a half-life of 5760 years. Living matter exchanges carbon with its surroundings (for example, through $\mathrm{CO}_2$ ) so that a constant level of ${ }^{14} \mathrm{C}$ is maintained, corresponding to 15.3 decay events per minute. Once living matter has died, carbon contained in the matter is not exchanged with the surroundings, and the amount of ${ }^{14} \mathrm{C}$ that remains in the dead material decreases with time due to radioactive decay. Consider a piece of fossilized wood that demonstrates $2.4{ }^{14} \mathrm{C}$ decay events per minute. How old is the wood?","The 774–775 carbon-14 spike is an observed increase of around 1.2% in the concentration of the radioactive carbon-14 isotope in tree rings dated to 774 or 775 CE, which is about 20 times higher than the normal year-to-year variation of radiocarbon in the atmosphere. The calculation of radiocarbon dates determines the age of an object containing organic material by using the properties of radiocarbon (also known as carbon-14), a radioactive isotope of carbon. Carbon-14 decays into nitrogen-14 () through beta decay. A calculation or (more accurately) a direct comparison of carbon-14 levels in a sample, with tree ring or cave-deposit carbon-14 levels of a known age, then gives the wood or animal sample age-since-formation. Carbon-14, C-14, or radiocarbon, is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. The presence of carbon-14 in the isotopic signature of a sample of carbonaceous material possibly indicates its contamination by biogenic sources or the decay of radioactive material in surrounding geologic strata. The following inventory of carbon-14 has been given: * Global inventory: ~8500 PBq (about 50 t) ** Atmosphere: 140 PBq (840 kg) ** Terrestrial materials: the balance * From nuclear testing (until 1990): 220 PBq (1.3 t) ===In fossil fuels=== Many man-made chemicals are derived from fossil fuels (such as petroleum or coal) in which is greatly depleted because the age of fossils far exceeds the half-life of . As a tree grows, only the outermost tree ring exchanges carbon with its environment, so the age measured for a wood sample depends on where the sample is taken from. Because decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less will be left. Accumulated dead organic matter, of both plants and animals, exceeds the mass of the biosphere by a factor of nearly 3, and since this matter is no longer exchanging carbon with its environment, it has a / ratio lower than that of the biosphere. ==Dating considerations== The variation in the / ratio in different parts of the carbon exchange reservoir means that a straightforward calculation of the age of a sample based on the amount of it contains will often give an incorrect result. This means that radiocarbon dates on wood samples can be older than the date at which the tree was felled. Cosmogenic nuclides are also used as proxy data to characterize cosmic particle and solar activity of the distant past. ==Origin== ===Natural production in the atmosphere=== right|thumb| 1: Formation of carbon-14 2: Decay of carbon-14 3: The ""equal"" equation is for living organisms, and the unequal one is for dead organisms, in which the C-14 then decays (See 2). Libby estimated that the radioactivity of exchangeable carbon-14 would be about 14 disintegrations per minute (dpm) per gram of pure carbon, and this is still used as the activity of the modern radiocarbon standard. Radiocarbon dating (also referred to as carbon dating or carbon-14 dating) is a method for determining the age of an object containing organic material by using the properties of radiocarbon, a radioactive isotope of carbon. If a sample that is 17,000 years old is contaminated so that 1% of the sample is modern carbon, it will appear to be 600 years younger; for a sample that is 34,000 years old, the same amount of contamination would cause an error of 4,000 years. This resemblance is used in chemical and biological research, in a technique called carbon labeling: carbon-14 atoms can be used to replace nonradioactive carbon, in order to trace chemical and biochemical reactions involving carbon atoms from any given organic compound. ==Radioactive decay and detection== Carbon-14 goes through radioactive beta decay: : → + + + 156.5 keV By emitting an electron and an electron antineutrino, one of the neutrons in the carbon-14 atom decays to a proton and the carbon-14 (half-life of 5,730 ± 40 years) decays into the stable (non- radioactive) isotope nitrogen-14. If the benzene sample contains carbon that is about 5,730 years old (the half-life of ), then there will only be half as many decay events per minute, but the same error term of 80 years could be obtained by doubling the counting time to 500 minutes.Bowman, Radiocarbon Dating, pp. 38-39.Taylor, Radiocarbon Dating, p. 124. Carbon-12 and carbon-13 are both stable, while carbon-14 is unstable and has a half-life of years. The fossil fuel effect was eliminated from the standard value by measuring wood from 1890, and using the radioactive decay equations to determine what the activity would have been at the year of growth. Calculating radiocarbon ages also requires the value of the half-life for . The old wood effect or old wood problem is a pitfall encountered in the archaeological technique of radiocarbon dating. The globally averaged production of carbon-14 for this event is . ==Hypotheses== Several possible causes of the event have been considered. ",4.86, 135.36,"""3.8""",1.5377,3920.70763168,A
+Determine the diffusion coefficient for Ar at $298 \mathrm{~K}$ and a pressure of $1.00 \mathrm{~atm}$.,"thumb|Thermal diffusion coefficients vs. temperature, for air at normal pressure The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. Some of these are (in approximate order of increasing accuracy): Name Formula Description ""Eq. 1"" (August equation) P = \exp\left(20.386 - \frac{5132}{T}\right) is the vapour pressure in mmHg and is the temperature in kelvins. On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). The Antoine equation \log_{10}P = A - \frac{B}{C + T} is in degrees Celsius (°C) and the vapour pressure is in mmHg. Vapour pressure of water (0–100 °C) T, °C T, °F P, kPa P, torr P, atm 0 32 0.6113 4.5851 0.0060 5 41 0.8726 6.5450 0.0086 10 50 1.2281 9.2115 0.0121 15 59 1.7056 12.7931 0.0168 20 68 2.3388 17.5424 0.0231 25 77 3.1690 23.7695 0.0313 30 86 4.2455 31.8439 0.0419 35 95 5.6267 42.2037 0.0555 40 104 7.3814 55.3651 0.0728 45 113 9.5898 71.9294 0.0946 50 122 12.3440 92.5876 0.1218 55 131 15.7520 118.1497 0.1555 60 140 19.9320 149.5023 0.1967 65 149 25.0220 187.6804 0.2469 70 158 31.1760 233.8392 0.3077 75 167 38.5630 289.2463 0.3806 80 176 47.3730 355.3267 0.4675 85 185 57.8150 433.6482 0.5706 90 194 70.1170 525.9208 0.6920 95 203 84.5290 634.0196 0.8342 100 212 101.3200 759.9625 1.0000 The vapour pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). Rearranging yields: \ln k = \frac{-E_{\rm a}}{R}\left(\frac{1}{T}\right) + \ln A. ""On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System)"". Alternatively, the equation may be expressed as k = Ae^\frac{-E_{\rm a}}{k_{\rm B}T}, where * is the activation energy for the reaction (in the same units as kBT), * is the Boltzmann constant. Suction pressure is also called Diffusion Pressure Deficit. The Boltzmann–Matano method is used to convert the partial differential equation resulting from Fick's law of diffusion into a more easily solved ordinary differential equation, which can then be applied to calculate the diffusion coefficient as a function of concentration. DPD decreases with dilution of the solution. == History == The term diffusion pressure deficit (DPD) was coined by B.S. Meyer in 1938. We can then rearrange: :D(c^*) = - \frac{1}{2 t} \frac{\int^{c^*}_{c_R} x \mathrm{d}c}{(\mathrm{d}c/\mathrm{d}x)_{c=c^*}} Knowing the concentration profile c(x) at annealing time t, and assuming it is invertible as x(c), we can then calculate the diffusion coefficient for all concentrations between cR and cL. === The Matano interface === The last formula has one significant shortcoming: no information is given about the reference according to which x should be measured. The difference between diffusion pressure of pure solvent and solution is called diffusion pressure deficit (DPD). The diffusion equation is a parabolic partial differential equation. McGraw-Hill == External links == * Diffusion Calculator for Impurities & Dopants in Silicon * A tutorial on the theory behind and solution of the Diffusion Equation. The Diffusion Handbook: Applied Solutions for Engineers. This form is significantly easier to solve numerically, and one only needs to perform a back-substitution of t or x into the definition of ξ to find the value of the other variable. === The parabolic law === Observing the previous equation, a trivial solution is found for the case dc/dξ = 0, that is when concentration is constant over ξ. Stefan: Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. The Mathematics of Diffusion. If some solute is dissolved in solvent, its diffusion pressure decreases. The suction pressure, along with the suction temperature the wet bulb temperature of the discharge air are used to determine the correct refrigerant charge in a system. == Further reading == # The measurement of Diffusion Pressure Deficit in plants by the method of Vapour Equilibrium (By R. O. SLATYER, 1958) == References == Category:Diffusion This can be interpreted as the rate of advancement of a concentration front being proportional to the square root of time (x \propto \sqrt t), or, equivalently, to the time necessary for a concentration front to arrive at a certain position being proportional to the square of the distance (t \propto x^2); the square term gives the name parabolic law.See an animation of the parabolic law. == Matano’s method == Chuijiro Matano applied Boltzmann's transformation to obtain a method to calculate diffusion coefficients as a function of concentration in metal alloys. ",+2.35,0.011,"""1.1""",35.2,0.132,C
+Gas cylinders of $\mathrm{CO}_2$ are sold in terms of weight of $\mathrm{CO}_2$. A cylinder contains $50 \mathrm{lb}$ (22.7 $\mathrm{kg}$ ) of $\mathrm{CO}_2$. How long can this cylinder be used in an experiment that requires flowing $\mathrm{CO}_2$ at $293 \mathrm{~K}(\eta=146 \mu \mathrm{P})$ through a 1.00-m-long tube (diameter $\left.=0.75 \mathrm{~mm}\right)$ with an input pressure of $1.05 \mathrm{~atm}$ and output pressure of $1.00 \mathrm{~atm}$ ? The flow is measured at the tube output.,"Fill the gas collecting tube with one of those fluids of given mass density and measure the overall mass, do the same with the second one giving the two mass values m_{fullgas}, m_{fullliquid}. Measure the overall mass m_{full} to calculate the mass of the fluid inside the tube m=m_{full}-m_{evactube} yielding the desired mass density \rho=\frac{m}{V}. ==Molar Mass from the Mass Density of a Gas== If the gas is a pure gaseous chemical substance (and not a mixture), with the mass density \rho=\frac{m}{V}, then using the ideal gas law permits to calculate the molar mass M of the gaseous chemical substance: :M = \frac{ m \cdot R \cdot T }{ p \cdot V}\, Where R represents the universal gas constant, T the absolute temperature at which the measurements took place. == References == ==External links== * Source of the notion ""gas collecting tube"", among others Category:Measuring instruments Category:Laboratory equipment Category:Laboratory glassware The gas collecting tube is weighed for a third and last time containing the liquid yielding the value m_{containing liquid}. The mass and volume of a displaced amount of gas are determined: At atmospheric pressure p, the gas collecting tube is filled with the gas to be investigated and the overall mass m_{full} is measured. ISBN 978-0-626-23561-1 == See also == * * * * * * * * * * * * * - a small, inexpensive, disposable metal gas cylinder for providing pneumatic power * * * * – a device for measuring leaf water potentials * * * or Knock-out drum * * * ==Notes== ==References== * A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed. * E.P. Popov, Engineering Mechanics of Solids, 1st ed. * Megyesy, Eugene F. ""Pressure Vessel Handbook, 14th Edition."" For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are: Spherical shells: Thickness has to be less than 0.356 times inner radius :\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE} Cylindrical shells: Thickness has to be less than 0.5 times inner radius :\sigma_\theta = \frac{p(r + 0.6t)}{tE} :\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE} where E is the joint efficiency, and all others variables as stated above. Consequently, for those two fluids, the definition of mass density can be rewritten: :\rho_{gas}=\frac{m_{fullgas}-m_{evactube}}{V}\, :\rho_{liquid}=\frac{m_{fullliquid}-m_{evactube}}{V} These two equations with two unknowns m_{evactube} and V can be solved by using elementary algebra: :V=\frac{m_{fullliquid}-m_{fullgas}}{\rho_{liquid}-\rho_{gas}}\, :m_{evactube}=\frac{\rho_{gas}\cdot m_{fullliquid}-\rho_{liquid} \cdot m_{fullgas}}{\rho_{liquid}-\rho_{gas}} (The relative error of the result significantly depends on the relative proportions of the given mass densities and the measured masses.) Each cylinder shall be painted externally in the colours corresponding to its gaseous contents. == Common sizes == The below are example cylinder sizes and do not constitute an industry standard. size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) Internal volume, 70 °F (21 °C), 1atm Internal volume, 70 °F (21 °C), 1atm U.S. DOT specs size Diameter × height, including 5.5 inches for valve and cap (inches) Nominal tare weight, including 4.5 lb for valve and cap (lb) Water capacity (lb) (liters) (cu.ft) U.S. DOT specs 2HP 43.3 1.53 3AA3500 K 110 49.9 1.76 3AA2400 A 96 43.8 1.55 3AA2015 B 37.9 17.2 0.61 3AA2015 C 15.2 6.88 0.24 3AA2015 D 4.9 2.24 0.08 3AA2015 AL 64.8 29.5 1.04 3AL2015 BL 34.6 15.7 0.55 3AL2216 CL 13 5.9 0.21 3AL2216 XL 238 108 3.83 4BA240 SSB 41.6 18.9 0.67 3A1800 10S 8.3 3.8 0.13 3A1800 LB 1 0.44 0.016 3E1800 XF 60.9 2.15 8AL XG 278 126.3 4.46 4AA480 XM 120 54.3 1.92 3A480 XP 124 55.7 1.98 4BA300 QT (includes 4.5 inches for valve) (includes 1.5 lb for valve) 2.0 0.900 0.0318 4B-240ET LP5 47.7 21.68 0.76 4BW240 Medical E (excludes valve and cap) (excludes valve and cap) 9.9 4.5 0.16 3AA2015 (US DOT specs define material, making, and maximum pressure in psi. (The difference of masses of the nearly evacuated tube and the liquid-containing tube gives the mass (m_{liquid}=m_{containing liquid}-m_{sucked}) of the sucked-in liquid, that took the place of the extracted amount of gas.) Hydraulic (filled with water) testing pressure is usually 50% higher than the working pressure. ==== Vessel thread ==== Until 1990, high pressure cylinders were produced with conical (tapered) threads. For a cylinder with hemispherical ends, :M = 2 \pi R^2 (R + W) P {\rho \over \sigma}, where *R is the Radius (m) *W is the middle cylinder width only, and the overall width is W + 2R (m) ====Cylindrical vessel with semi-elliptical ends==== In a vessel with an aspect ratio of middle cylinder width to radius of 2:1, :M = 6 \pi R^3 P {\rho \over \sigma}. ====Gas storage==== In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. * ISO 11439: Compressed natural gas (CNG) cylinders. For gases that remain gaseous under ambient storage conditions, the upstream pressure gauge can be used to estimate how much gas is left in the cylinder according to pressure. thumb|A 20 lb () steel propane cylinder. upright=1.4|thumb|A welded steel pressure vessel constructed as a horizontal cylinder with domed ends. The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. Because (for a given pressure) the thickness of the walls scales with the radius of the tank, the mass of a tank (which scales as the length times radius times thickness of the wall for a cylindrical tank) scales with the volume of the gas held (which scales as length times radius squared). Further the volume of the gas is (4πr3)/3. The vapor pressure in the cylinder is a function of temperature. Now fill the gas collecting tube with the fluid to be investigated. A gas cylinder is a pressure vessel for storage and containment of gases at above atmospheric pressure. New York:McGraw-Hill, , pg 108 thumb|300px|Stress in the cylinder body of a pressure vessel. ",3,0.166666666,"""4.738""",4.49,1260,D
+The vibrational frequency of $I_2$ is $208 \mathrm{~cm}^{-1}$. What is the probability of $I_2$ populating the $n=2$ vibrational level if the molecular temperature is $298 \mathrm{~K}$ ?,"A molecular vibration is excited when the molecule absorbs energy, ΔE, corresponding to the vibration's frequency, ν, according to the relation ΔE = hν, where h is Planck's constant. The probability of resonance absorption is called the resonance factor \psi, and the sum of the two factors is p + \psi = 1. The excess energy of the excited vibrational mode is transferred to the kinetic modes in the same molecule or to the surrounding molecules. The vibration frequencies, νi, are obtained from the eigenvalues, λi, of the matrix product GF. Molecule \tilde{v}(cm−1) \theta_{vib} (K) N2 2446 3521 O2 1568 2256 F2 917 1320 HF 4138 5957 HCl 2991 4303 ==References== Statistical thermodynamics University Arizona ==See also== *Rotational temperature *Rotational spectroscopy *Vibrational spectroscopy *Infrared spectroscopy *Spectroscopy Category:Atomic physics Category:Molecular physics For a diatomic molecule A−B, the vibrational frequency in s−1 is given by u = \frac{1}{2 \pi} \sqrt{k / \mu} , where k is the force constant in dyne/cm or erg/cm2 and μ is the reduced mass given by \frac{1}{\mu} = \frac{1}{m_A}+\frac{1}{m_B}. The vibrational temperature is used commonly when finding the vibrational partition function. Here, * N_\text{A} is the number of A molecules in the gas, * N_\text{B} is the number of B molecules in the gas, * \sigma_\text{AB} is the collision cross section, the ""effective area"" seen by two colliding molecules, simplified to \sigma_\text{AB} = \pi(r_\text{A}+r_\text{B})^2 , where r_\text{A} the radius of A and r_\text{B} the radius of B. * k_\text{B} is the Boltzmann constant, * T is the temperature, * \mu_\text{AB} is the reduced mass of the reactants A and B, \mu_\text{AB} = \frac{{m_\text{A}}{m_\text{B}}}{{m_\text{A}} + {m_\text{B}}} == Collision in diluted solution == In the case of equal-size particles at a concentration n in a solution of viscosity \eta , an expression for collision frequency Z=V u where V is the volume in question, and u is the number of collisions per second, can be written as: : u = \frac{8 k_\text{B} T}{3 \eta} n, Where: * k_B is the Boltzmann constant * T is the absolute temperature (unit K) * \eta is the viscosity of the solution (pascal seconds) * n is the concentration of particles per cm3 Here the frequency is independent of particle size, a result noted as counter-intuitive. A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. Vibrational energy relaxation, or vibrational population relaxation, is a process in which the population distribution of molecules in quantum states of high energy level caused by an external perturbation returns to the Maxwell–Boltzmann distribution. Through this process, the initially excited vibrational mode moves to a vibrational state of a lower energy. For a transition from level n to level n+1 due to absorption of a photon, the frequency of the photon is equal to the classical vibration frequency u (in the harmonic oscillator approximation). Vibrations of polyatomic molecules are described in terms of normal modes, which are independent of each other, but each normal mode involves simultaneous vibrations of different parts of the molecule. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1 and wavelengths of approximately 30 to 3 µm. In this approximation, the vibrational energy is a quadratic function (parabola) with respect to the atomic displacements and the first overtone has twice the frequency of the fundamental. The normal modes diagonalize the matrix governing the molecular vibrations, so that each normal mode is an independent molecular vibration. Collision frequency describes the rate of collisions between two atomic or molecular species in a given volume, per unit time. A diatomic molecule has one normal mode of vibration, since it can only stretch or compress the single bond. The combined excitation is known as a vibronic transition, giving vibrational fine structure to electronic transitions, particularly for molecules in the gas state. Intramolecular vibrational energy redistribution (IVR) is a process in which energy is redistributed between different quantum states of a vibrationally excited molecule, which is required by successful theories explaining unimolecular reaction rates such as RRKM theory. Vibrational excitation can occur in conjunction with electronic excitation in the ultraviolet-visible region. In addition, by the electronic transition, the molecule often moves to the vibrationally excited state of the electronic excited state. ",0,0.082,"""-2.5""",0.086,0.66666666666,D
+"The value of $\Delta G_f^{\circ}$ for $\mathrm{Fe}(g)$ is $370.7 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $298.15 \mathrm{~K}$, and $\Delta H_f^{\circ}$ for $\mathrm{Fe}(g)$ is $416.3 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at the same temperature. Assuming that $\Delta H_f^{\circ}$ is constant in the interval $250-400 \mathrm{~K}$, calculate $\Delta G_f^{\circ}$ for $\mathrm{Fe}(g)$ at 400. K.","The heat equation can be efficiently solved numerically using the implicit Crank–Nicolson method of . The behavior of the heat capacity near the peak is described by the formula C\approx A_\pm t^{-\alpha}+B_\pm where t=|1-T/T_c| is the reduced temperature, T_c is the Lambda point temperature, A_\pm,B_\pm are constants (different above and below the transition temperature), and is the critical exponent: \alpha=-0.0127(3). The same effect can be achieved with a low temperature and a long holding time, or with a higher temperature and a short holding time. ==Formula== In the Hollomon–Jaffe parameter, this exchangeability of time and temperature can be described by the following formula: :H_p = \frac {(273.15 + T)}{1000} \cdot (C + \log(t)) This formula is not consistent concerning the units; the parameters must be entered in a certain manner. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. In either case, one uses the heat equation :c_t = D \Delta c, or :P_t = D \Delta P. T is in degrees Celsius. The other (trivial) solution is for all spatial temperature gradients to disappear as well, in which case the temperature become uniform in space, as well. The ""diffusivity constant"" is often not present in mathematical studies of the heat equation, while its value can be very important in engineering. Then, according to the chain rule, one has Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of and solutions of the heat equation with . The equation becomes :q = -k \,\frac{\partial u}{\partial x} Let Q=Q(x,t) be the internal heat energy per unit volume of the bar at each point and time. The vibrational temperature is used commonly when finding the vibrational partition function. Thus instead of being for thermodynamic temperature (Kelvin - K), units of u should be J/L. Then the heat per unit volume u satisfies an equation : \frac{1}{\alpha} \frac{\partial u}{\partial t} = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + \frac{1}{k}q. The Hollomon–Jaffe parameter (HP), also generally known as the Larson–Miller parameter, describes the effect of a heat treatment at a temperature for a certain time. Since heat density is proportional to temperature in a homogeneous medium, the heat equation is still obeyed in the new units. Hence the heat capacity was measured within 2 nK below the transition in an experiment included in a Space Shuttle payload in 1992. thumb|250px|The plot of the specific heat capacity versus temperature. The temperature gradient is a dimensional quantity expressed in units of degrees (on a particular temperature scale) per unit length. This form is more general and particularly useful to recognize which property (e.g. cp or \rho) influences which term. :\rho c_p \frac{\partial T}{\partial t} - abla \cdot \left( k abla T \right) = \dot q_V where \dot q_V is the volumetric heat source. ===Three-dimensional problem=== In the special cases of propagation of heat in an isotropic and homogeneous medium in a 3-dimensional space, this equation is : \frac{\partial u}{\partial t} = \alpha abla^2 u = \alpha \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2 }\right) = \alpha \left( u_{xx} + u_{yy} + u_{zz} \right) where: * u = u(x, y, z, t) is temperature as a function of space and time; * \tfrac{\partial u}{\partial t} is the rate of change of temperature at a point over time; * u_{xx} , u_{yy} , and u_{zz} are the second spatial derivatives (thermal conductions) of temperature in the x , y , and z directions, respectively; * \alpha \equiv \tfrac{k}{c_p\rho} is the thermal diffusivity, a material-specific quantity depending on the thermal conductivity k , the specific heat capacity c_p , and the mass density \rho . Holloman and Jaffe determined the value of C experimentally by plotting hardness versus tempering time for a series of tempering temperatures of interest and interpolating the data to obtain the time necessary to yield a number of different hardness values. These authors derived an expression for the temperature at the center of a sphere :\frac{T_C - T_S}{T_0 - T_S} =2 \sum_{n = 1}^{\infty} (-1)^{n+1} \exp\left({-\frac{n^2 \pi^2 \alpha t}{L^2}}\right) where is the initial temperature of the sphere and the temperature at the surface of the sphere, of radius . A temperature gradient is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. ",+10,1.61,"""2598960.0""",92,355.1,E
+"The reactant 1,3-cyclohexadiene can be photochemically converted to cis-hexatriene. In an experiment, $2.5 \mathrm{mmol}$ of cyclohexadiene are converted to cis-hexatriene when irradiated with 100. W of 280. nm light for $27.0 \mathrm{~s}$. All of the light is absorbed by the sample. What is the overall quantum yield for this photochemical process?","A quantum yield of 1.0 (100%) describes a process where each photon absorbed results in a photon emitted. Since not all photons are absorbed productively, the typical quantum yield will be less than 1. The quantum yield of ANS is ~0.002 in aqueous buffer, but near 0.4 when bound to serum albumin. === Photochemical reactions === The quantum yield of a photochemical reaction describes the number of molecules undergoing a photochemical event per absorbed photon: \Phi=\frac{\rm \\#\ molecules\ undergoing\ reaction\ of\ interest}{\rm \\#\ photons\ absorbed\ by\ photoreactive\ substance} In a chemical photodegradation process, when a molecule dissociates after absorbing a light quantum, the quantum yield is the number of destroyed molecules divided by the number of photons absorbed by the system. In optical spectroscopy, the quantum yield is the probability that a given quantum state is formed from the system initially prepared in some other quantum state. Such effects can be studied with wavelength-tunable lasers and the resulting quantum yield data can help predict conversion and selectivity of photochemical reactions. For example, a singlet to triplet transition quantum yield is the fraction of molecules that, after being photoexcited into a singlet state, cross over to the triplet state. === Photosynthesis === Quantum yield is used in modeling photosynthesis: \Phi = \frac {\rm \mu mol\ CO_2 \ fixed} {\rm \mu mol\ photons \ absorbed} ==See also== *Quantum dot *Quantum efficiency == References == Category:Radiation Category:Spectroscopy Category:Photochemistry Photoexcitation is the production of an excited state of a quantum system by photon absorption. The quantum yield is then calculated by: \Phi = \Phi_\mathrm{R} \times \frac{\mathit{Int}}{\mathit{Int}_\mathrm{R}} \times \frac{1-10^{-A_\mathrm{R}}}{1-10^{-A}} \times \frac{{n}^2}{{n_\mathrm{R}}^2} where * is the quantum yield, * is the area under the emission peak (on a wavelength scale), * is absorbance (also called ""optical density"") at the excitation wavelength, * is the refractive index of the solvent. Quantum photoelectrochemistry in particular provides fundamental insight into basic light-harvesting and photoinduced electro-optical processes in several emerging solar energy conversion technologies for generation of both electricity (photovoltaics) and solar fuels.Ponseca Jr., Carlito S.; Chábera, Pavel; Uhlig, Jens; Persson, Petter; Sundström, Villy (August 2017). The absorption of photons with energy equal to or greater than the band gap of the semiconductor initiates photocatalytic reactions. Quantum photoelectrochemistry is the investigation of the quantum mechanical nature of photoelectrochemistry, the subfield of study within physical chemistry concerned with the interaction of light with electrochemical systems, typically through the application of quantum chemical calculations.Quantum Photoelectrochemistry - Theoretical Studies of Organic Adsorbates on Metal Oxide Surfaces, Petter Persson, Acta Univ. Upsaliensis., Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 544, 53 pp. Uppsala. . On the atomic and molecular scale photoexcitation is the photoelectrochemical process of electron excitation by photon absorption, when the energy of the photon is too low to cause photoionization. Fluorescence quantum yields are measured by comparison to a standard of known quantum yield. One example is the reaction of hydrogen with chlorine, in which as many as 106 molecules of hydrogen chloride can be formed per quantum of blue light absorbed.Laidler K.J., Chemical Kinetics (3rd ed., Harper & Row 1987) p.289 Quantum yields of photochemical reactions can be highly dependent on the structure, proximity and concentration of the reactive chromophores, the type of solvent environment as well as the wavelength of the incident light. Quantum photoelectrochemistry provides an expansion of quantum electrochemistry to processes involving also the interaction with light (photons). Key aspects of quantum photoelectrochemistry are calculations of optical excitations, photoinduced electron and energy transfer processes, excited state evolution, as well as interfacial charge separation and charge transport in nanoscale energy conversion systems.Multiscale Modelling of Interfacial Electron Transfer, Petter Persson, Chapter 3 in: Solar Energy Conversion – Dynamics of Electron and Excitation Transfer P. Piotrowiak (Ed.), RSC Energy and Environment Series (2013) thumbnail|Quantum photoelectrochemistry calculation of photoinduced interfacial electron transfer in a dye-sensitized solar cell. In particle physics, the quantum yield (denoted ) of a radiation-induced process is the number of times a specific event occurs per photon absorbed by the system. The term quantum efficiency (QE) may apply to incident photon to converted electron (IPCE) ratio of a photosensitive device, or it may refer to the TMR effect of a Magnetic Tunnel Junction. Thus, the fluorescence quantum yield is affected if the rate of any non-radiative pathway changes. Quantum yield is defined by the fraction of excited state fluorophores that decay through fluorescence: \Phi_f = \frac{k_f}{k_f + \sum k_\mathrm{nr}} where * is the fluorescence quantum yield, * is the rate constant for radiative relaxation (fluorescence), * is the rate constant for all non-radiative relaxation processes. This synthesis demonstrates that the thermal Diels–Alder reaction favors the undesired regioisomer, but the photoredox-catalyzed reaction gives the desired regioisomer in improved yield. 400px|frameless|center|Key photoredox cycloaddition in total synthesis of Heitziamide A === Photoredox organocatalysis === Organocatalysis is a subfield of catalysis that explores the potential of organic small molecules as catalysts, particularly for the enantioselective creation of chiral molecules. The photoexcitation causes the electrons in atoms to go to an excited state. ",1.4,0.5,"""243.0""",0.396,0.33333333,D
 "In this problem, $2.50 \mathrm{~mol}$ of $\mathrm{CO}_2$ gas is transformed from an initial state characterized by $T_i=450 . \mathrm{K}$ and $P_i=1.35$ bar to a final state characterized by $T_f=800 . \mathrm{K}$ and $P_f=$ 3.45 bar. Using Equation (5.23), calculate $\Delta S$ for this process. Assume ideal gas behavior and use the ideal gas value for $\beta$. For $\mathrm{CO}_2$,
 $$
 \frac{C_{P, m}}{\mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}}=18.86+7.937 \times 10^{-2} \frac{T}{\mathrm{~K}}-6.7834 \times 10^{-5} \frac{T^2}{\mathrm{~K}^2}+2.4426 \times 10^{-8} \frac{T^3}{\mathrm{~K}^3}
-$$","""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by : M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T and : c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. ==Definitions== The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 : \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} : c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} where : t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} measures the temperature relative to the critical point. ==Derivation== For the magnetic analogue of the Maxwell relations for the response functions, the relation : \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 follows, and with thermodynamic stability requiring that c_H, c_M\mbox{ and }\chi_T \geq 0 , one has : c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 which, under the conditions H=0, t>0 and the definition of the critical exponents gives : (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} which gives the Rushbrooke inequality : \alpha' + 2\beta + \gamma' \geq 2. The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup COSILAB is a software tool for solving complex chemical kinetics problems. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). Problems to be solved by COSILAB may involve thousands of reactions amongst hundreds of species for practically any mixture composition, pressure and temperature. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. ==See also== * F-distribution * Folded-t and half-t distributions * Hotelling's T-squared distribution * Multivariate Student distribution * Standard normal table (Z-distribution table) * t-statistic * Tau distribution, for internally studentized residuals * Wilks' lambda distribution * Wishart distribution * Modified half-normal distribution with the pdf on (0, \infty) is given as f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\\\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function. ==Notes== ==References== * * * * ==External links== * *Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term ""Student's distribution"") * First Students on page 112. If the flow velocity is negligible, the general equation of heat transfer reduces to the standard heat equation. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. In an ideal fluid, as described by the Euler equations, the conservation of energy is defined by the equation:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) \right] = 0 where h is the specific enthalpy. The Lee–Kesler method Lee B.I., Kesler M.G., This may also be written as :f(t) = \frac{1}{\sqrt{ u}\,\mathrm{B} (\frac{1}{2}, \frac{ u}{2})} \left(1+\frac{t^2} u \right)^{-( u+1)/2}, where B is the Beta function. Note that the t-distribution (red line) becomes closer to the normal distribution as u increases. ===Cumulative distribution function=== The cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function. It may also be extended to rotating, stratified flows, such as those encountered in geophysical fluid dynamics. == Derivation == === Extension of the ideal fluid energy equation === For a viscous, Newtonian fluid, the governing equations for mass conservation and momentum conservation are the continuity equation and the Navier-Stokes equations:\begin{aligned} {\partial \rho\over{\partial t}} &= - abla\cdot (\rho {\bf v}) \\\ \rho {D{\bf v}\over{Dt}} &= - abla p + abla \cdot \sigma \end{aligned}where p is the pressure and \sigma is the viscous stress tensor, with the components of the viscous stress tensor given by:\sigma_{ij} = \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right) + \zeta \delta_{ij} abla\cdot {\bf v} The energy of a unit volume of the fluid is the sum of the kinetic energy \rho v^{2}/2 \equiv \rho k and the internal energy \rho\varepsilon, where \varepsilon is the specific internal energy. In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature T. ",0,12,30.0,48.6,24,D
+$$","""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. Since the Helmholtz free energy is extensive, the normalization to free energy per site is given as : f = -kT \lim_{N \rightarrow \infty} \frac{1}{N}\log Z_N The magnetization M per site in the thermodynamic limit, depending on the external magnetic field H and temperature T is given by : M(T,H) \ \stackrel{\mathrm{def}}{=}\ \lim_{N \rightarrow \infty} \frac{1}{N} \left( \sum_i \sigma_i \right) = - \left( \frac{\partial f}{\partial H} \right)_T where \sigma_i is the spin at the i-th site, and the magnetic susceptibility and specific heat at constant temperature and field are given by, respectively : \chi_T(T,H) = \left( \frac{\partial M}{\partial H} \right)_T and : c_H = -T \left( \frac{\partial^2 f}{\partial T^2} \right)_H. ==Definitions== The critical exponents \alpha, \alpha', \beta, \gamma, \gamma' and \delta are defined in terms of the behaviour of the order parameters and response functions near the critical point as follows : M(t,0) \simeq (-t)^{\beta}\mbox{ for }t \uparrow 0 : M(0,H) \simeq |H|^{1/ \delta} \operatorname{sign}(H)\mbox{ for }H \rightarrow 0 : \chi_T(t,0) \simeq \begin{cases} (t)^{-\gamma}, & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\gamma'}, & \textrm{for} \ t \uparrow 0 \end{cases} : c_H(t,0) \simeq \begin{cases} (t)^{-\alpha} & \textrm{for} \ t \downarrow 0 \\\ (-t)^{-\alpha'} & \textrm{for} \ t \uparrow 0 \end{cases} where : t \ \stackrel{\mathrm{def}}{=}\ \frac{T-T_c}{T_c} measures the temperature relative to the critical point. ==Derivation== For the magnetic analogue of the Maxwell relations for the response functions, the relation : \chi_T (c_H -c_M) = T \left( \frac{\partial M}{\partial T} \right)_H^2 follows, and with thermodynamic stability requiring that c_H, c_M\mbox{ and }\chi_T \geq 0 , one has : c_H \geq \frac{T}{\chi_T} \left( \frac{\partial M}{\partial T} \right)_H^2 which, under the conditions H=0, t>0 and the definition of the critical exponents gives : (-t)^{-\alpha'} \geq \mathrm{constant}\cdot(-t)^{\gamma'}(-t)^{2(\beta-1)} which gives the Rushbrooke inequality : \alpha' + 2\beta + \gamma' \geq 2. The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup COSILAB is a software tool for solving complex chemical kinetics problems. It can be easily calculated from the cumulative distribution function Fν(t) of the t-distribution: :A(t\mid u) = F_ u(t) - F_ u(-t) = 1 - I_{\frac{ u}{ u +t^2}}\left(\frac{ u}{2},\frac{1}{2}\right), where Ix is the regularized incomplete beta function (a, b). Problems to be solved by COSILAB may involve thousands of reactions amongst hundreds of species for practically any mixture composition, pressure and temperature. For pressures above 1 bar, that means, above the normal boiling point, the typical errors are below 2%. Nowadays, statistical software, such as the R programming language, and functions available in many spreadsheet programs compute values of the t-distribution and its inverse without tables. ==See also== * F-distribution * Folded-t and half-t distributions * Hotelling's T-squared distribution * Multivariate Student distribution * Standard normal table (Z-distribution table) * t-statistic * Tau distribution, for internally studentized residuals * Wilks' lambda distribution * Wishart distribution * Modified half-normal distribution with the pdf on (0, \infty) is given as f(x)= \frac{2\beta^{\frac{\alpha}{2}} x^{\alpha-1} \exp(-\beta x^2+ \gamma x )}{\Psi{\left(\frac{\alpha}{2}, \frac{ \gamma}{\sqrt{\beta}}\right)}}, where \Psi(\alpha,z)={}_1\Psi_1\left(\begin{matrix}\left(\alpha,\frac{1}{2}\right)\\\\(1,0)\end{matrix};z \right) denotes the Fox–Wright Psi function. ==Notes== ==References== * * * * ==External links== * *Earliest Known Uses of Some of the Words of Mathematics (S) (Remarks on the history of the term ""Student's distribution"") * First Students on page 112. If the flow velocity is negligible, the general equation of heat transfer reduces to the standard heat equation. For information on its inverse cumulative distribution function, see . ===Special cases=== Certain values of u give a simple form for Student's t-distribution. u PDF CDF notes 1 \frac{1}{\pi (1+t^2)} \frac{1}{2} + \frac{1}{\pi}\arctan(t) See Cauchy distribution 2 \frac{1}{2\sqrt{2}\left(1+\frac{t^2}{2}\right)^{3/2}} \frac{1}{2}+\frac{t}{2\sqrt{2}\sqrt{1+\frac{t^2}{2}}} 3 \frac{2}{\pi\sqrt{3}\left(1+\frac{t^2}{3}\right)^2} \frac{1}{2}+\frac{1}{\pi}{\left[\frac{1}{\sqrt{3}}{\frac{t}{1+\frac{t^2}{3}}}+\arctan\left(\frac{t}{\sqrt{3}}\right)\right]} 4 \frac{3}{8\left(1+\frac{t^2}{4}\right)^{5/2}} \frac{1}{2}+\frac{3}{8}{\frac{t}{\sqrt{1+\frac{t^2}{4}}}}{\left[1-\frac{1}{12}{\frac{t^2}{1+\frac{t^2}{4}}}\right]} 5 \frac{8}{3\pi\sqrt{5}\left(1+\frac{t^2}{5}\right)^3} \frac{1}{2}+\frac{1}{\pi}{ \left[\frac{t}{\sqrt{5}\left(1+\frac{t^2}{5}\right)} \left(1+\frac{2}{3\left(1+\frac{t^2}{5}\right)}\right) +\arctan\left(\frac{t}{\sqrt{5}}\right)\right]} \infty \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \frac{1}{2}{\left[1+\operatorname{erf}\left(\frac{t}{\sqrt{2}}\right)\right]} See Normal distribution, Error function ===Moments=== For u > 1, the raw moments of the t-distribution are :\operatorname E(T^k)=\begin{cases} 0 & k \text{ odd},\quad 0 Moments of order u or higher do not exist. With :T \sim t_ u and location-scale family transformation :X = \mu + \tau T we get :X \sim lst(\mu, \tau^2, u) The resulting distribution is also called the non- standardized Student's t-distribution. ===Density and first two moments=== The location-scale t distribution has a density defined by: :p(x\mid u,\mu,\tau) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u}\tau\,} \left(1+\frac{1}{ u} \left( \frac{ x-\mu } {\tau } \right)^2\right)^{-( u+1)/2} Equivalently, the density can be written in terms of \tau^2: :p(x\mid u, \mu, \tau^2) = \frac{\Gamma(\frac{ u + 1}{2})}{\Gamma(\frac{ u}{2})\sqrt{\pi u\tau^2}} \left(1+\frac{1}{ u}\frac{(x-\mu)^2}{{\tau}^2}\right)^{-( u+1)/2} Other properties of this version of the distribution are: :\begin{align} \operatorname{E}(X) &= \mu & \text{ for } u > 1 \\\ \operatorname{var}(X) &= \tau^2\frac{ u}{ u-2} & \text{ for } u > 2 \\\ \operatorname{mode}(X) &= \mu \end{align} ===Special cases=== * If X follows a location-scale t-distribution X \sim \mathrm{lst}\left(\mu, \tau^2, u\right) then for u \rightarrow \infty X is normally distributed X \sim \mathrm{N}\left(\mu, \tau^2\right) with mean \mu and variance \tau^2. In an ideal fluid, as described by the Euler equations, the conservation of energy is defined by the equation:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) \right] = 0 where h is the specific enthalpy. The Lee–Kesler method Lee B.I., Kesler M.G., This may also be written as :f(t) = \frac{1}{\sqrt{ u}\,\mathrm{B} (\frac{1}{2}, \frac{ u}{2})} \left(1+\frac{t^2} u \right)^{-( u+1)/2}, where B is the Beta function. Note that the t-distribution (red line) becomes closer to the normal distribution as u increases. ===Cumulative distribution function=== The cumulative distribution function (CDF) can be written in terms of I, the regularized incomplete beta function. It may also be extended to rotating, stratified flows, such as those encountered in geophysical fluid dynamics. == Derivation == === Extension of the ideal fluid energy equation === For a viscous, Newtonian fluid, the governing equations for mass conservation and momentum conservation are the continuity equation and the Navier-Stokes equations:\begin{aligned} {\partial \rho\over{\partial t}} &= - abla\cdot (\rho {\bf v}) \\\ \rho {D{\bf v}\over{Dt}} &= - abla p + abla \cdot \sigma \end{aligned}where p is the pressure and \sigma is the viscous stress tensor, with the components of the viscous stress tensor given by:\sigma_{ij} = \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right) + \zeta \delta_{ij} abla\cdot {\bf v} The energy of a unit volume of the fluid is the sum of the kinetic energy \rho v^{2}/2 \equiv \rho k and the internal energy \rho\varepsilon, where \varepsilon is the specific internal energy. In statistical mechanics, the Rushbrooke inequality relates the critical exponents of a magnetic system which exhibits a first-order phase transition in the thermodynamic limit for non-zero temperature T. ",0,12,"""30.0""",48.6,24,D
 "One mole of $\mathrm{CO}$ gas is transformed from an initial state characterized by $T_i=320 . \mathrm{K}$ and $V_i=80.0 \mathrm{~L}$ to a final state characterized by $T_f=650 . \mathrm{K}$ and $V_f=120.0 \mathrm{~L}$. Using Equation (5.22), calculate $\Delta S$ for this process. Use the ideal gas values for $\beta$ and $\kappa$. For CO,
 $$
 \frac{C_{V, m}}{\mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}}=31.08-0.01452 \frac{T}{\mathrm{~K}}+3.1415 \times 10^{-5} \frac{T^2}{\mathrm{~K}^2}-1.4973 \times 10^{-8} \frac{T^3}{\mathrm{~K}^3}
-$$","""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Lange indirectly defines the values to be at a standard state pressure of ""1 atm (101325 Pa)"", although citing the same NBS and JANAF sources among others. K (? °C), ? K (? °C), ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? * J.D. Cox, DD., Wagman, and V.A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., * Values from CRC refer to ""100 kPa (1 bar or 0.987 standard atmospheres)"". J/(mol K) == Spectral data == UV- Vis Lambda-max ? nm Extinction coefficient ? ",537,4.946,-20.0,-0.16,24.4,E
+$$","""A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States"", AIChE J., 21(3), 510-527, 1975 allows the estimation of the saturated vapor pressure at a given temperature for all components for which the critical pressure Pc, the critical temperature Tc, and the acentric factor ω are known. == Equations == : \ln P_{\rm r} = f^{(0)} + \omega \cdot f^{(1)} : f^{(0)}=5.92714 - \frac{6.09648}{T_{\rm r}} - 1.28862 \cdot \ln T_{\rm r} + 0.169347 \cdot T_{\rm r}^6 : f^{(1)}=15.2518 - \frac{15.6875}{T_{\rm r}}-13.4721 \cdot \ln T_{\rm r} + 0.43577 \cdot T_{\rm r}^6 with :P_{\rm r}=\frac{P}{P_{\rm c}} (reduced pressure) and T_{\rm r}=\frac{T}{T_{\rm c}} (reduced temperature). == Typical errors == The prediction error can be up to 10% for polar components and small pressures and the calculated pressure is typically too low. J/(mol K) Gas properties Std enthalpy change of formation ΔfH ~~o~~ gas ? kJ/mol Standard molar entropy S ~~o~~ gas ? J/(mol·K) Std enthalpy change of vaporizationΔvapH ~~o~~ ? kJ/mol Std entropy change of vaporizationΔvapS ~~o~~ ? Auflage, McGraw-Hill, 1988 == Example calculation == For benzene with * Tc = 562.12 KBrunner E., Thies M.C., Schneider G.M., J.Supercrit.Fluids, 39(2), 160-173, 2006 * Pc = 4898 kPa * Tb = 353.15 KSilva L.M.C., Mattedi S., Gonzalez-Olmos R., Iglesias M., J.Chem.Thermodyn., 38(12), 1725-1736, 2006 * ω = 0.2120Dortmund Data Bank the following calculation for T=Tb results: * Tr = 353.15 / 562.12 = 0.628247 * f(0) = -3.167428 * f(1) = -3.429560 * Pr = exp( f(0) \+ ω f(1) ) = 0.020354 * P = Pr * Pc = 99.69 kPa The correct result would be P = 101.325 kPa, the normal (atmospheric) pressure. J/(mol·K) Solid properties Std enthalpy change of formation -ΔfH ~~o~~ solid 448.9 kcal/mol Std entropy change of formation -ΔfS ~~o~~ solid 71.9 kcal/mol Std Gibbs free energy change of formation -ΔfG ~~o~~ solid 427.5 kcal/mol Std enthalpy change of absorption -ΔaH ~~o~~ solid ? kJ/mol Standard molar entropy S ~~o~~ solid ? The unit system used (K or R for T) is irrelevant because of the usage of the reduced values Tr and Pr. ==See also== *Vapour pressure of water *Antoine equation *Tetens equation *Arden Buck equation *Goff–Gratch equation == References == Category:Thermodynamic models Pa Std enthalpy change of fusionΔfusH ~~o~~ ? kJ/mol Std entropy change of fusionΔfusS ~~o~~ ? == Specific heat capacity == J/(mol·K) J/(g·K) 1 H hydrogen (H2, gas) use 28.836 14.304 CRC 28.836 14.304 WEL 28.82 LNG 28.84 2 He helium (gas) use 20.786 5.193 CRC 20.786 5.193 WEL 20.786 LNG 20.786 3 Li lithium use 24.860 3.582 CRC 24.860 3.582 WEL 24.8 LNG 24.8 4 Be beryllium use 16.443 1.825 CRC 16.443 1.825 WEL 16.4 LNG 16.38 5 B boron (rhombic) use 11.087 1.026 CRC 11.087 1.026 WEL 11.1 LNG 11.1 6 C carbon (graphite) use 8.517 0.709 CRC 8.517 0.709 WEL 8.53 LNG 8.517 6 C carbon (diamond, nonstd state) use 6.115 0.509 WEL 6.115 LNG 6.116 7 N nitrogen (N2, gas) use 29.124 1.040 CRC 29.124 1.040 WEL 29.12 LNG 29.124 8 O oxygen (O2, gas) use 29.378 0.918 CRC 29.378 0.918 WEL 29.4 LNG 29.4 9 F fluorine (F2, gas) use 31.304 0.824 CRC 31.304 0.824 WEL 31.3 LNG 31.30 10 Ne neon (gas) use 20.786 1.030 CRC 20.786 1.030 WEL 20.786 LNG 20.786 11 Na sodium use 28.230 1.228 CRC 28.230 1.228 WEL 28.2 LNG 28.15 12 Mg magnesium use 24.869 1.023 CRC 24.869 1.023 WEL 24.9 LNG 24.87 13 Al aluminium use 24.200 0.897 CRC 24.200 0.897 WEL 24.4 LNG 24.4 14 Si silicon use 19.789 0.705 CRC 19.789 0.705 WEL 20.0 LNG 20.00 15 P phosphorus (white) use 23.824 0.769 CRC 23.824 0.769 WEL 23.84 LNG 23.83 15 P phosphorus (red, nonstd state) use 21.19 0.684 WEL 21.2 LNG 21.19 16 S sulfur (rhombic) use 22.75 0.710 CRC 22.75 0.710 WEL 22.6 LNG 22.60 16 S sulfur (monoclinic, nonstd state) use 23.23 0.724 LNG 23.23 17 Cl chlorine (Cl2 gas) use 33.949 0.479 CRC 33.949 0.479 WEL 33.91 LNG 33.95 18 Ar argon (gas) use 20.786 0.520 CRC 20.786 0.520 WEL 20.786 LNG 20.79 19 K potassium use 29.600 0.757 CRC 29.600 0.757 WEL 29.6 LNG 29.60 20 Ca calcium use 25.929 0.647 CRC 25.929 0.647 WEL 25.3 LNG 25.9 21 Sc scandium use 25.52 0.568 CRC 25.52 0.568 WEL 25.5 LNG 25.52 22 Ti titanium use 25.060 0.523 CRC 25.060 0.523 WEL 25.0 LNG 25.0 23 V vanadium use 24.89 0.489 CRC 24.89 0.489 WEL 24.9 LNG 24.90 24 Cr chromium use 23.35 0.449 CRC 23.35 0.449 WEL 23.3 LNG 23.43 25 Mn manganese use 26.32 0.479 CRC 26.32 0.479 WEL 26.3 LNG 26.30 26 Fe iron (alpha) use 25.10 0.449 CRC 25.10 0.449 WEL 25.1 LNG 25.09 27 Co cobalt use 24.81 0.421 CRC 24.81 0.421 WEL 24.8 LNG 24.8 28 Ni nickel use 26.07 0.444 CRC 26.07 0.444 WEL 26.1 LNG 26.1 29 Cu copper use 24.440 0.385 CRC 24.440 0.385 WEL 24.43 LNG 24.44 30 Zn zinc use 25.390 0.388 CRC 25.390 0.388 WEL 25.4 LNG 25.40 31 Ga gallium use 25.86 0.371 CRC 25.86 0.371 WEL 25.9 LNG 26.06 32 Ge germanium use 23.222 0.320 CRC 23.222 0.320 WEL 23.35 LNG 23.3 33 As arsenic (alpha, gray) use 24.64 0.329 CRC 24.64 0.329 WEL 24.6 LNG 24.64 34 Se selenium (hexagonal) use 25.363 0.321 CRC 25.363 0.321 WEL 25.36 LNG 24.98 35 Br bromine use (Br2) 75.69 0.474 CRC 36.057 0.226 WEL (liquid) 75.69 WEL (Br2, gas, nonstd state) 36.0 LNG (Br2, liquid) 75.67 36 Kr krypton (gas) use 20.786 0.248 CRC 20.786 0.248 WEL 20.786 LNG 20.786 37 Rb rubidium use 31.060 0.363 CRC 31.060 0.363 WEL 31.1 LNG 31.06 38 Sr strontium use 26.4 0.301 CRC 26.4 0.301 WEL 26 LNG 26.79 39 Y yttrium use 26.53 0.298 CRC 26.53 0.298 WEL 26.5 LNG 26.51 40 Zr zirconium use 25.36 0.278 CRC 25.36 0.278 WEL 25.4 LNG 25.40 41 Nb niobium use 24.60 0.265 CRC 24.60 0.265 WEL 24.6 LNG 24.67 42 Mo molybdenum use 24.06 0.251 CRC 24.06 0.251 WEL 24.1 LNG 24.13 43 Tc technetium use 24.27 LNG 24.27 44 Ru ruthenium use 24.06 0.238 CRC 24.06 0.238 WEL 24.1 LNG 24.1 45 Rh rhodium use 24.98 0.243 CRC 24.98 0.243 WEL 25.0 LNG 24.98 46 Pd palladium use 25.98 0.244 CRC 25.98 0.246 [sic] WEL 26.0 LNG 25.94 47 Ag silver use 25.350 0.235 CRC 25.350 0.235 WEL 25.4 LNG 25.4 48 Cd cadmium use 26.020 0.232 CRC 26.020 0.232 WEL 26.0 LNG 25.9 49 In indium use 26.74 0.233 CRC 26.74 0.233 WEL 26.7 LNG 26.7 50 Sn tin (white) use 27.112 0.228 CRC 27.112 0.228 WEL 27.0 LNG 26.99 50 Sn tin (gray, nonstd state) use 25.77 0.217 WEL 25.8 LNG 25.77 51 Sb antimony use 25.23 0.207 CRC 25.23 0.207 WEL 25.2 LNG 25.2 52 Te tellurium use 25.73 0.202 CRC 25.73 0.202 WEL 25.7 LNG 25.70 53 I iodine use (I2) 54.44 0.214 CRC 36.888 0.145 WEL (solid) 54.44 WEL (I2, gas, nonstd state) 36.9 LNG (I2, solid) 54.44 LNG (I2, gas, nonstd state) 36.86 54 Xe xenon (gas) use 20.786 0.158 CRC 20.786 0.158 WEL 20.786 LNG 20.786 55 Cs caesium use 32.210 0.242 CRC 32.210 0.242 WEL 32.2 LNG 32.20 56 Ba barium use 28.07 0.204 CRC 28.07 0.204 WEL 28.1 LNG 28.10 57 La lanthanum use 27.11 0.195 CRC 27.11 0.195 WEL 27.1 LNG 27.11 58 Ce cerium (gamma, fcc) use 26.94 0.192 CRC 26.94 0.192 WEL 26.9 LNG 26.9 59 Pr praseodymium use 27.20 0.193 CRC 27.20 0.193 WEL 27.2 LNG 27.20 60 Nd neodymium use 27.45 0.190 CRC 27.45 0.190 WEL 27.4 LNG 27.5 62 Sm samarium use 29.54 0.197 CRC 29.54 0.197 WEL 29.5 LNG 29.54 63 Eu europium use 27.66 0.182 CRC 27.66 0.182 WEL 27.7 LNG 27.66 64 Gd gadolinium use 37.03 0.236 CRC 37.03 0.236 WEL 37.0 LNG 37.03 65 Tb terbium use 28.91 0.182 CRC 28.91 0.182 WEL 28.9 LNG 28.91 66 Dy dysprosium use 27.7 0.170 CRC 27.7 0.170 WEL 27.2 LNG 27.7 67 Ho holmium use 27.15 0.165 CRC 27.15 0.165 WEL 27.2 LNG 27.15 68 Er erbium use 28.12 0.168 CRC 28.12 0.168 WEL 28.1 LNG 28.12 69 Tm thulium use 27.03 0.160 CRC 27.03 0.160 WEL 27.0 LNG 27.03 70 Yb ytterbium use 26.74 0.155 CRC 26.74 0.155 WEL 26.7 LNG 26.74 71 Lu lutetium use 26.86 0.154 CRC 26.86 0.154 WEL 26.9 LNG 26.86 72 Hf hafnium (hexagonal) use 25.73 0.144 CRC 25.73 0.144 WEL 25.7 LNG 25.69 73 Ta tantalum use 25.36 0.140 CRC 25.36 0.140 WEL 25.4 LNG 25.40 74 W tungsten use 24.27 0.132 CRC 24.27 0.132 WEL 24.3 LNG 24.3 75 Re rhenium use 25.48 0.137 CRC 25.48 0.137 WEL 25.5 LNG 25.5 76 Os osmium use 24.7 0.130 CRC 24.7 0.130 WEL 25 LNG 24.7 77 Ir iridium use 25.10 0.131 CRC 25.10 0.131 WEL 25.1 LNG 25.06 78 Pt platinum use 25.86 0.133 CRC 25.86 0.133 WEL 25.9 79 Au gold use 25.418 0.129 CRC 25.418 0.129 WEL 25.42 LNG 25.36 80 Hg mercury (liquid) use 27.983 0.140 CRC 27.983 0.140 WEL 27.98 LNG 28.00 81 Tl thallium use 26.32 0.129 CRC 26.32 0.129 WEL 26.3 LNG 26.32 82 Pb lead use 26.650 0.129 CRC 26.650 0.129 WEL 26.4 LNG 26.84 83 Bi bismuth use 25.52 0.122 CRC 25.52 0.122 WEL 25.5 LNG 25.5 84 Po polonium use 26.4 LNG 26.4 86 Rn radon (gas) use 20.786 0.094 CRC 20.786 0.094 WEL 20.79 LNG 20.79 87 Fr francium use 31.80 LNG 31.80 89 Ac actinium use 27.2 0.120 CRC 27.2 0.120 WEL 27.2 LNG 27.2 90 Th thorium use 26.230 0.113 CRC 26.230 0.113 WEL 27.3 LNG 27.32 92 U uranium use 27.665 0.116 CRC 27.665 0.116 WEL 27.7 LNG 27.66 93 Np neptunium use 29.46 LNG 29.46 94 Pu plutonium use 35.5 LNG 35.5 95 Am americium use 62.7 LNG 62.7 == Notes == * All values refer to 25 °C and to the thermodynamically stable standard state at that temperature unless noted. {{OrganicBox complete |wiki_name=Serine |image=110px|Skeletal structure of L-serine 130px|3D structure of L-serine |name=-2-amino-3-hydroxypropanoic acid |C=3 |H=7 |N=1 |O=3 |mass=105.09 |abbreviation=S, Ser |synonyms= |SMILES=OCC(N)C(=O)O |InChI=1/C3H7NO3/c4-2(1-5)3(6)7/h2,5H,1,4H2,(H,6,7)/f/h6H |CAS=56-45-1 |DrugBank= |EINECS= |PubChem= 5951 (D) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry=GMD MS Spectrum |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o_solid= |S_o_solid= |heat_capacity_solid= |density_solid=1.537 |melting_point_C=228 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.912 |isoelectric_point=5.68 |disociation_constant=2.13, 9.05 |tautomers=-3.539 |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # # # Category:Chemical data pages Category:Chemical data pages cleanup {{OrganicBox complete |wiki_name=Lysine |image=110px|Skeletal structure of L-lysine 130px|3D structure of L-lysine |name=-2,6-Diaminohexanoic acid |C=6 |H=14 |N=2 |O=2 |mass=146.19 |abbreviation=K, Lys |synonyms= |SMILES=NCCCCC(N)C(=O)O |InChI=1/C6H14N2O2/c7-4-2-1-3-5(8)6(9)10/h5H,1-4,7-8H2,(H,9,10)/f/h9H |CAS=56-87-1 |DrugBank= |EINECS=200-294-2 |PubChem=866 (DL) , 5962 (L) |index_of_refraction= |abbe_number= |dielectric_constant= |magnetic_susceptibility= |lambda_max= |extinction_coefficient= |absorption_bands= |proton_NMR= |carbon_NMR= |other_NMR= |mass_spectrometry= |triple_point_K= |triple_point_C= |triple_point_Pa= |criticle_point_K= |criticle_point_C= |criticle_point_Pa= |delta_fus_H_o= |delta_fus_S_o= |delta_vap_H_o= |delta_vap_S_o= |delta_f_H_o= |S_o_solid= |heat_capacity_solid= |density_solid= |melting_point_C=224 |melting_point_F= |melting_point_K= |delta_f_H_o_liquid= |S_o_liquid= |heat_capacity_liquid= |density_liquid= |viscosity_liquid= |boiling_point_C= |boiling_point_F= |boiling_point_K= |delta_f_H_o_gas= |S_o_gas= |heat_capacity_gas= |viscosity_gas= |MSDS= |main_hazards= |nfpa_health= |nfpa_flammability= |nfpa_reactivity= |nfpa_special= |flash_point= |r_phrases= |s_phrases= |RTECS_number= |XLogP=-2.926 |isoelectric_point=9.74 |disociation_constant=2.15, 9.16, 10.67 |tautomers= |H_bond_donor=3 |H_bond_acceptor=4 }} ==References== # (Lysine) # (DL-Lysine) # (L-Lysine) Category:Chemical data pages Category:Chemical data pages cleanup In fluid dynamics, the general equation of heat transfer is a nonlinear partial differential equation describing specific entropy production in a Newtonian fluid subject to thermal conduction and viscous forces:}_{\text{Heat Gain}} = \underbrace{ abla\cdot (\kappa abla T)}_{\text{Thermal Conduction}} + \underbrace{{\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla\cdot {\bf v})^{2}}_{\text{Viscous Dissipation}} |border colour=#0073CF|background colour=#F5FFFA}}where s is the specific entropy, \rho is the fluid's density, T is the fluid's temperature, D/Dt is the material derivative, \kappa is the thermal conductivity, \mu is the dynamic viscosity, \zeta is the second Lamé parameter, {\bf v} is the flow velocity, abla is the del operator used to characterize the gradient and divergence, and \delta_{ij} is the Kronecker delta. J/(mol K) Liquid properties Std enthalpy change of formation ΔfH ~~o~~ liquid ? kJ/mol Standard molar entropy S ~~o~~ liquid ? Then the general equation for conservation of energy is:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] + abla\cdot \left[ \rho {\bf v}(k+ h) - \kappa abla T - \sigma\cdot {\bf v} \right] = 0 === Equation for entropy production === Note that the thermodynamic relations for the internal energy and enthalpy are given by:\begin{aligned} \rho d\varepsilon &= \rho Tds + {p\over{\rho}}d\rho \\\ \rho dh &= \rho Tds + dp \end{aligned}We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity {\bf v} to yield:\rho {Dk\over{Dt}} = -{\bf v}\cdot abla p + v_{i}{\partial\sigma_{ij}\over{\partial x_{j}}} The second term on the righthand side may be expanded to read:\begin{aligned} v_{i} {\partial \sigma_{ij}\over{\partial x_{j}}} &= {\partial\over{\partial x_{j}}}\left(\sigma_{ij}v_{i} \right ) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ &\equiv abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \end{aligned} With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:\rho {Dk\over{Dt}} = -\rho {\bf v}\cdot abla h + \rho T {\bf v}\cdot abla s + abla\cdot (\sigma \cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now expanding the time derivative of the total energy, we have:{\partial\over{\partial t}}\left[ \rho (k+\varepsilon) \right] = \rho {\partial k\over{\partial t}} + \rho {\partial\varepsilon\over{\partial t}} + (k+\varepsilon) {\partial \rho\over{\partial t}} Then by expanding each of these terms, we find that:\begin{aligned} \rho {\partial k\over{\partial t}} &= -\rho {\bf v}\cdot abla k - \rho {\bf v}\cdot abla h + \rho T{\bf v}\cdot abla s + abla\cdot(\sigma\cdot {\bf v}) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} \\\ \rho {\partial\varepsilon\over{\partial t}} &= \rho T {\partial s\over{\partial t}} - {p\over{\rho}} abla\cdot(\rho {\bf v}) \\\ (k+\varepsilon){\partial\rho\over{\partial t}} &= -(k+\varepsilon) abla\cdot (\rho {\bf v}) \end{aligned} And collecting terms, we are left with:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:{\partial\over{\partial t}}\left[\rho(k+\varepsilon) \right ] + abla \cdot\left[\rho {\bf v}(k+h) - \kappa abla T - \sigma\cdot {\bf v} \right ] = \rho T {Ds\over{Dt}} - abla\cdot(\kappa abla T) - \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} The product of the viscous stress tensor and the velocity gradient can be expanded as:\begin{aligned} \sigma_{ij}{\partial v_{i}\over{\partial x_{j}}} &= \mu\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right){\partial v_{i}\over{\partial x_{j}}} + \zeta \delta_{ij}{\partial v_{i}\over{\partial x_{j}}} abla\cdot {\bf v} \\\ &= {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} \end{aligned} Thus leading to the final form of the equation for specific entropy production:\rho T {Ds\over{Dt}} = abla\cdot(\kappa abla T) + {\mu\over{2}}\left( {\partial v_{i}\over{\partial x_{j}}} + {\partial v_{j}\over{\partial x_{i}}} - {2\over{3}}\delta_{ij} abla\cdot {\bf v} \right)^{2} + \zeta( abla \cdot {\bf v})^{2} In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to Ds/Dt=0 \- showing that ideal fluid flow is isentropic. == Application == This equation is derived in Section 49, at the opening of the chapter on ""Thermal Conduction in Fluids"" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics. Lange indirectly defines the values to be at a standard state pressure of ""1 atm (101325 Pa)"", although citing the same NBS and JANAF sources among others. K (? °C), ? K (? °C), ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? J/(mol K) Heat capacity cp ? * J.D. Cox, DD., Wagman, and V.A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., * Values from CRC refer to ""100 kPa (1 bar or 0.987 standard atmospheres)"". J/(mol K) == Spectral data == UV- Vis Lambda-max ? nm Extinction coefficient ? ",537,4.946,"""-20.0""",-0.16,24.4,E
 "You are given the following reduction reactions and $E^{\circ}$ values:
 $$
 \begin{array}{ll}
@@ -1037,20 +1037,20 @@ $$
 \mathrm{Fe}^{2+}(a q)+2 \mathrm{e}^{-} \rightarrow \mathrm{Fe}(s) & E^{\circ}=-0.447 \mathrm{~V}
 \end{array}
 $$
-Calculate $E^{\circ}$ for the half-cell reaction $\mathrm{Fe}^{3+}(a q)+3 \mathrm{e}^{-} \rightarrow \mathrm{Fe}(s)$.","These changes can be represented in formulas by inserting appropriate electrons into each half reaction: :\begin{align} & \ce{Fe^2+ -> Fe^3+ + e-} \\\ & \ce{Cl2 + 2e- -> 2Cl-} \end{align} Given two half reactions it is possible, with knowledge of appropriate electrode potentials, to arrive at the complete (original) reaction the same way. The sum of these two half reactions is the oxidation–reduction reaction. ==Half-reaction balancing method== Consider the reaction below: :Cl2 + 2Fe^2+ -> 2Cl- + 2Fe^3+ The two elements involved, iron and chlorine, each change oxidation state; iron from +2 to +3, chlorine from 0 to −1\. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is the Faraday's constant. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is Faraday's constant. The determination of the formal reduction potential at pH = 7 for a given biochemical half-reaction requires thus to calculate it with the corresponding Nernst equation as a function of pH. The global reaction can thus be decomposed in half redox reactions as follows: :2 (Fe2+ → Fe3+ \+ e−) (oxidation of 2 iron(II) ions) :2 (H2O + e− → ½ H2 \+ OH−) (reduction of 2 water protons) to give: :2 Fe2+ \+ 2 H2O → 2 Fe3+ \+ H2 \+ 2 OH− Adding to this reaction one intact iron(II) ion for each two oxidized iron(II) ions leads to: :3 Fe2+ \+ 2 H2O → Fe2+ \+ 2 Fe3+ \+ H2 \+ 2 OH− Electroneutrality requires the iron cations on both sides of the equation to be counterbalanced by 6 hydroxyl anions (OH−): :3 Fe2+ \+ 6 OH− \+ 2 H2O → Fe2+ \+ 2 Fe3+ \+ H2 \+ 8 OH− :3 Fe(OH)2 \+ 2 H2O → Fe(OH)2 \+ 2 Fe(OH)3 \+ H2 For completing the main reaction, two companion reactions have still to be taken into account: The autoprotolysis of the hydroxyl anions; a proton exchange between two OH−, like in a classical acid–base reaction: :OH− \+ OH− → O2− \+ H2O :acid 1 + base 2 → base 1 + acid 2, or also, :2 OH− → O2− \+ H2O it is then possible to reorganize the global reaction as: :3 Fe(OH)2 \+ 2 H2O → (FeO + H2O) + (Fe2O3 \+ 3 H2O) + H2 :3 Fe(OH)2 \+ 2 H2O → FeO + Fe2O3 \+ 4 H2O + H2 :3 Fe(OH)2 → FeO + Fe2O3 \+ 2 H2O + H2 Considering then the formation reaction of iron(II,III) oxide: :Fe^{II}O + Fe^{III}2O3 -> Fe3O4 it is possible to write the balanced global reaction: :3 Fe(OH)2 → (FeO·Fe2O3) + 2 H2O + H2 in its final form, known as the Schikorr reaction: :3 Fe(OH)2 → Fe3O4 \+ 2 H2O + H2 == Occurrences == The Schikorr reaction can occur in the process of anaerobic corrosion of iron and carbon steel in various conditions. At pH = 7, \+ → P680 ~ +1.0 Half-reaction independent of pH as no is involved in the reaction ==See also== * Nernst equation * Electron bifurcation * Pourbaix diagram * Reduction potential ** Dependency of reduction potential on pH * Standard electrode potential * Standard reduction potential * Standard reduction potential (data page) * Standard state ==References== ==Bibliography== ;Electrochemistry * ;Bio-electrochemistry * * * * * * ;Microbiology * * Category:Biochemistry Standard reduction potentials for half-reactions important in biochemistry Category:Electrochemical potentials Category:Thermodynamics databases Category:Biochemistry databases The same also applies for the reduction potential of oxygen: : For , E^{\ominus}_\text{red} = 1.229 V, so, applying the Nernst equation for pH = 7 gives: : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 1.229 - \left(0.05916 \ \text{×} \ 7\right) = 0.815 \ V For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. Is it simply: * E_h = E_\text{red} calculated at pH 7 (with or without corrections for the activity coefficients), * E^{\ominus '}_\text{red}, a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it, * E^{\ominus '}_\text{red apparent at pH 7}, an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio \frac{h} {z} = \frac{\text{(number of involved protons)}} {\text{(number of exchanged electrons)}}. This immediately leads to the Nernst equation, which for an electrochemical half-cell is E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln Q_r=E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{a_\text{Red}}{a_\text{Ox}}. The values below are standard apparent reduction potentials for electro- biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. The offset of -414 mV in E_\text{red} is the same for both reduction reactions because they share the same linear relationship as a function of pH and the slopes of their lines are the same. Taking into account the activity coefficients (\gamma) the Nernst equation becomes: E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln\left(\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\frac{C_\text{Red}}{C_\text{Ox}}\right) E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \left(\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} + \ln\frac{C_\text{Red}}{C_\text{Ox}}\right) E_\text{red} = \underbrace{\left(E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\right)}_{E^{\ominus '}_\text{red}} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} Where the first term including the activity coefficients (\gamma) is denoted E^{\ominus '}_\text{red} and called the formal standard reduction potential, so that E_\text{red} can be directly expressed as a function of E^{\ominus '}_\text{red} and the concentrations in the simplest form of the Nernst equation: E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} ===Formal standard reduction potential=== When wishing to use simple concentrations in place of activities, but that the activity coefficients are far from unity and can no longer be neglected and are unknown or too difficult to determine, it can be convenient to introduce the notion of the ""so-called"" standard formal reduction potential (E^{\ominus '}_\text{red}) which is related to the standard reduction potential as follows: E^{\ominus '}_\text{red}=E^{\ominus}_\text{red}-\frac{RT}{zF}\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} So that the Nernst equation for the half-cell reaction can be correctly formally written in terms of concentrations as: E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} and likewise for the full cell expression. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives: : : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 0 - \left(0.05916 \ \text{×} \ 7\right) = -0.414 \ V In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons () into hydrogen gas is no longer zero as with the standard hydrogen electrode (SHE) at 1 M (pH = 0) in classical electrochemistry, but that E_\text{red} = -0.414\mathrm V versus the standard hydrogen electrode (SHE). In any given oxidation-reduction reaction, there are two half reactions—oxidation half reaction and reduction half reaction. For a complete electrochemical reaction (full cell), the equation can be written as E_\text{cell} = E^\ominus_\text{cell} - \frac{RT}{zF} \ln Q_r where: * is the half-cell reduction potential at the temperature of interest, * is the standard half-cell reduction potential, * is the cell potential (electromotive force) at the temperature of interest, * is the standard cell potential, * is the universal gas constant: , * is the temperature in kelvins, * is the number of electrons transferred in the cell reaction or half-reaction, * is the Faraday constant, the magnitude of charge (in coulombs) per mole of electrons: , * is the reaction quotient of the cell reaction, and * is the chemical activity for the relevant species, where is the activity of the reduced form and is the activity of the oxidized form. ===Thermal voltage=== At room temperature (25 °C), the thermal voltage V_T=\frac{RT}{F} is approximately 25.693 mV. In chemistry, a half reaction (or half-cell reaction) is either the oxidation or reduction reaction component of a redox reaction. So, : -zFE^\ominus_{cell} = -RT \ln{K} And therefore: : E^\ominus_{cell} = \frac{RT} {zF} \ln{K} Starting from the Nernst equation, one can also demonstrate the same relationship in the reverse way. The numerically simplified form of the Nernst equation is expressed as: :E_\text{red} = E^\ominus_\text{red} - \frac{0.059\ V}{z} \log_{10}\frac{a_\text{Red}}{a_\text{Ox}} Where E^\ominus_\text{red} is the standard reduction potential of the half-reaction expressed versus the standard reduction potential of hydrogen. This is represented in the following oxidation half reaction (note that the electrons are on the products side): :Zn_{(s)} -> Zn^2+ + 2e- At the Cu cathode, reduction takes place (electrons are accepted). Half reactions can be written to describe both the metal undergoing oxidation (known as the anode) and the metal undergoing reduction (known as the cathode). So, at pH = 7, E_\text{red} = -0.414 V for the reduction of protons. ",-0.041,1.06,27.0,56,0,A
-"At $298.15 \mathrm{~K}, \Delta G_f^{\circ}(\mathrm{C}$, graphite $)=0$, and $\Delta G_f^{\circ}(\mathrm{C}$, diamond $)=2.90 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Therefore, graphite is the more stable solid phase at this temperature at $P=P^{\circ}=1$ bar. Given that the densities of graphite and diamond are 2.25 and $3.52 \mathrm{~kg} / \mathrm{L}$, respectively, at what pressure will graphite and diamond be in equilibrium at $298.15 \mathrm{~K}$ ?","At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at , a pressure of is needed. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at , a pressure of is needed. ===Other properties=== thumb|upright=1.15|Molar volume against pressure at room temperature The acoustic and thermal properties of graphite are highly anisotropic, since phonons propagate quickly along the tightly bound planes, but are slower to travel from one plane to another. At surface air pressure (one atmosphere), diamonds are not as stable as graphite, and so the decay of diamond is thermodynamically favorable (δH = ). A similar proportion of diamonds comes from the lower mantle at depths between 660 and 800 km. Diamond is thermodynamically stable at high pressures and temperatures, with the phase transition from graphite occurring at greater temperatures as the pressure increases. The equilibrium pressure varies linearly with temperature, between at and at (the diamond/graphite/liquid triple point). It is chemically inert, not reacting with most corrosive substances, and has excellent biological compatibility. ===Thermodynamics=== The equilibrium pressure and temperature conditions for a transition between graphite and diamond are well established theoretically and experimentally. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, but diamond is metastable and converts to it at a negligible rate under those conditions. File:Graphite ambient STM.jpg|Scanning tunneling microscope image of graphite surface File:Graphite-layers- side-3D-balls.png|Side view of ABA layer stacking File:Graphite-layers- top-3D-balls.png|Plane view of layer stacking File:Graphite-unit- cell-3D-balls.png|Alpha graphite's unit cell ===Thermodynamics=== The equilibrium pressure and temperature conditions for a transition between graphite and diamond is well established theoretically and experimentally. However, at temperatures above about , diamond rapidly converts to graphite. However, at temperatures above about , diamond rapidly converts to graphite. Above the graphite- diamond-liquid carbon triple point, the melting point of diamond increases slowly with increasing pressure; but at pressures of hundreds of GPa, it decreases. At high pressure (~) diamond can be heated up to , and a report published in 2009 suggests that diamond can withstand temperatures of and above. Thus, graphite is much softer than diamond. The toughness of natural diamond has been measured as 7.5–10 MPa·m1/2. However, owing to a very large kinetic energy barrier, diamonds are metastable; they will not decay into graphite under normal conditions. ==See also== *Chemical vapor deposition of diamond *Crystallographic defects in diamond *Nitrogen-vacancy center *Synthetic diamond ==References== ==Further reading== *Pagel-Theisen, Verena. (2001). The pressure changes linearly between at and at (the diamond/graphite/liquid triple point). Sufficiently small diamonds can form in the cold of space because their lower surface energy makes them more stable than graphite. Diamonds are carbon crystals that form under high temperatures and extreme pressures such as deep within the Earth. Much higher pressures may be possible with nanocrystalline diamonds. ====Elasticity and tensile strength==== Usually, attempting to deform bulk diamond crystal by tension or bending results in brittle fracture. Research results published in an article in the scientific journal Nature Physics in 2010 suggest that at ultrahigh pressures and temperatures (about 10 million atmospheres or 1 TPa and 50,000 °C) diamond melts into a metallic fluid. ",3.8,4.0,0.54,-1.49,1.51,E
-"Imagine tossing a coin 50 times. What are the probabilities of observing heads 25 times (i.e., 25 successful experiments)?","Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. A fair coin has the probability of success 0.5 by definition. For comparison, we could define an event to occur when ""at least one 'heads'"" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. Cambridge University Press, New York (NY), 1995, p.67-68 ==Example: tossing coins== Consider the simple experiment where a fair coin is tossed four times. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses. ==Mathematical description== A random experiment is described or modeled by a mathematical construct known as a probability space. However, there are experiments that are not easily described by a set of equally likely outcomes-- for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. ==See also== * * * * * ==References== ==External links== * Category:Experiment (probability theory) In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, ""success"" and ""failure"", in which the probability of success is the same every time the experiment is conducted. In probability theory, an outcome is a possible result of an experiment or trial. The probability function P is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would approach agreement with the values P assigns them. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis. ==Experiments and trials== Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. The probability of exactly k successes in the experiment B(n,p) is given by: :P(k)={n \choose k} p^k q^{n-k} where {n \choose k} is a binomial coefficient. For the experiment where we flip a coin twice, the four possible outcomes that make up our sample space are (H, T), (T, H), (T, T) and (H, H), where ""H"" represents a ""heads"", and ""T"" represents a ""tails"". Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. As a simple experiment, we may flip a coin twice. For example, when tossing an ordinary coin, one typically assumes that the outcomes ""head"" and ""tail"" are equally likely to occur. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial. In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. When an experiment is conducted, one (and only one) outcome results-- although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. The probability of the outcome of an experiment is never negative, although a quasiprobability distribution allows a negative probability, or quasiprobability for some events. ",-1.5,+2.35,0.249,-1.49,0.11,E
-"In a rotational spectrum of $\operatorname{HBr}\left(B=8.46 \mathrm{~cm}^{-1}\right)$, the maximum intensity transition in the R-branch corresponds to the $J=4$ to 5 transition. At what temperature was the spectrum obtained?","That makes the reciprocal of the brightness temperature: ::T_b^{-1} = \frac{k}{h u}\, \text{ln}\left[1 + \frac{e^{\frac{h u}{kT}}-1}{\epsilon}\right] At low frequency and high temperatures, when h u \ll kT, we can use the Rayleigh–Jeans law: ::I_{ u} = \frac{2 u^2k T}{c^2} so that the brightness temperature can be simply written as: ::T_b=\epsilon T\, In general, the brightness temperature is a function of u, and only in the case of blackbody radiation it is the same at all frequencies. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation. == Calculating by frequency == The brightness temperature of a source with known spectral radiance can be expressed as: : T_b=\frac{h u}{k} \ln^{-1}\left( 1 + \frac{2h u^3}{I_{ u}c^2} \right) When h u \ll kT we can use the Rayleigh–Jeans law: : T_b=\frac{I_{ u}c^2}{2k u^2} For narrowband radiation with very low relative spectral linewidth \Delta u \ll u and known radiance I we can calculate the brightness temperature as: : T_b=\frac{I c^2}{2k u^2\Delta u} == Calculating by wavelength == Spectral radiance of black-body radiation is expressed by wavelength as: : I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1} So, the brightness temperature can be calculated as: : T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right) For long-wave radiation hc/\lambda \ll kT the brightness temperature is: : T_b=\frac{I_{\lambda}\lambda^4}{2kc} For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c: : T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} } ==In oceanography== In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature of the water. ==References== Category:Temperature Category:Radio astronomy Category:Planetary science Finally, rotational energy states describe semi-rigid rotation of the entire molecule and produce transition wavelengths in the far infrared and microwave regions (about 100-10,000 μm in wavelength). At high frequencies (short wavelengths) and low temperatures, the conversion must proceed through Planck's law. In agreement with this estimate, vibrational spectra show transitions in the near infrared (about 1 - 5 μm). The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. Typically, the largest fluctuations of the primordial CMB temperature occur on angular scales of about 1°. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. For the radiation of a helium–neon laser with a power of 1 mW, a frequency spread Δf = 1 GHz, an output aperture of 1 mm, and a beam dispersion half-angle of 0.56 mrad, the brightness temperature would be . It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. In spectroscopy, a Soret peak or Soret band is an intense peak in the blue wavelength region of the visible spectrum. The ""Cold Spot"" is approximately 70 µK (0.00007 K) colder than the average CMB temperature (approximately 2.7 K), whereas the root mean square of typical temperature variations is only 18 µK.After the dipole anisotropy, which is due to the Doppler shift of the microwave background radiation due to our peculiar velocity relative to the comoving cosmic rest frame, has been subtracted out. In this case, the brightness temperature is simply a measure of the intensity of the radiation as it would be measured at the origin of that radiation. Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. This combination will pass only a narrow (<0.1 nm) range of wavelengths of light centred on the H-alpha emission line. Also in . , vacuum (nm) 2 121.57 3 102.57 4 97.254 5 94.974 6 93.780 ∞ 91.175 Source: ===Balmer series ( = 2)=== 757px|thumb|center|The four visible hydrogen emission spectrum lines in the Balmer series. In the first year of data recorded by the Wilkinson Microwave Anisotropy Probe (WMAP), a region of sky in the constellation Eridanus was found to be cooler than the surrounding area. In particular, it is the temperature at which a black body would have to be in order to duplicate the observed intensity of a grey body object at a frequency u. For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity \epsilon. Four of the Balmer lines are in the technically ""visible"" part of the spectrum, with wavelengths longer than 400 nm and shorter than 700 nm. ",4943,-6.42,0.195,4.0,1.51,A
-"Given that the work function for sodium metal is $2.28 \mathrm{eV}$, what is the threshold frequency $v_0$ for sodium?","Using the equations given above one can then translate the electron energy E into the threshold energy T. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Therefore, datasheets will specify threshold voltage according to a specified measurable amount of current (commonly 250 μA or 1 mA). Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. Although an analytical description of the displacement is not possible, the ""sudden approximation"" gives fairly good approximations of the threshold displacement energies at least in covalent materials and low-index crystal directions An example molecular dynamics simulation of a threshold displacement event is available in 100_20eV.avi. The threshold energy should not be confused with the threshold displacement energy, which is the minimum energy needed to permanently displace an atom in a crystal to produce a crystal defect in radiation material science. ==Example of pion creation== Consider the collision of a mobile proton with a stationary proton so that a {\pi}^0 meson is produced: p^+ + p^+ \to p^+ + p^+ + \pi^0 We can calculate the minimum energy that the moving proton must have in order to create a pion. Thus, the thinner the oxide thickness, the lower the threshold voltage. The threshold limit value (TLV) is believed to be a level to which a worker can be exposed per shift in the worktime without adverse effects. If the desired result is to produce a third particle then the threshold energy is greater than or equal to the rest energy of the desired particle. Looking above, that the threshold voltage does not have a direct relationship but is not independent of the effects. The threshold voltage, commonly abbreviated as Vth or VGS(th), of a field-effect transistor (FET) is the minimum gate-to- source voltage (VGS) that is needed to create a conducting path between the source and drain terminals. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical or quantum mechanical molecular dynamics computer simulations. Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution. In particle physics, the threshold energy for production of a particle is the minimum kinetic energy that must be imparted to one of a pair of particles in order for their collision to produce a given result. Sodium vanadate can refer to: * Sodium metavanadate (sodium trioxovanadate(V)), NaVO3 * Sodium orthovanadate (sodium tetraoxovanadate(V)), Na3VO4 * Sodium decavanadate, Na6V10O28 The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . Using the body formulas above, V_{TN} is directly proportional to \gamma, and t_{OX}, which is the parameter for oxide thickness. This is the fundamental (""primary damage"") threshold displacement energy, and also the one usually simulated by molecular dynamics computer simulations. Accordingly, the term threshold voltage does not readily apply to turning such devices on, but is used instead to denote the voltage level at which the channel is wide enough to allow electrons to flow easily. The initial stage A. of defect creation, until all excess kinetic energy has dissipated in the lattice and it is back to its initial temperature T0, takes < 5 ps. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle. ",3.54,0.6321205588,5.51,8,1.7,C
-Calculate the de Broglie wavelength of an electron traveling at $1.00 \%$ of the speed of light.,"Free electron propagation (in vacuum) can be accurately described as a de Broglie matter wave with a wavelength inversely proportional to its longitudinal (possibly relativistic) momentum. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. The Compton wavelength for this particle is the wavelength of a photon of the same energy. The CODATA 2018 value for the Compton wavelength of the electron is .CODATA 2018 value for Compton wavelength for the electron from NIST. The standard Compton wavelength of a particle is given by \lambda = \frac{h}{m c}, while its frequency is given by f = \frac{m c^2}{h}, where is the Planck constant, is the particle's proper mass, and is the speed of light. Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. Thus the uncertainty in position must be greater than half of the reduced Compton wavelength . ==Relationship to other constants== Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron {2\pi}\simeq 386~\textrm{fm})}} and the electromagnetic fine-structure constant The Bohr radius is related to the Compton wavelength by: a_0 = \frac{1}{\alpha}\left(\frac{\lambda_\text{e}}{2\pi}\right) = \frac{\bar{\lambda}_\text{e}}{\alpha} \simeq 137\times\bar{\lambda}_\text{e}\simeq 5.29\times 10^4~\textrm{fm} The classical electron radius is about 3 times larger than the proton radius, and is written: r_\text{e} = \alpha\left(\frac{\lambda_\text{e}}{2\pi}\right) = \alpha\bar{\lambda}_\text{e} \simeq\frac{\bar{\lambda}_\text{e}}{137}\simeq 2.82~\textrm{fm} The Rydberg constant, having dimensions of linear wavenumber, is written: \frac{1}{R_\infty}=\frac{2\lambda_\text{e}}{\alpha^2} \simeq 91.1~\textrm{nm} \frac{1}{2\pi R_\infty} = \frac{2}{\alpha^2}\left(\frac{\lambda_\text{e}}{2\pi}\right) = 2 \frac{\bar{\lambda}_\text{e}}{\alpha^2} \simeq 14.5~\textrm{nm} This yields the sequence: r_{\text{e}} = \alpha \bar{\lambda}_{\text{e}} = \alpha^2 a_0 = \alpha^3 \frac{1}{4\pi R_\infty}. The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. For photons of frequency , energy is given by E = h f = \frac{h c}{\lambda} = m c^2, which yields the Compton wavelength formula if solved for . ==Limitation on measurement== The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: \sqrt{g_{kk}}=\lambda_c ==See also== * de Broglie wavelength * Planck–Einstein relation ==References== ==External links== * Length Scales in Physics: the Compton Wavelength Category:Atomic physics Category:Foundational quantum physics de:Compton-Effekt#Compton-Wellenlänge The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: \mathbf{ abla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. The Planck mass and length are defined by: m_{\rm P} = \sqrt{\hbar c/G} l_{\rm P} = \sqrt{\hbar G /c^3}. ==Geometrical interpretation== A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. ",243,+2.35,6.9,25,6.283185307,A
-"Show that $u(\theta, \phi)=Y_1^1(\theta, \phi)$ given in Example E-2 satisfies the equation $\nabla^2 u=\frac{c}{r^2} u$, where $c$ is a constant. What is the value of $c$ ?","For each angle \varphi the parameter :u = u(\varphi,m)=\int_0^\varphi r(\theta,m) \, d\theta (the incomplete elliptic integral of the first kind) is computed. In particular, :\theta =2\operatorname{am}\left(\frac{t\sqrt{2c}}{2},2\right)\rightarrow \frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0. The following table summarizes the expressions for all Jacobi elliptic functions pq(u,m) in the variables (x,y,r) and (φ,dn) with r = \sqrt{x^2+y^2} Jacobi elliptic functions pq[u,m] as functions of {x,y,r} and {φ,dn} q c s n d p c 1 x/y=\cot(\varphi) x/r=\cos(\varphi) x=\cos(\varphi)/\operatorname{dn} s y/x=\tan(\varphi) 1 y/r=\sin(\varphi) y=\sin(\varphi)/\operatorname{dn} n r/x=\sec(\varphi) r/y=\csc(\varphi) 1 r=1/\operatorname{dn} d 1/x=\sec(\varphi)\operatorname{dn} 1/y=\csc(\varphi)\operatorname{dn} 1/r=\operatorname{dn} 1 ==Definition in terms of Jacobi theta functions== ===Jacobi theta function description=== Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. Note that when \varphi=\pi/2, that u then equals the quarter period K. ==Definition as trigonometry: the Jacobi ellipse== \cos \varphi, \sin \varphi are defined on the unit circle, with radius r = 1 and angle \varphi = arc length of the unit circle measured from the positive x-axis. For the x and y value of the point P with u and parameter m we get, after inserting the relation: :r(\varphi,m) = \frac 1 {\operatorname{dn}(u,m)} into: x = r(\varphi,m) \cos (\varphi), y = r(\varphi,m) \sin (\varphi) that: : x = \frac{\operatorname{cn}(u,m)} {\operatorname{dn}(u,m)},\quad y = \frac{\operatorname{sn}(u,m)} {\operatorname{dn}(u,m)}. In short: : \operatorname{cn}(u,m) = \frac{x}{r(\varphi,m)}, \quad \operatorname{sn}(u,m) = \frac{y}{r(\varphi,m)}, \quad \operatorname{dn}(u,m) = \frac{1}{r(\varphi,m)}. The quantity u[\varphi,k]=u(\varphi,k^2) is related to the incomplete elliptic integral of the second kind (with modulus k) by :u[\varphi,k]=\frac{1}{\sqrt{1-k^2}}\left(\frac{1+\sqrt{1-k^2}}{2}\operatorname{E}\left(\varphi+\arctan\left(\sqrt{1-k^2}\tan \varphi\right),\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right)-\operatorname{E}(\varphi,k)+\frac{k^2 \sin\varphi\cos\varphi}{2\sqrt{1-k^2 \sin ^2\varphi}}\right), and therefore is related to the arc length of an ellipse. Therefore, \frac{\partial {\mathbf r}}{\partial u} = \frac{\partial{s}}{\partial u} \mathbf u where is the arc length parameter. Let : \begin{align} & x^2 + \frac{y^2}{b^2} = 1, \quad b > 1, \\\ & m = 1 - \frac{1}{b^2}, \quad 0 < m < 1, \\\ & x = r \cos \varphi, \quad y = r \sin \varphi \end{align} then: : r( \varphi,m) = \frac{1} {\sqrt {1-m \sin^2 \varphi}}\, . Elliptic functions are functions of two variables. Equivalent potential temperature, commonly referred to as theta-e \left( \theta_e \right), is a quantity that is conserved during changes to an air parcel's pressure (that is, during vertical motions in the atmosphere), even if water vapor condenses during that pressure change. Then the familiar relations from the unit circle: : x' = \cos \varphi, \quad y' = \sin \varphi read for the ellipse: :x' = \operatorname{cn}(u,m),\quad y' = \operatorname{sn}(u,m). The first variable might be given in terms of the amplitude \varphi, or more commonly, in terms of u given below. In the above, the value m is a free parameter, usually taken to be real such that 0\leq m \leq 1, and so the elliptic functions can be thought of as being given by two variables, u and the parameter m. Let P=(x,y)=(r \cos\varphi, r\sin\varphi) be a point on the ellipse, and let P'=(x',y')=(\cos\varphi,\sin\varphi) be the point where the unit circle intersects the line between P and the origin O. The \varphi that satisfies :u=\int_0^\varphi \frac{\mathrm d\theta} {\sqrt {1-m \sin^2 \theta}} is called the Jacobi amplitude: :\operatorname{am}(u,m)=\varphi. Similarly, Jacobi elliptic functions are defined on the unit ellipse, with a = 1\. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. == Notes == * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): ** The polar angle is denoted by \theta \in [0, \pi]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question. Then divide on both sides by \mathrm d u_i to get: \frac{\partial s}{\partial u_i} \hat{\mathbf u}_i = \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j. == See also == * Del * Orthogonal coordinates * Curvilinear coordinates * Vector fields in cylindrical and spherical coordinates == References == == External links == * Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates. Therefore, taking the partial derivative of this relation with respect to pressure yields: \left({\partial h \over \partial p}\right)_{S, \, \sigma} = \frac{1}{\rho} By integrating this equation, the potential enthalpy h^0 is defined as the enthalpy at a reference pressure p_r: h^0(S, \, \theta, \, p_r) = h(S, \, \theta, \, p) - \int^p_{p_r} \frac{1}{\rho(S, \, \theta, \, p')} dp' Here the enthalpy and density are defined in terms of the three state variables: salinity, potential temperature and pressure. === Conversion to conservative temperature === Conservative temperature \Theta is defined to be directly proportional to potential enthalpy. Damon Evans (born November 24, 1949) is an American actor best known as the second of two actors who portrayed Lionel Jefferson on the CBS sitcom The Jeffersons. ",0.0408,1260,6.6,-1.32,-2,E
-"The wave function $\Psi_2(1,2)$ given by Equation 9.39 is not normalized as it stands. Determine the normalization constant of $\Psi_2(1,2)$ given that the ""1s"" parts are normalized.","Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. thumb|250px|Mexican hat In mathematics and numerical analysis, the Ricker wavelet :\psi(t) = \frac{2}{\sqrt{3\sigma}\pi^{1/4}} \left(1 - \left(\frac{t}{\sigma}\right)^2 \right) e^{-\frac{t^2}{2\sigma^2}} is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function. A normalized 1s Slater-type function is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. The constant by which one multiplies a polynomial so its value at 1 is a normalizing constant. In this case, the reciprocal of the value :P(D)=\sum_i P(D|H_i)P(H_i) \; is the normalizing constant.Feller, 1968, p. 124. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. ==Definition== In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.Continuous Distributions at University of Alabama.Feller, 1968, p. Thus, \mathbf E_{1s} = \left\langle \left(\frac{\zeta^3}{\pi} \right)^{0.50} e^{-\zeta r} \right|\left. -\left(\frac{\zeta^3}{\pi} \right)^{0.50}e^{-\zeta r}\left[\frac{-2r\zeta+r^2\zeta^2}{2r^2}\right]\right\rangle+\langle\psi_{1s}| - \frac{\mathbf Z}{r}|\psi_{1s}\rangle \mathbf E_{1s} = \frac{\zeta^2}{2}-\zeta \mathbf Z. It has the form :\psi_{1s}(\zeta, \mathbf{r - R}) = \left(\frac{\zeta^3}{\pi}\right)^{1 \over 2} \, e^{-\zeta |\mathbf{r - R}|}. The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. Second normal form (2NF) is a normal form used in database normalization. 2NF was originally defined by E. F. Codd in 1971.Codd, E. F. And constant \frac{1}{\sqrt{2\pi}} is the normalizing constant of function p(x). In that context, the normalizing constant is called the partition function. ==Bayes' theorem== Bayes' theorem says that the posterior probability measure is proportional to the product of the prior probability measure and the likelihood function. For this reason { u} is also known as the normality parameter. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. Using the expression for Slater orbital, \mathbf \psi_{1s} = \left (\frac{\zeta^3}{\pi} \right ) ^{0.50}e^{-\zeta r} the integrals can be exactly solved. The normalization and the parameter χ have been obtained from data. is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background. == Definition == The probability density function (pdf) of the ARGUS distribution is: : f(x; \chi, c ) = \frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)} \cdot \frac{x}{c^2} \sqrt{1-\frac{x^2}{c^2}} \exp\bigg\\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\\}, for 0 \leq x < c. *A tutorial on the first 3 normal forms by Fred Coulson *Description of the database normalization basics by Microsoft 2NF de:Normalisierung (Datenbank)#Zweite Normalform (2NF) Orthonormal functions are normalized such that \langle f_i , \, f_j \rangle = \, \delta_{i,j} with respect to some inner product . For concreteness, there are many methods of estimating the normalizing constant for practical purposes. The constant is used to establish the hyperbolic functions cosh and sinh from the lengths of the adjacent and opposite sides of a hyperbolic triangle. ==See also== *Normalization (statistics) ==Notes== ==References== *Continuous Distributions at Department of Mathematical Sciences: University of Alabama in Huntsville * Category:Theory of probability distributions Category:1 (number) It is also known as the Marr wavelet for David Marr.http://www2.isye.gatech.edu/~brani/isyebayes/bank/handout20.pdf : \psi(x,y) = \frac{1}{\pi\sigma^4}\left(1-\frac{1}{2} \left(\frac{x^2+y^2}{\sigma^2}\right)\right) e^{-\frac{x^2+y^2}{2\sigma^2}} thumb|3D view of 2D Mexican hat wavelet The multidimensional generalization of this wavelet is called the Laplacian of Gaussian function. ",0.70710678,0.693150,17.4,420,14.80,A
-Find the bonding and antibonding Hückel molecular orbitals for ethene.,"A particular molecular orbital may be bonding with respect to some adjacent pairs of atoms and antibonding with respect to other pairs. Bonding and antibonding orbitals form when atoms combine into molecules. thumb|right|150px|H2 1sσ* antibonding molecular orbital In theoretical chemistry, an antibonding orbital is a type of molecular orbital that weakens the chemical bond between two atoms and helps to raise the energy of the molecule relative to the separated atoms. A molecular orbital becomes antibonding when there is less electron density between the two nuclei than there would be if there were no bonding interaction at all. If the bonding orbitals are filled, then any additional electrons will occupy antibonding orbitals. The four electrons occupy one bonding orbital at lower energy, and one antibonding orbital at higher energy than the atomic orbitals. Antibonding orbitals are also important for explaining chemical reactions in terms of molecular orbital theory. Antibonding orbitals are often labelled with an asterisk (*) on molecular orbital diagrams. Roald Hoffmann and Kenichi Fukui shared the 1981 Nobel Prize in Chemistry for their work and further development of qualitative molecular orbital explanations for chemical reactions. ==See also== *Bonding molecular orbital *Valence and conduction bands *Valence bond theory *Molecular orbital theory *Conjugated system ==References== ==Further reading== * Orchin, M. Jaffe, H.H. (1967) The Importance of Antibonding Orbitals. Similarly benzene with six carbon atoms has three bonding pi orbitals and three antibonding pi orbitals. When a molecular orbital changes sign (from positive to negative) at a nodal plane between two atoms, it is said to be antibonding with respect to those atoms. right|400px|the first four radialenes are alicyclic organic compounds containing n cross-conjugated exocyclic double bonds. There are two bonding pi orbitals which are occupied in the ground state: π1 is bonding between all carbons, while π2 is bonding between C1 and C2 and between C3 and C4, and antibonding between C2 and C3. If the bonding interactions outnumber the antibonding interactions, the MO is said to be bonding, whereas, if the antibonding interactions outnumber the bonding interactions, the molecular orbital is said to be antibonding. Since the antibonding orbital is more antibonding than the bonding orbital is bonding, the molecule has a higher energy than two separated helium atoms, and it is therefore unstable. ==Polyatomic molecules== thumb|right|200px|Butadiene pi molecular orbitals. There are also antibonding pi orbitals with two and three antibonding interactions as shown in the diagram; these are vacant in the ground state, but may be occupied in excited states. The higher-energy orbital is the antibonding orbital, which is less stable and opposes bonding if it is occupied. PLATO (Package for Linear-combination of ATomic Orbitals) is a suite of programs for electronic structure calculations. But-2-ene () is an acyclic alkene with four carbon atoms. The double bonds are commonly alkene groups but those with a carbonyl (C=O) group are also called radialenes. This overlap leads to the formation of a bonding molecular orbital with three nodal planes which contain the internuclear axis and go through both atoms. Since each carbon atom contributes one electron to the π-system of benzene, there are six pi electrons which fill the three lowest-energy pi molecular orbitals (the bonding pi orbitals). ",0.396,0.70710678,655.0,0.24995,91.17,B
-"Using the explicit formulas for the Hermite polynomials given in Table 5.3, show that a $0 \rightarrow 1$ vibrational transition is allowed and that a $0 \rightarrow 2$ transition is forbidden.","From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html The first Hermite numbers are: :H_0 = 1\, :H_1 = 0\, :H_2 = -2\, :H_3 = 0\, :H_4 = +12\, :H_5 = 0\, :H_6 = -120\, :H_7 = 0\, :H_8 = +1680\, :H_9 =0\, :H_{10} = -30240\, ==Recursion relations== Are obtained from recursion relations of Hermitian polynomials for x = 0: :H_{n} = -2(n-1)H_{n-2}.\,\\! } H_{n-2m}(x). ===Generating function=== The Hermite polynomials are given by the exponential generating function \begin{align} e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \mathit{He}_n(x) \frac{t^n}{n!}, \\\ e^{2xt - t^2} &= \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}. \end{align} This equality is valid for all complex values of and , and can be obtained by writing the Taylor expansion at of the entire function (in the physicist's case). H_{n-m}(x) &&= 2^m m! \binom{n}{m} H_{n-m}(x). \end{align} It follows that the Hermite polynomials also satisfy the recurrence relation \begin{align} \mathit{He}_{n+1}(x) &= x\mathit{He}_n(x) - n\mathit{He}_{n-1}(x), \\\ H_{n+1}(x) &= 2xH_n(x) - 2nH_{n-1}(x). \end{align} These last relations, together with the initial polynomials and , can be used in practice to compute the polynomials quickly. The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this: x^n = \frac{n!}{2^n} \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{m!(n-2m)! Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn: :H_n = \begin{cases} 0, & \mbox{if }n\mbox{ is odd} \\\ (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases} where (n - 1)!! = 1 × 3 × ... × (n - 1). ==Usage== From the generating function of Hermitian polynomials it follows that :\exp (-t^2 + 2tx) = \sum_{n=0}^\infty H_n (x) \frac {t^n}{n!}\,\\! In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. ==Definition== The polynomials are given in terms of basic hypergeometric functions by :H_n(x|q)=e^{in\theta}{}_2\phi_0\left[\begin{matrix} q^{-n},0\\\ -\end{matrix} ;q,q^n e^{-2i\theta}\right],\quad x=\cos\,\theta. ==Recurrence and difference relations== : 2x H_n(x\mid q) = H_{n+1} (x\mid q) + (1-q^n) H_{n-1} (x\mid q) with the initial conditions : H_0 (x\mid q) =1, H_{-1} (x\mid q) = 0 From the above, one can easily calculate: : \begin{align} H_0 (x\mid q) & = 1 \\\ H_1 (x\mid q) & = 2x \\\ H_2 (x\mid q) & = 4x^2 - (1-q) \\\ H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\\ H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1-q-q^3+q^4) \end{align} ==Generating function== : \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1} {\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty} where \textstyle x=\cos \theta. ==References== * * * * Category:Orthogonal polynomials Category:Q-analogs Category:Special hypergeometric functions Hermite polynomials were defined by Pierre-Simon Laplace in 1810, Collected in Œuvres complètes VII. though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. These numbers may also be expressed as a special value of the Hermite polynomials: T(n) = \frac{\mathit{He}_n(i)}{i^n}. === Completeness relation === The Christoffel–Darboux formula for Hermite polynomials reads \sum_{k=0}^n \frac{H_k(x) H_k(y)}{k!2^k} = \frac{1}{n!2^{n+1}}\,\frac{H_n(y) H_{n+1}(x) - H_n(x) H_{n+1}(y)}{x - y}. The Hermite functions satisfy the differential equation \psi_n(x) + \left(2n + 1 - x^2\right) \psi_n(x) = 0. Since they are an Appell sequence, they are a fortiori a Sheffer sequence. ==Contour-integral representation== From the generating- function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as \begin{align} \mathit{He}_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}\,dt, \\\ H_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}}\,dt, \end{align} with the contour encircling the origin. ==Generalizations== The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}, which has expected value 0 and variance 1. One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz. In terms of the probabilist's polynomials this translates to He_n(0) = \begin{cases} 0 & \text{for odd }n, \\\ (-1)^\frac{n}{2} (n-1)!! & \text{for even }n. \end{cases} ==Relations to other functions== ===Laguerre polynomials=== The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: \begin{align} H_{2n}(x) &= (-4)^n n! In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. The polynomials are sometimes denoted by , especially in probability theory, because \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} is the probability density function for the normal distribution with expected value 0 and standard deviation 1. thumb|right|390px|The first six probabilist's Hermite polynomials thumb|right|390px|The first six (physicist's) Hermite polynomials * The first eleven probabilist's Hermite polynomials are: \begin{align} \mathit{He}_0(x) &= 1, \\\ \mathit{He}_1(x) &= x, \\\ \mathit{He}_2(x) &= x^2 - 1, \\\ \mathit{He}_3(x) &= x^3 - 3x, \\\ \mathit{He}_4(x) &= x^4 - 6x^2 + 3, \\\ \mathit{He}_5(x) &= x^5 - 10x^3 + 15x, \\\ \mathit{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\\ \mathit{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\\ \mathit{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\\ \mathit{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\\ \mathit{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945. \end{align} * The first eleven physicist's Hermite polynomials are: \begin{align} H_0(x) &= 1, \\\ H_1(x) &= 2x, \\\ H_2(x) &= 4x^2 - 2, \\\ H_3(x) &= 8x^3 - 12x, \\\ H_4(x) &= 16x^4 - 48x^2 + 12, \\\ H_5(x) &= 32x^5 - 160x^3 + 120x, \\\ H_6(x) &= 64x^6 - 480x^4 + 720x^2 - 120, \\\ H_7(x) &= 128x^7 - 1344x^5 + 3360x^3 - 1680x, \\\ H_8(x) &= 256x^8 - 3584x^6 + 13440x^4 - 13440x^2 + 1680, \\\ H_9(x) &= 512x^9 - 9216x^7 + 48384x^5 - 80640x^3 + 30240x, \\\ H_{10}(x) &= 1024x^{10} - 23040x^8 + 161280x^6 - 403200x^4 + 302400x^2 - 30240. \end{align} ==Properties== The th-order Hermite polynomial is a polynomial of degree . The Hermite polynomial is then represented as H_n(x) = (-1)^n e^{x^2} \frac {d^n}{dx^n} \left( \frac {1}{2\sqrt{\pi}} \int e^{isx - \frac{s^2}{4}} \,ds \right) = (-1)^n e^{x^2}\frac{1}{2\sqrt{\pi}} \int (is)^n e^{isx - \frac{s^2}{4}} \,ds. A special case of the cross-sequence identity then says that \sum_{k=0}^n \binom{n}{k} \mathit{He}_k^{[\alpha]}(x) \mathit{He}_{n-k}^{[-\alpha]}(y) = \mathit{He}_n^{[0]}(x + y) = (x + y)^n. ==Applications== ===Hermite functions=== One can define the Hermite functions (often called Hermite- Gaussian functions) from the physicist's polynomials: \psi_n(x) = \left (2^n n! \sqrt{\pi} \right )^{-\frac12} e^{-\frac{x^2}{2}} H_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{-\frac12} e^{\frac{x^2}{2}}\frac{d^n}{dx^n} e^{-x^2}. In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(x;q) is also called discrete q-Hermite II polynomials. ==Definition== The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by :\displaystyle h_n(x;q)=q^{\binom{n}{2}}{}_2\phi_1(q^{-n},x^{-1};0;q,-qx) = x^n{}_2\phi_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2) = U_n^{(-1)}(x;q) :\displaystyle \hat h_n(x;q)=i^{-n}q^{-\binom{n}{2}}{}_2\phi_0(q^{-n},ix;;q,-q^n) = x^n{}_2\phi_1(q^{-n},q^{-n+1};0;q^2,-q^{2}/x^2) = i^{-n}V_n^{(-1)}(ix;q) and are related by :h_n(ix;q^{-1}) = i^n\hat h_n(x;q) ==References== * * * * * * Category:Orthogonal polynomials Category:Q-analogs Category:Special hypergeometric functions Similarly, \begin{align} H_{2n}(x) &= (-1)^n \frac{(2n)!}{n!} \,_1F_1\big(-n, \tfrac12; x^2\big), \\\ H_{2n+1}(x) &= (-1)^n \frac{(2n+1)!}{n!}\,2x \,_1F_1\big(-n, \tfrac32; x^2\big), \end{align} where is Kummer's confluent hypergeometric function. ==Differential-operator representation== The probabilist's Hermite polynomials satisfy the identity \mathit{He}_n(x) = e^{-\frac{D^2}{2}}x^n, where represents differentiation with respect to , and the exponential is interpreted by expanding it as a power series. In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. ==Definition== The polynomials are given in terms of basic hypergeometric functions. Reference gives a formal power series: :H_n (x) = (H+2x)^n\,\\! where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. The Hermite functions are closely related to the Whittaker function : D_n(z) = \left(n! \sqrt{\pi}\right)^{\frac12} \psi_n\left(\frac{z}{\sqrt 2}\right) = (-1)^n e^\frac{z^2}{4} \frac{d^n}{dz^n} e^\frac{-z^2}{2} and thereby to other parabolic cylinder functions. ",322,0,0.6749,-0.041,0.11,B
-"To a good approximation, the microwave spectrum of $\mathrm{H}^{35} \mathrm{Cl}$ consists of a series of equally spaced lines, separated by $6.26 \times 10^{11} \mathrm{~Hz}$. Calculate the bond length of $\mathrm{H}^{35} \mathrm{Cl}$.","Stretching or squeezing the same bond by 15 pm required an estimated 21.9 or 37.7 kJ/mol. Bond lengths in organic compounds C–H Length (pm) C–C Length (pm) Multiple-bonds Length (pm) sp3–H 110 sp3–sp3 154 Benzene 140 sp2–H 109 sp3–sp2 150 Alkene 134 sp–H 108 sp2–sp2 147 Alkyne 120 sp3–sp 146 Allene 130 sp2–sp 143 sp–sp 137 ==References== == External links == * Bond length tutorial Length Category:Molecular geometry Category:Length Since one atomic unit of length(i.e., a Bohr radius) is 52.9177 pm, the C–C bond length is 2.91 atomic units, or approximately three Bohr radii long. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. Bond distance of carbon to other elements Bonded element Bond length (pm) Group H 106–112 group 1 Be 193 group 2 Mg 207 group 2 B 156 group 13 Al 224 group 13 In 216 group 13 C 120–154 group 14 Si 186 group 14 Sn 214 group 14 Pb 229 group 14 N 147–210 group 15 P 187 group 15 As 198 group 15 Sb 220 group 15 Bi 230 group 15 O 143–215 group 16 S 181–255 group 16 Cr 192 group 6 Se 198–271 group 16 Te 205 group 16 Mo 208 group 6 W 206 group 6 F 134 group 17 Cl 176 group 17 Br 193 group 17 I 213 group 17 ==Bond lengths in organic compounds== The bond length between two atoms in a molecule depends not only on the atoms but also on such factors as the orbital hybridization and the electronic and steric nature of the substituents. Bond is located between carbons C1 and C2 as depicted in a picture below. thumb|center|150px| Hexaphenylethane skeleton based derivative containing longest known C-C bond between atoms C1 and C2 with a length of 186.2 pm Another notable compound with an extraordinary C-C bond length is tricyclobutabenzene, in which a bond length of 160 pm is reported. Unusually long bond lengths do exist. The carbon–carbon (C–C) bond length in diamond is 154 pm. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The bond lengths of these so- called ""pancake bonds"" are up to 305 pm. Shorter than average C–C bond distances are also possible: alkenes and alkynes have bond lengths of respectively 133 and 120 pm due to increased s-character of the sigma bond. Bond lengths are given in picometers. Current record holder for the longest C-C bond with a length of 186.2 pm is 1,8-Bis(5-hydroxydibenzo[a,d]cycloheptatrien-5-yl)naphthalene, one of many molecules within a category of hexaaryl ethanes, which are derivatives based on hexaphenylethane skeleton. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. The W band of the microwave part of the electromagnetic spectrum ranges from 75 to 110 GHz, wavelength ≈2.7–4 mm. In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. The smallest theoretical C–C single bond obtained in this study is 131 pm for a hypothetical tetrahedrane derivative. It is generally considered the average length for a carbon–carbon single bond, but is also the largest bond length that exists for ordinary carbon covalent bonds. In benzene all bonds have the same length: 139 pm. Carbon–carbon single bonds increased s-character is also notable in the central bond of diacetylene (137 pm) and that of a certain tetrahedrane dimer (144 pm). Structural formula 104x104px 100x100px 100x100px 107x107px Name Fluoromethane Chloromethane Bromomethane Iodomethane Melting point −137,8 °C −97,4 °C −93,7 °C −66 °C Boiling point −78,4 °C −23,8 °C 4,0 °C 42 °C Space-filling model 90x90px 110x110px 120x120px 130x130px The monohalomethanes are organic compounds in which a hydrogen atom in methane is replaced by a halogen. The 21 cm L/35 were a family of German naval artillery developed in the years before World War I and used in limited numbers. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. By approximation the bond distance between two different atoms is the sum of the individual covalent radii (these are given in the chemical element articles for each element). In this type of compound the cyclobutane ring would force 90° angles on the carbon atoms connected to the benzene ring where they ordinarily have angles of 120°. thumb|center|200px|Cyclobutabenzene with a bond length in red of 174 pm The existence of a very long C–C bond length of up to 290 pm is claimed in a dimer of two tetracyanoethylene dianions, although this concerns a 2-electron-4-center bond. ",129,0.375,11.0, -31.95,144,A
+Calculate $E^{\circ}$ for the half-cell reaction $\mathrm{Fe}^{3+}(a q)+3 \mathrm{e}^{-} \rightarrow \mathrm{Fe}(s)$.","These changes can be represented in formulas by inserting appropriate electrons into each half reaction: :\begin{align} & \ce{Fe^2+ -> Fe^3+ + e-} \\\ & \ce{Cl2 + 2e- -> 2Cl-} \end{align} Given two half reactions it is possible, with knowledge of appropriate electrode potentials, to arrive at the complete (original) reaction the same way. The sum of these two half reactions is the oxidation–reduction reaction. ==Half-reaction balancing method== Consider the reaction below: :Cl2 + 2Fe^2+ -> 2Cl- + 2Fe^3+ The two elements involved, iron and chlorine, each change oxidation state; iron from +2 to +3, chlorine from 0 to −1\. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is the Faraday's constant. For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side): : a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D The half-cell standard reduction potential E^{\ominus}_\text{red} is given by : E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF} where \Delta G^\ominus is the standard Gibbs free energy change, is the number of electrons involved, and is Faraday's constant. The determination of the formal reduction potential at pH = 7 for a given biochemical half-reaction requires thus to calculate it with the corresponding Nernst equation as a function of pH. The global reaction can thus be decomposed in half redox reactions as follows: :2 (Fe2+ → Fe3+ \+ e−) (oxidation of 2 iron(II) ions) :2 (H2O + e− → ½ H2 \+ OH−) (reduction of 2 water protons) to give: :2 Fe2+ \+ 2 H2O → 2 Fe3+ \+ H2 \+ 2 OH− Adding to this reaction one intact iron(II) ion for each two oxidized iron(II) ions leads to: :3 Fe2+ \+ 2 H2O → Fe2+ \+ 2 Fe3+ \+ H2 \+ 2 OH− Electroneutrality requires the iron cations on both sides of the equation to be counterbalanced by 6 hydroxyl anions (OH−): :3 Fe2+ \+ 6 OH− \+ 2 H2O → Fe2+ \+ 2 Fe3+ \+ H2 \+ 8 OH− :3 Fe(OH)2 \+ 2 H2O → Fe(OH)2 \+ 2 Fe(OH)3 \+ H2 For completing the main reaction, two companion reactions have still to be taken into account: The autoprotolysis of the hydroxyl anions; a proton exchange between two OH−, like in a classical acid–base reaction: :OH− \+ OH− → O2− \+ H2O :acid 1 + base 2 → base 1 + acid 2, or also, :2 OH− → O2− \+ H2O it is then possible to reorganize the global reaction as: :3 Fe(OH)2 \+ 2 H2O → (FeO + H2O) + (Fe2O3 \+ 3 H2O) + H2 :3 Fe(OH)2 \+ 2 H2O → FeO + Fe2O3 \+ 4 H2O + H2 :3 Fe(OH)2 → FeO + Fe2O3 \+ 2 H2O + H2 Considering then the formation reaction of iron(II,III) oxide: :Fe^{II}O + Fe^{III}2O3 -> Fe3O4 it is possible to write the balanced global reaction: :3 Fe(OH)2 → (FeO·Fe2O3) + 2 H2O + H2 in its final form, known as the Schikorr reaction: :3 Fe(OH)2 → Fe3O4 \+ 2 H2O + H2 == Occurrences == The Schikorr reaction can occur in the process of anaerobic corrosion of iron and carbon steel in various conditions. At pH = 7, \+ → P680 ~ +1.0 Half-reaction independent of pH as no is involved in the reaction ==See also== * Nernst equation * Electron bifurcation * Pourbaix diagram * Reduction potential ** Dependency of reduction potential on pH * Standard electrode potential * Standard reduction potential * Standard reduction potential (data page) * Standard state ==References== ==Bibliography== ;Electrochemistry * ;Bio-electrochemistry * * * * * * ;Microbiology * * Category:Biochemistry Standard reduction potentials for half-reactions important in biochemistry Category:Electrochemical potentials Category:Thermodynamics databases Category:Biochemistry databases The same also applies for the reduction potential of oxygen: : For , E^{\ominus}_\text{red} = 1.229 V, so, applying the Nernst equation for pH = 7 gives: : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 1.229 - \left(0.05916 \ \text{×} \ 7\right) = 0.815 \ V For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. Is it simply: * E_h = E_\text{red} calculated at pH 7 (with or without corrections for the activity coefficients), * E^{\ominus '}_\text{red}, a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it, * E^{\ominus '}_\text{red apparent at pH 7}, an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio \frac{h} {z} = \frac{\text{(number of involved protons)}} {\text{(number of exchanged electrons)}}. This immediately leads to the Nernst equation, which for an electrochemical half-cell is E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln Q_r=E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{a_\text{Red}}{a_\text{Ox}}. The values below are standard apparent reduction potentials for electro- biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. The offset of -414 mV in E_\text{red} is the same for both reduction reactions because they share the same linear relationship as a function of pH and the slopes of their lines are the same. Taking into account the activity coefficients (\gamma) the Nernst equation becomes: E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln\left(\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\frac{C_\text{Red}}{C_\text{Ox}}\right) E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \left(\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} + \ln\frac{C_\text{Red}}{C_\text{Ox}}\right) E_\text{red} = \underbrace{\left(E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\right)}_{E^{\ominus '}_\text{red}} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} Where the first term including the activity coefficients (\gamma) is denoted E^{\ominus '}_\text{red} and called the formal standard reduction potential, so that E_\text{red} can be directly expressed as a function of E^{\ominus '}_\text{red} and the concentrations in the simplest form of the Nernst equation: E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} ===Formal standard reduction potential=== When wishing to use simple concentrations in place of activities, but that the activity coefficients are far from unity and can no longer be neglected and are unknown or too difficult to determine, it can be convenient to introduce the notion of the ""so-called"" standard formal reduction potential (E^{\ominus '}_\text{red}) which is related to the standard reduction potential as follows: E^{\ominus '}_\text{red}=E^{\ominus}_\text{red}-\frac{RT}{zF}\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} So that the Nernst equation for the half-cell reaction can be correctly formally written in terms of concentrations as: E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}} and likewise for the full cell expression. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives: : : E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH : E_\text{red} = 0 - \left(0.05916 \ \text{×} \ 7\right) = -0.414 \ V In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons () into hydrogen gas is no longer zero as with the standard hydrogen electrode (SHE) at 1 M (pH = 0) in classical electrochemistry, but that E_\text{red} = -0.414\mathrm V versus the standard hydrogen electrode (SHE). In any given oxidation-reduction reaction, there are two half reactions—oxidation half reaction and reduction half reaction. For a complete electrochemical reaction (full cell), the equation can be written as E_\text{cell} = E^\ominus_\text{cell} - \frac{RT}{zF} \ln Q_r where: * is the half-cell reduction potential at the temperature of interest, * is the standard half-cell reduction potential, * is the cell potential (electromotive force) at the temperature of interest, * is the standard cell potential, * is the universal gas constant: , * is the temperature in kelvins, * is the number of electrons transferred in the cell reaction or half-reaction, * is the Faraday constant, the magnitude of charge (in coulombs) per mole of electrons: , * is the reaction quotient of the cell reaction, and * is the chemical activity for the relevant species, where is the activity of the reduced form and is the activity of the oxidized form. ===Thermal voltage=== At room temperature (25 °C), the thermal voltage V_T=\frac{RT}{F} is approximately 25.693 mV. In chemistry, a half reaction (or half-cell reaction) is either the oxidation or reduction reaction component of a redox reaction. So, : -zFE^\ominus_{cell} = -RT \ln{K} And therefore: : E^\ominus_{cell} = \frac{RT} {zF} \ln{K} Starting from the Nernst equation, one can also demonstrate the same relationship in the reverse way. The numerically simplified form of the Nernst equation is expressed as: :E_\text{red} = E^\ominus_\text{red} - \frac{0.059\ V}{z} \log_{10}\frac{a_\text{Red}}{a_\text{Ox}} Where E^\ominus_\text{red} is the standard reduction potential of the half-reaction expressed versus the standard reduction potential of hydrogen. This is represented in the following oxidation half reaction (note that the electrons are on the products side): :Zn_{(s)} -> Zn^2+ + 2e- At the Cu cathode, reduction takes place (electrons are accepted). Half reactions can be written to describe both the metal undergoing oxidation (known as the anode) and the metal undergoing reduction (known as the cathode). So, at pH = 7, E_\text{red} = -0.414 V for the reduction of protons. ",-0.041,1.06,"""27.0""",56,0,A
+"At $298.15 \mathrm{~K}, \Delta G_f^{\circ}(\mathrm{C}$, graphite $)=0$, and $\Delta G_f^{\circ}(\mathrm{C}$, diamond $)=2.90 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Therefore, graphite is the more stable solid phase at this temperature at $P=P^{\circ}=1$ bar. Given that the densities of graphite and diamond are 2.25 and $3.52 \mathrm{~kg} / \mathrm{L}$, respectively, at what pressure will graphite and diamond be in equilibrium at $298.15 \mathrm{~K}$ ?","At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. At normal temperature and pressure, and , the stable phase of carbon is graphite, but diamond is metastable and its rate of conversion to graphite is negligible. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at , a pressure of is needed. Rapid conversion of graphite to diamond requires pressures well above the equilibrium line: at , a pressure of is needed. ===Other properties=== thumb|upright=1.15|Molar volume against pressure at room temperature The acoustic and thermal properties of graphite are highly anisotropic, since phonons propagate quickly along the tightly bound planes, but are slower to travel from one plane to another. At surface air pressure (one atmosphere), diamonds are not as stable as graphite, and so the decay of diamond is thermodynamically favorable (δH = ). A similar proportion of diamonds comes from the lower mantle at depths between 660 and 800 km. Diamond is thermodynamically stable at high pressures and temperatures, with the phase transition from graphite occurring at greater temperatures as the pressure increases. The equilibrium pressure varies linearly with temperature, between at and at (the diamond/graphite/liquid triple point). It is chemically inert, not reacting with most corrosive substances, and has excellent biological compatibility. ===Thermodynamics=== The equilibrium pressure and temperature conditions for a transition between graphite and diamond are well established theoretically and experimentally. Another solid form of carbon known as graphite is the chemically stable form of carbon at room temperature and pressure, but diamond is metastable and converts to it at a negligible rate under those conditions. File:Graphite ambient STM.jpg|Scanning tunneling microscope image of graphite surface File:Graphite-layers- side-3D-balls.png|Side view of ABA layer stacking File:Graphite-layers- top-3D-balls.png|Plane view of layer stacking File:Graphite-unit- cell-3D-balls.png|Alpha graphite's unit cell ===Thermodynamics=== The equilibrium pressure and temperature conditions for a transition between graphite and diamond is well established theoretically and experimentally. However, at temperatures above about , diamond rapidly converts to graphite. However, at temperatures above about , diamond rapidly converts to graphite. Above the graphite- diamond-liquid carbon triple point, the melting point of diamond increases slowly with increasing pressure; but at pressures of hundreds of GPa, it decreases. At high pressure (~) diamond can be heated up to , and a report published in 2009 suggests that diamond can withstand temperatures of and above. Thus, graphite is much softer than diamond. The toughness of natural diamond has been measured as 7.5–10 MPa·m1/2. However, owing to a very large kinetic energy barrier, diamonds are metastable; they will not decay into graphite under normal conditions. ==See also== *Chemical vapor deposition of diamond *Crystallographic defects in diamond *Nitrogen-vacancy center *Synthetic diamond ==References== ==Further reading== *Pagel-Theisen, Verena. (2001). The pressure changes linearly between at and at (the diamond/graphite/liquid triple point). Sufficiently small diamonds can form in the cold of space because their lower surface energy makes them more stable than graphite. Diamonds are carbon crystals that form under high temperatures and extreme pressures such as deep within the Earth. Much higher pressures may be possible with nanocrystalline diamonds. ====Elasticity and tensile strength==== Usually, attempting to deform bulk diamond crystal by tension or bending results in brittle fracture. Research results published in an article in the scientific journal Nature Physics in 2010 suggest that at ultrahigh pressures and temperatures (about 10 million atmospheres or 1 TPa and 50,000 °C) diamond melts into a metallic fluid. ",3.8,4.0,"""0.54""",-1.49,1.51,E
+"Imagine tossing a coin 50 times. What are the probabilities of observing heads 25 times (i.e., 25 successful experiments)?","Find the probability that exactly two of the tosses result in heads. ===Solution=== For this experiment, let a heads be defined as a success and a tails as a failure. A fair coin has the probability of success 0.5 by definition. For comparison, we could define an event to occur when ""at least one 'heads'"" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. Because the coin is assumed to be fair, the probability of success is p = \tfrac{1}{2}. Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. Cambridge University Press, New York (NY), 1995, p.67-68 ==Example: tossing coins== Consider the simple experiment where a fair coin is tossed four times. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses. ==Mathematical description== A random experiment is described or modeled by a mathematical construct known as a probability space. However, there are experiments that are not easily described by a set of equally likely outcomes-- for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. ==See also== * * * * * ==References== ==External links== * Category:Experiment (probability theory) In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, ""success"" and ""failure"", in which the probability of success is the same every time the experiment is conducted. In probability theory, an outcome is a possible result of an experiment or trial. The probability function P is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would approach agreement with the values P assigns them. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis. ==Experiments and trials== Random experiments are often conducted repeatedly, so that the collective results may be subjected to statistical analysis. The probability of exactly k successes in the experiment B(n,p) is given by: :P(k)={n \choose k} p^k q^{n-k} where {n \choose k} is a binomial coefficient. For the experiment where we flip a coin twice, the four possible outcomes that make up our sample space are (H, T), (T, H), (T, T) and (H, H), where ""H"" represents a ""heads"", and ""T"" represents a ""tails"". Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number n of statistically independent Bernoulli trials, each with a probability of success p, and counts the number of successes. As a simple experiment, we may flip a coin twice. For example, when tossing an ordinary coin, one typically assumes that the outcomes ""head"" and ""tail"" are equally likely to occur. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial. In probability theory, an experiment or trial (see below) is any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. When an experiment is conducted, one (and only one) outcome results-- although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. The probability of the outcome of an experiment is never negative, although a quasiprobability distribution allows a negative probability, or quasiprobability for some events. ",-1.5,+2.35,"""0.249""",-1.49,0.11,E
+"In a rotational spectrum of $\operatorname{HBr}\left(B=8.46 \mathrm{~cm}^{-1}\right)$, the maximum intensity transition in the R-branch corresponds to the $J=4$ to 5 transition. At what temperature was the spectrum obtained?","That makes the reciprocal of the brightness temperature: ::T_b^{-1} = \frac{k}{h u}\, \text{ln}\left[1 + \frac{e^{\frac{h u}{kT}}-1}{\epsilon}\right] At low frequency and high temperatures, when h u \ll kT, we can use the Rayleigh–Jeans law: ::I_{ u} = \frac{2 u^2k T}{c^2} so that the brightness temperature can be simply written as: ::T_b=\epsilon T\, In general, the brightness temperature is a function of u, and only in the case of blackbody radiation it is the same at all frequencies. The brightness temperature can be used to calculate the spectral index of a body, in the case of non-thermal radiation. == Calculating by frequency == The brightness temperature of a source with known spectral radiance can be expressed as: : T_b=\frac{h u}{k} \ln^{-1}\left( 1 + \frac{2h u^3}{I_{ u}c^2} \right) When h u \ll kT we can use the Rayleigh–Jeans law: : T_b=\frac{I_{ u}c^2}{2k u^2} For narrowband radiation with very low relative spectral linewidth \Delta u \ll u and known radiance I we can calculate the brightness temperature as: : T_b=\frac{I c^2}{2k u^2\Delta u} == Calculating by wavelength == Spectral radiance of black-body radiation is expressed by wavelength as: : I_{\lambda}=\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{kT \lambda}} - 1} So, the brightness temperature can be calculated as: : T_b=\frac{hc}{k\lambda} \ln^{-1}\left(1 + \frac{2hc^2}{I_{\lambda}\lambda^5} \right) For long-wave radiation hc/\lambda \ll kT the brightness temperature is: : T_b=\frac{I_{\lambda}\lambda^4}{2kc} For almost monochromatic radiation, the brightness temperature can be expressed by the radiance I and the coherence length L_c: : T_b=\frac{\pi I \lambda^2 L_c}{4kc \ln{2} } ==In oceanography== In oceanography, the microwave brightness temperature, as measured by satellites looking at the ocean surface, depends on salinity as well as on the temperature of the water. ==References== Category:Temperature Category:Radio astronomy Category:Planetary science Finally, rotational energy states describe semi-rigid rotation of the entire molecule and produce transition wavelengths in the far infrared and microwave regions (about 100-10,000 μm in wavelength). At high frequencies (short wavelengths) and low temperatures, the conversion must proceed through Planck's law. In agreement with this estimate, vibrational spectra show transitions in the near infrared (about 1 - 5 μm). The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. Typically, the largest fluctuations of the primordial CMB temperature occur on angular scales of about 1°. H-alpha (Hα) is a deep-red visible spectral line of the hydrogen atom with a wavelength of 656.28 nm in air and 656.46 nm in vacuum. Approximately half the time, this cascade will include the n = 3 to n = 2 transition and the atom will emit H-alpha light. For the radiation of a helium–neon laser with a power of 1 mW, a frequency spread Δf = 1 GHz, an output aperture of 1 mm, and a beam dispersion half-angle of 0.56 mrad, the brightness temperature would be . It is the first spectral line in the Balmer series and is emitted when an electron falls from a hydrogen atom's third- to second-lowest energy level. thumb|In the Bohr model of the hydrogen atom, the electron transition from energy level n = 3 to n = 2 results in the emission of an H-alpha photon. In spectroscopy, a Soret peak or Soret band is an intense peak in the blue wavelength region of the visible spectrum. The ""Cold Spot"" is approximately 70 µK (0.00007 K) colder than the average CMB temperature (approximately 2.7 K), whereas the root mean square of typical temperature variations is only 18 µK.After the dipole anisotropy, which is due to the Doppler shift of the microwave background radiation due to our peculiar velocity relative to the comoving cosmic rest frame, has been subtracted out. In this case, the brightness temperature is simply a measure of the intensity of the radiation as it would be measured at the origin of that radiation. Since the emissivity is a value between 0 and 1, the real temperature will be greater than or equal to the brightness temperature. This combination will pass only a narrow (<0.1 nm) range of wavelengths of light centred on the H-alpha emission line. Also in . , vacuum (nm) 2 121.57 3 102.57 4 97.254 5 94.974 6 93.780 ∞ 91.175 Source: ===Balmer series ( = 2)=== 757px|thumb|center|The four visible hydrogen emission spectrum lines in the Balmer series. In the first year of data recorded by the Wilkinson Microwave Anisotropy Probe (WMAP), a region of sky in the constellation Eridanus was found to be cooler than the surrounding area. In particular, it is the temperature at which a black body would have to be in order to duplicate the observed intensity of a grey body object at a frequency u. For a grey body the spectral radiance is a portion of the black body radiance, determined by the emissivity \epsilon. Four of the Balmer lines are in the technically ""visible"" part of the spectrum, with wavelengths longer than 400 nm and shorter than 700 nm. ",4943,-6.42,"""0.195""",4.0,1.51,A
+"Given that the work function for sodium metal is $2.28 \mathrm{eV}$, what is the threshold frequency $v_0$ for sodium?","Using the equations given above one can then translate the electron energy E into the threshold energy T. In materials science, the threshold displacement energy () is the minimum kinetic energy that an atom in a solid needs to be permanently displaced from its site in the lattice to a defect position. It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Therefore, datasheets will specify threshold voltage according to a specified measurable amount of current (commonly 250 μA or 1 mA). Threshold displacement energies in typical solids are of the order of 10-50 eV. M. Nastasi, J. Mayer, and J. Hirvonen, Ion-Solid Interactions - Fundamentals and Applications, Cambridge University Press, Cambridge, Great Britain, 1996 P. Lucasson, The production of Frenkel defects in metals, in Fundamental Aspects of Radiation Damage in Metals, edited by M. T. Robinson and F. N. Young Jr., pages 42--65, Springfield, 1975, ORNL R. S. Averback and T. Diaz de la Rubia, Displacement damage in irradiated metals and semiconductors, in Solid State Physics, edited by H. Ehrenfest and F. Spaepen, volume 51, pages 281--402, Academic Press, New York, 1998.R. Smith (ed.), Atomic & ion collisions in solids and at surfaces: theory, simulation and applications, Cambridge University Press, Cambridge, UK, 1997 == Theory and simulation == The threshold displacement energy is a materials property relevant during high- energy particle radiation of materials. Although an analytical description of the displacement is not possible, the ""sudden approximation"" gives fairly good approximations of the threshold displacement energies at least in covalent materials and low-index crystal directions An example molecular dynamics simulation of a threshold displacement event is available in 100_20eV.avi. The threshold energy should not be confused with the threshold displacement energy, which is the minimum energy needed to permanently displace an atom in a crystal to produce a crystal defect in radiation material science. ==Example of pion creation== Consider the collision of a mobile proton with a stationary proton so that a {\pi}^0 meson is produced: p^+ + p^+ \to p^+ + p^+ + \pi^0 We can calculate the minimum energy that the moving proton must have in order to create a pion. Thus, the thinner the oxide thickness, the lower the threshold voltage. The threshold limit value (TLV) is believed to be a level to which a worker can be exposed per shift in the worktime without adverse effects. If the desired result is to produce a third particle then the threshold energy is greater than or equal to the rest energy of the desired particle. Looking above, that the threshold voltage does not have a direct relationship but is not independent of the effects. The threshold voltage, commonly abbreviated as Vth or VGS(th), of a field-effect transistor (FET) is the minimum gate-to- source voltage (VGS) that is needed to create a conducting path between the source and drain terminals. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical or quantum mechanical molecular dynamics computer simulations. Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution. In particle physics, the threshold energy for production of a particle is the minimum kinetic energy that must be imparted to one of a pair of particles in order for their collision to produce a given result. Sodium vanadate can refer to: * Sodium metavanadate (sodium trioxovanadate(V)), NaVO3 * Sodium orthovanadate (sodium tetraoxovanadate(V)), Na3VO4 * Sodium decavanadate, Na6V10O28 The data values of standard electrode potentials (E°) are given in the table below, in volts relative to the standard hydrogen electrode, and are for the following conditions: * A temperature of . Using the body formulas above, V_{TN} is directly proportional to \gamma, and t_{OX}, which is the parameter for oxide thickness. This is the fundamental (""primary damage"") threshold displacement energy, and also the one usually simulated by molecular dynamics computer simulations. Accordingly, the term threshold voltage does not readily apply to turning such devices on, but is used instead to denote the voltage level at which the channel is wide enough to allow electrons to flow easily. The initial stage A. of defect creation, until all excess kinetic energy has dissipated in the lattice and it is back to its initial temperature T0, takes < 5 ps. In most cases, since momentum is also conserved, the threshold energy is significantly greater than the rest energy of the desired particle. ",3.54,0.6321205588,"""5.51""",8,1.7,C
+Calculate the de Broglie wavelength of an electron traveling at $1.00 \%$ of the speed of light.,"Free electron propagation (in vacuum) can be accurately described as a de Broglie matter wave with a wavelength inversely proportional to its longitudinal (possibly relativistic) momentum. In physics, the thermal de Broglie wavelength (\lambda_{\mathrm{th}}, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. This explains why the 1-D derivation above agrees with the 3-D case. ==Examples== Some examples of the thermal de Broglie wavelength at 298 K are given below. The Compton wavelength for this particle is the wavelength of a photon of the same energy. The CODATA 2018 value for the Compton wavelength of the electron is .CODATA 2018 value for Compton wavelength for the electron from NIST. The standard Compton wavelength of a particle is given by \lambda = \frac{h}{m c}, while its frequency is given by f = \frac{m c^2}{h}, where is the Planck constant, is the particle's proper mass, and is the speed of light. Fortunately as long as the electromagnetic field traversed by the electron changes only slowly compared with this wavelength (see typical values in matter wave#Applications of matter waves), Kirchhoff's diffraction formula applies. This can also be expressed using the reduced Planck constant \hbar= \frac{h}{2\pi} as \lambda_{\mathrm{th}} = {\sqrt{\frac{2\pi\hbar^2}{ mk_{\mathrm B}T}}} . ==Massless particles== For massless (or highly relativistic) particles, the thermal wavelength is defined as \lambda_{\mathrm{th}}= \frac{hc}{2 \pi^{1/3} k_{\mathrm B} T} = \frac{\pi^{2/3}\hbar c}{ k_{\mathrm B} T} , where c is the speed of light. Thus the uncertainty in position must be greater than half of the reduced Compton wavelength . ==Relationship to other constants== Typical atomic lengths, wave numbers, and areas in physics can be related to the reduced Compton wavelength for the electron {2\pi}\simeq 386~\textrm{fm})}} and the electromagnetic fine-structure constant The Bohr radius is related to the Compton wavelength by: a_0 = \frac{1}{\alpha}\left(\frac{\lambda_\text{e}}{2\pi}\right) = \frac{\bar{\lambda}_\text{e}}{\alpha} \simeq 137\times\bar{\lambda}_\text{e}\simeq 5.29\times 10^4~\textrm{fm} The classical electron radius is about 3 times larger than the proton radius, and is written: r_\text{e} = \alpha\left(\frac{\lambda_\text{e}}{2\pi}\right) = \alpha\bar{\lambda}_\text{e} \simeq\frac{\bar{\lambda}_\text{e}}{137}\simeq 2.82~\textrm{fm} The Rydberg constant, having dimensions of linear wavenumber, is written: \frac{1}{R_\infty}=\frac{2\lambda_\text{e}}{\alpha^2} \simeq 91.1~\textrm{nm} \frac{1}{2\pi R_\infty} = \frac{2}{\alpha^2}\left(\frac{\lambda_\text{e}}{2\pi}\right) = 2 \frac{\bar{\lambda}_\text{e}}{\alpha^2} \simeq 14.5~\textrm{nm} This yields the sequence: r_{\text{e}} = \alpha \bar{\lambda}_{\text{e}} = \alpha^2 a_0 = \alpha^3 \frac{1}{4\pi R_\infty}. The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). Thus, \textstyle \frac{1}{\lambda}=\frac{E}{hc} (in this formula the h represents Planck's constant). The electron particle trajectory formula matches the formula for geometrical optics with a suitable electron-optical index of refraction. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. For photons of frequency , energy is given by E = h f = \frac{h c}{\lambda} = m c^2, which yields the Compton wavelength formula if solved for . ==Limitation on measurement== The Compton wavelength expresses a fundamental limitation on measuring the position of a particle, taking into account quantum mechanics and special relativity. Rydberg was trying: \textstyle n=n_0 - \frac{C_0}{\left(m+m'\right)^2} when he became aware of Balmer's formula for the hydrogen spectrum \textstyle \lambda={hm^2 \over m^2-4} In this equation, m is an integer and h is a constant (not to be confused with the later Planck constant). In this case, the Compton wavelength is equal to the square root of the quantum metric, a metric describing the quantum space: \sqrt{g_{kk}}=\lambda_c ==See also== * de Broglie wavelength * Planck–Einstein relation ==References== ==External links== * Length Scales in Physics: the Compton Wavelength Category:Atomic physics Category:Foundational quantum physics de:Compton-Effekt#Compton-Wellenlänge The reduced Compton wavelength appears in the relativistic Klein–Gordon equation for a free particle: \mathbf{ abla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \left(\frac{m c}{\hbar} \right)^2 \psi. thumb|Rydberg's formula as it appears in a November 1888 record In atomic physics, the Rydberg formula calculates the wavelengths of a spectral line in many chemical elements. Electron optics calculations are crucial for the design of electron microscopes and particle accelerators. But the Rydberg formula also provides correct wavelengths for distant electrons, where the effective nuclear charge can be estimated as the same as that for hydrogen, since all but one of the nuclear charges have been screened by other electrons, and the core of the atom has an effective positive charge of +1. The Planck mass and length are defined by: m_{\rm P} = \sqrt{\hbar c/G} l_{\rm P} = \sqrt{\hbar G /c^3}. ==Geometrical interpretation== A geometrical origin of the Compton wavelength has been demonstrated using semiclassical equations describing the motion of a wavepacket. As stressed by Niels Bohr, expressing results in terms of wavenumber, not wavelength, was the key to Rydberg's discovery. ",243,+2.35,"""6.9""",25,6.283185307,A
+"Show that $u(\theta, \phi)=Y_1^1(\theta, \phi)$ given in Example E-2 satisfies the equation $\nabla^2 u=\frac{c}{r^2} u$, where $c$ is a constant. What is the value of $c$ ?","For each angle \varphi the parameter :u = u(\varphi,m)=\int_0^\varphi r(\theta,m) \, d\theta (the incomplete elliptic integral of the first kind) is computed. In particular, :\theta =2\operatorname{am}\left(\frac{t\sqrt{2c}}{2},2\right)\rightarrow \frac{\mathrm d^2 \theta}{\mathrm dt^2}+c\sin \theta=0. The following table summarizes the expressions for all Jacobi elliptic functions pq(u,m) in the variables (x,y,r) and (φ,dn) with r = \sqrt{x^2+y^2} Jacobi elliptic functions pq[u,m] as functions of {x,y,r} and {φ,dn} q c s n d p c 1 x/y=\cot(\varphi) x/r=\cos(\varphi) x=\cos(\varphi)/\operatorname{dn} s y/x=\tan(\varphi) 1 y/r=\sin(\varphi) y=\sin(\varphi)/\operatorname{dn} n r/x=\sec(\varphi) r/y=\csc(\varphi) 1 r=1/\operatorname{dn} d 1/x=\sec(\varphi)\operatorname{dn} 1/y=\csc(\varphi)\operatorname{dn} 1/r=\operatorname{dn} 1 ==Definition in terms of Jacobi theta functions== ===Jacobi theta function description=== Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions. Note that when \varphi=\pi/2, that u then equals the quarter period K. ==Definition as trigonometry: the Jacobi ellipse== \cos \varphi, \sin \varphi are defined on the unit circle, with radius r = 1 and angle \varphi = arc length of the unit circle measured from the positive x-axis. For the x and y value of the point P with u and parameter m we get, after inserting the relation: :r(\varphi,m) = \frac 1 {\operatorname{dn}(u,m)} into: x = r(\varphi,m) \cos (\varphi), y = r(\varphi,m) \sin (\varphi) that: : x = \frac{\operatorname{cn}(u,m)} {\operatorname{dn}(u,m)},\quad y = \frac{\operatorname{sn}(u,m)} {\operatorname{dn}(u,m)}. In short: : \operatorname{cn}(u,m) = \frac{x}{r(\varphi,m)}, \quad \operatorname{sn}(u,m) = \frac{y}{r(\varphi,m)}, \quad \operatorname{dn}(u,m) = \frac{1}{r(\varphi,m)}. The quantity u[\varphi,k]=u(\varphi,k^2) is related to the incomplete elliptic integral of the second kind (with modulus k) by :u[\varphi,k]=\frac{1}{\sqrt{1-k^2}}\left(\frac{1+\sqrt{1-k^2}}{2}\operatorname{E}\left(\varphi+\arctan\left(\sqrt{1-k^2}\tan \varphi\right),\frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}}\right)-\operatorname{E}(\varphi,k)+\frac{k^2 \sin\varphi\cos\varphi}{2\sqrt{1-k^2 \sin ^2\varphi}}\right), and therefore is related to the arc length of an ellipse. Therefore, \frac{\partial {\mathbf r}}{\partial u} = \frac{\partial{s}}{\partial u} \mathbf u where is the arc length parameter. Let : \begin{align} & x^2 + \frac{y^2}{b^2} = 1, \quad b > 1, \\\ & m = 1 - \frac{1}{b^2}, \quad 0 < m < 1, \\\ & x = r \cos \varphi, \quad y = r \sin \varphi \end{align} then: : r( \varphi,m) = \frac{1} {\sqrt {1-m \sin^2 \varphi}}\, . Elliptic functions are functions of two variables. Equivalent potential temperature, commonly referred to as theta-e \left( \theta_e \right), is a quantity that is conserved during changes to an air parcel's pressure (that is, during vertical motions in the atmosphere), even if water vapor condenses during that pressure change. Then the familiar relations from the unit circle: : x' = \cos \varphi, \quad y' = \sin \varphi read for the ellipse: :x' = \operatorname{cn}(u,m),\quad y' = \operatorname{sn}(u,m). The first variable might be given in terms of the amplitude \varphi, or more commonly, in terms of u given below. In the above, the value m is a free parameter, usually taken to be real such that 0\leq m \leq 1, and so the elliptic functions can be thought of as being given by two variables, u and the parameter m. Let P=(x,y)=(r \cos\varphi, r\sin\varphi) be a point on the ellipse, and let P'=(x',y')=(\cos\varphi,\sin\varphi) be the point where the unit circle intersects the line between P and the origin O. The \varphi that satisfies :u=\int_0^\varphi \frac{\mathrm d\theta} {\sqrt {1-m \sin^2 \theta}} is called the Jacobi amplitude: :\operatorname{am}(u,m)=\varphi. Similarly, Jacobi elliptic functions are defined on the unit ellipse, with a = 1\. Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the z axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of the radius vector onto the x-y plane and the x axis. This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. == Notes == * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): ** The polar angle is denoted by \theta \in [0, \pi]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question. Then divide on both sides by \mathrm d u_i to get: \frac{\partial s}{\partial u_i} \hat{\mathbf u}_i = \sum_{j} \frac{\partial s}{\partial v_j} \frac{\partial v_j}{\partial u_i} \hat{\mathbf v}_j. == See also == * Del * Orthogonal coordinates * Curvilinear coordinates * Vector fields in cylindrical and spherical coordinates == References == == External links == * Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates. Therefore, taking the partial derivative of this relation with respect to pressure yields: \left({\partial h \over \partial p}\right)_{S, \, \sigma} = \frac{1}{\rho} By integrating this equation, the potential enthalpy h^0 is defined as the enthalpy at a reference pressure p_r: h^0(S, \, \theta, \, p_r) = h(S, \, \theta, \, p) - \int^p_{p_r} \frac{1}{\rho(S, \, \theta, \, p')} dp' Here the enthalpy and density are defined in terms of the three state variables: salinity, potential temperature and pressure. === Conversion to conservative temperature === Conservative temperature \Theta is defined to be directly proportional to potential enthalpy. Damon Evans (born November 24, 1949) is an American actor best known as the second of two actors who portrayed Lionel Jefferson on the CBS sitcom The Jeffersons. ",0.0408,1260,"""6.6""",-1.32,-2,E
+"The wave function $\Psi_2(1,2)$ given by Equation 9.39 is not normalized as it stands. Determine the normalization constant of $\Psi_2(1,2)$ given that the ""1s"" parts are normalized.","Note that if the probability density function is a function of various parameters, so too will be its normalizing constant. thumb|250px|Mexican hat In mathematics and numerical analysis, the Ricker wavelet :\psi(t) = \frac{2}{\sqrt{3\sigma}\pi^{1/4}} \left(1 - \left(\frac{t}{\sigma}\right)^2 \right) e^{-\frac{t^2}{2\sigma^2}} is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite function. A normalized 1s Slater-type function is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. The constant by which one multiplies a polynomial so its value at 1 is a normalizing constant. In this case, the reciprocal of the value :P(D)=\sum_i P(D|H_i)P(H_i) \; is the normalizing constant.Feller, 1968, p. 124. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one. ==Definition== In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.Continuous Distributions at University of Alabama.Feller, 1968, p. Thus, \mathbf E_{1s} = \left\langle \left(\frac{\zeta^3}{\pi} \right)^{0.50} e^{-\zeta r} \right|\left. -\left(\frac{\zeta^3}{\pi} \right)^{0.50}e^{-\zeta r}\left[\frac{-2r\zeta+r^2\zeta^2}{2r^2}\right]\right\rangle+\langle\psi_{1s}| - \frac{\mathbf Z}{r}|\psi_{1s}\rangle \mathbf E_{1s} = \frac{\zeta^2}{2}-\zeta \mathbf Z. It has the form :\psi_{1s}(\zeta, \mathbf{r - R}) = \left(\frac{\zeta^3}{\pi}\right)^{1 \over 2} \, e^{-\zeta |\mathbf{r - R}|}. The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. \\! | cdf =\begin{matrix} \frac{1}{2} + x \Gamma \left( \frac{ u+1}{2} \right) \times\\\\[0.5em] \frac{\,_2F_1 \left ( \frac{1}{2},\frac{ u+1}{2};\frac{3}{2}; -\frac{x^2}{ u} \right)} {\sqrt{\pi u}\,\Gamma \left(\frac{ u}{2}\right)} \end{matrix} where 2F1 is the hypergeometric function | mean =0 for u > 1, otherwise undefined | median =0 | mode =0 | variance =\textstyle\frac{ u}{ u-2} for u > 2, ∞ for 1 < u \le 2, otherwise undefined | skewness =0 for u > 3, otherwise undefined | kurtosis =\textstyle\frac{6}{ u-4} for u > 4, ∞ for 2 < u \le 4, otherwise undefined | entropy =\begin{matrix} \frac{ u+1}{2}\left[ \psi \left(\frac{1+ u}{2} \right) \- \psi \left(\frac{ u}{2} \right) \right] \\\\[0.5em] \+ \ln{\left[\sqrt{ u}B \left(\frac{ u}{2},\frac{1}{2} \right)\right]}\,{\scriptstyle\text{(nats)}} \end{matrix} * ψ: digamma function, * B: beta function | mgf = undefined | char =\textstyle\frac{K_{ u/2} \left(\sqrt{ u}|t|\right) \cdot \left(\sqrt{ u}|t| \right)^{ u/2}} {\Gamma( u/2)2^{ u/2-1}} for u > 0 * K_ u(x): modified Bessel function of the second kind | ES =\mu + s \left( \frac{ u + T^{-1}(1-p)^2 \tau(T^{-1}(1-p)^2 )} {( u-1)(1-p)} \right) Where T^{-1} is the inverse standardized student-t CDF, and \tau is the standardized student-t PDF. Second normal form (2NF) is a normal form used in database normalization. 2NF was originally defined by E. F. Codd in 1971.Codd, E. F. And constant \frac{1}{\sqrt{2\pi}} is the normalizing constant of function p(x). In that context, the normalizing constant is called the partition function. ==Bayes' theorem== Bayes' theorem says that the posterior probability measure is proportional to the product of the prior probability measure and the likelihood function. For this reason { u} is also known as the normality parameter. The parametrised normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics. Using the expression for Slater orbital, \mathbf \psi_{1s} = \left (\frac{\zeta^3}{\pi} \right ) ^{0.50}e^{-\zeta r} the integrals can be exactly solved. The normalization and the parameter χ have been obtained from data. is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background. == Definition == The probability density function (pdf) of the ARGUS distribution is: : f(x; \chi, c ) = \frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)} \cdot \frac{x}{c^2} \sqrt{1-\frac{x^2}{c^2}} \exp\bigg\\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\\}, for 0 \leq x < c. *A tutorial on the first 3 normal forms by Fred Coulson *Description of the database normalization basics by Microsoft 2NF de:Normalisierung (Datenbank)#Zweite Normalform (2NF) Orthonormal functions are normalized such that \langle f_i , \, f_j \rangle = \, \delta_{i,j} with respect to some inner product . For concreteness, there are many methods of estimating the normalizing constant for practical purposes. The constant is used to establish the hyperbolic functions cosh and sinh from the lengths of the adjacent and opposite sides of a hyperbolic triangle. ==See also== *Normalization (statistics) ==Notes== ==References== *Continuous Distributions at Department of Mathematical Sciences: University of Alabama in Huntsville * Category:Theory of probability distributions Category:1 (number) It is also known as the Marr wavelet for David Marr.http://www2.isye.gatech.edu/~brani/isyebayes/bank/handout20.pdf : \psi(x,y) = \frac{1}{\pi\sigma^4}\left(1-\frac{1}{2} \left(\frac{x^2+y^2}{\sigma^2}\right)\right) e^{-\frac{x^2+y^2}{2\sigma^2}} thumb|3D view of 2D Mexican hat wavelet The multidimensional generalization of this wavelet is called the Laplacian of Gaussian function. ",0.70710678,0.693150,"""17.4""",420,14.80,A
+Find the bonding and antibonding Hückel molecular orbitals for ethene.,"A particular molecular orbital may be bonding with respect to some adjacent pairs of atoms and antibonding with respect to other pairs. Bonding and antibonding orbitals form when atoms combine into molecules. thumb|right|150px|H2 1sσ* antibonding molecular orbital In theoretical chemistry, an antibonding orbital is a type of molecular orbital that weakens the chemical bond between two atoms and helps to raise the energy of the molecule relative to the separated atoms. A molecular orbital becomes antibonding when there is less electron density between the two nuclei than there would be if there were no bonding interaction at all. If the bonding orbitals are filled, then any additional electrons will occupy antibonding orbitals. The four electrons occupy one bonding orbital at lower energy, and one antibonding orbital at higher energy than the atomic orbitals. Antibonding orbitals are also important for explaining chemical reactions in terms of molecular orbital theory. Antibonding orbitals are often labelled with an asterisk (*) on molecular orbital diagrams. Roald Hoffmann and Kenichi Fukui shared the 1981 Nobel Prize in Chemistry for their work and further development of qualitative molecular orbital explanations for chemical reactions. ==See also== *Bonding molecular orbital *Valence and conduction bands *Valence bond theory *Molecular orbital theory *Conjugated system ==References== ==Further reading== * Orchin, M. Jaffe, H.H. (1967) The Importance of Antibonding Orbitals. Similarly benzene with six carbon atoms has three bonding pi orbitals and three antibonding pi orbitals. When a molecular orbital changes sign (from positive to negative) at a nodal plane between two atoms, it is said to be antibonding with respect to those atoms. right|400px|the first four radialenes are alicyclic organic compounds containing n cross-conjugated exocyclic double bonds. There are two bonding pi orbitals which are occupied in the ground state: π1 is bonding between all carbons, while π2 is bonding between C1 and C2 and between C3 and C4, and antibonding between C2 and C3. If the bonding interactions outnumber the antibonding interactions, the MO is said to be bonding, whereas, if the antibonding interactions outnumber the bonding interactions, the molecular orbital is said to be antibonding. Since the antibonding orbital is more antibonding than the bonding orbital is bonding, the molecule has a higher energy than two separated helium atoms, and it is therefore unstable. ==Polyatomic molecules== thumb|right|200px|Butadiene pi molecular orbitals. There are also antibonding pi orbitals with two and three antibonding interactions as shown in the diagram; these are vacant in the ground state, but may be occupied in excited states. The higher-energy orbital is the antibonding orbital, which is less stable and opposes bonding if it is occupied. PLATO (Package for Linear-combination of ATomic Orbitals) is a suite of programs for electronic structure calculations. But-2-ene () is an acyclic alkene with four carbon atoms. The double bonds are commonly alkene groups but those with a carbonyl (C=O) group are also called radialenes. This overlap leads to the formation of a bonding molecular orbital with three nodal planes which contain the internuclear axis and go through both atoms. Since each carbon atom contributes one electron to the π-system of benzene, there are six pi electrons which fill the three lowest-energy pi molecular orbitals (the bonding pi orbitals). ",0.396,0.70710678,"""655.0""",0.24995,91.17,B
+"Using the explicit formulas for the Hermite polynomials given in Table 5.3, show that a $0 \rightarrow 1$ vibrational transition is allowed and that a $0 \rightarrow 2$ transition is forbidden.","From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HermiteNumber.html The first Hermite numbers are: :H_0 = 1\, :H_1 = 0\, :H_2 = -2\, :H_3 = 0\, :H_4 = +12\, :H_5 = 0\, :H_6 = -120\, :H_7 = 0\, :H_8 = +1680\, :H_9 =0\, :H_{10} = -30240\, ==Recursion relations== Are obtained from recursion relations of Hermitian polynomials for x = 0: :H_{n} = -2(n-1)H_{n-2}.\,\\! } H_{n-2m}(x). ===Generating function=== The Hermite polynomials are given by the exponential generating function \begin{align} e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \mathit{He}_n(x) \frac{t^n}{n!}, \\\ e^{2xt - t^2} &= \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}. \end{align} This equality is valid for all complex values of and , and can be obtained by writing the Taylor expansion at of the entire function (in the physicist's case). H_{n-m}(x) &&= 2^m m! \binom{n}{m} H_{n-m}(x). \end{align} It follows that the Hermite polynomials also satisfy the recurrence relation \begin{align} \mathit{He}_{n+1}(x) &= x\mathit{He}_n(x) - n\mathit{He}_{n-1}(x), \\\ H_{n+1}(x) &= 2xH_n(x) - 2nH_{n-1}(x). \end{align} These last relations, together with the initial polynomials and , can be used in practice to compute the polynomials quickly. The corresponding expressions for the physicist's Hermite polynomials follow directly by properly scaling this: x^n = \frac{n!}{2^n} \sum_{m=0}^{\left\lfloor \tfrac{n}{2} \right\rfloor} \frac{1}{m!(n-2m)! Since H0 = 1 and H1 = 0 one can construct a closed formula for Hn: :H_n = \begin{cases} 0, & \mbox{if }n\mbox{ is odd} \\\ (-1)^{n/2} 2^{n/2} (n-1)!! , & \mbox{if }n\mbox{ is even} \end{cases} where (n - 1)!! = 1 × 3 × ... × (n - 1). ==Usage== From the generating function of Hermitian polynomials it follows that :\exp (-t^2 + 2tx) = \sum_{n=0}^\infty H_n (x) \frac {t^n}{n!}\,\\! In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. ==Definition== The polynomials are given in terms of basic hypergeometric functions by :H_n(x|q)=e^{in\theta}{}_2\phi_0\left[\begin{matrix} q^{-n},0\\\ -\end{matrix} ;q,q^n e^{-2i\theta}\right],\quad x=\cos\,\theta. ==Recurrence and difference relations== : 2x H_n(x\mid q) = H_{n+1} (x\mid q) + (1-q^n) H_{n-1} (x\mid q) with the initial conditions : H_0 (x\mid q) =1, H_{-1} (x\mid q) = 0 From the above, one can easily calculate: : \begin{align} H_0 (x\mid q) & = 1 \\\ H_1 (x\mid q) & = 2x \\\ H_2 (x\mid q) & = 4x^2 - (1-q) \\\ H_3 (x\mid q) & = 8x^3 - 2x(2-q-q^2) \\\ H_4 (x\mid q) & = 16x^4 - 4x^2(3-q-q^2-q^3) + (1-q-q^3+q^4) \end{align} ==Generating function== : \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1} {\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty} where \textstyle x=\cos \theta. ==References== * * * * Category:Orthogonal polynomials Category:Q-analogs Category:Special hypergeometric functions Hermite polynomials were defined by Pierre-Simon Laplace in 1810, Collected in Œuvres complètes VII. though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. These numbers may also be expressed as a special value of the Hermite polynomials: T(n) = \frac{\mathit{He}_n(i)}{i^n}. === Completeness relation === The Christoffel–Darboux formula for Hermite polynomials reads \sum_{k=0}^n \frac{H_k(x) H_k(y)}{k!2^k} = \frac{1}{n!2^{n+1}}\,\frac{H_n(y) H_{n+1}(x) - H_n(x) H_{n+1}(y)}{x - y}. The Hermite functions satisfy the differential equation \psi_n(x) + \left(2n + 1 - x^2\right) \psi_n(x) = 0. Since they are an Appell sequence, they are a fortiori a Sheffer sequence. ==Contour-integral representation== From the generating- function representation above, we see that the Hermite polynomials have a representation in terms of a contour integral, as \begin{align} \mathit{He}_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{tx-\frac{t^2}{2}}}{t^{n+1}}\,dt, \\\ H_n(x) &= \frac{n!}{2\pi i} \oint_C \frac{e^{2tx-t^2}}{t^{n+1}}\,dt, \end{align} with the contour encircling the origin. ==Generalizations== The probabilist's Hermite polynomials defined above are orthogonal with respect to the standard normal probability distribution, whose density function is \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}, which has expected value 0 and variance 1. One can also derive the (physicist's) generating function by using Cauchy's integral formula to write the Hermite polynomials as H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz. In terms of the probabilist's polynomials this translates to He_n(0) = \begin{cases} 0 & \text{for odd }n, \\\ (-1)^\frac{n}{2} (n-1)!! & \text{for even }n. \end{cases} ==Relations to other functions== ===Laguerre polynomials=== The Hermite polynomials can be expressed as a special case of the Laguerre polynomials: \begin{align} H_{2n}(x) &= (-4)^n n! In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. The polynomials are sometimes denoted by , especially in probability theory, because \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} is the probability density function for the normal distribution with expected value 0 and standard deviation 1. thumb|right|390px|The first six probabilist's Hermite polynomials thumb|right|390px|The first six (physicist's) Hermite polynomials * The first eleven probabilist's Hermite polynomials are: \begin{align} \mathit{He}_0(x) &= 1, \\\ \mathit{He}_1(x) &= x, \\\ \mathit{He}_2(x) &= x^2 - 1, \\\ \mathit{He}_3(x) &= x^3 - 3x, \\\ \mathit{He}_4(x) &= x^4 - 6x^2 + 3, \\\ \mathit{He}_5(x) &= x^5 - 10x^3 + 15x, \\\ \mathit{He}_6(x) &= x^6 - 15x^4 + 45x^2 - 15, \\\ \mathit{He}_7(x) &= x^7 - 21x^5 + 105x^3 - 105x, \\\ \mathit{He}_8(x) &= x^8 - 28x^6 + 210x^4 - 420x^2 + 105, \\\ \mathit{He}_9(x) &= x^9 - 36x^7 + 378x^5 - 1260x^3 + 945x, \\\ \mathit{He}_{10}(x) &= x^{10} - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945. \end{align} * The first eleven physicist's Hermite polynomials are: \begin{align} H_0(x) &= 1, \\\ H_1(x) &= 2x, \\\ H_2(x) &= 4x^2 - 2, \\\ H_3(x) &= 8x^3 - 12x, \\\ H_4(x) &= 16x^4 - 48x^2 + 12, \\\ H_5(x) &= 32x^5 - 160x^3 + 120x, \\\ H_6(x) &= 64x^6 - 480x^4 + 720x^2 - 120, \\\ H_7(x) &= 128x^7 - 1344x^5 + 3360x^3 - 1680x, \\\ H_8(x) &= 256x^8 - 3584x^6 + 13440x^4 - 13440x^2 + 1680, \\\ H_9(x) &= 512x^9 - 9216x^7 + 48384x^5 - 80640x^3 + 30240x, \\\ H_{10}(x) &= 1024x^{10} - 23040x^8 + 161280x^6 - 403200x^4 + 302400x^2 - 30240. \end{align} ==Properties== The th-order Hermite polynomial is a polynomial of degree . The Hermite polynomial is then represented as H_n(x) = (-1)^n e^{x^2} \frac {d^n}{dx^n} \left( \frac {1}{2\sqrt{\pi}} \int e^{isx - \frac{s^2}{4}} \,ds \right) = (-1)^n e^{x^2}\frac{1}{2\sqrt{\pi}} \int (is)^n e^{isx - \frac{s^2}{4}} \,ds. A special case of the cross-sequence identity then says that \sum_{k=0}^n \binom{n}{k} \mathit{He}_k^{[\alpha]}(x) \mathit{He}_{n-k}^{[-\alpha]}(y) = \mathit{He}_n^{[0]}(x + y) = (x + y)^n. ==Applications== ===Hermite functions=== One can define the Hermite functions (often called Hermite- Gaussian functions) from the physicist's polynomials: \psi_n(x) = \left (2^n n! \sqrt{\pi} \right )^{-\frac12} e^{-\frac{x^2}{2}} H_n(x) = (-1)^n \left (2^n n! \sqrt{\pi} \right)^{-\frac12} e^{\frac{x^2}{2}}\frac{d^n}{dx^n} e^{-x^2}. In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(x;q) is also called discrete q-Hermite II polynomials. ==Definition== The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by :\displaystyle h_n(x;q)=q^{\binom{n}{2}}{}_2\phi_1(q^{-n},x^{-1};0;q,-qx) = x^n{}_2\phi_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2) = U_n^{(-1)}(x;q) :\displaystyle \hat h_n(x;q)=i^{-n}q^{-\binom{n}{2}}{}_2\phi_0(q^{-n},ix;;q,-q^n) = x^n{}_2\phi_1(q^{-n},q^{-n+1};0;q^2,-q^{2}/x^2) = i^{-n}V_n^{(-1)}(ix;q) and are related by :h_n(ix;q^{-1}) = i^n\hat h_n(x;q) ==References== * * * * * * Category:Orthogonal polynomials Category:Q-analogs Category:Special hypergeometric functions Similarly, \begin{align} H_{2n}(x) &= (-1)^n \frac{(2n)!}{n!} \,_1F_1\big(-n, \tfrac12; x^2\big), \\\ H_{2n+1}(x) &= (-1)^n \frac{(2n+1)!}{n!}\,2x \,_1F_1\big(-n, \tfrac32; x^2\big), \end{align} where is Kummer's confluent hypergeometric function. ==Differential-operator representation== The probabilist's Hermite polynomials satisfy the identity \mathit{He}_n(x) = e^{-\frac{D^2}{2}}x^n, where represents differentiation with respect to , and the exponential is interpreted by expanding it as a power series. In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties. ==Definition== The polynomials are given in terms of basic hypergeometric functions. Reference gives a formal power series: :H_n (x) = (H+2x)^n\,\\! where formally the n-th power of H, Hn, is the n-th Hermite number, Hn. The Hermite functions are closely related to the Whittaker function : D_n(z) = \left(n! \sqrt{\pi}\right)^{\frac12} \psi_n\left(\frac{z}{\sqrt 2}\right) = (-1)^n e^\frac{z^2}{4} \frac{d^n}{dz^n} e^\frac{-z^2}{2} and thereby to other parabolic cylinder functions. ",322,0,"""0.6749""",-0.041,0.11,B
+"To a good approximation, the microwave spectrum of $\mathrm{H}^{35} \mathrm{Cl}$ consists of a series of equally spaced lines, separated by $6.26 \times 10^{11} \mathrm{~Hz}$. Calculate the bond length of $\mathrm{H}^{35} \mathrm{Cl}$.","Stretching or squeezing the same bond by 15 pm required an estimated 21.9 or 37.7 kJ/mol. Bond lengths in organic compounds C–H Length (pm) C–C Length (pm) Multiple-bonds Length (pm) sp3–H 110 sp3–sp3 154 Benzene 140 sp2–H 109 sp3–sp2 150 Alkene 134 sp–H 108 sp2–sp2 147 Alkyne 120 sp3–sp 146 Allene 130 sp2–sp 143 sp–sp 137 ==References== == External links == * Bond length tutorial Length Category:Molecular geometry Category:Length Since one atomic unit of length(i.e., a Bohr radius) is 52.9177 pm, the C–C bond length is 2.91 atomic units, or approximately three Bohr radii long. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy. Bond distance of carbon to other elements Bonded element Bond length (pm) Group H 106–112 group 1 Be 193 group 2 Mg 207 group 2 B 156 group 13 Al 224 group 13 In 216 group 13 C 120–154 group 14 Si 186 group 14 Sn 214 group 14 Pb 229 group 14 N 147–210 group 15 P 187 group 15 As 198 group 15 Sb 220 group 15 Bi 230 group 15 O 143–215 group 16 S 181–255 group 16 Cr 192 group 6 Se 198–271 group 16 Te 205 group 16 Mo 208 group 6 W 206 group 6 F 134 group 17 Cl 176 group 17 Br 193 group 17 I 213 group 17 ==Bond lengths in organic compounds== The bond length between two atoms in a molecule depends not only on the atoms but also on such factors as the orbital hybridization and the electronic and steric nature of the substituents. Bond is located between carbons C1 and C2 as depicted in a picture below. thumb|center|150px| Hexaphenylethane skeleton based derivative containing longest known C-C bond between atoms C1 and C2 with a length of 186.2 pm Another notable compound with an extraordinary C-C bond length is tricyclobutabenzene, in which a bond length of 160 pm is reported. Unusually long bond lengths do exist. The carbon–carbon (C–C) bond length in diamond is 154 pm. CL-35 or CL35 may refer to: * (CL-35), a United States Navy heavy cruiser * Chlorine-35 (Cl-35 or 35Cl), an isotope of chlorine The bond lengths of these so- called ""pancake bonds"" are up to 305 pm. Shorter than average C–C bond distances are also possible: alkenes and alkynes have bond lengths of respectively 133 and 120 pm due to increased s-character of the sigma bond. Bond lengths are given in picometers. Current record holder for the longest C-C bond with a length of 186.2 pm is 1,8-Bis(5-hydroxydibenzo[a,d]cycloheptatrien-5-yl)naphthalene, one of many molecules within a category of hexaaryl ethanes, which are derivatives based on hexaphenylethane skeleton. Longest C-C bond within the cyclobutabenzene category is 174 pm based on X-ray crystallography. The W band of the microwave part of the electromagnetic spectrum ranges from 75 to 110 GHz, wavelength ≈2.7–4 mm. In molecular geometry, bond length or bond distance is defined as the average distance between nuclei of two bonded atoms in a molecule. The smallest theoretical C–C single bond obtained in this study is 131 pm for a hypothetical tetrahedrane derivative. It is generally considered the average length for a carbon–carbon single bond, but is also the largest bond length that exists for ordinary carbon covalent bonds. In benzene all bonds have the same length: 139 pm. Carbon–carbon single bonds increased s-character is also notable in the central bond of diacetylene (137 pm) and that of a certain tetrahedrane dimer (144 pm). Structural formula 104x104px 100x100px 100x100px 107x107px Name Fluoromethane Chloromethane Bromomethane Iodomethane Melting point −137,8 °C −97,4 °C −93,7 °C −66 °C Boiling point −78,4 °C −23,8 °C 4,0 °C 42 °C Space-filling model 90x90px 110x110px 120x120px 130x130px The monohalomethanes are organic compounds in which a hydrogen atom in methane is replaced by a halogen. The 21 cm L/35 were a family of German naval artillery developed in the years before World War I and used in limited numbers. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. By approximation the bond distance between two different atoms is the sum of the individual covalent radii (these are given in the chemical element articles for each element). In this type of compound the cyclobutane ring would force 90° angles on the carbon atoms connected to the benzene ring where they ordinarily have angles of 120°. thumb|center|200px|Cyclobutabenzene with a bond length in red of 174 pm The existence of a very long C–C bond length of up to 290 pm is claimed in a dimer of two tetracyanoethylene dianions, although this concerns a 2-electron-4-center bond. ",129,0.375,"""11.0""", -31.95,144,A
 "The unit of energy in atomic units is given by
 $$
 1 E_{\mathrm{h}}=\frac{m_{\mathrm{e}} e^4}{16 \pi^2 \epsilon_0^2 \hbar^2}
 $$
-Express $1 E_{\mathrm{h}}$ in units of joules (J), kilojoules per mole $\left(\mathrm{kJ} \cdot \mathrm{mol}^{-1}\right)$, wave numbers $\left(\mathrm{cm}^{-1}\right)$, and electron volts $(\mathrm{eV})$.","Other units sometimes used to describe reaction energetics are kilocalories per mole (kcal·mol−1), electron volts per particle (eV), and wavenumbers in inverse centimeters (cm−1). 1 kJ·mol−1 is approximately equal to 1.04 eV per particle, 0.239 kcal·mol−1, or 83.6 cm−1. In slightly more fundamental terms, is equal to 1 newton metre and, in terms of SI base units :1\ \mathrm{J} = 1\ \mathrm{kg} \left( \frac{\mathrm{m}}{\mathrm{s}} \right ) ^ 2 = 1\ \frac{\mathrm{kg} \cdot \mathrm{m}^2}{\mathrm{s}^2} An energy unit that is used in atomic physics, particle physics and high energy physics is the electronvolt (eV). The electron relative atomic mass is an adjusted parameter in the CODATA set of fundamental physical constants, while the electron rest mass in kilograms is calculated from the values of the Planck constant, the fine-structure constant and the Rydberg constant, as detailed above. == Relationship to other physical constants == The electron mass is used to calculate the Avogadro constant : :N_{\rm A} = \frac{M_{\rm u} A_{\rm r}({\rm e})}{m_{\rm e}} = \frac{M_{\rm u} A_{\rm r}({\rm e})c\alpha^2}{2R_\infty h} . This list compares various energies in joules (J), organized by order of magnitude. ==Below 1 J== List of orders of magnitude for energy Factor (joules) SI prefix Value Item 10−34 6.626×10−34J Photon energy of a photon with a frequency of 1 hertz. 10−33 2×10−33J Average kinetic energy of translational motion of a molecule at the lowest temperature reached, 100 picokelvins Calculated: KE ≈ (3/2) × T × 1.38 = (3/2) × 1 × 1.38 ≈ 2.07J 10−30 quecto- (qJ) 10−28 6.6×10−28J Energy of a typical AM radio photon (1 MHz) (4×10−9 eV)Calculated: E = hν = 6.626J-s × 1 Hz = 6.6J. This very small amount of energy is often expressed in terms of an even smaller unit such as the kJ·mol−1, because of the typical order of magnitude for energy changes in chemical processes. In spectroscopy the unit cm−1 ≈ is used to represent energy since energy is inversely proportional to wavelength from the equation E = h u = h c/\lambda . In the European Union, food energy labeling in joules is mandatory, often with calories as supplementary information. ==Atom physics and chemistry== In physics and chemistry, it is common to measure energy on the atomic scale in the non-SI, but convenient, units electronvolts (eV). 1 eV is equivalent to the kinetic energy acquired by an electron in passing through a potential difference of 1 volt in a vacuum. The electron rest mass can be calculated from the Rydberg constant and the fine-structure constant obtained through spectroscopic measurements. In eV: 6.6J / 1.6J/eV = 4.1 eV. 10−27 ronto- (rJ) 10−24 yocto- (yJ) 1.6×10−24J Energy of a typical microwave oven photon (2.45 GHz) (1×10−5 eV)Calculated: E = hν = 6.626J-s × 2.45 Hz = 1.62J. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034 k_B T. == References == Category:SI derived units Hence it is also related to the atomic mass constant : :m_{\rm u} = \frac{M_{\rm u}}{N_{\rm A}} = \frac{m_{\rm e}}{A_{\rm r}({\rm e})} = \frac{2R_\infty h}{A_{\rm r}({\rm e})c\alpha^2} , where * is the molar mass constant (defined in SI); * is a directly measured quantity, the relative atomic mass of the electron. In eV: 13J / 6.022 molecules/mol / 1.6 eV/J = 0.13 eV. 10−20 4.5×10−20J Upper bound of the mass–energy of a neutrino in particle physics (0.28 eV)Calculated: 0.28 eV × 1.6J/eV = 4.5J 10−19 1.6×10−19J ≈1 electronvolt (eV) 10−19 3–5×10−19J Energy range of photons in visible light (≈1.6–3.1 eV)Calculated: E = hc/λ. In SI units, one kilocalorie per mole is equal to 4.184 kilojoules per mole (kJ/mol), which comes to approximately joules per molecule, or about 0.043 eV per molecule. Energy is defined via work, so the SI unit of energy is the same as the unit of work - the joule (J), named in honour of James Prescott Joule and his experiments on the mechanical equivalent of heat. For example, the Gibbs free energy of a compound in the area of thermochemistry is often quantified in units of kilojoules per mole (symbol: kJ·mol−1 or kJ/mol), with 1 kilojoule = 1000 joules. Historically Rydberg units have been used. ==Spectroscopy== In spectroscopy and related fields it is common to measure energy levels in units of reciprocal centimetres. These units (cm−1) are strictly speaking not energy units but units proportional to energies, with \ hc\sim 2\cdot 10^{-23}\ \mathrm{J}\ \mathrm{cm} being the proportionality constant. ==Explosions== A gram of TNT releases upon explosion. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. Constant Values Units kg Da MeV/c2 J MeV In particle physics, the electron mass (symbol: ) is the mass of a stationary electron, also known as the invariant mass of the electron. Note that is defined in terms of , and not the other way round, and so the name ""electron mass in atomic mass units"" for involves a circular definition (at least in terms of practical measurements). The joule per mole (symbol: J·mol−1 or J/mol) is the unit of energy per amount of substance in the International System of Units (SI), such that energy is measured in joules, and the amount of substance is measured in moles. ",92,15.425,27.211,22.2,311875200,C
-Calculate the probability that a particle in a one-dimensional box of length $a$ is found between 0 and $a / 2$.,"Finally, the unknown constant A may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. The potential energy in this model is given as V(x) = \begin{cases} 0, & x_c-\tfrac{L}{2} < x where L is the length of the box, xc is the location of the center of the box and x is the position of the particle within the box. The kinetic energy of a particle is given by E = p^2/(2m), and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above. ==Higher-dimensional boxes== ===(Hyper)rectangular walls=== thumb|320px|right|The wavefunction of a 2D well with nx=4 and ny=4 If a particle is trapped in a two-dimensional box, it may freely move in the x and y-directions, between barriers separated by lengths L_x and L_y respectively. However, the particle in a box may only have certain, discrete energy levels. For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by P_n(x,t) = \begin{cases} \frac{2}{L} \sin^2\left(k_n \left(x-x_c+\tfrac{L}{2}\right)\right), & x_c-\frac{L}{2} < x < x_c+\frac{L}{2},\\\ 0, & \text{otherwise,} \end{cases} Thus, for any value of n greater than one, there are regions within the box for which P(x)=0, indicating that spatial nodes exist at which the particle cannot be found. The boxcar function can be expressed in terms of the uniform distribution as \operatorname{boxcar}(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)), where is the uniform distribution of x for the interval and H(x) is the Heaviside step function. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by \Delta x\Delta p \geq \frac{\hbar}{2} It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.Davies, p. 15 Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies for a centered box are given respectively by \psi_{n_x,n_y} = \psi_{n_x}(x,t,L_x)\psi_{n_y}(y,t,L_y), E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m}, where the two-dimensional wavevector is given by \mathbf{k}_{n_x,n_y} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}}. For a centered box, the position wave function may be written including the length of the box as \psi_n(x,t,L). For a shifted box (xc = L/2), the solution is particularly simple. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.Davies, p.4 The walls of a one-dimensional box may be seen as regions of space with an infinitely large potential energy. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. This can be seen in the following equation, where m^*_e and m^*_h are the effective masses of the electron and hole, r is radius of the dot, and h is Planck's constant: \Delta E(r) = E_{\text{gap}}+\left ( \frac{h^2}{8r^2} \right ) \left( \frac{1}{m^*_e}+\frac{1}{m^*_h} \right) Hence, the energy gap of the quantum dot is inversely proportional to the square of the “length of the box,” i.e. the radius of the quantum dot. The Box is a 2007 American crime film starring Gabrielle Union, A.J. Buckley, RZA, Giancarlo Esposito, Jason Winston George, Brett Donowho and written and directed by A.J. Kparr. ==Plot== A disgraced former LAPD cop leads a home invasion in search of millions in stolen money. In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems. == One-dimensional solution == thumb|right|The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. Shown is the shifted well, with x_c = L/2 The simplest form of the particle in a box model considers a one-dimensional system. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as P(x) = |\psi(x)|^2. If we set the origin of coordinates to the center of the box, we can rewrite the spacial part of the wave function succinctly as: \psi_n (x) = \begin{cases} \sqrt{\frac{2}{L}} \sin(k_nx) \quad{} \text{for } n \text{ even} \\\ \sqrt{\frac{2}{L}} \cos(k_nx) \quad{} \text{for } n \text{ odd}. \end{cases} === Momentum wave function === The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. Mathematically, \int_0^L \left\vert \psi(x) \right\vert^2 dx = 1 (The particle must be somewhere) It follows that \left| A \right| = \sqrt{\frac{2 }{L}}. ",0.5,1.7,6.6,0.87,2283.63,A
+Express $1 E_{\mathrm{h}}$ in units of joules (J), kilojoules per mole $\left(\mathrm{kJ} \cdot \mathrm{mol}^{-1}\right)$, wave numbers $\left(\mathrm{cm}^{-1}\right)$, and electron volts $(\mathrm{eV})$.","Other units sometimes used to describe reaction energetics are kilocalories per mole (kcal·mol−1), electron volts per particle (eV), and wavenumbers in inverse centimeters (cm−1). 1 kJ·mol−1 is approximately equal to 1.04 eV per particle, 0.239 kcal·mol−1, or 83.6 cm−1. In slightly more fundamental terms, is equal to 1 newton metre and, in terms of SI base units :1\ \mathrm{J} = 1\ \mathrm{kg} \left( \frac{\mathrm{m}}{\mathrm{s}} \right ) ^ 2 = 1\ \frac{\mathrm{kg} \cdot \mathrm{m}^2}{\mathrm{s}^2} An energy unit that is used in atomic physics, particle physics and high energy physics is the electronvolt (eV). The electron relative atomic mass is an adjusted parameter in the CODATA set of fundamental physical constants, while the electron rest mass in kilograms is calculated from the values of the Planck constant, the fine-structure constant and the Rydberg constant, as detailed above. == Relationship to other physical constants == The electron mass is used to calculate the Avogadro constant : :N_{\rm A} = \frac{M_{\rm u} A_{\rm r}({\rm e})}{m_{\rm e}} = \frac{M_{\rm u} A_{\rm r}({\rm e})c\alpha^2}{2R_\infty h} . This list compares various energies in joules (J), organized by order of magnitude. ==Below 1 J== List of orders of magnitude for energy Factor (joules) SI prefix Value Item 10−34 6.626×10−34J Photon energy of a photon with a frequency of 1 hertz. 10−33 2×10−33J Average kinetic energy of translational motion of a molecule at the lowest temperature reached, 100 picokelvins Calculated: KE ≈ (3/2) × T × 1.38 = (3/2) × 1 × 1.38 ≈ 2.07J 10−30 quecto- (qJ) 10−28 6.6×10−28J Energy of a typical AM radio photon (1 MHz) (4×10−9 eV)Calculated: E = hν = 6.626J-s × 1 Hz = 6.6J. This very small amount of energy is often expressed in terms of an even smaller unit such as the kJ·mol−1, because of the typical order of magnitude for energy changes in chemical processes. In spectroscopy the unit cm−1 ≈ is used to represent energy since energy is inversely proportional to wavelength from the equation E = h u = h c/\lambda . In the European Union, food energy labeling in joules is mandatory, often with calories as supplementary information. ==Atom physics and chemistry== In physics and chemistry, it is common to measure energy on the atomic scale in the non-SI, but convenient, units electronvolts (eV). 1 eV is equivalent to the kinetic energy acquired by an electron in passing through a potential difference of 1 volt in a vacuum. The electron rest mass can be calculated from the Rydberg constant and the fine-structure constant obtained through spectroscopic measurements. In eV: 6.6J / 1.6J/eV = 4.1 eV. 10−27 ronto- (rJ) 10−24 yocto- (yJ) 1.6×10−24J Energy of a typical microwave oven photon (2.45 GHz) (1×10−5 eV)Calculated: E = hν = 6.626J-s × 2.45 Hz = 1.62J. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034 k_B T. == References == Category:SI derived units Hence it is also related to the atomic mass constant : :m_{\rm u} = \frac{M_{\rm u}}{N_{\rm A}} = \frac{m_{\rm e}}{A_{\rm r}({\rm e})} = \frac{2R_\infty h}{A_{\rm r}({\rm e})c\alpha^2} , where * is the molar mass constant (defined in SI); * is a directly measured quantity, the relative atomic mass of the electron. In eV: 13J / 6.022 molecules/mol / 1.6 eV/J = 0.13 eV. 10−20 4.5×10−20J Upper bound of the mass–energy of a neutrino in particle physics (0.28 eV)Calculated: 0.28 eV × 1.6J/eV = 4.5J 10−19 1.6×10−19J ≈1 electronvolt (eV) 10−19 3–5×10−19J Energy range of photons in visible light (≈1.6–3.1 eV)Calculated: E = hc/λ. In SI units, one kilocalorie per mole is equal to 4.184 kilojoules per mole (kJ/mol), which comes to approximately joules per molecule, or about 0.043 eV per molecule. Energy is defined via work, so the SI unit of energy is the same as the unit of work - the joule (J), named in honour of James Prescott Joule and his experiments on the mechanical equivalent of heat. For example, the Gibbs free energy of a compound in the area of thermochemistry is often quantified in units of kilojoules per mole (symbol: kJ·mol−1 or kJ/mol), with 1 kilojoule = 1000 joules. Historically Rydberg units have been used. ==Spectroscopy== In spectroscopy and related fields it is common to measure energy levels in units of reciprocal centimetres. These units (cm−1) are strictly speaking not energy units but units proportional to energies, with \ hc\sim 2\cdot 10^{-23}\ \mathrm{J}\ \mathrm{cm} being the proportionality constant. ==Explosions== A gram of TNT releases upon explosion. For ionization energies measured in the unit eV, see Ionization energies of the elements (data page). This is the energy per mole necessary to remove electrons from gaseous atoms or atomic ions. Constant Values Units kg Da MeV/c2 J MeV In particle physics, the electron mass (symbol: ) is the mass of a stationary electron, also known as the invariant mass of the electron. Note that is defined in terms of , and not the other way round, and so the name ""electron mass in atomic mass units"" for involves a circular definition (at least in terms of practical measurements). The joule per mole (symbol: J·mol−1 or J/mol) is the unit of energy per amount of substance in the International System of Units (SI), such that energy is measured in joules, and the amount of substance is measured in moles. ",92,15.425,"""27.211""",22.2,311875200,C
+Calculate the probability that a particle in a one-dimensional box of length $a$ is found between 0 and $a / 2$.,"Finally, the unknown constant A may be found by normalizing the wavefunction so that the total probability density of finding the particle in the system is 1. The particle in a box model is one of the very few problems in quantum mechanics which can be solved analytically, without approximations. The potential energy in this model is given as V(x) = \begin{cases} 0, & x_c-\tfrac{L}{2} < x where L is the length of the box, xc is the location of the center of the box and x is the position of the particle within the box. The kinetic energy of a particle is given by E = p^2/(2m), and hence the minimum kinetic energy of the particle in a box is inversely proportional to the mass and the square of the well width, in qualitative agreement with the calculation above. ==Higher-dimensional boxes== ===(Hyper)rectangular walls=== thumb|320px|right|The wavefunction of a 2D well with nx=4 and ny=4 If a particle is trapped in a two-dimensional box, it may freely move in the x and y-directions, between barriers separated by lengths L_x and L_y respectively. However, the particle in a box may only have certain, discrete energy levels. For the particle in a box, the probability density for finding the particle at a given position depends upon its state, and is given by P_n(x,t) = \begin{cases} \frac{2}{L} \sin^2\left(k_n \left(x-x_c+\tfrac{L}{2}\right)\right), & x_c-\frac{L}{2} < x < x_c+\frac{L}{2},\\\ 0, & \text{otherwise,} \end{cases} Thus, for any value of n greater than one, there are regions within the box for which P(x)=0, indicating that spatial nodes exist at which the particle cannot be found. The boxcar function can be expressed in terms of the uniform distribution as \operatorname{boxcar}(x)= (b-a)A\,f(a,b;x) = A(H(x-a) - H(x-b)), where is the uniform distribution of x for the interval and H(x) is the Heaviside step function. This can be explained in terms of the uncertainty principle, which states that the product of the uncertainties in the position and momentum of a particle is limited by \Delta x\Delta p \geq \frac{\hbar}{2} It can be shown that the uncertainty in the position of the particle is proportional to the width of the box.Davies, p. 15 Thus, the uncertainty in momentum is roughly inversely proportional to the width of the box. 300px|thumb|right|Rectangular function with a = 1 The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as \operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) = \left\\{\begin{array}{rl} 0, & \text{if } |t| > \frac{a}{2} \\\ \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\\ 1, & \text{if } |t| < \frac{a}{2}. \end{array}\right. Using a similar approach to that of the one-dimensional box, it can be shown that the wavefunctions and energies for a centered box are given respectively by \psi_{n_x,n_y} = \psi_{n_x}(x,t,L_x)\psi_{n_y}(y,t,L_y), E_{n_x,n_y} = \frac{\hbar^2 k_{n_x,n_y}^2}{2m}, where the two-dimensional wavevector is given by \mathbf{k}_{n_x,n_y} = k_{n_x}\mathbf{\hat{x}} + k_{n_y}\mathbf{\hat{y}} = \frac{n_x \pi }{L_x} \mathbf{\hat{x}} + \frac{n_y \pi }{L_y} \mathbf{\hat{y}}. For a centered box, the position wave function may be written including the length of the box as \psi_n(x,t,L). For a shifted box (xc = L/2), the solution is particularly simple. Here, the particle may only move backwards and forwards along a straight line with impenetrable barriers at either end.Davies, p.4 The walls of a one-dimensional box may be seen as regions of space with an infinitely large potential energy. However, infinitely large forces repel the particle if it touches the walls of the box, preventing it from escaping. This can be seen in the following equation, where m^*_e and m^*_h are the effective masses of the electron and hole, r is radius of the dot, and h is Planck's constant: \Delta E(r) = E_{\text{gap}}+\left ( \frac{h^2}{8r^2} \right ) \left( \frac{1}{m^*_e}+\frac{1}{m^*_h} \right) Hence, the energy gap of the quantum dot is inversely proportional to the square of the “length of the box,” i.e. the radius of the quantum dot. The Box is a 2007 American crime film starring Gabrielle Union, A.J. Buckley, RZA, Giancarlo Esposito, Jason Winston George, Brett Donowho and written and directed by A.J. Kparr. ==Plot== A disgraced former LAPD cop leads a home invasion in search of millions in stolen money. In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. It is one of the first quantum mechanics problems taught in undergraduate physics courses, and it is commonly used as an approximation for more complicated quantum systems. == One-dimensional solution == thumb|right|The barriers outside a one-dimensional box have infinitely large potential, while the interior of the box has a constant, zero potential. Shown is the shifted well, with x_c = L/2 The simplest form of the particle in a box model considers a one-dimensional system. In quantum mechanics, however, the probability density for finding a particle at a given position is derived from the wavefunction as P(x) = |\psi(x)|^2. If we set the origin of coordinates to the center of the box, we can rewrite the spacial part of the wave function succinctly as: \psi_n (x) = \begin{cases} \sqrt{\frac{2}{L}} \sin(k_nx) \quad{} \text{for } n \text{ even} \\\ \sqrt{\frac{2}{L}} \cos(k_nx) \quad{} \text{for } n \text{ odd}. \end{cases} === Momentum wave function === The momentum wavefunction is proportional to the Fourier transform of the position wavefunction. Mathematically, \int_0^L \left\vert \psi(x) \right\vert^2 dx = 1 (The particle must be somewhere) It follows that \left| A \right| = \sqrt{\frac{2 }{L}}. ",0.5,1.7,"""6.6""",0.87,2283.63,A